Introduction to Plastics Engineering 9781119536543, 1119536545

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Introduction to Plastics Engineering

Wiley-ASME Press Series Corrosion and Materials in Hydrocarbon Production: A Compendium of Operational and Engineering Aspects Bijan Kermani and Don Harrop Design and Analysis of Centrifugal Compressors Rene Van den Braembussche Case Studies in Fluid Mechanics with Sensitivities to Governing Variables M. Kemal Atesmen The Monte Carlo Ray-Trace Method in Radiation Heat Transfer and Applied Optics J. Robert Mahan Dynamics of Particles and Rigid Bodies: A Self-Learning Approach Mohammed F. Daqaq Primer on Engineering Standards, Expanded Textbook Edition Maan H. Jawad and Owen R. Greulich Engineering Optimization: Applications, Methods and Analysis R. Russell Rhinehart Compact Heat Exchangers: Analysis, Design and Optimization using FEM and CFD Approach C. Ranganayakulu and Kankanhalli N. Seetharamu Robust Adaptive Control for Fractional-Order Systems with Disturbance and Saturation Mou Chen, Shuyi Shao, and Peng Shi Robot Manipulator Redundancy Resolution Yunong Zhang and Long Jin Stress in ASME Pressure Vessels, Boilers, and Nuclear Components Maan H. Jawad Combined Cooling, Heating, and Power Systems: Modeling, Optimization, and Operation Yang Shi, Mingxi Liu, and Fang Fang Applications of Mathematical Heat Transfer and Fluid Flow Models in Engineering and Medicine Abram S. Dorfman Bioprocessing Piping and Equipment Design: A Companion Guide for the ASME BPE Standard William M. (Bill) Huitt Nonlinear Regression Modeling for Engineering Applications: Modeling, Model Validation, and Enabling Design of Experiments R. Russell Rhinehart Geothermal Heat Pump and Heat Engine Systems: Theory and Practice Andrew D. Chiasson Fundamentals of Mechanical Vibrations Liang-Wu Cai Introduction to Dynamics and Control in Mechanical Engineering Systems Cho W.S. To

Introduction to Plastics Engineering

Vijay K. Stokes

This Work is a co-publication between John Wiley & Sons Ltd and ASME Press

This edition first published 2020 © 2020 John Wiley & Sons Ltd This Work is a co-publication between John Wiley & Sons Ltd and ASME Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Vijay K. Stokes to be identified as the author of this work has been asserted in accordance with law. Registered Offices John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Office The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging-in-Publication Data Names: Stokes, Vijay Kumar, 1939- author. Title: Introduction to plastics engineering / Vijay K. Stokes. Other titles: Plastics engineering Description: First edition. | Hoboken, NJ : Wiley, [2020] | Includes bibliographical references and index. | Identifiers: LCCN 2018055488 (print) | LCCN 2019000594 (ebook) | ISBN 9781119536543 (AdobePDF) | ISBN 9781119536529 (ePub) | ISBN 9781119536574 (hardcover) Subjects: LCSH: Plastics. | Polymer engineering. Classification: LCC TA455.P5 (ebook) | LCC TA455.P5 S738 2019 (print) | DDC 620.1/923-- dc23 LC record available at https://lccn.loc.gov/2018055488 Cover Design: Wiley Cover Image: (top) courtesy of Professor J. Oliveira, University of Minho; (middle) courtesy Vijay K. Stokes; (bottom) courtesy SABIC Set in 10/12pt TimesLTStd by SPi Global, Chennai, India Printed in the UK by Bell & Bain Ltd, Glasgow. 10 9 8 7 6 5 4 3 2 1

dui biE dOsha bOrsha ka ubi bakO sata dENiE tEi jO pothi apNi ThOldi prOba lE meri Zani ka ubi behTa. This book is dedicated to my wife Prabha in recognition of over fifty years of loving support.

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Contents Series Preface xxix Preface xxxi

PART I

1

INTRODUCTION 1 Outlines for Chapters 1 and 2

Introductory Survey 3 1.1 Background 3 1.2 Synergy Between Materials Science and Engineering 4 1.3 Plastics Engineering as a Process (the Plastics Engineering Process) 7 1.4 Types of Plastics 9 1.4.1 Plastic Composites 9 1.4.2 Recycling of Plastics 10 1.5 Material Characteristics Determine Part Shapes 11 1.5.1 Stone as a Building Material 11 1.5.1.1 The Early Use of Stone 11 1.5.1.2 The Invention of the Arch 14 1.5.1.3 Vaults and Domes 14 1.5.1.4 Summary Comments 19 1.5.2 Cast Iron as a Building Material 19 1.5.3 Steel as a Building Material 20 1.5.3.1 Summary Comments 20 1.5.4 Shape Synthesis for Plastic Parts 21 1.5.4.1 Part Complexity and Consolidation 22 1.5.4.2 Plastic Hinges 24 1.5.4.3 Summary Comments 27 1.6 Part Fabrication (Part Processing) 27 1.7 Part Performance 28 1.7.1 The Role of Numerical Methods 29 1.7.2 Rapid Prototyping 31 1.8 Assembly 32 1.9 Concluding Remarks 33

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2

Evolving Applications of Plastics 35 2.1 Introduction 35 2.2 Consumer Applications 36 2.2.1 Clothing 36 2.2.1.1 Protective Clothing for Firefighters 36 2.2.1.2 Bulletproof Clothing 37 2.2.1.3 Hook-and-Loop Fasteners 39 2.2.2 Shoes 42 2.2.2.1 Athletic Shoes 42 2.2.2.2 Firefighters’ Boots 44 2.2.3 Toothbrushes 46 2.2.4 Disposable Razors 48 2.2.5 Eyewear 49 2.2.6 Contact Lenses 51 2.2.7 Bottle Caps 51 2.2.8 Drip-Proof Spouts 53 2.2.9 Plastic Tops for Paper Containers 53 2.2.9.1 Plastic Tops for Cardboard Salt Containers 2.2.9.2 Plastic Tops for Paper Juice Cartons 54 2.2.10 Toys 56 2.2.11 Consumer Audio 57 2.2.11.1 Recording Media 58 2.2.11.2 Audio Systems 60 2.2.12 Vacuum Cleaners 65 2.2.13 Small and Major Appliances 65 2.3 Medical Applications 67 2.3.1 Drip Bags and Accessories 67 2.3.2 Syringes 68 2.3.3 Medical Imaging Equipment 69 2.3.4 Plastic Models for Body Parts 70 2.4 Automotive Applications 70 2.4.1 Bumpers 71 2.4.2 Fenders 72 2.4.3 Throttle Bodies 72 2.4.4 Exhaust Manifolds 73 2.4.5 Gas Tanks 74 2.4.6 Door Modules 75 2.4.7 Boots for Constant-Velocity Joints 76 2.5 Infrastructure Applications 77 2.5.1 Glazing 77 2.5.2 Security Glazing 78 2.5.3 Water Management Systems 79 2.5.4 Large-Diameter Piping 83 2.5.5 Power Line Poles 84 2.5.6 Bridges 86 2.5.7 Composite Sheet Piling 86

54

Contents

2.6 2.7 2.8 2.9 2.10

Wind Energy 88 Airline Applications 90 Oil Extraction 91 Mining 92 Concluding Remarks 93

PART II MECHANICS 95 Outlines for Chapters 3 through 8 3

Introduction to Stress and Deformation 97 3.1 Introduction 97 3.2 Simple Measures for Load Transfer and Deformation 97 3.3 *Strains as Displacement Gradients 99 3.4 *Coupling Between Normal and Shear Stresses 101 3.5 *Coupling Between Normal and Shear Strains 102 3.6 **Two-Dimensional Stress 103 3.7 Concluding Remarks 105

4

Models for Solid Materials 107 4.1 Introduction 107 4.2 Simple Models for the Mechanical Behavior of Solids 4.3 Elastic Materials 108 4.4 *Anisotropic Materials 109 4.4.1 *Orthotropic Materials 109 4.5 Thermoelastic Effects 111 4.6 Plasticity 113 4.7 Concluding Remarks 116

5

107

Simple Structural Elements 119 5.1 Introduction 119 5.2 Bending of Beams 119 5.3 Deflection of Prismatic Beams 123 5.3.1 Deflection of a Cantilever Due to an End Load 123 5.3.2 Deflection of a Simply Supported Beam Due to a Central Load 124 5.3.3 Deflection of a Simply Supported Beam Due to a Noncentral Load 125 5.4 Torsion of Thin-Walled Circular Tubes 127 5.5 Torsion of Thin Rectangular Bars and Open Sections 129 5.6 Torsion of Thin-Walled Tubes 130 5.7 *Torsion of Multicellular Sections 131 5.8 Introduction to Elastic Stability 133 5.8.1 Concept of Stability 133 5.8.2 Stability of a Hinged Rigid Bar 134 5.8.3 *Spring-Supported Rigid Bar: Stability Above the Critical Load 136

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5.9

5.10 5.11 5.12 5.13

*Elastic Stability of an Axially Loaded Column 138 5.9.1 Buckling Load for a Pin-Jointed Column 139 5.9.2 Buckling of a Column Fixed at One End 140 Twist-Bend Buckling of a Cantilever 142 Stress Concentration 142 The Role of Numerical Methods 145 Concluding Remarks 145

6

Models for Liquids 147 6.1 Introduction 147 6.2 Simple Models for Heat Conduction 147 6.2.1 Steady-State Heat Conduction 148 6.2.2 Transient Heat Conduction 149 6.3 Kinematics of Fluid Flow 149 6.3.1 Measures for Deformation Rates 150 6.4 Equations Governing One-Dimensional Fluid Flow 151 6.4.1 One-Dimensional Continuity Equation 152 6.4.2 Balance of Linear Momentum in One Dimension 153 6.4.3 *Energy Balance in One Dimension 154 6.5 Simple Models for the Mechanical Behavior of Liquids 157 6.5.1 Newtonian Liquids 157 6.5.2 Non-Newtonian Liquids 157 6.5.3 Temperature-Dependent Viscosity Models 158 6.6 Simple One-Dimensional Flows 159 6.6.1 Surface-Driven One-Dimensional Steady Flow 159 6.6.2 Heat Generation in One-Dimensional Couette Flow 160 6.6.3 *One-Dimensional Couette Flow with Temperature-Dependent Viscosity 161 6.6.3.1 Linear Variation of Viscosity with Temperature 161 6.6.4 *Development of Couette Flow 162 6.6.5 Pressure-Driven One-Dimensional Steady Flow 162 6.6.6 Pressure-Driven Radial Flow 164 6.6.6.1 Continuity Equation for Radial Flow 165 6.6.6.2 Balance of Linear Momentum in Radial Flow 166 6.6.6.3 Incompressible Newtonian Radial Flow 168 6.7 Polymer Rheology 171 6.7.1 Die Swell 171 6.7.2 Tubeless Siphon 172 6.7.3 Vibration of a Ball Dropped in a Liquid 172 6.7.4 Weissenberg Effect 173 6.8 Concluding Remarks 173

7

Linear Viscoelasticity 175 7.1 Introduction 175 7.2 Phenomenology of Viscoelasticity 176

Contents

7.3

7.4

7.5

7.6

7.7

7.8 7.9 7.10 8

7.2.1 Stress Relaxation 176 7.2.2 Creep 176 Linear Viscoelasticity 179 7.3.1 Constitutive Equations 180 7.3.2 Stress-Relaxation Integral Form 181 7.3.3 Creep Integral Form 181 7.3.4 *Relationship Between the Relaxation Modulus and the Creep Compliance Simple Models for Stress Relaxation and Creep 182 7.4.1 Continuum Elastic Element (Elastic Spring) 183 7.4.2 Continuum Viscous Element (Dashpot) 183 7.4.3 Maxwell Model 184 7.4.3.1 Stress Relaxation 185 7.4.3.2 Creep 185 7.4.4 Kelvin-Voigt Model 185 7.4.4.1 Stress Relaxation 186 7.4.4.2 Creep 187 7.4.5 Standard Three-Parameter Model 187 Response for Constant Strain Rates 189 7.5.1 Maxwell Model 189 7.5.2 Kelvin–Voigt Model 190 7.5.3 Standard Three-Parameter Model 190 *Sinusoidal Shearing 190 7.6.1 Dynamic Mechanical Analysis (DMA) 191 7.6.1.1 DMA Curves for Three-Parameter Model 192 7.6.2 *Energy Storage and Loss 192 Isothermal Temperature Effects 193 7.7.1 Thermorheologically Simple Materials 194 7.7.2 Physical Interpretation for Time-Temperature Shift 195 *Variable Temperature Histories 195 *Cooling of a Constrained Bar 196 Concluding Remarks 196

Stiffening Mechanisms 199 8.1 Introduction 199 8.2 Continuous Fiber Reinforcement 199 8.2.1 Fiber-Matrix Interphase 202 8.3 Discontinuous Fiber Reinforcement 203 8.3.1 Load Transfer in a Discontinuous Fiber 8.3.2 Discontinuous Fiber Composite 208 8.3.3 Reinforcing Fillers 209 8.3.3.1 Spherical Fillers 209 8.3.3.2 Cylindrical Fillers 210 8.4 The Halpin–Tsai Equations 211 8.5 Reinforcing Materials 211

203

181

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8.6

8.5.1 Continuous Fibers 211 8.5.2 Chopped Fibers 212 8.5.3 Flakes 212 8.5.4 Particulates 212 8.5.5 Rubber Toughening 212 Concluding Remarks 213 Further Reading 213

PART III MATERIALS 215 Outlines for Chapters 9 through 15 9

Introduction to Polymers 217 9.1 Introduction 217 9.2 Thermoplastics 217 9.2.1 Polyethylene 217 9.2.1.1 Linear Polyethylene 218 9.2.1.2 Branched Polyethylene 220 9.2.2 Polypropylene 221 9.2.2.1 Tacticity 221 9.2.3 Cis and Trans Isomers 223 9.2.4 Polyisoprene 223 9.2.5 Homopolymers and Copolymers 224 9.2.6 Chain Entanglement 226 9.3 Molecular Weight Distributions 226 9.4 Thermosets 227 9.4.1 Phenolics 227 9.4.2 Elastomers 227 9.5 Concluding Remarks 227

10 Concepts from Polymer Physics 229

10.1 10.2

10.3

Introduction 229 Chain Conformations 229 10.2.1 *Freely Jointed Chain Models 230 10.2.2 *Effect of Bond Angle Restriction 231 10.2.3 *Effect of Steric Restrictions 232 Amorphous Polymers 234 10.3.1 Phenomenology of the Glass Transition 234 10.3.2 Physical Aging 236 10.3.3 Concept of Free Volume 236 10.3.4 Effect of Pressure on Glass Transition 238 10.3.5 Effect of Chemical Structure on Glass Transition 239 10.3.6 Effect of Molecular Weight on Glass Transition 240

Contents

10.4

10.5

10.6

Semicrystalline Polymers 240 10.4.1 Structure of Polymer Crystals 240 10.4.2 Melting Phenomenology of Semicrystalline Polymers 10.4.3 Degree of Crystallinity 242 Liquid Crystal Polymers 243 10.5.1 Liquid Crystal Phases and Transitions 244 10.5.2 Polymer Liquid Crystals 245 Concluding Remarks 245

242

11 Structure, Properties, and Applications of Plastics 247

11.1 11.2 11.3

11.4

11.5

11.6

Introduction 247 Resin Grades 248 Additives and Modifiers 248 11.3.1 Stabilizers 248 11.3.1.1 UV Stabilizers 249 11.3.1.2 Antioxidants 249 11.3.1.3 Thermal Stabilizers 249 11.3.1.4 Fire Retardants 249 11.3.2 Modifiers 250 11.3.2.1 Colorants 250 11.3.2.2 Fillers 250 11.3.2.3 Reinforcing Fibers 250 11.3.2.4 Impact Modifiers 251 11.3.2.5 Lubricants 251 11.3.2.6 Plasticizers 251 Polyolefins 251 11.4.1 Polyethylene 251 11.4.1.1 High-Strength Polyethylene Fibers 253 11.4.2 Polypropylene 253 11.4.3 Polybutylene 254 Vinyl Polymers 254 11.5.1 Poly(Vinyl Chloride) 254 11.5.1.1 Plastisol 255 11.5.2 Polyacrylonitrile 255 11.5.3 Polystyrene 256 11.5.3.1 Poly(Styrene-co-Acrylonitrile) (SAN) 256 11.5.3.2 Poly(Styrene-co-Maleic Anhydride) (SMA) 257 11.5.4 Poly(Methyl Methacrylate) 257 11.5.5 Poly(Ethylene-co-Vinyl Alcohol) 257 High-Performance Polymers 258 11.6.1 Polyoxymethylene 258 11.6.2 Poly(Phenylene Oxide) 259 11.6.3 Polyesters 259 11.6.4 Polycarbonate 260

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Contents

11.6.5 Polyamides 261 11.6.5.1 Semicrystalline Polyamides 261 11.6.5.2 Amorphous Polyamides 262 11.6.6 Fluoropolymers 263 11.6.6.1 Copolymers of Fluoropolymers 264 11.7 High-Temperature Polymers 265 11.7.1 Poly(Phenylene Sulfide) 265 11.7.2 Polyetherimide 266 11.7.3 Poly(Amide-Imide) 266 11.7.4 Polysulfones 267 11.7.4.1 Polysulfone 267 11.7.4.2 Polyethersulfone 268 11.7.4.3 Polyphenylsulfone (Polyarylethersulfone) 11.7.5 Polyketones 268 11.7.6 Liquid Crystalline Polyesters 269 11.7.7 Aromatic Polyamides (Aramids) 270 11.7.8 Polybenzimidazole 271 11.8 Cyclic Polymers 271 11.9 Thermoplastic Elastomers 272 11.9.1 Polypropylene-EPDM TPE 272 11.9.2 Thermoplastic Copolyester TPE 272 11.9.3 Thermoplastic Urethane (TPU) 273 11.10 Historical Notes 273 11.11 Concluding Remarks 274 277 Introduction 277 Blends 278 12.2.1 Acrylonitrile-Butadiene-Styrene 278 12.2.2 Acrylonitrile-Styrene-Acrylate 279 12.2.3 ABS/PVC Blends 279 12.2.4 Nylon/ABS Blends 279 12.2.5 Polycarbonate/ABS Blends 279 12.2.6 Poly(Phenylene Oxide)/Polystyrene Blends 12.2.7 Polycarbonate/PBT Blends 280 12.2.8 Nylon/PPO Blends 281 12.2.9 High-Temperature Blends 281 Historical Notes 282 Concluding Remarks 282

12 Blends and Alloys

12.1 12.2

12.3 12.4

13 Thermoset Materials 285

13.1 13.2

Introduction 285 Thermosetting Resins 285 13.2.1 Phenolics 286

280

268

Contents

13.3

13.4

13.5 13.6

13.2.1.1 Resole Resins 287 13.2.1.2 Novolak Resins 287 13.2.1.3 Applications of Phenolics 288 13.2.2 Urea-Aldehyde-Based Resins 288 13.2.2.1 Urea-Formaldehyde-Based Resin 288 13.2.2.2 Melamine-Aldehyde-Based Resins 289 13.2.2.3 Applications of Urea- and Melamine-Aldehyde Resins 291 13.2.3 Allyl Diglycol Carbonate (CR-39) 291 13.2.4 Thermosetting Polyesters 291 13.2.5 Vinyl Esters 293 13.2.5.1 Applications of Polyesters and Vinyl Esters 293 13.2.6 Epoxies 293 13.2.6.1 Applications of Epoxies 294 13.2.7 Sheet and Bulk Molding Compounds 294 13.2.8 Polyurethanes 295 13.2.8.1 Applications of Polyurethanes 296 High-Temperature Thermosets 296 13.3.1 Cyanate Esters 296 13.3.2 Bismaleimides 300 13.3.3 Polyimides 302 13.3.3.1 PMR-15 302 13.3.3.2 LaRC RP-46 303 13.3.4 Poly(Phenylene Benzobisoxazole) 303 Thermoset Elastomers 304 13.4.1 Diene Elastomers 304 13.4.1.1 Polyisoprene (Natural Rubber) 304 13.4.1.2 Polychloroprene (Neoprene) 305 13.4.1.3 Polybutadiene 306 13.4.1.4 Poly(Isobutylene-co-Isoprene) (Butyl Rubber) 306 13.4.1.5 Poly(Styrene-co-Butadiene) (SBR Rubber) 307 13.4.1.6 Poly(Acrylonitrile-co-Butadiene) (NBR Rubber) 307 13.4.2 Ethylene-Propylene Copolymer-Based Elastomers 308 13.4.2.1 Ethylene-Propylene Rubber (EPR) 308 13.4.2.2 Ethylene-Propylene-Diene Monomer (EPDM) Rubber 308 13.4.2.3 Silicone Elastomers 308 Historical Notes 309 Concluding Remarks 311

14 Polymer Viscoelasticity 313

14.1 14.2

Introduction 313 Phenomenology of Polymer Viscoelasticity 313 14.2.1 Relaxation Moduli at Constant Temperature 14.2.2 Relaxation Moduli at Constant Time 315 14.2.3 Relaxation Moduli of Several Resins 316

314

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14.3

14.4 14.5

14.2.3.1 Effect of Molecular Weight: Relaxation Moduli of Polystyrene 316 14.2.3.2 Effects of Crystallinity: Relaxation Moduli of Several Resins 317 14.2.3.3 Effects of Plasticizers: Relaxation Moduli of PVC 318 Time-Temperature Superposition 319 14.3.1 Experimental Characterization of the Master Curve 319 14.3.2 Corrections to the Time-Temperature Correspondence Relations 321 14.3.3 The WLF Equation 322 14.3.4 Physical Interpretation for the Time-Temperature Shift 322 14.3.5 Summary 322 Sinusoidal Oscillatory Tests 323 14.4.1 DMA Data for High-Performance Thermoplastics 324 Concluding Remarks 328

15 Mechanical Behavior of Plastics 331

15.1 15.2

15.3

15.4

15.5

15.6

Introduction 331 Deformation Phenomenology of Polycarbonate 332 15.2.1 Constant-Displacement-Rate Tensile Test 333 15.2.2 *Considère Treatment of Yield 336 15.2.3 *Uniaxial Extension of Wide PC Specimens 338 15.2.4 *Definition and Measurement of Initial Yielding 341 15.2.5 *Mechanical Behavior of Necked PC 342 15.2.6 *Composite Stress-Stretch Curve for PC 343 15.2.7 *Creep of PC at High Loads 343 15.2.8 *Deformation-Rate and Temperature Effects 346 15.2.9 *Biaxial Stretching of Clamped Circular PC Sheets by Fluid Pressure 349 15.2.10 Thermally Induced Recovery from a Mechanically Yielded State 355 15.2.11 Large-Deformation Applications 358 Tensile Characteristics of PEI 360 15.3.1 Constant-Displacement-Rate Tensile Test 360 15.3.2 *Deformation-Rate and Temperature Effects 362 Deformation Phenomenology of PBT 363 15.4.1 Constant-Displacement-Rate Tensile Test 363 15.4.2 *Definition and Measurement of Initial Yielding in PBT 364 15.4.3 *Mechanical Behavior of Necked PBT 366 15.4.4 *Composite Stress-Stretch Curve for PBT 367 15.4.5 *Deformation-Rate and Temperature Effects 368 15.4.6 *Post-Yield Behavior Prior to Necking 371 15.4.7 *Load History and Final Permanent Deformation 372 15.4.8 Large-Deformation Applications 375 Stress-Deformation Behavior of Several Plastics 376 15.5.1 Thermoplastics 376 15.5.2 Thermosets 380 15.5.3 Thermoplastic Elastomers 383 Phenomenon of Crazing 387

Contents

15.7

15.8 15.9

15.10

15.11 15.12 15.13

*Multiaxial Yield 393 15.7.1 Maximum Principal Stress Theory 394 15.7.2 Maximum Shear Stress Theory 394 15.7.3 Maximum Principle Strain Theory 396 15.7.4 Strain Energy of Distortion Theory 396 15.7.5 Comparison of Failure Theories 399 15.7.6 Failure Theories for Plastics 400 *Fracture 401 Fatigue 403 15.9.1 The S-N Curve 404 15.9.2 *Fatigue-Crack Propagation 406 15.9.3 The Role of Hysteretic Heating 412 Impact Loading 412 15.10.1 Instrumented Impact Test 412 15.10.2 Ductile-Brittle Transition 414 Creep 419 Stress-Deformation Behavior of Thermoset Elastomers 419 Concluding Remarks 420 Further Reading 420

PART IV PART PROCESSING AND ASSEMBLY

421

Outlines for Chapters 16 through 21 423 Introduction 423 Part Fabrication (Processing) Methods for Thermoplastics 424 16.2.1 Processes Using Double-Sided Molds 426 16.2.2 Processes Using Single-Sided Molds 426 Evolution of Part Shaping Methods 429 Effects of Processing on Part Performance 431 Bulk Processing Methods for Thermoplastics 439 16.5.1 Fiber Spinning 439 16.5.2 Film Blowing 439 16.5.3 Sheet Extrusion 439 16.5.4 Profile Extrusion 439 Part Processing Methods for Thermosets 440 16.6.1 Processes Using Double-Sided Molds 440 16.6.1.1 Processes Using Powder Resin 441 16.6.1.2 Processes Using Sheet and Bulk Molding Compounds 16.6.1.3 Processes Using Liquid Resin 442 16.6.2 Processes Using Single-Sided Molds 442 Part Processing Methods Advanced Composites 442 16.7.1 Pultrusion 442

16 Classification of Part Shaping Methods

16.1 16.2

16.3 16.4 16.5

16.6

16.7

442

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16.8

16.9

16.7.2 Filament Winding 442 16.7.3 Laminated Composites 443 16.7.3.1 Prepregs 443 16.7.3.2 Vacuum Bag Consolidation 443 16.7.3.3 Compression Molding 443 Processing Methods for Rubber Parts 443 16.8.1 Rubber Compounding 443 16.8.2 Dry Rubber Compounding 444 16.8.2.1 Molding Processes 444 16.8.2.2 Extrusion 444 16.8.2.3 Calendering 444 16.8.2.4 Reinforced and Coated Rubber Sheet 16.8.3 Wet Rubber Part Fabrication 444 16.8.3.1 Dip Molding 444 16.8.3.2 Dip Coating 444 Concluding Remarks 445

444

17 Injection Molding and Its Variants 447

17.1 17.2

17.3

17.4 17.5

Introduction 447 Process Elements 447 17.2.1 Mold Filling 453 17.2.1.1 Filling of an Off-Center Gated Mold Cavity 453 17.2.1.2 Filling of a Double-Gated Cavity 453 17.2.1.3 Effects of Material Differences on Flow in a Double-Gated Cavity 454 17.2.1.4 Effects of Slits in a Mold Cavity 455 17.2.1.5 Flow in a Double-Gated Cavity with Inserts 457 17.2.2 Part Thickness 458 17.2.3 Mold Clamp Forces 460 17.2.4 Mold Cooling 460 Fountain Flow 462 17.3.1 Meld Surfaces and Knit Lines 465 17.3.1.1 Head-on Welding of Two Flow Fronts 465 17.3.1.2 Melding of Flow Fronts Around a Pin 467 17.3.1.3 Effects of Gates, Part Geometries, and Materials on Knit Lines 469 17.3.2 The Role of Numerical Simulation 472 Part Morphology 473 Part Design 475 17.5.1 Part Stiffening Mechanisms 475 17.5.2 Molding-Driven Features 476 17.5.2.1 Part Thickness Distribution 476 17.5.2.2 Part Shrinkage 478 17.5.2.3 Part Warpage 480 17.5.2.4 Draft Angles 482 17.5.2.5 Boss Geometries 482

Contents

17.6

17.7

17.8

17.9

17.5.2.6 Molded-In Inserts 484 17.5.3 Plastic Hinges 485 Large- Versus Small-Part Molding 493 17.6.1 Thin-Wall Molding 493 17.6.2 Micromolding 496 Molding Practice 504 17.7.1 Two-Plate Cold-Runner Mold 505 17.7.2 Three-Plate Cold-Runner Mold 508 17.7.3 Molds for Parts with Undercuts 508 17.7.4 Molds with Collapsible Cores 508 17.7.5 Hot-Runner Molds 513 17.7.6 Sprues, Runners, and Gates 515 17.7.6.1 Runner Configurations 515 17.7.6.2 Imbalances from Flow Asymmetry 518 17.7.7 Gate Types 520 17.7.7.1 Sprue Gate 521 17.7.7.2 Edge Gate 521 17.7.7.3 Fan Gate 521 17.7.7.4 Diaphragm Gate 522 17.7.8 Jetting 522 17.7.9 Mold Venting 523 17.7.10 Mold Cooling 525 17.7.11 Summary Comments 525 Variants of Injection Molding 526 17.8.1 Methods for Reducing Injection Pressure 526 17.8.1.1 Sequential Gating 526 17.8.1.2 Injection-Compression Molding 527 17.8.2 Structural Foam Molding 529 17.8.2.1 Alternative Foam Molding Processes 533 17.8.2.2 Advantages, Disadvantages, and Applications 17.8.3 Microcellular Foam Molding 535 17.8.4 Multimaterial Molding 538 17.8.4.1 Coinjection Molding 538 17.8.4.2 Overmolding 538 17.8.5 Hollow Parts 540 17.8.5.1 Fusible-Core Molding 540 17.8.5.2 Gas-Assisted Injection Molding 541 17.8.5.3 Summary Comments 548 17.8.6 Knit and Meld Line Esthetics and Integrity 549 17.8.6.1 Multiple-Live-Feed Injection Molding 549 17.8.6.2 Push-Pull Injection Molding 550 17.8.7 In-Mold Decoration and Lamination 552 Concluding Remarks 553 References 553

534

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Contents

18 Dimensional Stability and Residual Stresses 555

18.1 18.2 18.3 18.4 18.5

18.6

18.7

18.8

18.9

Introduction 555 Problem Complexity 556 Shrinkage Phenomenology 556 Pressure-Temperature Volumetric Data 563 18.4.1 Quantification of PVT Data 564 Simple Model for How Processing Affects Shrinkage 567 18.5.1 Constant Packing-Pressure History 569 18.5.2 Effect of Gate Freeze-Off 572 18.5.3 Effect of Packing Duration 576 18.5.4 Summary Comments 577 *Solidification of a Molten Layer 578 18.6.1 *Freezing of a Molten Layer 578 18.6.2 *Fluid to Elastic-Solid Freezing Model 579 18.6.3 *Numerical Example for a 3-mm-Thick Plaque 581 18.6.4 *Effective Pressure as an Independent Variable 583 18.6.5 *Summary Comments 585 **Viscoelastic Solidification Model 585 18.7.1 Viscoelastic Material Model 585 18.7.2 Temperature Distribution in a Solidifying Melt 586 18.7.3 Evolution of Shrinkage and Residual Stresses 588 18.7.4 Effects of Packing-Pressure Level 590 18.7.5 Effect of Packing-Pressure Duration 593 18.7.6 Effect of Gate Freeze-Off Time 594 18.7.7 Summary Comments 599 **Warpage Induced by Differential Mold-Surface Temperatures 18.8.1 Temperature Distribution in a Solidifying Melt 602 18.8.2 Constant Packing-Pressure Level 602 18.8.3 Effect of Packing-Pressure Level 604 18.8.4 Effect of Gate Freeze-Off 605 18.8.5 Summary Comments 606 Concluding Remarks 609

19 Alternatives to Injection Molding 615

19.1 19.2

Introduction 615 Extrusion 615 19.2.1 Fiber Spinning 616 19.2.2 Film Blowing 618 19.2.3 Sheet Extrusion 618 19.2.3.1 Cast Film Extrusion 619 19.2.3.2 Calendered Sheet Extrusion 19.2.4 Profile Extrusion 620 19.2.4.1 Open Profiles 621 19.2.4.2 Closed Profiles 624

620

602

Contents

19.3

19.4

19.5

19.6

19.7 19.8

19.2.5 Coating 626 Blow Molding 627 19.3.1 Extrusion Blow Molding 627 19.3.1.1 Parison Programming 629 19.3.1.2 Deep-Draw Blow Molding 631 19.3.1.3 Flashless Blow Molding of Tubular Parts 633 19.3.1.4 Multilayer Extrusion Blow Molding 634 19.3.1.5 Blow Molding with Encased Modules 637 19.3.2 Injection Blow Molding 640 19.3.3 Part Stiffening 642 19.3.4 Summary Comments 642 Rotational Molding 643 19.4.1 Rock-and-Roll Rotational Molding 650 19.4.2 Advantages and Limitations 651 19.4.3 Part Morphology 654 19.4.4 Part Design 655 19.4.4.1 Approaches to Part Stiffening 655 Thermoforming 659 19.5.1 Vacuum Forming 659 19.5.2 Pressure Forming 662 19.5.3 Plug-Assisted Thermoforming 662 19.5.4 Twin-Sheet Forming 665 19.5.5 Advantages and Limitations 667 19.5.6 Part Stiffening 667 19.5.7 Mechanical Forming 668 Expanded Bead and Extruded Foam 669 19.6.1 Expanded Bead Foam Molding 669 19.6.2 Extruded Foam 670 3D Printing 670 Concluding Remarks 672

20 Fabrication Methods for Thermosets 675

20.1 20.2 20.3 20.4 20.5 20.6

20.7

Introduction 675 Gel Point and Curing 675 20.2.1 Shelf Life of Precursors 678 Compression Molding 678 20.3.1 Compression Molding of Thermoplastics 680 Transfer Molding 681 Injection Molding 681 20.5.1 Injection-Compression Molding 683 Reaction Injection Molding (RIM) 683 20.6.1 Reinforced Reaction Injection Molding (RRIM) 684 20.6.2 Structural Reaction Injection Molding (SRIM) 685 Open Mold Forming 685

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20.8

Fabrication of Advanced Composites 686 20.8.1 Pultrusion 687 20.8.2 Filament Winding 688 20.8.3 Laminated Composites 692 20.8.3.1 Prepregs 693 20.8.3.2 Vacuum Bag Consolidation 693 20.8.3.3 Compression Molding 693 20.8.3.4 Pressure Bag Molding 693 20.8.3.5 Liquid-Resin Transfer Molding 694 20.8.3.6 Sandwich Structures with Prepreg Skins 697 20.9 Fabrication of Rubber Parts 698 20.9.1 Rubber Compounding 699 20.9.2 Dry Rubber Part Fabrication 699 20.9.2.1 Molding Processes 699 20.9.2.2 Extrusion 699 20.9.2.3 Calendering 700 20.9.2.4 Reinforced and Coated Rubber Sheet 700 20.9.3 Wet Rubber Part Fabrication 700 20.9.3.1 Dip Molding 701 20.9.3.2 Dip Coating 703 20.9.4 Manufacture of Reinforced Rubber Parts 703 20.9.4.1 Tires 703 20.9.4.2 Conveyor Belts 708 20.9.4.3 Pressure Hoses 708 20.10 Concluding Remarks 708 711 Introduction 711 Classification of Joining Methods 712 Mechanical Fastening 713 21.3.1 Snap Fits 713 21.3.2 Use of Screws 715 Adhesive Bonding 721 21.4.1 Solvent Bonding 722 Welding 722 Thermal Bonding 723 21.6.1 Hot-Gas Welding 723 21.6.2 Extrusion Welding 723 21.6.3 Hot-Tool (Hot-Plate) Welding 723 21.6.3.1 Weld Morphology 729 21.6.3.2 Weld Strength 732 21.6.4 Infrared Welding 737 21.6.5 Laser Welding 738 Friction Welding 741

21 Joining of Plastics

21.1 21.2 21.3

21.4 21.5 21.6

21.7

Contents

21.8

21.9

21.7.1 Spin Welding 742 21.7.2 Vibration Welding 742 21.7.2.1 Weld Morphology 746 21.7.2.2 Weld Strength 749 21.7.3 Orbital Welding 753 21.7.4 Ultrasonic Welding 753 21.7.4.1 Ultrasonic Staking, Spot Welding, Swaging, Insertion, and Embedding 756 Electromagnetic Bonding 762 21.8.1 Resistance (Implant) Welding 762 21.8.2 Induction Welding 763 21.8.3 Dielectric Welding 770 Concluding Remarks 770

PART V MATERIAL SYSTEMS

771 Outlines for Chapters 22 through 25

22 Fiber-Filled Material Materials – Materials with Microstructure 773

22.1 22.2 22.3 22.4 22.5

22.6

22.7

Introduction 773 Fiber Types 773 Processing Issues 774 Material Complexity 774 Tensile and Flexural Moduli 780 22.5.1 Homogeneous Bar in Tension and Bending 780 22.5.2 Nonhomogeneous Bar in Tension 781 22.5.3 Bending of Nonhomogeneous Bar in the Lower Stiffness Mode 781 22.5.4 Bending of Nonhomogeneous Bar in the Higher Stiffness Mode 783 Short-Fiber-Filled Systems 784 22.6.1 Tensile Modulus 785 22.6.1.1 Test Procedures 785 22.6.1.2 Directional and Spatial Modulus Variation 787 22.6.1.3 Repeatability of Modulus Data 790 22.6.1.4 Effects of Plaque Thickness on the Tensile Modulus 791 22.6.1.5 Effects of Injection Speed on the Tensile Modulus 799 22.6.2 Tensile and Flexural Strength 801 22.6.2.1 Test Procedures 803 22.6.2.2 Directional Tensile and Flexural Strengths 805 22.6.2.3 Variations in Tensile and Flexural Strengths 808 22.6.3 Effects of Fiber Aspect Ratio 812 22.6.4 Effects of Matrix Resin 813 22.6.5 Summary of Mechanical Characteristics of Short-Fiber Systems 815 Long-Fiber Filled Systems 817 22.7.1 Tensile Modulus 819

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22.8

22.9

22.7.1.1 Test Procedures 819 22.7.1.2 Tensile and Flexural Tests 820 22.7.1.3 Strength Variation Study 822 22.7.1.4 In-Plane Tensile Modulus Variations 822 22.7.2 Spatial and Directional Variations of the Tensile Modulus 826 22.7.3 Flow and Cross-Flow Mechanical Properties of Injection-Molded Plaques 828 22.7.4 Variations in Strength 831 22.7.5 Mechanical Properties for Design 832 *Fiber Orientation 833 22.8.1 *Orientation of a Single Fiber 833 22.8.2 *Fiber Orientation Distribution Function 835 22.8.3 **Orientation Tensors 836 22.8.4 *Fiber Orientation Measurement 839 22.8.4.1 Direct Measurement 839 22.8.4.2 Through-Thickness Variations of Orientation Tensor Components 841 22.8.4.3 Indirect Measurement 844 22.8.5 **Fiber Orientation Models 846 22.8.5.1 Jeffery’s Model 847 22.8.5.2 Dinh–Armstrong Model 848 22.8.5.3 Folgar–Tucker Model 850 22.8.6 **Fiber Orientation Prediction 851 Concluding Remarks 851 853 Introduction 853 Material Complexity 855 Foams as Nonhomogeneous Continua 856 23.3.1 Nonhomogeneous Bar in Tension 856 23.3.2 Bending of a Nonhomogeneous Bar in the Stiff Mode 857 23.3.3 Bending of a Nonhomogeneous Bar in a Reduced Stiffness Mode 858 Effective Bending Modulus for Thin-Walled Prismatic Beams 860 23.4.1 I-Section Beam 862 23.4.2 T-Section Beam 862 Skin-Core Models for Structural Foams 863 23.5.1 Four-Parameter Model 863 23.5.2 Three-Parameter Model 864 Stiffness and Strength of Structural Foams 866 23.6.1 Test Procedure for Acquiring Stiffness and Strength Data 867 23.6.2 Plaque-to-Plaque and In-Plaque Variations of Material Properties 868 23.6.3 Effect of Density on Mechanical Properties 873 23.6.4 Dependence of Mechanical Properties on Plaque Thickness 875 23.6.5 Summary Comments 877 The Average Density and the Effective Tensile and Flexural Moduli of Foams 879 23.7.1 Test Procedure 879

23 Structural Foams – Materials with Millistructure

23.1 23.2 23.3

23.4

23.5

23.6

23.7

Contents

23.7.2 In-Plane Density Variations 881 Density and Modulus Variation Correlations 884 23.8.1 Density-Modulus Correlation for 6.35-mm Thick Foam 884 23.8.2 Density-Modulus Correlation for 4-mm Thick Foam 886 23.9 Flexural Modulus 887 23.10 **Torsion of Nonhomogeneous Bars 890 23.10.1 **Basic Equations for Modified Saint Venant’s Theory 891 23.10.2 **Torsion of Thin-Walled Rectangular Bars 893 23.10.3 **Torsion of Thin-Walled Open Prismatic Sections 895 23.10.4 **Torsion of Thin-Walled Tubes 895 23.11 Implications for Mechanical Design 898 23.12 Concluding Remarks 899

23.8

901 Introduction 901 GMT Processing 901 Problem Complexity 904 Effective Tensile and Flexural Moduli of Nonhomogeneous Materials 906 24.4.1 Tensile Test 906 24.4.2 Three-Point Flexural Test 908 24.5 Insights from Model Materials 909 24.5.1 Model Material with Sinusoidally Varying Modulus 909 24.5.1.1 Effective Tensile Modulus 909 24.5.1.2 Effective Flexural Modulus 911 24.5.1.3 Effect of Gauge Length on Modulus Distribution Measurement 913 24.5.2 Model Material with Rectangular Wave Modulus Variation 918 24.5.3 Summary of Lessons Learned from Model Materials 919 24.6 Characterization of the Tensile Modulus 921 24.6.1 Cross-Machine-Direction Tensile Moduli 921 24.7 Characterization of the Tensile Strength 924 24.7.1 Test Procedure 924 24.7.2 Machine-Direction Tensile Modulus and Strength Data 925 24.7.3 Cross-Machine-Direction Tensile Modulus and Strength Data 929 24.7.4 Comparison of Machine- and Cross-Machine Direction Strength Data 932 24.8 Statistical Characterization of the Tensile Modulus Experimental Data 934 24.8.1 Histograms for Tensile Modulus Data 935 24.8.2 *Moments of the Tensile Modulus Distributions 935 24.8.3 Probability Density Function for the Tensile Modulus 940 24.8.4 Higher Order Moments 941 24.9 Statistical Properties of Tensile Modulus Data Sets 943 24.9.1 Correlation Between the Left and Right Moduli 943 24.9.2 Linear Combination of Two Independent Random Variables 944 24.10 Gauge-Length Effects and Large-Scale Material Stiffness 946 24.10.1 Sample Size: Theoretical Considerations 947

24 Random Glass Mat Composites – Materials with Macrostructure

24.1 24.2 24.3 24.4

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24.10.2 Sample Size: Numerical Experiments 948 24.11 Methodology for Predicting the Stiffness of Parts 951 24.11.1 *Effective Structural Stiffness 957 24.11.2 Numerical Procedure 959 24.11.3 Some Numerical Results 960 24.12 *Statistical Approach to Strength 962 24.12.1 *State of Material Loading 962 24.12.2 *Interpretation of Measured Strains: Left and Right Moduli 963 24.12.3 *Correlation of Strength with Tensile Modulus 964 24.12.4 *Failure of Long Dog-Bone Tensile Samples 965 24.12.5 *Corrections for the Randomness of the Stress Field 967 24.12.6 Summary Comments 968 24.13 Implications for Mechanical Design 969 24.14 Concluding Remarks 969 25 Advanced Composites – Materials with Well-Defined Reinforcement Architectures 973

25.1 25.2

25.3

25.4

Introduction 973 Resins, Fibers, and Fabrics 974 25.2.1 Matrix Resins 974 25.2.2 Reinforcing Fibers 974 25.2.2.1 Glass Fibers 974 25.2.2.2 Carbon Fibers 975 25.2.2.3 Aramid Fibers 976 25.2.2.4 Polyethylene Fibers 976 25.2.2.5 Nylon Fibers 976 25.2.3 Reinforcing Tapes and Fabrics 976 Advanced Composites 977 25.3.1 Pultruded Composite Sections 977 25.3.2 Filament-Wound Composites 977 25.3.3 Laminated Composites 981 25.3.3.1 Mechanical Properties of a Laminae 981 25.3.3.2 Mechanical Properties of Laminae Stacks 982 25.3.3.3 Analysis of Laminate Structures 983 25.3.3.4 Defects and Failure Modes 984 25.3.4 Resin Transfer Molded Composites 985 25.3.5 Sandwich Structures 987 25.3.5.1 Defects and Failure Modes 987 25.3.6 Summary Comments 989 Rubber-Based Composites 990 25.4.1 Tires 990 25.4.1.1 Automotive Tires 990 25.4.1.2 Deformation of Tires 992 25.4.1.3 Tread Design 995 25.4.1.4 Large Heavy-Duty Tires 999

Contents

25.5

25.4.2 Reinforced Rubber Conveyor Belts 1000 25.4.3 Pressure Hoses 1003 25.4.4 Summary Comments 1008 Concluding Remarks 1008

Index 1011

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Series Preface The Wiley-ASME Press Series in Mechanical Engineering brings together two established leaders in mechanical engineering publishing to deliver high-quality, peer-reviewed books covering topics of current interest to engineers and researchers worldwide. The series publishes across the breadth of mechanical engineering, comprising research, design and development, and manufacturing. It includes monographs, references and course texts. Prospective topics include emerging and advanced technologies in Engineering Design; ComputerAided Design; Energy Conversion & Resources; Heat Transfer; Manufacturing & Processing; Systems & Devices; Renewable Energy; Robotics; and Biotechnology.

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Preface This book is an introduction to plastics engineering, the process for converting plastics into useful products. It is influenced by mechanical engineering that has evolved over a period of 150 years into an efficient process for converting materials, mainly metals, into a large number of useful products. A novel feature of this book is its synergistic focus on all aspects of materials use – concepts, mechanics, materials, part design, part fabrication, and assembly – required for converting plastic materials, mainly in the form of small pellets, into products. And it integrates the applications of four disparate polymer types – thermoplastics, thermosets, elastomers, and advanced composites – that normally are treated as separate subjects. The difference between plastics engineering as currently understood in the plastics industry – mainly focused on materials science aspects of plastics – and the paradigm developed in this book, is best explained by the synergy and differences between materials science and engineering. Materials science, a science-based discipline, is mainly concerned with synthesizing and modifying the properties of materials to obtain desirable characteristics. This process requires an understanding of how the molecular structure of a material affects its microstructure that, in turn, affects its macro continuum properties. In contrast, the engineering process, concerned with converting materials into useful products, uses the principles of mechanics, or engineering science, as the basic tool for understanding part design, part performance, and part fabrication issues. For using materials, materials science and engineering are separate but complementary disciplines that require different skill sets and training. The material aspects for metals were earlier covered by metallurgy, the discipline from which modern materials science has evolved. Mechanical engineering, concerned with the design and fabrication of structural components of the widest variety of product types, provided the engineering tools for rationally converting metals into useful products. In addition, electrical engineering developed to exploit the electrical and magnetic properties of materials. For historical reasons, the current paradigm for plastics engineering – which emphasizes materials aspects with some consideration of processing issues, and mainly draws on chemical engineering to provide rheological (fluid flow) principles for processing – is heavily biased toward materials science, and has not benefited from mechanics-based engineering (engineering science) principles that routinely are used for designing metal parts. Plastics engineering, as currently understood in the plastics industry, does not rationally address important issues for mechanical design, part performance, and part fabrication. Most applications have evolved by using iterative build-and-test methods. The higher tooling costs of larger parts and the need to reduce the product development cycle – especially higher load-bearing parts such as automotive bumpers – are driving the increasing use of basic engineering principles.

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This book provides a self-contained introduction to plastics engineering, synergistically encompassing materials and engineering aspects. This paradigm is an adaptation of mechanical engineering methodology to plastics materials, requiring a systematic incorporation of mechanics principles and a consideration of shape synthesis for plastic parts. Each facet of plastics engineering is presented at an elementary level, so that portions of the book may appear overly simplistic to specialists. However, although elementary, portions may be somewhat difficult for some. To facilitate the integration of materials and engineering aspects of plastics engineering the book is divided into five parts: Introduction – which includes a consideration of evolving concepts for plastics use – Mechanics, Materials, Part Processing and Assembly, and Material Systems. In addition, as a guide to the book, each part is preceded by a separator with outlines of the chapters in that part. In Part I, Chapter 1 provides a broad survey of plastics engineering, essentially a summary of the book: What plastics are and why they are becoming important engineering materials, the synergy between materials science and engineering, the effect of material type on part shape, the role of engineering science in part design and part processing, and the role of assembly methods. It also is a summary reference for readers as they progress through the succeeding chapters. The second chapter in this part focuses on evolving concepts for plastics use; a wide range of applications have been described to demonstrate how much plastics have penetrated almost every sphere of human activity. Understanding the process for selecting appropriate part shapes – a nontrivial exercise for plastics that requires looking beyond shapes appropriate for metals – is important because first attempts at using plastics consisted of one-for-one metal part replacement. This strategy failed because it did not account for the enormous complexity that can be built into single plastic parts, which is what makes the use of otherwise expensive plastics economical. Plastics used in an assembly can cost more than the cost of the metals replaced. It is through innovative designs and cost-effective fabrication (part processing) techniques that, in many applications, plastics have an edge over metals. Part II of the book is a self-contained introduction to all simple principles required for understanding the role of mechanics in plastics engineering. While mechanical engineers are well grounded in the fundamentals and applications of mechanics, the lack of an adequate exposure to mechanics can be a major barrier for plastics engineers to look beyond a materials mindset. Because an understanding of mechanics principles is an important aspect of the plastics engineering paradigm discussed in this book, concepts such as stress, strain, and constitutive relations are introduced at a very elementary, one-dimensional level. However, elementary differential and integral calculus are prerequisites. Some relatively advanced two-dimensional topics are preceded by an asterisk; still more advanced topics are indicated by two asterisks. While such topics are not essential for understanding the more elementary discussions, they do give an idea of the level of mathematics required for the next level of analyses for designing and processing actual parts. Simple material models are used to analyze the behavior of structural elements such as beams, torsion members, and columns. Another chapter addresses models for the behavior of fluids. And, in view of its importance and complexity, one-dimensional viscoelasticity is discussed in a separate chapter. The last chapter in Part II discusses mechanics principles underlying the stiffening of plastics by embedded fibers. For providing a better motivation, this chapter could also have been placed after the gross behavior of plastics has been introduced in Part III. Part III addresses materials issues: The chemical structure and classification of polymers, how structure affects properties, and the modification of properties through additives. However, polymer synthesis and related chemistry issues are not addressed, as are not methods for characterizing the properties of polymeric materials.

Preface

The first chapter in Part III introduces polymers; two simple polymers are used to illustrate basic concepts, and polymers are classified into three types – thermoplastics, thermosets, and elastomers. The next chapter introduces concepts from polymer physics to explain the macro behavior of plastics and to contrast their deformation mechanisms from those of metals. The following three chapters describe, respectively, the chemical structure and properties of industrially important plastics, plastic blends and alloys – the plastics equivalent of alloys in metals – and thermosets, including elastomers. Because of its importance for understanding mechanical behavior, one chapter in this section addresses the viscoelastic properties of plastics. In addition, a large chapter describes the phenomenology of the mechanical behavior of plastics. Electromagnetic and optical properties of plastics have not been addressed. Part IV addresses part shaping (part processing) and assembly issues: The first chapter discusses the evolution and classification of processing methods, and the coupling among part shape, part processing, and part performance. The second chapter discusses injection molding and its variants in detail. The third chapter is devoted to shrinkage and warpage issues. The fourth chapter describes several alternatives to injection molding of thermoplastics. The fifth chapter in this part describes fabrication methods for thermosets. And the last chapter addresses assembly issues. To highlight the system-like behavior of filled materials, Part V treats them as material systems in which the material behaves more like a part than as a homogeneous material. This distinction is important because the existing paradigm for plastics engineering treats filled materials as homogeneous, isotropic materials, the mechanical properties of which parallel those of unfilled plastics or metals. Quite to the contrary, in such filled systems the local properties can be nonhomogeneous and anisotropic, and these properties depend on the part shape and processing conditions. Separate chapters in this part discuss fiber-filled materials, which have a microstructure; structural foams, which have structures on a millimeter scale; random glass mat composites, which have a macrostructure on the scale of centimeters; and advanced composites, which use fibers in well-defined reinforcing geometries. Books on materials science and plastics engineering use elastomers (rubbers) as model materials to demonstrate the use of thermodynamics for predicting mechanical properties. This topic is chosen mainly because of the relative simplicity of rubber elasticity theories, and not because rubbers are technologically more important than plastics. This topic has not been addressed because of the scope of this book. Other topics not covered include polymer synthesis, polymer characterization methods such as differential scanning calorimetry (DSC) and nuclear magnetic resonance (NMR), polymer rheometry, and methods for characterizing polymer viscoelasticity. Also not addressed is the important topic of weathering, concerned with environmentally induced changes in the chemical and physical behavior of plastics, which can drastically reduce their mechanical performance and life. Some background on how this book came to be written may be of interest. On joining GE Corporate Research & Development in 1978, the author, a mechanical engineer specializing in continuum mechanics, was looking for a long-term project to work on. A chance visit to GE’s Major Appliance Division introduced him to a molded Permatuf ® (talc-filled polypropylene) dishwasher tub that replaced one made of welded, deep-drawn steel with an enameled interior. His understanding of how plastics were coming of age as engineering materials was reinforced by a demonstration of the enormous strength and ductility exhibited by polycarbonate, laminated sheets of which can stop bullets. At that time GE had a $250 million plastics business that, in addition to selling thermosets that had evolved from the company’s interest in developing better insulating material for its electrical businesses, was marketing several higher performance, relatively more expensive thermoplastics such as poly(phenylene ether), poly(butylene terephthalate), and polycarbonate.

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Of the several applications that can be used to mark the transition of plastics use from structurally non-demanding applications – in which esthetics and part cost are important – to demanding structural applications, the most noteworthy is the 1984 Ford Escort bumper designed to withstand 8-kmph (5-mph) impacts. Early meetings provided an interesting perspective on how existing plastics were being marketed for this application: One business division offered polycarbonate (PC) as a tough material that could stop bullets. Another division countered by stating that while PC was tough it would not withstand the chemical effects of gasoline, and offered poly(butylene terephthalate) (PBT) as an alternative material with good chemical resistance. If one material did not make the cut, there was a tendency to offer an alternative material, without understanding the engineering needs of the application. It took some time for the company to reorganize its marketing strategy to offer only the most appropriate material in its portfolio. In the end, a special blend of PC and PBT was developed for this application. The mechanical design of such demanding structural applications requires the use of mechanics principles, an understanding and the use of which, the author was surprised to learn, was conspicuously absent in what was called plastics engineering. Chemists and marketers, a smattering of chemical engineers, and the hands-on molding community – the expertise of which is what makes innovative products possible – dominated the plastics industry. The early 1980s were interesting times for plastics that were making inroads into the large automotive market. During this period, GE Plastics grew 20-fold in 5 years from a $250-million Plastics Business Operation to a $5-billion Plastics Division. A realization within its growing marketing operation, in that engineering principles could be used to market GE’s relatively more expensive plastics by demonstrating their use in demanding structural applications, opened new opportunities for mechanical engineers. This resulted in the author and his colleagues initiating a program to develop a comprehensive, mechanics-based technology for the mechanical design of plastic parts. The author worked on this exciting project with a team of talented mechanical, aerospace, and chemical engineers at GE Corporate Research & Development – G.G. Trantina, R.P. Nimmer, H.F. Nied, H. deLorenzi, A.J. Poslinski, and W.C. Bushko, ably assisted by H. Moran, L.P. Inzinna, and K.C. Conway – to understand what plastics were, how they could be characterized in an engineering sense, and how parts could be rationally designed, fabricated, and assembled. In particular, the author treasures memories of having collaborated with Dr. Wit Bushko on several projects. Chapter 18 is largely based on one such project; his critique of an early draft is very much appreciated. Engineers also had much to learn: With most engineers, there was a tendency to assume that plastics materials were like metals but with lower material properties; all that was needed was to measure their properties and to apply the design methodology used for metals to plastic parts. It soon became clear, however, that plastics were qualitatively different, so that their optimal use required a much deeper understanding of their mechanical behavior. For example, the plastics community was blindly applying standard materials characterization techniques for metals to all plastics. Such techniques cannot provide meaningful data for many plastics applications already in use, such as structural foams – nonhomogeneous “porous” materials with submillimeter-sized voids – and random glass mat composites, in which material properties can vary by a factor of two over a 15-mm length scale. Over a 15-year period, a large body of useful information was generated on different aspects of plastics. During this period the author was actively involved in research, working with GE’s plastics business – for which engineering mechanics was a marketing tool for their relatively more expensive plastics – and participating in technical conferences of the Society of Plastics Engineers, which was dominated by

Preface

marketing people, applied chemists, and the molding community. Struggling with the disparate perspectives of these three organizations helped the author to come up with a new paradigm for what plastics engineering ought to be. This book, the writing of which began in the year 2000, is the author’s attempt to put together an introductory course on plastics engineering. While it does have chapters on the constitution of polymers – normally, equivalent books in mechanical engineering do not include metallurgical aspects of metals – the focus of this book is on the processes required for converting plastics into useful products. Originally, the book was only to be about thermoplastics. When the resulting “story” seemed incomplete, thermosets, and then elastomers were included. Similarly, the last chapter on advanced composites – normally not included in plastics – was added to complete the story. The plastics industry really comprises four relatively disparate parts that this book presents in a unified manner as integral parts of plastics engineering: (i) Traditionally, the plastics industry refers to the manufacture of thermoplastics and thermosetting resins, and their conversion into parts using a host of part processing (fabrication) methods that have been developed for large-volume, low-cost part production. The success of this industry is based on part complexity and low-cost parts manufacture, both of which depend on innovations in tooling (molds) and fabrication methods. Part design is mostly based on build-and-test methods – mechanics principles are sparingly used. (ii) Paralleling the plastics industry is the older rubber industry, concerned with manufacturing elastomers, dominated by rubbers, and making rubber parts. (iii) At one extreme is the advanced composites industry that mainly caters to the aerospace industry. Advanced composites are made by embedding thin, very strong continuous glass or carbon fibers in a resin matrix, which mainly consist of high-performance, rigid thermosetting resins; except in niche applications, attempts at using the relatively more flexible thermoplastic resins have not been successful. In contrasts to plastics, the design of advanced-composite parts requires state-of-the-art mechanics. However, part manufacturing for the low-volume parts for the aerospace industry is very expensive. And (iv), the tire industry that parallels the advanced composites industry in the sense that it uses continuous fibers and rubber matrices, resulting in very flexible advanced composites. This book was written over a 17-year period. It took much effort to arrange, what at first seemed to be disparate areas, into a book on plastics engineering. It has been a lonely journey, first because the material is not arranged following the current trend; second, because this book has not been tried in a classroom setting – and has therefore not benefited from critical feedback; and third, because the chapters have not been reviewed by anyone. As such, this book is bound to have more errors than the norm for such books; the author would be very grateful for errors being pointed out and for suggestions for improving the book. Hopefully, with these caveats, this book will serve as a “straw man” for what plastics engineering ought to be. A note on style: To stimulate understanding, some statements end with “(Why?).” For example, in the sentence “While a rectangular beam cross section has been assumed, the assumptions for deriving this result, such as plane sections remaining plane, are valid for all prismatic beams with cross sections that are symmetric about the y-axis (Why?).,” the “(Why?)” is inserted to encourage the reader to figure out the basis for the statement. And, as pointed out earlier, some relatively advanced topics preceded by one or two asterisks are not parts of the main story, and can therefore be ignored. One reason for including such “advanced” material is to show that the next level of more detailed analyses requires a deeper understanding of mechanics, for which a much higher level of mathematics is a prerequisite. Permission from copyright holders of photos and diagrams now requires that each use be acknowledged in a specified format that varies with copyright holders. This should explain the rather long, “repetitive” acknowledgments at the ends of figure captions.

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1

Part I Introduction Besides introducing plastics, the two chapters in this part provide a comprehensive overview of plastics use.

Chapter 1 Introductory Survey The broad survey of plastics engineering in this chapter – essentially a roadmap for the book – also provides a summary reference for readers as they progress through the succeeding chapters: What plastics are and why they are becoming important engineering materials, the synergy between materials science and engineering, the effect of material type on part shape, the role of engineering science in part design, part processing, and assembly methods. The example of how the basic shape of a bridge changes with the material – stone, cast iron, wrought iron, and steel – is used to show how optimal shapes of objects depend on material properties. The complexity possible in a plastic part is illustrated by the example of a scissor-like pair of forceps made by a one-step injection-molding operation.

Chapter 2 Evolving Concepts for Plastics Use A large number of examples – such as bullet-proof vests, athletic shoes tooth brushes, safety razors, toys, audio equipment, household appliances, sewer systems, large pipes, glazing, many automotive applications, medical devices, large structures, large wind-turbine blades, and large rubber tires – are used to illustrate the very diverse, evolving concepts for plastics use. An understanding of the process for selecting appropriate concepts for parts – a nontrivial exercise for plastics that requires looking beyond shapes appropriate for metals – is important because first attempts at using plastics consisted of one-for-one metal part replacement. This strategy failed because it did not account for the enormous complexity that can be built into single plastic parts, which is what makes the use of otherwise expensive plastics economical.

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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1 Introductory Survey 1.1 Background Plastics are a relatively new class of materials invented during the past century, mostly over the past 75 years. Here are examples of some common plastics still in use together with the years they were invented: Bakelite (phenolic), 1909; PVC (polyvinyl chloride), 1933; nylon, 1934; Plexiglas® (polymethyl methacrylate), 1934; polystyrene, 1938; Teflon® (polytetrafluoroethylene), 1938; Orlon® (polyacrylonitrile), 1941; polyethylene, 1942; Araldite® (epoxy) 1943, Dacron® (poly(ethylene terephthalate)), 1953; polypropylene, 1954; PUR (polyurethane), 1954; LexanTM (polycarbonate), 1955; Delrin® (polyoxymethylene), 1959; Kapton® (polyimide), 1961; Udel® (polysulfone), 1965; NorylTM (poly(phenylene oxide)), 1966; Ryton® (poly(phenylene sulfide)), 1972; Kevlar® (aramid fibers), 1972; Torlon® (poly(amide-imide)), 1972; Ardel® (polyarylate), 1978; VictrexTM (polyetheretherketone, or PEEK) 1979; and UltemTM (polyetherimide), 1982. With the exception of advanced composites, mainly used in aerospace and defense applications, until recently, plastics were not considered engineering materials. Now plastics have become important for several reasons. First, their cost-effective use has been demonstrated in demanding automotive structural applications. An important example is the all-plastic bumper used in the 1984 Ford Escort, which was capable of withstanding an 8-kmph (5-mph) barrier impact (Figure 2.4.1). This bumper demonstrated that neat resins can be used in structural applications without the use of reinforcing fibers. Another example is that of injection molded fenders that resist denting (Figure 2.4.2). Second, their use allows styling freedom in design, in the sense that parts with complex surfaces can easily be made. Third, relative to metals, they make parts with enormous complexity possible, thereby cutting inventory and assembly costs while at the same time reducing part weight. First attempts at using plastics consisted of one-for-one metal part replacement. This approach did not succeed because it did not capitalize on the complexity that can be built into single parts, which is what makes the use of otherwise expensive plastics economical. Plastics used in an assembly tend to cost more than the cost of metals replaced. In many applications, it is innovative designs and cost-effective part fabrication (part processing) techniques that give plastics an edge over metals. For historical reasons discussed in the sequel, the use of plastics has not benefited from mechanics-based, or engineering-science-based, engineering principles that are routinely used for designing metal parts. Rather, much of the work in plastics use was carried out by applied chemists, materials scientists, and molders, with the main focus being on material properties. With the increasing use of plastics in engineering applications, they can benefit from the use of engineering principles. Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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This book is about plastics engineering, the process for converting plastics into useful products. The methodology is an adaptation of mechanical engineering that, over a period of 150 years, has evolved into an efficient process for converting materials – mainly metals – into a large number of useful products. This book focuses on all the processes involved in converting plastic materials, mainly in the form of small pellets, into products. To clarify the difference between plastics engineering as currently understood in the plastics industry and the paradigm followed in this book, it is necessary to first understand the synergy and differences between materials science and engineering.

1.2 Synergy Between Materials Science and Engineering Use of materials depends on the synergy between the disciplines of materials science and engineering as schematically shown in Figure 1.2.1. Materials science is mainly concerned with synthesizing and improving the properties of materials to obtain desirable characteristics. This process requires an understanding of how the molecular structure, or architecture, of the material – mainly on a nanometer scale – affects its microstructure on a micrometer scale that, in turn, affects its macro-continuum properties on an applications scale. Thus, this science-based discipline, which mainly uses chemistry and physics, spans the nanometer to millimeter scales. The focus of this activity is mainly on the microstructure of materials and on how this microstructure affects the continuum properties of materials. The tools used to elucidate the structure of plastics range from analytical chemical analysis tools such as nuclear magnetic resonance (NMR) to understand the molecular structure, to transmission electron microscopy (TEM) and optical microscopy to understand the microstructure of the material, and to rheometers and mechanical testing machines to study the continuum properties of the material. For correlating the macro behavior of materials to their molecular morphology, physical properties are used as measures for macro behavior. In contrast to materials science, engineering is concerned with converting materials into useful products. This process, which normally works in the millimeter to kilometer scales, is not concerned with developing or modifying the properties of materials. However, engineering processes are now being applied to devices on a micrometer scale, and nano-scale devices are under active consideration. The engineering process starts with materials characterization to establish the engineering properties of the material – numbers representing continuum properties that, in conjunction with mechanics-based algorithms, can be used for predicting part performance. This process uses continuum mechanics (applied mechanics) as the basic tool for predicting part performance. Thus, materials science and engineering are separate but complementary disciplines that require different skill sets and training. For metals, the material aspects were earlier covered by metallurgy, the discipline from which modern materials science evolved. And mechanical, aeronautical, and civil engineering are concerned with the design and fabrication of structural components of increasing size. Mechanical engineering covers the widest variety of product types and therefore offers a model paradigm for plastics engineering. Similarly, electrical engineering is concerned with exploiting the electrical and magnetic properties of materials. In the latter half of the nineteenth century, as the use of iron-based material grew, metallurgy played an important role in developing new steels and alloys that helped mechanical and civil engineers push the envelope of applications – larger and more rugged machines and longer bridges capable of supporting trains. For about 50 years, mechanical engineering and metallurgy progressed synergistically: The needs

Polymers

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Figure 1.2.1 Schematic diagram showing the synergy and differences between materials science and engineering.

of engineers spurred metallurgists to develop better materials, and newer materials developed by metallurgists helped engineers to push a wider range of applications. Once each understood the other, they began to grow independently. And, as ceramics and polymeric materials developed into engineering materials, metallurgy metamorphosed into material science. Because early plastics were used in non-structural, non-demanding applications, the material aspects of polymers evolved relatively independently of engineering aspects.

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Plastics represent a relatively new class of materials, the basic constitution of which as polymeric materials was not established till the mid-1920s. Most of the plastics in use today were developed only in the last 75 years. Developing plastics evolved as a discipline of applied chemistry. Later, techniques for converting (processing) plastics into parts evolved through innovation but without the full benefit of engineering principles: Early on, engineers did not consider plastics as engineering materials, those suitable for load-bearing applications. As a result, the use of plastics did not benefit from powerful mechanics-based engineering principles that evolved over the last 150 years. Most applications evolved by using iterative build-and-test methods. The growing use of plastics in structural applications, higher tooling costs of larger parts, and the need to drastically reduce the product development cycle are now driving the increasing use of engineering principles. As plastics began to be used in structural applications, especially in the automotive sector, the disconnect between materials and engineering aspects of plastics became clear in the 1980s: Of the several applications that can be used to mark the transition of plastics use from structurally non-demanding applications – in which esthetics and part cost are important – to demanding structural applications, the most noteworthy is the 1984 Ford Escort bumper mentioned previously. Early meetings provided an interesting perspective on how existing plastics were being marketed for this application by materials-oriented persons: One business division offered polycarbonate (PC) as a tough material that could stop bullets. Another division of the same company countered by stating that while PC was tough it would not withstand the chemical effects of gasoline, and offered poly(butylene terephthalate) (PBT) as an alternative material with good chemical resistance. If one material did not make the cut, there was a tendency to offer an alternative material without understanding the engineering needs of the application. Engineers who could not understand this approach asked an industrial designer to come up with the cartoon shown in Figure 1.2.2.

Figure 1.2.2 New materials as solutions to engineering design problems. (Cartoon courtesy of David Muyres.)

But engineers also had much to learn too: While they would change the “square wheel” to a circular one, it would still be of stone. Modern pneumatic tires were made possible by the development of rubber, clearly made possible by materials-oriented activity. With most engineers, there was a tendency

Introductory Survey

to assume that plastics materials were like metals but with lower material properties; all that was needed was to measure their properties and to apply the design methodology used for metals to plastic parts. It soon became clear, however, that plastics were qualitatively different, so that their optimal use required a much deeper understanding of their mechanical behavior. For example, the plastics community was blindly applying standard material characterization techniques for metals to all plastics. Such techniques cannot provide meaningful data for many plastics applications already in use, such as, structural foams – nonhomogeneous “porous” materials with submillimeter-sized holes – and random glass mat composites, in which material properties can vary by a factor of two over a 15-mm length scale. The older paradigm for plastics engineering combines materials and processing issues – mainly drawing on chemical engineering to provide rheological principles for processing. But it does not include the systematic use of mechanics principles for rational mechanical design, evaluation of part performance, and rational process design and evaluation.

1.3 Plastics Engineering as a Process (the Plastics Engineering Process) Learning to efficiently use materials has to be based on an understanding of material behavior and principles of part design, fabrication, and assembly – all four of which require an understanding of mechanics. The appropriate base discipline for this is mechanical engineering – essentially the process for converting materials into useful products – which is divided into a four-step process of materials characterization, part design, part fabrication, and assembly. These activities are supported by the engineering sciences, or mechanics, which, in turn are based on science. This mechanical engineering-based paradigm for plastics engineering is schematically shown in Figure 1.3.1. The materials box pertains to materials selection and characterization. The design box pertains to conceptual design (shape synthesis) of a part and evaluating its performance; it includes determining the part size and its features to meet performance goals. The processing, or fabrication, box covers shaping materials into geometries determined during the design phase. And the assembly box covers issues such as joining and welding. In using plastics, the four activities of materials characterization, part design, part fabrication, and assembly are interdependent and far more strongly coupled than in the use of metals. Plastics engineering comprises addressing all issues relevant to converting plastics into useful products. In general, the engineering process starts with a conceptual design aimed at fulfilling a specified need. The engineer conceives a device or machine that, in principle, will fulfill the need – with or without an external power source. Developing an appropriate conceptual design is a process of synthesis, and is, perhaps, one of the more creative aspects of engineering. This key process may result in a new invention. The next step involves selecting appropriate materials and part fabrication and assembly techniques for a physical realization of the concept. This is an iterative process in which the technical and economic feasibility of using a slate of materials and processing technologies is evaluated. Once materials and processes have been chosen, part performance is evaluated to ensure that performance goals – such as allowable deflections and load carrying capacity – are met. It may also involve an analysis of the fabrication process to ensure that the part can be formed to specifications. And since the functions that can be packaged into a part made in a one-step fabrication process are limited, the final functional part or device has to be assembled from subcomponents or subassemblies. Prediction of part and process performance requires the use of the engineering sciences, or mechanics – essentially consisting of classical physics grouped into discipline-based modules such as solid

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Figure 1.3.1 Paradigm for plastics engineering.

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mechanics, fluid mechanics, heat transfer, and thermodynamics – which is the analysis arm of the engineering process. While they are essential for the efficient and timely conversion of materials into useful products, by themselves the engineering sciences do not constitute engineering. Since engineering curricula rightly tend to emphasize fundamentals, that is, engineering sciences, it is sometimes easy to forget the distinction between engineering and engineering science – especially if their role in the engineering process is not emphasized.

PARADIGM FOR PLASTICS ENGINEERING

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1.4 Types of Plastics Plastics are polymeric materials that can be classified into two broad categories: Thermoplastics and thermosets. Thermoplastics can be melted and shaped without altering the chemical structure. They can further be divided into amorphous resins – glass-like materials that have no short-range structure – such as poly(vinyl chloride), polystyrene, poly(methyl methacrylate), polycarbonate, and polyetherimide; and semicrystalline resins – that consist both of crystalline and amorphous regions – such as polyethylene, polyoxymethylene, nylon, and PEEK. In contrast to thermoplastics, thermosets undergo chemical reactions during processing (fabrication) into parts that, once formed, cannot be further shaped except by machining. An increase in temperature results in material degradation rather than in melting or softening. Thermosets, such as phenolics and epoxies, have low ductility and tend to be brittle. Because of the chemical reactions that occur during part forming, thermoset molding usually involves longer cycle times. While thermoplastics can be economically recycled, thermosets cannot. Two or more thermoplastics can be combined to obtain materials with desired specific properties. Such materials are called blends; borrowing from metals technology, they are also referred to as alloys. Developing and commercializing new resins is an expensive process. Blending offers a faster and less expensive means for obtaining materials with desired properties. In contrast to thermoplastics that have very high melt viscosities, thermosets can more easily fill small pores and wet out fibers because of their low initial viscosities. Thermosets have therefore been used as matrix resins in highly filled continuous fiber and fabric reinforced composites. However, thermoplastic cyclic resins, consisting of small numbers of monomers joined end-to-end to form a ring, with low initial viscosities like those of thermosets have been developed. After impregnating fiber preforms with a cyclic resin, ring-opening polymerization induced by a thermally activated catalyst converts the cyclic resin into a high molecular weight thermoplastic. Another class of plastics is elastomers, rubberlike materials that have low elastic moduli, which can recover from large deformations. Most elastomers are thermosetting materials, but thermoplastic elastomers are now available. 1.4.1

Plastic Composites

In structurally demanding applications plastics are seldom used as neat resins. Instead, fillers are used for a variety of reasons, such as reducing cost, increasing stiffness, reducing shrinkage, increasing impact resistance, enhancing performance at low and elevated temperatures, and improving flame retardance. Plastics and their composites can be classified into several broad categories: • Neat resins, including blends and resins with small amounts of impact modifiers or flame retardants. • Structural foams (Figure 23.1.2). • Particulate-filled composites. • Chopped-fiber filled composites (Figure 22.4.5). • Random continuous fiber mat composites (Figure 24.2.1). • Fabric mat and braided composites (“advanced composites”) (Figure 25.3.1). • Unidirectional fiber composites (“advanced composites”) (Figures 25.3.1 and 25.4.3). This sequence represents a progression of increasing material stiffness and cost. Not included are microcellular foams – cellular materials having 2- to 25-μm-sized pores – and nanocomposites – in which

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nanometer-sized finely dispersed particles dramatically improve the properties of resins. Clearly, plastics and their composites cover a very broad range of material types, with very different morphologies and mechanical properties. Unidirectional fiber composites, which normally refer to materials with rigid thermoset matrices, provide the highest performance in terms of stiffness and strength. Parts are fabricated from unidirectional resin-impregnated fiber “pre pregs” or tapes, and therefore have a layered structure in which the predetermined orientation in each layer may be different; orientation can be used to attain desired directional properties. Parts can also be made by using filament wound preforms. The most commonly used fibers are of glass, graphite, and Kevlar. In the past, such high-performance composites have had thermoset matrices, the most common being epoxy. More recently, there has been a move to develop high-performance thermoplastic composites. Another class of advanced composites uses elastomers (rubbers) as the matrix. The largest use of such flexible composites is in tires for automotive and other transport vehicle applications. 1.4.2

Recycling of Plastics

During use, chemical reactions within the ambient degrade most metals, and the degraded material merges into the environment; heavy metals can have very harmful effects. In any case, metals used in parts can be separated and recycled into making virgin metal; recycling of metals from parts is a well-established industry. In contrast to metals, plastics do not degrade – or degrade very, very slowly – and are difficult to recycle. The pervasive use of plastics in almost all spheres of human activity – especially in produce and shopping bags, water and beverage bottles, throwaway plastic “crockery” and “cutlery,” and net bags – is resulting in non-degrading plastic ending up in landfills, lakes, and the sea, and also in litter along the countryside. One legislative remedy for the litter is to tax plastic containers – this practice already is in place for water and soda bottles – which can be refunded when the empty container is returned. But the real solution is recycling. Because the use of plastics grew very rapidly and pervasively – much faster than the ability of the public to have anticipated the effects of plastics on the environment – recycling of plastics was not seen as a viable business opportunity. But, as with metals, the environmental cost of plastics is driving the need for recycling; growing demand will eventually lead to system-oriented recycling businesses. Because thermoplastics melt on heating, they are relatively easy to recycle; the main barrier is separating different types of plastics from each other and from non-plastic materials. Depending on the purity of the recycled material, it can be used either as virgin resin or used as a filler inside multilayer plastic parts; in the latter case, the recycled plastic is referred to as regrind material. Thermosets do not melt, and only degrade on heating; chemical means for recycling them are very limited. Most recycling methods either involve (i) mechanical conversion into powdered fillers for other plastic products, or (ii) the use of thermal processes such as combustion to produce energy, using a fluidized bed to produce clean fibers and fillers together with energy recovery, and pyrolysis which results in chemical products, fibers, and fillers. Efforts are underway to produce more easily recyclable thermosets. Among elastomers, the largest recycling issue is with the very large volume of used tires. A large fraction of them are burned to produce energy. Shredded tires are used for back filling applications in civil engineering; other comminuted forms can be used in asphalt pavements, and even as aggregate for cement concrete.

Introductory Survey

1.5 Material Characteristics Determine Part Shapes During the conceptual design phase, initial part shapes are chosen to satisfy functional requirements. This choice requires an understanding of the mechanical properties of the materials under consideration and the characteristics of the part shaping methods available. For satisfying the same function (need), the shapes (architecture) of the part or device may be very different for different materials. This dependence of part shape on the material used is explored next by considering the evolution of materials use over the past 5,000 years. For over 5,000 years the main construction material was stone, which is difficult to shape. For about 100 years, in the nineteenth century, cast iron that can more easily be shaped into more complex parts, not only replaced stone in some applications, but made possible structures for which stone could not be used. Starting with the mid-nineteenth century, the large-scale production of steel made it the dominant material used in the broadest applications. The important link between shape and material used is illustrated next by the example of bridge construction using the three materials stone, cast iron, and steel. Plastics and advanced composites were invented and increasingly used in the twentieth century. The shape-materials link for plastics is briefly explained in Section 1.5.4; this aspect of plastics is explored in greater detail in Chapter 2 through examples from many different application types. 1.5.1

Stone as a Building Material

The earliest construction material was clay reinforced with straw. Much later, bricks made from fired clay provided uniform building construction material. But, as soon as metal chisels became available, for the longest time – well over 5,000 years – the most important building material was stone. Technology was developed for extracting large blocks of stone from quarries, transporting them to construction sites, and dressing them to desired shapes. Stone is very strong in compression – the “load” that one’s bottom feels while seated on a chair – but very weak in tension – the load an arm feels when pulled. (The difference between tensile and compressive loads, and measures for them, are discussed in Section 3.2.) Because the lower surfaces of loaded beams are in tension (Section 5.2), this lower tensile strength limits the lengths of stone beams. Stone is best used in compression. The invention of arched construction, perhaps the first major engineering advance in the use of materials, extended the use of stone by making larger spans possible. 1.5.1.1 The Early Use of Stone

As a material, stone is very strong in compression – as evidenced by the pyramids in Egypt in which blocks of stone are under enormous compressive loads. The first of these, the six-stepped Djoser pyramid in Saqqara, built in the twenty-seventh century BC, was originally 62 m (203 ft) high with a 109 × 125-m (358 × 410-f t) base. It may be considered as the earliest large-scale cut-stone construction. Figure 1.5.1 shows the largest of the many Egyptian pyramids, the Great Pyramid of Giza (also called the Pyramid of Khufu or Cheops), which was constructed in the twenty-sixth century BC. It is the only surviving Seven Ancient Wonders of the world. When built, it had a height of 146.5 m (481 ft) and was covered by a stone casing that formed a smooth outer surface; it remained the tallest man-made structure in the world for more than 3,800 years.

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Figure 1.5.1 Great pyramids at Giza. The largest of these is the pyramid of Cheops or Khufu built in the twenty-sixth century BC. (Photo courtesy of the Center for Documentation of Cultural and Natural Heritage, Bibliotheca Alexandrina.)

But stone is very weak in tension, which limits the length of an unsupported beam (lintel); the bottom surfaces of a loaded beam are subjected to tensile loads (see Section 5.2). Relatively short stone beams supported by tall stone columns show up in all early designs of large buildings, such as temples as enclosures for places of worship. The roof had to be supported by a large number of columns that interfered with vision in the enclosed space. As an example, Figure 1.5.2 shows the remains of the outside courtyard of the Garf Hussein Temple in Nubia – built in the thirteenth century BC during the reign of Ramesses II – before it was salvaged during the construction of the High Aswan Dam. While this courtyard has been relocated in New Kalabsha, the main rock-cut temple lies submerged in Lake Nasser. Early Greek civilization, which systematically explored esthetics and optimal shapes of buildings, had to work with the same limited tensile strength of stone. But their sense of proportion and esthetics – with the shapes of columns becoming an art form – resulted in buildings with beautiful facades, such as in the Parthenon, built in the mid-fifth century BC, shown in Figure 1.5.3. The base of the Parthenon is 69.5 × 30.9 (228 × 101 f t). The outer Doric columns, each with 20 vertical flutes, have a diameter of 1.9 m (6.2 ft) and a height of 10.4 m (34 ft). This majestic building is the most important surviving member of Classical Greek architecture, exemplifying the most advanced form of the Doric order, one of the three Greek architectural styles of columns. Considered one of the greatest cultural monuments in the world, the Parthenon is an enduring symbol of Ancient Greece, Athenian democracy and western civilization. Form and esthetics were very important to the Greeks. All dimensions, such as the column width to column distance, in the Parthenon follow a 4 : 9 proportion. The temple’s width to height is determined by the proportion 9 : 4, and the square of this proportion squared, 81 : 16, determines length of the temple to height. In addition to using pleasing proportions, the Greeks made minor modifications. To overcome the optical allusion of long straight surfaces appearing to sag at the ends, they raised the central portions.

Introductory Survey

Figure 1.5.2 Stone beams supported by stone columns. (Photo courtesy of the Center for Documentation of Cultural and Natural Heritage, Bibliotheca Alexandrina.)

Figure 1.5.3 Northwest view of the Parthenon. Built in the mid-fifth century BC, the esthetics of this majestic stone building raised architecture into an art form, which has been emulated since then into modern times. But, again, interior space has been created by using relatively short stone beams supported by stone columns. (© Greek Ministry of Culture and Sports – Acropolis Restoration Service Archive.)

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However, as esthetically pleasing as the Greek structures were, they too had to use columns to enclose large spaces. Their use of beautiful columns was so successful that such columns still are used in building facades even though they are not required for structural reasons. 1.5.1.2

The Invention of the Arch

The first great advance in engineering was the invention of the arch, which made it possible to span larger distances by using stone. Although arches appeared in Mesopotamian brick architecture the twentieth to eleventh centuries BC (2nd millennium BC), and arches were known to the Etruscans and ancient Greeks, it was the Ancient Romans who systematically applied them to a wide range of structures, including bridges, vaults and domes. An arch is a curved structure that, by using stone in compression, makes it possible to span large distances; that too by using much smaller blocks of stone. Growing needs for water in Ancient Rome led to the construction of the first aqueduct (an artificial channel for conveying water), the Aqua Apia, in 312 BC. Most of it was a buried in a conduit. Each day it brought about 75.5 × 103 m3 of water to Rome from a spring 16.4 km away by gravity, dropping 10 m over its length. Because water from elevated sources required aqueducts with uniform slopes to be raised above the ground in places, the Romans pioneered the use of stone aqueducts supported on a series of continuing arches. Many of these magnificent engineering feats can still be seen in Rome and in parts of its ancient empire, including France. An example of very early arched stone construction is the Roman Aqueduct (aqueduct bridge) in Segovia, shown in Figure 1.5.4. Built in the first century AD, it is part of an aqueduct that once transported water from the Rio Frio river in the mountains over a distance of 17 km (11 mi). Figure 1.5.4a shows two levels of stone arches supporting the water channel (Figure 1.5.4b). Figure 1.5.4c shows the design of the stone arches, which are constructed from wedge-shaped dressed stone blocks that transmit any vertical loads to the arch ends through compression. The use of arched construction using stone soon spread to different parts of the world. Figure 1.5.5 shows the oldest open-spandrel segmental stone arch bridge in the world, the Anji Bridge (“Safe Crossing” bridge) in China, built during 595 – 605 AD is. Also known as the Zhaozhou Bridge, or the Great Stone Bridge, it is the oldest standing bridge in China. The word spandrel refers to the openings piercing the regions between the lower supporting stone arch and the pedestrian stone roadway; in traditional stone arched bridges the entire structure is solid. The word segmental refers to the several segments (several openings). This bridge is about 50 m (160 ft) long with a central span of 37.37 m (122.6 ft). It is 7.3 m (24 ft) high and 9 m (30 ft) wide. In addition to its structural sophistication, note the embellishments to improve its esthetics. 1.5.1.3

Vaults and Domes

Vaults are features generated by the intersection one or more arches, and domes are “circular” arches. The need for large areas in places of worship, such as churches and cathedrals, with high walls and roofs supported on stone walls and columns, were fulfilled by using vaulted ceilings. Figure 1.5.6 shows the extensive use of arches and elegant vaulted ceilings in the ambulatory (continuation of the aisled spaces on either side of the nave, the central part of a church) of the massive Gothic-style Segovia Cathedral built in the mid-sixteenth century. This magnificent building has three tall vaults and an ambulatory, with fine tracery windows and many stained-glass windows. The Gothic vaults are 33 m (108 ft) high, 50 m (164 ft) wide, and 105 m (344 ft) long.

Introductory Survey

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Figure 1.5.4 Roman Aqueduct in Segovia. (a) View showing water channel supported by two rows stone arched structures. (b) Water channel at the top of the aqueduct. (c) Detail showing construction of arch from wedge-shaped stones, capped by the keystone. (Photos courtesy of Turismo de Segovia.)

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Figure 1.5.5 The pedestrian Anji (Zhaozhou) Bridge. (Photo courtesy of Wikipedia user Zhao 1974.)

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Figure 1.5.6 View of the ambulatory in Segovia Cathedral. (a) View showing stone arched structures and vaults. (b) Detail showing the elegant structure of the vaulted ceiling. Note the three stained glass windows. (Photos courtesy of Turismo de Segovia.)

Introductory Survey

An outstanding example of a stone building with a very large vaulted chamber is the Imambara in Lucknow, built by Nawab Asaf-ud-Daula in 1784. Figure 1.5.7a gives an overview of the complex, with the Imambara on a stepped platform on the left and one of the two large, imposing gates on the right. Figure 1.5.7b shows a frontal view of the main Imambara building, which has

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Figure 1.5.7 Views of the Imambara in Lucknow. (a) Overall view of the Imambara complex. Note the large gate on the right. (b) External view of the Imambara. (c) The main internal vaulted chamber. Notice the dark mezzanine balcony inside the chamber that could be accessed through doors at the upper level. (d) Long arched passage with access doors to the maze on the left. (Photos courtesy of UP Tourism.)

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a 50-m (160-ft) long, 16-m (52-ft) wide, and 15 m (49-ft) high vaulted central chamber – one of the largest such unsupported, arched structures in the world. Figure 1.5.7c shows the interior of this large chamber, with a tomb of Asaf-ud-Daula in the middle. Notice the dark mezzanine balcony below the ceiling; with access doors from the maze, it allowed people to observe activities in the camber. The space above eight surrounding chambers of different heights has been capped with a three-dimensional (3D) labyrinth of passages interconnected with 489 identical doorways (Figure 1.5.4d). These passages form a 3D maze, a popular attraction, designed as a hollow roof to reduce its weight. A dome is a 3D arch, essentially a circularly symmetric arched vault. Stone domes have been used in many churches and public buildings such as libraries and museums. In terms of shear ornateness and elegance, perhaps the most well-known domed structure in the world is the Taj Mahal in Agra, a front view of which is shown in Figure 1.5.8a. It is an immense mausoleum of ivory-white marble built between 1631 and 1648 by the Mughal emperor Shah Jahan in memory of his favorite wife, Mumtaz Mahal. It is built on top a 6.7-m (22-ft) high, 95-m (313-ft) square platform with four 41.5-m (137-ft) tall corner minarets. It has a 24.5-m (81-ft) high central inner dome with a diameter of 17.7 m (58 ft), which is surmounted by a marble outer shell that forms a 61-m (200-ft) high spectacular outer dome with a diameter of that sits on a 7-m (23-ft) high cylindrical drum. Figure 1.5.8b shows arched red sandstone structures inside the Mihman Khana (Assembly Hall) to the east of the Taj Mahal.

(a)

(b)

Figure 1.5.8 The Taj Mahal in Agra. (a) Frontal view of the marble structure. (b) Sandstone arches in the Mihman Khana (Assembly Hall). (Photos courtesy of UP Tourism.)

The white marble structure on a square plinth consists of a symmetrical building with arch-shaped doorway topped by a large dome and finial. The base structure is a large multi-chambered cube with chamfered corners forming an unequal eight-sided structure that is approximately 55 m (180 ft) on the four long sides. Four minarets frame the tomb. The false sarcophagi of Mumtaz Mahal and Shah Jahan are in the main chamber; the actual graves are at a lower level. In addition to the use of marble, the building has fine marble screens, and many panels are inlaid with precious and semi-precious stones.

Introductory Survey

1.5.1.4 Summary Comments

It took about 5,000 years for use technology for stones to mature. This required the invention of the arch and for better stone dressing technology to evolve with the availability of metal tools. Construction technology for bridges centered on using rectangular blocks because dressing (fabrication) technology for stone made it easier to make these shapes. For esthetic reasons, later designs were embellished with sculpted figures that did not serve any useful function. The invention of the arch had a far-reaching effect on the use of stone and bricks – and later even on the use of iron and steel – making it possible to enclose large covered areas, both rectangular and circular, without having to use columns; vaulted ceilings and domes used in churches are essentially 3D arches. This development also exemplifies the interplay between material characteristics and design: All materials have many characteristics that are attractive from an application standpoint, and other characteristics that are not desirable. One function of engineering design is to capitalize on the desirable properties and use design to circumvent the undesirable properties. 1.5.2

Cast Iron as a Building Material

Next, consider the use of cast iron in bridge construction. Like stone, cast iron is very strong in compression – much stronger than stone – but weak in tension. However, unlike stone, cast iron parts can easily be cast in more intricate shapes, so that the entire part does not have to be solid as in stone. The relatively high strength and easy shapability of cast iron make thin-walled parts of varying thicknesses possible, allowing it to be used, for example, for making decorative fences, components for spiral staircases, and other architectural features. Because of its low tensile strength, optimum cast iron bridges again make use of the principle of the arch to put most of the material in compression. The first cast iron arch bridge in the world, the Iron Bridge in Shropshire, England, which opened in 1781 AD and is still in use, is shown in Figure 1.5.9. Five cast iron ribs are used to form a span of 30.6 m (100 ft). It is assembled from about 1,700 individual components that were cast individually to fit with each other.

(a)

(b)

Figure 1.5.9 The Iron Bridge across the River Severn in Shropshire, England. (a) View showing the open girder construction. Note one of the two smaller cast iron arched spans on the left. (b) Close-up view of the arch ribs. (©Historic England Archive.)

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The decorative rings and ogees between the structural ribs do not serve any structural functions. Two similar cast iron supplemental arches carry a towpath on the south bank (left) and also act as flood arches. A stone arch carries a small path on the north (right) bank. 1.5.3

Steel as a Building Material

Finally, consider the use of steel in bridges. Steel is a very versatile material, equally strong in tension and compression, which can economically be rolled into thin-walled sections commonly used in construction. So, in principle, it can be used in both tension and compression. And steel bridges do come in very different shapes and forms, including arched bridges. However, because compression can cause thin-walled sections used in construction to buckle (Section 5.10) – to undergo an elastic instability – in long bridges steel is more efficiently used in tension; avoidance of buckling led to the invention of the suspension bridge. The first iron suspension bridges used pinned wrought-iron chains. Later, as technology made steel wire drawing possible, suspension bridges used cables made of multiple wire strands. Figure 1.5.10 parts a and b show two views of the iconic Clifton Suspension Bridge that spans the Avon Gorge and the River Avon in Bristol. Completed in 1864, this bridge is suspended from three independent sets wrought iron – forged steel – chains on each side (Figure 1.5.10c), assembled eyebars connected by bolts. To allow movement of the chains under load, the chains pass over roller-mounted “saddles” at the top of each stone tower. The bridge deck is suspended from the chains by 81 matching pairs of vertical wrought-iron rods. The bridge has a span of 214 m (702 ft), overall width of 9.45 m (31 ft), and an overall length of 412 m (1,352 ft). The towers are 26 m (86 ft) above deck. The bridge has a clearance of 75 m (245 ft) above the river high water level. Figure 1.5.11 shows views of the Akashi Kaikyo Bridge, the longest suspension bridge in the world; its suspension cables are made of steel wire strands. It spans the Akashi Strait and links the Kobe on Honshu to Iwaya on Awaji Island. It was completed in 1998. This marvel of engineering has three spans: At 1991 m (6,532 ft; 1.237 mi), it has the longest central span of any suspension bridge in the world. Two other sections are each 960 m (3,150 ft; 0.60 mi), resulting in an overall length of 3911 m (12,831 ft; 2.430 mi). The two towers are 1991 m (6,533 ft; 1.24 mi) apart. It is designed to withstand winds of 288 kph (179 mph), earthquakes up to a magnitude of 8.5, and harsh sea currents. The bridge has tuned mass dampers that operate at the resonance frequency of the bridge to dampen forces. The two main supporting towers rise 282.8 m (928 ft) above sea level. Because of heating, the bridge can expand by up to 2 m (6.6 ft) over the course of a day. Each suspension cable has a diameter of 112 cm (44 in) and contains 36,830 strands of wire, requiring about 150,000 km (95,000 mi) of steel wire. 1.5.3.1

Summary Comments

While stone arched bridges and steel suspension bridges serve the same function, they are conceptually very different. And no amount of optimization of one concept would result in the other, both representing different inventions. Thus, material characteristics have a large influence on how they can be shaped and on the optimum shape for a particular application. Choice of initial shapes – shape synthesis – is well understood for stone, metals, and even for advanced composites. However, this process has systematically not been studied for plastics.

Introductory Survey

(a)

(b)

(c)

Figure 1.5.10 The Clifton Suspension Bridge spanning the Avon Gorge and the River Avon in Bristol. (a) Overall view of the bridge and the gorge. (b) Closer view of the bridge. (c) Close-up view showing the chains made from bolted eyebar links. (Photos courtesy of Clifton Suspension Bridge Trust.)

1.5.4

Shape Synthesis for Plastic Parts

Plastics are relatively new materials that have been available for about a century. Early on they had limited uses, but the invention of a host to new plastics, and novel processes for converting them into useful products, has led to their rapid growth. In the beginning, plastic parts were shaped just like the metal parts they replaced. However, a realization that the formability of plastics into complex shapes was not being exploited gradually led to shapes suited to plastics. This section gives an introduction to how the unique features of plastics, such as the ease of making very complex shapes, is changing the shapes of plastic parts.

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(a)

(c)

(b)

(d)

Figure 1.5.11 Akashi Kaikyo Bridge spans the Akashi Strait and links the Kobe on Honshu to Iwaya on Awaji Island. (a) Panoramic view. (b) Closer view. (c) Multilayer deck with suspending cables. (d) View showing connection of vertical suspending cables to main cables. (Photos courtesy of Honshu-Shikoku Bridge Expressway Company Limited.)

Exploiting the properties and processability of plastics can be used to make parts with many more functional features than are possible in a metal part. The evolution of designs for more efficiently using plastics is addressed in Chapter 2. 1.5.4.1

Part Complexity and Consolidation

Advances in injection molding technology have made very complex parts possible. As an example, Figure 1.5.12 shows a thin-walled thermoplastic fuse box into which many fuses and electrical wires can easily be plugged in.

Introductory Survey

(a)

(b)

Figure 1.5.12 Automotive fuse box. (a) Front view. (b) Back view. (Photos courtesy of SABIC.)

Two examples of parts consolidation are shown in Figure 1.5.13. A very complex, molded thermoplastic computer housing from the 1990s is shown in Figure 1.5.13a; this single molding consolidated many metal housing parts and made assembly easier. Figure 1.5.13b shows a molded thermoset housing for window air conditioner from the 1970s. This single molding replaced more than 10 pressed steel parts and fasteners. In addition to saving on assembly costs, this molded part reduced inventory costs.

(a)

(b)

Figure 1.5.13 Parts consolidation. (a) Thermoplastic computer housing. (Photo courtesy of SABIC.) (b) Injection molded thermoset frame for a Carry Cool window air conditioner. (Photo courtesy of GE Appliances.)

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1.5.4.2

Plastic Hinges

Consider forceps and tweezers, several examples of which are shown in Figure 1.5.14. The long pair of tongs, made of a single piece of bent wire, is a tool for barbecuing. The long pair of steel tweezers is either made of a single piece of stamped and pressed steel or by brazing, or welding two pieces. And the two pairs of plastic tweezers are made by a single injection molding operation. Each of these four devices serves a similar function, that of manually picking up objects at the far “open” end by squeezing on the two halves near the joint or near the middle. The plastic tweezers would not work for picking up hot objects, but, because of the low thermal conductivity of plastics, could be used for picking up cold objects without the fingers being subjected to cold temperatures. Tweezers made from appropriate solvent resistant plastic would be useful for picking up objects immersed in corrosive liquids such as acids. While each of these devices can, in principle, be made in one operation, they have one shortcoming: the force exerted on the object being picked up cannot easily be controlled.

Figure 1.5.14 Examples of tongs (metal and wood), and tweezers of metal and plastic. Except for the wood used in the tongs, all functions are served by a single component, without any need for assembly.

The force or the amount of motion at the “picking up” end can be controlled by using the hinged, tong-like devices shown in Figure 1.5.15. The pivoted “scissors” action converts the motion of a pair of fingers at one end into a pick-up force at the other end. The level of this force or motion can be controlled by moving the position of the pivot, or fulcrum, relative to the two ends. The closer the fulcrum is to the pick-up end, the smaller the motion and larger the force exerted on the object. Thus, for example, in the steel forceps used for clamping blood vessels and tissues during surgery, the proximity of the fulcrum close to the clamping end assures a high clamp pressure. While in these devices the motion and force can be controlled more than in those shown in Figure 1.5.14, each such scissors-like device has to be made by assembling a minimum of three pieces – the two arms and a rivet – or even four pieces when a screw and nut are used for the fulcrum. So, adding more control function in this case results in more part complexity.

Introductory Survey

Figure 1.5.15 Hinged “scissor-like” tongs and forceps. The position of the hinge along the length is used to control the force exerted on the object being picked up clamped. These devices are made of a minimum of three parts.

Next consider the toggle jointed scissors-like device shown in Figure 1.5.16, which is made up of a minimum of 8 pieces, and up to 12 pieces if screws and nuts are used. Clearly, the simpler three-piece devices shown in Figure 1.5.15 can satisfy most of the functions served by this complex eight-piece device.

(a)

(b)

Figure 1.5.16 Scissor-like device with a toggle joint. (a) Overall view showing the four main parts connected at four joints. (b) Detail showing the four-pin toggle joint.

That is why such a device would not normally be made; this picture is an artist’s rendition. However, because “living hinges” – thin ligaments that can bend back and forth to provide hinge-like motion – can be molded into a plastic part, a one-piece version of the eight-piece device can be molded in a one-step

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operation! Figure 1.5.17 shows the resulting commercially available device made of polypropylene, which has good fatigue characteristics important for flexing the four living hinges. This example shows that a concept suited for serving a desired function can be greatly influenced by the characteristics of the material chosen.

(a)

(b)

(c)

Figure 1.5.17 A pair of plastic forceps. (a) Overall view of device in three positions. (b) Detail showing four “living hinges” in forceps-open position. (c) Detail showing four “living hinges” in forceps-closed position.

Introductory Survey

1.5.4.3 Summary Comments

The rapid growth in the use of plastics has mainly been fueled by the complexity that can be built into single parts, and the complex shapes that can easily be made to facilitate esthetic design freedom, all made possible by many low-cost part fabrication methods. Also, the use of tough thermoplastics, such as polycarbonate, has made possible transparent laminates for security glazing applications that, in addition to preventing intrusion, are bullet resistant (Section 2.5.2). True innovation in plastics products requires a keen understanding of designs that are tailored to capitalize on the properties of plastics. This is especially important for engineers – such as mechanical, aeronautical, and civil engineers – with a strong background in design and analysis of metal parts, who would initially tend to design plastic parts as “copies” of their metal counterparts. It is to highlight how plastics are used in a wide range of products that Chapter 2 explores the evolution of plastics use in many diverse applications.

1.6 Part Fabrication (Part Processing) Plastics processing has two connotations. The first is resin manufacturing, involving large-scale use of chemistries for making resins. This includes compounding, in which additives and fibers are mixed to form different resin grades. The end products of thermoplastic resin manufacture, which is essentially a chemical engineering process, are cylindrical pellets with nominal lengths and diameters of 4 and 2 mm, respectively. Second, the term processing is used for conversion of resins (pellets) into parts, which can further be divided into the manufacture of bulk products – such as film, sheet, and profile extrusion – and part shaping, the conversion of resins into diverse functional parts or components. The manufacture of bulk products is an important but specialized area requiring machines specifically built for this purpose. This book does not address the manufacture of resins. In the plastics industry, part fabrication is called part shaping, or part processing, and even simply processing, a classification of which is addressed in detail in Chapter 16. As shown in Figure 16.2.1, part shaping methods fall into two broad categories. The first comprises methods in which molten resin is injected into a cavity, formed by a two-sided mold, whose shape determines the final nominal part shape, including the part thickness. Examples of such processes include injection molding – the most commonly used method for making plastic parts – foam molding, compression molding, extrusion, and multiple live feed injection molding. The second category comprises processes in which a one-sided mold is used to shape the outer surface of the part. Here, the external shape is determined by the mold geometry, but the thickness distribution is not controlled by the mold – rather, it is determined by the process conditions. Examples of such processes include blow molding and thermoforming, rotational molding, and gas-assisted injection molding. Chapter 16 gives a classification of part shaping methods and how they evolved. Succeeding chapters describe specific shaping methods and the interplay between part shaping methods used and the resulting part shape.

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1.7 Part Performance Part performance can have several connotations such as esthetics – especially surface finish and color – mechanical performance, and dimensional stability. Surface finish of plastics products has always been important. While a large body of practical, empirical knowledge is available for avoiding surface defects – such as blush, splay, and streaks – the scientific basis for predicting, and thereby controlling, such defects is not well established. Dimensional stability refers to changes in size and shape caused by the large volumetric changes that molten plastics undergo on solidification – a 1-m panel can shrink by as much as 1 cm. The solidification process also results in residual stresses that, besides contributing to the overall stress level, can cause time-dependent dimensional changes. And since the shrinkage is not homogeneous, the part can warp. Dimensional stability issues are important for tight tolerance applications such as chassis for duplicators and scanners. Much attention has been paid to appearance and cost issues in processing. Control of sink marks, surface finish, and part warpage are examples of continuing concerns. Many of these issues have been addressed in the past by trial-and-error. Now that plastics are being used in more demanding applications, structural performance is becoming important. Also, competitive pressures are driving designs toward optimum materials usage. Designing to a material’s limits requires a sounder understanding of how processing affects material behavior in a part; mechanics plays a crucial role in developing this understanding. Mechanical or thermomechanical performance concerns the ability of a component to function without undergoing prescribed deflections, not undergoing permanent deformations, and not “failing” or fracturing under prescribed static or repeating loads and thermal histories. In principle, the procedure for predicting part performance starts with a definition of the part geometry and a specification of the local material properties across the part geometry. Next, the loads and displacements that the component is to be subjected to are specified; they constitute the boundary conditions for the model problem. Then, principles of engineering science (applied mechanics) are used to predict deflections, stresses, and whether or not the part will fail. Finally, an iterative process, involving changes in geometry and materials, is used to establish the optimum part geometry and material combination to satisfy the function for which the part is designed. In the past, the process of using mechanics principles (structural mechanics) to predict part performance involved solving complex differential equations representing the underlying physics; a tedious task requiring ingenuity that could only be used for parts having simplistic model geometries. Because of the inability to predict stresses and deflections in parts with complex geometries, experimental methods – such as photoelastic and strain gauge techniques – were developed to determine stresses in loaded parts. Later, numerical methods were developed to solve structural mechanics problems for more complex but realistic part geometries. In the early stages of development these numerical techniques were very tedious, initially requiring a large number of calculations by hand, which later were done by using mechanical calculators. The advent of digital computers made it possible to carry out very large number of numerical operations economically. Digital computers have made it possible to calculate the detailed stress and deformation field in complex parts subjected to complex loadings: The complex differential equations describing the mechanical behavior of complex structures and loadings can now be solved numerically by using finite element

Introductory Survey

methods that are now available in the form of robust, user-friendly, easy-to-use computer codes, at times generically referred to as software. The availability of such codes has revolutionized the prediction of deformation and stresses, and cut down on the amount of part testing. And such codes are not limited to the small deformation assumptions normally used for the analysis of parts. 1.7.1

The Role of Numerical Methods

Over the past 50 years, a powerful methodology, called the finite element method, which harnesses the computational power of digital computers, has been developed for very efficiently computing the displacements and stresses in parts with complex geometries. By enormously reducing the time for computing stresses and displacements, the finite element method has all but eliminated the need for extensive component tests for developing optimum designs. Also, by reducing the time, effort, and cost for obtaining the stress and deformation field in parts, engineers can once again focus on the main problem of synthesizing concepts for satisfying prescribed functional needs. The procedure for predicting part performance can schematically be described by the flow chart in Figure 1.7.1a: First, the part geometry, local material properties, and the loads and displacements (boundary conditions) applied on the part are defined. Then the finite element technique – which is essentially a numerical means for solving the differential equations of structural mechanics – is used to “automatically” predict the displacements and stresses throughout the part. This procedure is well suited to metals for which, to a good approximation, the part geometry – shape and thickness distribution – and local material properties are defined a priori. However, such is not the case with plastics, in which both the geometry and the local properties can be affected by processing, and may therefore not be known a priori. The geometry of a functional part comprises two elements: the shape of its surfaces (surface geometry) and the thickness distribution. For shaping processes using double-sided molds – such as injection molding – the mold cavity shape determines the final nominal part shape, including the part thickness distribution. However, in processes in which a one-sided mold is used to shape the outer surface of the part, while the external part shape is determined by the mold geometry the thickness distribution is not controlled by the mold – rather, it is determined by the process conditions. In this case, process mechanics – which accounts for all the processes that a material is subjected to during processing – has to be used to predict the final part thickness distribution. Also, in plastics the local morphology of the material in the final part can be greatly affected by the processing conditions; process mechanics is required for predicting the local morphology. And micromechanics is required for using the local morphology to predict the local material properties, such as stiffness and strength. The outputs from process mechanics and micromechanics analyses define the final part geometry – shape and thickness distributions – and the local property distribution – local material stiffness and strength – which complete the inputs required for the finite element analysis (structural mechanics). The flow chart for this complex process for plastics is shown in Figure 1.7.1b. Process mechanics is also used to simulate as to whether the part can be fully formed under a given set of processing conditions; an incompletely filled mold can result in an incomplete part, called a short shot. The flow chart in Figure 1.7.1b shows many corrective feedback loops for changes in geometry, materials, and processing conditions to ensure that the final part meets performance requirements. For different processing techniques and plastic types, Section 16.4 (Effects of Processing on Part Performance) discusses in detail the interactions among the different blocks in Figure 1.7.1b.

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element analysis of plastic parts.

PART

PART

MATERIAL

SHAPE PROCESS

MATERIAL

• Local stiffness • Local strength

GEOMETRY

• Shape • Thickness distribution

• Filling • Thickness • Morphology

PROCESS MECHANICS

• Thickness distribution

GEOMETRY

(b)

• Local mechanical properties

MICROMECHANICS

(a)

• Finite Element Analysis

STRUCTURAL MECHANICS

• Finite Element Analysis

STRUCTURAL MECHANICS

• Stiffness • Strength • Energy absorption

PART PERFORMANCE

• Stiffness • Strength • Energy absorption

PART PERFORMANCE

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Figure 1.7.1 Procedure for predicting structural performance of parts. (a) Finite element analysis of metal parts. (b) Finite

Introductory Survey

1.7.2

Rapid Prototyping

Plastic components tend to have complex shapes that are not easy to visualize. Prototyping methods were initially developed to make parts for verifying shapes and fits. While computer-generated models and advances in computer graphics made it possible to visualize the shape and look of a design with increasing realism, earlier rapid prototyping techniques could only make parts suitable for evaluating the look, feel, and fit of a part. Such parts were not strong enough for functional testing. Now, new techniques can be used to make parts that are strong enough for functional testing. These rapid prototyping techniques can be used to make prototype assemblies, or models, that can be used to evaluate the look and feel of parts, which is particularly important for plastic parts for which esthetics are important. Such early attempts have now evolved into the new discipline of 3D printing, also known as additive manufacturing, in which a digital model of the object is used to build or “print” 3D objects in a layer-by-layer process. It involves several steps: First, a complete digital model of the 3D part is constructed using a computer aided design (CAD) software program. The model is then digitally “sliced” into thin, flat layers. Finally, the digitally sliced CAD model is used with one of the many available 3D printing processes to sequentially lay down thin flat layers of material to build the desired solid shape. Figure 1.7.2 shows a multi-material, 3D-printed hand drill prototype made in a one-step printing process. It has four colors and varying textures, ranging from a hard, rigid plastic on the shell and base to

Figure 1.7.2 Hand drill prototype produced by multi-material 3D printing. Note the use of several materials. (Photo courtesy of Stratasys Ltd.)

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the softer, rubberlike hand grips. All of these characteristics are the result of using specific photopolymer resins deposited in the appropriate formulation, which is controlled by the multi-material 3D printer. Clearly, rapid advances in 3D printing technology have made possible quick evaluation of concepts through prototypes that can check fit and feel, esthetics, and function. 3D printing can also be used for making limited-use injection molds and functional parts (Section 19.8).

1.8 Assembly Plastics are relatively expensive materials. In most applications, their use has depended on fabrication techniques that reduce cost by building enormous complexity and increased function into a single component that, in metals, would have to be assembled from subcomponents. For this reason, for plastics one-step part fabrication methods have received much attention. In the plastics industry, anything beyond a one-step operation – such as surface finishing, painting, and joining – falls under the category of “secondary operations,” which normally are considered undesirable because they add cost. And joining methods are considered a very small part of secondary operations; the bulk of the secondary operations is concerned with producing good surfaces, such as by painting, or by decorating. One consequence of this neglect of assembly and joining issues is that parts are designed without having considered joining requirements, so that when joints have to be made, they are normally an afterthought, resulting in poorly designed parts. Assembly and joint design have to be considered concurrently with material selection, part design, and fabrication methods. In the engineering process schematically shown in Figure 1.3.1, the role of assembly is to combine a collection of parts or subassemblies into a desired functional device. The process of combining components to form a device or an assembly is called joining. And the interface at which subassemblies are joined is called a joint. Joints can be rigid – as in a welded joint – or may permit motion. Assemblies, and hence joints, may have to perform several functions. For example, a hinge is a joint that allows rotary motion while at the same time transmitting force. Because of the complexity that can be built into plastic parts, joining and joint design have not received the attention that their importance deserves. Joints in an assembly may be required for several reasons: • To provide function. Providing function can be in many forms. For example, a hinge on a lid to allow for easy and repeated opening and closing of a box – a joint that allows motion; and a screwed cover to permit access to subassemblies for repair – an example of a rigid joint that allows for repeated assembly and disassembly. • To achieve sufficient part complexity. This may be considered a subclass of providing function. The amount of complexity that can economically be built into a single component depends on the material and how easily it can be fabricated into parts. Increased part complexity can reduce cost by eliminating several fabrication and assembly steps and reduced inventory requirements. One example is a dishwasher tub. When made of sheet metal, the tub has to be assembled from several pieces because of the difficulty and expense of deep drawing it in one piece. And the interior surface of the tub has to be enameled to prevent corrosion. Using plastic, a talc-filled polypropylene in this case, the dishwasher tub can be injection molded in one piece; this material resists corrosion so that the tub interior does not have to be coated (Figure 2.2.28a).

Introductory Survey

• To permit the use of different materials – multi-material design. Sometimes a component has to be made up of different materials. For example, a bearing for a shaft can be made of a soft low-friction material that can easily conduct away heat generated by friction; the soft bearing material is imbedded in a steel casing to provide structural support. Another example is that of a clear glass or polycarbonate lens of an automotive headlight assembly that is supported in a metal or plastic structural casing. Still another example is that of a paper juice container to which a plastic spout with a screw-on lid is ultrasonically bonded (Figure 2.2.17). • To reduce cost. While building complexity into a single part generally reduces manufacturing cost – reduced part manufacturing, inventory and assembly costs – this is not always so. One example is that of a molded glass-filled nylon automotive inlet manifold. In earlier versions, the manifold was molded in one piece. But because of the geometric complexity of the manifold, fusible cores – of the type used in investment casting of metals – had to be used, thereby increasing the molding cost. Now these plastic manifolds are assembled by welding two molded halves (Figure 2.4.4). Joining of plastic materials and their composites can broadly be divided into mechanical fastening and bonding. Bonding can further be classified into adhesive bonding, solvent bonding, and welding. Mechanical fastening and adhesive bonding can be used for joining all materials, including metals. And the parts to be joined need not be of the same material. On the other hand, welding, which requires the materials at the joint interface to melt, is only applicable to thermoplastics because thermosets cannot be melted. Welding is also becoming important for emerging high-performance thermoplastic composites, the use of which may revolutionize assembly techniques in aerospace applications. More recently, recyclability considerations have made welding even more attractive because, in contrast to the use of adhesives, additional materials are not introduced into the assembly. All aspects of joining of plastic parts are addressed in Chapter 21.

1.9 Concluding Remarks The main purpose of this overview is to provide a perspective for the entire field of plastics engineering; something that a reader can come back to while assessing the role of individual chapters. This is important for understanding the new paradigm for plastics engineering proposed in this book – mapping plastics into useful products. An important message in this chapter pertains to how the characteristics of a material affect the shape of a part, and how engineering principles – such as the invention of the arch, which makes possible the use of brittle materials like stone, which are weak in tension – enhance the usefulness of materials. This theme is continued in Chapter 2, which explores the evolution of concepts for plastics use through examples drawn from different sectors.

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2 Evolving Applications of Plastics 2.1 Introduction The shape of an object designed to serve a particular function is determined by the properties and formability of the material of which it is made. The examples in Section 1.5 of bridge construction using three materials – stone, cast iron, and steel – showed that the basic shape of a part or device rendered in different materials can be very different. The choice of shapes – shape synthesis – is well understood for metals applications, and even for advanced composites. But this process has not been systematically explored for plastics – even though they are now routinely used in almost all spheres of human activity, from ubiquitous plastics bags, and water and beverage bottles, to plastic toys, to audio devices, to many medical devices including plastic knee replacement prostheses, to infrastructure items such as composite bridges and sewer systems, and to large-scale applications advanced composites in commercial aircraft. First attempts at using plastics, consisting of one-for-one metal part replacement, were not very successful as they failed to capitalize on the complexity that can be built into single parts, which is what makes the use of otherwise expensive plastics economical. Plastics used in an assembly tend to cost more than the metals replaced; it is innovative designs and cost-effective fabrication (processing) techniques that give plastics an edge over metals. The properties of thermoplastics and their formability affect the shapes of plastic parts in several ways: (i) The economics of part shaping require plastic parts to be thin-walled. (ii) The flexibility of thermoplastics and ease of forming into complex shapes make snap-fits possible for part assembly. (iii) The ability to elastically recover from large deformations, good fatigue properties, and the possibility of molding very thin sections make possible integrated molded “living hinges” and molded-in bistable spring mechanisms for bottle lids. And (iv), the integration of several functions into one component reduces part cost. True innovation in plastics products requires a keen understanding how to develop designs that are tailored to capitalize on the advantages of plastics. This is especially important for engineers – such as mechanical, aeronautical, and civil engineers – with a strong background in designing and analyzing the performance of metal parts, who would initially tend to design plastic parts as “copies” of their metal counterparts. It is for this reason that the remainder of this chapter reviews the evolution of plastics applications.

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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2.2 Consumer Applications Plastics are so ubiquitous in almost all items used by consumers – from the very wide variety of synthetic materials used in clothing and shoes; to diapers for children and the elderly; to writing pens; to protective plastic helmets used by bicycle and motorcycle riders, and baseball, football, and rugby players; to eyewear; and to packaging of solid and liquid foods – that they are oblivious to how much the use of plastics is affecting the consumer friendliness of products. The purpose of this section is to highlight how many consumer products have only been made possible by plastics. 2.2.1

Clothing

Successful substitutes for cotton, the workhorse material for clothes, began to appear in the beginning of the twentieth century. The first of these was a natural polymeric fiber comprising reconstituted cellulose, mainly derived from wood chips, which could be spun into very thin fibers. Commercialized in 1905 and still in use, this fiber was first called viscose and then rayon; it was also called “artificial silk” because of the shiny nature of fabric woven from it. The first synthetic fiber, nylon, was invented in the 1930s. Initially used for replacing silk in women’s stockings, in what came to be called “nylons,” it was also used for making parachutes and ropes. Polyester fibers, poly(ethylene terephthalate) (PET), which now account for about 60% of all synthetic fibers, became available in the 1940s. Acrylic fibers (polyacrylonitrile), commercially called OrlonTM , which is extensively used as a wool replacement for sweaters, became available in the 1950s. Spandex, an elastomeric polyurethane (PU) fiber, made it possible to produce synthetic elastic fabrics that, together with other synthetic fibers, are critical for making lingerie. At first, polyester fibers were mainly used as durable replacement for cotton; the shiny slippery texture of the fabric led to polyester-cotton blends. This successful application of synthetic fibers has spawned a remarkable revolution in the textile industry, both in terms of fiber types, such as fleece and microfibers, and novel fabrics used in many applications, from novel clothes that “breathe” and absorb moisture, to carpets. These materials are on the verge of making possible smart clothing that will monitor the body state. While synthetic fibers have successfully been used for making clothes for very low-temperature ambient conditions – replacing wool and animal fleece – two important applications only made possible by advanced synthetic fibers are high-temperature protective clothing for firefighters and bulletproof body armor for police and the military. 2.2.1.1

Protective Clothing for Firefighters

Firefighters require special clothing for protection against high-temperature, wet, and smoke-filled environments. The earliest firefighting gear was made of cotton canvas lined with quilted material for thermal insulation. Later, such outer wear, called turnout gear, was made with a layer of neoprene (Section 13.4.1.2) to provide waterproofing. However, even treated flame-retardant cotton clothing does not provide adequate protection to firefighters exposed to hazardous, high-temperature environments in residential, industrial, and forest fires. Heat- and flame-resistant Nomex® fibers (Section 11.7.7),

Evolving Applications of Plastics

specially developed for making protective fabrics and garments, are now routinely used in such high-temperature applications. Besides providing much better heat resistance, they have better abrasion resistance. Turnout gear is also available with fabrics made of still higher-temperature materials such as polybenzimidazole (PBI) (Section 11.7.8) and poly(phenylene benzobisoxazole) (PBO) (Section 13.3.4) National Fire Protection Association (NFPA) guidelines direct that firefighting turnout gear have three essential layers to provide protection: (i) The outer shell, provides tough, durable first line of defense against heat, flame, and abrasion; it also offers resistance to rips, cuts, and tears. The outer shell fabric facilitates shedding of water. (ii) The primary function of the thermal liner, typically a quilted fabric consisting of a woven face cloth and non-woven batt, provides a majority of the thermal insulation from the intense heat of firefighting. The face cloth is the woven fabric seen on the inside of a garment and is made of polymeric spun-yarn fabric; alternatively spun and filament yarns can be used in combination with FR rayon blends. The face cloth helps to protect the non-woven batt and to wick away the perspiration away from the body to keep firefighter drier and comfortable. Batt is a non-woven mechanically bonded textile structure that has loft, or air gaps, to maximize thermal insulation. Of the many material options, the face cloth can be of woven fibers, and the batting can be of layers of Nomex, Kevlar® (Section 11.7.7), and other high-temperature materials. And (iii), a moisture barrier to prevent liquids from entering the system while maintaining breathability. Clearly, modern firefighting turnout gear has complex fabric architecture that provides protection to firefighters in very hostile thermal environments. This level of protection has been made possible only by the development of high-temperature fibers such as Nomex. Figure 2.2.1a shows two firefighters wearing turnout gear in action. The self-contained breathing apparatus, gloves, and boots are mainly made of polymers. And the helmets are made of polymeric advanced composites. Figure 2.2.1b shows a firefighter in full turnout gear. Note the polymeric-fiber fire hose on his shoulder. Figure 2.2.1 parts c and d show a jacket and a pair of pants. Notice the reflective trim to help identify the presence firefighters in a smoke-filled environment. Also notice the extra protection provided for knees. 2.2.1.2 Bulletproof Clothing

Body armor, a form of protective clothing, was designed to protect against slashing, bludgeoning, and penetrating attacks by weapons. From earliest times, it evolved to protect soldiers in war: Starting from chainmail, comprising outer wear made of interlinked iron rings, body armor progressed to chainmail augmented by metal plates, to full-fledged metal body armor used by knights. Now, such armor has been replaced by polymer-fiber-based lightweight “bulletproof” or bullet-resistant outerwear that is routinely used by police and military personnel. Bullet-resistant, lightweight personal armor, such as combat vests, was originally made possible by the invention of Kevlar, a strong synthetic fiber that on a tensile strength-to-weight ratio is five times stronger than steel. Concealable body armor is typically made from multiple layers of woven Kevlar fabric or other ballistic-resistant materials, such as non-woven fabrics made of unidirectional ultra-high strength polyethylene fibers with crisscrossed layers. Such materials are used to form a ballistic panel, which is then inserted into a concealable or tactical vest. This combined body armor system is light and flexible, and provides good protection from ballistic objects.

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(a)

(c)

(b)

(d)

Figure 2.2.1 Nomex firefighting gear. (a) Firefighters in action in full Nomex gear. Note the steel cylinders that feed oxygen to gas masks (not shown). Also not shown are gloves and boots, all made of polymeric materials, as are the helmets. (b) Firefighter with full Nomex turnout gear. The double-jacket fire hose on the shoulder is also made of polymeric materials. (c) Nomex jacket with reflective horizontal identification stripes. (d) Nomex pants with additional knee protection. (Photos courtesy of Globe Manufacturing Company, LLC.)

Evolving Applications of Plastics

On striking body armor, a bullet or projectile is caught in a “web” of layered and stitched high-performance fibers, thereby helping meet protection specifications. At the impact point, the fibers absorb and disperse the impact energy transmitted to the vest by the bullet, causing it to deform or “mushroom” within the vest. Additional energy is absorbed by successive layers until the bullet is stopped. Because the fibers in the vest in individual and other layers work together, a large area of the garment is involved in dissipating the impact forces, eventually stopping the bullet. Besides preventing bullet penetration, this dissipation of impact force also prevents non-penetrating injuries to internal organs. Figure 2.2.2 shows different aspects of protective body armor: Figure 2.2.2a shows a test in which a ballistic panel is shot at close range. Figure 2.2.2b shows bullets stopped by a Kevlar ballistic panel. A concealable body armor carrier is shown in Figure 2.2.2c; note the adjustment straps on the sides for easy vest attachment to increase comfort. The ballistic panel is made of layers of woven Kevlar and high-performance, non-woven unidirectional polyethylene fibers in crisscrossed layers. The specially designed inner liner of the microfiber vest provides improved airflow and circulation for improved comfort. And Figure 2.2.2d shows a tactical body armor system consisting of base vest, collar/neck, biceps, and groin protection. The operator is also wearing a tactical helmet capable of stopping bullets. The shield is constructed of a high-performance, hybrid ballistic composite that is engineered to protect against high-velocity bullets. Its interior components include a stationary handle, a thick high-density foam pad, a forearm strap attachment system, and a viewport; it weighs less than 1.6 kg (35 lbs). 2.2.1.3 Hook-and-Loop Fasteners

Hook-and-loop fasteners consist of two polymer, fabric-based components: The first has tiny hooks, and the second has much thinner “hairier” loops. When pressed together, the hooks latch onto the loops, thereby temporarily binding the two parts together. The “joint” is opened by pulling or peeling the two surfaces apart, and is accompanied by a distinctive “ripping” sound. Both parts in the original version of these fasteners were made by a weaving process using nylon yarn. The hooks were formed in a subsequent loop slitting operation. Now the hook part is also made by an extrusion process that produces stronger hooks to increase the load carrying capacity of a joint. Hook-and-loop fasteners are now available in many plastics. Hook-and-loop fasteners have an interesting history. Based on observing that cocklebur flowers stuck to the fabric of his trousers and the fur of his dog, the idea for this invention came to Swiss electrical engineer George de Mestral in 1941. It took him about 10 years to perfect the hook-and-loop product, and in 1952 established the company VELCRO S.A. in Switzerland to manufacture the product. The trademark VELCRO® is derived from the French words “velour” (loop) and “crochet” (hook). Figure 2.2.3 shows the tape form of a Velcro brand plastic hook-and-loop fastener: In the macro view in Figure 2.2.3a, the upper curled back portion is the loop part with fuzzy fibers sticking out from the surface. On closing these fibers are snared by the hooks in the lower straight portion. The enlarged view in Figure 2.2.3b shows the hooks formed by slit loops. Many specific, stronger hooks (Figure 2.2.4) are made in a special purpose extrusion molding machine. Figure 2.2.4a shows one such double-headed design, a more refined version of which is shown in Figure 2.2.4a. The enlarged view in Figure 2.2.4c shows fibers from the loop portion interacting with molded hooks.

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(a)

(c)

(b)

(d)

Figure 2.2.2 Use of Kevlar in body armor. (a) Ballistic panel being shot at close range. (b) Bullet stopped by ballistic panel. (c) Concealable carrier, or “bulletproof” vest. (d) Tactical body armor system. (Photos courtesy of Point Blank Enterprises, Inc.)

Evolving Applications of Plastics

(b)

(a)

Figure 2.2.3 Typical hook-and-loop plastic fastener. (a) The top curled back part is the woven, loop part of the tape fastener; the bottom straight portion is part with hooks. (b) Enlarged view showing the slit hooks. (Photos courtesy of Velcro Companies.)

(b)

(a)

(c)

Figure 2.2.4 Extruded hook shapes. (a) Enlarged view of double hooks. (b) Enlarged view of an alternative double-hook design. (c) Close-up view of extruded hooks interacting with fibers from loop material. (Photos courtesy of Velcro Companies.)

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2.2.2

Shoes

Originally shoes were made of braided grass or from the surface layers of plants, such as from hemp used for making ropes. But, for most of human history, all practical shoes were made of leather. Then soles of rubber and crepe rubber were introduced. Later, lightweight athletic shoes, or “sneakers” with canvas uppers and rubber soles were introduced to serve an increasing market. Finally, in the latter half of the twentieth century, biomechanics principles were used to design more sophisticated shoes with synthetic uppers and complex soles of synthetic, shock-absorbing foam and solids. This technology has been applied to walking shoes. The emphasis on comfort and style have revolutionized shoe design – almost all the materials used for making shoes now are an array of synthetic materials. Leather shoes are relegated to formal wear – mainly worn on fewer and fewer occasions. 2.2.2.1

Athletic Shoes

All athletic and semi-formal shoes are now mostly made of synthetic materials: The uppers may be made of breathable woven fabric. The inner on which a foot rests is shaped to reduce the pressure across the foot – besides helping to spread the load across the foot surface a special soft material improves

(a)

(b) Figure 2.2.5 A modern shoe. (a) Profile view showing the use of different materials. (b) Components, made of different synthetic materials, used in assembling the shoe. (The Trefoil, 3-Stripes, adidas, and Boost trade marks, which are registered trademarks of the adidas Group, are used with permission from adidas AG.)

Evolving Applications of Plastics

the comfort level. The complex designs of the sole – which can have several components of different materials, including air-filled pockets to absorb shock – has been made possible by the ease with which they can be economically manufactured. How modern shoes are put together is illustrated in Figure 2.2.5: The shoe in Figure 2.2.5a is assembled from the components shown in Figure 2.2.5b, mostly with the use of adhesives. All the materials – with the possible exception of the upper surface, which could be of leather for “esthetic” reasons – are made of synthetic thermoplastic and thermoset materials. Figure 2.2.6 shows the complex design and materials used in a robust walking shoe for competition and practice on road and other hard surfaces. This shoe is a good example of how biomechanics can be been used to design sports shoes. Its profile is shown in Figure 2.2.6a. Figure 2.2.6b shows the complex design of the grippy and very durable rubber sole that has a combined sucking/leafy pattern. The cutaway view in Figure 2.2.6c shows the complex construction of the shoe. Different features in this

(a)

(b)

(c) Figure 2.2.6 Structure of a modern walking shoe. (a) Profile showing the use of different materials. (b) Complex structure of the sole. (c) Cutaway view showing the complex structure of the shoe. (The Trefoil, 3-Stripes, adidas and Boost trade marks are registered trade marks of the adidas Group, used with permission.)

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figure, as indexed by arabic numbers, are as follows: (1) A grippy and durable walking sole (marked A in Figure 2.2.6b), (2) Half shell sole for optimal connection between shaft and sole, and for better lateral movement. It has a triangular bending zone for easier rolling of the front of the foot. (3) Supporting rubber outside counterfort (buttress) with flattened heel spoiler to absorb the first impact and to facilitate the biomechanically correct start of the rolling of the foot. (4) Lightweight intermediate sole of Frelen with excellent damping properties. (Frelen is a lightweight, anatomically shaped insole, designed to provide extra support, cushioning, and shock absorption. These machine-washable insoles are made from dual-density polyethylene.) (5) Support made of harder Frelen to prevent over pronation – the inward roll of the foot during normal motion that occurs as the outer edge of the heel strikes the ground and the foot rolls inward and flattens out – of the inside, and to prevent over supination – or under-pronation, is the opposite of pronation and refers to the outward roll of the foot during normal motion – of the outside. (6) Lightweight nylon moccasin construction shaft. (7) Pre-shaped, exchangeable slip sole with terry top surface. (8) Vulnerable regions strengthened with gazelle suede (leather). (9) Extended counterfort for maximal heel stability. (10) Variable lace positions for optimal support for small or wide feet, including variable instep heights. Finally, (11) soft heel patch and padded entry. 2.2.2.2

Firefighters’ Boots

Nomex is now extensively used for providing thermal protection in firefighters’ turnout gear (Section 2.2.1.1). Together with other advanced polymeric materials, it is also used in firefighters’ boots, which have to withstand high temperatures, water, and other liquids commonly found on the fireground. Besides abrasion resistance, they must provide puncture resistance against sharp objects such as nails. Figure 2.2.7a shows a firefighters’ boot. The complexity of how such boots are constructed is shown in the exploded view in Figure 2.2.7b. It too makes extensive use of polymers in the form of Nomex and Kevlar fabrics, rigid plastic inserts for the heel, shin, sole, and toe. While the outer surface of the boot is most often of leather, the sole is always made of a polymeric material. Starting from the top left of the exploded view in Figure 2.2.7b and moving in a counterclockwise direction, the main construction features of the boot are: (i) The outside of the boot is often made of heavy-duty leather; the leather surface can be coated with a polymer to resist contamination. Leather resists cuts and has excellent abrasion resistance. (ii) Full-height bootie package of inner lining fabric, insulation, and in this boot Crosstech® moisture barrier. Crosstech provides protection against penetration of blood, body fluids, water, and common chemicals while still offering breathability and superior comfort. (iii) A Crosstech, Kevlar blend layer protects the moisture barrier, provides cut-resistance, and adds thermal protection. (iv) A molded piece is used to securely hold the heel to provide cushioning to the ankle without slipping. (v) A composite heel counter molded separately for each boot size. (vi) A thermally insulating composite shank that springs back to shape better. (vii) Vibram® contoured molded cup outsole wraps onto the leather upper for athletic shoe performance. (viii) Slip-resistant molded Vibram tread with thin slits cut into flat areas across the sole that opens up when flexed to provide additional traction on water and ice. Self-cleaning lugs and omnidirectional tread pattern designed for superior performance in all terrains and when working on ladders. (ix) More flexible than a steel plate, thermally insulating composite puncture protection. (x) A 3D composite board lasted to boot uppers with a built-in flex zone in the forefoot and a torsionally stable heel. (xi) A 3D molded removable footbed, contoured to cradle and cushion the bottom of foot and to provide arch support. (xii) Vibram toe bumper for abrasion resistance when crawling. (xiii) Lighter than steel, thermally insulating composite safety toe cap. (xiv) Flame-resistant, fluorescent, and reflective 3M ScotchliteTM reflective material to improve visibility. (xv) Padded composite shin guard for extra protection while working on a ladder. Finally, (xvi) a flexible Nomex webbing pull straps reinforced with leather easily slide under turnout pants.

Evolving Applications of Plastics

(a) FLAME- & WATER-RESISTANT HEAVY-DUTY LEATHER

PULL STRAPS 3D MOLDED & PADDED SHIN GUARD

FULL-HEIGHT INSULATION & MOISTURE BARRIER LINER

FLAME-RESISTANT REFLECTIVE MATERIAL

KEVLAR BLEND PROTECTIVE SHIELD

COMPOSITE SAFETY TOE CAP

INTERNAL FIT SYSTEM

TOE BUMPER COMPOSITE HEEL COUNTER

3D MOLDED REMOVABLE FOOTBED 3D COMPOSITE LASTING BOARD

COMPOSITE SHANK CONTOURED CUP OUTSOLE

SLIP-RESISTANT TREAD

COMPOSITE PUNCTURE PROTECTION

(b) Figure 2.2.7 Firefighters’ boot. (a) Profile showing the use of different materials. (b) Complex structure of the sole. (c) Cutaway view showing the complex structure of the shoe. (Photos courtesy of Globe Manufacturing Company, LLC.)

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2.2.3

Toothbrushes

The earliest toothbrushes had coarse hogs’ head hair attached to bone or bamboo handles. From 1938 onwards, nylon bristles replaced animal hair, and the handles are made of a hard thermoplastic such as polystyrene. Ridges to facilitate gripping can be molded into even the simplest toothbrush handles, as shown in Figure 2.2.8.

(a)

(b) Figure 2.2.8 (a) Top and (b) side views of a simple toothbrush with nylon bristles attached to a ridged plastic handle.

In a more ergonomic design, the handle is contoured and its surface is overmolded with a soft elastomer to facilitate gripping. In the design shown in Figure 2.2.9, the white and the darkest portions (top) are made of a rigid thermoplastic that has been overmolded with soft elastomers (light gray); different colors have been used for esthetics. Notice the circular patterns in which the nylon bristles have been arranged; the inner bristles are at a lower level. The bristles at the left-most edge are higher than the circular bristles. Also, the tip of the bottom surface has a serrated design for use as a tongue cleaner.

(a)

(b)

(c) Figure 2.2.9 Top (a), side (b), and bottom (c) views of an ergonomically improved toothbrush with nylon bristles of different color. The handle consists of a white rigid thermoplastic that has been overmolded with a grip of soft elastomers of different color to improve esthetics. Notice the circular arrangement of the nylon bristles in (a). The serrated elastomeric surface on the back tip is for use as a tongue cleaner.

Evolving Applications of Plastics

The first three photos in Figure 2.2.10 shows top, side, and bottom views of a toothbrush with a more substantial sculpted ergonomic handle overmolded with a grip (white) of soft elastomer. And the next three photos show three views of the brush head. In this design, the outer periphery has thicker “bristles” of the elastomer; these bristles help massage the gums. Also, the internal bristles are inclined at different angles to facilitate more cleaning action in each stroke.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2.2.10 Top (a), side (b), and bottom (c) views of a toothbrush with a more ergonomic handle with a soft elastomeric grip. Here again, the serrated elastomeric surface on the back tip is for use as a tongue cleaner. The last three figures show details of the bristles: notice the thicker (white) elastomeric “bristles”; on the outer periphery designed to gently massage the gums. Also notice that the interior bristles are attached at different angles to facilitate cleaning in each stroke.

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2.2.4

Disposable Razors

Before the twentieth century, the most commonly used razors were straight razors – also known as “cut-throats.” They consisted of a blade sharpened on one edge, made either of high-carbon steel that is easier to sharpen (hone) but blunts easily, or of stainless steel, which is more difficult to sharpen but holds its edge longer. The edges were honed on a stone and then “stropped” on a leather belt. The blade was pinned between two protective sides that protected the edge when folded. A safe shave required skilled use of the sharp edge, so they were mostly used by barbers. Safety razors with thin, disposable blades then dominated razors for over a century. After initial designs for specific razors, a standard blade design with a multi-faceted central channel was developed to fit different razors. In the latter half of the twentieth century, cartridge razors were developed in which a disposable cartridge with one to five blades snaps onto a more ergonomic handle and head that allows the cartridge to pivot to keep the blades angled to the skin at a predetermined angle. The most convenient to use are fully disposable safety razors in which two or more blades are molded together with the blade head and an integral plastic handle. In the early versions the blades were rigidly embedded in the handle assembly and therefore could not pivot. Now, two-piece injection-molded plastic designs that can snap-fit together allow the blade head to pivot. Such a disposable safety razor with a pivotable head that has two steel blades and a lubricating stick of a water-soluble lubricant is shown in Figure 2.2.11.

(a) Figure 2.2.11 (Continued)

(b)

(c)

(d)

Evolving Applications of Plastics

(e)

(f)

(g)

Figure 2.2.11 Photos of a plastic disposable safety razor. (a) Top view of a fully assembled, ready to use disposable safety razor with the transparent, snap-on protective cap in place. (b) Back view of the razor with the safety cap in place. Notice the two steel blades and the white strip of a water-double lubricating material. (c) Front view with the head removed from the shank. The left and right protrusions on the shank head slip into grooves in the head, and the thin central “tongue” provides a restoring spring action as the head pivots about the shank top. (d) Back view of the head and the shank. In addition to the blades, notice the white lubricating strip. (e) Detailed view of the back side of the molded head. Notice the four protrusions that stake the top and bottom parts of the plastic assembly used for encapsulating the steel blades. (f) Detail of the top view showing the left and right shank protrusions inside the corresponding grooves in the head. Notice the central shank “tongue” in contact with the back of the head surface. (g) Back view of the head-shank assembly. Note the position of the “tongue” that acts as a restoring spring.

2.2.5

Eyewear

Eyeglasses, invented in the thirteenth century, first worn by monks and scholars, were held in front of the eyes or balanced on the nose. The side or temple pieces that rest over the ear, and bifocals – a very important advance – were invented in the eighteenth century. Initially, because of personal vanity, eyeglasses with temples only were worn when necessary, especially by women. Scissor spectacles, in which the distance between the lenses could be changed, were developed in the nineteenth century for men who did not wish to be seen with spectacles – George Washington, US President, and Napoleon Bonaparte used them. During this period, lorgnettes – a hand-held design in which the lens frame is attached to a rod on one end – were used to avoid having glasses on the nose. Pince-nez (“pinch-nose”) glasses, which had no temples and in which the glasses sat on the nose, were introduced in the middle of the nineteenth century; US Presidents Teddy Roosevelt and Calvin Coolidge wore them regularly.

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Greater use of spectacles with frames with temples, or eyeglasses – glasses because the lenses were made of glass – began around the turn of the twentieth century. The lenses always were of glass; the frames were either of metal or turtle shell, referred to as horn-rimmed glasses. In the 1930s a Hollywood actor, Harold Lloyd, who wore tortoiseshell spectacles with large, round lenses, started a fashion craze for temple spectacles. The use of animal tissue such as horn or tortoise shell limited the production of spectacles. In the 1940s, the availability of new plastics resulted in widespread use of plastic “horn-rimmed” frames and, later, into fashionable frames, especially for women. While the demands of fashion have continually changed frame materials among plastics, stainless steel, titanium, and flex alloys, the most far-reaching change in spectacles has been the shift from glass lenses; while glass lenses have excellent scratch resistance, excellent optical clarity, and to which anti-reflective coatings adhere well, they are heavy and thick, and can chip or shatter more easily than plastic lenses. The first successful plastic material for lenses – which is still in use – is CR-39 (an abbreviation for Columbia Resin 39, 39th in a series of developmental materials) the name for a thermosetting monomer (allyl diglycol carbonate) that is polymerized (cured) to form the lens material. CR-39 lenses are 50% lighter than glass lenses, are far less likely to shatter, and have optical quality almost as good as glass. While they can be tinted easily, they require a scratch-resistant coating to improve durability, and special coatings for ultraviolet (UV) protection. In comparison to CR-39 lenses, those made of polycarbonate (PC), a thermoplastic, have higher impact resistance, are about 25% thinner, about 20% lighter, and block out all UV radiation. PC lenses too require scratch-resistant coatings and, being more reflective, may require anti-reflective coatings. Plastic lenses are thinner than glass ones owing to the higher refractive indexes of plastics. Because the densities of plastics are almost the same, plastics with higher refractive indices tend to be thinner. This feature is especially important for high-power glasses that would require thick glass lenses. The use of polycarbonate, which can easily be molded into complex shapes, makes possible high impact-resistant protective glasses that also are fashionable. Figure 2.2.12 shows lightweight protective goggles in which the clear “glass” is made of molded polycarbonate and the frame is of a different plastic. Note how the use of plastics helps with styling.

Figure 2.2.12 Protective goggles with polycarbonate eye “glasses.” (Photo courtesy of Covestro.)

Evolving Applications of Plastics

2.2.6

Contact Lenses

While special contact lenses that cover the sclera – the white portions of the eyeball – the bulk of all contact lenses only cover the cornea. All contact lenses are now made of plastics. The hard, or relatively more rigid, contact lenses, made of poly(methyl methacrylate) (PMMA), do not allow oxygen to diffuse through to the cornea and therefore cannot be worn for longer periods without affecting the eye. So-called soft, or more flexible, contact lenses are made with a variety of oxygen-permeable hydrogels that are more comfortable to wear. Silicone hydrogels have very high oxygen permeability, allowing them to be worn overnight; some disadvantages, such as having hydrophobic surfaces that affect wettability, have been overcome through development of newer materials and surface treatment. In addition to being used for eyesight correction, contact lenses of various hues are also used for cosmetic reasons. 2.2.7

Bottle Caps

One early application of plastics was to replace metal caps by molded plastic caps. Other than a knurled texture for ease of gripping, such caps did not have any additional functional features – they were designs optimized for metals applied to plastics. Now plastics caps include many functional features in a single molding. A good example is the one-piece, flip-top screw-on plastic molded cap for a blow-molded plastic ketchup bottle, shown in Figure 2.2.13, which has many features: A flip-top lid that is hinged to the screw-top portion by means of a living “hinge,” which is a very thin ligament that retains its elasticity during the bending involved in opening and closing the lid. It also has a pair of very thin molded webs that act as bistable springs, so that when the lid is opened beyond a certain angle, the lid pops open; just the reverse happens when closing this lid (Section 17.5.3). In addition, the inside surface of the lid has a molded-in cylindrical protrusion that, in the closed position, fits inside the cylindrical hole in the molded-in spout in the screw-top portion, thereby sealing the contents of the bottle. To facilitate opening, diametrically opposite the living hinge the lid has a small circumferential protrusion that, in the closed

(a)

(b)

Figure 2.2.13 Flip-top screw-on plastic molded cap. (a) Photos of closed and open cap. (b) View showing detail of hinge and screw thread (right-hand side of figure).

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position, is just above a depression molded into the screw-top portion, allowing the flip-top to be easily opened. To ensure that the lid does not open by itself due to looseness caused by poor tolerances, a small region of the outer cylindrical surface just above the molded-in depression, on which the inside rim of the flip-top seats, is provided with a thin molded protruding rim; on closing the lid “snaps” over this protrusion, thereby providing a locking mechanism. And, finally, the bottom portion of the screw-top cylindrical surface has fine, molded-in serrations that provide easy gripping while screwing or off the cap. Clearly, this bottle cap provides far more functionality than is possible with a metal cap. Further, molding-in these features is not significantly difficult from molding a simple metal-like cap; and the amount of material required is about the same. But all the molded functional features provide added value. The cap shown in this figure was designed for a ketchup bottle that can be squeezed to push ketchup through the spout. Contrast this with rigid glass bottles in which the ketchup has to be ejected by a jerking motion. Living hinges and web bistable springs used in this bottle cap, which makes this design possible, are now routinely used in many plastic caps, not all of which are screwed on. In one such example, shown in Figure 2.2.14a, instead of screwing on to the bottle spout, the elliptic cap has a cylindrical snap-fit that locks onto the blow-molded bottle spout; the inside of the cylindrical portion of the underside of the cap (top part of figure) has a groove that “snaps” around the protruding ring on the top of the bottle

(a)

(b)

(c)

Figure 2.2.14 Snap-fit plastic molded cap. (a) Underside of molded cap (top of figure) with cylindrical cap that snap-fits onto cylindrical spout on blow-molded bottle (bottom of figure). (b) Cap-bottle assembly with cap open. (c) Unsnapping cap by twisting over hump.

Evolving Applications of Plastics

spout (bottom part of figure) (Section 21.3.1). The resulting assembly, with the cap open, is shown in Figure 2.2.14b. The problem of removing the cap has been cleverly solved by seating the elliptic cap over a hump molded onto the bottle. Twisting the cap forces the cap onto the hump thereby pulling it off the spout (Figure 2.2.14c). In addition to facilitating cap removal the elliptic shape is esthetically pleasing. This type of cap is now being used in many applications. 2.2.8

Drip-Proof Spouts

After pouring liquid from a bottle some of the liquid flows down the outer surface of the bottle. With glass bottles, not much could be done to alleviate this problem – and the same problem continued with plastic bottles the tops of which had the same shapes as the glass bottles from which they were copied. This problem has successfully been solved by redesigning the top of plastic bottles. Figure 2.2.15 shows a blow-molded bottle for dispensing bleach. Figure 2.2.15a shows an external view of the bottle with the screw-on cap in position; notice the integrally molded handle. Figure 2.2.15b shows the bottle in the pour position; molded-on markings on the inner surface of the cap can be used to measure the amount of liquid poured. Figure 2.2.15c shows a close-up view of the spout; any liquid dripping on its outer surface will return to the bottle in its vertical position. (Figure 21.8.5 shows one way of attaching the molded spout to the blow-molded bottle.)

(a)

(b)

(c)

Figure 2.2.15 Drip-proof spout for liquid dispensing bottles. (a) blow-molded bottle with injection-molded cap. (b) Open bottle in pouring position into cap. (c) Detail showing spout design.

2.2.9

Plastic Tops for Paper Containers

Plane paper containers for powders – such as salt and pepper – or paper containers with coated surfaces – such as for milk and juices – are ideally suited for inexpensive short-term packaging of everyday household items. Early paper containers for powders had metal tops for dispensing the contents. And the tops of paper containers for liquids were earlier folded and sealed by adhesive bonding; they were not easy to open. Innovative use of plastics has replaced these methods by more user-friendly plastic carton tops. The combined use of paper and plastics involves joining considerations.

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2.2.9.1

Plastic Tops for Cardboard Salt Containers

In the past, 1-kg (2-lb), heavy-paper cylindrical containers for table salt had a metal spout for pouring salt. Figure 2.2.16 shows such a container with an innovative one-piece molded polyethylene top and a conventional metal bottom. The top has two molded-in living hinges – one for the lid for the molded screen opening and the other for the spout (Figure 2.2.16b) – but no bistable springs. The spout stays closed by friction acting on the sides; the screen lid stays closed by friction of the cylindrical protrusion seating inside the screen holes. Notice the two molded-in recesses to facilitate the opening of the overhanging lids. The bottom surface has ribs to stiffen the lid (Figure 2.2.16c). The bottom of the lid has a cylindrical protrusion of a smaller diameter to fit inside the cardboard cylinder to which it is attached by an adhesive. To better anchor the lid to the cardboard cylinder, the outer surface of the mating plastic lid has thin molded-in ribs. A noteworthy feature of this plastic lid is that it shows how much complexity can be molded into a single part in one operation. It is the added functionality that makes it economical to have a plastic lid. Because it is less expensive, notice that the featureless, flat bottom of the container is made of metal.

(a)

(b)

(c)

Figure 2.2.16 (a) Paper salt container with plastic lid. (b) Plastic top shown with the screen lid and the spout in the open position. Both of them are attached to the main molded body through living hinges. (c) Bottom surface of the molded plastic lid with the spout in the closed position.

2.2.9.2

Plastic Tops for Paper Juice Cartons

Coated paper cartons for milk, later adapted for juices, have been used for quite some time. In earlier versions the tops of these rectangular cartons were folded and adhesively bonded in such a way that a foldable portion acted as a spout. Now such paper cartons have closeable plastics tops. Figure 2.2.17a shows a juice paper carton with screw-on plastic cap that has a removable sealing diaphragm. The carton top with plastic cap and the sealing diaphragm removed is shown in

Evolving Applications of Plastics

(a)

(b)

(d)

(c)

(e)

(f)

(g)

Figure 2.2.17 (a) Juice carton with screw-on plastic cap. (b) Carton top with plastic cap removed. (c) Underside of the paper container top, with the plastic top with the sealing tab pulled up, and the plastic cap. (d) Plastic top with sealing diaphragm and pulling tab still in place. (e) Plastic top with partially removed diaphragm, and an alternative design with the top sealed with adhesively bonded aluminum cover (right). (f) Cutaway view of the plastic top showing the diaphragm attached to the side through a very thin molded-in connecting piece; note the pull tab attached to the diaphragm. (g) A view of the underside of the molded plastic top with two thin ridged rings used for ultrasonically bonding the plastic top to the paper. (Photos courtesy of PepsiCo.)

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Figure 2.2.17b. Figure 2.2.17c shows the underside of the paper container top, a view of the plastic top with the sealing tab pulled up, and the plastic cap. Figure 2.2.17d shows the plastic top with sealing diaphragm and pulling tab still in place. Figure 2.2.17e shows the plastic top with partially removed diaphragm, and an alternative design with the top sealed with adhesively bonded aluminum cover (right). Figure 2.2.17f is a cutaway view of the plastic top showing the diaphragm attached to the side through a very thin molded-in connecting piece; note the pull tab attached to the diaphragm. A view of the underside of the molded plastic top with two thin ridged rings used for ultrasonically bonding (Section 21.7.4) the plastic top to the paper is shown in Figure 2.2.17g. 2.2.10

Toys

Plastics have been used for toys for about a century. In the early years, celluloid was used for making blow-molded heads and bodies for dolls. Except for specialty toys made of natural materials such as wood, almost all toys for infants, toddlers, and older children are made of plastics. They can be “passive” toys with articulating parts that may even have audio prompts; tablet-based gaming toys with video screens, which come with educational software, are also available. Plastics are also used for “active” toys, such as “erector sets,” that require thought-based assembly. One of the most popular active toy systems is based on the iconic LEGO® brick (Figure 2.2.18a), which can be stacked (Figure 2.2.18b) by snap fitting to form complex objects such as buildings

(a)

(c)

(b)

(d)

Figure 2.2.18 LEGO building blocks. (a) Iconic LEGO brick. (b) Stacked bricks. (c) Bricks of different size and color. (d) Tire modules. (Photos courtesy of the LEGO Group.)

Evolving Applications of Plastics

and aircraft. The bricks come in different sizes and colors (Figure 2.2.18c), which have been augmented with different building blocks, such as tires (Figure 2.2.18d). The bricks and other building blocks are made of thermoplastic acrylonitrile-butadiene-styrene (ABS); the tires are made of an elastomer. These, and many other modules can be used to build many complex objects, requiring a significant amount mental effort; the assembly process helps with the development of three-dimensional spatial awareness and dexterity. Some examples of objects constructed from LEGO modules include the Sydney Opera HouseTM (Figure 2.2.19a), an airplane (Figure 2.2.19b), the Star WarsTM Death Star (Figure 2.2.19c), and the Star Wars Republic Gunship (Figure 2.2.19d).

(a)

(c)

(b)

(d)

Figure 2.2.19 Objects assembled from LEGO building blocks. (a) Sydney Opera House. (b) Airplane. (c) Star Wars Death Star. (d) Star Wars Republic Gunship. (Photos courtesy of the LEGO Group.)

2.2.11

Consumer Audio

The very large consumer audio market comprises two synergistic technologies: Recording audio and playing live or recorded audio. Casings of many audio recording and play devices, such as radios, Walkmans, disk players, and mobile phones, are made of plastics – as are casings of small and large speakers.

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In such applications the use of plastics helps to integrate many functions into a single molded casing that also provides an esthetically pleasing exterior – all at a reduced cost. With major advances in miniaturization of electronics, enormous advances in recording devices and media and in audio and video reproducing systems were made in the twentieth century. Small, portable devices for the mass market were only made possible by the use of plastics. While portable mobile devices such as cell phones – which make extensive use of plastics – can store large amount of audio, and can directly wirelessly stream audio – the quality of audio is limited by the size of speakers. This resulted in a new class of “portable” speaker systems – from simple stereo speakers to surround sound systems – that had to be connected by wires to the mobile device. With the advent of Bluetooth technology these devices can wirelessly connect with small, truly portable Bluetooth-enabled speakers encased in plastic enclosures. 2.2.11.1

Recording Media

Other than very early but limited use of recording audio on tin foil, audio recordings were made on wax cylinders. These recordings, which were difficult to duplicate, could be played back by amplifying the sound output by inactive horns. A major advance was in first directly record analog audio onto spiral grooves in a disk, using an electrodeposition process to make a “negative” image of the grooves onto a metallic master, and then using the master to press the grooves onto a plastic disk. The first lasting, successful disk material, shellac – a resin secreted by the female lac bug in the forests of India and Thailand – was used till the 1950s for making 30-cm (12-in) diameter, 78 rpm records, on which each side could record about 3.5 minutes of audio. These records were brittle, causing them to break easily. In the 1950s they were replaced by 25- or 30-cm (10- or 12-in), 33 rpm long playing records (LPs) made of polyvinyl chloride (PVC), or vinyl, a thermoplastic; they could record about 35- and 45-minutes of audio between the two sides, respectively. The ability to press microgrooves into these vinyl disks made stereo recordings possible. While these flexible disks did not break, they had a tendency to warp and scratched more easily than the shellac records. Another vinyl format was a 45-rpm, 18-cm (7-in) disk, mainly for recording a single song per side. In all the disk formats, in the play mode the stylus (transducer) moved along the spiral from the outside to the inside. The next advance in recording media was the use of thin, 0.25-in (6.35-mm) wide iron oxide coated polyester (polyethylene terephthalate, PET) films. In the record (or write) mode the signal to be recorded was used to vary the magnetic field on a stationary head, past which the tape was moved at a constant speed, to “write” the signal onto the tape by preferentially orienting the oxide crystals on the tape. In the play (read) mode, the same head would pick up the recorded crystal orientations to reconstruct the original signal. Reel-to-reel systems use a 6.35-mm (0.25-in) wide tape that normally moves at 19- or 9.5-cm (7.5- or 3.75-in) per second. The fidelity of the recorded signal could be improved by writing the same audio signal across more tape by increasing tape speed, longer tapes. The improvement in tape and media quality, and the use of electronic noise reduction systems, such as those from Dolby Labs, improved the quality of recordings. The main drawback of such reel-to-reel systems was the size of the desk-top tape decks required. Increasing tape and magnetic recording-head quality made possible prerecorded, 12.7 × 1.8-cm (5 × 7-in) cartridge tape recorders – in which the tapes did not have to be threaded – capable of recording eight tracks. But it was the invention of the two-track compact cassette, which uses 3.81-mm (0.15-in) wide tape at a constant tape speed of 4.75 (1.875 in s−1 ), that made stereo portable devices, such as the popular Sony Walkman, possible. With improvements in tape and media – which, in addition to iron oxide, were followed by chromium dioxide, and metal – cassette tapes became the medium of choice;

Evolving Applications of Plastics

high-performance cassette decks that could use the newer tapes, and were equipped with Dolby B and C noise reduction circuitry, were used for the highest fidelity recordings. Compact cassettes were available in play lengths of 60, 90, and 120 minutes in which the PET tapes have thicknesses of 15 –16, 10 –11, and 9 μm, respectively. The systems and media discussed thus far all used analog signals and recording systems. Although introduced after compact disks (CDs), digital audio tapes (DATs), for which the recorder player uses a rotating head as in video recorders, could make digital recordings at sampling rates of 48 kHz in comparison to the 44.1 kHz used in CDs. DAT cassettes used the same tape media as compact cassettes. A major advance in recording fidelity was a shift from analog to digital recording, made possible by plastic CDs in which a signal is recorded in the form of tiny indentations (pits), encoded in a spiral track molded on the top surface of a plastic disk. On CDs, the areas between pits are known as “lands.” Each pit is approximately 100 nm deep by 500 nm wide, and its length varies from 850 nm to 3.5 μm. The pitch, the distance between the tracks, is 1.6 μm. CDs have diameters of 120 mm (4.7-in), are made from 1.2-mm-thick polycarbonate disks, and can hold up to about 80 minutes of audio or about 700 MB of data (Figure 2.2.20).

Figure 2.2.20 Polycarbonate CDs, DVDs, and Blu-ray disks. (Photo courtesy of Covestro.)

In contrast to records, CDs are read outwards from the innermost groove. Also, while records rotate at a constant rpm during playback, so that the linear velocity at the stylus decreases as it spirals inwards, CDs are designed to be read at a constant surface velocity of 1.2 –1.4 m s−1 , approximately equivalent to 500 rpm at the inside to 200 rpm at the outside groove – as the disk is played, its rpm decreases from the beginning to the end. CDs are read by focusing a 780-nm (infrared) wavelength semiconductor laser within the CD player through the bottom of the polycarbonate layer. The change in heights between pits and lands results in a difference in the way the light is reflected; this information is converted into the encoded signal.

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The first enhancements in CD technology – essentially increasing the capacity of the 120-mm disks – was the digital video disk (DVD), which is read by 650-nm (red) diode lasers, allowing a smaller 0.74 μm pitch, resulting in an increased storage capacity of 4.37 GB. The next enhancement was the Blu-ray Disc, which uses a wavelength of 405 nm (violet), one dual-layer disc of which can store 50 GB of data. 2.2.11.2

Audio Systems

A knowledge of how audio playback devices work helps in understanding how plastics have made possible high-fidelity table-top radios and audio players. The most important component of any audio playback device is a speaker driver, a transducer that converts an electrical audio signal into audio sound waves. The functional components of a driver, schematically shown in Figure 2.2.21a, comprise the (i) speaker cone, also called a diaphragm, the linear oscillating motion of which causes pressure waves (air motion) that are perceived as sound by the ear, which is driven by (ii) the attached voice coil the oscillations of which are produced by the electrical audio signals reacting with the magnetic field in the gap, AB, in the yoke produced by permanent magnets. The cone-voice-coil assembly is connected to a rigid metal frame – consisting of an outer ring connected to the main supporting frame by means of legs – through a front surround suspension of a soft foam-like material and a rear, corrugated membrane called a spider; besides centering the assembly in the magnetic air gap, the tension in these suspension members centers the assembly in the air gap when the voice coil is not energized. The cone-voice-coil assembly is better visualized in the enlarged view in Figure 2.2.21b. The electromechanical motion in this system is similar to that of a solenoid, the difference being that in a solenoid the coil is fixed but the yoke moves. The motion of the voice coil requires that it be electrically connected through thin, flexible leads. In the vertical mode shown, the cone motion radiates sound in the vertical direction; in most speaker systems the driver is mounted with the cone axis being horizontal, so that the sound waves move normal to the speaker. The fidelity of the frequency range of sound reproduced by a driver depends on the diameter of the cone. High-frequency sound is best reproduced by small-diameter drivers called tweeters; very low-frequency sound requires large-diameter drivers called woofers. Intermediate frequencies are produced by mid-range drivers. High-fidelity speakers use a mix of such narrow frequency-range speakers to cover the wide frequency range perceived by human ears; the electrical signal to the speaker system is divided by crossover networks to feed individual drivers. The type of enclosure used for a driver affects sound quality. Because the outer ring of the metallic structure is connected to the main support structure through metallic legs, in addition to the forward pressure waves, the cone motion also produces backward waves. Since drivers are usually mounted in rigid enclosures of wood or plastic, the backward pressure waves are not utilized (Figure 2.2.22a). However, a vent in the enclosure (Figure 2.2.22b) makes it possible for the backward pressure waves to contribute to the sound generated by the driver; the volume of air in the enclosure and the size of the vent has to be “tuned” to obtain enhanced speaker performance in a desired frequency range. One shortcoming of small drivers, of the type used in small, table-top radios and music players, is loss of low-frequency fidelity – the low-frequency sounds have very low, almost inaudible amplitudes. One way of improving sound quality is to use an acoustic waveguide, examples of which are pipes in church organs and flutes. In both cases a small audio signal at one end – air from bellows in an organ, and air blown through lips in a flute – are magnified into rich, audible sound. The backward pressure waves could be used with an acoustic waveguide to improve low-frequency driver performance, as schematically indicated in Figure 2.2.23.

Evolving Applications of Plastics

CONE MOTION SPEAKER CONE

MOUNTING RING

FRONT SURROUND SUSPENSION

DUST COVER

SPIDER

B

A

FRAME LEG

B

VOICE COIL TERMINAL

A

VOICE COIL LEADS YOKE

SUPPORTING FRAME

VOICE COIL

PERMANENT MAGNET

(a) VOICE-COIL AND SPEAKER-CONE MOTION

SPEAKER CONE COMPLIANT CONNECTOR

VOICE COIL LEADS

SPIDER

A

B

B

A

COIL SUPPORT CYLINDER

PERMANENT MAGNET

YOKE VOICE COIL

(b) Figure 2.2.21 Functional components of a speaker driver. (a) A voice coil energized by an electrical audio signal reacts with the magnetic field in the air gap of a yoke to produce a to-and-fro motion in a cone, thereby generating pressure waves in air that are perceived as sound by ears. Note that the voice-coil-diaphragm assembly is connected to a rigid metallic support through a front (surround) and a rear, corrugated membrane (spider) suspensions. (b) An enlarged view of the voice-coil-diaphragm assembly in relation to the yoke air-gap region.

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(a)

(b)

Figure 2.2.22 Simple, rigid enclosures for a driver. (a) Driver mounted in an enclosed rigid box. (b) Driver mounted in a vented enclosure that allows backward pressure waves (indicated by arrow) to contribute to the sound in a desired frequency range. WAVEGUIDE FORWARD PRESSURE WAVES BACKWARD PRESSURE WAVES

Figure 2.2.23 Driver attached to a waveguide (tube) that magnifies the sound of the backward pressure waves in a tuned frequency band.

Using this concept requires overcoming two issues: First, organs require many pipes, each tuned to a specific frequency, in flutes the length of the waveguide is controlled by fingers opening vents. Second, the lengths of the waveguides a large, on the order of 65 cm (26 in). Through a groundbreaking invention, Bose Corporation designed a single waveguide to cover a wide frequency range. And Bose solved the waveguide length problem by folding it into a compact enclosure. The first commercial table-top radio using this acoustic waveguide speaker concept was the Bose Wave® radio, which while occupying a small space 10.6 × 36.9 × 21.9 cm (4.2 × 14.5 × 8.6in), produced rich, high-quality sound comparable to that of much larger speaker systems. In it two 65-cm (26-in) long waveguides are coiled to increase the amplitude of the small 5-cm (2-in) speakers at the front of the Wave radio. A version of this radio with a CD player is shown in Figure 2.2.24a; the detailed view of the plastic waveguide in Figure 2.2.24b shows where the two drivers are attached at the front ends.

Evolving Applications of Plastics

(a)

(b)

Figure 2.2.24 (a) Remotely controllable Bose Wave Music System. (b) View of the plastic waveguide showing attachment points for drivers at the left and right front ends. (Photos courtesy of Bose Corporation.)

Packaging this technology in such a small space would not have been possible without the use of plastics. Figure 2.2.25 shows the exploded view of the plastic casing and the waveguide.

Figure 2.2.25 Exploded view of the plastic casing and waveguide for a Bose Wave Music System. (Photo courtesy of Bose Corporation.)

The same technology is used in the Bose® SoundDock® audio player (Figure 2.2.26a) for portable music devices such as iPods. An exploded view of the internal waveguide system is shown in Figure 2.2.26b.

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(a)

(b) Figure 2.2.26 Remotely controllable Bose SoundDock audio player for portable phones and music devices such as iPods. (a) External view. (b) Exploded view of the plastic waveguide showing attachment points for drivers at the top left and right front ends. (Photos courtesy of Bose Corporation.)

Evolving Applications of Plastics

2.2.12

Vacuum Cleaners

Vacuum cleaners, first marketed by the Hoover Company in 1908, have existed for over a century. In the earliest models most parts were made of metal, and the suction hose was of rubber. The vacuumed particles were collected in cloth bags through which air could be sucked – these bags could be washed and reused. Later, cloth bags were replaced by disposable paper bags. In 1936 Hoover introduced the first vacuum cleaner with a plastic body of Bakelite (thermosetting phenolic). The increasing use of plastics made it possible to add more complexity and function, while at the same time significantly reducing weight and cost. Now a host of bagless vacuum cleaners, in which the collected particles can be emptied into a garbage bag, are available. Full-sized portable models that use rechargeable batteries eliminate the inconvenience of having to connect and pull electric cords around. Figure 2.2.27 shows views of a fully assembled upright, bagless Hoover portable vacuum cleaner (Figure 2.2.27a and b) in which almost all the parts – except the motor and shafts – are made of plastic; Figure 2.2.27c shows the filter; Figure 2.2.27 parts d, e, and f show the rotating hard brush on the vacuum head, a narrow dusting head, and a pivoting dusting head, respectively; and Figure 2.2.25g shows the battery pack and the charging box. Note that in the hard brush assembly, the shaft with a molded-in pinion for the belt drive and bristles are fabricated in a single injection molding operation. Note also that almost all the parts in this vacuum cleaner – including the bristles on the rotating head and on the dusting heads – are made of plastic. 2.2.13

Small and Major Appliances

The many features in most small appliances – electric toothbrushes, electric shavers, hair driers, electric irons, kitchen grinders, blenders, and mixers – would not be possible without the use of plastics. A good example is that of electric irons with water spray and steam options (see Figure 21.8.4). The use of plastics in these appliances has made possible many new features at reduced costs. Major appliances – window air conditioners, microwave ovens, dishwashers, clothes washers and driers, and refrigerators – have seen an increasing shift from the use of sheet steel to plastics. The two main drivers for the use of plastics in these appliances are: (i) parts consolidation that reduces assembly and inventory costs, and (ii) the addition of many features that would not be possible without the use of plastics. A good example of the benefits of parts consolidation is that of the molded Carry Cool window air conditioner housing, shown in Figure 1.5.11b, in which a single thermoset molding replaced more than 10 pressed steel parts and fasteners. Figure 21.7.4 shows the early version of a dishwasher-pump housing made by vibration welding two glass-filled polypropylene injection-molded parts. Another example is that of replacing deep-drawn sheet steel dishwasher tubs, which had to be enameled to prevent rusting, by the single injection-molded, talc-filled polypropylene tub shown in Figure 2.2.28a, which had more features that made attaching parts such as pumps easier. Still another example is that of refrigerator liners that are now exclusively made by thermoforming plastic sheets, mainly of ABS, that have complex internal features for movable shelves and trays. Figure 2.2.28b shows an early version of an ABS liner for a mid-sized refrigerator.

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(b)

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Figure 2.2.27 Full-sized, upright, bagless, portable Hoover vacuum cleaner that makes extensive use of plastics. (a, b) External views of the assembled vacuum cleaner. (c) Detail showing the filter. (d) Rotating hard brush. (e) narrow dusting head. (f) Pivoting dusting head. (g) Battery pack with charging box. (Photo courtesy of Techtronic Floor Care Technology Ltd.)

Evolving Applications of Plastics

(a)

(b)

Figure 2.2.28 Plastics in major appliances. (a) Injection-molded talc-filled polypropylene dishwasher tub. (b) Thermoformed ABS refrigerator liner. (Photos courtesy of GE Appliances.)

2.3 Medical Applications Plastics are now widely used in almost all spheres of medicine: Child-proof plastic caps for medicine bottles; PVC in catheters, drip bags and urine bags; silicones in contact lenses, breast implants and heart valves; ultrahigh molecular-weight polyethylene (UHMWPE) in total knee and hip replacement implants; and the casings for medical diagnostic and imaging equipment. The development of sterile-packed, single-use syringes have revolutionized the delivery of medical services. High-performance plastics are used in high-temperature applications, such as equipment requiring sterilization. 2.3.1

Drip Bags and Accessories

Intravascular therapy – direct injection of liquids into veins – is the most commonly method used in providing intensive care to a very large number of patients. This is mostly used for hydration, although medication can be combined with the fluid to reach body cells faster through veins. The earliest containers for intravenous infusion of medication consisted of drips from glass bottles with rubber caps that were kept in place by thin metal covers swaged onto the bottle opening. The central

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portion of the metal cap had an annular opening into which a needle could be pierced to allow the infusion liquid to drip and flow through rubber tubes. Later, such glass bottles were replaced by plastic ones, and rubber tubes were replaced by plastics tubes. Now all infusion liquids – such as blood for transfusion, glucose solution for hydration, and medication – are dispensed form transparent, sealed IV (intravascular) fluid bags (pouches) made of polyvinyl chloride. And the tubes used for transporting the fluid from the bag to the needle inserted into a vein are also made of PVC. More than 300 million IV bags are used annually in the US. Alternative polymers are being considered for IV bags because of toxicity concerns with PVC. 2.3.2

Syringes

The earliest syringes for medical applications were made of metal. In the next iteration, syringes consisted of glass cylinders, on one end of which was mounted a metal cap with a protrusion on which needles could be mounted. The glass barrel served two useful functions: to ensure that air bubbles were eliminated and a careful control of dosage through graduations on the outer surface. The metal needles used came with thin wires that could be used to clean the bore. To ensure sterility, before usage the syringe assembly and the needles were boiled in water. With the development of appropriate glasses and glass forming technology, all-glass syringes became possible. The outer barrel was molded with a cap for mounting needles and the plunger was also made of glass. The inner surface of the barrel and the outer surface of the hollow plunger had to be ground to provide a leak-proof interface. While the single piece barrels did cut cost, the inconvenience of having to sterilize the syringes continued. Injection-molded plastic syringes made it possible to make single-use, throw-away ready-for-use syringes in heat-sealed, sterile packs; the first such product was introduced by BD in 1961. Since then, many innovative advances in plastics syringe technology include safety variants that prevent accidental pricks from needles, automatic removal and disposal of used needles into biohazard containers, and sterilized syringe-needle assemblies with pre-filled dosage of injectables, which can be used for mass immunizations in remote areas. Figure 2.3.1 shows several polycarbonate injection-molded syringes for different applications.

Figure 2.3.1 Polycarbonate injection-molded syringes for several different applications. (Photo courtesy of Covestro.)

Evolving Applications of Plastics

A good example of a more complex “syringe” made possible by plastic is the insulin pen shown in Figure 2.3.2; the transparent portion is made of polycarbonate (PC), and the opaque portions of the housing are of a PC/ABS blend. The pen, which can be carried around in a shirt pocket just like a writing pen, is filled with an insulin charge. The dosage per shot can be dialed-in (controlled) – shown by the number 12 in this photo – by twisting the head, and injected through a needle at the head by pressing a “button” (plunger on the head).

Figure 2.3.2 Insulin pen for injecting controlled dosages of insulin. The transparent portion is made of polycarbonate (PC) and the opaque body is of a PC/ABS blend. (Photo courtesy of Covestro.)

2.3.3

Medical Imaging Equipment

For esthetics and cost reduction, the outer casings of almost all medical imaging equipment are made of plastic. In some equipment, for part consolidation, the use of plastics extends well beyond the outer casing. In MRI (Magnetic Resonance Imaging) equipment, in which high-intensity magnets are used for imaging, the outer casing has to be made of plastic. The photo in Figure 2.3.3 shows a Siemens Edge Computed Tomography (CT) scanner in which almost the entire external fascia parts are made of plastic. Notice the very large, annular plastic fascia covering the annular opening of the scanner.

Figure 2.3.3 Siemens Edge CT Scanner in which almost the entire external casing is made of plastic. (Photos courtesy of Siemens.)

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2.3.4

Plastic Models for Body Parts

Three-dimensional printing technology (Section 19.7) makes it possible to use CAT scan and MR images to prepare digital models that can be 3D printed into realistic, full-size models of human body parts. These models help surgeons prepare for interventional surgery. Three-dimensional printing technology is also revolutionizing how dental prostheses are made. New devices can accurately scan the interior of a patient’s mouth and convert it into a digital model that can be used to make 3D printed prosthetic devices. Because no two persons are alike, with each organ differing from those depicted in textbooks, surgeons have to develop detailed plans before complex operations. To improve patient outcomes, MR or CT imaging data are used to make 3D printed models of organs with all their intricacies, features, and fine detail. This significantly enhances surgical preparedness, reduces complications, and decreases operating time. Medical models can be created to simulate the same flexibility of human organs, allowing surgeons to accurately practice procedures. Three-dimensional printing offers a wide range of material properties, enabling anatomical models to accurately replicate organs and flesh or mimic the rigidity of bone. The photos in Figure 2.3.4 show three 3D printed heart models in which different plastics have been used to mimic the properties of different tissues.

(a)

(b)

(c)

Figure 2.3.4 Three 3D printed heart models in which different plastics have been used to mimic the properties of different tissues. (Photos courtesy of Stratasys Ltd.)

2.4 Automotive Applications Plastics have been used in automobiles almost from the beginning. Phenolics and other thermosets have been used for distributor caps and even for steering wheels. Later, fiberglass – a composite of glass fiber embedded in a thermoset matrix – was used for outer body panels of niche cars. And, of course, elastomers – in the form of natural and synthetic rubbers – used for tires and inner tubes, made possible automobiles as we know them. The more recent use of plastics in automobiles was first driven by weight reduction and aerodynamic styling to improve fuel efficiency. Safety considerations, integration of many parts into one, and the freedom provided to designers for using complex, curved shapes are additional drivers for the increasing

Evolving Applications of Plastics

use of plastics. Although plastics already make up about 50% of the volume of materials used in cars, they only account for about 10% of the weight. Much of the increasing use of plastics was made possible by a host of new thermoplastics that started to become available in the 1950s. Later, in the 1980s, special blends were developed for applications such as bumpers – which had to withstand 8-kph (5-mph) barrier impacts even at low temperatures – and fenders – for which the blends had to withstand the high paint-bake oven temperatures. The increasing use of plastics in these high-volume applications has led to major innovations in part fabrication methods for reducing cost. Although as many as 15 plastics are used in automobiles, polypropylene, polyurethane, and polyvinyl chloride account for about 30, 15, and 15%, respectively, of the plastics used. 2.4.1

Bumpers

The first major automotive structural (load-bearing) application of thermoplastics was an all-plastic bumper capable of withstanding an 8-kph (5-mph) front impact, which was developed in response to US new car bumper standards: The first US passenger car bumper standard, Federal Motor Vehicle Safety Standard (FMVSS) 215, for “Exterior Protection,” issued in 1971, called for passenger cars, beginning with model year (MY) 1973, to withstand 8-kph (5-mph) front and 3.2-kph (2-mph) rear impacts against a perpendicular barrier without damage to certain safety-related components such as headlamps and fuel systems. The 1972 Motor Vehicle Information and Cost Saving Act (MVICS Act) enacted by the US Congress mandated a bumper standard for yielding the maximum feasible overall cost reduction of to the public. In 1976, this new requirement was consolidated with FMVSS 215 into a new bumper standard, which applied to passenger cars beginning with MY 1979. While this standard was diluted in 1982 to reduce the front impact speed to 4-kph (2.5-mph), these requirements drove the increasing use of plastics in bumpers. At first these requirements were met by attaching the conventional metal bumper to the car frame through oil-filled “shock absorbers” that absorbed front and rear end impacts; the four shock absorbers added to the cost. The first, innovative, path-breaking all-plastic bumper, directly attached to the car frame, capable of an 8-kph (5-mph) barrier impact, was used in the 1984 Ford Escort shown in Figure 2.4.1; the car on the right has a conventional steel bumper beam.

Figure 2.4.1 1984 Ford Escort with the first all-plastic bumper capable of withstanding an 8-kph (5-mph) barrier impact. The car on the right has a conventional steel bumper. (Photo courtesy of Ford Motor Company.)

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Except at the two ends and the mounting points, for most of its length the bumper beam has a D-shaped cross section, made by welding a molded vertical back plate to a ribbed C-shaped molded section (see Figure 21.7.5). Of all later automotive applications, this bumper may be regarded as the first automotive full-sized structural application of an unreinforced thermoplastic. A blend of polycarbonate and poly(butylene terephthalate) (PBT) – with rubber modifiers to provide ductility even at low temperatures – was specially developed for this application (Section 12.2.7). This blend is now routinely used in many applications, such as lawnmower decks, which require impact resistance. 2.4.2

Fenders

Early use of thermoplastics in vertical body panels, such as fenders, was limited by the requirement that such panels be compatible with the high temperatures in the paint systems that the steel body, to which the panel was attached, had to go through. A special blend of modified polyphenylene ether (MPPE) and nylon was developed to address this application (Section 12.2.8). This material could endure the temperatures in online priming and painting, so that the panels could be assembled to the steel body-in-white (BIW) prior to painting. The first use of this thermoplastic blend on a vertical body panel, the fender shown in Figure 2.4.2, was on the General Motors 1987 year Buick LeSabre T-Type1 sports coupe. The switch from steel to thermoplastic enabled reduced the part weight by 45% – from 3.3 kg (7.3 lb) for steel to 1.8 kg (4 lb) for the thermoplastic blend.

FENDER

(a)

(b)

Figure 2.4.2 First online paintable thermoplastic fender. (a) Location of an automotive fender. (b) First MPPE/Nylon blend injection-molded fender. (Photo courtesy of SPE Automotive Division.)

This pioneering application resulted in the use of the MPPE/Nylon blend in more than 45 platforms on more than 20 million vehicles. Beyond automobiles, this material is used in tractors and lawnmowers for home and agricultural use. 2.4.3

Throttle Bodies

Throttle bodies for automotive engines were mainly made of die-cast aluminum. Driven by weight reduction requirements, one of the first cars to have injection-molded thermoplastic throttle bodies was the

Evolving Applications of Plastics

year 2000 Neon 2.0 L engine; its throttle body parts were made of 30 wt% glass-filled polyetherimide (30-GF-PEI), a high-temperature thermoplastic (Section 11.7.2). The series of photos of this first application in Figure 2.4.3 shows the complexity of these parts.

(a)

(b)

(d)

(c)

(e)

Figure 2.4.3 Several views of a molded throttle body made of a 30 wt% glass-filled polyetherimide, a high-temperature thermoplastic. (Photos courtesy of SABIC.)

2.4.4

Exhaust Manifolds

Originally exhaust manifolds were made of cast iron. Later, the push for weight reduction to improve gas mileage led to the use of aluminum and, for still more weight reduction, to the use of thermoplastics. Besides weight reduction, the smoother surfaces of plastic manifolds improved the efficiency of air flow. The first large-scale use of plastic exhaust manifolds required the use of the relatively complex, fusible core molding process (Section 17.8.5.1). Now plastic manifolds are made at much lower cost by vibration-welding injection-molded parts (Figure 21.7.6).

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Figure 2.4.4 shows a complete vibration-welded and bolted plastic manifold assembly for a Volkswagen engine. Notice the complexity in this assembly that vibration welding made possible.

Figure 2.4.4 Injection-molded automotive manifold assembly made by bolting together two vibration-welded 30 wt% glass-filled PA6,6 subassemblies. (With permission from Volkswagen AG, photo courtesy of Röchling Automotive SE & Co. KG.)

2.4.5

Gas Tanks

For the longest time, automotive gas tanks were made of terne-coated steel (an 8% tin-lead coating). Fabricating steel tanks requires welding; weld seams can be sources of leaks in crashes, during which sparks caused by impacting steel surfaces can cause explosions. And steel tanks were prone both to internal and external corrosion. A prototype nylon plastic fuel tank was built for Ford in 1964. A single-layer, high-density polyethylene (HDPE) was used in the VW Passat in 1972. In 1994, a six-layer co-extrusion blow-molding process (Section 19.3.1.4) made possible multilayer gas tanks incorporating a barrier layer to reduce emissions. In 2007, a new blow-molding process made it possible to integrate all interior components of the system into the tank, while at the same time reducing emissions by eliminating joints. Now well over 90% of all automotive gas tanks in Europe and the United states are made of such multilayer HDPE. Plastic gas tanks have many advantages: Lighter weight gives better fuel economy. Better safety in a crash because, in contrast to metal fuel tanks, plastic ones can bend and flatten, rather than rupturing and spilling gasoline, thereby eliminating fire or explosion resulting from fuel leakage. The molding process allows the tanks to be shaped to fit available space. They are corrosion resistant. The main disadvantage is material recyclability. Figure 2.4.5 shows a complex, blow-molded gas tank. It is now possible to blow-mold such tanks with several internal components such as rollover valves, fuel-delivery modules, and level sensors directly integrated into the tank during the molding process. The cutaway photo of one such gas tank with many internal features is shown in Figure 19.3.13.

Evolving Applications of Plastics

Figure 2.4.5 Photograph of a 55-liter, multilayer HDPE blow-molded gas tank. (Photo courtesy of Kautex Textron GMBH & Co. KG.)

2.4.6

Door Modules

The interior of conventional automotive doors had a large number of separate metal components and subassemblies that had to be individually bolted, screwed, welded, and/or riveted into place in a labor-intensive process. The first all-plastic integrated door-hardware module, the SuperPlug® integrated composite door-hardware module shown in Figure 2.4.6, first used in several 1997 model year General Motors Corporation (GM) vehicles, reduced components and fasteners by up to 75% by molding features into a single gas-assist injection-molded composite frame that “plugged-into” the inner door on the vehicle-assembly line.

Figure 2.4.6 Photograph of the first all-plastic door-hardware module, the SuperPlug integrated composite door-hardware module. Many key components – including wire harnesses, door handle, window guidance channels, stereo speakers, and electric motors – snap into place. After pretesting the assembly is shipped the car assembly lines. (Photo courtesy of SPE Automotive Division.)

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The success of this application can be gauged from estimates of over 250-million integrated composite door-hardware modules having been produced globally and used on over 60-million vehicles worldwide. At an average of 5 lb of plastic per module, this represents over a billion pounds of resin consumed in just this one automotive application. The SuperPlug module was made possible by three untried technologies: (i) It was the first complex part produced with gas-assist injection molding (Section 17.8.5.2), (ii) it used a new grade of glass-reinforced PC/PBT (Section 12.2.7), and (iii) it had the highest level of parts integration in door modules. Many key components – including wire harnesses, the door handle, window guidance channels, stereo speakers, and electric motors – snapped into place at the component assembly plant and pretested before being shipped to GM’s assembly lines. Molding features into the plastic carrier replaced over 40 metal parts and 15 – 25 fasteners with a single composite module and 5 – 6 fasteners. 2.4.7

Boots for Constant-Velocity Joints

A constant-velocity joints (CV joints, or CVJs) allows a drive shaft to transmit power through a variable angle at constant rotational speed. They are mainly used in front wheel drive vehicles. A CVJ boot is an elastomeric seal that serves two main functions. First, it protects bearings in the CV joint by keeping out dirt, water, salt, ice, snow, mud, and stones. Second, it keeps lubricating grease inside the CVJ as the drive axle rotates. Earlier, such boots were made of injection-molded rubber or hard plastic. Rubber boot failures accounted for the majority of drive axle repairs. Starting with model year 1984, General Motors Corporation introduced a new blow-molded, thermoplastic elastomer (TPE) CVJ half-shaft drive-axle boot seal. This new boot (Figure 2.4.7), made of a copolyester TPE (Hytrel®), provided a far more robust CVJ sealing solution; these new parts last the service life of the vehicle. Also, it was lighter, more durable, and less costly than the injection-molded rubber boots it replaced. It was 65 g (2.3 oz) lighter than the rubber part it replaced (85 versus 150 g). Continued innovations have resulted in TPE CVJ boots weighing as little as 40 g (1.4 oz).

Figure 2.4.7 First CV joint boot seal made of blow-molded copolyester TP E (Hytrel). (Photo courtesy of ABC Group Inc.)

Evolving Applications of Plastics

2.5 Infrastructure Applications Plastics are now being used in many infastructure applications both in the form of resins – such as in large-scale glazing applications – and in the form of advanced composites in utility poles, bridges, and for reinforcing old steel and steel-reinforced concrete structures. This section describes some large-scale applications of unfilled plastics, advanced composites, and rubber tires. These applications involve the use of thermoplastics, thermosets, and elastomers (rubber). 2.5.1

Glazing

Because of their optical clarity, polymethyl methacrylate (PMMA) and polycarbonate are (PC) used in glazing applications, but because it is somewhat brittle PMMA its use is limited. The very high toughness (impact strength) of PC makes it ideal for glazing and signage applications. Profile extrusions of PC, of the type shown in Figure 2.5.1a, are not fully transparent but allow sufficient light to pass through to form roofs of walkways and polyhouses. PC sheets with scratch-resistant coatings are extensively used in major glazing applications, such as in the roof of the stadium shown in Figure 2.5.1.

(a)

(b)

Figure 2.5.1 (a) Polycarbonate profile extrusion. (b) PC glazing in Helanshan Stadium, China. (Photos courtesy of Covestro.)

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2.5.2

Security Glazing

Modern buildings make extensive use of glass windows and doors that are vulnerable to intrusion. This has led to the formation of a host of non-glass alternatives for security glazing to prevent forced entry, and to provide bullet resistance. One such product line comprises laminates in which the outer layers are sheets of polycarbonate (PC) (Section 11.6.4) with scratch-resistant coatings that are laminated to sheets of PC or poly(methyl methacrylate) (PMMA) (Section 11.5.4) with bonding interlayers of polyurethane (PU). Figure 2.5.2 parts a and b show a 9-mm full metal copper jacketed bullet with

(a)

(c)

(b)

(d)

Figure 2.5.2 (a) Frontal view of 0.75-in thick PC-PMMA-PC laminate with embedded 9-mm bullet. (b) Rear view of 0.75-in thick plaque. (c) Frontal view of 1.25-in thick PC-PC-PC laminate with embedded 0.57 magnum bullet. (d) Rear view of 1.25-in thick plaque. (Photos courtesy of SABIC.)

Evolving Applications of Plastics

a lead core embedded in a 0.75-in thick Lexgard® MP750 laminate, that has two 0.125-in thick scratch-resistant coated PC outer sheets laminated to a 0.5-in thick PMMA core through PU interlayers. Figure 2.5.2a shows the bullet penetrated the ductile PC sheet through a small hole and then expanded in the thicker, more brittle PMMA sheet, the radial expansion causing star-shaped fractures in that sheet. The back surface of the laminate (Figure 2.5.2b) shows that, although there is a very slight bump on the surface, the bullet did not fully pierce the back sheet. The circular “bubble” behind star-shaped region shows delimitation at the PU layer between PMMA sheet and the rear PC sheet. In this laminate, the acrylic sheet absorbs the energy while the more flexible PC holds the laminate together. Figure 2.5.2 parts c and d show a 0.44 magnum lead semi-wadcutter bullet with a lead core embedded in a 1.25-in thick Lexgard SP1250 plaque, a 4-ply laminate that has two 0.125-in thick scratch-resistant coated PC outer sheets laminated to two inner 0.5-in thick PC cores through PU interlayers. Figure 2.5.2c shows the bullet penetrated into the ductile PC sheet through a small hole and then expanded in the two thicker PC sheets. In comparison to the MP750 laminate, the ductile PC cores do not develop star-shaped fracture surfaces. The back surface of the laminate (Figure 2.5.2d) exhibits a slightly bigger bump but, again, the bullet did not fully pierce the back sheet. The white circular region is caused by delamination at the PU interlayers. 2.5.3

Water Management Systems

Traditionally, water management has involved water delivery systems – with lead, cast iron, and steel pipes – and water and waste disposal systems, such as sewers – earlier made of stoneware and bricks, and later with concrete. Now plastics have made possible innovative systems that can be used for many applications such as rain water harvesting and attenuation (storing captured storm water to help control its flow back into a natural water course or drainage system), surface water drainage and disposal through fully integrated, easily installed plastic piping and tank systems. These systems are making it possible to capture, store, control, and reuse scarce water resources more efficiently. In contrast to older concrete manholes and culverts that have to be built up and cast in situ, their newer plastic replacements can be pre-fabricated off-site and delivered for on-site installation as modular units. The off-site construction of these complex pre-fabricated structures under factory-controlled conditions ensures high construction quality, and greatly reduces the time and amount of labor required for on-site installations. These plastic structures are typically up to 6% of the weight of their concrete equivalents, making them easier to transport and handle during installation. This weight difference is particularly important when installing larger items such as concrete culverts that require heavy lifting gear, resulting in longer lead times and added transportation costs. The following examples illustrate the innovative way in which Ridgistorm-XL large-diameter HDPE pipes, which are available in diameter ranges from 750 to 3000 mm, are used to handle the underlying complexity of such systems. The pipes are made by extruding polyethylene through custom built dies, which is then helically wound onto a mandrel. The pipes have a smooth inner surface that is supported by an external core tube structure to give the pipe its required stiffness. Their inherent strength and flexibility allow the loads from backfill to be distributed beyond the pipe, thereby reducing the risk of leaks and ingress. These pipes can be joined by quick, easy to install methods that do not require specialists – either by using the electrofusion process or a rubber seal system that uses socket and spigot jointing with two rubber seals installed into the spigot end of the pipe fitting. Figure 2.5.3 shows a pre-fabricated Ridgistorm-XL dual run 1500-mm diameter manhole, made by extrusion-welding, being lowered for installation into a trench.

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Figure 2.5.3 Pre-fabricated Ridgistorm-XL 1500-mm diameter, dual run manhole being lowered for installation into a trench. (Photo courtesy of Polypipe.)

Figure 2.5.4 shows a complex, underground large-scale multileg Ridgistorm-XL attenuation tank undergoing on-site installation. The pre-fabricated modules are made from 1200- and 1800-mm diameter pipes. This all-plastic tank has a capacity of 680,000 liters.

Figure 2.5.4 Complex, multileg Ridgistorm-XL attenuation tank undergoing on-site installation. (Photo courtesy of Polypipe.)

For efficient flow in sewers at low flow in dry weather conditions, traditional sewers have a narrow, deeper channel running along the lowest portion. In a Ridgistorm-XL system this is achieved by adding smooth bore shoulders at the bottom of the pipe, as shown in Figure 2.5.5a; this sewer pipe has

Evolving Applications of Plastics

an internal diameter of 1500 mm. The shoulders are hollow and, as shown in Figure 2.5.5b, the size of the deeper channel can be changed by varying the positions of these shoulders. The enlarged view in Figure 2.5.5c shows a resistance wire preform used for electrofusion welding of two pipe sections (Section 21.8.1).

RIDGISTORM-XL PIPE

SMOOTH BORE SHOULDERS

(a)

WIRE PREFORM FOR ELECTROFUSION WELDING

(c)

(b) Figure 2.5.5 Ridgistorm-XL pipe fitted with smooth bore shoulders to create a deeper channel for low flow in dry weather conditions. (Photo courtesy of Polypipe.)

Figure 2.5.6 shows a large-diameter, pre-fabricated 3000 mm Ridgistorm-XL chamber with benching and channeling. This chamber allows for the connection of multiple inlet and outlet pipes, as well as benching and channeling to allow for the flow and diversion of water within the chamber. This example shows the versatility with which complex systems can be fabricated with Ridgistorm-XL pipes. Plastics have made it possible to attenuate large amounts of stormwater underground, both for run-off management and for storing water for reuse, by means of hollow molded geocellular units that can easily be assembled to provide the space for water storage. In one such system, shown in Figure 2.5.7, the 1000 × 500 × 400-mm modular polypropylene Polystorm cells have void ratios of 95%, and can

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Figure 2.5.6 Complex Ridgistorm-XL pre-fabricated chamber with benching and channeling. Operator is extrusion-welding a large-diameter pipe to the main tank. (Photo courtesy of Polypipe.)

each store 190 liters of water. Clips are used for making horizontal connections among cells, and shear connectors are used for vertical location. POLYSTORM MODULAR CELL

PIPE CONNECTION PORT

VERTICAL SHEAR CONNECTOR

HORIZONTAL CONNECTION CLIP

Figure 2.5.7 Water storage polypropylene Polystorm modular cells with 95% voids connected horizontally with clips and located vertically through shear connectors. (Photo courtesy of Polypipe.)

Modular cells can be assembled to form structures of any shape and size. The structure is wrapped in a non-permeable, geomembrane that can attenuate rainwater collected from the roof gutter system or

Evolving Applications of Plastics

surface drains and either release the water within set discharge limits or, where soil conditions allow, be wrapped in a permeable geotextile to infiltrate water slowly back into the surrounding soil. Figure 2.5.8a shows a Polystorm water storage system being assembled. The detail in the lower half of Figure 2.5.8b shows another partially assembled system with an oval opening for horizontal access and maintenance purposes. Notice the vertical inlet pipe on the top part of the tank, this allows for vertical access for maintenance and inspection.

(a)

(b)

Figure 2.5.8 Assembly of water storage Polystorm modular cells for water storage. (a) View showing partial assembly. (b) Partially assembled system with oval opening for plastic pipe connections. (Photos courtesy of Polypipe.)

2.5.4

Large-Diameter Piping

Because of their excellent corrosion resistance, relative flexibility allowing wide bends, and weldability making possible very long pipes, extruded HDPE are used in many sectors such as agriculture, mining, water transportation, and the gas industry. In particular, large-diameter HDPE pipes are used for transporting potable water, gas, and sea water for cooling and desalination purposes. Weldability of HDPE allows for attachment of flanges and size reducers. And “saddle welds” on the side allow for lateral pipe connections, thereby making possible manhole systems for sewers. For a pipe with a specified outside diameter, the pipe wall thickness depends on design pressure – the thickness increases with the pressure. In the piping industry, the thickness is specified by the SDR number, which is the pipe diameter divided by the wall thickness. For each diameter, pipes are available for different SDR numbers, with smaller numbers corresponding to larger thicknesses and higher pressure ratings. A good example is that of a several kilometer-long plastic pipeline for pumping sea water from the Persian Gulf to near Assaluyeh in Iran for desalination and for use as cooling water in chemical

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plants. Because of the shallow seabed gradient, to get adequate suction, the pipeline had to be laid about 2 km into the sea. For this application a 1600 mm diameter HDPE pipe, with a wall thickness of 61.5 mm (SDR 26), was used. As the pipe factory is about 1,300 km from the Persian Gulf site, the pipes had to be transported on trucks, each of which could only carry only two 12 m pipe sections (Figure 2.5.9a). These 12-m sections were butt-welded at site (see Figure 21.6.5 in Section 21.6.3.) to form long pipes. Figure 2.5.9b shows long welded sections in an artificial dock to prevent the pipe being shifted during welding by wave action; the “black” rings at regular intervals along the pipes are the locations of the welds. The cranes and heavy equipment in the background give an idea of the size of the operation.

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Figure 2.5.9 Assembly of 1600 mm diameter HDPE pipe at Persian Gulf site. (a) Two 12 m length pipe sections on truck for transportation from factory to construction site 1,300 km away. (b) Welded pipe lowered into artificial dock. (Photos courtesy of SKZ – German Plastic Center.)

While a catalog lists an SDR 13.6, 1,600 mm diameter HDPE pipe with a 117.6-mm wall thickness, the extrusion of a 1600 mm diameter pipe with a 140-mm thickness (SDR 11.4), rated at a pressure of 14 bars (1.4 MPa), has been demonstrated; a cross section of the extruded pipe is shown in Figure 2.5.10. Pipe extrusion requires some of the largest extruders. Figure 2.5.11 shows the extrusion of a 2,400-mm diameter HDPE pipe with a wall thickness of about 90 mm. An idea of the size of this pipe can be had from the photo of a cross section shown in Figure 19.2.13. 2.5.5

Power Line Poles

Utility poles have traditionally been made of steel, concrete, and wood. Now, pultruded (Section 20.8.1) poles made from continuous glass fiber embedded in a vinyl ester thermosetting resin, which is corrosion and rot resistant, are providing a viable long-lasting alternative. Figure 2.5.12 shows such a pultruded utility pole four pultruded crossarms. Such pultruded poles are available in lengths up to 24 m (80 ft).

Evolving Applications of Plastics

(a)

(b)

Figure 2.5.10 (a) Section of a 140-mm thick, 1,600-mm diameter extruded HDPE pipe. (b) Thickness of cross section. (Photos courtesy of P.E.S Co.)

Figure 2.5.11 Extruded HDPE pipe with a diameter of 2,400 mm and a wall thickness of about 90 mm. (Photo courtesy of P.E.S Co.)

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Figure 2.5.12 Pultruded composite utility pole with pultruded crossarms. (Photos courtesy of Creative Pultrusion, Inc.)

2.5.6

Bridges

Modern bridges have traditionally been made of steel or reinforced concrete. Figure 2.5.13 shows a 25.9-m long by 1.2-m wide (85 × 4-f t) pedestrian bridge assembled by bolting together pultruded sections (Section 20.8.1). Besides saving weight, a bridge made from pultruded sections can be assembled and installed at the site in much less time than an equivalent steel bridge.

Figure 2.5.13 Pedestrian bridge assembled from pultruded composite structural profile. (Photos courtesy of Creative Pultrusion, Inc.)

2.5.7

Composite Sheet Piling

Traditionally, seawalls or bulkheads are made of wood that can rot, concrete that can spall, or steel that can rust. Pultruded composite fiberglass sheet piling – made in modular Z- and hat-sections with interlocking features along each edge allowing the panels to be joined together to form a single structure – overcomes the problems with traditional seawalls. They can be installed above (Figure 2.5.14a) or

Evolving Applications of Plastics

(a)

(b)

(c) Figure 2.5.14 Pultruded composite sheet piling for seawalls. (a) Installation above water line. (b) Below the waterline and in the transition area. (c) System accessories such as sheet pile corner connectors, sheet pile caps, walers, wale splice plates, and washers. (Photos courtesy of Creative Pultrusion, Inc.)

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below (Figure 2.5.14b) the waterline or in the transition area. This system comes with sheet pile corner connectors, sheet pile caps, wale splice plates, walers, and washers (Figure 2.5.14c).

2.6 Wind Energy The harnessing of wind energy is becoming an important source of renewable energy. A typical modern wind turbine (Figure 2.6.1a) consists of three blades mounted on a hub attached to a horizontal shaft; the rotating shaft drives an electrical generator, either directly or through a gear box, fitted inside a nacelle. The nacelle, mounted on a tall pole, can rotate about a vertical axis to enable the rotating blades to directly face the wind. Also, in order to extract the maximum energy from the wind, the blades can rotate about the hub mount to present the best angle to the wind direction. An active control system is required both to keep the rotor axis normal to the wind direction and to maintain the correct orientation of the blades. Typically, a large group of wind turbines is deployed in a “wind farm,” which can be on land or at sea. Figure 2.6.1b shows a land-based wind farm. An offshore wind farm is shown in Figure 2.6.1c.

(a)

(b)

(c)

Figure 2.6.1 Wind turbines (a) Single turbine with three blades. (b) Onshore wind farm. (c) Offshore wind farm. (Photos courtesy of LM Wind Power.)

The power P that can be extracted from wind is proportional to 𝜌Av 3 , in which 𝜌 is the density of air, A = 𝜋 r 2 is the area swept by the rotor, and v is the wind velocity. Higher power levels can be obtained by higher wind speeds and larger rotor diameters. Since the wind velocity at any location increases with the height, higher wind velocities can be obtained by mounting the rotor at larger heights. Larger rotor diameters require large blades, the lengths of which can be limited by structural requirements as well as by the difficulty of transporting the blades to the wind turbine site. Figure 2.6.2 shows the evolution in the sizes of wind turbines since 1978: The power generated by a turbine, the blade length used, and the tower height have increased, respectively, from 50 kW, 7.7, and 16 m in 1978 to 8 MW, 88.4, and 180 m in 2016. For comparison, this figure also shows the world’s largest commercial airplane, the AIRBUS A380, which has a wing span of 79.75 m, the Washington Monument that is 169 m high. At 180 m, the tower for the 8-MW turbine is taller than the Washington Monument. The 88.4 m blade of the 8-MW turbine is the longest blade in the world. The size of the blade can be gauged from its photos shown in Figure 2.6.3. Because of weight considerations, such blades can only be made of thermoset-based advanced composites (Chapter 25). Their large sizes require special manufacturing techniques that are described in Section 20.8.3.5.

Evolving Applications of Plastics

AIRBUS A380

2002

2004

2012

2016

POWER 50 kW 750 kW

1.5 MW

5 MW

5-6 MW

8 MW

BLADE 7.7 m LENGTH

23.5 m

34.0 m

61.5 m

73.5 m

88.4 m

TOWER HEIGHT

48.2 m

70 m

126 m

150 m

180 m

YEAR

1978

16 m

1998

WASHINGTON MONUMENT

Figure 2.6.2 Growth of power ratings and sizes of wind turbines. For comparison of the sizes involved, also shown is the world’s largest commercial airplane, the AIRBUS A380 that has a wing span of 79.75 m, and the Washington Monument that with a height of 169 m is shorter than the 180-m high tower of the 8-MW wind turbine. (Adapted from figure courtesy of LM Wind Power.)

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(b)

Figure 2.6.3 Photo of 88.4-long wind turbine blade. (a) Blade outside manufacturing building. (b) Blade transport from manufacturing facility to highway. (Photo courtesy of LM Wind Power.)

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(a)

(b)

Figure 2.6.4 Transportation of the world’s largest wind turbine blade. (a) Blade being hauled on two-lane highway. (b) Blade being transported straight across a roundabout. (Photos courtesy of LM Wind Power.)

The very large size of these blades also poses significant transportation problems, requiring temporary shutdown of roads and special arrangements at road crossings. Figure 2.6.4a shows a blade being hauled using two lanes of a temporarily closed two-lane highway, Figure 2.6.4b shows the blade being driven across a road roundabout rather than around it.

2.7 Airline Applications Advanced composites, made of high-strength fibers embedded in plastic matrices, were originally developed for high-stiffness, high-strength aerospace applications in which weight reduction is important. They are extensively used in large modern aircraft, such as the largest commercial airplane, the AIRBUS A380, shown in Figure 2.7.1. While advanced composites have been used in many of its structures, high-performance thermoplastics and thermosets have been used in the cabin interiors (Figure 2.7.1b) – such as windows, interior wall ceiling surfaces, and overhead storage compartments.

(a)

(b)

Figure 2.7.1 AIRBUS A380. (a) Photo of plane after takeoff. (b) View of seating in economy class cabin. (Photos courtesy of AIRBUS.)

Evolving Applications of Plastics

Depending on the seating configuration, in its two decks the A380 can carry between 544 and 853 passengers. It is 72.72 m (238.58 ft) long, 24.09 m (79.04 ft) high, has a wing span of 79.75 m (261.65 ft), and has a maximum takeoff weight of 575 metric tons (633.8 tons). Its four engines give it a range of 15,200 km at a maximum speed of Mach 0.89. This has been made possible using advanced aluminum alloys for the wing and fuselage, along with the extensive application of composite materials in the center wing box’s primary structure, wing ribs, and rear fuselage section. Advanced fiber-reinforced plastic composites make up about 22 wt% of the material used in this plane.

2.8 Oil Extraction Oil extraction – both from the ground and sea – requires large-diameter, high-pressure flexible hoses that can withstand abrasive corrosive environments. Figure 2.8.1a shows an offshore oil rig for pumping oil from the sea bed. Figure 2.8.1b shows high-performance pipes connected to an oil distribution head.

(a)

(b)

Figure 2.8.1 (a) Offshore oil rig. (b) High-performance hoses connected to a distribution head. (Photos courtesy ContiTech AG.)

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The oil industry uses the most demanding high-pressure hoses and flexible pipes, in which rubber layers are reinforced with robust layers that can include a stainless-steel interlock inner tube, and an outer stainless steel strip wound protection layer (Section 25.4.3).

2.9 Mining Advanced composites, made of high-strength fibers embedded in plastic matrices, were originally developed for high-stiffness, high-strength aerospace applications in which weight reduction is important. But rubber tires are also advanced composites with high-strength fibers embedded in rubber matrices; these advanced composites are highly flexible. The tires for mining equipment used in rough, challenging environments, can be very large. Figure 2.9.1 shows Titan LDR150 tires mounted on a Caterpillar Large Wheel Loader (CAT 994H). The LDR150 tire has an outside diameter of 3861 mm (12.67 ft), an unloaded section width of 1552 mm, and is rated for a payload of 125,000 kg at an inflation pressure of 7 bar (275,000 lbs at an inflation pressure of 102 psi).

(a)

(b)

Figure 2.9.1 (a) Titan LDR150 tire. (b) Tires on Caterpillar (CAT 994H) Large Wheel Loader. (Photos courtesy of Titan International, Inc.)

The “grooves” on either side of the tire (Figure 2.5.13a), called sipes, are only used in select mining applications, such as in the oil sands in Canada. They help to dissipate heat in a sand type application where the material engulfs the tire tread as it goes through its footprint. In many quarry and mining operations, chains are placed on the tires (Figure 2.5.13b) for protecting the tire surfaces from cuts and puncture damage from the rocks and jagged surfaces.

Evolving Applications of Plastics

As large as this tire is, the world’s largest tire, shown in Figure 25.4.14, has a diameter of 4,026 mm (13.21 ft) and is rated for a load of 105 kg at an inflation pressure of 7.5 bar (2.21 × 105 pounds at an inflation pressure of 110 psi).

2.10 Concluding Remarks Clearly, plastics are now being used in almost all products used by society – from thin films used for grocery bags and in various packaging applications; to high-temperature resistant clothing for firefighters and “bulletproof” body armor; to all types of shoes; to bottles for water, soda, shampoo, and medicine; to squeezable tubes for dispensing toothpaste and medicine; to toys; to audio and video devices; to IV bags and syringes for medical use; to vacuum cleaners, dishwasher tubs, and refrigerator liners; and to many parts in automobiles. While in most of these applications plastic resins without fillers are used, many applications require the use of particulate or chopped glass fiber fillers to increase stiffness and strength. The highest stiffness and strengths are obtained by using thermoset plastic matrices to bind high-stiffness and high-strength glass or carbon fibers into the desired shapes of parts; such materials are called advanced composites (Chapter 25). An important class of very flexible, yet very strong advanced composites uses rubber (elastomers) as the matrix material. The largest such application is the use of continuous fiber-reinforced rubber in tires. Belts and hoses also use such materials. A perusal of the representative applications described in this chapter shows that an important reason for the growing use of plastics is the availability of a host of part processing (fabrication) methods that allow very complex shapes, which combine many functions that in metal would require several metal parts, to be made in one manufacturing operation. The ease with which very complex shapes can be made allows products to be made with pleasing esthetic shapes.

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Part II Mechanics

This part has six chapters that introduce all simple principles required for understanding the role of mechanics in plastics engineering. Mathematics is limited to elementary differential and integral calculus.

Chapter 3 Introduction to Stress and Deformation Because an understanding of mechanics principles is an important aspect of the new plastics engineering paradigm, concepts such as stress and strain are introduced at a very elementary, one-dimensional level. However, some relatively advanced two-dimensional topics are indicated by an asterisk; still more advanced topics are indicated by two asterisks. Such relatively advanced topics are not essential for understanding the more elementary discussions, but give an idea of the analyses required for a more complete understanding of stresses and strains under more complex loading.

Chapter 4 Models for Solid Materials This chapter provides an elementary, mostly one-dimensional introduction to constitutive equations that relate stress to strain, or deformation. Included are elastic materials that fully recover to their original state on unloading, and plastic solids that undergo permanent deformation. Thermally induced strains and their effects are also considered.

Chapter 5 Simple Structural Elements Elastic material models are used to analyze the behavior of structural elements, such as beams and torsion members, subjected to simple loads. The phenomenon of buckling, which is important for most thin-walled plastic parts, has been explained in simple, understandable steps. The phenomenon of stress concentration is discussed. Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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Chapter 6 Models for Liquids The discussion of fluids in this chapter is preceded by an introduction to conductive heat transfer, and how simple results can be used for estimating some transient thermal events. Here again, the focus is mainly on one-dimensional analyses of problems that really require a two-dimensional treatment. Starred sections provide insights into the level of mathematics necessary for a better understanding of fluid mechanics. Several phenomena exhibited by polymer solutions and melts, not seen in gases and simple liquids like water, are described.

Chapter 7 Linear Viscoelasticity This chapter provides a one-dimensional analytical foundation for viscoelasticity that is important for understanding materials, such as polymers, which exhibit both solid- and fluid-like behavior.

Chapter 8 Strengthening and Stiffening Mechanisms The last chapter in this part discusses mechanics principles underlying the stiffening of plastics by embedded fibers, which are important for understanding particulate- and chopped-glass-filled plastics, and advanced composites.

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3 Introduction to Stress and Deformation 3.1 Introduction Prediction of part performance and processing requires models that characterize the response of plastics to thermal, mechanical, and electromagnetic stimuli. In mechanics, or engineering science, the relationships between imposed stimuli and the resulting response are called constitutive equations. Because of the complex response of plastics – both in the solid and liquid phases – constitutive equations for plastics are more complex than those for metals and ceramics. Simple models for the mechanical behavior of plastics are discussed in Chapter 4. To develop a feel for how materials are characterized, material characteristics important for structural applications and measures for such characteristics, or properties, are first explored. To establish the terminology and to provide continuity, the basic principles of mechanics are first briefly reviewed.

3.2 Simple Measures for Load Transfer and Deformation A material body subjected to loads experiences mechanical interactions representing transfer of the loads through the body; such loads can induce motion in the body. Mechanical interactions are those that can be measured by forces and moments. A loaded body undergoes deformation: Changes in geometry measured by changes in lengths of material line elements and changes in the angles between pairs of material line elements. The simplest mode of load transfer occurs in a slender bar pulled by an axial force (Figure 3.2.1a), called a tensile load because it causes “tension” in the bar. In this one-dimensional loading the axial tensile load P is uniformly distributed across the cross section having area A, resulting in a uniform load per unit area, 𝜎 = P∕A, called tensile normal stress, which is a measure of the intensity of force transfer. In this simple case the normal stress 𝜎 is uniform. This uniaxial tensile load, or stress, tends to “stretch” the bar in the longitudinal direction, resulting in an increase Δl in its original length l. This stretch Δl is accompanied by a decrease Δw in its lateral, or transverse, width w. Because each section of the bar is subjected to the same load, the stretch Δl is uniformly distributed along the length of the bar. This normal stretch is therefore measured by the stretch per unit length 𝜀L = Δl∕l, called tensile normal strain. In general, normal strains will be denoted by the symbol 𝜀. The subscript L in 𝜀L is used to indicate that this strain is in the longitudinal direction. Similarly, the reduction in width Δw, which is uniformly distributed across the width w, is measured by the width reduction per Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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P

P

𝜎=

P A

𝜀L =

P

∆l l

𝜎= −

P

P A

𝜀L = −

𝜀T = ∆w

𝜀T = − ∆ww

P

P

∆l l

w

P

P

Tensile Length l increases by ∆l Load Width w decreases by ∆w

Compressive Length l decreases by ∆l Width w increases by ∆w Load

Tensile Normal Stress

Compressive Normal Stress

(a)

(b)

Figure 3.2.1 Slender bar subjected to an axial load P. (a) Tensile load increases bar length but decreases width. (b) Compressive load decreases bar length but increases width.

unit width 𝜀T = − Δw∕w, called transverse (lateral) stain. The negative sign in this expression indicates a decrease in width. Instead of a tensile stretching force, the forces can be applied in a pushing sense – called a compressive load (Figure 3.2.1b). By convention, tensile loads and stresses are considered positive, while compressive ones are considered negative. A compressive stress results in a length reduction and an increase in width, so that, while the compressive longitudinal strain 𝜀L is negative, the transverse strain 𝜀T is positive. Thus, for both tensile and compressive loads, the longitudinal and lateral strains have opposite signs. In the second simple mode, the load acts parallel to the surface on which it is applied. Consider such a tangential load P applied to a rectangular block ABCD (Figure 3.2.2). In this case the tangential force is uniformly distributed across the area A, and the load per unit area, P∕A, is called the shear stress 𝜏 . This tangential stress tends to deform the block into the rectangular parallelepiped A′B ′C ′D ′, such that planes of the material parallel to the tangential load slide over each other – in a manner similar to cards in a deck of cards sliding over each other when subjected to a tangential load – without any change in dimensions in the direction of sliding. In this case the deformation, called shear deformation, is measured by the decrease 𝛾 in radians of the right angle enclosed by a rectangular material element of which one of the sides is parallel to the shear stress. This measure is called the shear strain 𝛾 .

Introduction to Stress and Deformation

P

P C

D

D'

τ = A

γ

P A

B

C'

B'

A' P

P

Figure 3.2.2 Deformation caused by a tangential load acting on the surface of a rectangular block.

The normal stress 𝜎 and the shear stress 𝜏 have units of force per unit area, which in SI and inch-pound systems are called Pascals (Pa) and pounds-per-square-inch (psi), respectively. The normal strain 𝜀 and the shear strain 𝛾 are dimensionless quantities. Note that these two measures for deformation are useful mainly when the strains are very small in comparison to unity, that is, when 𝜀 and 𝛾 ≪ 1.

3.3 *Strains as Displacement Gradients The definition for longitudinal strain, 𝜀L = Δl∕l, assumes that the stretch Δl is uniformly distributed over the length l. However, in general, the stretch, and hence the strain, will not be uniform. Consider a small portion AB, of length Δ x, of the unloaded bar, located between positions x and x + Δ x as shown Figure 3.3.1. On the application of a longitudinal load, let the material point A located at x move to A ′ at position x + u, and B move to B ′ at x + Δ x + u + Δ u. Then the length of A ′B ′ will be ( x + Δ x + u + Δ u) − ( x + u) = Δ x + Δ u (Why?), so that stretch in AB caused by the load will be Δ u. It follows that the longitudinal strain in the bar element of length Δx will be Δu∕Δx. In the limit this may be written as 𝜀x = du∕dx, where the subscript x in 𝜀x indicates a longitudinal strain along the coordinate direction x. Expressing shear strain in terms of displacement requires the consideration of two-dimensional deformations. Recall that the shear strain is the reduction, in radians, of the angle between two material line elements that initially are at right angles to each other. With reference to Figure 3.3.2, at the point A located at ( x, y) consider two material line elements AB of length Δ x and AC of length Δ y, aligned along the x- and y-axes, respectively. On deformation, let the point A move to A′ located at ( x + ux , y + uy ), so that (ux , uy ) is the displacement of A. Then, as shown, the point B will move to B ′, located at the point ( x + Δ x + ux + Δ uy , y + Δ uy ) (Why?). In a similar manner, C will move to C ′ located at ( x + Δ ux , y + Δ y + uy + Δ uy ). It follows that the lengths of B ′B ′′ and C ′C ′′ are, respectively, Δ uy and Δ ux . And the lengths of A ′B ′ and A ′C ′ are, respectively, Δ x + Δ ux and Δ y + Δ uy . For small deformations, for which Δ uy ∕Δ x is small in comparison to unity, the angle 𝜃 1 is given in radians by (Why?)

𝜃1 =

Δ uy Δ x + Δ ux

=

Δ uy ∕Δ x 1 + Δ ux ∕Δ x

≈

Δ uy Δx

Similarly,

𝜃2 =

Δ ux Δ ux ∕Δy Δ ux = ≈ Δy + Δ uy 1 + Δ uy ∕Δy Δy

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B'

A'

∆ x + ∆u

x+u

O

x + ∆ x + u + ∆u

B

A

∆x

x

O

x + ∆x

Figure 3.3.1 Differential deformation of a material element AB.

∆u x C'

C''

θ2

C

θ1

∆y

B''

A' B

A

y O

B'

x

∆u y

∆x x + ∆x x + ux

∆ x + ∆ux

x + ∆ x + u x + ∆ux

Figure 3.3.2 Shear deformation between two orthogonal material elements AB and AC.

Introduction to Stress and Deformation

It follows that the shear strain 𝛾 , which is the reduction in the right angle between the material line elements AB and AC, is given by

𝛾 = 𝜃1 + 𝜃2 ≈

Δ uy Δx

+

Δ ux Δy

Now, both ux = ux ( x, y) and uy = uy ( x, y) are functions of both x and y. Also, the variation Δ uy in uy occurs over a distance Δ x over which y does not vary, and the variation Δ ux in ux occurs over a distance Δ y over which x does not vary. Therefore, while taking limits, partial derivatives are used. It then follows that the shear strain is given by

𝛾xy =

𝜕 uy 𝜕 ux + 𝜕x 𝜕y

(3.3.1)

where the subscripts x and y on 𝛾 xy are used to indicate that the shear strain occurs between material line elements that are initially aligned along the x- and y-coordinate directions. In terms of the two-dimensional displacements ux = ux ( x, y) and uy = uy ( x, y), the two-dimensional strain field is given by

𝜀x =

𝜕 uy 𝜕 ux , 𝜀y = , 𝜕x 𝜕y

and 𝛾xy =

𝜕 uy 𝜕 ux + 𝜕x 𝜕y

(3.3.2)

Thus, while it is easier to conceptualize normal strains in one dimension, shear strains involve deformations that at least extend over two dimensions.

3.4 *Coupling Between Normal and Shear Stresses Notice that although material elements in planes parallel to the shear stress direction do not undergo any changes in length, material elements in other planes do (Figure 3.2.2). For example, a material line element along the diagonal AC increases in length, that is, undergoes a positive normal strain. Similarly, a material line element originally along the diagonal DB becomes shorter, that is, undergoes a compressive, or negative, normal strain. Thus, although a shear stress causes planes of material parallel to the shear stress to slide over each other, material elements in other planes undergo normal strains, which are generally associated with normal stresses. To understand how normal stresses may be present even when the applied load is a shear force, consider a portion ABCD of the bar in Figure 3.2.1a, cut normal to the axis at the bottom and cut at the top at an angle 𝜃 to the base AB. For equilibrium, force balance requires that the same axial force P act on both the faces AB and DC. On the top face this force P, which acts at an angle 𝜃 to the local surface normal DC, can be resolved into a force Pn = P sin 𝜃 normal to the surface DC and a force Ps = P cos 𝜃 parallel to this surface. Since the area of surface DC is A∕sin 𝜃 (Why?), these two force components are equivalent to a normal stress 𝜎 n = (P∕A) sin2 𝜃 = 𝜎 sin2 𝜃 and a shear stress 𝜏 = (P∕A) sin 𝜃 cos 𝜃 = 𝜎 sin 𝜃 cos 𝜃 acting on the inclined surface DC. Thus, although all transverse cross sections of a bar subjected to an axial load 𝜃 are subjected to a normal stress 𝜎 = P∕A, material planes inclined to the axis at an angle 𝜃 are subjected to both a normal stress 𝜎 n = 𝜎 sin2 𝜃 and a shear stress 𝜏 = (𝜎 sin 2𝜃 )∕2. The maximum shear stress occurs on 𝜃 = 𝜋 ∕4 and is given by 𝜏 max = 𝜎 ∕2. Note that this plane also has a normal stress 𝜎 n = 𝜎 ∕2. The shear stress on the plane with 𝜃 = 3𝜋 ∕4 has a shear stress 𝜏 max = − 𝜎 ∕2, the negative sign indicating a change in direction of the shear stress. This plane also has a normal stress

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Ps P

D

θ

θ

Pn n

θ C

B

A P = Aσ

Figure 3.4.1 Force P acting on an inclined surface CD resolved into a normal force Pn and a shear force Ps .

compressive 𝜎 n = 𝜎 ∕2. Thus a normal force, or stress, applied along the longitudinal axis of a bar causes a combination of normal and shear stresses on planes inclined to the bar axis. The two most important physical measures for load transfer are normal and shear stresses. In general, the local force transfer at a point in a material is equivalent to a normal stress and a shear stress, the magnitudes of which depend on the orientation n of the area across which the load transfer is being considered. Thus, at any point in a loaded body, the load transmitted across an area depends on the orientation of the area and, in general, the stress on that area consists of both a normal and a shear stress, the magnitudes of which depend on this orientation. And, of course, just as loads, these stresses at a point may also vary with time. The two most important measures for deformation are normal and shear strains. As previously discussed, material elements undergoing shear strains in some planes may even be subject to normal strains in other planes. Thus, at any point in a deforming body, the deformation at a point will consist of a combination of normal and shear strains, each of which can vary with time. A complete understanding of stress – internal load transfer in a body – and strain – deformations in a body – requires a three-dimensional treatment. Even a two-dimensional analysis can provide more insight than is available from the one-dimensional treatment in this section.

3.5 *Coupling Between Normal and Shear Strains The block A ′B ′C ′D ′ in Figure 3.2.2 has undergone a shear strain 𝛾 . As mentioned previously, in this deformation mode material elements in planes parallel to the shear stress direction do not undergo any

Introduction to Stress and Deformation

changes in length. However, a material line element along the diagonal AC increases in length; it undergoes a positive normal strain. Similarly, a material line element originally along the diagonal DB becomes shorter; it undergoes a compressive, or negative, normal strain. Thus, although a shear stress causes planes of material parallel to the shear stress to slide over each other, material elements in other planes undergo normal strains, which are caused by normal stresses. ′ ′ In Figures 3.2.2 and √ 3.5.1, let the lengths of the various line segments be: AB = A B = a, BC = ′ ′ ′ ′ ′ 2 2 B C = b, AC = c = a + b , and A C = c . Also, let angle CAB be 𝜃 , so that tan 𝜃 = b∕a, sin 𝜃 = b∕c, cos 𝜃 = a∕c, B ′C ′′ = b sin 𝛾 and C ′C ′′ = b cos 𝛾 . Then, for the right triangle A′C ′C ′′ c ′ = [(a + b sin 𝛾 )2 + (b cos 𝛾 )2 ]1∕2 = [c2 + 2ab sin 𝛾 cos 𝛾 ]1∕2 so that

[ ]1∕2 ab c ′∕c = 1 + 2 2 sin 𝛾 cos 𝛾 = [1 + 2 sin 𝜃 cos 𝜃 sin 𝛾 cos 𝛾 ]1∕2 c Now for small 𝛾 , for which sin 𝛾 ≈ 𝛾 and cos 𝛾 ≈ 1, it follows that c ′∕c = 1 + (sin 𝜃 cos 𝜃 ) 𝛾 . The normal strain 𝜀𝜃 = (c ′ − c)∕c along AC is then given by ) ( 1 𝜀𝜃 = sin 2𝜃 𝛾 2 The maximum normal strain along AC occurs for 𝜃 = 𝜋 ∕4 and is given by 𝜀max = 𝛾 ∕2. For 𝜃 = 3𝜋 ∕4, which corresponds to the direction DB, the normal strain is given by 𝜀min = − 𝛾 ∕2, the negative sign indicating a compressive normal strain. Thus a shear stain 𝛾 on two orthogonal planes results in normal strains of ± 𝛾 on planes that bisect the planes of shear, planes inclined at 𝜋 ∕4 and 3𝜋 ∕4 to the planes of shear. While the normal strains caused by shear have been determined, such normal strains are, in general, accompanied by shear strains. A complete understanding of this coupling requires a two-dimensional treatment of deformations.

P

P C

D c

A

θ

D'

τ =

b

a B

P A

γ

C' c'

b γ

a B'

A'

P

C''

P

Figure 3.5.1 Deformed geometry of a rectangular block subjected to a tangential load.

3.6 **Two-Dimensional Stress Consider the triangular portion ABC (Figure 3.6.1) of a rectangular plate of uniform thickness, h, subjected to a uniformly distributed normal force in the y-coordinate direction. The resulting uniform normal

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Ps Py

C

θ

Pn n

θ

θ

A

B Py = Ay σy

Figure 3.6.1 Stresses caused by a y-direction force acting on a triangular material element.

stress 𝜎 y in the y-coordinate direction will be equivalent to a force Py = 𝜎 y Ay on the face AB, where Ay = h × AB is the cross-sectional area of the plate in the y-direction. Let the normal n to the face CB be inclined at an angle 𝜃 to the face AB (x axis), with cross-sectional area ACB = AAB ∕sin 𝜃 = Ay ∕sin 𝜃 . Then, just as in Section 3.4, equilibrium requires that the force interaction across the plane CB must equal the force Py = 𝜎 y Ay , which can be resolved into a normal force, Pn = Py sin 𝜃 = 𝜎 y Ay sin 𝜃 , and a tangential force, Ps = Py cos 𝜃 = 𝜎 y Ay cos 𝜃 , acting on CB. It follows that 𝜎 n = Pn ∕ACB = Pn ∕(Ay ∕sin 𝜃 ) = 𝜎 y sin2 𝜃 and 𝜏 = Ps ∕ACB = Ps ∕(Ax ∕sin 𝜃 ) = 𝜎 y sin 𝜃 cos 𝜃 . Next consider the case in which the rectangular plate is subjected to a uniformly distributed load in the x-direction. Then the resulting uniform normal stress 𝜎 x in the in the x-coordinate direction will be equivalent to a force Px = 𝜎 x Ax on the face CA, which can be resolved into a normal force, Pn = Px cos 𝜃 = 𝜎 x Ax cos 𝜃 , and a tangential force, Ps = − Px sin 𝜃 = − 𝜎 x Ax sin 𝜃 , acting on BC. The negative sign in the expression for the tangential force is a consequence of the convention by which the shear stress is considered to be positive in the BC direction, that is, in the direction of increasing 𝜃 , or in a counterclockwise sense; for this geometry, the tangential force Ps acts along the CB direction (contrast this with the absence of the negative sign in the expression for the shear stress 𝜏 caused by 𝜎 y ). From the geometry shown in Figure 3.6.2 it follows that the cross-sectional area of the face CB is ACB = ACA ∕cos 𝜃 = Ax ∕cos 𝜃 , and that 𝜎 n = Pn ∕(Ax ∕cos 𝜃 ) = 𝜎 x cos2 𝜃 and 𝜏 = Ps ∕(Ax ∕cos 𝜃 ) = − 𝜎 x sin 𝜃 cos 𝜃 .

Pn

C

θ

n

Ps

θ Px

Px = Ax σx A

B

Figure 3.6.2 Stresses caused by an x-direction force acting on a triangular material element.

Introduction to Stress and Deformation

The stress resulting on the face CB from uniform loads acting on a rectangular plate in both the xand y-directions can be obtained by the stresses on CB caused by the uniform stresses 𝜎 x and 𝜎 y acting individually. With reference to Figure 3.6.3 it then follows that the normal and shear stresses acting on a plane inclined at an angle 𝜃 to the x-axis are given by

𝜎n = 𝜎x cos2 𝜃 + 𝜎y sin2 𝜃 𝜏=

1 (𝜎 − 𝜎x ) sin 2𝜃 2 y Ps

θ Pn C

θ

P

n

Px = Ax σx A

B Py = Ay σy

Figure 3.6.3 Stresses caused by x- and y-direction forces acting on a triangular material element.

As expected the shear stress is zero for 𝜃 = 0 and 𝜋 ∕2, the normal stresses on these two planes being 𝜎 x and 𝜎 y , respectively. Clearly, the shear stress is a maximum for the planes with 𝜃 = ± 𝜋 ∕4, on which the shear stress is 𝜏 = ±(𝜎y − 𝜎x )∕2; this maximum shear stress is accompanied by normal stresses 𝜎n = (𝜎x + 𝜎y )∕2. It can be shown that, for the two-dimensional case, at least two mutually orthogonal planes exist on which the shear stresses are zero. And the normal stresses on such planes take on their maximum and minimum values. When 𝜎 x ≠ 𝜎 y , only two such planes exist. However, when 𝜎 x = 𝜎 y , the shear stress is zero on all planes, and the normal stress on every plane is 𝜎 n = 𝜎 x = 𝜎 y .

3.7 Concluding Remarks This chapter has shown that an understanding of load transfer within a material body requires the concepts of normal and shear stresses, and that a description of the resulting deformations requires the concepts of normal and shear strains. In the simplest case of a bar subjected to an axial load, the load transfer through the body and the resulting deformation are adequately described by means of normal stresses and normal strains. Also, a pure shear stress induces a pure shear strain. However, even in a uniaxially loaded body such as a bar, a plane inclined to the normal cross section has both normal and shear stresses. And even in a block subjected to a pure shear stress, other planes have both normal and shear stresses.

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Section 3.6 shows how normal stresses acting on two orthogonal planes can be used determine the normal and shear stresses on a plane inclined to these two planes. This result can be extended to the case when the two orthogonal planes carry shear stresses in addition to the normal stresses: The physics is the same, only additional components must be considered in the force balances. While simple examples can be used – as in this chapter – the description of load transfer within a body is at least a two-dimensional problem. In full generality, the definition of stress requires a balance of forces in three dimensions. Again, the physics is straightforward, requiring simple geometrical analyses for carrying out force balances. A description of nonuniform deformation – one in which the strains vary from point to point – requires relating the local strain to the displacement gradients, mathematical quantities such as those in the expressions 𝜀x = 𝜕 ux ∕𝜕 x, 𝜀y = 𝜕 uy ∕𝜕 y, and 𝛾 xy = 𝜕 uy ∕𝜕 x + 𝜕 ux ∕𝜕 y. This involves the use of partial derivatives, which are beyond the stated requirements for this book. For a generalization of the simple results in this chapter the physics remains the same: an application of Newton’s laws of motion. But a more sophisticated understanding of partial differentiation and differential equations is necessary for understanding deformation. While concepts of two-dimensional stress and strain have been introduced in this chapter, most analyses in the following chapter only make use of the very simple concepts introduced in the nonstarred section in this chapter.

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4 Models for Solid Materials 4.1 Introduction Prediction of part processing and performance requires models that characterize the response of plastics to thermal, mechanical, and electromagnetic stimuli. In mechanics, or engineering science, the equations that describe the response of a material to imposed stimuli are called constitutive equations. The complex behavior of plastics – both in the solid and liquid phases – requires constitutive equations that are more complex than those used for metals and ceramics. This chapter describes simple models for the mechanical behavior of solids.

4.2 Simple Models for the Mechanical Behavior of Solids In most applications of solid bodies at rest or in motion, the bodies are considered rigid; they are assumed not to deform. However, as mentioned before, a body subjected to loads, or a stressed body, undergoes deformations. For the same deformation state, as measured by strains or strain rates, the stresses in the body will be different for different materials. Stresses are related to strains through constitutive equations that define the characteristic response of each material. In response to applied loads, materials can exhibit very complex deformation behavior. Many solids can recover completely from small deformations when the loads causing them are removed; the solid goes back to its original undeformed state. A rubber band exhibits this kind of behavior. In some solids, such as clay, loads can cause permanent deformations in the sense that the material does not return to its undeformed state when the loads are removed. In other solids the deformations continue to change even under a constant load. Similarly, fluids can exhibit very complex deformation behavior. In a very broad sense the deformations in a loaded “solid” are well described by strains. However, in “liquids,” in which applied loads induce large, continuing deformations, the deformation patterns are well described by time rates of strain. Because of the complexity of the deformations induced in materials by applied loads, the relations between stress and deformation for materials are best studied by means of idealized models that attempt to mimic specific types of phenomenology. This chapter focuses on time-independent models, which approximate, to varying degrees, the behavior of materials in which the deformations induced by constant, time-independent, loads are time-independent.

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

Introduction to Plastics Engineering

4.3 Elastic Materials The simplest type of solid is an elastic solid, one in which the deformations induced by an applied load disappear when the load is removed. A rubber band exhibits this type of behavior. The simplest model for such solids is that of a linear elastic solid, in which the stress is linearly related to the strain,

𝜎 = E 𝜀, and 𝜏 = G 𝛾

(4.3.1)

E and G are called, respectively, the Young’s modulus and the shear modulus; they both have the dimensions of stress. The stress-strain curve for this simple model is shown in Figure 4.3.1: On loading the stress increases linearly with the strain along OA; unloading occurs along AO.

A

STRESS

108

STRESS-STRAIN CURVE

O STRAIN Figure 4.3.1 Stress-strain curve for a simple elastic solid.

A bar pulled longitudinally to a strain 𝜀L is generally accompanied by a transverse strain 𝜀T of the opposite sign, given by 𝜀T = − 𝜈𝜀L , in which 𝜈 is called the Poisson’s ratio. For most materials the Poisson’s ratio varies between 0.2 and 0.35. For an incompressible material, one which does not undergo any volumetric changes, 𝜈 = 0.5. Consider again the case of a uniaxially loaded bar. Let the longitudinal axis of the bar be the x-axis. Then the stresses and strains will be

𝜎x = E 𝜀x , and 𝜀y = 𝜀z = − 𝜈𝜀x

(4.3.2)

In the two-dimensional case, with 𝜎 z = 0, the stress-strain relations for a linear elastic material have the form 1 𝜀x = (𝜎x − 𝜈 𝜎y ) E

𝜀y =

1 ( 𝜎 − 𝜈 𝜎x ) E y

𝜈 𝜀z = − (𝜎x + 𝜎y )

(4.3.3)

1 𝜏 G xy

(4.3.4)

E

and

𝛾xy =

Models for Solid Materials

And, in the more general three-dimensional case,

𝜀x =

1 [𝜎 − 𝜈 (𝜎y + 𝜎z )] E x

𝜀y =

1 [𝜎 − 𝜈 (𝜎z + 𝜎x )] E y

𝜀z =

1 [𝜎 − 𝜈 (𝜎x + 𝜎y )] E z

(4.3.5)

and 1 1 1 𝜏xy , 𝛾yz = 𝜏yz , 𝛾zx = 𝜏zx (4.3.6) G G G These relations are for isotropic materials, in which the material properties at any point are the same in all directions. It can be shown that isotropic materials have only two independent elastic moduli, the three previously used ones are related through

𝛾xy =

E = 2 (1 + 𝜈 ) G

(4.3.7)

The Young’s modulus E is very different for different materials: Its values for the metals aluminum, brass, copper, and steel are given, respectively, by 69, 104, 117, and 207 GPa (10 ×, 15 ×, 17 ×, and 30 × 106 psi). Glass has a Young’s modulus of about 69 GPa (106 psi). Thus, the Young’s moduli of metals and glass lie in the range of 70 – 210 GPa (10 – 30.5 × 106 psi). In contrast to these high values, the Young’s moduli for unfilled thermoplastics lie in the much lower range of 1 – 15 GPa (145 – 2.175 × 103 psi). Unfilled thermoset plastics can have somewhat higher moduli. Rubbers have very low moduli of only about 7 MPa (103 psi). For most materials the Poisson’s ratio 𝜈 varies between 0.2 and 0.35. Its value for aluminum, brass, and copper is around 0.34. For an incompressible material, which does not undergo any volumetric changes, 𝜈 = 0.5.

4.4 *Anisotropic Materials The previous section has discussed isotropic materials, ones in which the material properties at a point are independent of the direction. In such materials the principal axes of stress – along which shear stresses are zero – are parallel to the principal axes of strain. While this is a good approximation for homogenous materials such as metals, ceramics, and plastics, many materials exhibit anisotropy. In such materials the local mechanical properties are direction-dependent. Such materials can exhibit several types of anisotropic behavior. One of the simplest subclasses is orthotropic materials that have three, orthogonal preferred directions, such that if these directions are chosen as the coordinate axes then the principal axes of stress and strain are parallel. 4.4.1

*Orthotropic Materials

Wooden planks are approximately orthotropic. They have certain properties “along the grain” and different properties “across the grain.” Another example of an approximately orthotropic material is a lamina of aligned unidirectional fibers embedded in a polymeric matrix (Section 25.3.1.1), the mechanical analyses

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for which require an orthotropic material model. A wooden plank and an idealization of it are shown in Figure 4.4.1, in which the x-, y-, and z-axes are along the preferred directions of the orthotropic material.

z = x3 y = x2

(a)

(b)

x = x1

Figure 4.4.1 (a) Wooden plank. (b) Idealized orthotropic model for plank.

Then, with respect to the preferred axes x = x1 , y = x2 , and z = x3 , the stress-strain relations for an orthotropic material, in terms of the 12 elastic constants E1 , E2 , E3 , 𝜈 12 , 𝜈 21 , 𝜈 23 , 𝜈 32 , 𝜈 31 , 𝜈 13 , G1 , G2 , and G3 , are

𝜀x = 𝜀x1 = 𝜀y = 𝜀x2 = 𝜀z = 𝜀x3 =

𝜎x E1

𝜎y E2

𝜎z E3

+ − −

𝜈12 E2

𝜈23 E3

𝜈31 E1

𝜎y − 𝜎z − 𝜎x −

𝜈13 E3

𝜈21 E1

𝜈32 E2

𝜎z 𝜎x 𝜎y

(4.4.1)

and

𝛾yz =

1 1 1 𝜏 , 𝛾zx = 𝜏 , 𝛾xy = 𝜏 G1 yz G2 zx G3 xy

(4.4.2)

Not all the 12 elastic constants are independent; they are related through E1 𝜈12 = E2 𝜈21 , E2 𝜈23 = E3 𝜈32 , E3 𝜈31 = E1 𝜈13

(4.4.3)

so that orthotropic materials are described by nine independent elastic constants. In contrast to this, an isotropic elastic solid is described by just two independent elastic moduli E and G. Furthermore, while for an isotropic solid the principal axes of stress and strain are parallel in all coordinate systems, for an orthotropic material this is true only in the preferred coordinate system. This points to the complexity in analyzing orthotropic parts.

Models for Solid Materials

For the two-dimensional case, in which in the preferred coordinate system the only non-zero stress components are 𝜎 x , 𝜎 y , and 𝜏 xy , the stress-strain equations for an orthotropic material reduce to

𝜀x = 𝜀y =

𝜎x E1

𝜎y E2

𝜀z = − and

𝛾xy =

+ −

𝜈31 E1

𝜈12 E2

𝜈21 E1

𝜎y 𝜎x

𝜎x −

𝜈32 E2

𝜎y

𝜏xy G1

(4.4.4)

(4.4.5)

with E1 𝜈12 = E2 𝜈21 , E2 𝜈23 = E3 𝜈32 , E3 𝜈31 = E1 𝜈13

(4.4.6)

Thus the stress-strain relations for the in-plane stresses 𝜎 x , 𝜎 y , and 𝜏 xy , and the in-plane strains 𝜀x , 𝜀y , and 𝛾 xy , are related through the five elastic constants E1 , E2 , 𝜈 12 , 𝜈 21 , and G1 , of which only four are independent because of the relationship E1 𝜈 12 = E2 𝜈 21 . Determining the strain 𝜀z requires the two additional constants 𝜈 31 and 𝜈 32 .

4.5 Thermoelastic Effects It is well known that on heating most materials expand; that is, undergo an increase in volume. In isotropic materials this expansion occurs equally in all directions and does not affect shear. Clearly, the simple linear elastic solid considered before does not account for such temperature induced expansion effects. For isotropic solids, such thermal expansion effects are accounted for by adding “thermal strains” to the normal strains induced by stresses. In the simplest, linear theory, such thermal strains are given by 𝛼 ΔT, in which 𝛼 is the coefficient of thermal expansion, having dimensions of reciprocal temperature, and ΔT is the local increase in temperature. Thus, as indicated in Figure 4.5.1a, the thermal strains induced in an unloaded bar by a temperature increase ΔT are given by

𝜀x = 𝛼 ΔT, and 𝜀y = 𝜀z = 𝛼 ΔT Then, by adding the stresses 𝜀x = 𝜎 x ∕E and 𝜀y = 𝜀z = − 𝜈𝜀x caused by a load P acting on an unheated bar (Figure 4.4.1b), the stresses in a heated bar pulled in uniaxial tension (Figure 4.5.1c) are given by 1 𝜎 + 𝛼 ΔT, and 𝜀y = 𝜀z = − 𝜈 𝜀x + 𝛼 ΔT (4.5.1) E x This model is adequate for explaining the phenomenology that occurs when a bar is heated or cooled. Let the length of the bar be l. First consider an unloaded bar, so that the stress in it is zero. A temperature increase of ΔT will induce a thermal strain 𝛼 ΔT, resulting in a length increase of l (𝛼 ΔT). Next suppose that the bar is fixed at the ends and is not allowed to expand or

𝜀x =

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P

𝜎x =

𝜀x = 𝛼 ΔT

ΔT

P A

𝜀x =

𝜎x

𝜀x =

E

𝜀y = 𝜀z

𝜀y = 𝜀z

=−𝜈

= 𝛼 ΔT

P

Tensile Bar

P

Normal stress Normal strain

𝜎x E

+ 𝛼 ΔT

𝜀y = 𝜀z

= − Δ 𝜀x

= 𝜀x

Heated No stress Bar Normal strain

P

P

= − 𝜈 𝜀x + 𝛼 ΔT

𝜎x

=−𝜈

E

𝜎x E

+ 𝛼 ΔT

P Normal stress Normal strain

Heated bar; not loaded

Loaded bar; not heated

Heated and loaded bar

(a)

(b)

(c)

Figure 4.5.1 (a) Heated but not loaded bar. (b) Axially loaded but not heated bar. (c) Heated and loaded bar.

contract longitudinally, so that 𝜀x = 0. It follows that the bar will develop a (compressive) stress 𝜎 x = − E (𝛼 ΔT). If the constraints are removed – which is equivalent to applying a tensile stress 𝜎 x = E (𝛼 ΔT) to reduce the total stress to zero, then the bar will expand by l (𝛼 ΔT) corresponding to a thermal strain of 𝛼 ΔT. Thus a knowledge of the Young’s modulus E, the coefficient of thermal expansion 𝛼 , and the temperature differential ΔT is sufficient for explaining the behavior of a loaded or unloaded bar when it is heated or cooled with or without constraints. Note that the final state only depends on the initial and final temperatures. What the intermediate temperatures were, or the temperature history, has no effect on the final state. In the two-dimensional case, with 𝜎 z = 0, the stress-strain relations for a linear elastic material have the form

𝜀x =

1 (𝜎 − 𝜈 𝜎y ) + 𝛼 ΔT E x

Models for Solid Materials

𝜀y =

1 (𝜎 − 𝜈 𝜎x ) + 𝛼 ΔT E y

𝜈 𝜀z = − (𝜎x + 𝜎y ) + 𝛼 ΔT

(4.5.2)

E

and 1 𝜏 (4.5.3) G xy And, for the general three-dimensional case, the constitutive equations for thermoelasticity are 1 𝜀x = [𝜎x − 𝜈 (𝜎y + 𝜎z )] + 𝛼 ΔT E

𝛾xy =

𝜀y =

1 [𝜎 − 𝜈 (𝜎z + 𝜎x )] + 𝛼 ΔT E y

𝜀z =

1 [𝜎 − 𝜈 (𝜎x + 𝜎y )] + 𝛼 ΔT E z

(4.5.4)

𝛾xy =

1 1 1 𝜏 , 𝛾 = 𝜏 , 𝛾 = 𝜏 G xy yz G yz zx G zx

(4.5.5)

and

4.6 Plasticity Many materials undergo permanent deformation (strain) when the stress exceeds a level called the yield stress. This behavior is called plasticity. A highly idealized model for an elastic-plastic solid is shown in Figure 4.6.1. The material behaves elastically along the loading path OA; along this path the strain becomes zero once the stress is removed, that is, on unloading. Loading beyond point A, at which the stress is the yield stress 𝜎 Y , results in two changes. First, the slope of the stress-strain line ABD becomes smaller than for the elastic line. Second, a permanent strain remains when the load is removed. In this

D

σY STRESS

A

B WORK HARDENING ALONG PATH ABD

UNLOADING & RELOADING PATH

O

C

STRAIN

Figure 4.6.1 Idealized simple stress-strain curve for a linear elastic and linearly work-hardening elastic-plastic solid.

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model, on unloading from a point such as B, the stress-strain path BC is parallel to the elastic line OA, resulting in a strain OC, called the plastic strain. If the load is increased at any point along the path CB, the stress and deformation vary along the unloading line CB till point B, after which deformation continues along the original loading path ABD. On loading from the unloaded state at point C, the material behaves elastically up to the point B, to a stress higher than the initial yield stress 𝜎 Y . For this reason the material is said to have “work hardened,” because unrecoverable work has to be done to attain the state represented by point B. With the work-hardening model, consider, again, an elastic-plastic bar loaded in tension, for which the detailed stress-strain path is shown in Figure 4.6.2. For loads below the yield point, the deformation behavior (along OA) is elastic. On unloading the strain drops to zero as the bar reverts to its initial undeformed state. If the bar is loaded to a stress 𝜎 beyond yield, that is, if 𝜎 ≥ 𝜎 Y , then from the geometry shown in this figure, 𝜎 𝜎 − 𝜎Y OK = Y , KL = E EP so that 𝜀x = OK + KL is given by

𝜀x =

𝜎Y E

+

𝜎 − 𝜎Y EP

and, since CL = 𝜎 ∕E, that the plastic strain 𝜀P = OL − CL is given by [ ] 1 1 𝜀P = − ( 𝜎 − 𝜎Y ) EP E

D B

σ A

σY

F

H

STRESS

114

O

K

C

L STRAIN

Figure 4.6.2 Loading-unloading-loading path for a work-hardening elastic-plastic solid.

Models for Solid Materials

Unloading a bar stressed beyond yield (to point B) results in a residual permanent plastic strain (OC). Thus, in a work-hardening material the residual strain depends on the level to which the bar was stressed. The stress in the bar on loading beyond yield, and then unloading, are indicated in Figure 4.6.3. In the yielded, plastic domain the material is essentially incompressible with a Poisson’s ratio 𝜈 = 0.5, which is larger than the value for elastic deformations. As a result, the final values of the lateral strains 𝜀y and 𝜀z require these changes to be accounted for. The question marks next to expressions for these lateral strains in Figure 4.6.3 indicate that their values have not been evaluated.

𝜎x ≥ 𝜎Y

𝜎x =

P A

𝜀x =

𝜎x E

+

𝜀y = 𝜀z = ?

P

𝜎x − 𝜎Y EP

𝜀x =

1 1 (𝜎 − 𝜎Y) − E EP

𝜀y = 𝜀z = ?

𝜎x ≥ 𝜎Y Residual plastic strain

Bar loaded beyond yield

Bar unloaded after yielding

(a)

(b)

Figure 4.6.3 Stresses in a work-hardening bar: (a) On loading beyond yield. (b) On unloading.

Actually, plasticity is a far more complex phenomenon. The stress-strain variation is not bilinear as assumed in the previous model. Rather, it is a continuous nonlinear curve. Furthermore, the unloading-reloading paths, represented by the line BC in this figure, are different. However, this simple bilinear model does capture many of the complexities of plasticity. An even simpler model for plasticity is shown in Figure 4.6.4a. In this model, called the elasticperfectly-plastic model, there is no work hardening and, beyond yield, the stress remains constant at the yield stress, 𝜎 Y , also called the flow stress. The simplest model for plasticity shown in Figure 4.6.4b, called the rigid perfectly plastic model, assumes no elastic deformation. In this model, a body does not deform until the stress reaches the yield or flow stress, after which the stress remains constant at the yield

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stress. This model is useful for studying large process-induced deformations in metal forming in which the elastic strains, typically on the order of 0.002, are very small in comparison to the large plastic strains. Note that in each of these three simple plasticity models, unloading from a plastic state occurs along a path (BC) parallel to the elastic loading path (OA), resulting in a residual strain (OC). On reloading, the stress-deformation path (CB) first retraces the unloading path (BC) and then continues along the original loading path (BD). YIELD STRESS

σY A

D

STRESS

B

UNLOADING & RELOADING PATH

C

O

STRAIN (a) YIELD STRESS

σY

A

D

B

STRESS

116

NO ELASTIC DEFORMATION

UNLOADING & RELOADING PATH

O

C

STRAIN (b)

Figure 4.6.4 Simple stress-strain curves: (a) For a linearly elastic nonhardening solid. (b) For an ideally plastic solid – no elastic deformation.

4.7 Concluding Remarks The study of the mechanical behavior of loaded solids is called solid mechanics; it includes the study of elastic, plastic, and viscoelastic materials.

Models for Solid Materials

This chapter has discussed simple material models for solids: The first is a linear elastic material that has a perfect memory in the sense that on unloading the material returns to its initial state before loading. This is then extended to include thermal expansion effects in a loaded material; this model too has a perfect memory. Finally, the linear elastic model is extended to include plastic effects in which the material does not return to its initial undeformed state when the loads are removed. Clearly, elastic-plastic models are the most complex of the constitutive models discussed in this chapter; such materials do not have a perfect memory. Thermal expansion effects can be added to such elastic-plastic models. Although these are simple, highly idealized models for actual material behavior, they are useful for evaluating the response of objects to applied loads. Clearly, because of the nonlinear loading-unloading path-dependence of elastic-plastic materials, predicting part performance will involve greater mathematical complexity. All the models discussed in this chapter are time-independent in the sense that time-independent loads result in time-independent deformations. This is a reasonable approximation for the behavior of solid metals and ceramics for fairly large loads and temperature ranges – although, at high temperatures, time-independent loads can cause slow, time-dependent deformations. The response of simple, elastic structural elements – such as beams, columns, and torsion members – is addressed in Chapter 5. In contrast to solids, constant loads cause continuous motion in liquids. Because of this, instead of using strains as deformation measures, rates of deformation, measured by time rates of strain are more appropriate. Constitutive models for fluids are discussed in Chapter 6. Plastic materials exhibit both solid- and liquid-like behavior, called viscoelasticity, which requires the use of both strains and strain rates as deformation measures. Simple constitutive models for such viscoelastic materials are addressed in Chapter 7.

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5 Simple Structural Elements 5.1 Introduction While parts may be designed to serve several functions, structural performance is often the most important requirement. It addresses the twin issues of stiffness and strength: How much will a part deform for a given load, and will the material of which the part is made fail – break, or fracture? Evaluating structural performance is one of the most important aspects of part design. To develop a first-level feel for structural issues – the factors that control stiffness and strength – the responses of simple structural parts subject to simple loads, are first explored. The simplest structural element is a bar loaded in tension, the stress deformation relations for which have been addressed in Chapter 4. While the nominal compressive stress in a bar loaded in tension is easily evaluated, compressive loads in slender bars can result in an instability that causes the bar to buckle – “bend” sideways. The analysis of this instability requires an understanding of beam bending. This chapter shows how mechanics principles can be used to evaluate the bending, torsion (twisting), and elastic stability of simple structures. However, most plastics structures have very complex geometries, the mechanical performance of which can only be determined by means of computer-based numerical codes. Also, because of their importance to the mechanical design process, results for some topics, such as the twisting of thin-walled tubes and multicellular sections, and the phenomenon of stress concentration are presented without analyses, which are well beyond the scope of this book.

5.2 Bending of Beams In contrast to bars that are designed to carry axial loads, a beam is a structural element that carries transverse loads and bending moments. Under such loads a beam “bends,” that is, it deforms into an arcuate shape. One important difference between an axially loaded bar and a transversely loaded beam is that the stress distribution on the cross section of a beam is not uniform. The simplest example of a loaded beam is a prismatic beam subjected to a bending moment. Consider then a prismatic beam with a rectangular cross section of width b and depth d, subjected to a bending moment Mz . Figure 5.2.1 shows a portion of the bent beam, in which every longitudinal fiber bends into a circular arc. Because of the symmetric geometry and the uniform loading, initially plane transverse sections, such as KCL, will remain plane after bending (Why?), and each straight longitudinal fiber must deform into a circular arc (Why?). As will be shown, the deformations for this simple loading Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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O

θ R Mz •

K F

D A

•

y

Mz •

x C

B

εx

b

d

•

y

L

Figure 5.2.1 Deformed geometry of a rectangular prismatic beam subjected to a uniform bending moment.

are consistent with some fibers not undergoing a length change. Let the y-axis be measured from such fibers, ABC, and the radius of curvature of these fibers be R, as shown. The length of these fibers, and the undeformed lengths of all longitudinal fibers, is then R 𝜃 . From the geometry shown in the figure, the deformed length of a fiber DF at a distance y from ABC will then be (R − y) 𝜃 . It then follows that the strain in the fibers DF is given by (R − y) 𝜃 − R 𝜃 y =− R R𝜃 so that the stress is given by

𝜀x =

E (5.2.1) R Now this stress distribution must equal the imposed pure bending moment Mz . Since there is no longitudinal force, it follows (Why?) that

𝜎x = E 𝜀x = − y

∫

𝜎x b dy = 0 ⇒

∫

y b dy = 0 ⇒ y = 0

so that y = 0 is the centroid of the cross section. Finally, since the stress distribution must equal the moment Mz , Mz =

∫

−y (𝜎x b dy) =

Mz E E = , where Iyy = y2 b dy y2 b dy ⇒ ∫ R∫ Iyy R

which can be rewritten as Mz 𝜎 E = x = , Iyy = y2 b dy ∫ Iyy −y R

(5.2.2)

While a rectangular beam cross section has been assumed, the assumptions for deriving this result, such as plane sections remaining plane, are valid for all prismatic beams with cross sections that are

Simple Structural Elements

symmetric about the y-axis (Why?). For such prismatic beams Eq. 5.2.2 is valid with b = b(y). In this equation Iyy , which is a geometric property of the cross section, is called the second moment of area because the expression for it can be written as Iyy =

∫

y2 b dy =

∫

y2 dA

(5.2.3)

where dA is an element of area at a distance y from the centroid. For a rectangular cross section 1 (5.2.4) bd3 12 This expression shows that while the second moment of area Mz , and hence the beam stiffness, is proportional to the width b, it is proportional to the cube of the depth d. Therefore, beam stiffness is better achieved by increasing the beam depth than its width. Thus, the stresses and deformations caused by the pure bending of a prismatic beam with a symmetric cross section is governed by Eq. 5.2.2, in which the general expression for the second moment of area is given in Eq. 5.2.3. Note that while y = 0 is located at the center for a rectangular section, in the general case y = 0 goes through the centroid of the section. Equation 5.2.2 shows that the curvature 1∕R of the beam – which is a measure of the beam stiffness (smaller curvature means higher stiffness) – is directly proportional to the applied bending moment Mz , but inversely proportional to the product E Iyy , which is called the beam stiffness. Clearly, both the strain and stress vary linearly with y, being zero along the surface y = 0; the stress is compressive for y > 0 and tensile for y < 0. For this reason, the line y = 0 on a cross section is called the neutral axis of the cross section; the surface y = 0 is called the neutral surface. The stiffness of a beam can be increased by using a stiffer material, that is, one with a higher Young’s modulus E. A more efficient way is to increase the second moment of area Iyy of the cross section. This is best achieved by moving material away from the neutral axis. To get a better feel for how distribution of the cross-sectional area affects the second moment of area, consider the I-beam shown in Figure 5.2.2a. Assume that the width b of the flange and the depth d of the web are much larger than the thickness t. Then, Iyy =

1 3 1 3 (5.2.5) t d + 2bt (d∕2)2 = t d (1 + 6 b∕d ) 12 12 Note that when b and d are about the same, the two flanges contribute about six times more to Iyy than does the web. It is for this reason that steel beams, or girders, commonly have thin-walled I-sections. Iyy =

b

b

b

d

d 2t

(a)

(b)

t /2

(c)

Figure 5.2.2 Dimensions of I-, cross-, and box-section beams having the same cross-sectional areas.

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Next consider the cross-shaped section shown in Figure 5.2.2b, which has the same cross-sectional area as the I-beam (Figure 5.2.2a) – the horizontal arm of the cross has twice the area of each flange. For this cross section, [ ] b 1 1 1 3 (5.2.6) [t d3 + b(2t) 2 ] = td 1 + 8 (t∕d ) 2 ≈ t d3 Iyy = 12 12 d 12 because the term (t∕d) 2 is very small and can be neglected. It then follows that II-section b =1+6 ICross-section d

(5.2.7)

Then, for comparable b and d, Iyy for the I-section beam is about seven times that for the cross-section beam – even though they have the same cross-sectional areas. The box beam shown in Figure 5.2.2c, in which each web has half the thickness of the I-section in Figure 5.2.2a, has the same cross-sectional area as well as the same Iyy as the I-section beam (Why?). However, if the bending moment is applied about the y-axis, then the box-sectioned beam would be much stiffer, having a much larger Izz than the I-sectioned beam. The three cross sections in Figure 5.2.2a – c are symmetric about both the x- and y-axes, so that the distance of the outermost fibers from the centroid is the same, resulting in the extreme tensile and compressive stresses having the same magnitude. While the channel section in Figure 5.2.3a is symmetric about the z-axis – so that the tensile and compressive stresses in the flanges have the same magnitude – it is not symmetric about the y-axis, so that the centroid is closer to the web rather than at the center of the flanges as in the three cases in Figure 5.2.2. As a result, a bending moment My will cause a lower stress in the flange than at the ends of the “C.” Note that a moment Mz will result in the same stress distributions in the two cross sections shown in Figures 5.2.2a,c and 5.2.3a (Why?). The Z-section is neither symmetric about the y-axis nor about z-axis, even though the centroid is located at the “center” of the section. However, there is an important difference between this section and the four other sections (Figures 5.2.2a – c and 5.2.3a) – in which a moment Mz causes the beam to bend in the x-y plane – in the Z-section a moment Mz will not cause the beam to bend in the x-y plane. Rather, the plane of bending will lie between the y- and z-planes. The same will hold true for bending caused by a moment My . This phenomenon is called unsymmetrical bending and requires a consideration of the simultaneous effects of My and Mz , which is well beyond the scope of this book.

b

b

d

b (a)

(b)

Figure 5.2.3 Dimensions of C- and Z-section beams having the same cross-sectional areas.

Simple Structural Elements

5.3 Deflection of Prismatic Beams The previous section has related the stress and the radius of curvature induced in a prismatic beam by an imposed bending moment. This section addresses the resulting deflection of the beam axis. The analysis is confined to the small-deflection case in which the slope of the beam is so small that its square can be neglected in comparison to unity. Let the deflected shape of the neutral axis, having radius of curvature R, be v = v (x), as shown in Figure 5.3.1. Now Eq. 5.2.2 relates the bending moment Mz to the radius of curvature R of the neutral axis through Mz d2 v∕dx2 1 v ′′ = = = 2 3∕2 EI yy R [1 + (dv∕dx) ] [1 + (v ′ ) 2 ]3∕2 On neglecting (v ′ )2 in comparison to unity in the expression for the curvature 1∕R in this equation, the differential equation governing beam deflection is obtained as EI yy v ′′ = Mz

(5.3.1)

v

• R v = v (x) x

O

Figure 5.3.1 Deformed (deflected) centroidal axis of a beam.

5.3.1

Deflection of a Cantilever Due to an End Load

Consider the cantilever AB, of length l, fixed at the end A and subjected to an end load P, as shown in Figure 5.3.2. With the origin of the coordinate system being at A as shown, the bending moment Mz at distance x from A is given by (Why the minus sign?) Mz = − P(l − x) It follows that, for small deflections, the deflection of a cantilever is governed by d2 v P =− (l − x) 2 EI yy dx

(5.3.2)

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y P

v

l

•

A

x

− P(l − x)

B

Mz

Figure 5.3.2 Cantilever subjected to an end load P.

The appropriate boundary conditions for the fixed end A are v(0) = 0 and v ′ (0) = 0 (Why?). An integration of this equation gives ( ) dv x2 P lx − =− + C1 dx EI yy 2 where the constant of integration C1 must be zero to satisfy v ′ (0) = 0. An integration of ( ) x2 P dv lx − =− dx EI yy 2 subject to the boundary condition v(0) = 0 then gives v=−

Px2 (3l − x) 6EI yy

The deflection 𝛿 = − v(l) under the load P, in the direction of the load, is then given by

𝛿=

5.3.2

Pl 3 3EI yy

(5.3.3)

Deflection of a Simply Supported Beam Due to a Central Load

Consider the beam ABC of length l, simply supported at the ends A and C, subjected to a central load P, as shown in the top half (above the arrow) of Figure 5.3.3. Clearly, the central load P gives rise to equal reactions, or support loads, P∕2 at A and C. While the deflection under the load may be obtained by integrating Eq. 5.3.1 subject to the boundary conditions that the deflections must be zero at the two supports at A and C, the desired result can more easily be obtained by exploiting the symmetry of the problem, which requires that the slope of the deflected load be zero under the central load. Then, with reference to the bottom half (below the arrow) of Figure 5.3.3,

Simple Structural Elements

y

v

P

A

•

C

B x l /2

l /2 l

Mz

Pl 4

B

B

A

C l /2

l /2

P /2

P /2

Figure 5.3.3 Simply supported beam subjected to a central concentrated load P.

the deflection under the load equals the deflection of a cantilever of length l∕2 subjected to an end load P∕2 (Why?). It then follows from Eq. 5.3.3 that the deflection under the central load is given by

𝛿=

(P∕2) (l∕2)3 Pl 3 = 3EI yy 48EI yy

(5.3.4)

This equation forms the basis for bending tests for determining the flexural modulus of a material. 5.3.3

Deflection of a Simply Supported Beam Due to a Noncentral Load

Consider the beam ABC of length l, simply supported at the ends A and C, subjected to a noncentral load P, as shown in Figure 5.3.4. With the origin of the coordinate system being at A as shown, the bending moment Mz at distance x from A is given by (Why?) ⎧ ⎪ Mz = ⎨ ⎪ ⎩

Pb x , l Pa (l − x) , l

0≤x≤a (5.3.5) a≤x≤l

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y

v

P

A

B x a

b l Mz

ab −P l

Figure 5.3.4 Simply supported beam subjected to a noncentral concentrated load P.

To obtain the beam deflection, Eq. 5.3.1 has to solved for the simply supported boundary conditions v(0) = v (l) = 0. Because of the shape of the bending moment diagram, solutions for the segments 0 ≤ x ≤ a and a ≤ x ≤ l have first to be obtained separately; by then matching the deflections v(a) and the slopes v ′ (a) for both solutions at x = a. For segment 0 ≤ x ≤ a, a double integration of v ′′ = (Pb∕l ) x gives v = (Pb∕6E Iyy l ) x3 + C1 x + C2 , where C1 and C2 are constants of integration. The condition v(0) = 0 implies that C2 = 0. The solution for the deflection and slope for segment a ≤ x ≤ l is then Pb x3 + C1 x, 6E Iyy l Pb 2 v′ = x + C1 , 2E Il v=

Pb a3 + C1 a 6E Iyy l Pb 2 v ′ (a) = a + C1 2E Il

v(a) =

(5.3.6)

For segment a ≤ x ≤ l, a double integration of v ′′ = (Pa∕l) (l − x) gives v = (Pb∕6E Il ) (l − x)3 + C3 (l − x) + C4 where C3 and C4 are constants of integration. The condition v (l) = 0 implies that C4 = 0. The solution for the deflection and slope for segment a ≤ x ≤ l is then Pa (l − x)3 + C3 (l − x), 6E Il Pa v′ = − (l − x)2 − C3 , 2E Iyy l v=

Pa 3 b + C3 b 6E Il Pa v ′ (a) = − b2 − C3 2E Iyy l v(a) =

(5.3.7)

By equating respective values of v(a) and v ′ (a) as given by Eqs. 5.3.6 and 5.3.7, it follows that C1 and C3 must satisfy Pa Pb a3 + C1 a = b3 + C3 b 6E Iyy l 6E Iyy l

Simple Structural Elements

and Pb Pa a2 + C1 = − b2 − C3 2E Iyy l 2E Iyy l From these two simultaneous equations, the values of C1 and C3 are given by C1 =

Pab Pab (a + 2b) and C3 = (b + 2a) 6E Iyy l 6E Iyy l

The deflection and slope for the beam are then given by ⎧ Pb , 0≤x≤a [ x3 − a(a + 2b) x] ⎪ 6EI l ⎪ yy v(x) = ⎨ ⎪ Pa [(l − x)3 − b(b + 2a) (l − x)] , a ≤ x ≤ l ⎪ 6EI yy l ⎩

(5.3.8)

⎧ Pb , 0≤x≤a [3x2 − a(a + 2b) x ] ⎪ 6EI l ⎪ yy v ′ (x) = ⎨ ⎪ Pa [−3(l − x)2 + b(b + 2a)] , a ≤ x ≤ l ⎪ 6EI yy l ⎩

(5.3.9)

and

The deflection 𝛿 = − v(a) under the load P, in the direction of the load, is then given by

𝛿=

Pa 2 b 2 3E Iyy l

(5.3.10)

For a = b = l∕2, Eq. 5.3.10 gives the deflection for a centrally loaded beam that, in Eq. 5.3.4 was obtained by a symmetry argument.

5.4 Torsion of Thin-Walled Circular Tubes Torsion refers to the twisting action caused by moments, called twisting moments, acting about the axis of a bar. Common examples of bars subjected to torsion, or twisting moments, are shafts used in machines for transmitting rotary power. Consider the thin-walled circular tube of radius a, thickness t, with a ≫ t, shown in Figure 5.4.1a, subjected to a twisting moment T. The resulting torsion, or twisting, results in a symmetric deformation about the longitudinal axis of the tube. Because of the symmetry, all plane transverse sections must remain plane after twisting (Why?) and retain their circular shapes; essentially, circular sections “rotate” relative to adjacent sections in a uniform manner. And because of uniform twisting about the longitudinal tube axis, longitudinal fibers such as AB on the tube surface must deform into helices AB1 , as shown in Figure 5.4.1a. As a result, the deformation of the tube is equivalent to making a longitudinal slit along AB1 on the tube surface and developing the thin wall onto a flat surface, resulting in a thin rectangular slab that undergoes uniform shear, as indicated in Figure 5.4.1b; note that the helix angle 𝛾 is the same as the shear strain in the developed surface. Let the twist over a tube length l be 𝜃 ; the uniform twist per

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a

•

lγ = a θ

T B1

θ B

Shear Stress τ

B1 l

l

γ

γ A A T

2π a Shear Stress τ

(a)

(b)

Figure 5.4.1 (a) Geometry of a thin-walled circular tube subjected to a twisting moment. (b) Equivalent thin-walled rectangular slab subjected to a shear stress.

unit length is then given by 𝜃 0 = 𝜃 ∕l. From the geometry shown in the figure it follows that l 𝛾 = a 𝜃 , so that 𝛾 = a 𝜃 0 . The shear stress on the top surface of the developed surface, which is the same as the shear stress acting on the circular tube cross section, is then given by 𝜏 = G 𝛾 = G a 𝜃 0 . Since the action of this shear stress acting on the cross section must be equivalent to the applied twisting moment T, it follows that T = a (2𝜋 at) 𝜏 = (2𝜋 a3 t) G 𝜃 0 . The results for the twisting of a thin-walled circular tube may be summarized as J = 2𝜋 a3 t,

T = JG 𝜃0 ,

𝜏 = G a 𝜃0

(5.4.1)

Thus, the torsional stiffness T∕𝜃 0 = JG can be increased by choosing a material with a higher shear modulus G, but a more efficient means is to increase J = 2𝜋 a3 t, which is proportional to the cube of the tube radius but only linearly proportional to the tube thickness. Note that the shear stress 𝜏 = Ga 𝜃 0 , which is constant through the tube thickness, only increases linearly with the tube radius.

Simple Structural Elements

5.5 Torsion of Thin Rectangular Bars and Open Sections The cross section ABCD of a thin rectangular bar, in which the width b is much larger than the thickness t, is shown in Figure 5.5.1. Characterizing the torsional stiffness of this bar and the stress distribution caused by a twisting moment acting about the bar axis, that is, acting about the normal to the cross section, is a more complex task than for the torsion of a thin-walled circular tube, and requires a two-dimensional analysis, which is outside the scope of this book (Section 23.10.3 presents such a two-dimensional analysis for the more general case of a nonhomogeneous bar.). In this case plane transverse sections do not remain plane as in the case of thin-walled tubes, but warp, that is, undergo out-of-plane deformations. With reference to this figure, a two-dimensional analysis shows that the torsional rigidity and the stress distribution are given by J=

1 3 bt , 3

T = JG𝜃0 ,

𝜏 = (2G𝜃0 ) x

(5.5.1)

x B

A

t

•

C

D b

Figure 5.5.1 Geometry of the cross section of a thin rectangular prismatic bar.

The torsion of thin-walled rectangular bars has three noteworthy aspects. First, the shear stress is not constant through the thickness but varies linearly through the thickness and the shear stress is always parallel to the boundary at the bar surface. Second, the torsional stiffness J = bt3∕3 varies as the cube of the thickness t, and is therefore small. And third, the shear stresses at the corners A, B, C, and D of the bar cross section are identically zero. It turns out that the torsion of all thin-walled open sections having uniform thicknesses can be simply expressed in terms of the torsion of thin-walled rectangular bars. The expressions for J, T, and 𝜏 are the same as in Eq. 5.5.1, with b being the peripheral length, and x being measured along the local normal to the boundary. For example, for the angle section, C-section, and the slit circular tube shown in Figure 5.5.2a – c, b is given, respectively, by b = b1 + b2 , b = b1 + b2 + b3 , and b = 2𝜋 a. As a result, prismatic bars of thin-walled open sections normally have low torsional rigidities. An idea of by how much the torsional rigidity of a tube is reduced by cutting a thin longitudinal slit on the surface – which converts the closed tube into a thin-walled open section – can be had by comparing the torsional rigidity of a thin-walled circular tube, JClosed Tube = 2𝜋 a3 t, with that for the tube with a slit, JOpen Tube = (2𝜋 ∕ 3) at3 . It follows that JClosed Tube = 3 (a∕t) 2 JOpen Tube For a∕t = 10, the closed tube has about 300 times the torsional stiffness of the same tube with a longitudinal slit. One reason for this difference is that the cross sections of open sections have less constraint and can warp more easily, resulting in reduced torsional stiffness.

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b1

b1

a

b2

b2

•

b3 (a)

(b)

(c)

Figure 5.5.2 Geometry of the cross section of (a) a thin angle-sectioned prismatic bar, (b) a thin C-sectioned prismatic bar, and (c) a thin-walled circular tube with a longitudinal slit.

5.6 Torsion of Thin-Walled Tubes Figure 5.6.1 shows the cross section of a thin-walled prismatic tube of thickness having a t, a perimeter P, enclosing an area A. A two-dimensional analysis shows that the torsion of such tubes is governed by J = 4A2 t∕P,

T = JG𝜃0 ,

𝜏 = (2G𝜃0 ) A∕P

(5.6.1)

Perimeter: P

Enclosed Area: A Figure 5.6.1 The cross section of a thin-walled prismatic tube having a perimeter P enclosing an area A.

This result shows that, for a given wall thickness t a perimeter P, the largest torsional stiffness is attained for the largest enclosed area A, which is known to be a circle. Thus, circular tubes provide the highest possible torsional stiffness (Section 23.10.4 presents a two-dimensional analysis for the more general case of the torsion of thin-walled nonhomogeneous tubes.). As in the case of circular tubes, a small longitudinal slit, which converts the tube into a thin-walled open section, reduces the torsional stiffness considerably. From the results for thin-walled open sections

Simple Structural Elements

it follows that JClosed Tube = 12 (A∕Pt ) 2 JOpen Tube

(5.6.2)

This result shows that, for a given wall thickness t a perimeter P, a longitudinal slit causes the largest reduction in torsional stiffness for the largest enclosed area A, that is, for a circular cross section. Thus, although a circular tube provides the highest possible torsional stiffness, a longitudinal slit causes the largest reduction in its torsional stiffness. As an example, consider the circular tube of diameter 2a and the square tube of side 2a shown in Figure 5.6.2. It follows that JSquare Tube = 8 a3 t, which is different from the torsional stiffness JCircular Tube = 2𝜋 a3 t for a circular tube. The torsional stiffness for the square tube with a slit, an open section is given by (Why?) JOpen Square Tube = (8 a3 t∕3), so that JClosed Squae Tube JOpen

= 3 (a∕t )2

(5.6.3)

Squae Tube

which in this case is the same as that for a circular tube.

2a

Figure 5.6.2 Cross sections of thin-walled circular and square prismatic tubes.

5.7 *Torsion of Multicellular Sections The discussions in the previous two sections have established that thin-walled closed sections have significantly larger torsional stiffnesses than the corresponding open sections. Multicellular thin-walled sections, first successfully used in aircraft design in which thin aluminum sheets are used to form sections, provide excellent torsional stiffness, while at the same time providing enhanced bending stiffness to structures. Such multicellular sections are used in many thin-walled plastics applications. While the analyses of the structural performance of such sections are well beyond the scope of this book, results for some simple sections are discussed to illustrate some stiffening issues. The previous section has shown that the torsional rigidity of a thin-walled closed section of wall thickness t, having a perimeter P, enclosing an area A, is given by J = 4A2 t∕P. This result does not depend on other geometrical parameters such as the aspect ratio of the section. While overly elongated sections can provide the desired torsional rigidity, bending can induce buckling in the long unsupported skins. Lateral ribs are added to overcome this shortcoming. As an example, the cross section shown in Figure 5.7.1a can be stiffened by a central rib (Figure 5.7.1b) that will stiffen the two skins of length 2a. It turns out that because of the symmetry of the two square cells in Figure 5.7.1b, the two cross sections shown in Figure 5.7.1a,b have the same torsional rigidity J = (8∕3) a3 t. But, while the outer skins carry a shear stress 𝜏 = (2∕3) G 𝜃 0 a, the central rib does not have any shear stress.

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2a

2a

2a

(a)

(b)

(c)

(d)

Figure 5.7.1 Four thin-walled rectangular cross sections.

Next consider these two sections with a small slit in the right vertical sides, shown in Figure 5.7.1c,d. The open section in Figure 5.7.1c has the torsional stiffness J = 2at3 . It can be shown that the section shown in Figure 5.7.1d, consisting of two open sections, each of length 3a∕2, attached to a square closed section, has the torsional rigidity J = at (a2 + t2 )∕2. Clearly, the cross section in Figure 5.7.1d is much stiffer than the section in Figure 5.7.1c. Now the ratio of the torsional rigidity of the section in Figure 5.7.1a to that in Figure 5.7.1c, J = (4∕3) (a∕t)2 , is very large. On the other hand, the corresponding ratio of the rigidities of the sections shown in Figure 5.7.1b,d, J = (16∕3) [1 + (t∕a)2 ]−1 ≈ 16∕3, is much smaller. Such changes are not intuitively obvious, and an understanding of how geometry affects stiffness can only be obtained through mechanics-based analyses. While the central rib in the cross section in Figure 5.7.1b does not have any shear stress, this is not necessarily true for all multicellular sections. For example, in the section shown in Figure 5.7.2, the two central ribs have a shear stress 𝜏 12 = 𝜏 23 = (1∕7) G𝜃 0 a, in which the subscript 12 on 𝜏 indicates that this stress is in the leg between cells 1 and 2; the subscript 23 has a similar interpretation. The three outermost legs of cells 1 and 2 have the same shear stress 𝜏 1 = 𝜏 3 = (5∕7) G𝜃 0 a, while the shear stress in the outermost skins of cell 2 have the shear stress 𝜏 1 = 𝜏 3 = (6∕7) G𝜃 0 a. This three-celled section can be shown to have the torsional rigidity J = (32∕7) a3 t.

Figure 5.7.2 A three-celled thin-walled rectangular cross section.

Simple Structural Elements

5.8 Introduction to Elastic Stability Elastic stability is an important consideration in the design process. A structure is said to be elastically stable under a given load system if small external disturbances, such as a force, a moment, or the displacement at some point, cause small deflections in the structure. It is said to be neutrally stable, if it remains in equilibrium at continuously different configurations under the same load system. Finally, it is said to be elastically unstable if small disturbances cause large deflections. Another way of describing these three conditions is as follows: Consider a structure that is in equilibrium under the action of a system of forces. Let a small disturbance, say a small displacement, be applied to the structure at some point by means of a disturbing force that, in general, will result in the structure deforming. If the structure returns to its original equilibrium configuration when the disturbing force is removed, it is said to be elastically stable. If it remains in the disturbed position, caused by the external disturbance, then it is said to be neutrally stable. However, if very large deformations are caused by the external disturbance, then the structure is said to be elastically unstable. It is not easy to visualize the concept of elastic stability. The fundamental ideas concerning stability will first be illustrated by means of rigid bodies, and then by elastic members attached to a rigid body. Finally, the stability of elastic members will be considered. 5.8.1

Concept of Stability

Consider a rigid sphere resting on the three surfaces shown in Figure 5.8.1a – c, in each configuration of which the sphere is in equilibrium. If the sphere is displaced by a small amount on the supporting surface and let go, then, for the case shown in Figure 5.8.1a, the sphere will return to its original position. This configuration of the sphere is therefore said to be stable. For the configuration shown in Figure 5.8.1b the displaced sphere will remain wherever it is placed on the plane; this configuration is therefore said to be neutrally stable. Finally, since a small disturbance will cause the sphere to roll down the slope, the configuration shown in Figure 5.8.1c is said to be unstable.

(a)

(b)

(c)

Figure 5.8.1 Diagram illustrating the states of (a) stability, (b) neutral stability, and (c) instability of a rigid body.

In an unstable system it is not necessary that the ball will continue to move as in Figure 5.8.1c. It may move into another stable or neutrally stable position as indicated in Figure 5.8.2 parts a and b, respectively. Position A is unstable. However, a disturbance in case (a) will cause the sphere to shift to the stable position B, whereas in case (b) it will shift into the neutrally stable positions indicated by B. More generally, a system is said to be stable if small external disturbances cause small reversible changes in the configuration of the system. It is said to be neutrally stable, if small externally imposed disturbances cause it to remain in equilibrium at continuously different configurations. And it is said to be unstable if small disturbances cause large shifts in the configuration.

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A

A

B

B (a)

B

(b)

Figure 5.8.2 Possible terminal states of a body after passing through an instability; (a) stable state and (b) neutrally stable state.

5.8.2

Stability of a Hinged Rigid Bar

Consider a system consisting of a rigid bar hinged at the lower end A. This can be kept in equilibrium in a vertical position by either (a) a tensile force P, applied to the upper end B, as shown in Figure 5.8.3a, or (b) by a compressive force P, as shown in Figure 5.8.3b. If a small lateral force F is applied at B, then in the case of part (a), the beam will undergo a small deflection v = Fl∕P, where l is the length of the beam (Why?). Further, if F is removed, the beam will revert to its original vertical position. Therefore, the load system shown in Figure 5.8.3a is stable. On the other hand, a small force F applied to case (b) would cause the beam to rotate continuously about the hinge at A, so that this is an example of an unstable system.

v P

P

P B B'

•

(a)

A

B

F

•

A

(b)

Figure 5.8.3 Stability of a hinged rigid rod subjected to (a) a tensile force and (b) a compressive force.

Simple Structural Elements

P

•

•

v B •

B•

P y

•

B'

F

u

x

θ •

A

(a)

A

•

(b)

Figure 5.8.4 Hinged rigid rod stabilized by a spring.

Next consider the case when the unstable system shown in Figure 5.8.3b is stabilized by means of a linear spring of stiffness k (Figure 5.8.4a). It will be shown that, depending upon the magnitude of P, this system can be stable, neutrally stable, or unstable. Introduce the coordinate axes, (x, y), with origin at B. Under the action of the disturbing force F, let the deflections, the displacement components, of the point B be u and v in the x and y directions, respectively. Then, since the sum of the moments about A must be zero, it follows that for equilibrium (F − k v) (l − u) + Pv = 0

(5.8.1)

where − kv is the spring force generated by the elongation of the spring. From the geometry of deformation, it follows that u = l (1 − cos 𝜃 ), v = l sin 𝜃

(5.8.2)

Substituting these expressions for u and v in Eq. 5.8.1 then gives (F − kl sin 𝜃 ) l cos 𝜃 + Pl sin 𝜃 = 0 or sin 𝜃 − 𝛼 tan 𝜃 = 𝛽

(5.8.3)

in which 𝛼 = P∕kl and 𝛽 = F∕kl. Thus, for particular values of 𝛼 = P∕kl and 𝛽 = F∕kl, the deflections u and v are given by Eq. 5.8.2, where 𝜃 is given by the smallest root of Eq. 5.8.3. If it is assumed that the deflections are small, so that sin 𝜃 ≈ tan 𝜃 ≈ 𝜃 and cos 𝜃 ≈ 1, then from Eq. 5.8.3 𝜃 = 𝛽 ∕(1 − 𝛼 ), so that ⎫ ⎪ ⎬ F 1 v≃ k (1 − 𝛼 ) ⎪ ⎭ u≃0

(5.8.4)

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Thus, for a given value of 𝛼 , for small deflections v is directly proportional to F. As 𝛼 approaches unity from below, the load F required to cause a deflection v decreases. For 𝛼 = 1, v can only be finite if F is zero. In fact, for small deflections, 𝛼 = 1 corresponds to the neutrally stable case. This can be seen as follows: For small deflections, u can be neglected. Then for equilibrium, since the sum of the moments about A must vanish, it follows that

or

⎫ Pv + (F − k v) l = 0 ⎪ ⎬ F (𝛼 − 1) v + = 0 ⎪ k ⎭

(5.8.5)

This equation is satisfied for all (small) values of v when 𝛼 = 1 if F = 0; so the system will remain in equilibrium for all (small) values of v. Thus 𝛼 = 1, or P = k l, corresponds to the load at which the system is neutrally stable. For 𝛼 < 1, for small deflections, Eq. 5.8.4 shows that the system is stable since the deflection is proportional to the load F, so that v will become zero if F is removed. The “small-deflection” equation, v = (F∕k)∕(1 − 𝛼 ), fails to give an answer when 𝛼 > 1 since it predicts a negative value of v, which is not admissible on physical grounds. Of course, what happens is that the system becomes unstable for 𝛼 > 1, so that the small-deflection assumption, given by Eq. 5.8.2, does not hold. In the foregoing problem, the system is stable for small deflections as long as P < kl. When P is increased to a value P = kl, the system becomes neutrally stable. Any further increase in the value of P causes the system to become unstable. Thus, the load P = kl, or 𝛼 = 1, has a special significance and is referred to as the critical load. More generally, the load at which a system becomes neutrally stable for small deflections is called the critical load. The concept of stability discussed before holds only for small deflections. For loads smaller than the critical load, the system is stable for small deflections. For loads greater than the critical load, the system becomes unstable for small deflections. However, when an unstable system is disturbed, the deflections do not necessarily increase unboundedly; instead, the system may stabilize at a different stable state. This phenomenon is discussed in the following section. Thus, while a small deformation analysis gives the critical load at which the system becomes neutrally stable for small deflections, it cannot predict the equilibrium position once the critical load is exceeded. 5.8.3

*Spring-Supported Rigid Bar: Stability Above the Critical Load

In the problem discussed before, the results of the small-deflection analysis, given by Eq. 5.8.4, do not give the value of v for 𝛼 = 1. Once 𝛼 > 1, there are no solutions to the equilibrium equation, sin 𝜃 − 𝛼 tan 𝜃 = 𝛽 , in the quadrant 0 ≤ 𝜃 ≤ 𝜋 ∕2. For, in that quadrant, sin 𝜃 ≤ tan 𝜃 , so that sin 𝜃 − 𝛼 tan 𝜃 ≤ 0 for 𝛼 ≥ 1, whereas the right-hand side of the equation is positive. Now the solutions of the equilibrium equation are given by the points of intersection of the curves y = sin 𝜃 − 𝛼 tan 𝜃 and y = 𝛽 . The graph of y = sin 𝜃 − 𝛼 tan 𝜃 for 𝛼 = 0.1 is shown in Figure 5.8.5. For 𝛽 = 0, that is for no lateral force F, three equilibrium states are possible, namely 𝜃 = 0, 𝜃 ≃ 15𝜋 ∕32 and 𝜃 = 𝜋 . Let the lateral force be increased to 𝛽 = 0.1, say. The equilibrium values of 𝜃 then correspond, respectively, to the points marked A, B, and C. Thus, if the initial equilibrium state with 𝛽 = 0 is 𝜃 = 0, then for 𝛽 = 0.1, the equilibrium point will shift to 𝜃 A . Similarly, if the initial state for 𝛽 = 0 is 𝜃 ≃ 15𝜋 ∕32 or 𝜋 then, for 𝛽 = 0.1, the final states will be 𝜃 B and 𝜃 C , respectively. As the value of 𝛽 is increased, the equilibrium points shift along

Simple Structural Elements

the curves. For example, for 𝛽 = 0.5, the equilibrium values 𝜃 D , 𝜃 E , and 𝜃 F correspond, respectively, to the points D, E, and F. As the value of 𝛽 is increased further, a stage is reached, at 𝛽 = 0.78, when two of the possible solutions merge into a single point G. Then, instead of three solutions, only two solutions 𝜃 G and 𝜃 H are possible. For any further increase in 𝛽 , only one solution is possible. For example, for 𝛽 = 1.0, the only possible solution is 𝜃 I . With increasing 𝛽 the only equilibrium solution will approach 𝜃 = 𝜋 ∕2 from above.

π /4

0

3 π /4

π /2

π

1.5

y = sin θ − α tan θ

α = 0.1 I

•

y = β = 0.1

1.0

H

G D

•

E

y = β = 0.5

•

•

0.5

•

A

•

•

B

F

•

C

y = β = 0.2

•

2.0

3.0

0

α = 0.1 − 0.5 0

1.0

π /2

ANGULAR ROD POSITION θ (radians) Figure 5.8.5 Diagram for obtaining the equilibrium angle 𝜃 for a non-dimensional load 𝛼 = P∕k l = 0.1 for different values of the non-dimensional lateral force 𝛽 = F∕k l. (Adapted from Figure 8.1.5 in “Mechanics of Elastic Solids,” V.K. Stokes © 1975.)

For the initial equilibrium configuration 𝜃 = 0, for 𝛽 = 0 small increases in 𝛽 cause small increases in 𝜃 . For example, 𝜃 = 𝜃 A for 𝛽 = 0.1. If 𝛽 is reduced to zero, 𝜃 A will shift to 𝜃 = 0. This holds for increasing values of 𝛽 until the point G is reached. Next consider the case for which 𝛽 > 0.78, say 𝛽 = 1.0. Let the initial equilibrium state for 𝛽 = 0 be 𝜃 = 0. For an applied lateral force is 𝛽 = 1.0, the equilibrium state is given by 𝜃 I . If the lateral force is then decreased to 𝛽 = 0, the equilibrium point will shift to 𝜃 = 𝜋 ! Thus, for 𝛽 = 0.1, the system is stable for small lateral forces 𝛽 < 0.78 and unstable for 𝛽 > 0.78. Note that two equilibrium solutions are also possible for negative values of 𝛽 – that is for the force F in Figure 5.8.5 acting from right to left – one just below 𝜃 = 𝜋 ∕2 and the other above 𝜃 = 𝜋 . As Figure 5.8.6 shows, the system exhibits the same behavior for larger values of 𝛼 as long as 𝛼 < 1. That is, there are three possible solutions for low values of 𝛽 . As 𝛽 is increased, two of the solutions merge at a certain value of 𝛽 . For values of 𝛽 above this critical value, only one solution is possible. As 𝛼 is increased, this critical value decreases. Further, as 𝛼 approaches 1 from below, the two solutions in the

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first quadrant for 𝛽 = 0, and the solution corresponding to the critical value of 𝛽 , all approach 𝜃 = 0. For 𝛼 = 1, all these three solutions merge into 𝜃 = 0. This follows by showing that y(0) = y ′ (0) = y ′′ (0) = 0. Thus, for 𝛼 = 1, there are several equilibrium states, which are infinitesimally close to each other and, therefore, this gives the condition for neutral stability. If the initial configuration is 𝜃 = 0, even the slightest increase in 𝛽 pushes the equilibrium state to the neighborhood of 𝜃 = 𝜋 .

π /4

0

3 π /4

π /2

π

1.5

α = 0.1

y = sin θ − α tan θ

138

y = β = 0.1

1.0

0.3

D

•

•

0.5 A

B

•

0

5

2

1

y = β = 0.5

•

5

•

H E

2

I

G

•

0.5

1

•

F

y = β = 0.1

• C

•

0.3 0.5 α = 0.1

− 0.5 0

1.0

π /2

2.0

3.0

ANGULAR ROD POSITION θ (radians) Figure 5.8.6 Diagram showing how different values of the non-dimensional load 𝛼 = P∕kl and the non-dimensional lateral force 𝛽 = F∕kl affect the equilibrium angle 𝜃 . (Adapted from Figure 8.1.5 in “Mechanics of Elastic Solids,” V.K. Stokes © 1975.)

And again, for 𝛼 > 1 all the equilibrium solutions for positive values of 𝛽 lie in the second quadrant and approach 𝜃 = 𝜋 ∕2 from above with increasing values of 𝛽 . Also, again, two equilibrium solutions are possible for negative values of 𝛽 – that is, for the force F in Figure 5.8.5 acting from right to left – one below 𝜃 = 𝜋 ∕2 and the other above 𝜃 = 𝜋 .

5.9 *Elastic Stability of an Axially Loaded Column So far, only the stability of rigid structures connected by a spring has been considered. A similar phenomenon of instability is exhibited by elastic members such as beams and columns. The formulation of stability problems of such elastic structures is based on determining the critical load, the load at which the system becomes neutrally stable for small deflections. The critical load is then obtained by requiring that the small-deflection solution to the deflection problem have continuously many equilibrium solutions.

Simple Structural Elements

A column is a structural member designed to carry axial compressive loads. A beam-column carries both lateral and axial loads. At a critical load, a column subjected to compressive loads can also exhibit the phenomenon of elastic instability, also referred to as buckling. The critical load is also called the buckling load. Let an axial compressive load P acting on the column cause it to deflect in the x-y plane into the neutrally stable shape v = v( x), a segment ABC of which is shown in Figure 5.9.1. For the axial load P at x = l applied at a height v = v(l ) the bending moment at point B is Mz = P [v(l ) − v( x)]. It then follows (Why?) from Section 5.3 that the deflected shape is governed by the differential equation EI yy v ′′ = Mz = P [v(l ) − v( x)] which can be rewritten as d2 v + m2 v = m2 v(l ) , where m2 = P∕EI yy dx2 This has the general solution

(5.9.1)

v = v(l ) + A sin mx + B cos mx

(5.9.2)

where A and B are arbitrary constants to be determined by the boundary conditions at the column ends.

v

C B

•

A v O

P v(l ) − v(x) v(l )

= v(x) x x=l

Figure 5.9.1 Geometry of a segment of a deformed column subjected to an axial load.

5.9.1

Buckling Load for a Pin-Jointed Column

Consider a pin-jointed column, one in which while the deflections at the ends are constrained to be zero; the ends are free to rotate about the z-axis, as shown schematically in Figure 5.9.2a. Figure 5.9.2b shows the column with the boundary conditions v(0) = 0 and v (l ) = 0. With v(l ) = 0, the first of these two conditions requires that in the general solution for buckling (Eq. 5.9.2) B = 0, so that v = A sin mx. The second condition then requires A sin ml = 0, which can be satisfied by A = 0, but this corresponds to the solution v(0) ≡ 0. However, for all small values of A, a neutrally stable solution v = A sin mx satisfying all the boundary conditions exists provided sin ml = 0, which requires that ml = n𝜋 , n = 0, 1, 2, …. Since the critical, or buckling, load is the smallest load that results

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in a neutrally stable solution, ml = 𝜋 . It follows that the buckling load for a pin-jointed column is given by Pc =

𝜋 2 EI yy

(5.9.3)

l2

v •

x

P

•

A

B (a)

v P

•

x

A

l

B

P

(b) Figure 5.9.2 Pin-jointed column subjected to axial end loads.

For historical reasons this load is called the Euler critical load for a pin-jointed column. For P < Pc , the column will be elastically stable, that is, it will not buckle. For P > Pc , the small-deflection analysis, which has been used, will predict infinitely large deflections. However, a complete analysis, in which Mz = EIz v ′′∕ [1 + (v ′ ) 2 ] 3/2 , will show that even for P > Pc there is a unique equilibrium position. The resulting deflections will be disproportionately large. Thus, in most applications, an important design criterion is that the loads on columns should not exceed the critical load. 5.9.2

Buckling of a Column Fixed at One End

Consider a column BC of length l, which is fixed at the end B and is subjected to an axial load P at the free end C, as shown in Figure 5.9.3a. The boundary conditions for this column are that the deflection v(0) and the slope v ′ (0) both be zero at B (at x = 0). Next consider the two columns shown in Figure 5.9.3b, in both of which the slope at the fixed end must be zero. By combining these two columns it can be seen that the buckling load P for the column shown in Figure 5.8.3a is the same as the buckling load P for the pin-jointed column of length 2l, shown in Figure 5.9.3c (Why?). It then follows from Eq. 5.8.2 that the buckling load for the column in Figure 5.9.3a is Pc =

𝜋 2 EI yy

=

𝜋 2 EI yy

(5.9.4) (2l ) 2 4l 2 Clearly, the buckling, or critical, load for a column fixed at one end with the other end being free is smaller than that for a pin-jointed column of the same length. In general, more constraints on a structure will result in larger buckling loads.

Simple Structural Elements

v C

B

•

P

x l (a)

P

B

B

A

C

l

P

l (b)

v

P

A

B

C

P

l

l 2l

(c) Figure 5.9.3 Column fixed at one end subjected to an axial end load.

Symmetry arguments have been used to simply obtain the buckling load for the column shown in Figure 5.9.3a. Of course, this critical load can be obtained from the general solution for buckling given in Eq. 5.9.1. The boundary condition v (0) = 0 requires B = − v(l ), so that the buckled shape has the form v = v(l ) (1 − cos mx)

(5.9.5)

which also satisfies the boundary condition v ′ (0) = 0. Imposing the condition v = v(l ) at x = l then requires that cos ml = 0, which has the solution ml = n𝜋 ∕2, n = 1, 3, 5, …, the smallest of which, ml = 𝜋 ∕2, then gives the buckling load in Eq. 5.9.3.

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5.10 Twist-Bend Buckling of a Cantilever A thin cantilever subjected to an end load, as shown in Figure 5.10.1a, instead of bending in the x-y plane can buckle “sideways” due to a combination of bending and twisting as shown in Figure 5.10.1b.

P

P y

z

l

•

(a)

x

(b)

Figure 5.10.1 (a) Geometry of a thin rectangular beam subjected to a transverse bending load P. (b) Out-of-plane deformation resulting from a combination of bending and twisting.

For a thin, not necessarily rectangular-sectioned, cantilever carrying an end transverse load P to resist the maximum stiffness of the cantilever, let the second moments of area of the cross section about the stiff and flexible axes be Iyy = Istiff and Izz = Iflex , respectively. Then a stability analysis shows that the critical buckling load for out-of-plane deformations of the cantilever is √ GE JIflex Pc = 4.01 l2 where G is the modulus of rigidity of the material and J the torsional rigidity of the cross section. For a rectangular-sectioned cantilever of thickness t and depth d, Iflex = Izz = dt 3∕12 and J = Izz = dt 3∕3, so that √ dt 3 E dt 3 Pc = 0.67 GE 2 = 0.67 √ 2 l 2 (1 + 𝜈 ) l

5.11 Stress Concentration In the simple structural analyses considered in the previous sections, the stresses are either uniform or vary linearly across the cross section. However, small changes in geometry can result in very high local stresses, well beyond the nominal stresses in the surrounding regions. This phenomenon is referred to as stress concentration, and the ratio of the high local stress to the nominal stress is called the stress concentration factor. Regions with such stress concentrations can act as nuclei for failure initiation. In elementary mechanical design the nominal stresses are first calculated by simple analyses that do not account for such stress concentrations, and the stress at known points of stress concentration are calculated by multiplying the nominal stress by the appropriate stress concentration factor.

Simple Structural Elements

The principles of mechanics can be used to accurately predict the stress concentration for any given geometry and loading. However, such analyses require full two- or three-dimensional analyses that are well beyond the scope of this book. Several examples are used to illustrate stress concentration effects. The first is the case of a flat bar, with a tooth BAB, pulled by a uniform tensile stress 𝜎 0 , shown in Figure 5.11.1. Away from the tooth the bar has the uniform stress 𝜎 0 . However, at the reentrant corners B – where the included angle at the material boundary is greater than 𝜋 – the stresses are very high. Interestingly though, the stress at the convex corner A is zero!

A

B

B

σ0

σ0

Figure 5.11.1 Geometry of a thin rectangular bar, with an integral tooth BAB, subjected to a uniform tensile stress 𝜎 0 .

As a second case, consider the twisting of the L-shaped prismatic bar shown in Figure 5.11.2. It can be shown that at the convex corners A, B, C, D, and E the stresses are identically zero. But the reentrant corner F has very high stresses.

B

A E F

C

D

Figure 5.11.2 Geometry of a prismatic bar having an L-shaped cross section.

A general rule of thumb is that stress concentration mainly occurs at sharp reentrant corners. In such cases, the stress concentration can be reduced by rounding such corners through fillet curves. Stress concentrations are important for many thin-walled plastic parts that are stiffened by ribs that result in many reentrant corners. Finally, consider a large plate (of infinite extent in the detailed analysis) with a hole of radius a, subjected to a uniform tensile stress 𝜎 0 (Figure 5.11.3). It can be shown that at points on the hole surface

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marked A, with coordinates ( x = ± a, y = 0), the only component of stress is 𝜎 y = − 𝜎 0 , that is, the tensile stress 𝜎 x = 𝜎 0 , acting on the plate induces a compressive stress 𝜎 y = − 𝜎 0 at these two locations. Even more interesting and significant is the result that at the points on the hole surface marked B, with coordinates ( x = 0, y = ± a ), the only component of stress is 𝜎 x = 3 𝜎 0 – the local stress is three times the nominal stress applied on the plate.

5 4

y/a

3 2 B

σ0

A

•

y

• • •

•

2 A

x

3

σx / σ0 σ0

B

Figure 5.11.3 An infinite plate with a with a hole of radius a, subjected to a uniform tensile stress 𝜎 0 .

It can be shown that the variation of the stress 𝜎 x along the y-axis is given by ) ( 𝜎 a2 a4 𝜎x = 0 2 + 2 + 4 2 r r

(5.11.1)

For r∕a = 1, that is, at the points marked B on the hole surface, this equation gives 𝜎 x = 3 𝜎 0 , or 𝜎 x ∕𝜎 0 = 3. The variation of 𝜎 x ∕𝜎 0 versus r∕a is shown in the upper half of Figure 5.11.3. For r∕a = 2, 3, and 4, 𝜎 x ∕𝜎 0 has the values 1.219, 1.0741, and 1.0371, respectively. Clearly, the stress concentration effect is limited to a distance of within a few multiples of the radius, after which the stress settles down to the applied nominal stress 𝜎 x = 𝜎 0 . It is worth pointing out that the stress concentration factor of 3 is independent of the hole size. Even the smallest hole has a stress concentration factor of 3, which can result in the nucleation of a failure site.

Simple Structural Elements

5.12 The Role of Numerical Methods This chapter has mainly focused on problems in which only one stress component varies. Most practical problems involve at least two-dimensional stresses that require more complex analyses. But even they, requiring the solution of partial differential equations, are not adequate for predicting the deformation and stresses in geometrically complex structures, as most load-bearing plastic parts are. Only numerical methods can provide such information for mechanical design. With the advent of digital computers, the complex differential equations describing the mechanical behavior of complex structures and loadings can now be solved numerically by using finite element methods that are now available in the form of robust, user-friendly computer codes, at times generically referred to as software. The availability of such codes has revolutionized the prediction of deformation and stresses. And such codes are not limited to the small deformation assumptions normally used for the analysis of parts. However, such codes should not be used without an understanding of the underlying basic mechanics phenomena such as bending, torsion, buckling, and stress concentration. Most plastic structures are thin-walled. As such, they are treated as shells, and shell elements are used in the finite element analyses. While this treatment is numerically efficient and adequate for predicting deformations – including elastic stability – and stresses in the thin-walled sections, it is inadequate for predicting the two- or three-dimensional stresses at the intersections of ribs with the shell structure, where the stress concentrations are likely to be. A second level analysis is required to address such high-stress regions.

5.13 Concluding Remarks This chapter has shown how mechanics principles can be used to determine the stiffness of, and the stresses in, simple structural members. Many non-intuitive results have been quantified: For example, (i) in bending, the stiffness of a beam increases linearly with the width but with the cube of the depth. (ii) A thin close-walled cross section has a significantly larger torsional stiffness than the same section with a slit. (iii) Under an axial compressive load a column can become elastically unstable (buckle). Such instabilities are particularly important for plastics structures that tend to be thin-walled. (iv) The phenomenon of stress concentration. While plastics structures have far more complex geometries, the stiffness of, and the stresses in, which can only be determined by numerical methods, the insights provided by such simple analyses help engineers to develop an “intuitive” understanding of how the initial geometry of a structure should be chosen. For optimizing the geometry of a structure to meet performance criteria, while its stiffness and the stresses induced in it by a given load can be evaluated through numerical codes, determining the stresses that will cause the structure to break, or fracture, require criteria that define the conditions under which the material will fail. Such failure criteria are expressed in terms of three-dimensional stresses. Some simple aspects of failure of plastics under two-dimensional stress fields are addressed in Sections 15.6 to 15.10. The simple analyses in this chapter have assumed that material response is time-independent. That is, for an applied time-independent or static load, the response of the material is also time-independent. This is a good assumption for metals and ceramics in most applications at normal use temperatures. However, except at very low loads, plastic parts subjected to static loads undergo slow, continuing time-dependent deformations. This requires an understanding of time-dependent material models, the simplest linear versions of which are discussed in Chapter 7.

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6 Models for Liquids 6.1 Introduction As discussed in the last chapter, a given load or stress acting on a body results in deformations that for ideal solids are adequately measured by strains. In such solids, the application of a time-independent load on a body results in an instantaneous, time-independent strain. For an ideally elastic material, on removing the load the body reverts to its original unloaded state, the strains dropping to zero. However, in a plastic material, unloading can result in a residual strain called plastic strain. In contrast to solids, the application of time-independent stresses on liquids, or more generally on fluids, results in time-dependent deformations. Fluids tend to continue to deform, or flow, under constant stresses, that is, the strains continue to increase with time. For this reason, the deformations in a fluid are better measured by deformation rates rather than by deformations; instead of using strains, the rates of change of strains with time, called strain rates, are used. The study of fluid flow is called fluid mechanics. Rheology is a subset concerned with the flow of high-viscosity fluids that also exhibit solid-like behavior. The resistance to fluid flow is measured by the viscosity of the material, which for polymer melts is strongly temperature-dependent. As a result, the analysis of fluid flow requires a determination of the temperature, requiring an analysis of heat transfer. The simplest form of heat transfer is that of heat conduction in solids, which is considered next.

6.2 Simple Models for Heat Conduction In addition to mechanical interactions that cause deformations and motion in material bodies, exchanges occur among different forms of energy. Of the work done on a body, some is transformed into kinetic energy, some is stored as potential energy – such as elastic energy, which can be recovered on unloading – and some changes the internal energy of the body through thermal interactions involving temperature changes. Thermal energy transfer within a material body, which causes temperature changes, is important for understanding material behavior. For characterizing such interactions, consider a small portion ABCD, of length Δx, of a bar of cross-sectional area A, located between positions x and x + Δx as shown Figure 6.2.1. Let the amount of energy flowing out of ABCD across the area A at AD be Q. Then, the amount of heat outflow per unit area per unit time, called the heat flux, is given by q (x, t) = Q ∕A. The heat flux at BC – the amount of heat leaving ABCD at BC – will then be A (q + Δq) (Why?). It follows that the net amount of heat flowing into ABCD in time Δt will be − A (Δq) Δt. Let this heat inflow in Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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D

C Q + ∆Q = A(q + ∆q)

Q = Aq A ∆x

x

O

B

x + ∆x

Figure 6.2.1 Energy flow in a prismatic bar.

time Δt cause the temperature of ABCD to increase from T to T + ΔT, so that the increase in the internal energy of ABCD will be 𝜌 (A Δx) c ΔT (Why?), where 𝜌 and c are, respectively, the density and the specific heat of the material. Since this increase in internal energy must equal the net inflow of energy, it follows that 𝜌 (A Δx) c ΔT = − A (Δq) Δt. This expression simplifies to 𝜌c (ΔT∕Δt) = − Δq ∕Δx. Notice that both the heat flux f = f (x, t) and the temperature T = T (x, t) are functions of x and t, so that while taking limits, partial derivatives must be used. It then follows that one-dimensional energy interactions in a bar are governed by

𝜌c

𝜕q 𝜕T =− 𝜕t 𝜕x

(6.2.1)

Just as in the case of stress, where how the stress depends on the strain is a material property characterized by the constitutive equation for the material, so also how the heat flux q = q (x, t) depends on the temperature T = T (x, t) is different for different materials. In the simplest model, the heat flux is linearly related to the spatial temperature gradient through q = −k

𝜕T 𝜕x

(6.2.2)

in which the constant k is called the thermal conductivity of the material. Substituting this expression in the energy balance equation gives the equation governing the one-dimensional conduction of heat as

𝜌c 𝜕 T 𝜕 2 T 1 𝜕T 𝜕2T = 2 or = k 𝜕t 𝛼 𝜕t 𝜕x 𝜕 x2 in which 𝛼 = 𝜌c∕ k is called the thermal diffusivity of the material. 6.2.1

(6.2.3)

Steady-State Heat Conduction

Under steady-state conditions Eq. 6.2.3 reduces to d2 T∕dx2 = 0 (Why?), which has the general solution T = Ax + B, where A and B are arbitrary constants to be determined by boundary conditions. Consider a slab of solid material of thickness h bounded by x = 0 and x = h, in which the temperatures at x = 0 and

Models for Liquids

x = h are maintained at T = T1 and T = T2 , respectively. An application of these boundary conditions to the general solution gives A = (T2 − T1 )∕h and B = T1 , so that the temperature distribution in the slab is T − T1 T = T1 + 2 x (6.2.4) h It then follows from Eq. 6.2.2 that the heat transfer per unit area across the slab is given by q= 6.2.2

k (T − T2 ) h 1

(6.2.5)

Transient Heat Conduction

Of the many possible solutions of the differential equation in Eq. 6.2.3, the transient one-dimensional temperature distribution T = T (x, t) due to conduction into a stationary material is especially useful. Consider an infinitely long bar, one end of which is located at x = 0, the other end being at x = ∞. At time t < 0, let the initial temperature of the bar, located along 0 ≤ x ≤ ∞, be T0 . At time t = 0, let a constant temperature T1 be imposed at the bar face at x = 0, so that T (0, t) = T1 for t ≥ 0. It can be shown that the temperature distribution in the bar is then given by ( √ ) ( √ ) T(x, t) − T0 𝜗 (x, t) = = 1 − erf x∕ 4𝛼 t = cerf x ∕ 4𝛼 t (6.2.6) T1 − T0 where the error function, erf (𝜂 ), is defined by 2 erf(𝜂 ) = √

𝜂

𝜋 ∫0

2

e−y dy

(6.2.7)

Now erf (0) = 0 and erf (∞) = 1, which gives 𝜗 (0, t) = 1 − erf (0) = 1 and 𝜗 (x, 0) = 1 − erf (∞) = 0. The variations of the error function, erf (𝜂 ), and the complementary error function, cerf (𝜂 ), are shown in Figure 6.2.2. Although erf (∞) = 1, this function becomes very close to unity at very small values of the argument 𝜂 . For instance, the values of this function for 𝜂 = 1.5, 2, 2.5, and 3 are, respectively, 0.966105, 0.995322, 0.999593, and 0.999978. Therefore, for all practical purposes, the error function may be assumed to be one for 𝜂 = 1, and with even more precision for 𝜂 = 2, or 𝜂 = 3. This result has important implications for phenomena described by the error function. As an example, from the solution in √ 𝜗 (x, t) is zero for 𝜗 (x, t) = 𝜂 = x ∕ 4 𝛼 t = 1, Eq. 6.2.6, it follows that, for all practical purposes, √ that is , for x = 4𝛼 t. Thus, the effect of a temperature change on the surface is felt at a depth x at time t = x2 ∕ 4 𝛼 . Or, at time t, the effect of a temperature change on the surface has penetrated a distance √ (6.2.8) x = 4𝛼 t

6.3 Kinematics of Fluid Flow The description of fluid flow is inherently more complex than the deformation of solids: While in solids neighboring particles stay in close proximity even after deformation, those in fluids tend to drift far apart. As a result, using local spatial coordinates to tag material particles is adequate for the small deformations

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ERF AND CERF FUNCTIONS

150

1.0 erf(η) =

2

η

2

e − y dy π ∫0

0.5 cerf(η) = 1 − erf(η)

0 0

1.0

2.0 DISTANCE

3.0

η

Figure 6.2.2 Error and complementary error functions.

that solids normally undergo. In fluids, a particle in one position in space at one instant of time can be far away at a later time, so that a tagging of particles by local spatial coordinates is not practical: In describing fluid flow, local coordinates characterize the properties of the particle that happens to be occupying that position at a particular instant of time. The use of such spatial coordinates describes events at spatial points at different times, but this description does not focus on the processes that individual particles are undergoing, which requires the use of coordinates embedded on particles, called material coordinates. Although material coordinates better describe the histories of individual particles – for example, while the rate of time change of the velocity of a particle gives its acceleration, the time rate of the local velocity in spatial coordinates is not the acceleration – they are mathematically more cumbersome to use. A link between these two types of coordinates systems is required for linking the actual properties of moving particles to properties of particles that happen to be occupying a particular position at a particular instant of time. A general exposition of the fluid flow is well beyond the scope of this book. However, the elementary forms of the three basic sets of equations – the continuity equation that expresses the conservation of mass, the momentum equations that express the balance of linear momentum, and the energy equation that is an expression for the balance of energy – are discussed. But first, measures for deformation in fluids, of which the most important are shear rates, must be considered 6.3.1

Measures for Deformation Rates

Since fluids undergo continuing, large deformations under a constant stress, it is more appropriate to use the velocity of a deforming particle rather than the displacement used for solids. By far, shearing is the most important deformation mode for liquids.

Models for Liquids

u + ∆u

P C

D

D' γ

∆y A

B

B'

A'

u

y O

C'

τ =

P A

P

x

Figure 6.3.1 Shearing of a fluid element.

With reference to Figure 6.3.1, consider a liquid element ABCD of thickness Δy, subjected to a shear stress 𝜏 . Under this constant shear stress, the liquid will flow in the x-direction. Let the velocity along AB be u = u (y), then that along DC will be u + Δu. At time t the material points at A and D will move, respectively, to A ′ and D ′. It follows that the x-direction displacement of D ′ relative to A ′ will be (Δu) t (Why?), so that the shear strain at time t will be 𝛾 (t) = (Δu∕Δy) t, which in the limit becomes 𝛾 (t) = (du∕dy) t. The time rate of strain, or the strain rate, is then given by

𝛾̇ (t) =

du(y) d 𝛾 (t) = dt dy

(6.3.1)

Strain rate has the dimensions of inverse time (one over time) and is normally expressed in inverse seconds (s−1 ). Notice that the shear strain rate 𝛾̇ (t) is the variation of the x-direction velocity in the y-direction. As with shear strain, this strain rate should be referred to as 𝛾̇ xy (t). In general the local velocity of a material particle will have both x- and y-components, vx (x, y) and vy (x, y). Then, as with the strain-displacement relations in Eq. 3.3.2, the two-dimensional components of the strain rate can be shown to be

𝜀̇ x =

𝜕 vy 𝜕 vx , 𝜀̇ y = , 𝜕x 𝜕y

and

𝛾̇ xy =

𝜕 vy 𝜕 vx + 𝜕x 𝜕y

(6.3.2)

6.4 Equations Governing One-Dimensional Fluid Flow This section introduces the three basic sets of equations for one-dimensional flow: The continuity equation that expresses the conservation of mass, the momentum equations that express the balance of linear momentum, and the energy equation that is an expression for the balance of energy. This case has only one component of the velocity that in the sequel will be assumed to be the x-direction velocity component vx = u.

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6.4.1

One-Dimensional Continuity Equation

Inside a rectangular channel consider the balance of fluid mass in a small region, ABCD, of length AB = CD = Δx, height AD = BC = Δy and depth d, as shown in Figure 6.4.1. Assume that all the flow occurs only in the x-direction, so that the only non-zero component of velocity is vx = vx (x, y). Then, in Δt, time the mass of fluid entering the element through the side AD is ( 𝜌 vx ) Δt (Δy d ) (Why?), and the mass leaving the side BC is ( 𝜌 vx + Δ𝜌 vx ) Δt (Δy d ), so that the net mass leaving this element across the sides AD and BC equals (Δ𝜌 vx ) Δt (Δyd ) (Why?), which can be written as (Δ𝜌 vx ∕Δx) Δt (Δx Δy d ).

D

C

( ρ vx )∆t(∆y d )

∆y

( ρ vx + ∆ ρ vx )∆t(∆y d ) A

B

y

∆x

x

O

x + ∆x

Figure 6.4.1 Geometry for one-dimensional flow along a rectangular channel.

Now at time t the mass of fluid in the element ABCD is 𝜌 (Δx Δy d ), and that at time t + Δt must be ( 𝜌 + Δ𝜌) (Δx Δy d ), so that the net increase in mass in time Δt equals Δ𝜌 (Δx Δy d ) = (Δ𝜌∕Δt) Δt (Δx Δy d ), which, because of conservation of mass, must equal the negative (Why?) of the net mass leaving the element during this time interval. By equating these two sets of quantities it follows that Δ𝜌 vx Δ𝜌 =− Δt Δx or, in the limit,

𝜕𝜌 𝜕 (𝜌 vx ) + =0 𝜕t 𝜕x

(6.4.1)

which is called the continuity equation. Note that the velocity can change along the channel if the density is not constant. However, for an incompressible fluid – which is a good approximation for most polymeric materials, except for some processing applications, such as injection molding, in which very high pressures can require consideration of material compressibility effects – the density is constant so that the one-dimensional continuity equation reduces to

𝜕 vx =0 𝜕x

(6.4.2)

Models for Liquids

This implies that the velocity vx is only a function of y and the time t. In the sequel this velocity component will be denoted by vx = u (y, t). 6.4.2

Balance of Linear Momentum in One Dimension

Now consider the balance of momentum for the one-dimensional case in which material elements only see a pressure and one component shear stress. Figure 6.4.2 shows a material element ABCD subjected to a pressure p and a shear stress 𝜏 . The pressure p on the face AD generates a rightward acting force p (Δy d ), where d is the depth, or thickness, of the element ABCD. The leftward acting force on face BC is then ( p + Δp) (Δy d ), so that net rightward force acting on the element due to the pressure force on faces AD and BC is − Δp (Δy d ) = − (Δp∕Δx) (Δx Δy d ) (Why?).

( p + ∆p)(∆x d ) (τ + ∆τ )(∆x d ) C

D ∆y

( p + ∆p)(∆yd )

p(∆yd )

(τ + ∆τ )(∆yd )

τ (∆yd ) A

B p(∆x d )

y

x

O

τ (∆x d )

∆x

Figure 6.4.2 Normal pressure and shear stresses acting on a small rectangular fluid element.

Next consider the rightward force contributed by the shear stress 𝜏 acting on the faces AB and CD: The shear stress 𝜏 on the face AB generates a leftward acting force 𝜏 (Δx d ). The rightward acting force on face DC is then (𝜏 + Δ𝜏 ) (Δx d ), so that the net rightward force acting on the element due to the shear stresses on the faces AB and CD is (Δ𝜏 ) (Δxd ) = (Δ𝜏 ∕Δy) (Δy Δx d ). It follows that the net x-direction force on the fluid element ABCD due to the combined effects of pressure and shear stresses is − (Δp∕Δx + Δ𝜏 ∕Δy) (Δx Δy d ). Similarly, the net upward force acting on ABCD due to y-direction stresses is − (Δp∕Δy + Δ𝜏 ∕Δx) (Δx Δy d ). Now the mass of ABCD is 𝜌 (Δx Δy d ). Let the x- and y-direction components of the acceleration of the material element ABCD be ax and ay , respectively. Then, by applying the principle that the mass times the acceleration in any direction must equal the force in that direction to the x-direction components results in 𝜌ax (Δx Δy d ) = − (Δp∕Δx + Δ𝜏 ∕Δy) (Δx Δy d ), or

𝜌ax = −

Δp Δ𝜏 + Δx Δy

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which, in the limit, gives the equation of motion for the x-direction as

𝜌ax = −

𝜕 p 𝜕𝜏 + 𝜕x 𝜕y

(6.4.3)

Similarly, the equation of motion for the y-direction as

𝜌ay = −

𝜕 p 𝜕𝜏 + 𝜕y 𝜕x

But because the only component of velocity is vx = u (y, t), the acceleration ay = 0 (Why?). It follows that the x-direction equation of motion is −

𝜕 p 𝜕𝜏 + =0 𝜕y 𝜕x

(6.4.4)

In the case of an incompressible fluid, since the shear stress must be the same for all values of x for any fixed y, it follows (Why?) that 𝜕𝜏 ∕𝜕 x = 0, so that 𝜕 p∕𝜕 y = 0. Thus, for incompressible fluids, 𝜏 = 𝜏 (y, t), p = p (x, t), and the equation of motion reduces to

𝜌ax = −

𝜕 p(x, t) 𝜕𝜏 (y, t) + 𝜕x 𝜕y

(6.4.5)

For the steady-state case this equation reduces to dp(x) d𝜏 (y) = dx dy

(6.4.6)

in which the pressure gradient dp(x)∕dx is constant. 6.4.3

*Energy Balance in One Dimension

The stresses acting on a fluid element do work that contributes to the balance of energy in the element. In Figure 6.4.3 the forces on the element ABCD caused by the shear stresses and pressures are indicated by arrows. And the arrows with dashed lines indicate the velocities, vx = u (y, t), at the surfaces. Now, in time Δt, the shearing force − 𝜏 (Δx d ) acting on the surface AB causes a displacement u Δt in the fluid element at AB, thereby doing work − 𝜏 (Δx d ) (u Δt) = − 𝜏 u (Δx d Δt) (Why?); the minus sign indicates that the force and the displacement are in opposite directions, so that the fluid in ABCD does work on the surrounding. Similarly, in time Δt, the shearing force on CD does work 𝜏 (Δx d ) (u Δt ) = {𝜏 u + [Δ(𝜏 u)∕Δy] Δy (Δx d Δt)} (Why?). It follows that the net work done on the fluid element ABCD by the shear stresses at AB and CD equals [Δ(𝜏 u)∕Δy] (Δx Δy d Δt). Since the shear stresses on AD and BC act at right angles to the displacements of the fluid elements in time Δt, they do no work. Thus, the total work done on the fluid element ABCD by the shear stresses acting on the faces AB, BC, CD, and DA equals [Δ(𝜏 u)∕Δy] (Δx Δy d Δt). Also, in time Δt, the normal force p (Δy d ) acting on the surface DA causes a displacement u Δt in the fluid element at DA, thereby doing work p (Δy d ) (u Δt) = p u (Δy d Δt) on the fluid in ABCD. Similarly, in time Δt, the normal force [ p + (Δp∕Δx) Δx] (Δy d ) acting on the surface BC causes a displacement u + (Δu∕Δx) Δx on CD, doing work − { p u + [Δ ( pu)∕Δx] Δx (Δy d Δt)} (Why?); the minus sign results from the force and the displacement being in opposite directions, indicating that the fluid in ABCD does work on the surrounding. It follows that the net work done on the fluid element ABCD by the normal (pressure) stresses on DA and BC equals − [Δ( pu)∕Δy] (Δx Δy d Δt). And since the normal stresses on

Models for Liquids

[p + (∆p /∆y) ∆y] (∆x d ) (τ + ∆τ )(∆x d )

τ (∆yd )

u + (∆u /∆y) ∆y C

D

u + (∆u /∆x) ∆x

u ∆y

[p + (∆p /∆x) ∆x](∆y d )

p(∆yd ) A

B

τ (∆x d )

y

(τ + ∆τ )(∆y d )

u p(∆x d )

O

x

∆x

Figure 6.4.3 Normal and shear stresses, and velocities contributing to the increase in energy of a small rectangular fluid element.

AB and CD act at right angles to the displacements of the fluid elements in time Δt, they do no work. Thus, the total work done on the fluid element ABCD by the normal stresses (pressures) acting on the faces AB, BC, CD, and DA equals − [Δ(pu)∕Δx] (Δx Δy d Δt). It follows that the work done by the normal and shear stresses acting on ABCD in time Δt is ] [ Δ( pu) Δ(𝜏 u) + (Δx Δy d Δt) (6.4.7) − Δx Δy Besides the mechanical energy inflow into the element ABCD resulting from the work done by the normal and shear stresses, non-mechanical work, such as heat, can also flow in across the surfaces of the element. Let the energy flux (flow) per unit area out of a surface be q as shown by the dash-dotted arrows in Figure 6.4.4. Then the net outflow of energy in time Δt from the surfaces AB, BC, CD, and DA must equal (Why?) ( ) Δqy Δqx Δqy Δqx Δx Δt (Δy d) − Δy Δt (Δx d ) = − + Δt (Δx Δy d ) (6.4.8) Δx Δy Δx Δy By combining this expression with the energy inflow from shear stresses it follows that the net energy flowing into ABCD in time Δt is [ ( )] Δqx Δqy Δ( pu) Δ(𝜏 u) − + − + Δt (Δx Δy d ) (6.4.9) Δx Δy Δx Δy This must equal the increase in the energy of the fluid in ABCD. The energy of the fluid in ABCD consists of two parts: Its kinetic energy 𝜌 (Δx Δy d ) (u2 ∕ 2) (Why?), and its internal energy 𝜌 (Δx Δy d ) 𝜀, in which 𝜀 is the internal energy per unit mass of the fluid. Because the mass 𝜌 (Δx Δy d ) of the fluid

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q + (∆q /∆y) ∆y

C

D ∆y

q + (∆q /∆x) ∆ x

q

A q

y

O

B

x

∆x

Figure 6.4.4 Non-mechanical energy fluxes contributing to increase in energy of a small rectangular fluid element.

element remains constant (Why?), the increase in the energy of ABCD in time Δt is ( ) Δ𝜀 1 Δu2 𝜌 (Δx Δy d ) + Δt Δt 2 Δt

(6.4.10)

By equating the expressions in Eqs. 6.4.7 and 6.4.8 and taking the limit, the energy balance equation takes the form ) ) ( ( 𝜕 qx 𝜕 qy 𝜕 ( pu) 𝜕 (𝜏 u) 𝜕𝜀 1 𝜕 u2 =− (6.4.11) + + − + 𝜌 𝜕t 2 𝜕t 𝜕x 𝜕y 𝜕x 𝜕y which can be rewritten as ( ) ( ) 𝜕 qx 𝜕 qy 𝜕𝜀 𝜕u 𝜕 u 𝜕 p 𝜕𝜏 =𝜏 − + + − −u 𝜌 𝜌 𝜕t 𝜕y 𝜕x 𝜕y 𝜕t 𝜕x 𝜕y

(6.4.12)

It follows from Eq. 6.4.3 that the terms in the second set of parentheses in Eq. 6.4.12 must be zero, so that the one-dimensional energy balance equation is ( ) 𝜕 qx 𝜕 qy 𝜕𝜀 𝜕u =𝜏 − + (6.4.13) 𝜌 𝜕t 𝜕y 𝜕x 𝜕y Following the discussion in Section 6.2, with 𝜀 = c T, qx = − k 𝜕 T∕𝜕 x, and qy = − k 𝜕 T∕𝜕 y, Eq. 6.4.1 takes the form ( 2 ) 𝜕T 𝜕 T 𝜕2T 𝜕u =𝜏 +k 𝜌c + (6.4.14) 𝜕t 𝜕y 𝜕 x2 𝜕 y2

Models for Liquids

which, for the steady state case, with no variations in the x-direction, reduces to

𝜏

d2 T du +k 2 =0 dy dy

(6.4.15)

in which the smaller term 𝜕 2 T∕𝜕 x2 has been neglected.

6.5 Simple Models for the Mechanical Behavior of Liquids In the simplest models for solids (Chapter 4) the stress in a material depends linearly on the deformation as measured by strains. In fluids, stress causes the material to flow continuously, resulting in ever increasing strains. For such materials it is more appropriate to use time rates of deformation, as characterized by strain rates, as measures of deformation induced by stresses. While the stress-strain-rate relations for air and water are adequately described by simple linear relationships between stress and strain rate, those for polymers are far more complex: First, the dependence of stress on strain rate is far from linear, and second, such relationships are highly temperature-dependent. The following subsections address this increasing complexity. And how this increasing complexity affects actual flows is discussed in Section 6.6. 6.5.1

Newtonian Liquids

The simplest type of fluid, in which the shear stress 𝜏 is linearly related to the shear strain rate 𝛾̇ (t), is called a Newtonian fluid. For such fluids, the constitutive equation is then

𝜏 = 𝜇 𝛾̇ = 𝜇

du dy

(6.5.1)

in which the ratio, 𝜇 , of the shear stress to the shear strain rate is called the viscosity of the material. Thus, the viscosity of a fluid is a measure of the fluid’s resistance to flow. It has dimensions of stress multiplied by time; the SI units for viscosity are Pascal-seconds (Pa.s). The viscosities of air, water, and oil are about 1.8 × 10−5 , 10−3 , and 3 × 10−2 Pa s, respectively. In contrast to these relatively low viscosities, polymer melts have very high viscosities in the range 101 – 103 Pa s. For most practical purposes, most liquids may be considered to be incompressible. 6.5.2

Non-Newtonian Liquids

The Newtonian model does not describe the highly nonlinear behavior of polymer melts. To develop more appropriate models, in the next level of complexity in stress-strain-rate relations the stress is assumed to depend nonlinearly on the strain rate. In the simplest such model, called the power law model, this relationship is written as

𝜏 = K |𝛾̇ |n−1 𝛾̇ or 𝜇 = K |𝛾̇ |n−1

(6.5.2)

in which the absolute value of the shear rate is used to ensure a positive viscosity, that is, to ensure that the shear stress and the shear rate have the same sign. This equation can be used for empirical fits to

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melt-flow data. However, it can only provide fits over several decades of strain-rate variations. While the power law index n is relatively insensitive to the temperature, K, called the consistency index, is highly temperature-dependent. In another empirical non-Newtonian viscosity model, called the Carreau–Yasuda model, the viscosity has the form ( )(n−1)∕𝛼 𝜇 = 𝜇0 1 + 𝛽 |𝛾̇ |𝛼 (6.5.3) In this model 𝜇 → 𝜇 0 as 𝛾̇ → 0, and for large shear rates (𝛽 |𝛾̇ |𝛼 ≫ 1 or |𝛾̇ | ≫ 𝛽 −𝛼 ), 𝜇 → 𝜇0 𝛽 (n−1)∕𝛼 |𝛾̇ |n−1 , so that it approaches the power law model with K = 𝜇0 𝛽 (n−1)/𝛼 . The Cross model is a particular case of the Carreau–Yasuda model with 𝛼 = 1 − n, and therefore has the form ( )−1 𝜇 = 𝜇0 1 + 𝛽 |𝛾̇ |1−n (6.5.4) Let 𝜇 0 and 𝜇 ∞ be the viscosities at zero and infinite shear rates, respectively. The Carreau–Yasuda and Cross models can then be written in the respective forms

𝜇 = 𝜇∞ + (𝜇0 − 𝜇∞ ) (1 + K|𝛾̇ |𝛼 )(n−1)∕𝛼 and

𝜇 = 𝜇∞ + 6.5.3

𝜇0 − 𝜇∞ 1 + K|𝛾̇ |m

Temperature-Dependent Viscosity Models

Temperature has a dramatic effect on the viscosities of polymer melts: a 10°C increase in the temperature of a melt can reduce the viscosity by a factor of two. This temperature dependence is normally accounted for by either using an exponential model – essentially an Arrhenius-type model – or by the more commonly used Williams-Landel-Ferry (WLF) model. The Exponential Model has the form

𝜇 (T) = 𝜇r exp[− b(T − Tr )] in which b is a material constant and 𝜇 r is the viscosity at a reference temperature Tr . The WLF model, more extensively discussed in Section 7.7.1, has the form [ ] −C1 (T − Tr ) 𝜇 (T) = 𝜇r exp C2 + (T − Tr )

(6.5.5)

(6.5.6)

in which C1 and C2 are material constants and 𝜇 r is the viscosity at a reference temperature Tr . These temperature-dependent viscosity models are used in conjunction with the Non-Newtonian models (Section 6.5.2) to model both strain-rate and temperature-dependent nonlinearities with the zero-shear 𝜇 0 viscosity in Eq. 6.5.2 being replaced by 𝜇 (T ) from either Eqs. 6.5.5 or 6.5.6. One commonly used viscosity model is the generalized Cross model [ ](n−1)∕𝛼 𝜇 = 𝜇0 1 + (𝜇0 𝛾̇ ∕𝜏 ∗ )𝛼 (6.5.7) in which 𝜏 ∗ , n, and 𝛼 are material constants and 𝜇 0 is the viscosity 𝜇 (T ) from either Eqs. 6.5.5 or 6.5.6. This generalized Cross model is essentially the same as the Carreau–Yasuda model in Eq. 6.5.3.

Models for Liquids

6.6 Simple One-Dimensional Flows A substitution for the stress 𝜏 in Eq. 6.4.3 from the stress-deformation-rate models (constitutive equations) in Sections 6.5.1 or 6.5.2 results in an equation of motion in terms of the velocity. The determination of the velocity distribution from this equation requires boundary conditions. The most commonly used boundary conditions for fluid flow are that the relative velocity at a solid interface must be zero. However, in polymer melt flows, boundary conditions with slip at such interfaces – the relative velocity at a solid-liquid interface is not zero – are also considered. The constitutive equations in Sections 6.5.1 and 6.5.2 are temperature-independent. That is, the viscosities are assumed not to depend on the temperature. While this assumption simplifies the equation of motion by decoupling it from the energy balance equation (Eq. 6.4.14), thereby facilitating the determination of the velocity distribution, it does not account for the very strong temperature dependence of the viscosity. The analysis of “real” polymer melt flows require the simultaneous solution of the equation of motion (Eq. 6.4.3) and the energy balance equation (Eq. 6.4.14) that are coupled by the temperature-dependent viscosity models in Section 6.5.3. However, the uncoupled, temperature-independent flow models do provide an insight into the parameters affecting flows. This section analyzes two of the simplest one-dimensional fluid flows that are relevant to polymer processing phenomenon: In the first of these the pressure does not vary and the flow is generated by a moving solid-liquid interfaced. In the second example, the solid-liquid interfaces are not moving and the flow is generated by pressure differences. 6.6.1

Surface-Driven One-Dimensional Steady Flow

Consider the simple case of a flow of a fluid between two parallel plates AB and CD distance h apart, as shown in Figure 6.6.1, in which the upper plate moves rightward with a constant velocity V. In this simple case, called Couette flow, the pressure is constant and the flow is only induced by the motion of the upper plate CD. For this case the equation of motion (Eq. 6.4.3) reduces to 𝜕𝜏 ∕𝜕 y = 0, which implies a constant shear stress 𝜏 = 𝜏 0 throughout the fluid.

D

C

y=h

u=V

y O

x

y=0

u=0 A

B

Figure 6.6.1 One-dimensional Couette flow.

Case 1 Newtonian Fluid

For a Newtonian fluid it follows from Eq. 6.5.1 that (Why?) u = 𝜇 𝛾̇ =

𝜏0 y+C 𝜇

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where C is a constant of integration. The boundary condition u = 0 at y = 0 requires that C be zero. And the condition u = V at y = h finally results in u=

V y , h

𝜏0 = 𝜇

V h

(6.6.1)

This linear variation of the velocity with the distance is indicated by the arrows in Figure 6.6.1. Let the mass flow rate across the channel height per unit depth in the z direction be Q. Then (Why?) h

Q=

∫0

𝜌udy =

𝜌V 2h

(6.6.2)

Case 2 Non-Newtonian Fluid

For a Non-Newtonian fluid, it follows from Eq. 6.5.2 that ( )n ( 𝜏 )1∕n du du 𝜏0 = K or = 0 dy dy K It follows (Why?) that u=

V y, h

𝜏0 = K

( )n V V =𝜇 h h

(6.6.3)

and that the mass flow rate across the channel height per unit depth in the z-direction is given by Eq. 6.6.2. Thus, for plane Couette flow the expressions for the velocities and the shear stresses are the same for both Newtonian and Non-Newtonian flows.

6.6.2

Heat Generation in One-Dimensional Couette Flow

For the linear velocity field in Eq. 6.6.1, du∕dy = V∕h, so that the steady-state energy equation (Eq. 6.4.13) reduces to

𝜇 d2 T = − (V∕h)2 k dy2

(6.6.4)

Assuming that the surfaces at y = 0 and y = h are both at the same temperature T0 , an integration of this equation twice gives T − T0 =

𝜇 2k

[ ] (V∕h)2 y(h − y)

(6.6.5)

From this equation it follows that the maximum temperature, which occurs at y = h∕2, is given by Tmax − T0 =

𝜇 8k

V2

(6.6.6)

Models for Liquids

6.6.3

*One-Dimensional Couette Flow with Temperature-Dependent Viscosity

For this case the equation of motion reduces to a constant 𝜏 = 𝜏 0 . Assuming that in the simple temperature-dependent viscosity model in Eq. 6.5.5 the temperature is measured from the reference temperature Tr , which is equivalent to defining a new temperature variable (T − Tr ) that does not affect the energy equation (Eq. 6.4.15) (Why?), the viscosity variation has the form

𝜇 (T) = 𝜇0 exp (−bT )

(6.6.7)

the use of which in conjunction with 𝜏 0 = 𝜇 du∕dy and the energy balance equation (Eq. 6.4.15) results in

𝜏 d2 T d2 T 𝜏 d2 T du + 𝜏0 = k 2 + 0 = k 2 + 0 exp (bT ) = 0 2 dy 𝜇 (T ) 𝜇0 dy dy dy 2

k

2

which can be written as

𝜏 d2 T + 𝛼 exp (bT ) = 0, 𝛼 = 0 k𝜇0 dy2 2

(6.6.8)

A solution of this nonlinear differential equations subject to appropriate boundary conditions at the surfaces at y = 0 and y = h will determine the temperature distribution in the fluid. The velocity distribution will then have to be determined by solving the differential equation

𝜏 𝜏 du = 0 = 0 exp (bT ) dy 𝜇 (T ) 𝜇0 2

(6.6.9)

subject to the boundary conditions u = 0 at y = 0 and u = V at y = h. Since Eq. 6.6.9 is of first order, only one boundary condition is required; the second condition will determine the value of the shear stress 𝜏 0 . Clearly, the equation of motion and the energy balance equation are coupled by the temperature dependence of the viscosity and cannot be solved independent of each other as in the case of temperature-independent Newtonian and Non-Newtonian liquids. A closed-form solution of Eq. 6.6.9 is not available and it and the more general cases have to be solved numerically. A simple case, in which a closed-form solution can be obtained, is discussed next. 6.6.3.1 Linear Variation of Viscosity with Temperature

When bT ≪ 1, 𝜇 (T ) = 𝜇 0 exp (−bT ) ≃ 𝜇 0 (1 − bT ), so that a linear variation of the viscosity can be obtained from the exponential variation approximated by imposing the condition bT ≪ 1. It then follows from Eq. 6.6.7 that b𝜏0 2 d2 T 2 2 + 𝛽 T = − 𝛼 , 𝛽 = 𝛼 b = k𝜇0 dy2

(6.6.10)

which has the general solution 1 (6.6.11) b By imposing the boundary conditions T = T0 at y = 0 and T = T0 at y = h, this equation can be shown to have the solution ) )( 1 − cos 𝛽 h ( 1 sin 𝛽 y + cos 𝛽 y − 1 (6.6.12) T − T0 = T0 + b sin 𝛽 h T = A sin 𝛽 y + B cos 𝛽 y −

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The maximum temperature, which occurs at y = h∕2, is given by Tmax − T0 1 −1 = T0 + 1∕b cos 𝛽 h∕2 6.6.4

(6.6.13)

*Development of Couette Flow

Consider a fluid at rest in the region 0 ≤ y < ∞, so that all the components of the velocity, in particular u(y, 0), are identically zero. At time t = 0, let the surface at y = 0 be subject to a constant velocity u(y, t) = V. In the absence of a pressure gradient (dp∕dx = 0), the only velocity component will be u(y, t). In this case, since the velocity u(y, t) is not zero, the acceleration component in Eq. 6.4.5 will be ax = 𝜕 u∕𝜕 t. Then, with the Newtonian constitutive equation in Eq. 6.5.1, the equation of motion (Eq. 6.4.5) for this time-dependent flow reduces to

𝜕u 𝜕2u 1 𝜕u 𝜕2u = 𝜇 2 , or 𝜈 = (6.6.14) 𝜕t 𝜕y 𝜕 t 𝜕 y2 in which 𝜈 = 𝜇 ∕𝜌 is called the kinematic viscosity of the fluid. Then, with the initial condition u(y, t) ≡ 0 𝜌

for t < 0, and the boundary condition u(y, t) = V for t ≥ 0, discussed before, the solution of Eq. 6.6.2 is the same as that for transient heat conduction (Eq. 6.2.3) given in Eq. 6.2.6. It follows that the solution for flow development in Couette flow is given by ( √ ) ( √ ) u( y, t) (6.6.15) = 1 − erf y∕ 4𝜈 t = cerf y∕ 4𝜈 t V The variation of cerf (𝜂 ) is shown in Figure 6.2.2. Since erf (3) ≈ 1, or cerf (3) √≈ 0, the effect of the velocity V imposed on the fluid at y = 0 is felt only as far as 𝜂 ≈ 3, or y = 6 𝜈 t. So, for the Couette flow geometry shown in Figure 6.6.1, the action of the velocity V at y = 0 is felt at y = h at time tsteady ≈ h2 ∕36 𝜈 . 6.6.5

Pressure-Driven One-Dimensional Steady Flow

Consider the simple case of the steady flow of a fluid between two stationary parallel plates AB and CD distance h apart, as shown in Figure 6.6.2, in which the flow is driven by a pressure gradient in the x-direction. For this simple case, called Poiseuille flow, the equation of motion (Eq. 6.4.3) reduces to 𝜕 p 𝜕𝜏 = (6.6.16) 𝜕x 𝜕y with the pressure gradient dp∕dx being constant throughout the fluid.

y=h

D

C u=0

y O

x

y=0

u=0 A

Figure 6.6.2 One-dimensional Poiseuille flow.

B

Models for Liquids

Case 1 Newtonian Fluid

For a Newtonian fluid it follows from Eq. 6.5.1 that (Why?) d2 u 1 dp = (6.6.17) 𝜇 dx dy2 which, subject to the boundary conditions that u = 0 at y = 0 and y = h, integrates to the parabolic profile 1 dp [ y(h − y)] (6.6.18) 2𝜇 dx From this equation it follows that the maximum velocity, which occurs at y = h∕2, is given by u(y) = −

h2 dp (6.6.19) 8𝜇 dx The shear stress distribution then is 1 dp 𝜏 (y) = − (2y − h) (6.6.20) 2𝜇 dx The magnitude of the maximum shear stress, which occurs at the surfaces at y = 0 and y = h, is given by umax = −

h || dp || 2 || dx || The mass flow rate across the channel height per unit depth in the z-direction is given by ( ) h dp 𝜌h3 − Q = 𝜌udy = ∫0 12𝜇 dx

𝜏max =

So that the mass average velocity defined by 𝜌 huav = Q is given by ( ) dp h2 uav = − 12𝜇 dx

(6.6.21)

(6.6.22)

(6.6.23)

Case 2 Non-Newtonian Fluid

Because in the Non-Newtonian case the absolute value of the shear rate has to be used, the regions with du∕dy > 0 and du∕dy < 0 have to be treated separately (Why?). However, since the velocity distribution must be symmetric about the centerline y = h∕2 (Why?), it is sufficient to focus on the region 0 ≤ y ≤ h∕2 in which du∕dy > 0, so that no adjustment of sign is necessary. Since the pressure gradient dp(x)∕dx is a constant, an integration of Eq. 6.4.6 across the region 0 ≤ y ≤ h∕2 gives dp (y + A) dx in which A is a constant of integration that, on using the boundary condition 𝜏 = 0 at y = h∕2 (Why?) results in ) dp ( h 𝜏= −y (6.6.24) dx 2 Then, by using the expression for 𝜏 Eq. 6.5.2 it follows (Why?) that

𝜏=

du = dy

( )1 ( )1n 1 dp n h − −y K dx 2

(6.6.25)

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An integration of which gives ] ( )1 [( )n+1 n 1 dp n h n −B −y − u= n+1 K dx 2 in which the constant of integration is determined by imposing the boundary condition u = 0 at y = 0, finally resulting in ] ( )1 [( )n+1 ( )n+1 n 1 dp n h n n h , 0 ≤ y ≤ h∕ 2 (6.6.26) − −y − u= n+1 K dx 2 2 The maximum velocity, which occurs at y = h∕2, is then given by ( )1 ( )n+1 n 1 dp n h n umax = (6.6.27) − n+1 K dx 2 From Eq. 6.6.21, the magnitude of the maximum shear stress, which occurs at y = 0 and y = h, is then given by h | dp | 𝜏max = || || (6.6.28) 2 | dx | 6.6.6

Pressure-Driven Radial Flow

Another class of basic flows with only one velocity component is axisymmetric radial flow, the model geometry for which is shown in Figure 6.6.3. This type of flow is generated by injecting fluid through a hole in one surface of two parallel plates. In the continuing flow, the fluid advances across a circular flow AXISYMMETRIC RADIAL FLOW-FRONT

INFLOW

H

y

R(t) r

(a)

(b)

Figure 6.6.3 Axisymmetric radial flow. (a) Top view showing radial flow field and axisymmetric flow front generated between two parallel plates by a fluid injected into a hole. (b) Side view of a section through the hole H showing division of the flow in radial directions. The flow front at time t has advanced to a radius R (t).

Models for Liquids

front such that each point moves in a radial direction (Figure 6.6.3a). The axial flow entering through the hole H in Figure 6.6.3b divides into flow in the radial directions. As such, the flow is truly radial only away from the injection point r = 0; the flow near r = 0 will be axisymmetric two-dimensional and not radial. In this figure, the flow front has been shown to have advanced to a radius R (t). Because of symmetry, this axisymmetric flow will only have one velocity (radial) component and only one component of the (radial) shear stress. With the origin of the radial coordinate system at the center of the bottom plate, and the y-coordinate measured upward from this origin, so that the lower and upper plates are at y = 0 and y = h, respectively, the velocity field has the components vr = u(r, y, t), v𝜃 ≡ 0, and vy ≡ 0; the only non-zero component of the shear stress is 𝜏 yr = 𝜏 (r, y, t). 6.6.6.1 Continuity Equation for Radial Flow

Inside the channel shown before, consider the balance of fluid mass passing through a volume element ABCD within a small angle Δ𝜃 , radii r and r + Δr, and height y and y + Δy, shown in Figure 6.6.4. Then, with the radial lengths AB = CD = Δr, height AD = BC = Δy, the volume of this radial element

D

C

A

B

∆y

y

r

O

∆r r + ∆r (a)

F

E

ρ(∆y r∆θ )(v r ∆t) O

∆θ

ρ(∆y r∆θ )(vr ∆t)

+ D

∂ [ ρ(∆y r∆θ )(vr ∆t)] ∆r ∂r

C (b)

Figure 6.6.4 Radial flow through a small volume element. (a) Front view of a small volume element ABCD inside a channel. (b) Top view of a radial volume element with radial flow through.

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is given by ΔV = rΔ𝜃 Δr Δy. Because of axisymmetry, all the flow occurs only in the radial direction, so that the only non-zero component of velocity is vr = u(r, y, t). Then, in time Δt, the mass of fluid entering the element through the side AD is 𝜌 (u Δt) (rΔ𝜃 Δy) (Why?), and the mass leaving the side BC is 𝜌 (u Δt) (rΔ𝜃 Δy) + {𝜕 [ 𝜌 (u Δt) (rΔ𝜃 Δy)]∕𝜕 r}Δr, so that the net mass leaving this element across the sides AD and BC equals {𝜕 [ 𝜌 (u Δt) (rΔ𝜃 Δy)]∕𝜕 r}Δr = (Δr Δ𝜃 Δy Δt)[𝜕 ( 𝜌 ru)∕𝜕 r] (Why?). At time t the mass of fluid in the element ABCD is 𝜌 ΔV, and that at time t + Δt must be [ 𝜌 + (𝜕𝜌∕𝜕 t) Δt] ΔV, so that the net increase in mass in time Δt equals (𝜕𝜌∕𝜕 t) Δt ΔV = (Δr Δ𝜃 Δy Δt) r (𝜕𝜌∕𝜕 t) that, because of conservation of mass, must equal the negative (Why?) of the net mass leaving the element during this time interval. By equating these two sets of quantities, it follows that 𝜕𝜌 𝜕 ( 𝜌 ru) (Δr Δ𝜃 Δy Δt) r = − (Δr Δ𝜃 ΔyΔt) 𝜕t 𝜕r so the continuity equation is

𝜕𝜌 1 𝜕 + ( 𝜌 r u) = 0 𝜕 t r 𝜕r

(6.6.29)

For an incompressible fluid, this one-dimensional continuity equation for radial flow reduces to

𝜕 (r u) = 0 𝜕r

(6.6.30)

which has the solution vr (r, y, t) = u(r, y, t) =

f (y, t) r

(6.6.31)

Notice the singularity at r = 0, at which the radial velocity is infinite. 6.6.6.2

Balance of Linear Momentum in Radial Flow

Now consider the balance of linear momentum for the radial-flow material element ABCD shown in Figure 6.6.5a, which is only acted upon by a pressure p and one component shear stress 𝜏 . Because of axisymmetry, momentum balance only needs to be considered in the r- and y-directions. Radial-Direction Momentum Balance The pressure p on the face AD generates a rightward act-

ing force p (rΔ𝜃 Δy) on this material element. The leftward acting force on face BC is then p (rΔ𝜃 Δy) + {𝜕 [ p (rΔ𝜃 Δy)]∕𝜕 r}Δr so that net rightward force acting on the element due to the pressure forces on faces AD and BC is − {𝜕 [ p(rΔ𝜃 Δy)]∕𝜕 r}Δr = − (Δr Δ𝜃 Δy) [𝜕 (rp)∕𝜕 r] (Why?). In addition, the pressure p acting on the lateral faces CD and EF (Figure 6.6.5b) also generate normal forces p (Δr Δy). Each of these two forces contribute p (Δr Δ𝜃 Δy)∕2 – total force of p (Δr Δ𝜃 Δy) – to the rightward forces acting on this material element. (Notice that the vertical components of the normal forces on the faces CD and EF cancel out.) The net rightward force acting on this material element due to the pressure forces is then (Δr Δ𝜃 Δy)[ p − 𝜕 (rp)∕𝜕 r] = − (Δr Δ𝜃 Δy) (r 𝜕 p∕𝜕 r). Next consider the rightward force contributed by the shear stress 𝜏 acting on the faces AB and CD: The shear stress 𝜏 on the face AB generates a leftward acting force (rΔ𝜃 Δr) 𝜏 . The rightward acting force on face DC is then (rΔ𝜃 Δr) 𝜏 + {𝜕 [(rΔ𝜃 Δr) 𝜏 ]∕𝜕 y}Δy, so that the net rightward force acting on the element due to the shear stresses on the faces AB and CD is [𝜕 {(rΔ𝜃 Δr) 𝜏 }∕𝜕 y] Δy = (Δ𝜃 Δr Δy) r 𝜕𝜏 ∕𝜕 y. It follows that the net r-direction force on the fluid element ABCD due to the combined effects of pressure and shear stresses is (r Δr Δ𝜃 Δy) (− 𝜕 p∕𝜕 r + 𝜕𝜏 ∕𝜕 y).

Models for Liquids

p(rΔθΔr) ∂ [ p(rΔθΔr)] Δy + ∂y

τ (rΔθΔr) ∂ [τ(rΔθΔr)]Δy + ∂y τ (rΔθΔy)

C

D

p(rΔθΔy) ∂ [ p(rΔθΔy)]Δr + ∂r

Δy p (rΔθΔy)

τ (rΔθΔr)

y

O

B

A

τ (rΔθΔy) ∂ [τ(rΔθΔy)]Δr + ∂r

p (rΔθΔr) Δr

r

(a) p (rΔθΔy)/2 p(ΔrΔy)

G E

H

D

H G

K

Δθ/2 F

C K

p (rΔθΔy)/2 (b) Figure 6.6.5 Normal pressure and shear stresses acting on a small radial fluid element.

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Now the mass of material element ABCD is 𝜌 (rΔ𝜃 Δy Δy). Let the r-component of the acceleration of the material element ABCD be ar . Then, by applying the principle that the mass times the acceleration in any direction must equal the force in that direction to the r-direction components results in 𝜌 (rΔ𝜃 Δy Δy) ar = − (r Δr Δ𝜃 Δy) (𝜕 p∕𝜕 r + 𝜕𝜏 ∕𝜕 y). It follows that the equation of motion for the radial direction is 𝜕 p 𝜕𝜏 + (6.6.32) 𝜌 ar = − 𝜕r 𝜕y Vertical-Direction Momentum Balance Next consider the balance of forces in the balance of forces

in the y-direction: The pressure forces on faces AB and CD contribute forces p (rΔ𝜃 Δr) and − p (rΔ𝜃 Δr) − {𝜕 [ p(rΔ𝜃 Δr)]∕𝜕 y}Δy, respectively, resulting in a net upward force of − {𝜕 [ p (rΔ𝜃 Δr)]∕𝜕 y}Δy = − (rΔ𝜃 Δr Δy) 𝜕 p∕𝜕 y. The shear stresses acting on faces AD and BC of the material element contribute vertical forces 𝜏 (rΔ𝜃 Δy) and 𝜏 (rΔ𝜃 Δy) + {𝜕 [𝜏 (rΔ𝜃 Δy)]∕𝜕 r}Δr, resulting in a net upward force of {𝜕 [𝜏 (rΔ𝜃 Δy)]∕𝜕 r}Δr = (Δr Δ𝜃 Δy) 𝜕 (r𝜏 )∕𝜕 r. It follows that the net y-direction force on the fluid element ABCD due to the combined effects of pressure and shear stresses is (Δr Δ𝜃 Δy)[− r 𝜕 p∕𝜕 y + 𝜕 (r𝜏 )∕𝜕 r]. Let the acceleration in the y-direction be ay . It then follows that the equation of motion in the y-direction is (Why?) 𝜕p 1 𝜕 + (r𝜏 ) 𝜌 ay = − (6.6.33) 𝜕y r 𝜕r Since for radial flow vy ≡ 0, it follows that ay ≡ 0, so that

𝜕p 1 𝜕 = (r𝜏 ) 𝜕y r 𝜕r 6.6.6.3

(6.6.34)

Incompressible Newtonian Radial Flow

For Newtonian flow 𝜏 = 𝜇 𝜕 u∕𝜕 y, so that from Eq. 6.6.31 r𝜏 = 𝜇 𝜕 (ru)∕𝜕 y = 𝜇 𝜕 f (y)∕𝜕 y. It then follows from Eq. 6.6.34 that 𝜕 p∕𝜕 y = (𝜇 ∕r) 𝜕 [𝜕 f (y, t)∕𝜕 y]∕𝜕 r = 0, which implies p = p(r, t)

(6.6.35)

The complexities of radial flow are best shown through the two special cases of steady flow and creeping flow. Steady Radial Flow In steady flow time-dependence is absent. And, because it can be shown that ar =

u (𝜕 u∕𝜕 r), Eq. 6.6.32 can be written as dp 𝜇 𝜕 2 u 𝜕u = − 𝜌u + dr r 𝜕 y2 𝜕r where, for this case, p = p(r) and ru = f (y). Eq. 6.6.36 can then be written as

(6.6.36)

d2 f 𝜌 f2 r dp + = (6.6.37) 𝜇 dr dy2 𝜇 r2 If the pressure distribution, p = p(r), were known, Eq. 6.6.37 would still be a nonlinear differential equation for f = f (y). However, p = p(r) is not known a priori and has to be determined as a part of the solution. This demonstrates how complex pressure-driven radial flows are in comparison to the equivalent problems for a rectangular geometry, for which it is possible to obtain closed-form solutions (Section 6.6.5).

Models for Liquids

Creeping Flow Approximation Because of the very high viscosities of polymer melts, the acceleration term

on the left-hand-sides of the momentum balance equations are very small in comparison to the force terms (right-hand-sides). In the creeping flow approximation – which should apply to polymer melts – the acceleration terms are set to zero. For the special case of radial flow being considered, the radial momentum balance equation (Eq. 6.6.32) then has the form 𝜕p 𝜕𝜏 =− (6.6.38) 𝜕y 𝜕r with p = p (r, t) and ru = f (y, t). It follows that Eq. 6.6.38 can be written as

𝜇

𝜕p 𝜕2 u , = 2 𝜕y 𝜕r

or as

𝜕 2 f (y, t) r 𝜕 p(r, t) = 2 𝜇 𝜕r 𝜕y

(6.6.39)

Note that time, which normally enters the momentum balance equations through the time rate of change of density or velocity in the inertia terms (acceleration term on the left-hand side of the momentum balance equations), is not present in the creeping flow approximation because that term is set to zero in this approximation. But time does appear in an implicit way. Because of the absence of the explicit presence of time, Eq. 6.6.39 can be integrated without considering any time variations. Then the equation to be integrated is d2 f r dp = = C(t) (6.6.40) 2 𝜇 dr dy in which the two left-most terms in this equation have been set equal to a function, C(t), of time because, while the left-most term in this equation is only a function of the vertical position and time, the second (pressure) term is only a function of the radius and time. An integration of this equation results in the general solution ( 2 ) y f (y, t) = C(t) + Ay + B 2 where A and B are constants of integration. The boundary condition u(y, t) = 0 at y = 0 requires that B = 0. And u(y, t) = 0 at y = h results in A = − h∕2. It follows that ) y h2 C(t) ( 𝜂 − 𝜂2 , 𝜂 = (6.6.41) f (y, t) = − 2 h Let Q (t) be the volumetric flow through the radial gap. Then (Why?) h

1

𝜋 h3

6 (6.6.42) C(t) ⇒ C(t) = − 3 Q (t) ∫0 ∫0 6 𝜋h so that from Eq. 6.6.40 dp 6𝜇 1 (6.6.43) = − 3 Q (t) r dr 𝜋h Finally, from Eqs. 6.6.41 and 6.6.42 f (y, t) y(h − y) 3 u(y, t) = = (6.6.44) Q (t) 3 r r 𝜋h Thus, for this creeping flow, the velocity variation at any radius is parabolic across the thickness, but varies inversely with the radius. And the magnitude of the velocity is directly proportional to the flow through Q (t), through which time is introduced in an implicit manner. The pressure drop varies as the inverse of the radius, and is directly proportional to the through-flow Q (t). Q (t) =

2𝜋 ru dy = 2𝜋 h

f (𝜂 , t) d𝜂 = −

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Estimate for Mold Fill Time Consider the flow of molten plastic through a hole into the space between two parallel plates (Figure 6.6.3). Right at the entrance to the hole the initially axial inlet flow will divide into all radial directions, thereby forming an axisymmetric flow front. At a small distance from r = 0, say at r = rG , (the subscript G suggesting gate), the flow will become radial. Let the pressure at this radius be pG . At time t let the flow front have advanced to radius R(t), at which in a vented mold the pressure may be assumed to be zero (atmospheric pressure). Then, from and integration of Eq. 6.6.43 it follows that 0

∫p

dp = − G

R (t) 6𝜇 dr Q (t) r ∫r 𝜋 h3 G

which results in pG =

[ ] 6𝜇 R(t) Q (t) ln , rG 𝜋 h3

or

Q (t) =

𝜋 h3 pG

(6.6.45)

6𝜇 ln [R(t)∕rG ]

Now the volume of melt contained within this flow front will be V(t) = 𝜋 R2 (t)h. It then follows from Q (t) = dV(t)∕dt and Eq. 6.6.45 that Q (t) =

𝜋 h3 pG dV(t) dR(t) = 2𝜋 hR = dt dt 6𝜇 ln [R(t)∕rG ]

which can be written as [ ] dR(t) h2 pG d𝜉 R ln R(t)∕rG = rG 2 𝜉 ln 𝜉 , = dt 12𝜇 dt

R(t)

𝜉= r G

(6.6.46)

Let the time to fill, or fill to the radius R(tF ), be tF . Then, from Eq. 6.6.46, R(tF )∕rG h2 pG tF dt = rG 2 𝜉 ln 𝜉 d𝜉 ∫1 12𝜇 ∫0

which, on using the result ∫

𝜉 ln 𝜉

d𝜉 𝜉2 = (2 ln 𝜉 − 1) dt 4

finally gives TF =

tF 1 = 2 ln 𝜉 − 1 + 2 , (3𝜇 R2 )∕(h2 pG ) 𝜉

where

R(t )

𝜉 = rF G

(6.6.47)

The variation of the nondimensional time TF versus the nondimensional fill radius 𝜉 = R(tF )∕rG is shown in Figure 6.6.6. Clearly, for 𝜉 greater than about 100, TF varies very slowly with increases in 𝜉 : it varies from TF = 8.2 for 𝜉 = 100, to TF = 11.4 for 𝜉 = 500. Using a value of TF = 10, an estimate for the time to fill up to a radius R is given by tF ≈ 30

𝜇 R2 h2 pG

(6.6.48)

NONDIMENSIONAL FILL TIME TF

Models for Liquids

15

10

TF =

5

tF

(3μ

R 2 ) / (h 2 pG )

= 2 ln ξ − 1 +

1

ξ

2

0 0

100

200

300

400

500

NONDIMENSIONAL FILL RADIUS ξ Figure 6.6.6 Variation of the nondimensional time TF to fill up to nondimensional fill radius 𝜉 = RF ∕rG .

6.7 Polymer Rheology Many polymer melts and solutions exhibit very complex Non-Newtonian behavior beyond that discussed in Section 6.5.2. First, they can exhibit viscoelasticity in which the stress depends both on the strain – as in solids – and on the strain rate – as in viscous fluids. Such viscoelastic fluids, the subject of Chapter 7, exhibit phenomena not shown by flows of water and air. Second, the nonlinear dependence of the stress on finite, large strain rates results in “normal stress” effects, not present in the linear models appropriate for water and air: in the linear models, the three normal stresses are equal, so that the differences among any two of them is the same, namely zero. But in the nonlinear model the differences among any two of the three normal stresses is not the same. The understanding and modeling of such “anomalous” flow behavior is covered in polymer rheology – the fluid mechanics of polymer flows – a very important topic that is beyond the scope of this book. Some examples of such “anomalous” behavior are described in the next few subsections. 6.7.1

Die Swell

An important polymer processing method is extrusion, in which a melt is forced through a die. In many polymer melts and solutions, the extrudate swells considerably, that is, the diameter of the extrudate can be much larger than that of the die orifice: Figure 6.7.1 shows a fluid exiting at the die face BC after flowing in the passage ABCD. On exiting at BC, the extrudate diameter increases to a maximum at EH and then decreases as shown. Because of the zero-velocity condition at the walls, the velocity distribution in the flow path ahead of the die has the near parabolic shape shown. On exiting from the die, the lateral fluid surfaces are no longer subject to any constraints so that the velocity distribution throughout the extrudate BEFGHC is uniform; the velocity being lower at larger diameters.

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A

D

B

C H

E

VELOCITY DISTRIBUTION ALONG EH

F

G

Figure 6.7.1 Die swell in an extrudate.

This phenomenon, called die swell, or extrudate swell, can be attributed to the viscoelasticity of the fluid: The compressed viscoelastic liquid in the flow path ahead of the die is under large compressive stresses that are relieved on exiting the die, resulting on die swell. Higher extrusion rates cause higher compressive stresses that result in larger die swell. 6.7.2

Tubeless Siphon

When a nozzle on a tube connected to a vacuum source is brought in contact with a liquid surface, it begins to suck fluid; this system is referred to as a siphon. In Newtonian liquids such as water, the siphon (suction of liquid) works only so long as the nozzle surface is in contact with the liquid surface, and suction stops when the nozzle surface is raised above the liquid surface. If the same experiment is done with a Non-Newtonian liquid, the siphoning action continues even when the nozzle surface is raised above the fluid free surface, as shown in Figure 6.7.2. Evidently, viscoelastic stresses resulting from the stretching of the fluid in the column are able to support the weight the fluid in between the nozzle tip and the free fluid surface. This system is referred to as a tubeless or open siphon. 6.7.3

Vibration of a Ball Dropped in a Liquid

When a small metal ball, such as a ball bearing, is dropped in a Newtonian liquid, after disrupting the fluid surface, possibly resulting in an ejection of fluid droplets, the ball smoothly sinks in the fluid. But in the case of a viscoelastic fluid, the ball first sinks, then vibrates within the fluid and then, after some time, sinks smoothly.

Models for Liquids

SUCTION

SUCTION TUBE

SUCTION TUBE TUBELESS SIPHON

(a)

(b)

Figure 6.7.2 (a) Nozzle applying suction on fluid surface. (b) Nozzle moved above surface.

6.7.4

Weissenberg Effect

After some time, a spinning rod placed in water in a beaker will cause the water to rotate about the rod axis. This rotation will cause the initially flat water surface to have the concave upward shape shown in Figure 6.7.3a. However, in some polymer melts and solutions, the rotating fluid begins to climb up the rotating rod and takes on the shape shown in Figure 6.7.3b. This phenomenon is called the Weissenberg effect, and is attributed to the nonlinear normal stress effect briefly mentioned before.

6.8 Concluding Remarks Clearly, even the simplest Newtonian, one-dimensional fluid-flow problems require more mathematics than for one-dimensional deformation problems of solids. While linear stress-strain models work well for solids, the stress-strain-rate relationships for polymeric materials are inherently nonlinear. And, because of the strong dependence of the viscosity of polymer melts on temperature, the momentum (flow) balance equations are coupled to the energy equation. As a result, in contrast to the fluid mechanics of water and air – which are essentially Newtonian fluids – the flow of polymer melts is governed by coupled nonlinear equations. In the fluid mechanics of water and air, while the terms representing the work done by stresses – which are linearly related to the strain rates – are linear, the terms representing the inertia of motion give rise to nonlinearities that, among other effects, are responsible for modeling the phenomenon of flow turbulence.

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WEISSENBERG EFFECT

ROTATING ROD

(a)

(b)

Figure 6.7.3 (a) Rod spinning in a Newtonian liquid. (b) Weissenberg effect.

In contrast to this, in the flow of polymer melts the inertia terms are insignificant in comparison to terms representing the work done by the stresses, in which the highly nonlinear stress-strain-rate relationships result in the flow equations being nonlinear. Another difference relates to polymer melts exhibiting elastic effects that are not present in water and air. The simultaneous modeling of elastic effects – which use strain – and viscous flow effects – which involve the use of strain rate effects – requires even more complex mathematics. The simplest model for such materials, called linear viscoelasticity, is discussed in Chapter 7. There is another important difference between the normal flow problems associated with water and air and those for polymer processing problems: Most problems relating to the flow of water and air are steady-state problems, in which the flow does not vary with time. Examples of such fully developed flows are the one-dimensional, surface-driven and pressure-driven flows discussed, respectively, in Sections 6.6.1 and 6.6.4. In contrast to this, many important flows of polymers, such as those associated with injection molding, are developing flows that require the solution of time-dependent differential equations. As a result, only very highly idealized flow problems associated with polymer melts can be solved in closed form. Most meaningful problems require the use of numerical codes that, in addition to the nonlinear behavior of polymers, can handle the complex geometries through which such flows occur.

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7 Linear Viscoelasticity 7.1 Introduction Chapter 4 discussed simple models for solids – in which the stress is determined by the strain – and Chapter 6 discussed fluids – in which the stress is determined by the strain rate. Both these sets of models for solids and fluids are time-independent in the sense that the stress at any instant of time is determined, respectively, by the values of the strain and the strain rate at that instant of time; the response of the material to any stimulus (stress or deformation) is instantaneous. In essence, the past histories of the strain and strain rate have no effect on the current stress. Many materials, such as polymers, exhibit both solid- and fluid-like behavior: Solid-like in the sense that the sudden imposition of a stress results in an instantaneous strain, and fluid-like in the sense that even when the stress is maintained at a constant value, the strain continues to change with time, that is, the material “flows.” In such materials the stress can depend both on the deformation (strain) – as in elastic and plastic solids – and the deformation rate (strain rate) – as in fluids. Such materials exhibit time-dependent behavior in which the response of the material to an applied stress or deformation is time-dependent. Since such materials exhibit both elastic and viscous flow behavior, they are generically referred to as viscoelastic materials. Viscoelastic materials comprise the subset of time-dependent materials in which elastic and viscous effects dominate; permanent set as in plasticity are not accounted for. Viscoelasticity is far more complex than elasticity and fluid flow – both in terms of the phenomenology as well as in its mathematical structure. This chapter discusses the simplest model, called linear viscoelasticity, for such materials. Even the simplest models for viscoelasticity require the use of more mathematics than those for elasticity and fluid flow. But because viscoelasticity is important to many facets of plastics – such as materials characterization, long-term structural part performance, processing, and residual stresses and shrinkage and warpage – it is necessary to develop a feel for the underlying structure of the theory and its applications. An effort has been made to provide understandable, qualitative explanations to elucidate theoretical concepts. This chapter provides simple mathematical models for understanding the phenomenology of the time-dependent behavior of polymeric materials discussed in Chapter 14.

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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7.2 Phenomenology of Viscoelasticity Two types of model experiments are useful for understanding the behavior of time-dependent materials. In the first, called a stress-relaxation experiment, a constant deformation is imposed on an otherwise unstressed body and the resulting stress is monitored. In the second, called a creep experiment, a constant stress is imposed on the body and the evolution of deformation with time is monitored. 7.2.1

Stress Relaxation

Consider an unstressed body on which a constant deformation is imposed at time zero. For simplicity, only the one-dimensional case shown in Figure 7.2.1a, in which the imposed deformation is either a constant normal strain 𝜀0 or a constant shear strain 𝛾 0 , is considered. As discussed in Chapter 6, the most important deformation mode for fluids is shear deformation, which also could have been used to describe the behavior of solids. As such, although the deformation of solids usually is discussed in terms of normal strains, in Figure 7.2.1a both normal and shear strains are considered as alternative stimuli. The time history of this strain – 𝜀(t) = 0 for t < 0, and 𝜀(t) = 𝜀0 for t ≥ 0 – can conveniently be written as 𝜀(t) = 𝜀0 H(t), where H(t) is the Heaviside unit step function: H(t) = 0 for t < 0, and H(t) = 1 for t ≥ 0. Because this function has constant values for t < 0 and for t > 0, its derivative in this time domain is zero. However, since H(t) jumps from 0 at t = 0− to 1 at t = 0, its time derivative at t = 0 is infinite. This time derivative of H(t) with a “spike” at t = 0, written as H ′(t) = 𝛿 (t), is called Dirac’s delta function, which has the following properties: +∞

∫−∞

f (t) 𝛿 (t − t0 ) dt = f (t0 ),

+∞

∫−∞

𝛿 (t) dt = 1

(7.2.1)

The strain rate corresponding to this step change in deformation shown in Figure 7.2.1b can then be written as 𝛾̇ (t) = 𝛾0 𝛿 (t). The stress history 𝜎 (t) = E 𝜀0 H(t) for an elastic solid, corresponding to 𝜀(t) = 𝜀0 H(t), is shown in Figure 7.2.1c. And the stress history 𝜏 (t) = E 𝛾 0 𝛿 (t) for a fluid, for 𝛾 (t) = 𝛾 0 H(t), is shown in Figure 7.2.1d; the “spike” in the stress is caused by the infinite strain rate at t = 0. So, a step change in strain causes a step change in the stress in an elastic solid and a “spike” in the stress in a fluid. In some materials the response to constant strain is time-dependent. With reference to Figure 7.2.1e, the step change in strain first causes an instantaneous increase in stress – just as in an elastic material. But the stress then begins to decay, or relax, with time. This phenomenon is called stress relaxation. The material is said to be a viscoelastic solid if the stress stabilizes to a constant value (Figure 7.2.1e). It is called a viscoelastic fluid if the stress relaxes to zero, as shown in Figure 7.2.1f. These are idealized models for material behavior, and the distinction between a viscoelastic solid and a viscoelastic fluid is somewhat arbitrary. Over sufficiently long periods of time, a solid can behave as a fluid. And over very short time scales the behavior of a fluid will be like that of a solid. 7.2.2

Creep

Next consider an unstressed body on which a constant stress is imposed at time zero. In the one-dimensional case shown in Figure 7.2.2a, the imposed stress is either a constant normal stress 𝜎 0 or a constant shear stress 𝜏 0 . In this figure, the response of solids and liquids are described, respectively, in terms of normal and shear strains.

Linear Viscoelasticity

ε0 or γ0

•

ε = ε0 H(t)

ε· = ε0 δ (t)

γ = γ0 H(t)

γ· = γ0 δ (t)

t

O

t

O

Strain

Strain Rate

(a)

(b) Stimulus

σ (t) = E ε0 H(t)

σ0 Stress

τ (t) = μγ0 δ (t) Stress

t

O

t

O

Elastic Solid

Viscous Fluid

(c)

(d) Time-Independent Response

σ0

σ (t) = ε0 E (t)

Stress

τ0

τ (t) = γ0 G (t)

Stress t

O

t

O

Viscoelastic Solid

Viscoelastic Fluid

(e)

(f) Time-Dependent Response

Figure 7.2.1 (a) Imposed strain history. (b) Corresponding strain rate history. (c) Time-independent stress developed in an elastic solid. (d) Time-independent stress developed in a viscous fluid. (e) Time-dependent stress developed in a viscoelastic solid. (f) Time-dependent stress developed in a viscoelastic fluid.

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σ = σ0 H(t)

σ0 or τ0

τ = τ0 H(t)

Stress

t

O Stimulus (a)

ε (t) = (σ0 /E ) H(t)

ε0

γ (t) = (τ0 / μ )t Strain

Strain t

O

t

O

Elastic Solid

Viscous Fluid

(b)

(c)

Time-Independent Response

ε (t) = σ0 C(t) Strain

γ (t) = τ0 J(t) Strain

t

O

t

O

Viscoelastic Solid

Viscoelastic Fluid

(d)

(e) Time-Dependent Response

Figure 7.2.2 (a) Imposed stress history. (b) Resulting time-independent strain history in an elastic solid. (c) Resulting time-independent strain history a viscous fluid. (d) Resulting time-dependent strain history in a viscoelastic solid. (e) Resulting time-dependent strain history in a viscoelastic fluid.

The time history of this stress can be written as 𝜎 (t) = 𝜎 0 H(t) or as 𝜏 (t) = 𝜏 0 H(t). For this stress history, the strains 𝜀(t) = (𝜎0 ∕E ) H(t) in an elastic solid and 𝛾 (t) = (𝜏0 ∕𝜇 ) t H(t) in a viscous fluid are shown, respectively, in Figures 7.2.2b,c. In a viscoelastic material, on the application of a step change in stress, the strain first increases instantaneously, just as in an elastic solid, but then increases with time. This time-dependent increase in strain

Linear Viscoelasticity

is called creep. In a viscoelastic solid the strain stabilizes to a constant value (Figure 7.2.2d). In a viscoelastic fluid the strain continues to increase with time (Figure 7.2.2e). As with stress relaxation, the long-term behavior of strain depends on the time scale under consideration.

7.3 Linear Viscoelasticity In idealized linear elastic solids, the linear dependence of the stress on the strain implies that the ratio of the applied stress to the strain, called the elastic modulus, E (or G ), is independent of the applied stress level. And the ratio of the applied strain to the resulting stress, called the compliance, C (or J ), is independent of the level of strain imposed. Furthermore, the modulus and the compliance are reciprocals of each other; that is, C = 1∕E and J = 1∕G. In a similar manner, in a linear viscoelastic material the ratio of the constant stress applied in a stress-relaxation experiment to the resulting strain is independent of the applied stress. This time-dependent ratio, shown as G (t) in Figure 7.3.1a, is called the relaxation modulus of the material. In such materials, the ratio of the constant strain applied in a creep experiment to the resulting stress is also independent of the imposed strain. This time-dependent ratio, shown as J (t) in Figure 7.3.1b,

Gg

G(t)

Je

J(t) Jg

Ge tr

O

t

t

O

Linear Stress Relaxation Modulus

Linear Creep Compliance

(a)

(b) Viscoelastic Solid

G(t)

Gg

J(t)

η0

Jg O tr

t

O

t

Linear Stress Relaxation Modulus

Linear Creep Compliance

(c)

(d) Viscoelastic Fluid

Figure 7.3.1 (a) Linear stress-relaxation modulus for a viscoelastic solid. (b) Linear creep compliance for a viscoelastic solid. (c) Linear stress-relaxation modulus for a viscoelastic fluid. (d) Linear creep compliance for a viscoelastic fluid.

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is called the creep compliance of the material. In contrast to elastic solids, it later will be shown that J(t) ≠ 1∕G(t). In the stress-relaxation modulus for a viscoelastic solid, shown in Figure 7.3.1a, the value of G (0), which describes the initial instantaneous response, is shown as Gg = G (0). The subscript g refers to the “glassy” modulus of polymers, the significance of which is discussed in Chapters 10 and 14. The long-time value of the relaxation modulus is written as Ge = G(∞). Similarly, for the creep compliance, Jg = J(0) and Je = J(∞). How rapidly G (t) decays from Gg to Ge depends on the initial slope G ′ (0) of the relaxation modulus. The intercept of the tangent to the relaxation curve on the time axis is called the relaxation time, tr , of the material. From the geometry shown in Figure 7.3.1a it follows that tr = − Gg ∕G ′(0), so that a larger initial slope corresponds to a shorter relaxation time. In the relaxation modulus and the creep compliance for a viscoelastic fluid, shown in Figure 7.3.1 parts c and d, respectively, Gg , Je , and tr play a similar role. Simple, intuitive arguments can be used for relating some of the parameters defining the relaxation modulus and the creep compliance: When a step change in strain is imposed in a stress-relaxation experiment, the instantaneous response of the material is given by 𝜏 (0) = Gg 𝛾 (0). Since this instantaneous response should be independent of the future strain history, this instantaneous response should be the same as when a step change in stress is applied; that is, the strain 𝛾 (0) = Jg 𝜏 (0) corresponding to the stress 𝜏 (0) in a creep experiment should be the same as stress 𝜏 (0) corresponding to the imposed strain 𝛾 (0) in a stress-relaxation experiment. It then follows from 𝜏 (0) = Gg 𝛾 (0) = Gg Jg 𝜏 (0) that Gg Jg = 1. Similarly, if the stress and the strain at very long times approach constant, limiting values 𝛾 (∞) and 𝜏 (∞), respectively, the material will not “remember” whether these values were obtained by either holding the strain constant at 𝛾 (∞) (stress-relaxation experiment) or holding the stress constant at 𝜏 (∞). It then follows from 𝜏 (∞) = Ge 𝛾 (∞) = Ge Je 𝜏 (∞) that Ge Je = 1. The discussion on linear viscoelasticity thus far has focused on the response of the material to step changes in deformation or stress histories. This discussion needs to be extended to the responses to time-varying strain and stress histories. 7.3.1

Constitutive Equations

Just as in the small-deformation (strain) theory of elastic solids, it can be shown that for small deformations the stress caused in a linear viscoelastic material by two step increases of deformation is the sum of the stresses resulting from the individual strain increments. Thus for step changes 𝛾 1 and 𝛾 2 , applied, respectively, at times t1 and t2 ,

𝛾 (t) = H(t − t1 ) 𝛾1 + H(t − t2 ) 𝛾2

⇒

𝜎 (t) = G(t − t1 ) 𝛾1 + G(t − t2 ) 𝛾2

and more generally, for n steps in strain

𝛾 (t) =

𝛼∑ =n 𝛼 =1

H(t − t𝛼 ) 𝛾𝛼

⇒

𝜎 (t) =

𝛼∑ =n 𝛼 =1

G(t − t𝛼 ) 𝛾𝛼

By considering infinitesimally small strain increments, d𝛾 , it follows in the limit that

𝛾 (t) = H(t − 𝜉 ) d 𝛾 (𝜉 ) ∫

⇒

𝜎 (t) = G(t − 𝜉 ) d 𝛾 (𝜉 ) ∫

This basic linear constitutive equation can be written in two equivalent ways.

(7.3.1)

Linear Viscoelasticity

7.3.2

Stress-Relaxation Integral Form

In this form, the equation

𝜎 (t) =

t

∫−∞

G(t − 𝜉 ) d 𝛾 (𝜉 )

(7.3.2)

can be expressed in several alternative forms

𝜎 (t) =

t

∫−∞

G(t − 𝜉 ) 𝛾 ′(𝜉 ) d 𝜉 =

∞

∫0

G(𝜉 ) 𝛾 ′(t − 𝜉 ) d 𝜉

t

=

d G(t − 𝜉 ) 𝛾 (𝜉 ) d 𝜉 dt ∫−∞

= Gg 𝛾 (t) +

7.3.3

t

∫−∞

G ′ (t − 𝜉 ) 𝛾 (𝜉 ) d 𝜉

(7.3.3)

Creep Integral Form

In this form,

𝛾 (t) =

t

∫−∞

J(t − 𝜉 ) d 𝜎 (𝜉 )

(7.3.4)

For tensile response, in term of the tensile stress-relaxation modulus E(t) and the tensile creep compliance D(t), the stress-relaxation integral and the creep integral equations have the form

𝜎 (t) =

7.3.4

t

∫−∞

E(t − 𝜉 ) d 𝜀(𝜉 ),

𝜀(t) =

t

∫−∞

D(t − 𝜉 ) d 𝜎 (𝜉 )

(7.3.5)

*Relationship Between the Relaxation Modulus and the Creep Compliance

When Jg ≠ 0, it can be shown that t

G(t) =

1 1 − J ′ (t − 𝜉 ) G(𝜉 ) d 𝜉 Jg Jg ∫0

(7.3.6)

If J(t) is known, then this integral equation can be solved to determine G(t). Similarly, if G(t) is given and Gg ≠ ∞ , then J(t) can be obtained from the integral equation t

J(t) =

1 1 − G ′ (t − 𝜉 ) J(𝜉 ) d 𝜉 Gg Gg ∫0

(7.3.7)

By assuming that J(t) is an increasing function of t and that G(t) is a decreasing function of the time t, it can be shown that J(t) G(t) ≤ 1

(7.3.8)

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7.4 Simple Models for Stress Relaxation and Creep It is instructive to develop a feel for the relaxation modulus and the creep compliance by studying very simple models. An intuitive feel for such simple models is facilitated by considering the stiffness of two springs arranged in series or parallel. In a simple spring, the force, F, is related to the resulting extension, x, by F = k x, where k is the spring constant, or stiffness. Figure 7.4.1a show two springs, having stiffnesses k1 and k2 arranged in series. Clearly, while the extensions x1 and x2 in the two springs are different, each spring is subjected to the same force F = k1 x1 = k2 x2 , so that x1 = F∕k1 and x2 = F∕k2 . In terms of the total extension x = x1 + x2 of the two-spring system, its effective stiffness k is given by F∕k = x = x1 + x2 = F∕k1 + F∕k2 , from which it follows that 1 1 1 + = k k1 k2

(7.4.1)

Now the reciprocal 1∕ k of the spring stiffness is called its compliance. Thus, the compliance of two springs connected in series is the sum of their compliances. Next, consider two springs connected in parallel (Figure 7.4.1b). In this case, the two springs undergo the same extension x, but the forces in these springs are F1 = k1 x and F2 = k2 x, respectively. Then, from F = F1 + F1 = k1 x + k2 x = (k1 + k2 ) x, it follows that k = k1 + k2

(7.4.2)

x = x1 + x2

x1 F = k2 x 2

F = k1 x1 A

•

k1

•

F = kx

C

B k2 (a) x

F1 = k1 x

•

F=kx

F2 = k2 x (b) Figure 7.4.1 (a) Two linear springs connected in series. (b) Two linear springs connected in parallel.

Linear Viscoelasticity

Thus, the stiffness of two springs connected in parallel is the sum of their stiffnesses. This discussion on discreet springs can be extended to one-dimensional continuous systems. The continuum elements will be called springs and dashpots. 7.4.1

Continuum Elastic Element (Elastic Spring)

For a “continuum spring,” represented in Figure 7.4.2, the stress and deformation, related through 𝜏 (t) = G 𝛾 (t), the relaxation modulus and the creep compliance are related through G(t) = G H(t) J(t) = J H(t), where J = 1∕G

(7.4.3)

where G is the modulus of rigidity. Equivalently, in terms of the normal stress, strain, and Young’s modulus, for which 𝜎 (t) = E 𝜀(t), E(t) = E H(t) D(t) = D H(t), where C = 1∕E

(7.4.4)

where C is the compliance.

γ (t)

•

•

τ (t)

Figure 7.4.2 Continuum spring element.

7.4.2

Continuum Viscous Element (Dashpot)

In a viscous element, or a “dashpot,” represented in Figure 7.4.3a, the stress is linearly related to the strain rate through 𝜏 (t) = 𝜇 𝛾̇ (t). When a constant strain 𝛾 0 is imposed at time zero, the strain rate is 𝛾̇ (t) = 𝛿 (t) (Why?), so that 𝜏 (t) = G(t) 𝛾̇ (t) = 𝜇𝛿 (t) implies G(t) = 𝜇𝛿 (t). Also, the imposition of a constant stress 𝜏 0 , gives 𝛾̇ (t) = 𝜏0 ∕𝜇 , which on integration results in J (t) = (t∕𝜇 )H(t) (Why?). Thus, for a dashpot, G(t) = 𝜇 𝛿 (t) t H(t) J(t) =

𝜇

the variations of which are shown in Figure 7.4.3b,c.

(7.4.5)

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γ (t)

•

τ (t)

• (a)

J (t)

G (t)

μ t

O

t

O

(b)

1

(c)

Figure 7.4.3 Continuum viscous element, also called a dashpot. (a) Schematic representation. (b) Relaxation modulus. (c) Creep compliance.

7.4.3

Maxwell Model

The Maxwell model comprises a spring and viscous element in series (Figure 7.4.4). The argument used for springs in series – in which the system compliance is the sum of their compliances – can be extended to obtain the creep compliance and the relaxation modulus for the model. For this model, the spring and dashpot are subjected to the same stress 𝜏 (t), so that in terms of the strains 𝛾 1 (t) and 𝛾 2 (t) in these two elements, the stress and the total strain for this model are

𝜏 (t) = Gg 𝛾1 (t) = 𝜇 𝛾̇ 2 (t) 𝛾 (t) = 𝛾1 (t) + 𝛾2 (t)

γ1 (t)

(7.4.6)

γ (t) = γ1 (t) + γ2 (t)

γ2 (t)

• Gg

τ (t)

μ

Figure 7.4.4 Maxwell model: Continuum spring and dashpot elements connected in series.

Linear Viscoelasticity

7.4.3.1 Stress Relaxation

For a stress-relaxation experiment in which a constant strain 𝛾 (t) = 𝛾 0 is imposed on the system, 𝛾 1 (t) + 𝛾 2 (t) = 𝛾 0 , so that from the first of the previous two equations

𝛾̇ 2 (t) =

Gg

𝜇

𝛾1 (t) =

Gg

𝜇

[𝛾0 − 𝛾2 (t)] ,

or 𝛾̇ 2 (t) +

Gg

𝜇

𝛾2 (t) =

Gg

𝜇

𝛾0

(7.4.7)

which has the general solution

𝛾2 (t) = A exp [(−Gg ∕𝜇 ) t] + 𝛾0 where A is a constant that, with the initial condition 𝛾 2 (0) = 0, is given by A = − 𝛾 0 , so that

𝛾2 (t) = 𝛾0 [1 − exp (− Gg ∕𝜇 ) t] ,

𝛾1 (t) = 𝛾0 − 𝛾2 = 𝛾0 exp (−Gg ∕𝜇 ) t

(7.4.8)

Then, from 𝜏 (t) = Gg 𝛾 1 (t) and 𝜏 (t) = G(t) 𝛾 0 it follows that G(t) = Gg exp (− t∕tr )

(7.4.9)

in which tr = 𝜇 ∕Gg = 𝜇 Jg is called the relaxation time. 7.4.3.2 Creep

For a creep experiment, in which a constant stress 𝜏 (t) = 𝜏 0 is imposed on the system, from Eq. 7.4.6, 𝛾 1 (t) = (𝜏 0 ∕Gg ) H(t) = (𝜏 0 Jg ) H(t) (Why?) and 𝛾̇ 2 (t) = 𝜏0 ∕𝜇 , which results in 𝛾 2 (t) = (𝜏 0 ∕𝜇) t H(t), so that from J(t) = 𝛾 (t)∕𝜏 0 ( ) ( ) t t J(t) = Jg + H(t) = Jg 1 + H(t) (7.4.10) 𝜇 tr By using normalized time T = t∕tr = t∕(𝜇 Jg ) the expressions for the relaxation modulus and the creep compliance in Eqs. 7.4.9 and 7.4.10, respectively, reduce to G(T)∕Gg = exp (− T ) J(T)∕Jg = (1 + T) H(T )

(7.4.11)

the variations of which are shown in Figure 7.4.5.

7.4.4

Kelvin-Voigt Model

This model comprises a spring and viscous element in parallel (Figure 7.4.6). The argument used for springs in parallel – in which the system stiffness is the sum of their stiffnesses – can be extended to obtain the relaxation modulus and the creep compliance for this model.

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1.0 G (T ) = exp(− T ) Gg

3

G (T ) 0.5 Gg

2 J (T ) Jg

0

J (T ) = (1 + T ) H(t) Jg

1 0

0

1

3

2

4

5

0

1

2

T = t/tr

T = t/tr

(a)

(b)

3

4

Figure 7.4.5 Maxwell model. (a) Relaxation modulus. (b) Creep compliance.

Ge

γ (t) τ1 (t)

•

τ (t) = τ1 (t) + τ2 (t)

τ2 (t) μ Figure 7.4.6 Kelvin–Voigt model: Continuum spring and dashpot elements connected in parallel.

For this model,

𝜏1 (t) = Gg 𝛾1 (t) , 𝜏2 (t) = 𝜇 𝛾̇ 2 (t) 𝜏 (t) = 𝜏1 (t) + 𝜏2 (t) 7.4.4.1

(7.4.12)

Stress Relaxation

For a stress-relaxation experiment in which a constant strain 𝛾 (t) = 𝛾 0 is imposed on the system, 𝜏 1 (t) + 𝜏 2 (t) = 𝜏 0 , so that from the first of the previous two equations

𝜏1 (t) = Ge 𝛾 (t) ,

𝜏2 (t) = 𝜇 𝛿 (t) (Why?)

and

𝜏 (t) = Ge 𝛾0 H(t) + 𝜇 𝛿 (t) from which

] [ 𝜇 G(t) = Ge H(t) + 𝛿 (t) Ge

(7.4.13)

Linear Viscoelasticity

or G(t) = H(t) + tr′ 𝛿 (t) , Ge

tr′ =

𝜇

(7.4.14)

Ge

in which tr′ = 𝜇 ∕Ge = 𝜇 Je is called the retardation time. 7.4.4.2 Creep

For a creep experiment in which a constant stress 𝜏 (t) = 𝜏 0 is imposed on the system, 𝜏 1 (t) + 𝜏 2 (t) = 𝜏 0 , so that from Eq. 7.4.12,

𝜏1 (t) = Ge 𝛾 (t) ,

𝜏2 (t) = 𝜇 𝛾̇ (t) = 𝜏0 − Ge 𝛾 (t)

from which

𝜇 𝛾̇ (t) + Ge 𝛾 (t) = 𝜏0 ,

or

𝛾̇ (t) +

Ge

𝜇

𝛾 (t) =

𝜏0 , 𝜇

or

𝛾̇ 2 (t) +

Gg

𝜇

𝛾2 (t) =

Gg

𝜇

𝛾0

(7.4.15)

which has the general solution

𝛾 (t) = B exp[(−Ge ∕𝜇 ) t] + 𝜏0 ∕𝜇 With 𝛾 (0) = 0 (Why?), B = − 𝜏 0 ∕Ge , so that with J(t) = 𝛾 (t)∕𝜏 0 , J(t) = Je [1 − exp (−Ge ∕𝜇 ) t]

(7.4.16)

In terms of the normalized time T ′ = t∕tr′ = (Ge ∕𝜇 ) t = t∕( 𝜇 Je ), the expressions in Eqs. 7.4.14 and 7.4.16 can be written as G(T ′) = H(T ′ ) + tr′ 𝛿 (T ′) , Ge

tr′ =

𝜇 Ge

J(T ′) = [1 − exp (−T ′ )] Je

(7.4.17)

the variations of which are shown in Figure 7.4.7.

7.4.5

Standard Three-Parameter Model

This model comprises a Maxwell element in parallel with a spring (Figure 7.4.8). The extra spring element in series with the dashpot removes the sharp spike in the relaxation modulus for the Kelvin-Voigt model. By following the procedures used in the previous two sections, the relaxation modulus and the creep compliance for this three-parameter model are given by G(t) = Ge + (Gg − Ge ) exp (− t∕tr ) , where tr = J(t) = Je − (Je − Jg ) exp (− t∕tr′ ) , where tr′ =

𝜇 Gg − Ge

Gg Ge

tr

(7.4.18)

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1.0

tr' δ (T ')

G (T ') Ge

J (T ') 0.5 Je J (T ') = 1 − exp (− T ') Je

1 0

0

0

0

1

2

3

T ' = t / tr'

T ' = t / tr'

(a)

(b)

4

5

Figure 7.4.7 Kelvin–Voigt model. (a) Relaxation modulus. (b) Creep compliance.

μ

(Gg − Ge )

γ(t)

τ1 (t)

•

τ(t) = τ1 (t) + τ2 (t)

τ2 (t) Ge Figure 7.4.8 Standard three-parameter model.

With T = t∕tr = [(Gg − Ge )∕𝜇 ] t, and T ′ = t∕tr′ = (Ge ∕𝜇 ) (1 − Ge ∕Gg ) t, the relaxation modulus and the creep compliance can be written as G (t)∕Gg = Ge ∕Gg + (1 − Ge ∕Gg ) exp (−T ) J(t)∕Je = 1 − (1 − Jg ∕Je ) exp (− T ′)

(7.4.19)

These equations can further be reduced to the compact forms G(t) − Ge = exp (− T ) Gg − Ge Je − J(t) = exp (− T ′) Je − Jg the variations of which are shown in Figure 7.4.9.

(7.4.20)

Linear Viscoelasticity

1.0 G (T ) − Ge = exp( − T ) Gg − Ge G (T ) − Ge Gg − Ge

Je − J (T ') = exp( − T ') Je − Jg 0.5

Je − J (T ') Je − Jg

T= T'=

Gg − Ge

μ

t

Ge Gg − Ge t μ Gg

0 1

0

2

3

4

5

T , T' Figure 7.4.9 Variations of the relaxation modulus and the creep compliance for the standard three-parameter model.

7.5 Response for Constant Strain Rates The relaxation modulus and creep compliance are defined, respectively, by responses to step changes in strain and stress. And superposition gives the responses for small deformations. Responses to constant strain rates are important because of their application to experimental characterization of material properties. For a constant strain rate 𝛾̇ 0 imposed t = 0, at it follows from Eq. 7.5.2 that

𝜎 (t) =

t

∫−∞

G(t − 𝜉 ) 𝛾 ′(𝜉 ) d 𝜉 = 𝛾̇ 0

t

∫0

G(t − 𝜉 ) d 𝜉 = 𝛾̇ 0

t

∫0

G(𝜉 ) d 𝜉

(7.5.1)

For a constant strain rate 𝛾̇ 0 the strain is 𝛾 = 𝛾̇ 0 t, so that the stress can be expressed as

𝜎 (𝛾 ) = 𝛾̇ 0 7.5.1

𝛾 ∕𝛾̇ 0

∫0

G(𝜉 ) d 𝜉

(7.5.2)

Maxwell Model

For the Maxwell model, an integration of the expression for G(t) results in

𝜎 (t) = 𝜇 𝛾0 {1 − exp [− 𝛾∕( 𝛾0 tr )]}

(7.5.3)

which can be rewritten as

𝜎 (𝛾 ) = Gg 𝛾0 [1 − exp (− 𝛾∕𝛾0 )] in which 𝛾0 = 𝛾̇ tr is the shear strain at time t = tr .

(7.5.4)

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For small 𝛾 , exp (− 𝛾∕𝛾 0 ) ≃ 1 − 𝛾∕𝛾 0 , so that 𝜎 ( 𝛾 ) ≃ Gg 𝛾 . For large 𝛾 , 𝜎 ( 𝛾 ) ≃ Gg 𝛾 0 . From d𝜎 = Gg exp (− 𝛾∕𝛾0 ) d𝛾 it then follows that, for all 𝛾̇ 0 and 𝛾 0 , d𝜎 || d𝜎 || = Gg , and =0 d𝛾 ||𝛾 =0 d𝛾 ||𝛾 →∞ 7.5.2

(7.5.5)

Kelvin–Voigt Model

For this model

𝜎 (t) = 𝛾̇ 0

t

∫0

G(𝜉 ) d 𝜉 = Ge 𝛾̇ 0 (t + tr′ ) , where tr′ =

𝜇 Ge

= 𝜇 Je

(7.5.6)

which can be rewritten as

𝜎 (𝛾 ) = Ge ( 𝛾 + 𝛾0 ) , where 𝛾0 = 𝛾̇ 0 tr′ 7.5.3

(7.5.7)

Standard Three-Parameter Model

For this model t

∫0

G(𝜉 ) d 𝜉 = Ge t + (Gg − Ge ) tr [1 − exp (− t∕tr )] ,

where tr =

𝜇 Gg − Ge

(7.5.8)

which can be rewritten as

𝜎 (𝛾 ) = Ge 𝛾 + (Gg − Ge ) 𝛾0 [1 − exp (− 𝛾∕𝛾0 )] ,

where 𝛾0 = 𝛾̇ 0 tr

(7.5.9)

For small 𝜎 (𝛾 ) ≃ Gg 𝛾 . For large 𝛾 , 𝜎 (𝛾 ) ≃ Ge 𝛾 + (Gg − Ge ) 𝛾 0 . From d𝜎 = Ge + (Gg − Ge ) exp (− 𝛾∕𝛾0 ) d𝛾 it follows that, for all 𝛾̇ 0 and 𝛾 0 , d𝜎 || = Gg , d𝛾 ||𝛾 =0

and

d𝜎 || = Ge d𝛾 ||𝛾 →∞

(7.5.10)

7.6 *Sinusoidal Shearing Consider a sinusoidal shearing motion in which the shear strain varies as 𝛾 (t) = 𝛾 0 cos (𝜔 t), with amplitude 𝛾 0 and circular frequency 𝜔 = 2𝜋 n, where n is the frequency of the oscillatory motion. For mathematical convenience, this shearing motion can be interpreted as being the real part of 𝛾 (t) = 𝛾 0 exp (i𝜔t).

Linear Viscoelasticity

For this strain history, the stress is then the real part of

𝜎 (t) =

∞

∫0

G(𝜉 ) 𝛾 ′(t − 𝜉 ) d 𝜉

= G(t) 𝛾 (t) +

∞

∫0

[G(𝜉 ) − G(t)] 𝛾 ′(t − 𝜉 ) d 𝜉

(7.6.1)

And for this strain history to start at t, t

𝜎 (t) = G(t) 𝛾0 exp (i𝜔t) + 𝛾0 exp (i𝜔t) [G(𝜉 ) − G(t)] exp (−i𝜔t) d 𝜉 ∫0

Assuming that this integral exists for t → ∞, which requires that G(t) approach a limit Ge fast enough, it follows that for large times, say a few relaxation times, the stress will approach a sinusoidal form

𝜎 (t) = G ∗ (𝜔) 𝛾0 exp (i𝜔t)

(7.6.2)

G *(𝜔) is called the dynamic modulus, or complex modulus, and is given by G ∗ (𝜔) = Ge + i𝜔

t

∫0

[G(𝜉 ) − Ge ] exp (−i𝜔 t) d𝜉

(7.6.3)

The relation between stress and strain in sinusoidal shearing can also be written as

𝛾 (t) = J ∗ (𝜔) 𝜎 (t)

(7.6.4)

where J *(𝜔), is called the complex compliance. Then, from 𝛾 (t) = 𝛾 0 exp (i𝜔t) it follows that J ∗ (𝜔) G ∗ (𝜔) = 1

(7.6.5)

Let G *(𝜔) = |G *(𝜔)| exp(i𝛿 ). The magnitude |G *(𝜔)| of G *(𝜔) is clearly the ratio of the stress amplitude to the shearing strain amplitude, and 𝛿 , the phase angle by which the strain lags the stress, is called the loss angle. Empirically, 0 ≤ 𝛿 ≤ 𝜋∕2. The complex modulus and the complex compliance can be written in terms of their real and imaginary parts as G *(𝜔) = G1 (𝜔) + i G2 (𝜔), and J *(𝜔) = J1 (𝜔) − i J2 (𝜔). Each of these four real components has a special name: G1 and G2 are called, respectively, the storage modulus and the loss modulus. And J1 and J2 are called, respectively, the storage compliance and the loss compliance. Also, tan 𝛿 , called the loss tangent, is related to these four quantities through tan 𝛿 =

G2 (𝜔) J2 (𝜔) = G1 (𝜔) J1 (𝜔)

(7.6.6)

Two additional quantities also have special names: G2 (𝜔)∕𝜔 is the dynamic viscosity and G *(𝜔)∕(i𝜔) is the complex viscosity. 7.6.1

Dynamic Mechanical Analysis (DMA)

In polymer viscoelasticity (Section 14.4), using such oscillatory tests to characterize G1 (𝜔), G2 (𝜔), tan 𝛿 is called DMA. The variations of these quantities with the frequency, 𝜔, at constant temperature, or the variations with the temperature at constant frequency, provide insights into the molecular constitution of polymers.

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In polymer viscoelasticity, the variables used are G ′ = G1 , G ′′ = G2 , J ′ = J1 , and J ′′ = J2 . Subscripted variables have been used in this chapter because primes and double primes are also used to denote differential of variables. 7.6.1.1

DMA Curves for Three-Parameter Model

By using Eq. 7.6.3 in conjunction with the definitions for G ′ and G ′′, it can be shown that for the standard three-parameter model discussed in Section 7.4.5 G ′(𝜔) = Ge + G ′′(𝜔) = tan 𝛿 =

(Gg − Ge ) 𝜔2 tr 2 1 + 𝜔2 tr 2

=

Ge + Gg 𝜔2 tr 2 1 + 𝜔2 tr 2

(Gg − Ge ) 𝜔 tr

(7.6.7) (7.6.8)

1 + 𝜔2 tr 2

G ′′(𝜔) (Gg − Ge ) 𝜔 tr = Ge + Gg 𝜔2 tr 2 G ′(𝜔)

(7.6.9)

By using 𝜃 = 𝜔tr = 𝜇𝜔∕(Gg − Ge ) in place of 𝜔tr , the expressions in Eqs. 7.6.7, 7.6.8, and 7.6.9 can be written as G ′(𝜔) − Ge 𝜃2 = Gg − Ge 1 + 𝜃2

(7.6.10)

G ′′(𝜔) 𝜃 = Gg − Ge 1 + 𝜃2

(7.6.11)

tan 𝛿 =

(Gg − Ge ) 𝜃 Ge + Gg 𝜃 2

=

(1 − 𝛽 ) 𝜃 , 𝛽 + 𝜃2

𝛽=

Ge Gg

(7.6.12)

The variable 𝜃 is the angle that the circular frequency 𝜔 covers in relaxation time 𝜔 tr . The variations of the nondimensional storage and loss moduli and the loss tangent are shown in Figure 7.6.1. Note that while the use of this variable collapses the variations of the nondimensional storage and loss moduli into single curves, the loss tangent depends on the additional parameter 𝛽 = Ge ∕Gg . 7.6.2

*Energy Storage and Loss

With 𝛾 (t) = 𝛾 0 exp (i𝜔t), 𝛾 ′(t) = d𝛾 ∕dt = i𝜔 𝛾0 exp (i𝜔t) = i𝜔𝛾 (t), so that

𝜎 (t) = G ∗ (𝜔) 𝛾 (t) = (G1 + i G2 ) 𝛾 (t) = G1 𝛾 (t) +

G2

(7.6.13) 𝛾 ′(t) 𝜔 In this expression, G1 𝛾 (t) is the elastic part of stress with modulus G1 , and (G2 ∕𝜔) 𝛾 ′ (t) is the viscous part of stress with viscosity G2 (𝜔)∕𝜔. For a small increment in strain, d 𝛾 (t), the incremental work done is dW = 𝜎 d𝛾 = d

(

) G 1 G1 𝛾 2 + 2 ( 𝛾 ′ ) 2 dt 2 𝜔

(7.6.14)

Linear Viscoelasticity

1.0 G ' (ω) − Ge Gg − Ge G '(ω) − Ge Gg − Ge G ''(ω) Gg − Ge

G ''(ω) Gg − Ge

tan δ

G

β = Ge g

0.5

β = 0.25

tan δ

β = 0.5 β = 0.75 0 0.001

0.01

0.1

1

10

100

1000

θ = ωtr Figure 7.6.1 Variations of the storage modulus, the loss modulus, and the loss tangent (dashed curves) with the shearing frequency.

The first term on the right-hand side is a perfect differential that can be interpreted as an increment in stored elastic energy, which will average to zero over a complete cycle. By integrating the expression in Eq. 7.6.14, the work done per cycle is then ∮

dW =

G2

𝜔 ∮

( 𝛾 ′ ) 2 dt

(7.6.15)

which is like a rate of viscous dissipation.

7.7 Isothermal Temperature Effects All the preceding discussion has focused on linear viscoelasticity at a fixed temperature. Just as the effect of varying strain or stress were considered in two steps – first those due to step changes and then those due to continuous variations – so also the effects of varying temperature will be considered in two steps. This section will focus on the possibility of relating the relaxation moduli and creep compliances at different, isothermal conditions. The dependence of these functions on the temperature can be expressed by explicitly writing them as G = G(t, T ) and J = J(t, T ). The dependence of these functions on both time and temperature makes the task of experimentally determining them for particular materials a daunting task. In some materials the values of these functions at two different temperatures are simply related. An idealized model for this dependence is that of a thermorheologically simple material.

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7.7.1

Thermorheologically Simple Materials

In many materials – especially polymers – the relaxation modulus at higher temperatures, when plotted against log time, just shifts to the left by an amount that is approximately only a function of the temperature T. Thermorheologically simple materials are idealized materials in which this is exactly true. Figure 7.7.1 is a plot of the relaxation modulus G = G(t, T ) versus log time for a thermorheologically simple material. The right-hand curve, G = G(t, Tg ), shows how the relaxation modulus varies at a fixed reference temperature Tg . The left-hand curve, which is obtained by a leftward shift of G = G(t, Tg ) by an amount ln [1∕a (T )], shows the variation at a different fixed temperature T, such that T > Tg . This shift causes point P moves to point Q. Because the segment PQ = ln[1∕a (T)], the time coordinate of point P is lnt + ln[1∕a (T )] = ln[t∕a (T )], so that with tP = t, tQ = t∕a(T ). Then, from G(tQ , T ) = G(tP , Tg ), it follows that G(t, T ) = G [t∕a(T ), Tg ]

T > Tg G (t, T )

G (t) = G (t,Tg )

G (t) Q

•

P

•

ln [t/a(T )]

• ln t

ln [t/a (T )]

Figure 7.7.1 Time-temperature shift of the relaxation modulus for a thermorheologically simple material.

In this formulation the equilibrium moduli are not affected by the time-temperature shift. The equilibrium moduli of some materials are affected by temperature. For example, the equilibrium modulus of rubber increases with temperature. This effect is much smaller than the time-temperature shift, and can be compensated for as follows T G [t∕a(T ), Tg ] (7.7.1) G(t, T ) = Tg For the tensile relaxation modulus T E [t∕a(T ), Tg ] E(t, T ) = Tg

(7.7.2)

For a thermorheologically simple material the leftward shift, determined by a(T ), called the shift function, is only a function of the temperature. One good model for a(T ) is the Williams-Landel-Ferry equation, commonly referred to as the WLF equation, in which − C1 (T − Tg ) (7.7.3) log a(T ) = C2 + (T − Tg )

Linear Viscoelasticity

has the form shown in Figure 7.7.2. In this equation, Tg is a characteristic temperature such as the glass transition temperature. Originally, the constants C1 and C2 were thought to be universal constants valid for all polymers. However, they do vary from polymer to polymer.

2 1 ln a(T )

•

0 −1 −2

• −10

0

10

20

30

(T − Tg ) Figure 7.7.2 Shape of the shift function for the WLF Model.

7.7.2

Physical Interpretation for Time-Temperature Shift

In the polymer community, the contention is that the stress relaxation in many materials proceeds at a rate determined by a material’s internal clock. The hotter the material gets, the faster the motion at a molecular level becomes, and the faster the internal clock runs. In one tick, or unit time, on the material’s internal clock, the external observer’s elapsed time is a(T ), say, an amount that decreases rapidly with increase in temperature.

7.8 *Variable Temperature Histories This section considers viscoelasticity for the general case of time-varying temperatures. Intuitively, the main effect of temperature is to speed up or slow down the internal processes within a material. In a sense, the material’s internal clock speeds up (slows down) as the temperature is increased (decreased). Let 𝜉 and t measure the times in the material’s and external observer’s clocks, respectively. Then, the observer’s elapsed time dt during a small time interval d 𝜉 on the material’s clock may be related by d𝜉 =

1 dt = 𝜑 (T ) dt , a(T )

𝜑 (T ) =

1 a(T )

(7.8.1)

The usefulness of such scaled times, in which the scaling function is a function of the temperature, has only been established for isothermal experiments. For time-varying temperatures it will be assumed that

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viscoelastic processes occur at a rate determined by the reduced time 𝜉 (t) defined by

𝜉 (t) =

t

∫0

𝜑 [T (𝜁 )] d𝜁

(7.8.2)

and that these processes are governed by the constitutive equation

𝜎 (t) =

t

∫−∞

E [t(𝜉 ) − t (𝜆)]

d [𝜀(𝜆) − 𝛼 ΔT(𝜆)] d𝜆 , d𝜆

ΔT = T (𝜆) − T (0)

(7.8.3)

7.9 *Cooling of a Constrained Bar Paralleling the discussion on the cooling of a thermoelastic bar (Section 4.4), consider the development of normal stress in a bar that undergoes the simple cooling history { Th − r t , 0 ≤ t ≤ tc T(t) = (7.9.2) Tc , t ≥ tc Let the relaxation modulus be represented by the standard three-parameter model E(t) = Ee + (Eg − Ee ) [1 − exp (−t∕tr )] Further assume that the coefficient of thermal expansion is a constant 𝛼 (T ) = 𝛼 0 , and that the shift function is obtained by linearizing the WLF equation around Tg , resulting in

𝜑 (t) = 𝜑0 exp [𝛽 (T − Tg )]

(7.9.3)

It can then be shown that the normal stress history of the bar is 𝜓 (0 ) −x ⎧ 𝛼 e dx , ⎪ 𝛼0 Ee rt + (Eg − Ee ) exp 𝜓 (t) ∫ 𝛽 𝜓 (t ) x ⎪ [ ] ⎪ 𝜓 (0 ) −x 𝜎 (t) = ⎨ 𝛼 e 𝛼 Ee rtc + (Eg − Ee ) exp 𝜓 (tc ) dx exp[−𝜑0 (t − tc )∕tr ] , 0 ⎪ ∫𝜓 (t ) x 𝛽 c ⎪ ⎪ where 𝜓 ( t ) = (𝜑 ∕𝛽 r t ) exp (−𝛽 r t) 0 r ⎩

0 ≤ t ≤ tc t ≥ tc

(7.9.4) Clearly, the normal stress history in this cooling viscoelastic bar is far more complex than its thermoelastic counterpart. Furthermore, as opposed to the thermoelastic case – in which the stress is independent of the temperature history, and only depends on the initial and final temperatures – the stress in the thermoviscoelastic case depends on the entire temperature history.

7.10 Concluding Remarks Clearly, viscoelastic materials exhibit time-dependent behavior that is far more complex than that of fluids; even the linear, small-deformation theory requires substantially more mathematical sophistication.

Linear Viscoelasticity

Linear viscoelasticity theory provides the simplest model for analyzing the critical role of temperature history in the behavior of polymeric materials The linear theory is important for three reasons: (i) Polymer science uses experimentally observed viscoelastic data for understanding phase transitions, for which small-deformation data are adequate. This aspect has been extensively documented. (ii) Viscoelastic behavior of polymer melts is very important for understanding many processing phenomena such as melt fracture. Substantial progress has been made in understanding the effects of melt elasticity on processing. And (iii), since solid polymeric materials undergo time-dependent deformations even under constant loads, viscoelastic analysis should be important for predicting the time-dependent response of polymeric parts. Unfortunately, this aspect is the least developed: Instead of accounting for time-temperature effects from the beginning, time-independent elastic models are used first for analyzing the performance of parts subjected to loads; viscoelastic effects are then accounted for as an afterthought by using creep analysis techniques, developed for the high-temperature behavior of metals, which are not well grounded in mechanics. The use of viscoelasticity theory in polymer science for understanding phase transitions has dominated the development of test equipment and the interpretation of data, for which a large body of literature is available. This methodology has not been extended for the use of viscoelasticity for predicting the time-dependent response of structures to static loads, for which purpose viscoelasticity data may be required to be less precise but cover larger strains. The linear theory does not include normal stress effects of the type required for explaining the Weissenberg effect (Section 6.7.4). The inclusion of the normal stress phenomenon requires constitutive equations, not necessarily including elastic effects, in which the stress is a function of large deformation rates. A comprehensive theory would synergistically include finite deformation and deformation rates, elastic effects, and temperature effects.

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8 Stiffening Mechanisms 8.1 Introduction While plastics have many desirable properties, such as the ease with which they can be used to make complex parts, their relatively low stiffness and strength can limit their structural applications. However, this limitation can be overcome by forming composites in which the plastic matrix acts as a glue that holds stiffer and stronger fibers together. This chapter discusses several stiffening mechanisms, including the use of aligned fibers in advanced composites, and small chopped fibers, flakes, and particulates in injection moldable compounds. Characteristics of the commonly used fibers, flakes, and particulates are also described.

8.2 Continuous Fiber Reinforcement Consider a continuous circular fiber, of cross-sectional area Af , embedded in a larger circular cylinder, having a cross-sectional area Ac , as shown in Figure 8.2.1a. Then the cross-sectional area of the annular region occupied by the matrix material of the cylinder is Am = Ac − Af . Assume that the surface of the fiber is bonded to the matrix material of the cylinder. An axial load P acting on this system will stretch both the fiber and the matrix material by equal amounts (Why?), resulting in the same uniform tensile strain 𝜀 in both. While the uniform strains in the fiber and the matrix material are the same, the stresses will be different. Let the Young’s modulus of the fiber material and the stress in it be Ef and 𝜎 f , respectively, and let the corresponding values of these quantities for the matrix material be Em and 𝜎 m . The average stress on the system is 𝜎 c = P∕Ac . Since the load P must equal the sum of the load Pm on the annular matrix and the load Pf on the fiber, it follows that 𝜎 c Ac = P = Pf + Pm = 𝜎 f af + 𝜎 m am , which can be rewritten as

𝜎c = 𝜎f af + 𝜎m am ,

af + am = 1

(8.2.1)

where af = Af ∕Ac and am = Am ∕Ac are, respectively, the fractions of the composite system crosssectional areas occupied by the fiber and the matrix. Since the geometry does not change along the system axis, the area fractions are the same as the volume fractions of the material (Why?), so that in terms of the volume fractions vf and vm Eq. 8.2.1 can be written as

𝜎c = 𝜎f vf + 𝜎m vm ,

vf + vm = 1

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

(8.2.2)

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P P

P P (a)

(b)

Figure 8.2.1 (a) Circular fiber embedded in a larger circular cylinder. (b) Noncircular cylinder filled with noncircular fibers.

With Em = 𝜎 m ∕𝜀c and Ef = 𝜎 f ∕𝜀c it then follows that the effective longitudinal composite modulus, EL = 𝜎 c ∕𝜀c , is given by EL = vf Ef + vm Em = vf Ef + (1 − vf ) Em

(8.2.3)

Thus, the effective composite modulus along the fiber direction is the volume-fraction weighted arithmetic mean of the matrix modulus and the fiber modulus. For historical reasons, mainly stemming from chemistry usage, this formula is referred to as following the rule of mixtures. The same argument applies to the noncircular cylinder filled with noncircular cylindrical fibers shown in Figure 8.2.1b, so that Eqs. 8.2.1 and 8.2.2 are valid for this system, with Ac being the cross-sectional area of the cylinder, Af being the total cross-sectional area of the fibers, and af = Af ∕Ac and am = Am ∕Ac being, respectively, the fractions of the composite system crosssectional areas occupied by the fibers and the matrix. Similarly, for the rectangular geometry shown in Figure 8.2.2a, in which instead of stiff fibers the matrix is stiffened by a plate of a stiffer material with elastic modulus Ef , Eqs. 8.2.1 and 8.2.2 apply with Ac = bt, Af = btf , Am = b(tm1 + tm2 ), af = Af ∕Ac = tf ∕t, and am = Am ∕Ac = (tm1 + tm2 )∕t, and P = 𝜎 Ac = 𝜎 f Af + 𝜎 m Am . More generally, these two equations also apply to the multiple reinforcing-plate composite geometry shown in Figure 8.2.2b.

Stiffening Mechanisms

P

P t

h

P P (a)

(b)

Figure 8.2.2 (a) Matrix stiffened by a plate of a stiffer material subjected to longitudinal load. (b) Matrix stiffened by several plates of stiffer materials subjected to longitudinal load.

The reason for considering this rectangular, reinforcing-plate geometry is that it can more easily be analyzed for loads acting normal to the fiber/plate axes, as shown in Figure 8.2.3a, in which the same stress 𝜎 c = P∕Ac = P∕bh is applied to all the layers. However, in Figure 8.2.3a, while the strains 𝜀m1 = Δtm1 ∕tm1 = 𝜎c ∕Em and 𝜀m2 = Δtm2 ∕tm2 = 𝜎 ∕Em = 𝜀m1 in the matrix material will be the same, they will differ from the strain 𝜀f = Δtf ∕tf = 𝜎∕Ef in the reinforcing plate. It follows (Why?) that the extensions of the three layers in the direction of the load will, respectively, be Δtm1 = 𝜎 (tm1 ∕Em ), Δtm2 = 𝜎c (tm2 ∕Em ), and Δtf = 𝜎 c (tf ∕Ef ). The total extension is then given by ( ) ( ) tm1 + tm2 tf tf tm + + Δt = Δtm1 + Δtf + Δtm2 = 𝜎 =𝜎 Em Ef Em Ef in which tm = tm1 + tm2 is the total thickness of the matrix material, so that the average strain in the composite will be ( ) ( ) ( ) tf btf af tm b tm am Δt 𝜀c = + + + = 𝜎c =𝜎 = 𝜎c t t Em t Ef bt Em bt Ef Em Ef

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tf

tm2 b

tm1

P P

P

P

(a)

(b)

Figure 8.2.3 (a) Matrix stiffened by a plate of a stiffer material subjected to transverse load. (b) Matrix stiffened by several plates of stiffer materials subjected to transverse load.

It then follows that the effective transverse composite modulus, ET = 𝜎 c ∕𝜀c , is given by a a v v v 1 − vf 1 = f + m = f + m = f + ET Ef Em Ef Em Ef Em

(8.2.4)

Thus, the effective composite modulus transverse to the fiber direction is the volume-fraction weighted harmonic mean of the matrix and fiber moduli. 8.2.1

Fiber-Matrix Interphase

While the stiffness and strength of a composite are mainly determined by the properties of the matrix, fiber, and fiber concentration, they also depend on efficient load transfer at the fiber-matrix interface. In the analyses in the previous sections perfect bonding between the two phases is assumed at the fiber-matrix interface. Because of the very different compositions of the matrix and fiber surfaces at the interface, the natural tendency is for them to not form strong interfacial bonds. Improved bonding requires the use of a compatible coating on the fiber surface; such coatings are referred to as sizing. Thus, besides the matrix and fiber, sizing comprises a thin interphase that, while improving the bonding at the fiber-matrix interface, can affect nucleation and growth of cracks, ultimately causing debonding at the fiber-matrix interface. The analysis of how cracks nucleate and grow at the interface, and the role of an interphase of a third material, requires the use of advanced mechanics concepts that are well beyond the scope of this book.

Stiffening Mechanisms

8.3 Discontinuous Fiber Reinforcement Industrial applications use plastics filled with short fibers, resulting in discontinuities at the matrix-fiber interfaces. In contrast to the continuous fiber case – in which the fiber and matrix strains are the same as the strain in the composite – differences in the mechanical properties of the fiber and matrix materials result in different strains in the fiber and the surrounding matrix. These differing deformations result in stress gradients in the fiber and the immediate surrounding matrix – even when the assembly is pulled by a uniform stress. In this case, load transfer from the matrix to the fiber occurs through an interfacial shear stress that balances the variations in the fiber stress. 8.3.1

Load Transfer in a Discontinuous Fiber

Figure 8.3.1 shows a small element, ABCD, of a fiber of radius rf and length dx that is subjected to mean axial tensile stresses 𝜎 f and 𝜎 f + (d𝜎 f ∕dx) dx on the ends AB and BC, respectively, and a shear stress 𝜏 f on its cylindrical surface (AB, DC).

τi

D

C

σf +

2rf

σf A

τi

d σf dx dx

B

dx x

x + dx

Figure 8.3.1 Normal and shear stresses acting on a small fiber element embedded in a matrix.

Equilibrium of all the axial forces on this element requires ( ) d𝜎f 2 𝜋 r f 𝜎f + dx − 𝜋 rf 2 𝜎f + 2𝜋 rf 𝜏i = 0 dx resulting in d𝜎f 2 + 𝜏i = 0 dx rf

(8.3.1)

A determination of the stresses in the fiber requires approximations: In the shear lag model, it is assumed that the fibers are evenly spaced at regular intervals of 2R, so that the composite is made up of regular hexagonal cylinders with embedded fibers at the centers, as shown in the composite cross section in Figure 8.3.2a. The hexagonal cylinders are then approximated by inscribed circular cylinders of radius R, and it is further assumed that the displacement at the surface of the cylinder at r = R is the

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displacement uc of the composite. A cross section of this model composite is shown in Figure 8.3.2b. Figure 8.3.2c shows a thin slice EFGH of this composite, of length dx, on which shear stress at radius r is 𝜏 . MATRIX

COMPOSITE

FIBER

2R

2R

(a)

(b)

H

G

τ

N

M

τi

σf

σf

τi K

r

R

L

τ

E

F dx x

x + dx (c)

Figure 8.3.2 Shear lag approximation. (a) Cross section of composite showing fibers arranged in a regular pattern placed at a distance of 2R from each other. (b) Composite near a fiber approximated as a fiber embedded in a matrix cylinder of radius R. (c) Shear variation within model cylindrical composite; shear stress at radius r is 𝜏 .

If variations in the axial normal stress in the matrix material in this slice are neglected, as indicated in Figure 8.3.2c, then equilibrium in the axial direction requires the shear force, (2𝜋 r dx) 𝜏 , acting on the

Stiffening Mechanisms

cylindrical matrix surface KLMN of radius r to equal the shear force, (2𝜋 rf dx) 𝜏 i , acting on the fiber surface, resulting in

𝜏m =

rf 𝜏 r i

(8.3.2)

From 𝜏 m = Gm 𝛾 m = Gm (du∕dr), it then follows that

𝜏 r du = f f dr Gm r

(8.3.3)

An integration of this equation from r = rf , at which u(rf ) = uf , to r = R, at which u(R) = uc , results in uc − uf =

𝜏i Gm

rf ln(R∕rf )

(8.3.4)

With 𝜀c = duc ∕dx and 𝜀f = duf ∕dx, a differentiation of Eq. 8.3.4 gives uc − 𝜀f =

rf d𝜏 ln (R∕rf ) f Gm dx

(8.3.5)

A differentiation of Eq. 8.3.1 results in ) ( d2 𝜎f 𝜎f 2Gm 2Gm 2 d𝜏f (𝜀 − 𝜀f ) = − 2 =− 𝜀 − =− 2 rf dx dx2 rf ln(R∕rf ) c rf ln(R∕rf ) c Ef which can be written as [ 2 [ 2 ] ] 2 2Gm 2Gm l 2 d 𝜎f l l − 𝜎 =− E𝜀 4 dx2 4rf 2 Ef ln(R∕rf ) f 4rf 2 Ef ln(R∕rf ) f c This differential equation has the general solution

𝜎f = Ef 𝜀c + A sinh (2𝛼 x∕l ) + B cosh (2𝛼 x∕l ) where 𝛼 is the nondimensional number √ 2Gm l 𝛼 = a𝛽 , a = , 𝛽= 2rf Ef ln(R∕rf )

(8.3.6)

(8.3.7)

in which a = l∕(2rf ) is the fiber aspect ratio, and the nondimensional number 𝛽 = [2Gm ∕Ef 1 ln(R∕rf )] ∕2 depends on the ratio of the material property of the matrix to that of the fiber, and on the ratio R∕rf that is related to the fiber volume fraction of the composite. The boundary conditions that 𝜎 f = 0 at x = ± l∕2 require A = 0 and B = − Ef 𝜀c ∕cosh 𝛼 , so that, finally

𝜎f

Ef 𝜀c

=

𝜀f cosh 2𝛼 x∕l =1− 𝜀c cosh 𝛼

(8.3.8)

and from Eq. 8.3.1 that 2𝜏i sinh 2𝛼 x∕l = cosh 𝛼 𝛽 Ef 𝜀c

(8.3.9)

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Clearly, for a given composite strain 𝜀c , the tensile and interfacial shear stress distributions in the fiber depend on 𝛼 . The variations of the nondimensional axial fiber stress (Eq. 8.3.8) and the nondimensional interfacial shear stress (Eq. 8.3.9) along the fiber, with 𝛼 as parameter, are shown, respectively, in Figures 8.3.3 and 8.3.4.

1.0

NONDIMENSIONAL TENSILE FIBER STRESS

206

25

10

5

3 2

0.5 .

α =1

0 − 0.5

0

0.5 .

ξ = x /l NONDIMENSIONAL DISTANCE ALONG FIBER Figure 8.3.3 Variation of the nondimensional axial tensile fiber stress with the nondimensional distance along the fiber, with α as parameter.

From Eqs. 8.3.8 and 8.3.9, 𝜎 f (−x) = 𝜎 f (x) and 𝜏 f (−x) = −𝜏 f (x), with 𝜏 f (0) = 0. The maximum values of 𝜎 f and 𝜏 f occur, respectively, at x = 0 and x = l∕2, and are given by (𝜎f )max 1 = 1− Ef 𝜀c cosh 𝛼

(8.3.10)

|(𝜏i )max | = tanh(𝛼 ) (𝛽 Ef 𝜀c )∕2

(8.3.11)

The variations of the nondimensional maximum fiber tensile stress (Eq. 8.3.10) and the maximum nondimensional interfacial shear stress (Eq. 8.3.11) with the parameter 𝛼 are shown in Figure 8.3.5.

Stiffening Mechanisms

NONDIMENSIONAL INTRFACIAL SHEAR STRESS

0.5

0

25

2

10

5

3

α =1

. − 0.5 − 0.5

0.5 .

0

ξ = x/l NONDIMENSIONAL DISTANCE ALONG FIBER Figure 8.3.4 Variation of the nondimensional interfacial shear stress with the nondimensional distance along the fiber, with α as parameter.

NONDIMENSIONAL MAXIMUM STRESS

1.0 (τi ) max 1βE ε f c 2

= tanh(α)

0.5 (σf ) max = (1 − 1/cosh α) Ef εc

0 0

5

10

15

20

25

α NONDIMENSIONAL PARAMETER Figure 8.3.5 Variations of the nondimensional maximum normal and interfacial shear stresses with the parameter α.

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8.3.2

Discontinuous Fiber Composite

In the simplest model for a short-fiber-filled composite, fibers of the same length are assumed to be aligned along axes parallel to the load axis, but distributed randomly along the length, as shown in Figure 8.3.6. In this case at any cross section, such as the one shown in marked by the line XX, the fibers are intercepted at different points along their lengths, so that the stresses at the ends will be different. The load borne by the fibers at this cross section will be Σ Af 𝜎 f = Af Σ 𝜎 f . For random fiber placement along the length, Σ 𝜎 f will be the average stress in a fiber (Why?). This average stress 𝜎 f is given by

𝜎f =

l∕2

∫−l∕2

𝜎f dx

(8.3.12)

X

σc

σc

X Figure 8.3.6 Randomly distributed aligned fibers embedded in a matrix. Notice that at any cross section, such as XX, the fibers (here shown filled) are cut off at different points along their lengths. (Adapted with permission from, “Principles of Polymer Engineering,” N.G. McCrum, C.P. Buckley, C.B. Bucknall, Oxford University Press, New York, 1988.)

For the stress distribution given in Eq. 8.3.8, Eq. 8.3.12 results in

𝜎f

𝜀f tanh 𝛼 =1− (8.3.13) 𝜀c 𝛼 in which 𝜀f = 𝜎 f ∕Ef is the average strain in the fiber. The variation of 𝜎 f ∕Ef 𝜀c with α is shown in Ef 𝜀c

=

Figure 8.3.7. By applying the argument used in Section 8.2 for obtaining the modulus of a continuous fiber composite (Eqs. 8.2.2 and 8.2.3) to the present case, it follows that the stress in the composite and the longitudinal composite modulus are given, respectively, by

𝜎c = vf 𝜎f + (1 − vf ) 𝜎m

(8.3.14)

Ec = (1 − tanh 𝛼∕𝛼 ) vf Ef + (1 − vf ) Em

(8.3.15)

and In this expression the factor 𝜂 c = (1 − tanh 𝛼∕𝛼 ), the variation of which is shown in Figure 8.3.7, is a measure of the efficiency of reinforcement. 𝜂 c varies from 0 for 𝛼 = 0 to 1 for 𝛼 = ∞. Actually, 𝜂 c is very close to unity for much smaller values of 𝛼 . For example, 𝜂 c = 0.905, 0.935, 0.952, 0.962, and 0.982 for 𝛼 = 10, 15, 20, 25, and 50, respectively.

Stiffening Mechanisms

NONDIMENSIONAL AVERAGE FIBER STRESS

1.0

0.5

σf

Ef εc

= (1 − tanh ( α) / α )

0 0.1

10

1

100

α NONDIMENSIONAL PARAMETER Figure 8.3.7 Variation of the nondimensional average normal tensile stress with the parameter α.

8.3.3

Reinforcing Fillers

In plastics applications, many different filler types are commonly used: particulates, chopped fibers, chopped ribbons, and flakes. For such fillers, the effectiveness of reinforcement is characterized by the ratio aAR = A∕V of the filler surface area A to its volume V; higher values of aAR being more desirable (Why?). The sequel considers two questions about fillers: (i) How does particle size affect the area ratio aAR ? Spherical particles are used to address this question. And (ii), how does the particle aspect ratio affect the area ratio? This issue is addressed by varying the aspect ratios of cylindrical short fibers. 8.3.3.1 Spherical Fillers

With A = 4𝜋 r2p and V = 4𝜋 r3p ∕3, the surface-to-volume ratio for spherical particles of radius rp is aAR = 3∕rp . Clearly, this surface-to-volume ratio is very large for small particles. For a fixed filler volume fraction, let the volume of spherical particles be V0 . If this volume fraction is made up of just one spherical particle, then its radius r1 is given by r1 = (3V0 ∕4𝜋 )1∕3

(8.3.16)

so that the surface-to-volume ratio is AAR-1 = 3∕r1 = 3(4𝜋 ∕3V0 )1∕3

(8.3.17)

If this volume is divided among n smaller particles, each of radius rn , then rn = (3V0 ∕4𝜋 n)1∕3 = r1 ∕n1∕3

(8.3.18)

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and the surface-to-volume ratio for all the n smaller particles is (Why?) AAR-n =

n(4𝜋 rn 2 ) n(4𝜋 rn 2 ) 3n(4𝜋 r1 2 ∕n2∕3 ) 3n1∕3 = = = = n1∕3 AAR-1 3 3 V0 r1 4𝜋 r1 ∕3 4𝜋 r1

(8.3.19)

so that dividing the filler into n smaller particles increases the surface-to-volume ratio by a factor of n1/3 . 8.3.3.2

Cylindrical Fillers

For given matrix and fiber material, the stiffening effect of short fibers for a fixed volume fraction depends on the fiber aspect ratio a = l∕(2rf ). For a cylindrical particle of radius rf and length l, A = 2𝜋 rp (rp + l) and V = 2𝜋 rp l, so that aAR = 2𝜋 (1∕rp + 1∕l ), which, in terms of the fiber aspect ratio a = l∕(2rp ), can be written as aAR 1 = 2a1∕3 + 2∕3 (2𝜋∕V)1∕3 a

(8.3.20)

The variation of aAR versus the aspect ratio a is shown in Figure 8.3.8; aAR has a minimum of aAR = 3 (2𝜋∕V)1/3 at a = 1. Clearly, aAR is large either for large a – which corresponds to fibers – or for very small values of a – which corresponds to flakes, or platelets.

NONDIMENSIONAL AREA RATIO

210

20

FLAKE

10

aAR

FIBER

(2𝜋 /V )

1/3

1/3 = 2a +

1 a 2/3

0 0.01

0.1

10

1

100

1000

α NONDIMENSIONAL PARAMETER Figure 8.3.8 Variations of the nondimensional area ratio with the parameter α. (Adapted with permission from, “Principles of Polymer Engineering,” N.G. McCrum, C.P. Buckley, C.B. Bucknall, Oxford University Press, New York, 1988.)

Stiffening Mechanisms

8.4 The Halpin–Tsai Equations The very simple models for continuous fiber filled composites in Section 8.2 showed how the longitudinal (along the fiber direction) and transverse (normal to the fiber direction) elastic moduli of continuous fiber filled composites depend on the fiber volume fraction. While such fibers are very efficient in increasing the stiffness in the fiber direction, the transverse stiffness is much, much lower. Because aligned short fibers are not uniformly stressed, they are not as efficient at stiffening composites. Even the simple, approximate analysis in Section 8.3 for the stiffening effect of a single short fiber, and the use of that result to estimate the stiffening effect of homogeneously distributed aligned short fibers, requires a significantly more complex mechanics analysis. More accurate results have been obtained through numerical analyses of model geometries, a discussion of which is beyond the scope of this book. Many different filler types are used in plastics applications: particulates, chopped fibers, chopped ribbons, and flakes. In such applications, while particulates are homogeneously distributed in the matrix and result in isotropic materials, the nonaligned distribution of chopped fibers, chopped ribbon, and flake fillers results in anisotropic materials. Determining the stiffening effect of a filler is a difficult task, even when the distribution and orientation of such fillers is given. But the determination of the stiffening effect is even more difficult in molded parts because the distribution and orientation of the filler are strongly dependent on the processing conditions and on the part geometry. A practical means for estimating the stiffening effects of different types of filler types is provided by the Halpin–Tsai equations. These semi-empirical equations are based on extrapolations of accurate solutions for simple filler geometries. For any elastic moduli pf and pm of the fiber and matrix, respectively, the Halpin–Tsai equations give expressions for the corresponding composite modulus pc as pc 1 + 𝜁 𝜂 vf = , pm 1 − 𝜂 vf

𝜂=

MR − 1 , MR + 𝜁

MR =

pf pm

(8.4.1)

in which 𝜁 , a measure of the reinforcing filler geometry, depends on the loading conditions. Some limiting cases give an insight into the role of the parameters in the Halpin–Tsai equations: Rigid Inclusions. In this case the fiber modulus pf → ∞, so that MR → ∞, and 𝜂 = 1. Homogeneous Material. In this case the fiber modulus pf = pm , so that MR = 1, and 𝜂 = 0. Voids in Place of Fillers. In this case the fiber modulus pf = 0, so that MR = 0, and 𝜂 = −1∕𝜁 .

8.5 Reinforcing Materials Although in most applications fillers are used mainly to increase the rigidity of plastics, in many applications particulate fillers are also used for reducing cost. And specially tailored rubber particles are used to toughen brittle plastics. This section summarizes the characteristics of the fillers used for enhancing mechanical properties. 8.5.1

Continuous Fibers

The highest performance rigid, advanced composites are reinforced by aligned, continuous graphite fibers, available in both high-modulus and high strength versions, with diameters in the 5 – 7 μm range.

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Glass fibers, with diameters in the 3 – 20 μm range are also used. More detailed information on these and other fibers are given in Section 25.2.2. Flexible advanced composites, such as in tires, use continuous polyester, nylon, fiber, and steel wires. Such applications are discussed in Section 25.4. Another class of composites with thermoplastic matrices is reinforced with randomly distributed nonaligned continuous glass fibers. These materials, called random glass mat reinforced thermoplastics (GMTs), also called random continuous glass fiber thermoplastic composites, come in the form of thermoplastic sheets, which can be inexpensively thermostamped into parts. The characteristics of such materials are discussed in Chapter 24. 8.5.2

Chopped Fibers

Thin chopped fibers – most commonly made of glass but sometimes of graphite – are often used as fillers to increase the stiffness and strength of injection molded plastic parts, and to reduce the shrinkage and thermal expansion. Stainless steel and other metal fibers are used to enhance electromagnetic properties of the based resin. In most applications the glass fibers have diameters and lengths in the ranges of 7 – 18 μm and 0.2 – 3 mm, respectively, for which the fiber ratio a = l∕(2rf ) varies in the range of about 10 – 500. For obtaining higher stiffness and strength, longer fibers with lengths on the order of 10 mm are used, for which the fiber aspect ratio is on the order of 500 – 1,500. The use of chopped-fiber filled plastics is addressed in Chapter 22. 8.5.3

Flakes

Depending on the thickness, thin plate-like fillers are called flakes (thinner) or platelets (thicker). With nondimensional area ratios (Figure 8.3.8) in the 20 – 40 range, these relatively flat, high-aspect-ratio fillers have lower aspect ratios than of chopped fibers (10 – 1,500). The two most used examples of such fillers are mica flakes and talc platelets. 8.5.4

Particulates

Particulate fillers, with low aspect ratios in the 1 – 3 range have low stiffening and strengthening effects. They are mainly used as extenders to reduce cost and to improve surface characteristics. The most commonly used particulate filler is ground (coarser) or precipitated (finer) calcium carbonate. 8.5.5

Rubber Toughening

Many plastics are brittle – when stretched in tension, they fracture at relatively low strains without undergoing any plastic deformation. One way of avoiding brittle behavior in such materials is to chemically incorporate small amounts of elastomers into the chain, which form phases that prevent brittle failure through the formation of crazes (Section 15.6). Another successful method is to incorporate 10 – 25% surface-compatibilized rubber particles into rigid plastic. While they increase the toughness – avoid sudden brittle failure through plastic deformation – the much lower tensile modulus of the rubber particles reduces the stiffness of the material. An understanding of the underlying toughening mechanisms requires the use of advanced mechanics.

Stiffening Mechanisms

8.6 Concluding Remarks The low stiffness and strengths of plastics can limit their use in structural applications; this limitation can be overcome by forming composites in which the plastic matrix acts as a glue, holding stiffer and stronger fibers together. Aligned continuous fibers in rigid thermoset matrices result in the highest performance parts; such composites are referred to as advanced composites. Discontinuous reinforcements, such as chopped fibers and particulates, used both in thermoplastic and thermoset matrices, result in lower performance, but such materials are easy to process into parts with complex geometries. Establishing the principles of stiffening by aligned continuous fibers (Section 8.2) requires the use of mechanics. Even for the simple geometries discussed in Section 8.2 the analyses are not complete – how the strength of an advanced composite depends on the properties of the fibers and the matrix have not been addressed. The geometry of fibers in an actual advanced composite is far more complex, and predicting their performance requires the use of advanced mechanics, which is well beyond the scope of this book. While the term “advanced composites” normally refers to relatively rigid structures made from rigid thermoset matrices, tires (Section 25.4.1) with rubber matrices are examples of flexible advanced composites. Because the flexibility of such composites results in large deflections, the prediction of part performance requires deeper mechanics analyses than for advanced composites. Even the simplest analysis of the stiffening provided by discontinuous fibers (Section 8.3) required substantially more mechanics than for the continuous fiber counterparts. In actual applications, the local orientation of the fibers is influenced by the part geometry and processing, the very complex analysis for even the simplest models for which is described in Section 22.8. Once the fiber orientation has been established, more mechanics is required to predict the local mechanical properties. Random glass mat composite sheets, in which randomly oriented continuous fibers are embedded in thermoplastic matrices, are a class of materials that can be thermostamped into complex parts. The randomly oriented fibers uncoil during forming, which is not possible with continuous aligned fiber thermoplastic composites. While such materials are easy to form into parts, their mechanical properties vary randomly on a macroscopic scale, some consequences of which are addressed in Chapter 24.

Further Reading Many issues regarding fiber reinforcement are discussed in Chapter 6 in the first and subsequent editions of the book, Principles of Polymer Engineering, by N.G. McCrum, C.P. Buckley, C.B. Bucknall, Oxford University Press, 1988.

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This part has seven chapters that systematically describe the classification and chemical structure of polymers, how structure affects properties, the modification of properties through additives, and their mechanical behavior. However, polymer synthesis and related chemistry issues are not addressed, nor are methods for characterizing the properties of polymeric materials.

Chapter 9 Introduction to Polymers Two simple polymers, polyethylene and polypropylene, are used to introduce basic polymer structures and terminology. Polymers are classified into three types – thermoplastics, thermosets, and elastomers.

Chapter 10 Concepts from Polymer Physics Polymer physics concepts are used to explain the macro behavior of plastics and to contrast their deformation mechanisms from those of metals. The glass transition temperature is explained in terms of changes in polymer chain motion with temperature.

Chapter 11 Structure, Properties, and Applications of Plastics The chemical structure and properties of industrially important plastics, the use of additives to modify their properties, and typical applications are presented.

Chapter 12 Blends and Alloys Industrially important plastic blends and alloys – the plastics equivalent of alloys in metals – and their properties and typical applications are given. Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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Chapter 13 Thermoset Materials The structure, properties, and applications of thermosets, including elastomers, are discussed.

Chapter 14 Polymer Viscoelasticity Viscoelastic properties of plastics – the analytical models for which are addressed in Chapter 7 – important for understanding their mechanical behavior, are described in detail. Dynamic mechanical analysis (DMA) data for several industrially important plastics are given.

Chapter 15 Mechanical Behavior of Plastics This large chapter describes the phenomenology of the mechanical behavior of plastics, and provides an important link between materials and engineering aspects of plastics. The types of data required for mechanical design of parts, including failure mechanisms, are provided for a wide range of industrially important plastics, including thermoplastics, thermosets, and elastomers.

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9 Introduction to Polymers 9.1 Introduction Industrial plastics are basically polymeric materials, comprising very large molecules consisting of repeating chemical structural units connected by chemical covalent bonds. The word polymer is made up of the Greek roots poly (many) and meros (part). Most common polymers are based on carbon and hydrogen chemistry, and their chemical constitution and synthesis form a branch of organic chemistry. The chemical structure of such polymers is dominated by the valencies of four and one of carbon and hydrogen, respectively. An understanding of plastic materials requires knowledge of their chemical structure (polymer chemistry) and the behavior of very large molecules (polymer physics). Polymers can be classified into two broad categories: thermoplastics – which soften and melt on heating, and parts made of which can be melted and reshaped – and thermosets, which undergo chemical reactions during forming, so that parts made of such materials cannot be reshaped other than by machining processes. Most elastomers, or rubbers, are thermosets. This chapter uses the simple examples of polyethylene and polypropylene, two very widely used polymers, to introduce the chemical structure of thermoplastics and the terminology used for characterizing different aspects of polymer behavior. Thermosets are briefly discussed. The synthesis and manufacture of polymers is not addressed.

9.2 Thermoplastics Polyethylene, polypropylene, and polybutylene are the three simplest thermoplastic polymers in a series called polyolefins that are based on olefins, unsaturated hydrocarbons molecules having the general formula Cn H2n+2 , in which adjacent carbon atoms are connected by double bonds. 9.2.1

Polyethylene

Polyethylene, obtained by polymerizing ethylene, C2 H4 , may be linear – in which the polymerization proceeds along a continuous carbon chain that forms the backbone of the molecule – or it may be branched – in which, in addition to the carbon backbone, polymerization forms side chains.

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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9.2.1.1

Linear Polyethylene

The polymer polyethylene is obtained from ethylene, C2 H4 , an unsaturated hydrocarbon molecule in which the two carbon atoms are connected by a double bond, as indicated in its structural formula shown in Figure 9.2.1.

H

H

C

C

H

H

Figure 9.2.1 Structural formula for ethylene.

The process of polymerization involves opening up the double bond and creating increasingly larger molecules, such as in Figure 9.2.2.

H

H

H

H

H

C

C

C

C

H

H

H

H

H

H

H

H

H

H

H

H

C

C

C

C

C

C

H

H

H

H

H

H

(a)

H

(b)

Figure 9.2.2 Oligomers of ethylene: (a) Dimer and (b) trimer.

These increasingly larger molecules made up of two, three, four … mers are referred to, respectively, as dimers, trimers, tetramers, and so on. Collectively, such relatively small molecules, say comprising about 25 mers, are called oligomers. Dimers and trimers of polyethylene are shown, respectively, in Figure 9.2.2a and b. This process can be continued to form very large molecules, called polyethylene, the structural formula for which can be written as shown in Figure 9.2.3.

H

H

H

C

C

H

H

H , or

H

CH2

CH2

n

H

n

Figure 9.2.3 Structural formula for polyethylene.

For convenience this may also be written as H (CH2 CH2 )n H. The larger n is, the larger the molecular weight of the particular polyethylene will be. Because the C2 H2 units are connected in a continuous chain, this polymer is called linear polyethylene.

Introduction to Polymers

This example illustrates three important aspects of all linear polymers, the repeating unit mer, C2 H2 in this case, and n the number of repeat units, which is normally very large – on the order of 104 – 10 6 , and the end caps that begin and end the chain, which in this case are both hydrogen atoms. Because of their very large sizes, such molecules are called macromolecules. The degree of polymerization, which has a marked effect on the physical characteristics of the polymer, is the number of monomeric units in a polymer chain. One measure for this is the number-average degree of polymerization DPn = Mn ∕M0 , in which Mn is the molecular weight of the polymer and M0 is the molecular weight of the monomer unit. But in some polymers the repeat unit may consist of several monomer units, in which case the degree of polymerization can be defined as the ratio of the molecular weight of the polymer to that of the repeat unit; this measure would be smaller than DPn . However, for simple polymers such as polyethylene, in which the repeat unit is the same as the monomer, this measure reduce to DPn . Whichever measure is used, it is very large for all industrially important polymers. While the structural formulae show how different parts of a molecule are bonded, they do not show the spatial connection of the atoms. For example in methane, CH4 , the four atoms of hydrogen are located at the four vertices of a regular tetrahedron with the carbon atom at its center, as indicated in the structure in Figure 9.2.4, in which the carbon atom is on the plane of the paper, the dark CH bonds stick out from this plane and the hashed bonds stick into the paper. The angle between any two carbon bonds connected to two hydrogen atoms is 109° 28 ′. H C

H H

H

Figure 9.2.4 Three-dimensional structure of methane. The hydrogen atoms are at the vertices of a regular tetrahedron.

In polyethylene, the carbon chain backbone lies in a plane (i.e. all the carbon atoms are in one plane), with the CC bond angle being 109° 28 ′, so that the spatial arrangements of the atoms in the straight chain has the structure as shown in Figure 9.2.5.

HH C

HH C

HH

C

HH C

HH

C

HH C

HH

C

HH C

HH

C

C HH

Figure 9.2.5 Three-dimensional structure of polyethylene.

In this configuration the bond length between two carbon atoms is 0.154 nm, so that the distance between two adjacent carbon atoms projected along the chain length is 0.127 nm. As such, chains with 10 4 – 10 6 repeat units will have stretched out lengths of 1.27 μm to 0.127 mm. Of course, in general, the

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polymer chains are not stretched out; rather, they are balled up and entangled, so that the end-to-end distances are much shorter. However, they can attain sizes that can be observed under a microscope. Chain coiling and end-to-end distances of polymer chains are discussed in Section 10.2. With the molecular weights of hydrogen and carbon being 1 and 12 units, respectively, the molecular weight of this polyethylene macromolecule, H (CH2 CH2 )n H, is n × (molecular weight of 2 C2 H2 ) + 2 × (molecular weight of H) = 28n + 2 units. Since n is on the order of 104 – 106 , the molecular weight of H (CH2 CH2 )n H is essentially 28n. The molecular weight of a polymer, a measure of the chain length, or the number of repeat units n, is an important characteristic as it determines many physical properties of the material, such as its viscosity, elastic modulus, and strength. More generally, a linear polymer with mer A, chain length n, and end caps E can be written as E (A)n E, or, as shown in Figure 9.2.6, in which the first chain has n repeat units of the mer A. For large n, the molecular weight of E (A)n E will be n × (molecular weight of A).

E

A

A

A

A

A

E , or E

A

n

E

Figure 9.2.6 Linear polymer formed with mer A.

The properties of a linear polymer are mainly determined by the chemical structure of the mer, A, and the molecular weight, nmA , of the polymer – where mA is the molecular weight of the mer – and, to a lesser extent, by the chemical structure of the end cap E. The chemical structure of the mer determines the characteristic properties of the resulting polymers, such as its stiffness and resistance to heat. The molecular weight, or chain length, has a large effect on the physical properties. For example, for a given mer, the melt viscosity and the stiffness of the polymer increase with increasing molecular weights. The end caps do not affect the main characteristics or the physical properties; they mainly affect how the polymer responds to water and other solvents. 9.2.1.2

Branched Polyethylene

Instead of forming linear polyethylene, during polymerization some C2 H2 mers can form short or longer side chains, such as the C2 H2 H, C2 H2 C2 H2 H, and C2 H2 C2 H2 C2 H2 C2 H2 C2 H2 H side chains shown in the structure in Figure 9.2.7. Such side chains can be much longer.

H H

CH2

C

H CH2

C

H CH2

C

CH2

CH2

CH2

H

CH2

CH2

H

CH2

CH2

CH2

CH2

CH2

H

H

Figure 9.2.7 Branched polyethylene with side chains.

More generally, a branched polymer with mer A and end caps E has the chain structure shown in Figure 9.2.8.

Introduction to Polymers

E

A

A

A

A

A

A

A

A

A

E

A

A

E

A

A

A

A

E

E

Figure 9.2.8 Branched polymer formed with mer A.

In general, both the lengths of the side chains and their locations along the main chain are randomly distributed. 9.2.2

Polypropylene

Polypropylene is obtained by polymerizing propylene, CH2 CHCH3 , again an unsaturated hydrocarbon molecule, which has the structural formula

H

H

C

C

H

CH3

to give the structural formula in Figure 9.2.9.

H

H

H

C

C

H

CH3

H , or H

CH2

CHCH3

H n

n

Figure 9.2.9 Structural formula for polypropylene.

In contrast to polyethylene, polypropylene can have several different structural variants that affect its properties. The first difference is in the way the successive mers CH2 CHCH3 are connected. The head-to-toe connection results in the form H (CH2 CHCH3 )n H. However, the mers can be connected in a head-to-head mode in which adjacent mers are connected as — CH2 CHCH3 — CHCH3 CH2 —. And along the chain in different segments the mers can be connected in the head-to-toe and head-to-head modes. Just as with polyethylene, polypropylene can be linear or branched. 9.2.2.1 Tacticity

The other difference of polypropylene from the simple structure of polyethylene relates to the location of the methyl group CH3 (Me) along the C-C backbone, which in the simplest form is always on the same side as shown in the structure in Figure 9.2.10.

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H H

H H C

H H

C

C

C

C

C

C

H H C

C

C

H Me H Me

H Me

H Me

H Me

H H

Figure 9.2.10 Structural formula for isotactic polypropylene.

In another variant, the location of the methyl group along the chain can alternate as shown in the structure in Figure 9.2.11.

C

H H

H H

H H

C

C

C

C

H H C

C

H H C

C

H Me

H Me Me H

H Me Me H

C

Figure 9.2.11 Structural formula for syndiotactic polypropylene.

And the placement of Me can vary randomly, as shown in Figure 9.2.12.

H H C

H H

H H C

C

C

C

H H C

C

C

C H Me

H Me

H Me Me H

H H C H Me

Figure 9.2.12 Structural formula for atactic H(CH2 CHR)n H.

This type of ordering of a chemical group R, such as CH3 in the case of propylene, is referred to as tacticity. A polymer in which the mers are connected in the same way along the chain is said to be isotactic. When the successive units in a chain vary in a regular way the polymer is said to be syndiotactic. In an atactic polymer, the mers are connected in an irregular manner. More generally, isotactic, syndiotactic, and atactic ordering of polymers H (CH2 CHR)n H will have the structures shown, respectively, in Figures 9.2.13, 9.2.14, and 9.2.15.

H H

H H C

C H R

C

H H C

HR

C

H H C

HR

C

H H C

H R

C

C H R

Figure 9.2.13 Structural formula for isotactic H(CH2 CHR)n H.

Introduction to Polymers

H H

H H C

C

C

H H C

C

C

C

H H C

C

C H R

R H

H R

R H

H R

H H

Figure 9.2.14 Structural formula for syndiotactic H(CH2 CHR)n H.

H H

H H C

C

C

H H C

C

RH

H R

H H C

C

HR

H H C

C

H R

C H R

Figure 9.2.15 Structural formula for atactic H(CH2 CHR)n H.

9.2.3

Cis and Trans Isomers

In contrast to the propylene molecule, which has just one configuration, 2-butene has the two distinct isomers shown in Figure 9.2.16a and b. These two molecules are distinct because the double bond cannot rotate.

CH3

C

C

CH3 H

H

(a)

CH3

H C

C CH3

H

(b)

Figure 9.2.16 Two isomers of butene: (a) cis-2-butene and (b) trans-2-butene.

In the first molecule, called cis-2-butene, the CH3 groups are on the same side of the carbon double bond. In the first molecule, called trans-2-butene, the CH3 groups are on opposite sides of the carbon double bond. The terms cis and trans come from Latin, in which cis means “on this side,” or “on the same side,” and trans means “across,” or “on the other side.” While such isomers occur in molecules with double bonds, they can also occur in other structures with restricted bond rotations. Polymers chains containing double bonds can form cis and trans versions that can have very different properties, as exhibited by the different polymers based on isoprene. 9.2.4

Polyisoprene

The isoprene molecule, which has two double bonds, has the structure

CH2

CH

C CH3

CH2

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The three-dimensional structure of this molecule, also called 2-methyl-1,3-butadiene, can be written as the two seemingly different isomers

CH2

C

C

H

CH2 CH3

and

CH2

C

C

CH3 CH2

H

While at first glance these two molecules may appear to be different isomers, because of the rotations possible about the CC bond they are the same, as can be seen by rotating —CCH2 CH3 by 180° about the central CC bond. However, when a molecule with two double bonds polymerizes, one of the double bonds can remain in the chain, and can result in the polymer having several different isomers. Polyisoprene can have the four different isomers shown in Figure 9.2.17a – d.

CH2

C

C

H

CH2 CH3

CH2

n

C

C

CH2

H

(a)

(b)

CH3

H

CH2 C

CH2 C

C

C

CH2

H

n

(c)

CH3

CH2

CH3

n

n

(d)

Figure 9.2.17 Four isomers of polyisoprene: (a) cis-1,4-polyisoprene and (b) trans-1,4-polyisoprene.

Of these, cis-1,4-polyisoprene is an amorphous, rubbery polymer that is the major component of natural rubber. In contrast, trans-1,4-polyisoprene, which has a more symmetric structure, is a rigid semicrystalline material that is a major component of naturally occurring gutta percha. Thus, different isomers of polymers based on the same molecule can have very different properties. 9.2.5

Homopolymers and Copolymers

So far, simple polymers, each formed from a single mer, have been considered; they are called homopolymers. For enhancing properties, polymers having two or more different mers along the chain can be synthesized; the resulting polymers are called copolymers. An example is the ethylene-propylene copolymer, which is a thermoplastic elastomer called EPM rubber that has the structure shown in Figure 9.2.18, in which the numbers m and n determine the sizes of the repeat units that determine the mechanical characteristics of the material.

Introduction to Polymers

H

CH2

CH2

m

CH2

CHCH3

H

n

Figure 9.2.18 Copolymer of ethylene and polypropylene.

More generally, a two-component or binary copolymer of two mers (monomers) A and B can be connected in many ways: In an alternating copolymer the two mers are distributed in a regularly alternating manner, as shown in Figure 9.2.19. E

A

B

A

B

A

A

B

A

E

Figure 9.2.19 Alternating copolymer of mers A and B.

In a statistical polymer, the two components are distributed unevenly, perhaps randomly, as shown in Figure 9.2.20. E

A

BB

AAA

B

AA

BBB

A

BBB

AA

B

AAAAA

Figure 9.2.20 Statistical copolymer of mers A and B.

In block copolymers, a long sequence of connected mers of one monomer, called a block, are connected to a block of the second monomer, as illustrated in Figure 9.2.21.

E

AAAA

BBBBBBB

AAAAAAA

BBBB

AAAA

BBBBBBBB

Figure 9.2.21 Block copolymer of mers A and B.

In a graft copolymer, one component is attached as side chains to the main chain as shown in Figure 9.2.22.

E

A

A

A

A

A

A

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

Figure 9.2.22 Graft copolymer of mers of B grafted as side chains on linear polymer of mer A.

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9.2.6

Chain Entanglement

In a thermoplastic the polymeric chains are not stretched out and aligned. Rather, the chains are randomly coiled. In a molten or solid state these coils interpenetrate each other. In amorphous polymers the chains are randomly oriented with no internal structure. In semicrystalline materials, such as polyethylene and polypropylene, aligned segments can form crystals that are embedded in an amorphous matrix. Aspects of polymer chain structure, and the amorphous and semicrystalline states are addressed in Chapter 10. For sufficiently long chains, as in industrial applications, these coils can become physically entangled. But the entanglements are so spread out that the chains segments between entanglements are long enough to behave like individual chains. However, by preventing chain alignment, such chain entanglements do hinder crystallization. Also, they result in an increase in the melt viscosity.

9.3 Molecular Weight Distributions The molecular weight of a polymer chain of prescribed length is easily calculated. However, all commercial polymers come with a wide distribution of chain lengths and hence a wide distribution of molecular weights. Such a mixture does not have a molecular weight since this definition does not apply to mixtures. Instead, their size is characterized by an average molecular weight. Of the several different possible averages the two most commonly used ones are the number-average molecular weight and the weight-average molecular weight. The variation of molecular weights can be described by a molecular distribution curve as schematically shown in Figure 9.3.1, in which the number of molecules having molecular weight Mi is Ni , so that their weight is Wi = Ni Mi . Let the total number of molecules in the sample of total weight W be N. It then

NUMBER OF MOLECULES N

226

Mn Ni Mw

0 Mi MOLECULAR WEIGHT M Figure 9.3.1 Molecular weight distribution curve.

Introduction to Polymers

follows that ∞ ∞ ∞ ∑ ∑ ∑ N= Ni and W = Wi = Ni Mi i =1

i =1

(9.3.1)

i =1

The number-average molecular weight, M n , and the weight-average molecular weight, M w , are then defined by ∞ ∑

W Mn = = N

∞ ∑

Ni Mi

i =1 ∞

∑

and M w = Ni

i =1

∞ ∑

Wi Mi

i =1

W

=

Ni Mi 2

i =1 ∞

∑

(9.3.2) Ni Mi

i =1

Polymers having chains of the same length are said to be monodisperse, while those having a distribution of different chain lengths are said to be polydisperse. In general, M w ≥ M n , with the two being equal only for monodisperse systems. The ratio M w ∕M n is a measure for polydispersity.

9.4 Thermosets In contrast to thermoplastics, in thermosets the coiled, entangled chains are physically pinned together by irreversible, crosslinking chemical reactions to form relatively rigid parts that do not soften on heating. As a result, once formed, a part can only be reshaped by machining processes. 9.4.1

Phenolics

The first, and commonest, thermosetting plastic polymers are phenolics. They are made by reacting phenol and formaldehyde to form thermosetting resin oligomers, called novolacs and resoles, which are mixed with hexamethylene tetramine, heated, and formed into powders. During part forming these powders are molded under compression at higher temperatures in heated molds, when constituents in the powder react (cure), forming a highly crosslinked, rigid structure. The curing reaction produce gases and water that have to diffuse from the part interior to the surface, resulting in a significant shrinkage in the part size. Phenolics and other thermoset materials are discussed in more detail in Chapter 13. 9.4.2

Elastomers

Elastomers are elastic polymers that can be thermoplastics (Section 11.9) or thermosets (Section 13.4), such as the rubber used in automotive tires.

9.5 Concluding Remarks The examples of two very simple polymers, polyethylene, and polypropylene, have been used to introduce the definitions and terminology used for polymers. These two polymers are widely used, but in modified forms to stabilize them and to obtain desirable physical characteristics. Such modifications and a large

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number of industrially important thermoplastic polymers are discussed in Chapter 11. Blends (mixtures) of two or more polymers are discussed in Chapter 12, and Chapter 13 addresses thermoset polymers. The discussion in this chapter has described the chemical architecture of polymer chains, but has not addressed how they physically occupy space in a liquid or solid state. How chains occupy space, the motion of chains, and how they affect the properties of polymers forms the subject of polymer physics, some aspects of which, including the difference between amorphous and semicrystalline polymers, are addressed in Chapter 10.

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10 Concepts from Polymer Physics 10.1 Introduction The structure of polymers, which are built from basic units of long chain macromolecules, differs very significantly from that of metals, the basic units of which are atoms and molecules. Most metals are polycrystalline materials while most polymers are amorphous, in which the chains are not arranged in a regular, ordered manner. Polymers exhibiting crystallinity are really semicrystalline materials in which crystalline structures are embedded in amorphous matrices. And, in contrast to metals, the crystalline structures of polymers are very complex. These basic differences between metals and polymers translate into significant qualitative and quantitative differences between their properties. While the chemical structures of polymers do affect their properties, their behavior is mainly governed by how the constituent long molecular chains are organized and entangled. Polymer physics addresses how chain architecture affects properties. This highly mathematical discipline, which has evolved in the last 75 years, is concerned with developing conceptual frameworks for predicting the macroscopic behavior of polymers in the liquid and solid states on the basis of the behavior of molecular chains. The insights provided by polymer physics into the behavior of polymers are important for developing and modifying new classes of polymers. The focus of this book is not on understanding why polymers behave the way they do. Rather, it is on the use of polymers for which a phenomenological characterization of polymer behavior is adequate. Nevertheless, an acquaintance with the physics underlying polymer behavior is useful for a use focus, especially since engineers are mostly familiar with metals, the underlying physics for which is very different from that for polymeric materials. In view of the scope of this book, with mathematical analysis limited to basic calculus, this chapter focuses on concepts relating to polymer chains and summarizes results on properties of polymer chain aggregates.

10.2 Chain Conformations Thus far the main characteristics of a polymeric macromolecule discussed have been the chemical structure of the mers and the chain length or its molecular weight. Other than the length of a polyethylene chain, the physical configurations occupied by chains have not been considered. Rotations about carbon-carbon bonds cause the chains to twist around, resulting in twisted chains. Each chain can twist in different ways to take on different shapes, each of which is referred to as a conformation. Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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For different conformations of the same chain the end-to-end distance can be different. And, a fixed end-to-end distance smaller than the chain length, can result from many different conformations. Of all possible conformations not all are equally likely: Consider a simple model for a molecular chain of length L = nl consisting of n freely jointed links, each of length l, such that at each joint adjacent links are free to rotate in any direction. Then, there is only one conformation in which the links are stretched out in a straight line. In this unique conformation the distance r between its two ends is r = L = nl. If the two ends are fixed such that end-to-end distance is less than the chain length, that is, for r < L, there will be many more possible conformations. And the number of possible conformations will increase as r decreases. Not all rotations about a bond are equally likely because of interatomic repulsion among atoms on a chain, and such restrictions must be accounted for in a statistical description of chains. As an example, because the bond angle between three carbon atoms is 109.466° ≈ 109.5°, with l = 0.154 nm, the end-to-end distance for a zigzag chain with n mers would be r = 0.127 n nm instead of 0.154 n nm, resulting in a 17.5% reduction in chain length. The very large number possible conformations of a chain requires a statistical description of the chain shapes. Consider a material sample consisting of a very large number of identical chains. Each chain in the sample can take on many different conformations over time, so that one way of characterizing its shape is through a time average over these conformations. Another way of obtaining average characteristics is to obtain an average over conformations of a large number of different chains in the sample at one time instant. In the liquid state, since each chain can take on all possible conformations over time, both these averaging methods give the same result. However, in the solid state, in which the chains are not free to rapidly take on all conformations, only the instantaneous spatial average is appropriate for a statistical characterization of the chain behavior. In the following sections, some results, which explain important concepts, are described. 10.2.1

*Freely Jointed Chain Models

The simplest model is that of a freely jointed chain, in which n links, each of length l, are joined such that adjacent links at a joint are free to rotate in any direction. Instead of a complete mathematical analysis, a brief description of how of how the final result is obtained will be outlined. Consider a very large number, say N, of conformations of the chain. Let the end-to-end distance of the 𝛼 th chain be r𝛼 . Then, the square of the mean end-to-end distance, characterized by the RMS (root mean square) value R, is given by R2 = r 2 = < r 2 > =

N 1∑ 2 r N 𝛼 =1 𝛼

(10.2.1)

In the 𝛼 th conformation, that is, in the 𝛼 th chain, let the spatial position of the ith link be described by a vector l𝛼 , i of length l. Then, the end-to-end position for this chain, described by a vector r𝛼 , having length r𝛼 , is given by r𝛼 =

n ∑

l𝛼 , i

i =1

After some mathematical manipulation, the end-to-end distance is given by r𝛼2 = nl 2 + 2l 2 (cos 𝜃𝛼 ,12 + cos 𝜃𝛼 ,13 + · · · + cos 𝜃𝛼 , n, n−1 )

Concepts from Polymer Physics

in which 𝜃 𝛼 ,ij is the angle between the ith and the jth links in the 𝛼 th chain. It then follows from Eq. 10.2.1 that r2 = < r2 > =

N ( )] 1 ∑[ 2 nl + 2l 2 cos 𝜃𝛼 ,12 + cos 𝜃𝛼 ,13 + · · · + cos 𝜃𝛼 , n, n−1 N 𝛼 =1

N ) 2l 2 ∑ ( = nl + cos 𝜃𝛼 ,12 + cos 𝜃𝛼 ,13 + · · · + cos 𝜃𝛼 , n, n−1 N 𝛼 =1 2

(10.2.2)

Since the angles 𝜃 𝛼 , ij vary randomly over all possible values, 𝜋 + 𝜃 𝛼 , ij will occur with equal frequency for very large values of N, so that for every term cos 𝜃 𝛼 , ij will be a corresponding term − cos 𝜃 𝛼 , ij . As a result, for very large N, N ∑ ( ) cos 𝜃𝛼 ,12 + cos 𝜃𝛼 ,13 + · · · + cos 𝜃𝛼 , n, n−1 = 0

(10.2.3)

rfree 2 = r 2 = < r 2 > = nl 2

(10.2.4)

𝛼 =1

so that

wherein r has been replaced by rfree to emphasize that this value of corresponds to a freely jointed chain. It follows that RMS end-to-end distance is Rfree = n1∕2 l

(10.2.5)

and that Rfree 1 = 1∕2 L n

(10.2.6)

This simple equation shows that for long chains, that is, large n, the chain RMS end-to-end distance, R, is very small in comparison to the chain length L. For example for n = 104 and n = 106 , R is 1% and 0.1% of the chain length L, respectively. As a result, even though some chains may be in long extended conformations, most of them are in closely coiled states.

10.2.2

*Effect of Bond Angle Restriction

In the freely jointed chain model adjacent links are free to rotate in any direction. However, if the rotational motion of two adjacent links is constrained to a fixed included angle 𝜃 then the cos 𝜃 𝛼 , ij terms in Eq. 10.2.2 do not add up to zero. It can be shown that for this case, called the valence angle model, N ∑ ( ) 2 cos 𝜃 cos 𝜃𝛼 ,12 + cos 𝜃𝛼 ,13 + · · · + cos 𝜃𝛼 , n, n−1 ≈ − nl 2 1 + cos 𝜃 𝛼 =1

so that the RMS end-to-end distance is ( ) 2 cos 𝜃 2 2 2 rrba = < r > = nl 1 − 1 + cos 𝜃

(10.2.7)

(10.2.8)

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For a CC bond angle of 𝜃 = 109.5°, with cos 𝜃 = − 0.333, (−2 cos 𝜃 )∕(1 + cos 𝜃 ) is very close to 1, so that rrba 2 = 2nl 2

(10.2.9)

where the subscript rrba is used to emphasize that this result holds for a fixed bond angle restriction. The RMS end-to-end distance Rrba is then 1.4 times longer than that for a freely jointed chain. 10.2.3

*Effect of Steric Restrictions

In the previous section although the bond angle between two links is assumed to be constant, the second link can rotate freely about the first link while maintaining this fixed included angle 𝜃 . However, as the links rotate the distances between the side atoms change. As a result, the potential energy of the molecule changes, thereby rendering some positions more stable than others. For example, consider ethane, C2 H6 . Figure 10.2.1a shows the cis conformation in which the two carbon atoms and two hydrogen atoms are in a plane with the hydrogen atoms located on the same side of the CC bond. In this conformation one pair of hydrogen atoms, shown by dark arrows, stick out of the paper, while the remaining pair, shown by dashed arrows, sticks into the paper. In this eclipsed conformation the hydrogen atoms are so aligned that, when viewed along the CC bond, the triad of hydrogen atoms on the right are eclipsed by the triad on the left. This is shown in Figure 10.2.1b, in which the hashed bonds, here shown slightly rotated clockwise for purposes of visualization, actually align with the dark bonds that totally eclipse the former. It turns out that the repulsions among the hydrogen atoms causes this conformation to have the maximum potential energy, thereby rendering it less likely. Figure 10.2.1 parts c and d show a conformation in which the hydrogen atoms on the right have been rotated in a clockwise direction by an angle 𝜙. And Figure 10.2.1 parts e and f show the conformation with 𝜙 = 60°, in which the two carbon atoms and two hydrogen atoms are in a plane with the hydrogen atoms located on opposite sides of the CC bond. This staggered conformation, called the trans conformation, has the minimum potential energy, and is therefore more likely.

H

H C

H

C

H

H H

H

H C

H

C

(a)

(c)

H H

HH

H H

HH (b)

H H

H

H H

H (d)

H

H H

C

H

C

H

H (e) H H

H

H

H

H H (f)

Figure 10.2.1 Conformations of ethane molecules; (a) cis conformation, (b) eclipsed hydrogen atoms in cis conformation, (c) and (d) conformation in which the hydrogen atoms on the right of (c) have been rotated in a clockwise direction by an angle 𝜙. (d) and (e) trans conformation for which 𝜙 = 60°.

Concepts from Polymer Physics

For ethane, the variation of the potential energy is described by the approximation u𝜙 = u60 (1 + cos 3 𝜙) Some conformations of n-butane are shown in Figure 10.2.2, in which Me stands for the methyl group CH3 . Here again, the planar cis conformation (𝜙 = 0°), in which the two carbon atoms and two methyl groups are in a plane with the methyl groups located on the same side of the CC bond, has the highest potential energy. And the trans conformation (𝜙 = 180°) has the lowest potential energy. This molecule also exhibits two equal local minima at (𝜙 = 60°), and (𝜙 = − 60°) (or 𝜙 = 300°); these two conformations are, respectively, referred to as gauche+ and gauche− . This molecule also exhibits two equal local maxima at 𝜙 = 120° and 𝜙 = −120° (or 𝜙 = 240°).

Me C

H

Me

Me

H

H

C

H

H

C

H

H

(a)

H

Me H

C

HH (b)

H (d)

Me (e)

Me Me H H

H

C

H

(c)

Me Me H H

Me C H

H

Me H

H

H

H Me (f)

Figure 10.2.2 Some conformations of n-butane molecules; (a) and (b) cis conformation (𝜙 = 0°). (c) and (d) conformation in which the atoms on the right of (c) have been rotated in a clockwise direction by an angle 𝜙. (d) and (e) trans conformation for which 𝜙 = 180°.

In longer chains, such steric interaction will be more complex. In a simplified analysis of the polyethylene chain, the rotational motions of each set of three adjacent bonds in a conformation are restricted to those that correspond to energy minima, resulting in the following expression for the mean square end-to-end ) ( )( 1+ < cos 𝜙 > 2 cos 𝜃 2 2 2 (10.2.10) rsteric = < r > = nl 1 − 1 + cos 𝜃 1− < cos 𝜙 > in which < cos 𝜙 >, the average value of cos𝜙, is positive for the preferred minimum energy configurations, so (1+ < cos 𝜙 >)∕(1− < cos 𝜙 >) ≥ 1. It follows that Rsteric > Rrba > Rfree

(10.2.11)

The greater the restrictions on the chain link model conformations the larger is the chain RMS end-to-end distance R. For example, the phenyl side groups in polystyrene result in a significant increase in R.

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10.3 Amorphous Polymers On cooling, while some polymer melts partially crystallize at relatively slower cooling rates, the bulk of polymers solidify into unstructured, amorphous conformations. Amorphous polymers do not crystallize no matter how slow the cooling process is. In the molten state, polymer chains are in a state of constant, thermally activated motion; the intensity of which increases with temperature. The chains are in random conformations that do not exhibit an ordered structure; the melt is an amorphous state with no long-range structure. On cooling, the decrease in thermal motions results in the melt becoming more viscous. Eventually, at a low enough temperature, the viscosity becomes so high that the material is in a solid state in which the chains are still in an unstructured, amorphous state. The structure is frozen in such that thermal motions are absent. This amorphous, unstructured state is called the glassy state. Thermal transitions in amorphous polymers are phenomenologically different from those of small-molecule materials such as ice and metals, in which, at a constant pressure, sharp transitions at fixed temperatures occur from the solid to the liquid state at the melting point and from the liquid to the vapor state at the boiling point. First solid polymers do not vaporize; they tend to disintegrate before vaporization. And the solid-liquid transition in polymers is not sharp; it occurs over a range of temperatures that depend on the degree of polydispersity. At low molecular melts the chains in a melt or solution tend to act independently. At high molecular weights the longer chains tend to entangle, thereby hindering the motion of other chains. The transition from independently acting chains to interacting ones occurs at the entanglement molecular weight. This shift has a marked effect on the behavior of the material: The dependence of the viscosity on molecular weight becomes stronger above this molecular weight. And the mechanical properties, such as the elastic moduli, increase significantly. Modeling of how the properties of a polymer depend on its molecular weight is an important part of polymer physics. Such models, in which chains move relative to each other by snake-like, slithering motion called reptation, have provided mechanisms for explaining observed phenomena. Because of the introductory nature of this book, the level of mathematics required precludes a quantitative discussion of such theories. 10.3.1

Phenomenology of the Glass Transition

In an amorphous polymer, the transition from the liquid to the solid state with decreasing temperature occurs very gradually, going through an intermediate “rubbery” state. The transition from the hard, brittle glass to a softer rubbery state occurs over a narrow temperature range referred to as the glass transition; this range is characterized by the glass transition temperature, Tg . As such, the glass transition temperature is the single most important characteristic defining the regime of polymer behavior. Much work has been done in developing theories that describe glass transition in terms of the motion of the constituent chains. A discussion of such theories is well beyond the scope of this book. Instead, phenomenology is used to introduce the concept of glass transition. The glass transition is best described by the changes in the specific volume of the material with decreasing temperature as a melt is cooled to the solid state. As shown in Figure 10.3.1, on cooling the melt at a

Concepts from Polymer Physics

constant rate along ABC, the specific volume of the melt decreases steadily – for all practical purposes, linearly – and then, over a very narrow temperature range the rate of decrease of the specific volume decreases substantially. As indicated in the figure, the glass transition temperature, Tg , is defined by the intersection at G of the asymptotes of the curves on either side of this transition.

SPECIFIC VOLUME v

A

B C

•

G

Tg TEMPERATURE T Figure 10.3.1 Definition of the glass transition temperature Tg .

The glass transition temperature depends on the cooling rate: Figure 10.3.2 shows two cooling curves, A1 B1 C1 and A2 B2 C2 , at cooling rates r1 and r2 , respectively, with r1 > r2 ; higher cooling rates result in higher transition temperatures. The glass transition temperature also depends on the pressure: as indicated in Figure 10.3.3, higher pressures result in higher transition temperatures; as the melt pressure increasing from p0 to p1 , the locus of Tg is along S0 S1 S2 . Normally, a cooling rate of 1°C per minute is used to determine Tg . However, in industrial processes, such as injection molding, the cooling rates are much higher, in the range of 10 – 50°C s−1 . The large variations in cooling rates and pressures in such processes require data that are normally not considered in the literature on polymers, which is mainly concerned with explaining and quantifying the origins of such phenomenon.

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A1

SPECIFIC VOLUME v

236

COOLING RATE r

A2

r1 > r2

B1 C1 C2

r1 r2

• G1 B2

• G2

Tg

2

Tg

1

TEMPERATURE T Figure 10.3.2 Effect of the cooling rate on the glass transition temperature Tg .

10.3.2

Physical Aging

The dependence of the glass transition temperature on the cooling rate indicates the nonequilibrium nature of the solidification process. More rapid cooling “freezes in” higher-temperature equilibrium structures than when the melt is cooled more slowly, allowing the high-temperature structures more time to relax. Consider the cooling of an amorphous melt at a constant cooling rate that would normally follow the locus ABC in Figure 10.3.4. If, instead, the melt is first cooled at the same rate till point I located a few degrees below Tg , and then kept at the same temperature TI , thereafter, the specific volume will slowly decrease along the path IE till it reaches the “equilibrium” volume at E. This phenomenon, in which the material slowly relaxes to an equilibrium state even at constant temperatures, is called physical aging. The rate of this relaxation, resulting in a slow but continuous decrease of specific volume, decreases with decreasing temperature. 10.3.3

Concept of Free Volume

Let the volume in space occupied by a sample of the material be V, and the actual volume of the polymer molecules in that sample be Vp . Then the free volume Vf is defined by Vf = V − Vp

(10.3.1)

In a simplistic sense, the free volume may be thought of as the space available for polymer molecules to move around in. At high temperatures, in the liquid state Vf is assumed to be large enough to allow ample space for the thermally activated molecules to move around in. With decreasing temperature, as the thermally activated motions become slower, Vf is assumed to decrease, thereby decreasing the space

Concepts from Polymer Physics

SPECIFIC VOLUME v

A0 p0

p1 > p 2 > p3

S0

•

B0 B1 B2

p1 S1

•

Tg ( p0 )

•S

p2

A1 A2

2

Tg ( p 2 )

TEMPERATURE T Figure 10.3.3 Effect of the pressure on the glass transition temperature Tg .

available for the motion of the molecules. Eventually, at some temperature, Vf will have become small enough to prevent all motions of the molecules; this temperature is assumed to be Tg , since below this temperature the molecules are effectively frozen in a glassy state. This conceptual model provides the rationale for the increase in the viscosity of a melt with decreasing temperature. A simple model for free volume variation with temperature is shown in Figure 10.3.5, in which the line ABC shows the variation of the total volume V, and line FGH shows the variation of the polymer sample volume Vp , assumed to vary linearly with the temperature T. In this figure the cross-hatched region then represents the free volume Vf = V − Vp , and the line segment BG is the free volume Vf (Tg ) at Tg . For this linear model, above Tg , Vf = Vf (Tg ) + 𝛼f (T − Tg )V

(10.3.2)

in which the constant 𝛼 f is the rate of thermal expansion per unit sample volume with temperature difference T − Tg . This linear model is better described by defining the fractional free volume by f = Vf ∕V, in terms of which f = fg + 𝛼f (T − Tg )

(10.3.3)

where fg is the fractional free volume at Tg ; 𝛼 f may be called the thermal expansion coefficient of f above Tg . Empirical viscosity models can be used to develop models for Tg . A good empirical model for the variation of the viscosity, 𝜂 , of polymers with the free volume is the Doolittle equation ( ) 1 ln 𝜂 = ln A + B −1 (10.3.4) f

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A

SPECIFIC VOLUME v

238

C

B

I

•G

• •E TI

Tg

TEMPERATURE T Figure 10.3.4 Physical aging: Specific volume change along IE due to relaxation at constant temperature TI .

in which A and B are constants. A substitution in this equation from the linear, free-volume model in Eq. 10.3.3 gives [ ] 1 ln 𝜂 = ln A + B −1 fg + 𝛼f (T − Tg ) Subtracting from this equation its value at Tg , ln 𝜂g = ln A + B ( 1∕fg − 1), results in ] [ [ ] 𝜂 (T) 1 B 1 1 =− =B − ln 𝜂 (Tg ) fg + 𝛼f (T − Tg ) fg fg 1 + ( fg ∕𝛼f )∕(T − Tg ) which can be written as [ ] 𝜂 (T) B 1 1 log = log aT = − 𝜂 (Tg ) 2.303 fg 1 + ( fg ∕𝛼f )∕(T − Tg )

(10.3.5)

With C1 = B∕(2.303 fg ) and C2 = fg ∕𝛼f , this equation is the same as the Williams–Landel–Ferry (WLF) equation discussed in Chapter 7. 10.3.4

Effect of Pressure on Glass Transition

The free-volume concept can be used to model the effect of pressure on Tg . Increasing pressure can be thought as causing a decrease in the available free volume, resulting in lack of mobility associated the

Concepts from Polymer Physics

SAMPLE VOLUME v

A

B C H

G

• •

F

Tg TEMPERATURE T Figure 10.3.5 Simple model for the variation of free volume with temperature.

glass transition occurring at a higher temperature, that is, causing Tg to increase. The decrease in free volume due to an increase in pressure may be described by the linear model f = f0 − 𝛽f (p − p0 )

(10.3.6)

in which f0 is the free volume at a reference pressure p0 , and the coefficient 𝛽 f is a measure of the “compressibility” of the free volume. The effects of the temperature and the pressure on the free volume can then be modeled by f = f0 + 𝛼f (T − T0 ) − 𝛽f (p − p0 )

10.3.5

(10.3.7)

Effect of Chemical Structure on Glass Transition

The chemical structure of the polymer does effect glass transition. Flexible chain backbones, in which rotation about the main-chain bonds is easier, such as those of polyethylene, result in low glass transition temperatures. The addition of structures such as that of benzene decreases chain flexibility, resulting in higher transition temperatures. High-temperature polymers, which are discussed in Section 11.6, have very stiff backbones that hinder rotations about the main-chain bonds. Side chains also decrease chain flexibility by hindering rotations about the main chin bonds, and thereby increase the glass transition temperature; stiffer side chains have a larger effect than more flexible ones.

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10.3.6

Effect of Molecular Weight on Glass Transition

Increasing molecular weight results in increasing Tg . This effect can be modeled by using the free-volume concept. By assuming that chain free ends have greater mobility, or more free volume available, it can be shown that Tg = Tg∞ − c∕M

(10.3.8)

in which c is a constant and Tg∞ is the glass transition temperature at very high molecular weights.

10.4 Semicrystalline Polymers Crystallization – which involves solidification from a polymer solution or polymer melt into well-defined, geometrically regular shapes called crystals – is facilitated by stereoregular chain structures; the most regular of which, isotactic polymers, always result in crystalline solids. Depending on the amount of regularity, syndiotactic polymers can also be crystalline. But atactic polymers do not solidify into crystalline states. However, crystalline polymers consist of crystals embedded in amorphous matrices: All crystalline polymers are really semicrystalline materials. And the degree of crystallinity depends on the cooling rate; lower cooling rates result in higher crystallinity. In some semicrystalline polymers, rapid cooling can result in a purely amorphous state. All commercial polymers have chains of varying lengths, that is, they come with a distribution of molecular weights. For any polymer, a narrower molecular weight distribution – in which the lengths of chains are more uniform – promotes faster and more uniform crystallization. Polyethylene and polypropylene, discussed in Chapter 9, are both examples of semicrystalline polymers.

10.4.1

Structure of Polymer Crystals

The structures of polymer crystals are very complex and involve regular chain folding. A detailed exposition of this important topic is well beyond the scope of this book; only a very broad overview is presented. Two types of polymer crystal structures are important: those grown from solutions and those grown from melts. Single crystals can be grown from very dilute solutions. As an example, those of linear polyethylene can be grown from very dilute solutions in xylene. Such single crystals have lamellar shapes (flat platelets) with thicknesses on the order of 10 nm and lateral (“in-plane”) dimensions on the order of a micrometer, so that the ratio of the lateral dimension to the thickness is on the order of 1,000. The polymer chains within the lamellae are aligned normal to the lateral surface. As the distance between carbon atoms along the chain direction is 0.25 nm, a 10 – 15 nm crystal thickness implies that only about 50 – 60 carbon pairs (about 100 – 120 carbon atoms) span the crystal thickness. It follows that a polymer chain having 1,000 – 10,000 carbon atoms must fold many times within this thickness. Some portion of the chain may fold into an adjoining crystal after passing through an amorphous region. Polyethylene crystals grown in a dilute solution actually have a hollow pyramidal shape that can flatten on drying. More concentrated solutions result in lamellae with a spiral overgrowth pattern.

Concepts from Polymer Physics

Single crystals provide important insights into the basic mechanisms of crystal formation in polymers, such as in elucidating the mechanism of chain folding. However, crystals in engineering applications – in which they crystallize from the melt – have very different complex structures. Rather than platelets, the melt-grown crystals have complex, spherical crystalline structures called spherulites, with diameters in the 0.1 – 1.0 mm range. Spherulites are formed from lamellae the structures of which have similarities with those of single crystals grown from solution, but are less perfect than single crystals. For example, chain folding is less regular; and they often branch as in polypropylene or twist as in polyethylene. The lamellae include some amorphous regions (chain ends or irregularities in the chain) that mingle with the external amorphous zones. As crystallization progresses, low molecular weight material, irregular chains, and additives present in the polymeric material are pushed away from the lamellae to their boundaries into the amorphous zones. The concept of spherulites embedded in an amorphous matrix only appears to be valid for low crystallinity levels. During crystallization, spherulites nucleating at different times at different points in the melt will have different diameters. At some point the growing spherulites begin to contact neighboring spherulites, causing the contact surfaces to flatten, resulting in crystal structures that exhibit polygonal boundaries. The sequence of micrographs in Figure 10.4.1 show the process of growth and coalescence of spherulites during the crystallization of polypropylene at 130°C in a microscope hot stage, a flat furnace with transparent windows that fits on top of a microscope stage. A controller makes it possible to heat

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10.4.1 Evolution of the spherulitic microstructure in a polypropylene melt at 130°C in a microscope hot stage. Note that

each white length marker in this figure is 0.1 mm (100 μm) long. (a) Spherulite 4 minutes after nucleation. (b) Spherulite six minutes after nucleation. Notice a second smaller spherulite. (c) Spherulites 8 minutes after nucleation of first spherulite. (d) Spherulites 12 minutes after nucleation of first spherulite. (e) Spherulites 14 minutes after nucleation of first spherulite. Notice that the two spherulites have just come into contact. (f) Polygonal boundaries formed by contacting spherulites. (Photos courtesy of Professor J. Oliveira, University of Minho.)

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or cool material placed inside the furnace between a glass slide and a cover slip at a pre-set rate or at a fixed temperature. Note that each white length marker in this figure is 0.1 mm (100 μm) long. Figure 10.4.1a shows the micrograph of a single spherulite with a diameter of about 0.1 mm (100 μm), about 4 minutes after nucleation. The photo in Figure 10.4.1b, taken after 2 minutes, shows the increase in its size and a second, smaller spherulite. The views in Figure 10.4.1c – e track the growth of these two spherulites after additional time intervals of 2, 6, and 10 minutes, respectively; notice that in the last image the two spherulites are just coming into contact. Eventually, the growing spherulites push into each other, resulting in the relatively homogeneous crystallized structure shown in Figure 10.4.1f. However, because of intense shear and differential cooling rates, the crystalline morphology in a molded part is nonhomogeneous, and the size and shape of the deformed spherulitic structure in a part varies through the part thickness (Figures 17.4.1 and 17.4.2). 10.4.2

Melting Phenomenology of Semicrystalline Polymers

Polymer crystals have regular structures having sharply defined melting points at which the molten material is in an amorphous state. In contrast to the amorphous phase, the regular structures of the crystalline phase result in the same amount of crystalline material occupying less space. As a result, on melting the specific volume increases sharply. Conversely, when the melt is cooled, at the melting point the specific volume of the material undergoes a sharp reduction. This phenomenon is illustrated in Figure 10.4.2, in which the amorphous phase cools along ABC, and undergoes a glass transition at point G. If the polymer does not crystallize easily, then, for sufficiently rapid cooling the specific volume will undergo this variation. If the cooling rate is sufficiently slow, the amorphous melt will begin to crystallize at the melting point Tm , which is higher than Tg . Because of crystallization, at Tm the material will undergo a large reduction in the specific volume along M1 M2 . The drop M1 M2 is the contribution of the crystallized phase. On further cooling, the semi crystalline phase – the mixture of amorphous and crystalline phases – follow the path M2 M3 M4 . The volumetric drop M1 M2, and consequently the drop CM4 , will be larger for larger degrees of crystallization. Empirically, it has been found that Tg ∕Tm varies in the range of 0.5 – 0.8, when the temperature is measured in Kelvin. 10.4.3

Degree of Crystallinity

The rate of cooling effects crystallization from a melt in two ways: Faster cooling allows less time for nucleation and growth of crystals; in some polymers faster cooling can result in a purely amorphous solid. And, slower cooling rates allow the crystals to grow. In any semicrystalline material the physical properties depend both on the size of the crystals and on the degree of crystallinity or the overall crystal content. One measure for the degree of crystallization is the ratio of the volume of crystals in a sample to its total volume. Let the volumes of the amorphous and crystal phases in a volume VSC of a semicrystalline polymer be VA and VC , respectively, so that the degree of crystallization is given by VC ∕VSC . And let the respective masses of the materials in these volumes be mSC , mA , and mC . Then, VSC = VA + VC and mSC = mA + mC

(10.4.1)

With m = 𝜌V applying to each of the three phases, the mass balance equation (Eq. 10.4.1) gives

𝜌SC VSC = 𝜌A VA + 𝜌C VC = 𝜌A (VSC − VC ) + 𝜌C VC

Concepts from Polymer Physics

SPECIFIC VOLUME v

A

M1 M2 B AMORPHOUS

C M4

•G M3

SEMICRYSTALLINE

Tg

Tm

TEMPERATURE T Figure 10.4.2 Comparison of cooling curves for amorphous and semicrystalline materials.

which, on simplification, gives the degree of crystallization as VC 𝜌 − 𝜌A = SC VSC 𝜌C − 𝜌A

(10.4.2)

and, with the expression for the specific volume v = 1∕𝜌 applying to each of the three phases, results in the alternative expression VC v v − vSC = C A VSC vSC vA − vC

(10.4.3)

In Eq. 10.4.3, the specific volumes vA and vSC for a polymer at any temperature can be obtained from cooling curves of the type shown in Figure 10.4.2. For determining the degree of crystallization, the density 𝜌C of the crystals has to be determined independently.

10.5 Liquid Crystal Polymers Matter mostly comes in three states: Solid, liquid, and gaseous. In the solid-state constituent atoms, molecules, or chains cannot move about freely; the only motion is thermal vibrations about the equilibrium position. A solid may be crystalline – in which on a microscopic level its constituents are organized

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in a three-dimensional, long-range positional, and orientational order in a periodic manner – or amorphous – in which the constituents do not exhibit any long-range order. Aspects of the amorphous and crystalline states of polymers have been discussed, respectively, in Sections 10.3 and 10.4. With increasing temperature, the thermal vibrations in the solid state become stronger and, at the melting temperature, the crystals melt into an isotropic-liquid state that has no long-range orientational order. And, at still higher temperatures, the liquid passes into the gaseous phase in which the constituent molecules are not positionally bound, and are free to move around randomly. However, in addition to these three common states there exists another state, generally referred to as the mesomorphic state, in which the degree of order is intermediate between that of crystals and liquids. A subset of this state, called the liquid crystal (LC) state, has long-range orientational order and either partial positional order or complete positional disorder. This LC state exists between the crystalline solid and the isotropic-liquid states; among other stimuli, its form is controlled by temperature, pressure, and electric fields. Many LCs can have more than one phase in the LC state, each of which is referred to as a mesophase. 10.5.1

Liquid Crystal Phases and Transitions

In the simplest phase of a liquid crystal (LC), called a nematic LC, the microscopic constituents are aligned along a preferred direction, called the director. They have long-range orientational order but no positional order, which can be altered, for example, by the application of an electric or magnetic field. Such phases are highly anisotropic. In the smectic phase, in addition to maintaining the orientational order of the nematic phase, the microscopic constituents tend to align themselves in layers or planes to which the motion is restricted. And different planes “slide” past each other. Because of this increased order, in comparison to the nematic phase, the smectic phase is more solid-like. Some smectic phases exhibit chirality (a chiral molecule is one that is not superimposable on its mirror image). The cholesteric (or chiral nematic) phase consists of nematic constituents containing a chiral center, resulting in a structure consisting of a stack of thin two-dimensional nematic layers with the director in a layer being rotated in its plane relative to the directors of the adjacent planes; the directors actually form in a helix along the normal to the layers. This chiral structure results in cholesteric LCs exhibiting interesting optical effects. As the temperature of an LC solid is increased, at some temperature it goes into a smectic mesophase, which, at a higher temperature goes into a nematic phase. Finally, at a still higher temperature, the material becomes an isotropic liquid. Thus, the transition of an LC from the solid to the isotropic-liquid state involves several phase transition temperatures: The first from the solid to the smectic phase, the second from the smectic to the nematic stage, and the third from the nematic to the isotropic-liquid state. Such LCs, in which the phase transitions are induced thermally are said to be thermotropic. If the solid to smectic to nematic to isotropic-liquid transitions induced by increasing temperatures are reversed on cooling the isotropic-liquid phase, the LC is said to be an enantiotropic LC. If such transitions are not reversible but are only caused either by increasing temperatures or on cooling the material is called a monotropic LC. Another class of LCs, called lyotropic LCs, differs from thermotropic LCs in that the phase transitions in lyotropic materials are induced by solvent-induced aggregation of the constituents into micellar structures, which, with increasing concentration increase in size coalesce and separate from the solvent into an LC state.

Concepts from Polymer Physics

10.5.2

Polymer Liquid Crystals

Polymer liquid crystals (PLCs), which can have the same mesophases of LCs described in the previous section, combine the useful properties of polymers with those of LCs. PLCs are made by including rod-like or disk-like mesogens in the backbone of flexible chain polymers. As shown in Figure 10.5.1, PLCs are of two types: In main-chain PLCs the mesogens are a part of the main chain. This is effected either by the backbone being made up by stiff rod-like monomers – which can include several aromatic rings – or, as in most such materials, by inserting stiff mesogens in the main chain. In the second type, side-chain PLCs, the mesogens are inserted as side chains to the main chain.

n n n

(a)

(b)

(c)

Figure 10.5.1 Liquid crystal polymers (PLCs) in which the main polymer backbone is shown by zigzag lines and the mesogens are shown by slim, cross-hatched rectangular objects. (a) Main-chain PLC. (b) and (c) Side-chain PLCs.

10.6 Concluding Remarks The discussion in this chapter has highlighted how the major differences between the molecular architecture of polymeric materials – that consist of large molecules in the form of chains – and metals and ceramics – that are made up of molecular aggregates – result in the very large differences observed in their macroscopic behavior. Although the dynamics of chains and how they affect properties is beyond the scope of this book, some aspects have been discussed to provide insights of how observed macroscopic phenomena exhibited by polymers can be understood in terms of how chains behave. This is particularly true of the key concept of the glass transition temperature, which is important for understanding the behavior of amorphous polymers. The brief discussions on semicrystalline and liquid crystal polymers highlight the complexity of the behavior of these materials, an appreciation of which is important for understanding issues relating to the technical uses of these materials.

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11 Structure, Properties, and Applications of Plastics 11.1 Introduction This chapter describes the chemical structure and properties of many industrially important thermoplastics used in a wide variety of applications. The word plastic comes from roots that refer to “of molding,” “to mold,” and “form.” It can refer to materials that can be molded, such as clay, which can be deformed continuously and permanently without rupture. In engineering, the word plastic connotes a host of organic synthetic or processed materials – mostly thermoplastic or thermosetting polymers that can be molded, cast, extruded, or drawn; or laminated into parts; or formed into films or fibers. Most industrial plastics are synthetic polymers modified by additives or fillers for several reasons: such as to reduce degradation caused by ultraviolet (UV) radiation; improve chemical resistance, fire retardance, and processability; attain desirable color and surface finish; reduce cost through inexpensive fillers; improve mechanical properties such as stiffness, strength, fatigue, and impact resistance; and attain desirable dielectric properties to improve antistatic properties, electrical insulation, and electromagnetic shielding. While modifiers can provide improved performance, the properties of the resulting plastic are mainly governed by those of the base polymer, which in industry is referred to as the base resin. The base polymer, or resin, can be a thermoplastic – which softens and melts on heating, and parts can be melted and reshaped – or a thermoset, which undergoes a chemical reaction during forming, so that a part cannot be reshaped other than by machining processes. Even base resins come with different molecular-weight distributions having different physical properties, so that their engineering properties have to be specified separately for each material grade. In contrast to data on simple resins published in science journals, much of the compositional data on commercial grades of plastics is proprietary. And, in contrast to information on metals, the data on the properties of plastics materials published by manufacturers is not necessarily of a scientific nature. This is partially a reflection on the state of the plastics industry, which still focuses mainly on manufacturing and marketing plastics. As such, the physical and mechanical properties of the plastics discussed in the sequel should be treated as representatives for each plastic type. It is worth reiterating that modifiers can even effect a qualitative change in properties. For example, the use of chopped fibers in a plastic makes it anisotropic.

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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11.2 Resin Grades An industrial resin is normally made up of polymeric chains having a wide distribution of molecular weights characterized by an average molecular-weight. They are sold in unfilled and filled grades that may contain many additives. The process of mechanically mixing all the ingredients together in a melt phase to produce moldable pellets is called compounding. Different unfilled grades of a resin are tailored for specific application types, and mainly differ in their average molecular weights. Lower molecular-weight grades tend to have lower viscosities and are therefore easier to mold, but this results in lower toughness, that is, in more brittle resins. Higher molecular-weight grades are tougher, but can require much higher molding temperatures and pressures. Special grades for blow molding applications require higher melt strengths. For injection molding, the resins are sold in the form of small, cylindrical pellets having nominal diameters and lengths in the 3 – 4 and 3 – 6 mm ranges, respectively. For foam molding these pellets may contain a blowing agent that releases a gas during molding, resulting in a bubble-filled structure. Rotational molding requires resins in a powder form. Particulate- and glass-filled resins can have filler contents as high as 70 wt%. In most applications the glass fibers have diameters and lengths in the ranges of 7 – 18 μm and 0.2 – 3 mm, respectively. Typically, the use of 10, 20, and 30 wt% of glass in a plastic increases the elastic modulus and the strength of a resin by about 50, 100, and 150%, respectively. Higher stiffness and strength can be obtained by using longer fibers with lengths on the order of 10 mm, for which the pellet lengths are of this order. For glazing applications transparent plastics are sold in sheet form. Sheet material is also used for vacuum and thermoforming applications. And sheets with high loadings of random glass-fiber mats are available for thermostamping applications.

11.3 Additives and Modifiers Synthetic polymers are seldom used as chemically homogeneous resins. Rather, they are compounded with two types of ingredients: Additives, mainly stabilizers – added in very small amounts, thereby having very small effects on mechanical properties – are used to slow down or prevent chemical degradation and to improve processability. Modifiers are used in larger proportions, mainly to improve the esthetics and physical properties of resins. The development of appropriate additives and modifiers is a vast, important subject in itself. And the companies that develop them tend to be different from those that manufacture and market resins. The purpose of this section is to create an awareness of the complexity of compounded plastic resins; they can be substantially different from the base resins from which they are compounded. While the responsibility of developing appropriate grades rests with companies that market plastics, it is important for users to understand the advantages and limitations of the resulting additive containing materials. 11.3.1

Stabilizers

Stabilizers are used to ameliorate four types of chemically induced degradation. The first three address chemical degradation caused by UV radiation, oxidation by oxygen in the air, and high-temperature degradation during the processing of resin into parts. The fourth relates to fire retardance.

Structure, Properties, and Applications of Plastics

11.3.1.1 UV Stabilizers

Photons from ambient UV radiation in the 290 – 400 nm range absorbed by chemical groups in a polymer can cause chain scission, that is, a separation of a polymer at a C — C bond, resulting in shorter chains with lower molecular weights, which results in performance degradation such as embrittlement. This type of degradation is especially important for outdoor application of plastics; the ability of plastics to withstand ambient degradation is referred to as weatherability. Two types of stabilizers are used to improve the long-term stability of polymers. The first, are better UV absorbers than the polymer, and are therefore referred to as absorbers. This absorbed radiation is converted into harmless infrared radiation. The second kind, called quenchers, exchange energy from the excited molecules, thereby diffusing the effect of the UV radiation. Pigments used for imparting desired colors offer UV protection by absorbing UV radiation. Carbon black, which absorbs all visible radiation, is also an excellent UV absorber, but its limitation is its color. Other types of stabilizers have to be used when transparency is required. 11.3.1.2 Antioxidants

Even at moderate temperatures, reactions with ambient oxygen and ozone can form free radicals that, by shortening chains, can result in polymer degradation. Two types of antioxidants are used: Primary antioxidants are free-radical scavengers that inhibit oxidation by reacting with radicals that would otherwise continue to degenerate polymer chains. Secondary antioxidants break down peroxides formed by oxidation. 11.3.1.3 Thermal Stabilizers

During processing, such as in injection molding, a resin sees much higher temperatures than during applications. This can result in degradation, requiring the use of thermal stabilizers. This is of particular importance for polyvinyl chloride, a commonly used plastic. 11.3.1.4 Fire Retardants

The burn cycle of a polymer subjected to heat in the presence of oxygen involves the formation of combustible volatiles, inert gases, and smoke. The combustion of the volatiles provides the heat for continuing the burn cycle and for flame spreading. Some of the combustion products may be toxic. Fire retardants are used reduce these effects in a variety of ways. Although reactive flame retardants, in which the retarding compounds are part of the polymer structure, are sometimes used for thermoplastics – they are mainly used with thermosets – the most commonly used flame retardants are additives that contain organic compounds of bromine, chlorine, and phosphorus. Inorganic compounds, such as alumina trihydrate and magnesium hydroxide are also used. Different types and amounts of flame retardants are used for different resins. And the mechanisms by which they provide retardance are also different: In halogen (bromine or chlorine)-based organic retardants the burn cycle is interrupted by the halogen radicals generated by combining with the radicals produced by the burning resin. Their effectiveness is enhanced by the use of antimony oxide. Phosphorus-based retardants generate phosphorus oxides that form a glass-like coating on the burning surface, thereby depriving it of oxygen. And alumina trihydrate releases water to inhibit the burn cycle.

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11.3.2

Modifiers

Modifiers are used to improve the appearance and physical properties of a base resin. Clearly, because processability of plastics allows for esthetic designs, color is an important consideration. And enhancement of physical properties is always desirable. 11.3.2.1

Colorants

Desirable color can be obtained by either using dyes or pigments. Dyes are compounds that are soluble in the base resin. Because of easy dispersion, they provide means for achieving both good color and transparency. But they can suffer from thermal stability and can react with other degradation products. And, because of solubility, they can migrate into surrounding objects in contact with a part. The bulk of color modification in plastics is achieved through the use of pigments, which are insoluble particulates. They must be capable of withstanding high processing temperatures. The mixing of finely ground pigments involves high shear rates, which can necessitate the use of special purpose high-speed, high-intensity mixers. Desirable properties of pigments include chemical inertness and thermal stability. Although many of the pigments are metal oxides, the most common of which, titanium dioxide, is used for obtaining white color; metal salts are also used. Carbon black is used for obtaining black color. Organic pigments are also available. Fine flakes of metals, such as aluminum, are used for creating a metallic appearance. Special transparent pigment flakes are available for creating a pearlescent effect. 11.3.2.2

Fillers

Fillers are essentially additives used to reduce the cost of resins. Their shapes can be spherical – which helps in maintaining the isotropy of the resin – short and long fibers, ribbons, and flakes, all of which result in anisotropy. Fillers can also be classified as extenders – which mainly reduce cost; increasing concentrations of such fillers while reducing cost result in property degradation – and functional fillers which also improve properties of the material. Finely ground limestone, mainly comprising calcium carbonate, is a commonly used inexpensive filler that does not affect surface appearance. The use of stearate-coated fine particles prevents particle agglomeration and in reduced abrasion of processing equipment. Other fillers include fine Kaolin clay – a hydrous aluminosilicate mineral in the form of platelets – and talc, a hydrated magnesium silicate. Carbon black is an important filler, especially for rubber, which besides acting as a black pigment also provides UV protection. 11.3.2.3

Reinforcing Fibers

Chopped or milled glass fibers are used to improve the stiffness and strengths of resins. The diameters of the fibers are in the 10 – 13 μm range. For injection molding applications 1 – 2 mm long short fibers are used resulting in length-to-diameter ratios in the 50 – 100 range, which is important for achieving the reinforcing effect. The use of longer fibers does not help because of the length reduction that the fibers undergo in the injection molding extruder. However, longer fibers, with enhanced reinforcing effect, can be used with molding machines with special nozzles. Much longer fibers, with lengths up to 50 mm, can be used in compression molding applications. The fibers normally are coated with coupling agents that help the resin to adhere to the fiber surface, which is important for achieving the reinforcing effect. While the use of fibers improves some of the

Structure, Properties, and Applications of Plastics

mechanical properties of the resin, the impact strength is reduced, and molded parts no longer have a glossy surface. And they introduce a high degree of anisotropy. They are also more difficult to process and cause abrasive wear of the molding machine surfaces. For special applications, such as for electromagnetic interference (EMI) shielding, carbon and metal fibers can be used. 11.3.2.4 Impact Modifiers

Most polymeric resins are brittle at low temperatures and at high deformation rates that occur during impact, that is, they fail at relatively low strains without undergoing plastic deformation. As a manifestation of this, for example, a plastic panel impacted by a projectile – such as a stone impacting an automobile body – could shatter into shards instead of undergoing a local puncture. This lack of ductility – the ability to undergo large deformations prior to failure – can be overcome by using impact modifiers that, essentially are complex elastomeric (rubber) particles. How impact modifiers work and how they are made is a subject in itself. The elastomers used tend to be block copolymers with hard and soft segments. And, by a process of successive polymerizations, the particles can have a core-shell structure in which the main elastomer is encapsulated in a shell designed to adhere well to the resin being modified. The particles have to remain stable during the high temperatures and shear rates that occur in many part manufacturing processes. 11.3.2.5 Lubricants

Lubricant additives, used in small amounts, serve a twofold purpose. First, by facilitating motions among neighboring polymeric chains they are used to reduce the resin viscosity during process operations. And second, they are used to reduce the friction between the resin and the processing machine and mold surfaces during processing operations. 11.3.2.6 Plasticizers

Plasticizers are added to resins to improve their flexibility and processability. They are either nonvolatile organic liquids or organic solids with low melting points that tend to lower the glass transition temperature of a resin. A resin is rigid at room temperature if its Tg is higher than the room temperature. It can be made flexible by using a plasticizer to lower its Tg below room temperature. Plasticizers improve the processability of a resin by decreasing its viscosity.

11.4 Polyolefins The chemical structures of the two most commonly used polyolefins – polyethylene (PE) and polypropylene (PP) both of which are semicrystalline polymers – have already been discussed in Chapter 9. The third in this series, polybutylene (PB), which is also semicrystalline, is discussed in Section 11.4.3. Of the three, the symmetric nature of the polyethylene chain results in polyethylene being a highly crystalline material; PB is the least crystalline of the three. 11.4.1

Polyethylene

Polyethylene (PE) is available in many forms and hundreds of grades, both as homopolymers and copolymers; glass-fiber filled and impact modified grades are also available. PE is chemically very inert, does

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not dissolve in any solvent at room temperature, and resists alkalis and acids. It has low coefficients of friction. Without treating its surface, PE is very difficult to paint or print on. It has excellent dielectric properties, making it very useful as an insulating material for wires and cables. PE thin films are transparent but the material becomes progressively translucent with increasing thickness. PE is normally classified into the following categories: low-density (branched) polyethylene (LDPE), linear low-density polyethylene (LLDPE), high-density (linear) polyethylene (HDPE), high molecular-weight polyethylene (HMWPE), and ultrahigh molecular-weight polyethylene (UHMWPE). Some properties of the different types of PEs are listed in Table 11.4.1. Table 11.4.1 Some properties of polyethylenes. Property

LDPE

LLDPE

HDPE

HMWPE

UHMWPE

Density

g cm−3

0.917 – 0.932

0.918 – 0.940

0.952 – 0.965

0.947 – 0.955

0.94

Melting point Tm

°C (°F)

98 – 115 (208 – 239)

122 – 124 (251 – 255)

130 – 137 (266 – 279)

125 – 135 (257 – 275)

125 – 138 (257 – 280)

Tensile modulusGPa (103 psi)

0.17 – 0.28 (25 – 41)

0.26 – 0.52 (38 – 75)

1.07 – 1.09 (155 – 158)

0.94 (136)

0.83 (120)

Tensile yield strength

9 – 14.5 (1.3 – 2.1)

9.7 – 19.3 (1.4 – 2.8)

26.2 – 33.1 (3.8 – 4.8)

19.3 – 26.9 (2.8 – 3.9)

21.4 – 27.6 (3.1 – 4.0)

MPa (103 psi)

LDPE homopolymer, polymerized in a high-pressure process was the first polyethylene to be commercialized. Because of its resistance to chemicals and moisture, high tear strength, and local nicks not propagating, it is the material of choice for making thin films for packaging, with thicknesses in the 25 – 125-μm (0.001 – 0.005-in) range. Hindrance from long-chain branching results in it having relatively low crystallinity levels in the 50 – 60% range, making transparent films possible. It is also used in injection-, blow-, and rotational-molding applications to produce housewares, containers, and toys. Low-density copolymers of ethylene with vinyl acetate, ethyl acrylate, and methyl acrylate are also available. LLDPE is a linear copolymer of ethylene with 5 – 12 wt% butane, octane, or hexane. Because of very short side chains this material has very high crystallinity levels over 90%. While LLDPE has slightly better properties than LDPE, tougher films made of this material are less transparent than LDPE films. Like LLDPE, all the higher density PEs – such as HDPE, HMWPE, and UHMWPE – are linear polymers with very little side branching, having high crystallinity levels above 90%. The molecular weights of the linear PEs have weight-averaged molecular weights, M w , in the range of 105 to 2 × 105 . HDPE, having better properties than LDPE, is used for making stronger films, sheets, blow-molded bottles, injection-molded parts, such as pails, crates, bottle caps, houseware items, wire and cable insulation, conduits, and pipes. It is also used for making blow-molded automotive fuel tanks. The molecular weights of HMWPE are in a higher range of M w = 3 × 105 to 5 × 105 . It has higher strength and better resistance to environmental stress-cracking. It can be processed by normal plastic processing techniques, and is used for making large blow-molded containers and pipes. UHMWPE has molecular weights in the range M w = 3 × 105 to 6 × 105 . Its very high abrasion resistance and high-impact resistance makes it suitable for products such as bearings, sprockets, and conveyor-belt parts. Because of its very high viscosity – too high for normal molding processes – parts are made by compression molding.

Structure, Properties, and Applications of Plastics

11.4.1.1 High-Strength Polyethylene Fibers

Ultra-high-strength polyethylene fibers are made from UHMWPE by a gel spinning process that orients the chains to develop a parallel orientation in excess of 95%, and crystallinity levels in the 39 – 75% range, resulting in lightweight (density of 0.97 g cm−3 ), high-performance fibers with tensile strengths and tensile moduli as high as 3.25 and 118 GPa, respectively. These values compare favorably with those of high-strength steels, but the strengths of UHMWPE fibers are much higher than those of low-carbon steels. With an approximate density of steel of 7.7 g cm−3 , the strength-to-weight ratios for UHMWPE fibers are about 8 – 15 times higher than for steel. Also, the strength-to-weight ratios for these fibers are about 40% higher than those for aramid fibers (Section 11.7.7). Although UHMWPE has a melting temperature in the 130 – 136°C (266 – 277°F), because of excessive creep the use temperature for these fibers is restricted to the 80 – 100°C (176 – 212°F); it becomes brittle below −150°C (−240°F). UHMWPE fibers are used in personal body armor – sometimes even in vehicle armor, cut-resistant gloves, bow strings, climbing equipment, fishing lines, high-performance sails, suspension lines on sport parachutes and paragliders, and rigging in yachts. They are also used for cables and hawsers for ships, and for industrial heavy-duty slings.

11.4.2

Polypropylene

Most of the commercially available polypropylene (PP) homopolymer is isotactic, which has high crystallinity levels. With a density in the range of 0.90 – 0.91 g cm−3 , it is the lightest of the commercial plastics. Qualitatively, many of its properties are similar to those of polyethylene: High chemical and solvent resistance, excellent dielectric properties, low coefficient of friction, and high abrasion resistance. But, in contrast to PE, it is not prone to environmental stress cracking. It has a much higher melting point Tm of 165°C (197°F) so that it can be used in higher-temperature applications, such as for making electric kettle housings. With tensile moduli and strengths in the ranges of 1.14 – 1.55 GPa (165 − 225 × 103 psi) and 31 – 37.2 MPa (4.5 − 5.4 × 103 psi), respectively, it is stiffer and stronger than PE. But, in contrast to PE, which retains its impact resistance at low temperatures that of PP falls off. And, PP is less stable in terms oxidation and UV attack, requiring the use of appropriate stabilizers. PP has the highest fatigue resistance – resistance to cyclic loads – of all polymers, which makes possible the use of thin, molded-in “living hinges” of the types used in making single-piece molded scissor-like clamps (Figure 1.5.17) and bottle lids (Figure 2.2.13). The homopolymer is used for making films and sheet, injection-molded luggage shells, battery cases, and washing machine parts. Filled PP homopolymer is available in a wide variety of grades: 10 – 40 wt% talc-filled; 10 – 40 wt% calcium carbonate-filled; 10 – 40 wt% short-glass fiber-reinforced; 20 – 40% long glass-fiber reinforced, 10 – 50 wt% mica-filled; impact modified, 40 wt% mica-filled; 30 – 40 wt% of random glass mat reinforced thermo-stampable sheet; and 30 wt% polyacrylonitrile (PAN) carbon-fiber-filled material for EMI applications. Copolymers of PP with PE have better impact properties than PP; they are also available as talc-filled, calcium carbonate-filled, glass-fiber-reinforced grades. Polypropylene, with its high, stiffness, strength, chemical and solvent resistance, and relatively high melting temperature, is a versatile material that is finding increasing use in demanding automotive parts, such as 5-mph impact resistant bumpers. A talc-filled grade is used for one-piece molded dishwasher tubs (Figure 2.2.28a).

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11.4.3

Polybutylene

Polybutylene, the third in the series of polyolefins is obtained by polymerizing butylene, CH2 CHCH2 CH2 , to give

H

H

H

C

C

H

or

H,

H

CHCH2CH3

CH2

H n

CH2 CH3 n

PB crystallizes in several different forms. A metastable version forms on crystallization from the melt, as in injection molding. Subsequently, this form irreversibly changes to a stable form in several days. An extrusion grade of PB has a density in the range of 0.91 – 0.925 g cm−3 , and a melting point of 126°C (259°F). Its tensile moduli and tensile yield stresses in the ranges of 0.21 – 0.27 GPa (30 − 40 × 103 psi) and 11.7 – 17.2 MPa (1.7 − 2.5 × 103 psi), respectively, are comparable to those of low-density polyethylenes. Among flexible plastics, it has the highest rating for hydrostatic stress and is used for extruded pipes for cold and hot water applications.

11.5 Vinyl Polymers Vinyl materials are obtained by replacing one hydrogen atom in ethylene by a chemical group R; the resulting compound is called vinyl R. Polymerization of vinyl R gives poly(vinyl R):

H

H

C

C

H

R

⇒

H

H

H

C

C

H

R

H n

Examples of several important vinyl polymers follow. 11.5.1

Poly(Vinyl Chloride)

Poly(vinyl chloride), H (CH2 CHCCl)n H, referred to as PVC, obtained by polymerizing vinyl chloride,

H

H

C

C

H

Cl

⇒

H

H

H

C

C

H

Cl

H n

Structure, Properties, and Applications of Plastics

is a colorless, rigid, amorphous polymer with a glass transition temperature, Tg in the range of 75 – 105°C (167 – 221°F). It is believed to have a head-to-toe, partially syndiotactic structure. It has a relatively high density in the range of 01.30 – 1.58 g cm−3 , and its tensile moduli and tensile yield stresses are in the ranges of 2.41 – 4.14 GPa (350 − 600 × 103 psi) and 40.7 – 44.8 MPa (5.9 − 6.5 × 103 psi), respectively. It is a relatively inert material that resists acids and alkalis. It has good dielectric properties. PVC has poor thermal and UV stability, so much so that it degrades under normal molding conditions. However, adequate stabilizers are available for correcting these shortcomings. Extruded products represent the largest use of rigid PVC, the largest application being pipes and conduits. Extruded PVC is also extensively used in the construction industry for vinyl siding, window frames, and doors. Fittings and electrical outlet boxes are injection molded. Flexible forms of PVC are obtained by using plasticizers, such as dioctyl phthalate, which reduce the Tg of the material below room temperature. Applications of flexible PVC include flexible sheet, floor tiles, and electric wire insulation and tape. Rigid and flexible PVC are available in many unfilled and filled grades, including one with 20 wt% of short-glass fiber reinforcement. Several copolymers with improved moldability characteristics, such as copolymers with vinyl acetate – which were used for making vinyl music long playing (LP) records – are also available. PVC is used in many medical applications, such gloves, blood bags, and tubing. Much of the clear thin-film used by supermarkets and consumers for wrapping food is made of a copolymer of PVC with the highly crystalline poly(vinylidene chloride) that has the chemical structure

H

H

Cl

C

C

H

Cl

H n

11.5.1.1 Plastisol

Plastisol is a suspension of PVC particles in a liquid plasticizer, which is a liquid at room temperature. On heating to about 177°C (350°F) the resin and plasticizer combine to form plasticized PVC. On cooling below 60°C (140°F) this material becomes a flexible, plasticized solid. Besides molding applications, plastisol is used for coating items such as furniture and metal tools. It is also used as a textile ink in screen-printing. 11.5.2

Polyacrylonitrile

Polyacrylonitrile (PAN), H (CH2 CHCCN)n H, obtained by polymerizing acrylonitrile,

H

H

C

C

H

CN

⇒

H

H

H

C

C

H

CN

H n

is a semicrystalline polymer with a Tg of 105°C (221°F), a Tm of 318°C (604°F), and has an amorphous density of 1.184 g cm−3 at 25°C.

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Copolymers with PAN as the base component are used to spin synthetic wool fibers, which are referred to as “acrylic” fibers. PAN forms industrially important copolymers with styrene, butadiene, and an acrylic ester, such as styrene-acrylonitrile (SAN), acrylonitrile-butadiene-styrene (ABS), and acrylonitrile-styrene-acrylate (ASA). PAN fiber is used as a precursor for making high-strength and high-modulus carbon fibers. 11.5.3

Polystyrene

Polystyrene (PS), H (CH2 CHCC2 H5 )n H, obtained by polymerizing styrene, or vinyl benzene,

H

H

C

C

⇒

H

H

H

H

C

C

H

H n

is a clear, odorless, and tasteless rigid amorphous thermoplastic with a density of 1.05 g cm−3 , and tensile moduli and tensile yield stresses in the ranges of 2.76 – 3.45 GPa (400 − 500 × 103 psi) and 41.4 – 55.2 MPa (6 − 8 × 103 psi), respectively. It is available in grades with Tg in the range of 100 – 110°C (212 – 230°F). Although clear, the brittleness of the homopolymer is a barrier to its application when impact resistance is important. By dissolving polybutadiene in the styrene during the polymerization, a rubber modified PS, called high-impact polystyrene (HIPS), is produced. While HIPS is not brittle, it is translucent and its tensile moduli and tensile yield stresses are reduced to the ranges of 1.72 – 2.41 GPa (250 − 350 × 103 psi) and 13.8 – 34.5 MPa (2 − 5 × 103 psi), respectively. A very commonly used PS is expandable polystyrene (EPS) in which a blowing agent is added during the polymerization process. When expanded, by several available methods such as by the use of steam during a molding process, EPS results in lightweight, low-density cellular structures that are used in a wide variety of applications such as “foam” cups, molded forms to provide cushioning for packaging, backing for bumper systems, and foam peanuts – also called packing peanuts or packing noodles – to fill voids in packaging. 11.5.3.1

Poly(Styrene-co-Acrylonitrile) (SAN)

Poly(styrene-co-acrylonitrile), abbreviated as SAN, has the structure

H

H

H

H

C

C

C

C

H

CN

H n

m

It is a hard, rigid, transparent material, which transmits more than 90% of light. The n : m ratio of PS to acrylonitrile varies from 70 : 30 to 80 : 20. While higher acrylonitrile content improves mechanical properties and chemical resistance, it adds a yellow tint to the transparent plastic.

Structure, Properties, and Applications of Plastics

Because it retains its mechanical properties at higher temperature (Tg > 100°C), it is widely used in place of polystyrene. Its resistance to food stains (fats and oils) and to cleaning agents, and its appealing texture, has made it attractive for many home products and containers for cosmetics. It is also used in applications such as battery cases, plastic optical fibers, and cylindrical impellers for airconditioners. Its main limitation is brittleness, which led to the development of a SAN-rubber copolymer called ABS (Section 12.2.1). 11.5.3.2 Poly(Styrene-co-Maleic Anhydride) (SMA)

Styrene maleic anhydride, also known as SMA, is a near alternating copolymer of styrene and maleic anhydride having the structure

H

H

H

H

C

C

C

C C

C

H n

O

O

O

m

SMA is available in a broad range of molecular weights and maleic anhydride (MA) content. Its transparency offers design flexibility. It is available in impact-modified and fiber-filled grades. 11.5.4

Poly(Methyl Methacrylate)

Poly(methyl methacrylate) (PMMA), H (CH2 CHCH3 COOCH3 )n H, obtained by polymerizing methyl methacrylate,

H

CH3

C

C

H

COOCH3

⇒

H

H

CH3

C

C

H

COOCH3

H n

is a transparent, colorless amorphous thermoplastic with a glass transition temperature, Tg in the range of 85 – 105°C (185 – 221°F). With a density of 1.17 – 1.20 g cm−3 , which is less than half that of glass, PMMA often used for glazing and other optical applications. Its impact strength is higher than of both glass and polystyrene. It is available in both pellet and sheet forms. 11.5.5

Poly(Ethylene-co-Vinyl Alcohol)

Poly(ethylene vinyl alcohol), or poly(ethylene-co-vinyl alcohol), (EVOH), H(CH2 CH2 )m (CH2 CHOH)n H, a copolymer of polyethylene and vinyl alcohol, has the structure

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H

H

H

H

C

C

C

C

H

H

H

OH

n

m

It is a highly transparent, weather resistant, oil and solvent resistant polymer that exhibits excellent oxygen barrier properties – the gas barrier properties EVOH are 4,400 times better than those of HDPE. The ethylene content of the copolymer varies in the range of 27 – 44 mol%. The composition with 32 mol% of polyethylene is a semi-crystalline polymer with a Tg of 69°C (156°F), and a Tm of 183°C (361°F). Its barrier properties decrease with increasing polyethylene content. An EVOH layer is used in multilayer packaging films and bottles to act as an oxygen barrier for food packaging applications. It is also used as a barrier layer for gasoline in multilayer blow-molded automotive gas tanks.

11.6 High-Performance Polymers Most polymers considered thus far are relatively inexpensive polyolefin-based materials, except for polystyrene that has a benzene radical. These materials soften at relatively low temperatures. Materials for higher mechanical performance and for high-temperature applications have larger, more rigid mers or benzene rings in the backbone; they are therefore more expensive. The structures of several such materials are described next. 11.6.1

Polyoxymethylene

Polyoxymethylene (POM), also known as acetal, polyacetal, and polyformaldehyde, is a thermoplastic having the structure

H C H

O n

This semicrystalline polymer has a Tm of 172 – 184°C (342 – 363°F), and a density of 1.41 – 1.42 g cm−3 . It has a tensile modulus in the range 2.76 – 3.59 GPa (400 × 103 – 520 × 103 psi) and a tensile yield strength in the range 65.5 – 82.7 MPa (9.5 × 103 – 12 × 103 psi). Because of its high crystallinity (75 – 85%), the base resin has an opaque white color. Acetal is used for making high-performance engineering components such as gear wheels. It is also widely used in the automotive and consumer electronics industries. It is also available as a copolymer with dioxolane or ethylene oxide, which has a Tm of about 165°C (329°F) density of 1.40 g cm−3 , and somewhat lower mechanical properties: tensile modulus in the range

Structure, Properties, and Applications of Plastics

2.56 – 3.2 GPa (400 × 103 – 520 × 103 psi) and a tensile yield strength in the range 57.2 – 71.7 MPa (8.3 × 103 – 10.4 × 103 psi). 11.6.2

Poly(Phenylene Oxide)

Poly(2,6-dimethyl-1,4-phenylene ether), commonly known as poly(phenylene oxide), or PPO, has the structure

CH3 O CH3

n

With a high Tg of 215°C (419°F), this amorphous resin has a very high viscosity and is very difficult to mold. It is widely used in the form of blends with polystyrene with which it is fully miscible. Such miscible blends are discussed in Section 12.2.6. 11.6.3

Polyesters

The two most important thermoplastic polyesters are poly(ethylene terephthalate), commonly known as PET, which has the structure

O

O

O

C

C

O

CH2CH2 n

and poly(butylene terephthalate), commonly known as PBT, which has the structure

O

O

O

C

C

O

(CH2)4 n

Some of the properties of these two polyesters are listed in Table 11.6.1. Note that the densities of these two semicrystalline polymers vary over a range of values: The densities of the amorphous and crystalline phases – 1.33 and 1.50 g cm−3 , respectively, for PET – are different and the final density depends on the degree of crystallinity.

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Table 11.6.1 Some properties of PET and PBT. Property

PET

PBT

1.31 – 1.38

Density (semicrystalline)

g cm−3

1.29 – 1.40

Density (amorphous)

g cm−3

1.33

Density (crystalline)

g cm−3

1.50

Glass transition Tg

°C (°F)

68 – 80 (154 – 176)

66 (151)

Melting point Tm

°C (°F)

212 – 265 (414 – 509)

232 – 267 (450 – 513)

Tensile modulus

GPa (103 psi)

2.76 – 414 (400 – 600)

1.93 – 3.0 (280 – 4.36)

Tensile yield strength

MPa (103 psi)

59.3 (8.6)

56.5 – 60 (8.2 – 8.7)

The largest use of PET is for making textile fibers and tire cord. The next largest use is for making blow-molded bottles for carbonated soda, water, beer, juice, and detergents. Unoriented and biaxially oriented PET films find many uses: microfilm, drawing office drafting film, overhead transparencies, X-ray films, audio cassette tape, and videotape. It is also widely used for thermoformed packaging applications. Recycled PET bottles are used for making polyester carpets. It is also a base for several important blends (Chapter 12). The lower melting temperature of PBT, which makes it easier to mold than PET, is used in many injection-molded applications. It is extensively used for electrical fittings. The properties of PBT result from its high crystallinity levels. It comes in many filled grades. A heavy, barium sulfate-filled grade is used to mold chinaware. PBT is also used as a base for several important blends (Chapter 12). 11.6.4

Polycarbonate

Bisphenol A polycarbonate (PC), the most used polycarbonate, which is commonly referred to as polycarbonate, having the structure

O

CH3

O

C

C

CH3

O n

is a clear – clearer than many glasses – amorphous thermoplastic with a glass transition temperature of about 150°C (302°F), and a density 1.2 g cm−3 . It has a tensile modulus of 2.38 GPa (345 × 103 psi) and a tensile yield strength of 62.1 MPa (9 × 103 psi). PC is a highly ductile material with exceptional impact resistance: optical quality PC has been used for making single-piece cockpit canopies for fighter jets; it is also used for thick, transparent barriers for bank-teller windows. PC laminates are used to make bullet-resistant windows (Figure 2.5.2). Although tough, PC has low scratch resistance so that for glazing applications it is coated with scratch-resistant coatings. Because of its impact resistance and optical clarity, it is widely used for making molded auto-

Structure, Properties, and Applications of Plastics

motive headlamp lenses, sunglass and eyeglass lenses, and safety glasses. Compact CDs, DVDs, and Blu-ray discs are molded from this material. While most beverage and water bottles are molded from PET, the large bottles used in water dispensers are molded from PC. PC is also used as a base for several important blends (Chapter 12).

11.6.5

Polyamides

Polyamides are a class of polymers, commonly called nylons, in which the mers contain the amide group CONH. Nylons are commercially available both in the semicrystalline and amorphous forms, the former being the more common. All nylons are easy to process. But their tendency to absorb moisture causes dimensional stability problems and the degradation of electrical and mechanical properties. 11.6.5.1 Semicrystalline Polyamides

This is an important class of versatile polymers with good chemical, abrasion, and flammability resistance. Rubber modified versions also have good impact resistance. But being semicrystalline materials, they are not transparent and have dimensional stability (shrinkage and warpage) issues. They are used in a wide range of applications: automotive and transportation, electrical and electronics, appliances, business equipment, machinery, construction materials, film, sheet, and pipes. In addition to the aliphatic semicrystalline polyamides discussed in this section, aromatic semicrystalline polyamides with very high-temperature performance are available; they are discussed in the section on high-temperature materials. The two most common nylons, both of which are semi-crystalline, are poly(hexamethylene adipamide), or nylon 6,6, which has the structure

H N

(CH2)6

H

O

N

C

O (CH2)4

C n

and polycaprolactam, or nylon 6, which has the structure

O

H N

(CH2)5

C n

They are a part of a series of commercially available semicrystalline nylons having the structures

H N

(CH2)p

H

O

N

C

O (CH2)q

C n

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called nylon p, q, and

H

O

N

(CH2)p

C n

called nylon p. Their properties are listed in Tables 11.6.2 and 11.6.3. Table 11.6.2 Some properties of ( p, q) nylons. Property

Nylon 4,6

Values of p, q

Nylon 6,6

Nylon 6,9

Nylon 6,10

Nylon 6,12

4, 4

6, 4

6, 7

6, 8

6, 10

1.10 – 1.18

1.13 – 1.15

1.08 – 1.10

1.08

1.06 – 1.10

Glass transition °C (°F) Tg

40 (104)

51 (124)

55 (131)

46 (115)

40 (104)

Melting point Tm

295 (563)

255 – 265 (491 – 509)

205 (401)

277 – 299 (530 – 570)

195 – 219 (383 – 426)

1.72 – 3.0 (250 – 435)

1.59 – 3.8 (230 – 550)

1.9 (275)

2.4 (350)

1.5 – 2.0 (218 – 290)

30.0 – 237 (4.35 – 34.4)

40.0 – 57.9 (5.8 – 8.4)

g cm−3

Density

°C (°F)

Tensile modulusGPa (103 psi) Tensile yield strength

55.2 – 82.7 (8 – 12)

MPa (103 psi)

(Adapted with permission from Table 10.5, pp. 298 – 299, in “Nylon Plastics Handbook,” M.I. Kohan (Ed.), Hanser Publishers, Munich, 1995.)

Table 11.6.3 Some properties of p nylons. Property

Values of p −3

Nylon 6

Nylon 11

Nylon 12

5

10

11 1.01 – 1.02

Density

g cm

1.12 – 1.14

1.03 – 1.05

Glass transition Tg

51 (124)

44 (111)

40 (104)

Melting point Tm

°C (°F) °C (°F)

210 – 220 (410 – 428)

180 – 190 (356 – 374)

160 – 209 (320 – 408)

Tensile modulus

GPa (103 psi)

2.62 – 3.2 (380 – 464)

1.28 (185)

0.25 – 1.24 (36 – 180)

90.3 (13.1)

15.0 – 44.0 (2.1 – 6.38)

20.7 – 42.1 (3 – 6.1)

Tensile yield strength

3

MPa (10 psi)

(Adapted with permission from Table 10.5, pp. 298 – 299, in “Nylon Plastics Handbook,” M.I. Kohan (Ed.), Hanser Publishers, Munich, 1995.)

11.6.5.2

Amorphous Polyamides

The two main advantages of these materials over the semicrystalline forms are transparency, or clarity, and better dimensional stability. But their flammability resistance is not as good. Examples of several commercially available materials follow.

Structure, Properties, and Applications of Plastics

In addition to transportation, electrical and electronics, and business machine applications, the first two resins listed next, having glass transition temperatures Tg of 130 and 148°C, respectively, are also used in chemical processing applications.

H N

(CH2)10

O

CH3 H

C

N

CH3 CH2

H

O

N

C

C O n

H N

CH3 (CH2)2

C

CH3 CH2

C

CH2

H

O

O

N

C

C

CH3

n

With a Tg of 161°C a rubber modified version of the following resin is especially suited for automotive applications:

H

H O

O H

N (CH2)6

N C

C N

H O CH2

N C C O

n

With a Tg of 130°C, and good barrier properties, several versions of the following resin are suited for packaging applications:

H N

(CH2)6

H

O

N

C C O

11.6.6

n

Fluoropolymers

The most common fluoropolymers are obtained by polymerizing ethylene in which one or more of the hydrogen atoms have been substituted by fluorine. Polytetrafluoroethylene (PTFE), F (CF2 CF2 )n F, the

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most common fluoropolymer, is obtained by polymerizing tetrafluoroethylene:

F

F

C

C

F

F

⇒

F

F

F

C

C

F

F

F n

PTFE comes as a white powder having a density of about 2.2 g cm−3 . Because of its low coefficient of friction (0.05 – 0.10), it is used for parts, such as bearings and gears, in which mating surfaces need to slide. Its tendency to creep makes it useful for seals as it helps the seal to conform to the mating surface. Since PTFE does not melt like other thermoplastics, it has to be formed into parts by using special techniques such as compression molding; such parts are therefore made by specialty molders. Because of its inertness and low coefficient of friction it is used in non-stick coatings for cookware and as a stain-resistant coating for fabrics. This non-stick property has led to the use of the word Teflon, borrowed from Teflon®, DuPont’s registered trade for this material, to describe “slippery” characteristics of people. Besides poly(vinyl fluoride) (PVF), which has the structure (CH2 –CHF)n , other fluoropolymers include polyvinylidene fluoride (PVDF) having the structure (CH2 –CF2 )n , and polychlorotrifluoroethylene (PCTFE) that has the structure (CClF–CF2 )n . In contrast to PTFE these resins are melt processable. However, because of the backbones of these polymers, they do not have the chemical resistance of PTFE. 11.6.6.1

Copolymers of Fluoropolymers

Many copolymers of fluoropolymers were developed mainly to improve processability. Two important melt copolymers that can be processed in conventional injection molding machines are perfluoroalkoxy polymer (PFA), which has the structure

F

F

F

F

C

C

C

C

F

F

F

O

n

CF3

m

and fluorinated ethylene-propylene (FEP), which has the structure

F

F

F

F

C

C

C

C

F

F

F

CF3

n

m

Structure, Properties, and Applications of Plastics

Another such copolymer is poly(ethylene-co-tetrafluoroethylene), also called polyethylenetetrafluoroethylene (ETFE), is

H

H

F

F

C

C

C

C

H

H

F

F

n

This copolymer has a very high melting temperature and has excellent resistance to chemical, electrical, and high energy radiation.

11.7 High-Temperature Polymers Some representative high-temperature polymers are described below. 11.7.1

Poly(Phenylene Sulfide)

Poly(p-phenylene sulfide), commonly referred to as poly(phenylene sulfide) (PPS), is a semicrystalline polymer having the structure

S n

PPS has a glass transition temperature Tg of 85°C (185°F), a melting temperature Tm of 285°C (545°F), a density of 1.35 g cm−3 , a tensile modulus of 3.45 GPa (500 × 103 psi), and a tensile yield strength of 93.1 MPa (13.5 × 103 psi). The neat resin has a light tan color. PPS has no known solvents, even for long-term exposure at temperatures up to 200°C. Its chemical resistance includes all automotive fluids; it is also resistant to organic and inorganic solutions, acids and alkali solutions. In addition, PPS is inherently flame resistant. It comes in two forms: The older, branched (cross-linked and cured) and the newer linear form which is easier to process. PPS is available in a wide range of molecular weights – making it possible to injection mold thin-walled parts and to make blow-molded products – and with a large array of glass-fiber-filled (as high as 40% glass content) and glass-fiber-mineral combinations (with up to 70 wt% of fillers). Specially lubricated grades are available for bearing applications. PPS grades are also available for blow-molding applications, and fiber spinning for filters for chemical applications. It is also available in solid moldings that can be machined to close tolerances. The applications of PPS include many automotive applications such as powertrain components and pumps; blower and pump parts, impellers, and flow meters for industrial applications; and electric heater grills, power tool parts, microwave components, and insulators for consumer and appliance components.

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11.7.2

Polyetherimide

Polyetherimide (PEI), an amorphous polymer having the structure

O

O

C

C

N

N

CH3 C

O

C

O

C

CH3

O

O n

has a glass transition temperature Tg of 210°C (410°F), a density of 1.28 g cm−3 , a tensile modulus of 3.45 GPa (500 × 103 psi), and a tensile yield strength of 113.8 MPa (16.5 × 103 psi). The neat resin is a translucent material with an amber color. Besides an unfilled grade for injection molding, blow molding, foam molding, and extrusion applications, PEI is available in glass-filled grades with glass contents of 10, 20, 30, and 40 wt%. Special grades are available for bearing applications. Because of its heat and radiation resistance, hydrolytic stability, and transparency, PEI is used in medical applications. Its applications in the electronics include burn-in sockets, bobbins, and printed circuit substrates. Its automotive applications include lamp sockets and underhood temperature sensors. PEI plastic sheets are used in aircraft interiors. 11.7.3

Poly(Amide-Imide)

Poly(amide-imide) (PAI) is an amorphous thermoplastic having the structure

H N

CH2

O

O

C

C

N C O

n

It has a glass transition temperature Tg of 280°C (537°F), a density of 1.41 g cm−3 , a tensile modulus of 4.14 GPa (600 × 103 psi), and a tensile yield strength of 124.1 MPa (18 × 103 psi). PAI is the highest performing thermoplastic in the sense that it has higher stiffness and strength up to 275°C (525°F) than all other thermoplastics. PAI resins for extrusion and injection molding applications have relatively low molecular weights. They include unreinforced, glass-fiber reinforced, carbon-fiber reinforced, and wear resistant grades.

Structure, Properties, and Applications of Plastics

Because of their thermal stability, and chemical and abrasion resistance, special grades of PAI in powder form are used for magnet wire enamel.

11.7.4

Polysulfones

Polysulfones (PSUs) are thermoplastics containing aryl-SO2 -aryl groups (aryl stands for the benzene ring) in their backbone. The simplest polysulfone, having the structure

O S O

n

decomposes on melting at of 520°C (968°F), and is therefore not a useful thermoplastic. To make materials that can be processed at lower temperatures in conventional injection molding equipment, the stiff backbone of the polysulfone needs to be made more flexible. This is achieved by inserting ether links in the chain; the resulting materials are called polyethersulfones (PESs), all commercial versions of which are amorphous thermoplastics. Unfortunately, a unique set of generic names do not exist for these commercially available materials. The three polyethersulfones described below are amorphous resins that exhibit excellent thermal stability, high strength and toughness, excellent hydrolytic stability, transparency, and good resistance to environmental stress cracking. 11.7.4.1 Polysulfone

Polysulfone (PSU) is a polyethersulfone having the structure

CH3 O

C CH3

O O

S O

n

This material has a glass transition temperature Tg of 190°C (374°F), a density of 1.24 g cm−3 , a tensile modulus of 2.48 GPa (360 × 103 psi), and a tensile yield strength of 70 MPa (10.2 × 103 psi). Of the three polyethersulfones discussed in this section, it is the least expensive and has the lowest thermal performance, lowest hydrolytic stability, and the lowest resistance (environmental stress cracking) to organic solvents. Membranes of this resin with pores in the nanometer range are used in applications such as hemodialysis, waste water recovery, food and beverage processing, and gas separation. These polymers are also used in high-temperature automotive and electronic applications. Its high hydrolytic stability at high temperatures allows parts made of this material to be sterilized with steam or in an autoclave.

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11.7.4.2

Polyethersulfone

One commercial polysulfone, called polyethersulfone (PES), has the structure

O S

O

O

n

This amorphous resin has a glass transition temperature Tg of 220°C (428°F), a density of 1.37 g cm−3 , a tensile modulus of 2.65 GPa (385 × 103 psi), and a tensile yield strength of 83 MPa (12 × 103 psi). 11.7.4.3

Polyphenylsulfone (Polyarylethersulfone)

Polyphenylsulfone (PPSF), also called polyarylethersulfone, has the structure

O O

S

O

O

n

This amorphous thermoplastic has a glass transition temperature Tg of 220°C (428°F), a density of 1.29 g cm−3 , a tensile modulus of 2.65 GPa (385 × 103 psi), and a tensile yield strength of 70 MPa (10.1 × 103 psi). Of the three polyethersulfones discussed in this section, this resin is the most expensive and has the highest thermal performance, highest hydrolytic stability, and the highest resistance (environmental stress cracking) to organic solvents. 11.7.5

Polyketones

Two industrially important polyketones are: polyaryletherketone, which has the structure

O O

C n

and polyetheretherketone, commonly called PEEK, which has the structure

O O

O

C n

Structure, Properties, and Applications of Plastics

11.7.6

Liquid Crystalline Polyesters

Liquid crystal polymers (LCPs) are a class of polymers that, on melting, exhibit phases (mesophases) that exhibit characteristics of both solids and liquids. The general structure of such materials has been described in Section 10.5. While processing does align polymer chains, they tend to entangle and fold in amorphous and semicrystalline polymers. However, in LCPs the chains tend to align in the stretching direction, so that in fiber drawing processes they tend to form fibers that, because of the strong alignment of chains, have very high strengths. Two commercial liquid crystal (LC) polyesters, both of which are nematic LCs (Section 10.5.1), are essentially main chain aromatic thermotropic LCs, that are available in many neat and mineral-, graphite, glass- and carbon-fiber- and PTFE-filled grades. These LCPs are shear thinning materials, that is, their viscosities decrease with increasing shear rates, and their relatively low melt viscosities make it possible to mold intricate thin-walled parts. The main monomer used in all commercial thermotropic LCPs is 4-hydroxybenzoic acid (HBA). Although the homopolymer of HBA is an LC, it does not flow below 500°C (932°F). To reduce the processing temperature and to modify its properties it is copolymerized with several other monomers, such as 2,6-hydroxynaphthoic acid (HNA), 4,4 ′-biphenol (BP), and terephthalic acid (TA). While the exact structure of the backbones of these commercial LCPs are not known, they are thought to consist of HBA copolymerized with varying amounts HNA, BP, and TA. One such commercial LCP, called Xydar®, is said to have the structure of the type

O O

O

C

O

O

O C

C y

x

in which the chain in the first bracket is made up of oligomers of HBA and the chain in the second bracket is made up of contributions from BP and TA. The ratios x and y of these two chains control the properties of the proprietary resins; for one member of this family the ratio x : y is 2 : 1. The base resin is available in three grades with melting temperatures in the respective ranges of 415 – 430°C (780 – 805°F), 375 – 390°C (707 – 735°F), and 352 – 367°C (666 – 693°F). This material can be injection molded into thin-walled components and has high strength at high temperatures to 300°C (572°F). It is inherently flame resistant, transparent to microwave radiation, and resistant to virtually all chemicals. It is used in the automotive – device connectors and electronic component housings; business machines – ink-jet printer cartridges, and internal printer components; and in the electrical and electronics sectors – capacitor and chip carriers, connectors, disc drives, heat sinks, optoelectronic devices, and switches. Another commercial LCP, a nematic LCP (Section 10.4.1) called Vectra®, is said to have the structures of the type

O O O

C O

C x

y

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in which the second chain section [y] is formed from HNA. The ratio of HBA to HNA in a commercial resin of this material is thought to be 73 : 27. Most of the commercial grades of this material have mineral, fiber, and other fillers for obtaining desirable properties. The unfilled and filled grades are anisotropic. Unfilled resin of this material has a Tg in the 120 – 155°C range, a melting temperature Tm of 285°C (545°F), a density of 1.35 g cm−3 , a flow-direction tensile modulus of 1 GPa (1.5 × 103 psi), and a flow-direction tensile yield strength of 159 MPa (23 × 103 psi). The cross-flow-direction strength and tensile modulus are lower by factors of about 2.3 and 3.3, respectively. Furthermore, because of changes in the through-thickness chain orientation morphology, the strengths and tensile moduli are thickness -dependent. This material is used for manufacturing electrical and electronic components for fiber optics, computers, telecommunication devices, chemical processing equipment, medical devices, in the automotive and machine construction industries, and in air- and space-craft technology. 11.7.7

Aromatic Polyamides (Aramids)

As the name suggests, aromatic polyamides, also called aramids (from aromatic polyamides), are polyamides with benzene rings in the polymer backbone. There are two high-temperature, high-performance liquid crystalline aramids, with widely recognized trade names. The first, poly(paraphenylene terephthalamide), called Kevlar®, is a lyotropic LC (Section 10.4.1) having a Tg = 327°C (621°F). It does not melt, but decomposes in air at 427 – 482°C (800 – 900°F). It has the structure

O

O

H

H

C

C

N

N n

This LCP can be spun into very high-strength fibers that, among other applications, are used for making bullet-proof body armor (Section 2.2.1.2). The second, poly(metaphenylene isoterephthalamide), called Nomex®, has a Tg = 264°C; it does not melt but chars at temperatures well above 370°C (700°F). It has the structure

O

O

H

H

C

C

N

N n

Nomex can be formed into fibers and sheets, and is used in high-temperature applications such as protective outerwear for firefighters (Section 2.2.1.1).

Structure, Properties, and Applications of Plastics

11.7.8

Polybenzimidazole

Polybenzimidazole (PBI) has the structure

H

H

N

N C

C N

N n

This resin has a glass transition temperature Tg of 427°C (800°F), a density of 1.3 g cm−3 , a tensile modulus of 5.9 GPa (855 × 103 psi), and a tensile yield strength of 160 MPa (23.2 × 103 psi). It does not melt; it begins to decompose at 540°C (1,004°F). PBI has the highest heat resistance of any unfilled plastic, in the sense that it does not melt and retains its mechanical properties over 205°C (400°F). Because of its very high transition temperature, stock materials, in the form of tubes, rods, and plates made by powder sintering processes, are supplied by polymer manufacturers. Parts can then be made by standard machining processes used for metals. PBI is used to fabricate high-performance protective apparel such as firefighter turnout coats and suits, astronaut space suits, high-temperature protective gloves, race driver suits, and aircraft wall fabrics. It can be made in films and fibers used for making braided packings.

11.8 Cyclic Polymers Very high stiffness and strength can be achieved in glass- and carbon-fiber continuous, random glass mat, and woven composites in a plastic matrix. The high fiber loading in these composites require that the matrix resin, which must infiltrate the small interstitial regions between the fibers, have very low viscosities. In advanced composites this is achieved by using very low viscosity thermoset resins that, after infiltrating the fiber mat, are cured into a high molecular-weight thermoset matrix in a heated mold. The curing process involves chemical reactions that generate water and gases that have to be removed. The brittle nature of thermosets results in the composites being brittle. While thermoplastic matrices for advance composites offer the potential for more ductile composites, their realization is limited by the extremely high viscosities of molten thermoplastics, because of which the melt cannot infiltrate the very small inter fiber spaces. This limitation of thermoplastics can be overcome by using low viscosity oligomers, containing a catalyst, to infiltrate the inter fiber spaces and then to in-situ polymerize the resin by the application of heat. One such product uses cyclic (ring-shaped) oligomers of PBT containing 2 – 20 repeat units of butylene terephthalate, called cyclic PBT (CBT) that has a very low melt viscosity, to infiltrate the fiber mat. In situ

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polymerization by the application of high-temperature then results in a very high fiber content composite in a PBT matrix. CBT powder grades are also available for rotational-molding applications.

11.9 Thermoplastic Elastomers Most elastomers (rubbers) are thermoset materials, discussed in Section 13.4, in which the irreversible cross-linking reaction is referred to as vulcanization. In contrast, in thermoplastic elastomers (TPEs) the elastic network is formed by reversible entanglements, so that these elastomers can be recycled. The main difference between thermoplastics and TPEs is that the latter can be stretched to much larger elongations from which they recover elastically on unloading. Besides use in many geometrically complex applications for which parts can be injection molded or extruded, TPEs can be overmolded onto stiffer thermoplastic parts to provide a soft touch feel. TPEs are either copolymers or physical mixtures of polymers. They are now widely used in the automotive, household appliance, and medical sectors. TPEs can be classified in several ways, one of which divides them into six groups: (i) Styrenic block copolymers, such as the triblock block copolymers of polystyrene and polybutadiene (SBS); (ii) polyolefin blends (blends and alloys are defined in Section 12.1); (iii) elastomeric alloys; (iv) thermoplastic polyurethanes; (v) thermoplastic copolyesters; and (vi) thermoplastic polyamides. A detailed discussion of the chemical and physical structures of these materials is beyond the scope of this book. However, three widely used TPEs are described in the sequel. 11.9.1

Polypropylene-EPDM TPE

An important class of TPEs, called thermoplastic vulcanizates (TPVs), consists of a cross-linked rubber phase dispersed within a thermoplastic polymer phase. A commercially important TPV, polypropylene ethylene-propylene-diene rubber (PP-EPDM), is a dynamically vulcanized blend of ethylene-propylene-diene rubber (EPDM) rubber particles encapsulated in a polypropylene (PP) matrix (SantopreneTM ). This material is used in a wide range of applications: For weatherseals, interior, underhood and under-car components in the automotive sector; syringe tips, peristaltic tubes, seals and gaskets, and grips in the healthcare sector. 11.9.2

Thermoplastic Copolyester TPE

TPEs based on thermoplastic copolyesters constitute another important class. One commercially important member of this class is Hytrel®. This material has high toughness (fracture resistance) and resilience (elastic recovery), and high resistance to creep, impact, and flexural fatigue. It retains its flexibility at low temperatures, and performs well at elevated temperatures. This material can be processed into products by several thermoplastic processing techniques, including injection molding, extrusion, blow molding, rotational molding, and melt casting. It is used in demanding flexible-component applications requiring mechanical strength and durability. Examples include seals, belts, bushings, pump diaphragms, gears, protective boots, hose and tubing, springs, and impact-absorbing devices. In many such applications, this material allows a multi-piece rubber, plastic, or even metal composite assembly to be replaced with a single part.

Structure, Properties, and Applications of Plastics

11.9.3

Thermoplastic Urethane (TPU)

Thermoplastic urethane (TPU) is a linear segmented block copolymer composed of hard and soft segments. The hard segment can be either aromatic or aliphatic. Some TPUs are made by reacting polyols (long-chain diols), diisocyanates, and short-chain diols. The reaction of the polyol with the diisocyanate creates the flexible segments, and the combination of the diisocyanate with short-chain diol produces the rigid segments. TPUs are flexible without having to use of plasticizers, and they cover a broad range of hardness and can be highly elastic. TPUs bridge the gap between rubbers and plastics. They can be used both as a hard rubber and as a soft thermoplastic. They can be sterilized, welded, and easily processed. They are used in a wide variety of applications including auto-body side molding, cattle tags, drive belts, hydraulic hoses, hydraulic seals, inflatable rafts, medical tubing and biomedical devices, swim fins and goggles, and wire and cable coating.

11.10 Historical Notes Michael Faraday isolated benzene from lighting gas in 1825. It was shown to have the empirical formula CH in 1834. In 1865 August Kekule proposed the ring structure for benzene. The first thermoplastic, made by combining nitrocellulose with camphor (Celluloid) was invented in 1868. Among other uses it was used for movie and photographic films till the 1950s; it is still used for making table tennis balls. The first major advance in synthetic plastics was the invention of a thermosetting phenol formaldehyde resin by Leo Baekeland in 1909. Known as Bakelite, it was made by reacting phenol with formaldehyde, and was used for a host of products from telephone casings, to kitchenware to electrical insulators, to toys for children. It is still in use. Early on, the high molecular weights of polymers such as rubber, starch, and cellulose were attributed aggregation of small molecules into colloids. In 1920 Hermann Staudinger proposed that these materials actually were long-chain molecules formed by short chains linked by covalent bonds. At first his revolutionary proposal was met with much skepticism, and was not accepted by the chemistry establishment. Staudinger coined the term macromolecules for these large molecules. He received the 1953 Nobel Prize in Chemistry for his pioneering work in macromolecular chemistry, the foundation for polymer chemistry. Because of the efforts of several researchers, there was a spurt of invention of thermoplastics in the 1930s, starting with PVC (1933), PMMA (1934), PA (1934), PTFE (1938), PS (1938), PAN (1941), and PE (1942). Of these, one major advance was the invention in 1934 of polyamide 6,6 (PA6,6), at the DuPont Experimental Station, which was given the name Nylon. This name was neither registered nor trademarked, so that nylon is now used as generic name for all polyamides. As a replacement for silk, fibers of this material were used with great success for weaving stockings for women, which came to be known as nylons. Another noteworthy invention from DuPont in 1938 was PTFE (polytetrafluoroethylene), known as Teflon. There was another spurt of inventions in the 1950s, during which several of the workhorse “engineering thermoplastics” were developed: polyurethane (PU) (1954), PP (1954), PET (1954), PC (1955), POM (1959), PI (1961), PSU (1965), and PPO (1966). Most of the high-temperature thermoplastics – PPS (1972), PAI (1972), Aramid fibers (1972), Polyarylate (PAR) (1978), PEEK (1979), PEI (1982), and PBI (1983) – were invented in the 1970s.

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The progress of invention of thermoplastics in the twentieth century is shown in the chart in Figure 11.10.1. It also includes two important thermosets (boxes with dashed lines): the first commercially important plastic (phenolic, 1909) and epoxies, which still are the most often used matrix resins for advanced composites.

PHENOLIC 1909 PU 1954

PPS 1972

PMMA 1934

PP 1954

PAI 1972

PA 1934

PET 1954

ARAMID FIBERS 1972

PVC 1933

PTFE 1938

PC 1955

PS 1938

POM 1959 PI 1961

PAN 1941 STAUDINGER advances macromolecular hypothesis 1920

10

20

30

1900

PE 1942

PSU 1965

EPOXIES 1942

PPO 1966

60

40

1950

70

PAR 1978 PEEK 1979 PEI 1982 PBI 1983

80

90

2000

Figure 11.10.1 Diagram charting the invention of thermoplastics in the twentieth century. The two boxes with dashed lines are for thermosets: phenolic, the first commercially important plastic, and epoxies that still are the most used matrix resins for advanced composites.

11.11 Concluding Remarks This chapter has provided a survey of the large number of commercially available thermoplastic resins, including thermoplastic elastomers, why and how they are modified by additives and modifiers, and some typical uses. Because of the very large number of grades, for which properties of interest are not always available, the properties in this chapter, such as the melting and glass transition temperatures, and the elastic moduli and strength, should only be considered as generic values; actual values for a particular grade have to be obtained from manufacturers or, when accurate data are needed, have to be obtained through in-house tests.

Structure, Properties, and Applications of Plastics

It is worth emphasizing that these plastics (modified polymers) are complex materials, which do not behave like the pure polymers studied in polymer science. The objectives of polymer science and plastics engineering are different: The former is concerned with understanding how the chemistry and architecture of polymer chains affects their macroscopic properties, and with using this understanding to develop polymers with desirable properties. In contrast, the latter is concerned with issues relating to how these materials are used in applications – thermomechanical material characterization, part design, and fabrication and assembly methods. As mentioned in Chapter 1, the applications of plastics grew rapidly as a marketing driven industry. One legacy of this approach is the use of the term engineering plastic, or high-performance plastic, began to be used for higher strength and elastic modulus plastics – such as polycarbonate (PC), poly(butylene terephthalate) (PBT), and nylon (PA) – that retain their mechanical properties above higher temperatures of about 100°C (212°F). And the high-temperature subcategory of which applies to plastics that can be used at temperatures above 200°C (392°F). In contrast to engineering plastics, the lower-cost, lower performance plastics – such as polyethylene (PE), polypropylene (PP), poly(vinyl chloride) (PVC), polystyrene (PS), and poly(methyl methacrylate) (PMMA) – are called commodity plastics. However, with the use of better designs, some of the cheaper materials, such as PP, can be used in demanding engineering applications, so that the distinction between commodity and engineering plastics is not so clear.

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12 Blends and Alloys 12.1 Introduction Thus far, the focus has been on different classes of thermoplastics and their properties, which differ considerably among each other. While new polymeric materials are being synthesized, scaling up and marketing of new polymers can be so expensive that, even when superior properties have been demonstrated, they may not be commercialized. An alternative strategy for developing new materials with desired characteristics – such as better mechanical properties, including impact performance; higher thermal performance; improved processability, especially for high-temperature materials; and improved surface appearance – at lower costs is to mix different polymers. Such mixtures are called polymer blends. The major push for the use of plastics in load-bearing automotive applications in the 1980s was marketing driven by plastics manufacturers. As high-performance polymers began to penetrate this market, borrowing from metals, blends were called alloys to give them a high-tech image; all polymer alloys are polymer blends. In this spirit, the term engineering plastic, or high-performance plastic, began to be used for higher strength and elastic modulus plastics – such as polycarbonate (PC), polybutylene terephthalate (PBT), nylon (PA) – that retain their mechanical properties above higher temperatures of about 100°C (212°F). The term high-temperature plastics refers to a subcategory that can be used at temperatures above 200°C (392°F). In contrast to engineering plastics, the lower-cost, lower performance plastics – such as polyethylene (PE), polypropylene (PP), poly(vinyl chloride) (PVC), polystyrene (PS), and poly(methyl methacrylate) (PMMA) – are called commodity plastics. However, with the use of better designs, PP is penetrating many applications in which engineering plastics were initially used. As might be expected, the commodity plastics are cheaper than the engineering plastics, which, in turn, are cheaper than the high-temperature plastics. In the 1980s and 1990s several blends of engineering thermoplastics were developed to address the needs of demanding automotive applications – such as for bumpers and vertical side panels. The rapidity with which these tailor-made blends were developed shows one advantage of blends over having to develop new resins for specific types of applications. Also, blending offers a means for recycling thermoplastics.

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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This chapter describes the constituents and properties of many industrially important blends. Unfortunately, the exact compositions and additives of these mostly proprietary blends are not available. And, as with plastic resins, users must work with plastics manufacturers and blenders in choosing suitable materials for their applications. And, the data on the properties of these blends published by manufacturers are even sparser than for resins and copolymer systems. So, the physical and mechanical properties of the plastics discussed in this chapter should be considered as representative values.

12.2 Blends While most blends are made from two active constituents, they can have more than two. AcrylonitrileButadiene-Styrene (ABS), an important blend, is made from three polymers, and is therefore called a ternary blend. Blends may broadly be classified into two groups: (i) In miscible blends the constituents are miscible in varying degrees. The most notable fully miscible blends are those of poly(phenylene oxide) with polystyrene; these blends behave like resins. (ii) In immiscible blends the constituents tend to segregate into separate clusters – somewhat like oil drops in water – which can consist of large chunks, so that compatibilizers have to be used to achieve materials with evenly distributed, more homogeneous morphologies. In the design of immiscible blends, changes in morphology induced during part processing have to be an important consideration. Inert compatibilizers work by changing the interfacial properties of one or more of the constituents. Reactive blending, which is most useful for highly incompatible polymers, uses an agent that reacts with the materials during blending to make them more compatible. Compounding of blends – especially incompatible blends for which compatibilizers have to be used – to achieve the right type of microstructures requires as much art as science. 12.2.1

Acrylonitrile-Butadiene-Styrene

Acrylonitrile-Butadiene-Styrene (ABS) is an amorphous blend consisting of a continuous phase of the copolymer styrene-acrylonitrile (SAN) having a dispersed phase of polybutadiene (PBD) particles, which have SAN layers grafted onto their surfaces to make them compatible with the continuous phase. ABS owes most of its properties – such as rigidity, hardness, and heat and chemical resistance – to that of the continuous SAN phase with the PBD particles contributing to the impact resistance. The double bond in the PBD makes ABS vulnerable to weathering by ultraviolet (UV) radiation and heat, requiring the use of stabilizers. ABS is a versatile, widely used thermoplastic at the low end of the properties-price index. It is available in a wide range of unfilled and filled injection-molding and extrusion grades, the properties of which depend on the contents of acrylonitrile (15 – 35%), butadiene (5 – 30%), and styrene (40 – 60%). Its nominal properties are: a density of 1.05 g cm−3 , a tensile modulus of 2.1 GPa (0.3 × 106 psi), and a tensile yield strength of 34.5 MPa (5.0 × 103 psi). For most applications, ABS can be used over a temperature range of −25 to 80°C (−13 to 176°F). This versatile thermoplastic is used in many common applications: Interior thermoformed refrigerator liners and pans; housings for appliances such as hair driers, coffee makers, food processors, and vacuum cleaners; telephone and cell phone housings and computer keyboards; computer, printer and copier housings; and molded briefcases and hard-sided luggage. Most Lego® bricks are made of ABS. Finely ground ABS is even used as a pigment in tattoo inks.

Blends and Alloys

12.2.2

Acrylonitrile-Styrene-Acrylate

Like ABS, Acrylonitrile-Styrene-Acrylate (ASA) is an amorphous blend consisting of a continuous phase of the copolymer styrene-acrylonitrile (SAN) having a dispersed phase of saturated acrylic rubber particles. In contrast to the unsaturated butadiene rubber used in ABS, the saturated rubber used in ASA gives it excellent long-term UV stability, making it useful for outdoor applications. It can be injection molded, extruded into sheets and profiles. It can also be coextruded over ABS and PVC. ASA has the nominal properties: a density of 1.07 g cm−3 , a tensile modulus of 2.2 GPa (0.32 × 106 psi), and a tensile yield strength of 46.9 MPa (6.8 × 103 psi). ASA is used in a wide array of applications requiring better weathering than ABS.

12.2.3

ABS/PVC Blends

An important limitation of the otherwise versatile ABS is its poor flame retardancy, which requires the use of relatively expensive flame retardants. In blends of ABS/PVC the PVC provides an inexpensive flame retardant and also adds to the UV stability of the blend. As such, this relatively inexpensive blend has displaced ABS in applications requiring fire retardancy. It is used in electrical components, appliances, and business machine housings.

12.2.4

Nylon/ABS Blends

Nylon 6/ABS blends having good impact performance at both low and high temperatures are available. In these materials the ABS reduces moisture absorption associated with PA6 and improves impact strength. PA6 provides high-temperature performance and chemical resistance to this blend. Since PA6 and ABS are not compatible, a compatibilizer such as SMA has to be used. Because of its high-temperature capability, this material has been used for automotive body panels and in underhood applications. It is also used in the consumer sector in applications such as vacuum cleaner housings.

12.2.5

Polycarbonate/ABS Blends

PC/ABS blends are a class of relatively low-cost materials in which PC provides the mechanical performance, especially impact properties, and better temperature capabilities, while the cheaper ABS contributes better chemical resistance and flow properties of the blend. Both the performance and the cost of the PC/ABS blends are intermediate between those of PC and ABS. This versatile blend is available with a variety of compositions and additives to serve a wide range of applications. The nominal compositions, elastic moduli, and strengths of two commercial PC/ABS blends are shown in Table 12.2.1. The first PC/ABS-1, developed for automotive instrument panels, does not contain a fire retardant. The second PC/ABS-2, specially developed for computer housings, is a low viscosity grade containing a fire retardant. Note that, in addition to PC and ABS, both these blends also contain SAN. PC/ABS are used in a very wide array of applications including portable appliances and telephones, laptop computer housings, keyboards, printer housings, automotive wheel covers, and hoods for small tractors.

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Table 12.2.1 Nominal compositions, elastic moduli, and strengths of two PC/ABS blends. Blend Property

PC/ABS-1

PC/ABS-2

PC

65

82

ABS

19

8

SAN

16

2

FR Agent

—

8

3

2.65 (385)

3.09 (447)

3

56.5 (8.2)

60.0 (8.7)

Nominal composition wt%

Tensile modulus GPa (10 psi) Tensile strength MPa (10 psi)

12.2.6

Poly(Phenylene Oxide)/Polystyrene Blends

With a high Tg of 215°C (419°F), the amorphous resin poly(phenylene oxide), or PPO (Section 11.6.2), has a very high viscosity and is very difficult to mold. It is widely used in the form of blends with polystyrene with which it is fully miscible. High-impact polystyrene (HIPS) is used to impart impact resistance. This PPO/HIPS is called modified polyphenylene oxide (M-PPO). Depending on amount of polystyrene in the blend this material has a Tg in the range of 100 – 190°C (212 – 374°F), a density of 1.1.04 – 1.10 g cm−3 , a tensile modulus in the range 2.14 – 2.62 GPa (310 × 103 – 380 × 103 psi), and a tensile yield strength in the range 44.8 – 62.1 MPa (6.5 × 103 – 9 × 103 psi). M-PPO is a versatile, easily moldable plastic with relatively high-temperature performance, which is available in unfilled and reinforced grades, with and without flame retardants. It is widely used for a wide variety of products in the automotive, electronics – TV and computer housings – and industrial sectors. M-PPO is also available in structural foam grades. 12.2.7

Polycarbonate/PBT Blends

The push for the use of thermoplastics in demanding structural applications in automobiles started in the late 1970s with the initial focus on bumpers, for which impact resistance is important. While PC did have the requisite impact resistance, thermal performance, stiffness, and dimensional stability, it lacked the required chemical resistance needed for an automotive environment. And while semicrystalline thermoplastics such as PBT and poly(ethylene terephthalate) (PET) did have the required chemical resistance and mechanical properties, they lacked the required impact properties and have poor dimensional stability associated with all semicrystalline thermoplastics. This led to the development of PC/PBT blends – which synergistically combined the impact resistance and dimensional stability of PC with the chemical resistance and lower melt viscosities (better flow) of PBT – specifically targeted at bumpers. And PC helped to ameliorate the moisture sensitivity associated with PBT. Rubber toughening was required to provide adequate impact resistance over the normal use temperature. The first PC/PBT rubber-toughened blend, PC/PBT-1 (with about 45 wt% each of PC and PBT, and about 10 wt% of impact modifier), having a tensile strength of 52.7 MPa (7.6 × 103 psi), did not perform well at the very low winter temperatures seen by bumpers, causing them to shatter, that is, these blends

Blends and Alloys

lacked low-temperature ductility. A second version, PC/PBT-2 (with about 42.5 wt% each of PC and PBT, and about 15 wt% of impact modifier), having a tensile strength of 50.1 MPa (7.3 × 103 psi), was quickly developed to successfully address this issue, resulting in a widespread use of such blends. The first all-plastic thermoplastic bumpers made of this blend, capable of an 8-kph (5-mph) barrier impacts, were used in the 1984 Ford Escort (see Figure 2.4.1). Blends of PC with PET are also available: While PC/PBT blends exhibit better toughness (impact performance), chemical resistance, and better flow properties, PC/PET blends have better dimensional stability and thermal performance. PC/PBT blends are now extensively used in applications requiring low-temperature ductility (failure without shattering) and chemical resistance, such as in automotive bumpers and side moldings, lawnmower decks and garden tractor hoods, hand tool housings, and irrigation components. PC/PET blends are preferred for electrical wiring connectors. 12.2.8

Nylon/PPO Blends

After bumpers, the next big push was for body side panels for which PC/PBT blends provided adequate performance. However, body panels have a peculiar requirement based on how automobiles are assembled: body panels are attached to the frame and, during the painting cycle, the assembly is subjected to high temperatures during the paint bake cycle. Such painting cycles were developed for metal body panels and the automotive industry was not about to change the paint cycle to suit yet untried plastic side panels. So, a thermoplastic material had to be developed for retaining part shapes during the paint bake high temperatures, even though the parts would subsequently not see such high temperatures! For this application a class of nylon (PA) and modified PPO were developed in the 1980s, the most successful of which had about 40 wt% of PPO, about 50 wt% of polyamide 6,6 (PA 6,6), about 10 wt% of an impact modifier, and a small amount of a compatibilizer additive. It had a tensile modulus of 1.94 GPa (282 × 103 psi) and a tensile strength of 58.4 MPa (8.47 × 103 psi). The continuous phase in this blend was PA 6,6 and the impact modifier was distributed in the PPO phase. The first thermoplastic automotive vertical body panels – the front fenders on the 1987 model-year Buick LeSabre T-Type1 sports coupe – were made of this material (see Figure 2.4.2). Eventually, this material was used on vertical body panels in over 20 million vehicles. PPO/PA blends are now available in many unfilled and filled grades for injection-molding, extrusion, and blow molding applications. The polyamide component can be PA6, PA 6,6, and even an amorphous PA. And the blends may also contain PS. Special conductive grades have been developed for powder coat painting applications. 12.2.9

High-Temperature Blends

There is a host of expensive, high-temperature blends designed for applications requiring mechanical performance at high temperatures. Besides blending to improve the properties of the individual components, using a cheaper component to reduce cost is another objective for these blends. Most of these high-temperature blends are immiscible. One important counterexample is that of polyetherimide (PEI) and polyetheretherketone (PEEK) which are miscible and form a PEI/PEEK blend that exhibits a single Tg . Other examples include blends such as PPS/polytetrafluoroethylene (PTFE), polyarylate/PET, and PPS/liquid crystal polymer (LCP) in which the LCP is Vectra®.

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12.3 Historical Notes Section 12.1 provides the rationale for developing thermoplastic blends. The first commercial blend, PVC/ABS, was developed in 1951. The highly successful high-performance compatible blend PPO/HIPS of PPO and HIPS was developed in 1966; in this book it is referred to as M-PPO (modified PPO). The next spurt in new blends was PC/ABS (1977), PA/ABS (1981), PC/PBT (1982), and PA/PPO (1983). Later, two more blends, ABS/thermoplastic urethane (TPU) (1990) and PEI/PC (1992) were developed in the 1990s. The progress of invention of thermoplastic blends in the twentieth century is shown in the chart in Figure 12.3.1. In comparison to thermoplastics (Figure 11.10.1), the bulk of the blends were developed in the last quarter of the twentieth century.

PVC/ABS 1951 PPO/HIPS 1966

ABS/TPU 1990

PC/ABS 1977

PEI/PC 1992

PA/ABS 1981 PC/PBT 1982 PA/PPO 1983

10

20

30

1900

60

40

70

80

1950

90

2000

Figure 12.3.1 Diagram charting the invention of thermoplastics blends in the twentieth century.

12.4 Concluding Remarks This chapter has surveyed commercially available thermoplastic blends. Because of the speed with new blends can be formulated, the plastics industry is quickly able to respond to new needs. Compounding of blends – especially incompatible blends for which compatibilizers have to be used – to achieve the right

Blends and Alloys

type of microstructures requires as much art as science. And the processing (molding) of these resins to achieve desirable microstructures in parts requires more attention. In comparison to larger data sets for resin systems, the lack of a complete set of properties for blends is a reflection on how plastics are used in applications – in which analyses for thermomechanical and electromagnetic design often rather crude. As with resins systems, when needed accurate data have to be obtained through in-house tests.

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13 Thermoset Materials 13.1 Introduction The bulk of the material discussed thus far in this book has been concerned with the structure and properties of, and component design and fabrication with, thermoplastics – materials that soften and melt on heating and can be reshaped by heating and cooling operations. In contrast, in thermosets the coiled, entangled chains are physically pinned together by irreversible, cross-linking chemical reactions to form relatively rigid parts that do not soften on heating. As a result, once formed, a part can only be reshaped by machining processes. In thermoplastics, other than the changes in the microstructure induced during processing into parts, the thermoplastic remains unchanged. Even in the molten state thermoplastics tend to be very viscous, requiring high molding pressures and, therefore heavy molds. In contrast, in thermosetting materials, the initial material – oligomers called a prepolymer or resin, which may be powders or low-viscosity liquids – is different from the material in the part; during the molding process an exothermic irreversible reaction in the prepolymer causes the oligomers to form a cross-linked, interconnected rigid structure. The first commercial applications of polymeric materials were thermosetting rubbers – which are called elastomers – and rigid thermosetting resins. While elastomers continue to increase in importance – the largest applications of rubbers continue to be for automotive tires – the early applications of rigid thermosetting plastics have now moved to more easily processed thermoplastics. However, thermosets are of great importance for advanced composites, which originally were used for weight-saving aerospace applications – for which specialty high-temperature materials have been developed – but now are now extensively used in many consumer sporting goods such as fishing rods, high-performance light-weight bicycles, and compound bows. In this chapter the chemical structures of thermosets, including common elastomers, and their applications are briefly described. Part fabrication methods for these materials are addressed in Chapter 20.

13.2 Thermosetting Resins Thermosetting resins are low molecular weight oligomers that react during the part forming phase to form three-dimensional structures, or networks, that are strongly cross-linked across the part. These oligomers are not long-chain polymers and are therefore referred to as prepolymers or resins. Because the strongly cross-linked polymer networks do not have the mobility of chains in thermoplastics, thermosets result in more rigid parts. Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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During part forming the resins undergo exothermic chemical reactions resulting in continually growing cross-linked polymer networks; this phase is called curing. The point at which the chains begin to form polymer networks across the part is called the gel point. At this point the reacting material has a low-number average molecular weight but an extremely high weight-average molecular weight – essentially a small number of very large three-dimensional cross-linked chains. The gel point marks the conversion of low-viscosity reactants into a highly viscous liquid or solid. Continuing polymerization results in essentially all of the reacting material transforming into a collection of rigid networks. The change in viscosity with time during the curing process is schematically shown in Figure 20.2.1. 13.2.1

Phenolics

The first, and commonest, thermosetting plastic polymers are phenolics that are made from phenol and formaldehyde. The first products of reactions are methylolphenols; the reaction presented here shows formation of ortho-methylolphenol

OH

OH CH2OH

H +

C

O

H Other reaction products shown next are para-methylolphenol, 2,4-dimethylolphenol, 2,6dimethylolphenol, and 2,4,6-trimethylolphenol

OH

OH CH2OH

OH HOCH2

CH2OH

CH2OH OH CH2OH

HOH2C

CH2OH

CH2OH

Thermoset Materials

These products react with each other and with phenol to produce a series of products with methylene (CH2 ) bridges, such as 2,2 ′dihydroxydiphenylmethane

OH

OH

OH

OH

CH2OH

CH2

+

and

OH

OH

OH

OH

CH2OH

CH2

CH2OH

CH2

and products with dimethylene ether (CH2 O CH2 ) links, such as

OH

OH CH2OCH2

OH CH2OH

OH CH2OCH2

CH2OH Depending on the conditions under which phenol and formaldehyde are reacted, two types of phenol-formaldehyde resin oligomers are formed: Resoles and Novolak resins. 13.2.1.1 Resole Resins

Resoles are formed when the reaction occurs with an excess of formaldehyde under alkaline conditions. The reaction is continued until the solid resin produced has a melting point in the range 45 – 50°C. A powdered form of this solid then provides the base resin for molding applications. The material consists of a mixture of the methylol (CH2 OH) group-rich compounds described in the previous section. On heating in a mold, these methylol groups react to provide the cross-linking that results in a non-fusible rigid structure. 13.2.1.2 Novolak Resins

When the phenol-formaldehyde reaction occurs with an excess of phenol under acidic conditions, the compounds shown in Section 13.2.1 react among each other till all the methylol groups have been consumed. The end product is a brittle solid with no methylol links, having a melting point in the range of 65 – 75°C. Such materials are called Novolaks and consist of a mixture of oligomers of the form

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OH CH2

CH2

HO

HO

CH2

CH2 OH

OH

CH2

OH with varying numbers of phenyl rings, with average being about six rings. Because of the absence of methylol links Novolaks do not react and cross-link on further heating, as in a mold. To promote cross-linking the material is mixed with a catalyst – the most commonly used being hexamethylene tetramine – heated, and formed into powders. During the part forming, or molding, phase these powders, mixed with fillers – such as sawdust to reduce cost, and mica flakes and chopped glass fibers to improve mechanical properties – and pigments are molded under compression at higher temperatures in heated molds, when constituents in the powder react (cure), forming a highly cross-linked, rigid structure. The curing reaction produces gases and water, which have to diffuse from the part interior to the surface, resulting in a significant shrinkage in the part size. 13.2.1.3

Applications of Phenolics

Phenolics tend to be dark and opaque. They are far more rigid than thermoplastics, that is, they have much higher elastic moduli, are more heat resistant, and parts maintain their shapes. They resist water, most organic solvents, and most chemicals, but are attacked by sodium hydroxide solutions. Phenolics are used both in unfilled and filled forms. Fillers can be sawdust to reduce cost, and cellulose and glass fibers to enhance mechanical properties. They were earlier used in consumer electronic products such as telephones and radios. But are now seldom used in such applications, partly because brittle and because of higher molding costs. However, they continue to be used in small components requiring dimensional precision. An exception to the overall decline is the use in small precision-shaped components where their specific properties are required: molded disc brake cylinders, handles for saucepans and frying pans, electrical plugs and switches, and electrical iron parts. In sheet and rod form they are widely used in many industrial applications. 13.2.2

Urea-Aldehyde-Based Resins

Urea-formaldehyde and melamine-formaldehyde form the bases for aminopolymers that are formed by reactions between amines with aldehydes. 13.2.2.1

Urea-Formaldehyde-Based Resin

Urea-formaldehyde resins are made by reacting urea (NH2 CONH2 ) with formaldehyde, which results in various methylolureas such as

Thermoset Materials

O

C

NH2CH2OH

O

NH2CH2OH

C

NH2CH2OH

NH2

Further reactions among urea and these methylolureas result in methylene-linked compounds of the form

N

CO

NH2

N

CH2

NH2

CO

NH2

CH2 CO

N

CO

NH2

N

Continued reactions then result in urea-formaldehyde resins, or prepolymers, with structures of the form

HOCH2

NH

CO

NH

CH2

NH

CO

NH

CH2OH

n

During molding in a heated mold, these resins cross-link to form three-dimensional networks with structures of the form

CH2

CH2 N

CO

N

CH2

N

CO

CO

N

CH2

N

CO

N

CH2

CO

N

CH2

CH2

CH2 N

CO

N

CH2

N

CH2

CH2 N

CH2

CH2

CH2 N

N

CO

N

CH2

N

CO

N

CH2

CH2

CH2

13.2.2.2 Melamine-Aldehyde-Based Resins

Melamine reacts with formaldehyde to give methylolmelamines such as dimethylolmelamine

NH2 N H2N

C

C

N

NH2

N C

H +

NH2

C H

N

O HOCH2NH

C

C

N

N C

NHCH2OH

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and trimethylolmelamine and tetramethylolmelamine

HOCH2

NHCH2OH N C

HOCH2NH

C

N

N

N C

HOH2C

NHCH2OH

HOH2C

N

C

N C

N

CH2OH

N C

CH2OH

N

CH2OH

Curing during molding results in these compounds forming into three-dimensional networks of the form

NH

C N

N C

C

NHCH2OCH2NH

C N

N

NCH2OH

N C

N C

C

NHCH2OCH2NH

N

C N

NH

N C

C N

NH

CH2

CH2

O

O

CH2

CH2

NH

NH

C N

N C

N NHCH2OCH2NH

C N

N C

C

NHCH2OCH2NH

N

NH CH2 NCH2OH N NH

C

C N

NH

N C

HN

C

C N

N C

NH

Thermoset Materials

13.2.2.3 Applications of Urea- and Melamine-Aldehyde Resins

Urea- and melamine-formaldehyde resins are clear and colorless and can be used to produce light color parts. Molding powders of these materials can be used to make switch cover plates, handles for kitchen pots, electric mixer housings, radio cabinets, coffee makers, and door knobs.

13.2.3

Allyl Diglycol Carbonate (CR-39)

An important thermosetting plastic, widely used for making eyeglass lenses, is made by cross-linking the monomer diethyleneglycol bis allylcarbonate, or allyl diglycol carbonate (ADC), which has the structure

O CH2

CH2

O

C

O

CH2

CH

CH2

O

CH2

CH

CH2

O

O CH2

CH2

O

C

The solid form of this thermosetting polymer is made by polymerizing ADC with diisopropyl peroxydicarbonate (IPP) as an initiator; the double bonds in the allyl groups (—CH=CH2 ) make cross-linking possible. This material is referred to as CR-39, an abbreviation for Columbia Resin 39, being the 39th in a series of developmental materials. It can be used continuously for temperatures up to 100°C, and to 130°C for 1 hour. Originally, this material was used for making fiberglass-reinforced plastic fuel tanks for aircraft bomber in World War II to reduce weight. It has also been used to measure ionizing neutron radiation. CR-39, the first successful plastic for lenses, is still extensively used for eyeglass lenses: CR-39 is transparent to the visible spectrum, is almost opaque to ultraviolet (UV) radiation – thereby providing UV protection for the eyes – and, because of the much lower density, eyeglass lenses made of this material weigh about half of equivalent glass lenses. And the optical properties of the lenses are comparable to those made of glass. It has higher scratch and abrasion resistance than all other uncoated plastics used in optical applications. Also, it is resistant to most solvents and other chemicals. The material is cast into optically clear disks that are ground to form lenses, including bi and trifocal lenses.

13.2.4

Thermosetting Polyesters

Unsaturated polyester resins form an important class of thermosetting resins used in a variety of applications in which large amounts of glass fibers are used to obtain rigid parts. Of the several types of unsaturated polyester resin systems, the most common is type is obtained by reacting propylene glycol, maleic anhydride, and phthalic anhydride to give

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CH3 CH

CO

O

CH

CH2

OH

CH

CO

O

CH

CH2

OH

CH3 which is reacted with polystyrene to give prepolymers, or resins, having linear structures of the type

CH3 OC

CH

CH

CH3

COOCHCH2OOC COOCHCH2OOC

CH

CH

COOCHCH2O

This liquid prepolymer mixed with excess styrene forms the polyester resin to which is added an inhibitor, such as hydroquinone, to prevent premature cross-linking. Cross-linking of the unsaturated linear polyester resin during curing is effected in two ways by using initiators: 1. For room-temperature applications peroxy compounds are used as initiators together with an accelerator that activates the action of the peroxy compounds. 2. For cross-linking at higher temperatures, such as in molding operations, peroxides are used as initiators; benzoyl peroxide is the most commonly used initiator. During cross-linking, chains of polystyrene of different lengths react with the unsaturated bonds to form three-dimensional networks of the form

COOH COOH

COOH

CH

CH

CH

CH2

CH

CH CH2

CH

CH

CH COOH

COOH

COOH n

m

where the long bonds at the two ends indicate the connections of these polystyrene bonds with the polyester chains.

Thermoset Materials

13.2.5

Vinyl Esters

Vinyl esters are unsaturated resins with the unsaturated, reactive double bonds located at the ends of the resin. One common form, obtained by reacting methacrylic acid with a bisphenol-A epoxy resin, has the structure:

CH3 CH2 C

C O CH2CHOHCH2 O

C

O CH2CHOHCH2 O C C CH2

CH3

CH3 O

O CH3

n

which has methacrylic end groups. The relatively high viscosity of these resins at ambient temperatures is lowered by using a reactive monomer, such as styrene, which during curing results in cross-linking just as with unsaturated polyesters. Since the unsaturated cross-linking sites are at the ends of the resin chains, shorter chains (lower values of n) result in higher cross-linking densities in the cured resin, which has higher heat and chemical resistance and better dimensional stability. 13.2.5.1 Applications of Polyesters and Vinyl Esters

These resins are extensively used as the matrix resin for making large hand layup glass-filled parts such as boat hulls. Filled with long, chopped glass fibers, they are used to make molding precursors such as sheet molding compound (SMC) – which can be compression molded – and bulk molding compound (BMC) that can be injection molded (Section 20.5). 13.2.6

Epoxies

Epoxies are thermosetting polymers that require a second compound, called a hardener, for effecting the curing reaction. They are so named because the oligomers of the resin, or prepolymer, contain the epoxide end group

O H2C

CH2

The oligomers of the bulk of the commercial epoxies, prepared by reacting epichlorhydrin with bisphenol-A, have structures of the form

CH3

O H2C

CHCH2

O

C CH3

CH3

OH OCH2CHCH2

O n

C

O OCH2HC

CH2

CH3

The oligomers are liquids for solids n = 0, 1, and 2, and solids for n > 2. Curing is achieved by mixing the epoxy resin with a cross-linking agent – which can be aliphatic or aromatic polyamines, acid anhydrides, polysulfides, and phenol-formaldehydes – that reacts with the epox-

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ide groups, finally resulting in a three-dimensional network. For example, curing with trimethylenetetramine results in the network structure

O H2C

CH

+

H2N

CH2CH2 NH CH2CH2 NH CH2CH2 NH

OH CH CH

OH CH2 N CH2

CH2CH2

OH

13.2.6.1

N

CH2CH2

CH2CH2 N

OH

CH

CH2 CH CH2 CH

CH2

CH2 CH

N

OH

OH

Applications of Epoxies

Epoxies have a wide range of industrial applications in electronic and electrical components, high tension electrical insulators, and as structural adhesives. But the most important technical use of epoxies is as a matrix material for advanced composites using continuous filament glass and carbon fibers. However, even the most advanced epoxies cannot be used in applications requiring exposure to continuous temperatures above 130°C (266°F). 13.2.7

Sheet and Bulk Molding Compounds

Sheet molding compound (SMC), also called sheet molding composite, comprises a thermosetting resin – such as unsaturated polyesters and vinyl esters (Sections 13.2.4 and 13.2.5), an epoxy (Section 13.2.6), or a phenolic (Section 13.2.1) – filled with fibers, fillers, and additives. While the reinforcing fibers could be of carbon or Kevlar, chopped glass fibers are used in most applications. The most commonly used filler is calcium carbonate; hollow glass microspheres and alumina trihydrate are also used. A plethora of additives in the form of inhibitors and initiators, low-profile additives, thickening and mold release agents, viscosity reducers, and wetting agents are used. Most SMC parts are made by using an unsaturated polymer in combination with styrene as the thermosetting resin, chopped glass fibers longer than 25 mm as reinforcement, calcium carbonate as the filler, polyvinyl acetate as a low-profile additive, magnesium oxide as a thickener, zinc stearate as a mold release agent, p-benzoquinone as an inhibitor, and t-butyl peroxybenzoate as an initiator.

Thermoset Materials

SMC is made by first separately mixing two batches of material. The first consists of styrene, low-profile additive, thickener, color pigment, and filler. And the second consists of the polyester resin with a small amount of styrene, the filler, a small amount of the low-profile additive, traces of the inhibitor, initiator, wetting agent, and mold release agent. Predetermined amounts of these batches are thoroughly mixed to form a paste that is applied to two carrier films with the glass fiber in the middle. This compounded sandwich material is stored for several days in a controlled environment to increase its initial viscosity in the range of 10 – 40 Pa s to the molding viscosity in the range of 20 – 40 kPa s. SMC parts are made by compression molding (Section 20.3). Bulk molding compound (BMC) uses pastes similar to that used in SMC, with shorter glass fibers that, for reducing fiber breakage, are added during the last stages of the paste mixing process. Aged BMC is in the form of bulky short-glass-filled chunks that can be used to injection mold parts (Section 20.5); it can also be compression (Section 20.3) or injection-compression molded (Section 20.5.1). 13.2.8

Polyurethanes

Polyurethanes are polymers containing the urethane group NH–CO–O—, and polyurethane parts are made by reacting diisocyanates (having two —N=C=O groups) with glycols (organic alcohols) mixed just prior to insertion in a mold. As such, just like with epoxies, polyurethane parts are made by reacting two different compounds. While polyurethanes can be molded into solid parts, the main applications are as foams, which have flexible or rigid forms. Other applications include surface coatings, adhesives, and elastomers. Gas bubbles have to be generated during the curing reaction to form foams. Depending on the degree of cross-linking allowed, the foams can be flexible (lower cross-linking) or rigid (high degree of cross-linking). In forming foams, the volume of the solid material (in the absence of foaming) is less than the final volume achieved by the foam bubbles expanding the material to fill the desired volume. There are three types of foams. Flexible foams may be produced by using a mixture polyol (organic compound with multiple hydroxyl groups) and water to react with a diisocyante. Besides reacting with the polyol to produce polyurethane, the diisocyante reacts with the water to generate carbon dioxide, the bubbles of which generate the foam. Let a volume V0 of the urethane expand into a volume of the V of the foam. Then, because they have the same mass, the corresponding densities are related through 𝜌∕𝜌0 = V 0 ∕V. The degree of foaming may be described by (V − V0 )∕V0 = (𝜌 0 ∕𝜌 − 1), which can also be expressed as a percentage density reduction. Lighter foams have higher density reductions. For very small density reductions individual bubbles will be dispersed in the matrix. As the degree of foaming increases, the bubbles come into contact to form larger bubbles that are totally enclosed by the matrix. Such foams, in which such larger bubbles are totally surrounded by the matrix, are referred to as closed-cell foams. At higher density reductions after making contact the walls of the adjacent bubbles puncture to form continuous paths through the matrix to form open-cell foams. Polyurethanes can also be used to from structural, or integral, foams, in which a part with a thin solid skin enclosing a porous interior are made in a single molding operation. To form such parts the mixture of the diisocyanate and polyol and water are injected into a cooled mold. On coming in contact with the

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mold wall the material sets before the evolution of gas to form the skin, which is kept under pressure by the expanding interior, thereby preventing bubble formation in the skin. 13.2.8.1

Applications of Polyurethanes

Most applications of polyurethanes are in the form of flexible or rigid foams, which form the interiors of parts: flexible foams are used in upholstery; rigid foams are used as insulation inside refrigerators and in building construction – as insulation and for door frames. But it is also used in solid form for large Reaction Injection Molding (RIM) panels – such as for large CT scanners (Section 20.6) – and thicker Reinforced Reaction Injection Molding (RRIM) fiber-filled parts (Section 20.6.1).

13.3 High-Temperature Thermosets The development of high-temperature thermosets was driven by the ever-increasing temperatures at which advanced composites have to operate. Epoxies, the main resins used in advanced composites, cannot be used at continuous temperatures above 130°C (266°F). A large number of high-temperature thermosetting resins are now available. Their high cost can only be justified in aerospace applications of advanced composites and in niche applications. The rest of this section discusses three representative resins: The newest class, cyanate esters, can operate continuously up to 177°C (350°F). The next class, bismaleimide (BMI), can be used at continuous temperatures of 180°C (356°F). And the third in this class is the polyimide PMR-15, which can withstand continuous operating temperatures up to 357°C (675°F); this was the first high-temperature thermoset developed. 13.3.1

Cyanate Esters

This family of thermoset monomers comprises esters of cyanic acid, HCN, and bisphenols, HO—R—OH, in which the hydrogen in the hydroxyl group OH is substituted by the cyanide group —CN, resulting in the cyanate ester NCO—R—OCN. Cyanate esters can be cured by heating at high temperatures or at lower temperatures by using a catalyst. A widely used cyanate ester is based on the bisphenol-A based dicyanate monomer

CH3 N

C

O

C

O

C

N

CH3 Curing takes place in two steps. First a cyclotrimerization reaction results in the formation of a bisphenol-A backboned triazine prepolymer

Thermoset Materials

N C O

H3C

C

CH3

O N O

C

C

N

N C

O

H3C

CH3 C

C CH3

N

C

O

H3C O

C

N

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Further heating causes this prepolymer to polymerize to form a three-dimensional network resulting in the thermoset polymer polycyanate, having a glass transition temperature of Tg = 289°C,

O O N C

O

C

N

N N C

O

R

O

C

C

N

N C

O R O

N

O O C

N O

N

C

N

N C

O

R

O

C

C

N

R

O

C

C

N

N C

O

N

N C

R

O

O

C

O C

N

N C

O

in which the R stands for

CH3 C CH3 Many cyanate ester thermosets are commercially available, such as those based on the following dicyanate monomers, for which the glass transitions temperatures listed are for the cured resins

CH3 N

C

O

C H

O

C

N, Tg = 258°C

Thermoset Materials

CH3

CH3 CH3

N

C

C

O

CH3

CH3

N

C

O

O

N, Tg = 252°C

CH3

CH3

CH3

C

C

CH3

CH3

OCN

C

O

OCN

CH2

C

N, Tg = 192°C

OCN

CH2 n

OCN

NCO

OCN

0.2

CF3 N

C

O

C CF3

O

C

N, Tg = 270°C

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Of the many other cyanate esters, two containing sulfur and phosphorus in the backbone are, respectively,

O NCO

O

S

O

OCN

O

OCN

O and

O NCO

O

P

Cyanate esters offer many advantages: the processability of epoxies, the high-temperature capabilities of bismaleimides – they can operate at continuous temperatures up to 177°C (350°F) – thermal and fire resistance of phenolics, toughness, low moisture absorption, good dielectric properties, and radar transparency. They are replacing epoxies in high-speed electronics applications. They are expected to replace epoxies in many advanced composites applications. 13.3.2

Bismaleimides

Bismaleimides are high-temperature thermosetting materials that can operate at continuous temperatures of 180°C (356°F) in moist environments required in aerospace applications. A bismaleimide (BMI) is prepared by reacting maleic anhydride with a diamine:

O

O

O

C

C

C

O

+

H2N

R

N

NH2

R

N

C

C

C

O

O

O

A commonly used BMI is obtained by using methylene dianiline (MDA)

H H2N

C H

as the diamine to give the BMI

NH2

Thermoset Materials

O

O H

C N

C N

C H

C

C O

O

BMIs can be cured in two ways to form three-dimensional thermoset polymeric networks. The first is by heating the material at a high-temperature which causes the material to form a homopolymer:

O

O N

O

N

R

O

O

O

O

N

O

O

N

O

O

N

O

R

R

R

N

N

O

O

N

O

O

Alternatively, they may be cured by using a curing reactant. For example, curing with an olefin results in copolymers:

R1 R 3 C O

O N R

+

R

N O

R1

O

2

R3 C

C

4

R

O

N

O

C

R2 R 4

R O

N

O R1 R 3 C

C

R2 R 4 Many different types of BMIs are available for addressing different needs. They are used as matrix materials for making laminates and prepregs for high-temperature, high-performance structural composites requiring high toughness. Their processing characteristics are similar to epoxy resins.

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13.3.3

Polyimides

Because of the demands of the aerospace industry, a large number of high-temperature polyimides have been developed and commercialized to address many requirements such as, high continuous operating temperatures, toughness, low radar absorption, and ease of part processing. This section discusses two polyimides, called PMR-15 and LaRC RP-46, which are still in use. 13.3.3.1

PMR-15

The best known and first applied high-temperature thermosetting polyimide is PMR-15, in which PMR is the acronym for “in-situ polymerization of monomer reactants”; the 15 in PMR-15 stands for a formulated molecular weight of 1,500. PMR-15 is made by reacting three monomers – the dimethyl ester of 3,3 ′4,4 ′-benzophenone tetracarboxylic anhydride (BTDE), 4,4 ′-methylene dianiline (MDA), and the monomethyl ester of 5-norbornene-2,3-dicarboxylic anhydride (NE) – resulting in the nadic end-capped molecule O

O

O

O

C

C

C

C

N

CH2

O C

N

N

CH2

N

C

C

C

C

O

O

O

O n

in which n is a small number. To achieve the 1,500 molecular weight, the BTDE, MDA, and NE are combined in the molar ratio of n:(n + 1):2, where n = 2.087. The basic starting monomer for this resin is then O

O

O

O

C

C

C

C

N

CH2

O C

N

N

CH2

N

C

C

C

C

O

O

O

O

2.087

During curing the nadic end caps undergo complex reactions that are not fully understood. In the addition-cure reaction the norbornene endcap is believed to produce cyclopentadiene or cyclopentadienyl segments shown here: O

O

O

O

C

C

C

C

N

CH2

N

O C N

CH2

N

C

C

C

C

O

O

O

O

2.087

n

These segments recombine with maleimide segments to complete the cross-linking reaction, to eventually form three-dimensional networks. PMR-15 is mainly used in the form of prepregs (Section 20.8.3.1) – resin impregnated fiber or fabric laminae – that can be laminated in several ways (Section 20.8.3).

Thermoset Materials

While this material has been used in many demanding high-temperature aerospace applications, the carcinogenic nature of the ingredient MDA requires careful handling that adds to the product cost. 13.3.3.2 LaRC RP-46

The search for a less toxic alternative to PMR-15 led to the development of LaRC RP-46; LaRC is an acronym for “Langley Research Center,” where it was developed. It has properties similar to those of PMR-15 without having its toxic ingredients. It is made from three monomers – the dimethyl ester of 3,3 ′4,4 ′-benzophenonetetracarboxylic acid (BTDA), 3,4 ′-oxydianile (ODA), and the monomethyl ester of 5-norbornene-2,3-dicarboxylic anhydride (NE) as an end-capping agent – resulting in the nadic end-capped molecule O C N C

O

O

O

C

C

C

N

O

O

O C

N C

C

O

O

O

N C

n

O

To achieve the 1,500 molecular weight, the BTDE, ODA, and NE are combined in the molar ratio of n:(n + 1):2, where n = 2.073. O

O

O

O

C

C

C

C

N

CH2

O C

N

N

CH2

N

C

C

C

C

O

O

O

O

O

O

O

O

C

C

C

C

N

CH2

N

2.087

O C N

CH2

N

C

C

C

C

O

O

O

O

2.087

n

13.3.4

Poly(Phenylene Benzobisoxazole)

Poly(phenylene benzobisoxazole) (PBO) is a rigid-rod isotropic liquid crystal polymer, having the chemical structure, poly(p-phenylene-2,6-benzobisoxazole),

N

N

O

O

n

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A commercial fiber version (Zylon®) of this material has a high tensile strength, 5.8 GPa (841 × 103 psi), which is 1.6 times that of p-Aramid fibers (Kevlar®, Section 11.7.7), and a tensile modulus of 180 GPa. It exhibits very low creep, low moisture absorption and is dimensionally stable against humidity, high chemical resistance, and high abrasion resistance – even at high temperatures – and has high flame resistance and thermal stability. The performance of this thermoset material exceeds that of thermoplastic aramids (Kevlar) across a wide spectrum of physical and chemical properties. PBO can be processed into continuous filament, staple fiber, spun yarn, and woven and knitted fabrics. Besides the use of its fibers as a unidirectional reinforcement in high-temperature, advanced composites (Chapter 25), PBO fibers have been used in snow-mobile drive-belts, tennis and table tennis racquets, medical devices, snowboards, and firefighting gear.

13.4 Thermoset Elastomers Elastomers are non-crystalline, low-modulus polymers used above their glass transition temperatures, at which they exhibit rubber-like behavior of recovering from large deformations. They may be thermoplastic elastomers (Section 11.9) or thermosetting elastomers that may be classified into three categories: (i) Those in which the precursor thermoplastic chains are unsaturated and are cross-linked to form thermoset elastomers by a process called vulcanization. (ii) Those made from saturated polymers. And (iii), inorganic elastomers, such as silicones. 13.4.1

Diene Elastomers

Diene elastomers are unsaturated polymers having carbon double bonds in the main chain. Common elastomers of this family are based on the dienes isoprene, butadiene, and their derivatives. 13.4.1.1

Polyisoprene (Natural Rubber)

The first industrially important elastomers were various forms of rubber – so named because it could be used for rubbing out marks made by lead pencils – made from the sticky, milky latex tapped from incisions in the barks of para rubber trees (Hevea brasiliensis). While polyisoprene has four isomers (Section 9.2.4), the main ingredient of natural rubber is the thermoplastic cis-1,4-polyisoprene having the structure

CH2 H

C

C

CH2 CH3

n

which can now be synthesized from isoprene:

CH2

CH

C CH3

CH2

CH2

CH

C CH3

CH2 n

Thermoset Materials

Synthetic isoprene has better consistency and much tighter control of molecular weight distributions than that derived from natural latex. Natural rubber is tacky (sticky); it deforms easily when warm but is brittle when cold. Its hardness and elastic properties can be improved by using another chain, such as sulfur, to cross-link the thermoplastic isoprene chains at some of the carbon double-bond sites. This process of cross-linking, or curing, is called vulcanization. Use of sulfur to effect vulcanization in polyisoprene results in cross-linked networks of the type

S CH3

CH3 S CH2

CH2

CH

C

CH2

CH2

C

CH

CH2 CH2

CH

CH2

S S CH3

CH3 S CH

CH2

CH2

CH

C

CH2

CH2

C

S S C

CH3

CH3 S

CH3 CH

CH2

CH2

C

CH

CH2

CH2

C

CH

S

The sulfur bridges between chains, here shown with three sulfur atoms, can contain between one and eight sulfur atoms. The degree of cross-linking results in different grades of harder rubber. It is often compounded with carbon black to form the type of rubber used in automotive tires. 13.4.1.2 Polychloroprene (Neoprene)

In polychloroprene, the methyl (CH3 ) group in the polyisoprene chain is replaced by chlorine, resulting in the structure

CH2

CH2 C

H

C Cl

n

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Because of the presence of chlorine, sulfur is ineffective in vulcanizing polychloroprene. Instead metallic oxides, such as zinc oxide, are used to provide oxygen to open and link carbon double bonds. Neoprene is a relatively expensive, general-purpose synthetic rubber, having high tensile strength, and oil and flame resistance. It resists degradation by oxygen and ozone well. And it is very resilient; that is, it exhibits excellent elastic recovery from deformations. Neoprene is used in applications that require resistance to oil, heat, flame, and abrasion. Industrial applications include wire and cable insulation, hoses, automotive fan belts, springs, flexible mounts, gaskets, and adhesives. Non-industrial uses of this material include inner liners for knee and elbow pads; laptop sleeves, holders for smart phones, and remote controls; life jackets; and professional diving suits.

13.4.1.3

Polybutadiene

Polybutadiene, obtained by polymerizing butadiene

CH2

CH

CH

CH2

CH2

CH

CH

CH2

n

has cis, trans, and vinyl forms, of which the vinyl form only accounts for a few percent. The other two forms have the structures

CH2 C H

CH2

CH2

H C

C

H

H

C CH2 n

n

cis 1,4 (1,3 butadiene)

trans 1,4 (1,3 butadiene)

The trans material has a straighter backbone and forms a semicrystalline material; the cis material has much larger amorphous regions and forms highly resilient (elastic) rubber. Polybutadiene is cured (vulcanized) using sulfur in the same way as for polyisoprene. Over 70% of polybutadiene, a majority of it in the cis form, is used in tires, for which it competes mainly with natural rubber and styrene-butadiene rubber SBR). Because of its resilience the inner cores of most golf balls are made of polybutadiene; this core is encased in a dimpled core of a harder material.

13.4.1.4

Poly(Isobutylene-co-Isoprene) (Butyl Rubber)

Poly(isobutylene-co-isoprene), often referred to as, butyl rubber, is a copolymer of isobutylene

CH3 CH2

CH CH3

Thermoset Materials

and isoprene, and has the structure

CH3 CH2

CH2

C CH3

CH

C

CH2

CH3

n

m

in which the ratio of isobutylene to isoprene is about 98 : 2. The double bond in the isoprene unit makes it possible to cross-link chains of this copolymer through the process of vulcanization used for cross-linking polyisoprene. Because of its excellent flexibility and impermeability to air, this material is widely used in applications requiring an airtight rubber, such as in inner tubes for tires, inflatable bladders for balls used in different sports, gas masks, and even as a base for chewing gum. 13.4.1.5 Poly(Styrene-co-Butadiene) (SBR Rubber)

This is a copolymer of styrene and butadiene, normally called poly(styrene-butadiene) and has the structure

H

H

C

C

CH2

CH

CH

CH2 m

H

n

The ratio of styrene to butadiene is most commonly about 25 : 75. Depending on the styrene content it has a Tg of about −55°C (−67°F), and finds use over a temperature range of about −40 – 100°C (−40 – 212°F). As with other rubbers, it is mainly used as a thermoset vulcanized by using sulfur. Oxidation by oxygen and ozone causes increased cross-linking so that this material tends to harden with age; instead, natural rubber softens. This polymer is also cured (vulcanized) using sulfur. It is by far the most widely used rubber. Filled with carbon black its abrasion resistance makes it the rubber of choice for pneumatic tires for automobiles. 13.4.1.6 Poly(Acrylonitrile-co-Butadiene) (NBR Rubber)

This is a copolymer of acrylonitrile and butadiene has the structure H

H

C

C

H

CN

CH2

CH

CH

CH2 m

n

which, on curing, is referred to as nitrile rubber, nitrile-butadiene rubber (NBR). The ratio of acrylonitrile to butadiene in NBR varies from 15 : 85 to 50 : 50. With increasing acrylonitrile content, while the strength increases, the tendency to swell from hydrocarbons decreases, and the permeability to gases decreases, the material exhibits lower flexible at lower temperatures. It can be used in the temperature range of −40 – 108°C.

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It is used in the automotive and aeronautical industries to make fuel and oil handling hoses, O-rings, gaskets, transmission belts. It is also used in disposable non-latex gloves, footwear, adhesives, sealants, sponges, expanded foams, and floor mats. 13.4.2

Ethylene-Propylene Copolymer-Based Elastomers

There are two types of important ethylene-propylene based rubbers: Ethylene-propylene rubber (EPR) is obtained by curing ethylene-propylene copolymers, and ethylene-propylene-diene monomer rubber (EPDM) is obtained by curing ethylene-propylene-diene terpolymers. Although they do have differences, their performance characteristics are similar in most applications. While both EPR and EPDM work well in water, steam, and polar solvent environments, they swell dramatically in petroleum products. A notable difference is that some EPDM compounds tolerate a concentrated acetic acid environment far better than most EPRs. 13.4.2.1

Ethylene-Propylene Rubber (EPR)

EPR cannot be vulcanized by using sulfur and require a peroxide-based cure system. Its stable, saturated backbone makes it resistant to heat (up to 160°C) and ozone, and it weathers well. It has very high electrical resistance. EPR has many applications: Automotive weather-stripping and seals, garden and appliance hoses, tubing, belts, and roofing membranes. Because of its high electrical resistance, it is used as an insulator for high-voltage cables. 13.4.2.2

Ethylene-Propylene-Diene Monomer (EPDM) Rubber

In comparison to EPRs, while peroxides can also be used, the additional diene monomer in ethylene-propylene-diene allows curing through sulfur. Several different dienes are used in EPDM. EPDM is most commonly used in automotive door seals, window seals, and trunk seals. It is used in automotive paint spray, industrial-respirator face seals for which silicones cannot be used. Besides garden and appliance hose, pond liners, washers, belts, electrical insulation, and O-rings, it is also used in solar panel heat collectors and for speaker cone surrounds. Since EPDM does not pollute water, it is used for roofing membranes, especially for run-off rainwater harvesting. It is widely used for geomembranes. 13.4.2.3

Silicone Elastomers

Instead of the carbon-carbon backbone of organic polymers, silicone polymers have a silicon-oxygensilicon backbone that provides a high level of resistance to degradation by oxygen, ozone, UV radiation, and moisture. The main silicone polymer is poly(dimethylsiloxane) having the structure

CH3 Si CH3

O n

This polymer can be cross-linked in two ways to obtain silicone rubber: (i) Thermosetting rubbers are made by using peroxides to cross-link the poly(dimethyl siloxane) chains at the vinyl (methyl) groups;

Thermoset Materials

cross-linking, or vulcanization, requires heating. The material normally comes in the form of two separate components that are mixed just before use. They are also available as a one-component material containing an inhibitor that is deactivated by heat. (ii) Room-temperature vulcanizing (RTV) silicones are one-component low molecular weight poly(dimethyl siloxane) liquids with reactive end groups, which cross-link by reacting with ambient moisture. Silicone rubber has very low strength in comparison to the organic rubbers. However, it can be used at extreme temperatures, −55 – 300°C, at which it retains its mechanical properties. Because of their high-temperature properties retention, they are used in many automotive under-hood applications, such as for ignition cables, coolant and heater hoses, O-rings, and seals. These materials are inherently inert and biocompatible, because of which they are used in food and medical products, including baby bottle nipples, surgical tubing, subdermal implants, and prosthetic devices.

13.5 Historical Notes The discovery of vulcanization of natural rubber in 1839 led to the first commercial thermosetting material, rubber. The next major advance in synthetic plastics was the invention of a thermosetting phenol-formaldehyde resin by Leo Baekeland in 1909, named Bakelite, which dominated plastics technology till the invention of alkyds (1926) and urea-formaldehyde (1928). Alkyds, developed in the 1920s, and not discussed elsewhere in this book, are polyesters modified by adding fatty acids and other components. They are derived from polyols and a dicarboxylic acid or carboxylic acid anhydride. Alkyd is a modification of “alcid,” derived from alcohol and organic acids. The fatty acids help to improve the flexibility of coatings, thereby making alkyds useful additives for paints and for binding sand molds for castings. The original alkyds, compounds of glycerol and phthalic acid, were sold under the name Glyptal as substitutes for the darker colored Copal resins, creating alkyd varnishes that were much paler in color. Unsaturated polyester resins were invented in 1933. Melamine-formaldehyde was invented in 1935, and commercially introduced in 1938. The first (German) patent on epoxies was issued in 1939, and a second important patent was issued to Castan in 1943. Finally, epoxies were commercially introduced in 1947. Silicones were invented in 1942. Polyurethane elastomers were invented in 1946. And polyurethanes were introduced in the US in 1954. Several high-temperature thermosets were developed as resins for high-temperature advanced composites: polyimide products in 1964, cyanate esters during 1970 – 1980, and bismaleimides during 1980 – 1990. Poly(phenylene benzobisoxazole) (PBO), used for making high-temperature, high-strength thermoset fibers for reinforcing advanced composites was invented ad developed in the 1980s. Figure 13.5.1 shows the timeline for the invention of most of the commercially important thermoset resins in the twentieth century. The discovery of vulcanization of natural rubber in 1839 set the stage for the large-scale use of natural rubber, which originally was only available from the latex of rubber trees in the Amazon. The first synthetic rubber, methyl isoprene, was synthesized in 1909. The first rubber from butadiene was synthesized in 1910. Much of the research on synthetic rubbers was catalyzed by the unavailability of natural rubber during the two World Wars for making rubber hoses, belts, gaskets, and tires.

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PHENOLIC 1909 ALKYD 1926

PU (US) 1954 POLYIMIDE PRODUCTS 1964

NEOPRENE 1930

UNSATURATED POLYESTER 1933

CYANATE ESTERS 1970 –1980

MELAMINE INTRODUCED COMMERCIALLY 1938 VULCANIZATION OF NATURAL RUBBER 1839

URETHANE-POLYESTER (GERMANY) 1941

STAUDINGER advances macromolecular hypothesis 1920

SILICONE 1942

PBO 1980

EPOXIES 1942

BMI 1980 –1990

PU ELASTOMERS 1946 EPOXY INTRODUCED COMMERCIALLY 1947

10

1900

20

30

60

40

70

1950

80

90

2000

Figure 13.5.1 Diagram charting the invention of thermoset resins in the twentieth century.

Commercial production of thiokol, a rubber produced from the mixture of ethylene dichloride and polysulfate, began in 1930. Because of it resisted oil and solvents far better than natural rubber, it sold as a premium product. DuPont commercialized polychloroprene (Neoprene) in 1933. IG Farben in Germany polymerized Buna-S, styrene-butadiene rubber (SBR) in the 1930s; it was being produced in large quantities by 1935. IG Farben developed acrylonitrile butadiene rubber (Buna-N), nitrile rubber (NBR) in 1931, and began its mass production in 1935. Hypalon synthetic rubbers and EPDM were first introduced into the United States between 1955 and 1965. Hypalon has a polymer polyethylene backbone that is inert and converted to a semicrystalline or an amorphous polymer with elastomeric characteristics through chlorosulfonation and chlorination. EPDM and EPDM rubbers are made of copolymers and ethylene-propylene (EP), which are all polymers. These polymers undergo compounding with fillers, plasticizers, and catalysts to form sheets that are vulcanized afterward to produce cured elastomeric membranes.

Thermoset Materials

Prysmian pioneered the development of Ethylene Propylene (EP) based insulation compounds in the late 1950s. These EP compounds became commonly referred to as Ethylene Propylene Rubber (EPR).

13.6 Concluding Remarks Because of ease of part fabrication methods, when they satisfy part performance requirements, thermoplastics are the preferred materials for making parts in large volumes. Although their part processing costs are higher, thermosets are used when high rigidity with very little creep is required of a part. And thermoset parts can be used up to much higher temperatures. The thermosets discussed in this section cover three distinct industries: (i) The thermosetting resins in Section 13.2 are used in the traditional plastics industry, which uses relatively low-cost resins to fabricate parts using low-cost manufacturing processes. Thermosets form a small part of this industry segment that is dominated by thermoplastics. (ii) All the high-temperature materials in Section 13.3 and the epoxies (Section 13.2.6) are used as matrix materials in advance composites (Chapter 25). The advanced composites industry uses relatively expensive materials to fabricate parts using specialized machines to produce smaller number of parts; as a result, manufacturing costs tend to be high. In contrast to the plastics industry, the advanced composites industry uses sophisticated mechanics for designing parts to meet demanding performance requirements. And (iii), the materials in Section 13.4 are used in the rubber industry, which is dominated by automobile tires. As in advanced composites, tire performance prediction also requires the use of sophisticated mechanics tools.

311

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14 Polymer Viscoelasticity 14.1 Introduction Under normal conditions the response of metals and ceramics to external stimuli such as stress or deformation is instantaneous and time-independent, so that a constant load results in a constant, time-independent deformation. For these materials the theory of elasticity provides an adequate model for characterizing mechanical properties and mechanical design. In contrast, polymeric materials exhibit time-dependent behavior for the understanding of which linear viscoelasticity theory, discussed in Chapter 7, provides the simplest model. In this class of materials, the deformation can continue to change even under a constant load. Not only does this make materials characterization more difficult, it requires a complete rethinking of mechanical design methodologies that have yet to be adequately defined. This chapter focuses on phenomenological aspects of the time-dependent behavior of polymeric materials. The simplest theoretical framework for this class of materials has been discussed in some detail in Chapter 7 (Linear Viscoelasticity), a perusal of the introductory portions of which will aid in understanding the material in this chapter.

14.2 Phenomenology of Polymer Viscoelasticity Two types of model experiments discussed in Chapter 7 are useful for understanding the timedependent behavior of polymeric materials: In a stress relaxation experiment the stress induced by a constant deformation is monitored; the ratio of the time-dependent stress 𝜎 (t) to the constant strain 𝜀0 is the time-dependent relaxation modulus E (t) = 𝜎 (t)∕𝜀0 . In a creep experiment, the strain induced by a constant stress is monitored; the ratio of the time-dependent stress 𝜀(t) to the constant stress 𝜎 0 is the time-dependent creep compliance C (t) = 𝜀(t)∕𝜎 0 . In addition to the moduli depending on time, they are also strongly affected by the temperature. Thus, in contrast to metals and ceramics – which for most applications are essentially elastic materials with weak dependence on temperature and very weak dependence on time, so that their elastic moduli are constants – in plastics the relaxation moduli, such as E (t, T ) or G (t, T ), are strong functions of both time and temperature, resulting in enormous complexity, both in terms of materials characterization and in terms of how such material properties are to be used for part performance prediction.

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

Introduction to Plastics Engineering

In polymer materials discussions, this time and temperature dependence is introduced by considering how properties vary with time at a fixed temperature and with temperature at a fixed time. Fortunately, the time and temperature dependence can be collapsed into a single variable. The following sections address these issues. 14.2.1

Relaxation Moduli at Constant Temperature

Because polymers exhibit very large variations in modulus over long time periods, logarithmic scales have to be used to exhibit modulus-time variation data. Figure 14.2.1 shows the schematic variation of the normalized tensile modulus, E (t, T )∕E (0, T ), at a fixed temperature, T, versus log t, in which the time is measured in seconds. In this schematic, the normalized tensile modulus is sensibly constant till about 10 seconds (log t = 1), and then begins to decline rapidly, dropping by slightly over three orders of magnitude by log t = 4, or till t = 10 4 seconds (about 2.8 h). The normalized modulus then remains constant at about 10 −3.5 till about log t = 9, or till t = 10 9 s (about 32 years). After that the modulus begins another rapid decline from about 10 −3.5 , attaining a value of about 10 −7 by log t = 12, or by t = 1012 s (about 32,000 years).

LOG NORMALIZED MODULUS

314

0 E (t,T0 )/(0,T0 )

–5

–8 0

5

10

LOG TIME Figure 14.2.1 Schematic variation of the normalized tensile modulus, E (t, T0 )∕E (0, T0 ), at a fixed temperature, T0 , versus log (t), in which the time is measured in seconds.

Clearly, then, for an experiment lasting about 10 seconds, this material would be considered elastic with a normalized modulus of unity. But for an experiment lasting from about 3 hours to about 32 years, the same material would have a much smaller constant normalized modulus of about 10 −3.5 . The material behaves elastically in both plateaus but with very different elastic moduli. And for experiments lasting over thousands of years the material can exhibit a normalized modulus of 10 −7 .

Polymer Viscoelasticity

14.2.2

Relaxation Moduli at Constant Time

LOG NORMALIZED MODULUS

In another class of experiments the relaxation modulus is measured at different temperatures but at a fixed time. The schematic variations of the normalized tensile modulus with the temperature, T, varying over the range of −140 to 140°C, at a fixed time, t0 , which for purposes of this discussion could be 10 seconds, say, for two types of materials are shown in Figure 14.2.2, in which the shaded region represents the temperature range of 20 – 50°C. Rubbers are materials that, at this ambient range, behave elastically with a normalized elastic modulus of about 10 −6 in the lower plateau. In contrast to this, in this temperature range a plastic has a normalized elastic modulus of about unity in the upper plateau. Of course, for the examples shown in this figure, for lower temperatures, say below −10°C, the rubber will have a normalized modulus of unity in the upper plateau region; and at much higher temperatures, say in the range of 100 – 115°C, the plastic will behave like a rubber with a normalized modulus of about 10 −4.5 .

E (t0 ,T )/(t0 ,TLow )

–5 RUBBER PLASTIC –8

–100

0

100

TEMPERATURE (°C) Figure 14.2.2 Schematic variation of the normalized tensile modulus, E (t0 , T )∕E (t0 , TLow ), for a fixed test time, t0 , versus log t. The shaded region represents the temperature range of 20 – 50°C; the transition temperatures of rubbers and plastics are, respectively, below and above this temperature range.

Thus, in terms of elastic response in the “normal” ambient temperature range of 20 – 50°C, rubbers are relatively soft polymeric materials with low normalized elastic moduli, of about 10 −6 ; they exhibit relatively large strains or elongations under stress. In this temperature range a plastic is a relatively hard polymeric material with a high elastic modulus, so that they exhibit relatively low elongations under stress. The large reduction in the elastic modulus from the upper, hard high-modulus region characteristic of a plastic to the lower, low-modulus region characteristic of a rubber occurs over a very narrow temperature range which coincides with the glass transition region (discussed in Section 10.3.1) of the material. Every

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material behaves like a hard glass above its glass transition temperature, and like a soft rubber below this temperature. Using the elastic fixed-time (say 10 seconds) modulus-temperature schematic in Figure 14.2.3, the viscoelastic behavior of typical linear and lightly crosslinked amorphous polymers may be divided into four regions, of which the first three are common to both types of materials: (i) the glassy region in which the material behaves like a hard rigid glass with an elastic modulus Eg , which decreases slowly with increases in temperature, (ii) the transition region in which the modulus drops by about three orders of magnitude over a narrow temperature range of about 5 – 25°C, and (iii) the rubbery region with a low rubbery modulus Er , which also decreases slowly with temperature increases. Beyond this rubbery region a lightly crosslinked polymer continues to behave as a rubber. But the modulus of a linear polymer continues to decrease.

0 GLASSY

–5

TRANSITION

LOG NORMALIZED MODULUS

316

RUBBERY

LIGHTLY CROSSLINKED

LINEAR

–8

TEMPERATURE Figure 14.2.3 Schematic variation of the normalized tensile modulus, E (t0 , T )∕E (t0 , TLow ) versus log t illustrating the four typical regimes of polymer viscoelastic behavior.

14.2.3

Relaxation Moduli of Several Resins

This subsection illustrates the effects of several parameters – such as the molecular weight, differences in materials, and additives such as plasticizers – on the relaxation modulus. All comparisons are based on 10-second moduli. 14.2.3.1

Effect of Molecular Weight: Relaxation Moduli of Polystyrene

Figure 14.2.4 shows the 10-second relaxation moduli of three polystyrene (PS) samples. The first, with a molecular weight of about M n = 10 3 , exhibits a very rapid reduction in the modulus with increases in

Polymer Viscoelasticity

temperature, and does not exhibit a rubbery plateau. In contrast, the second and third samples, with molecular weights of M n = 1.4 × 10 5 and Mn = 2.17 × 10 5 , respectively, exhibit well-defined glassy and rubbery plateaus. Moreover, the low-molecular weight sample starts its decline in modulus at a lower temperature than the higher molecular weight samples.

POLYSTYRENE

LOG E(10) (Pa)

1.0

Mn = 2.17 × 10 5 Mn = 10 3 0.5

Mn = 1.4 × 10 5 100

150

200

TEMPERATURE (°C) Figure 14.2.4 Variation of the 10-second modulus with temperature of polystyrene for three molecular weights. (Adapted with permission from “Introduction to Polymer Viscoelasticity,” by J.J. Aklonis, W.J. MacKnight, and M. Shen, Wiley-Interscience, 1972.)

The relaxation moduli of the higher molecular weight samples are almost indistinguishable till about 120°C; for higher temperatures, the higher molecular weight sample exhibits a more pronounced rubbery plateau. These phenomenological differences in the behavior of polystyrene as a function of the molecular weight can be explained in terms of the constraints imposed on the motion of chains the lengths of which increase with molecular weight. But a discussion of the effects of chain lengths and entanglements is beyond the scope of this book. In general, increasing molecular weights tend to increase the glass transition temperature of a material. 14.2.3.2 Effects of Crystallinity: Relaxation Moduli of Several Resins

Figure 14.2.5 compares the 10-second relaxation modulus of amorphous polystyrene with those of the two semicrystalline polymers PVC and polyethylene (PE).

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PVC

PE

LOG E(10) (Pa)

318

PS

1.0

PVC PE PS 0.5 – 50

0

50

100

150

200

TEMPERATURE (°C) Figure 14.2.5 Variation of the 10-second modulus with temperature of polyethylene, poly(vinyl chloride), and polystyrene. (Adapted with permission from “Introduction to Polymer Viscoelasticity,” by J.J. Aklonis, W.J. MacKnight, and M. Shen, Wiley-Interscience, 1972.)

Polyethylene is a highly crystalline polymer with a glass transition temperature of about Tg = 75°C and a crystalline melting temperature of Tm = 125°C. The amorphous matrix begins to soften above Tg , but the decline in modulus is tempered by the strengthening effect of the crystallites. Then, at the melting temperature, Tm , the modulus drops rapidly to that at the rubbery plateau. Further increases in temperature will result in a viscous “liquid” state, just as for a non-crosslinked polymer, in which the material will flow. In contrast to the sudden reduction of the modulus of PS at its Tg , PE exhibits a gradual decline in the modulus in the temperature range between its Tg and Tm . Because PVC has a much lower degree of crystallinity than PE, it has a sharper modulus drop at its Tg = 78°C; similar to that characteristic of amorphous polymers. Also, its rubbery plateau is more pronounced and extends to its melting temperature of Tm = 180°C. 14.2.3.3

Effects of Plasticizers: Relaxation Moduli of PVC

Plasticizers (Section 11.3.2.6) are added to resins to improve their flexibility; they achieve this by lowering the glass transition temperature of a resin to below room temperature. An important example is that of PVC (Section 11.5.1), a rigid resin that is made flexible using plasticizers such as dioctyl phthalate (DOP). Figure 14.2.6 compares the relaxation modulus of PVC with that of PVC plasticized with 30 wt% of DOP. Notice the reduction in Tg and broadening of the glass transition region. Notice also the reduction in the effective melting temperature, indicating fluid-like flow occurring at a lower temperature.

LOG E(10) (Pa)

Polymer Viscoelasticity

1.0 PVC

PVC / 30 wt% DOP

0.5 – 50

0

50

100

150

200

TEMPERATURE (°C) Figure 14.2.6 Effect of plasticizer on the relaxation of PVC. (Adapted with permission from M.C. Shen and A.V. Tobolsky, in “Plasticization and Plasticizer Processes,” Norbert A. J. Platzer (Ed.), Advances in Chemistry, Volume 48, pp. 118 – 124, 1965. Copyright 1965, American Chemical Society.)

14.3 Time-Temperature Superposition The strong dependence of the relaxation moduli, such as E (t, T ) or G(t, T ), on both time and temperature, pose several challenges in terms of materials characterization: (i) for each temperature, how to conduct relaxation experiments over very, very long times, (ii) for each time, how to obtain the variations of such moduli with temperature, and (iii) how to report this time-temperature data. Fortunately, experiments on a wide range of plastics have shown that they exhibit a time-temperature equivalence that collapses data onto a single, master modulus-time curve at a reference temperature from which the moduli at any other temperature can easily be obtained. The characterization of this experimental behavior is referred to as time-temperature superposition. 14.3.1

Experimental Characterization of the Master Curve

This section first gives an overview of how the full stress-time relaxation curve over the full time range at a fixed temperature is constructed from stress-time relaxation data obtained over relatively short time intervals at different fixed temperatures. This process will be described by using Figure 14.3.1, in which part (a) schematically shows 10-second relaxation test data for five fixed temperatures. Curves I, II, III, IV, and V represent 10-second relaxation data obtained at fixed temperatures T1 , T2 , T3 , T4 , and T5 , where T1 > T2 > T3 > T4 > T5 . Part (b)

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of this figure shows how these data are used to construct the relaxation curve at temperature T1 over the full time range: The process starts with Curve I for temperature T1 being slid to the right onto the modulus-time frame in part (b) of the figure. Then, Curve II, corresponding to fixed temperature T2 , is shifted to the right till its head melds with the tail of Curve I. As shown in the figure, let this rightward shift on the logarithmic axis be ln[1∕a(T2 )]. The factor a(T2 ), is a temperature-dependent, dimensionless material property. In a similar manner Curve III is shifted rightward, by amount ln[1∕a(T2 )], till its head melds with the tail of Curve II. Finally, this process results in the composite Curve I-II-III-IV-V, called the master curve at temperature T1 . So far there is nothing to indicate that this composite curve has any physical significance. But what is remarkable is that experiments on a wide array of polymers have shown that this composite curve gives the complete relaxation modulus at temperature T1 .

LOG NORMALIZED MODULUS

320

0

I

T1

0

I II

II

MASTER CURVE

T2 III IV

T3 T4 –5

–5

III

ln[1/a(T2 )]

IV

ln[1/a(T3 )]

V

ln[1/a(T4 )]

V

ln[1/a(T5 )] –8

T5 –2

0

–8

0

5

10

LOG TIME (a)

(b)

Figure 14.3.1 (a) Schematic, constant-temperature 10-second stress relaxation data at five different temperatures. (b) Construction of the complete master curve for stress relaxation at a fixed temperature over the full time range, showing horizontal shift of experimental data and the definition of the shift factor a (T).

With reference to the enlarged view in Figure 14.3.2, which only shows Curves I and II, what this experimental finding means is that the relaxation modulus E (t, T2 ), at time t and temperature T2 , at point A in part (a) of this figure is the same as the relaxation modulus at point B in part (b) of this figure at temperature T1 and at logarithmic time ln[1∕a(T2 )] + ln t = ln[t∕a(T2 )], that is at time t, so that E (t, T2 ) = E [t∕a(T2 ), T1 ]. Similarly, for any point on Curve III, E (t, T3 ) = E [t∕a(T3 ), T1 ], and so on.

LOG NORMALIZED MODULUS

Polymer Viscoelasticity

0

I II

T1

0

I II

A

B T2

ln t –5

–5

–8

–8

–2

ln t

ln[1/a(T2 )]

0

ln[1/a(T2 )] + ln t

0

5 LOG TIME

(a)

(b)

Figure 14.3.2 Enlarged view of Figure 14.3.1 with only Curves I and II shown. Construction illustrates the experimentally observed time-temperature correspondence that allows stress relaxation data over short time periods at different temperatures to be used for constructing the complete stress relaxation curve at a constant temperature.

For any temperature T, this result can be written as E (t, T ) = E [t∕a(T ), T1 ] which relates the modulus at temperature T to that at a fixed temperature T1 through the shift function a (T ). The fixed temperature to which the moduli at all other temperatures are related is referred to as the reference temperature TRef , so that the experimentally established time-temperature correspondence can be written as E (t, T ) = E [t∕a(T, TRef ), TRef ]

(14.3.1)

where the shift factor a(T, TRef ) is a function of both T and TRef . This equation can be rewritten (how?) as E [a(T, TRef ) t, T ] = E (t, TRef )

(14.3.2)

from which it follows that the relaxation moduli at two temperatures T1 and T2 are related through E [a(T1 , TRef ) t, T1 ] = E [a(T2 , TRef ) t, T2 ]

(14.3.3)

One often used practice is to use the glass transition temperature Tg as the reference temperature. While the relationships among the relaxation moduli have been discussed in terms of the tensile relaxation moduli, they apply equally to the shear relaxation moduli. That is, all the equations are valid with E (t, T ) replaced by G (t, T ). 14.3.2

Corrections to the Time-Temperature Correspondence Relations

The simple time-temperature superposition discussed in the previous section does not account for two effects: First, changes in temperature have a direct effect on the modulus that is not accounted for the time-temperature shift. Second, there is a temperature-dependent change in volume, or density, which is also not accounted for in this process. To account for these effects, in addition to the horizontal shifts for

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obtaining the master curve, the short-time experimental data also have to undergo small vertical shifts. One adjustment used for these effects is to modify Eq. 14.3.1 to E (t, T ) E [t∕a(T, TRef ), TRef ] (14.3.4) = 𝜌 (T ) T 𝜌 (TRef ) TRef which, for any two temperatures T1 and T2 results in E [a(T1 , TRef ) t, T1 ] E [a(T2 , TRef ) t, T2 ] = (14.3.5) 𝜌 (T1 ) T1 𝜌 (T2 ) T2 14.3.3

The WLF Equation

The shift factor a(T, TRef ), which is determined during the process of converting short-time, modulus-time experimental data into a master curve representing modulus-time data over a large time range, is an experimentally determined material property. When it is understood from the context what the reference temperature is, the shift factor is shortened to aT = a(T ). When the glass transition temperature is used as the reference temperature, M.L. Williams, R.F. Landel, and J.D. Ferry showed that the shift factor for amorphous polymers is well approximated by the WLF (Williams–Landel–Ferry) equation −C1 (T − Tg ) (14.3.6) log aT = log a(T, Tg ) = C2 + (T − Tg ) in which C1 and C2 originally were thought to be universal constants, C1 = 17.4 and C2 = 51.6, valid for all polymers. However, they do vary by small amounts from polymer to polymer. For example, for polystyrene and poly(methyl methacrylate), the values of C1 , C2 are 14.5, 50.4, and 17.6, 65.5, respectively. The WLF equation is valid only for T ≥ Tg , the regime from which data must be used for determining the constants C1 and C2 . As such, this equation is useful only in the flow regime of the material, with the viscosity decreasing with increasing temperature. Use of data below Tg results in negative values of the constants C1 and C2 , and the resulting equation does not exhibit the Arrhenius-like behavior above Tg . The WLF equation is therefore not useful for structural applications of plastics, in which plastics always are used below Tg . Clearly, for T = Tg , the shift factor has the value aTg = a(Tg , Tg ) = 1. The variation of the shift factor in the WLF equation with the temperature difference (T − Tg ) is shown in Figure 14.3.3. The apparently slow variation of log aT with increasing (T − Tg ) is deceptive, as can be seen from the variation of aT with (T − Tg ) indicated by the dashed line, which shows that aT is essentially zero for as small a temperature difference as (T − Tg ) = 10°C. 14.3.4

Physical Interpretation for the Time-Temperature Shift

While the time-temperature correspondence model is not fully valid over all temperature and time ranges, it does appear to apply to excursions around the glass transition temperature. This model provides a useful interpretation of polymer behavior: Stress relaxation is determined by a material’s internal clock and proceeds at the same rate when measured by this clock. The hotter the material gets the faster the motion at a molecular level becomes and faster the internal clock runs. In one tick, or unit time, on the material’s internal clock, the external observer’s elapsed time is a(T ), an amount that decreases rapidly with increase in temperature. 14.3.5

Summary

Although not fully correct, the time-temperature superposition principle provides a very useful model for interpreting stress relaxation data, as measured by the stress relaxation modulus, at different temperatures.

Polymer Viscoelasticity

Without the insights provided by this model, managing and interpreting stress relaxation data for different temperatures would be unmanageable. The bulk of the work on polymer viscoelasticity has been done by materials oriented, polymer scientists who are mainly interested in using relaxation data, a form of “mechanical spectrometry,” to develop models for, and correlate the behavior of, chain architecture and motion with macroscopic behavior. For that activity, even small variations and nuances are important for understanding polymer behavior. But viscoelastic relaxation data and the time-temperature shift model are also very important for development of viscoelasticity models for predicting the behavior of components under time-varying loads and temperature histories. The discussions in Section 14.3 form the basis for the theory of the idealized model of thermorheologically simple materials discussed in Section 7.7.1.

0

1.0

log a T = log a(T, Tg) =

– C1 (T– Tg ) C2 + (T– Tg )

–5

log a T

0.5

aT

– 10

aT 0

– 15 0

50

100

TEMPERATURE DIFFERENCE (T – Tg ) (°C) Figure 14.3.3 Variation of the shift function aT with the temperature difference (T − Tg ).

14.4 Sinusoidal Oscillatory Tests While the tensile relaxation moduli are important from the standpoint predicting part stiffness, and are conceptually appealing, determining them for wide temperature ranges over long time periods is a tedious process – even if the time-temperature superposition discussed in the previous section is valid. It is easier to do dynamic experiments in which the test specimen is subject to a sinusoidally varying load, as for example by subjecting the end of a tube to a sinusoidally varying twisting motion that induces a sinusoidally varying stress.

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The theoretical framework for such oscillatory stress tests on viscoelastic materials, called dynamical mechanical analysis (DMA), is developed in Section 7.6: A sinusoidal shear strain 𝛾 (t) = 𝛾 0 cos (𝜔 t), which for mathematical can be interpreted as being the real part of 𝛾 (t) = 𝛾 0 exp (i𝜔 t), results in time-varying stress, which is the real part of 𝜎 (t) = G*(𝜔) 𝛾 0 exp (i𝜔 t). The dynamic modulus, or complex modulus G*(𝜔), is written as G* (𝜔) = G ′(𝜔) + i G ′′(𝜔), in which the real quantities G ′(𝜔) and G ′′(𝜔) are called, respectively, the storage modulus and the loss modulus. With the dynamic modulus written as G*(𝜔) = |G*(𝜔)| exp(i𝛿 ), the magnitude |G*(𝜔)| of G*(𝜔) is the ratio of the stress amplitude to the shearing strain amplitude, and 𝛿 , the phase angle by which the strain lags the stress, is called the loss angle. Empirically, 0 ≤ 𝛿 ≤ 𝜋 ∕2. Also, tan 𝛿 , called the loss tangent, is related to the storage and loss moduli through tan 𝛿 = G ′′(𝜔)∕G ′(𝜔). 14.4.1

DMA Data for High-Performance Thermoplastics

While it is easier to experimentally characterize G ′ and G ′′ as a function of the frequency 𝜔 at constant temperature, data on commercial resins is most often presented as curves of G ′ and G ′′ as a function of the temperature at a fixed frequency. Besides indicating the temperature range over which G ′ and G ′′ are stable, in such graphs the glass transition range shows up as a region in which G ′, G ′′, and tan 𝛿 undergo very rapid changes. Because of the large variations in properties and temperature, the data are presented in the form of log (property) versus log (temperature). Figure 14.4.1 shows the variations of logG ′, logG ′′, and log(tan 𝛿 ) versus log(temperatures) for the amorphous plastics polycarbonate (PC), polyetherimide (PEI), modified poly(phenylene oxide) 1 G'

10

G" 0 log (tan δ )

log G ' , log G " (Pa)

324

PC (Lexan® 100) –1

5 tan δ

–2

2 0 (a)

10

100

1000

LOG TEMPERATURE (°C)

Figure 14.4.1 Variations of G ′, G ′′, and tan 𝛿 versus the temperature at a constant frequency. (a) Polycarbonate. (b) Polyetherimide. (c) Modified poly(phenylene oxide). (d) ABS. (e) Poly(butylene terephthalate). (f) PC/PBT blend. (g) PC/ABS blend. (h) M-PPO/PA blend. (Adapted from figures courtesy of SABIC.)

Polymer Viscoelasticity

1 G'

10

0 log (tan δ )

log G' , log G" (Pa)

G"

PEI (ULTEM® 1000) –1

5 tan δ

–2

2 0

10

(b)

100

1000

LOG TEMPERATURE (°C) 1 G'

10

0 log (tan δ )

log G ' , log G " (Pa)

G"

M-PPO (Noryl® PX0844) –1

5 tan δ

–2

2 0 (c) Figure 14.4.1 (Continued)

10

100

LOG TEMPERATURE (°C)

1000

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1 G'

G"

0 log (tan δ )

log G' , log G " (Pa)

10

tan δ –1

5

ABS (Cycolac ® MG37) –2

2 0

10

(d)

100

1000

LOG TEMPERATURE (°C) 1 G'

10

G"

0 log (tan δ )

log G' , log G " (Pa)

326

PBT (Valox ® 310) –1

5

tan δ –2

2 0 (e) Figure 14.4.1 (Continued)

10

100

LOG TEMPERATURE (°C)

1000

Polymer Viscoelasticity

1 G'

10

0 log (tan δ )

log G ' , log G " (Pa)

G"

PC/PBT (Xenoy ® 1102) –1

5 tan δ

–2

2 0

10

(f)

100

1000

LOG TEMPERATURE (°C) 1 G'

G"

0 log (tan δ )

log G' , log G " (Pa)

10

PC/ABS (Cycoloy ® 1200HF) –1

5 tan δ

–2

2 0 (g) Figure 14.4.1 (Continued)

10

100

LOG TEMPERATURE (°C)

1000

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1 G'

10

G" 0 log (tan δ )

log G ' , log G " (Pa)

328

M-PPO/PA (Noryl GTX ® 910) –1

5

tan δ

–2

2 0 (h)

10

100

1000

LOG TEMPERATURE (°C)

Figure 14.4.1 (Continued)

(M-PPO), and acrylonitrile-butadiene-styrene (ABS); the semicrystalline plastic poly(butylene terephthalate) (PBT); blends PC with PBT and ABS; and a blend of M-PPO with polyamide (PA). For the amorphous plastics PC, PEI, M-PPO, and ABS, notice the rapid changes in G ′, G ′′, and tan 𝛿 near the glass transition temperature. Also notice how the variations in these quantities for the semicrystalline plastic PBT are qualitatively different; this difference can also be seen in the curves for the PC/PBT blend.

14.5 Concluding Remarks This chapter has focused on a phenomenological description of polymer viscoelasticity. After a brief description of the complex dependence of the properties of plastics on time and temperature, empirically based time-temperature superposition, which considerably simplifies materials characterization, is discussed. Experimental data are used to describe how molecular weight, crystallinity, and plasticizers affect stress relaxation. Oscillatory tests, in which sinusoidally varying stresses are imposed on test samples, are described, and DMA data have been presented for several high-performance amorphous and semicrystalline resins and thermoplastic blends.

Polymer Viscoelasticity

Much of the material in this chapter is an outcome of efforts to understand how the molecular structure and chain dynamics affect the mechanical behavior of polymers, a sound basis for which is provided by polymer physics. In a broad sense, such topics come under the purview of polymer science. Now that plastics are being used in structurally demanding applications, a new focus for such activity will be in materials characterization that can be used for predicting part performance.

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15 Mechanical Behavior of Plastics 15.1 Introduction The mechanical behavior of plastics can be approached from two different perspectives: In polymer science, which focuses on understanding the relation of the chemical structure of a material to its macroscopic mechanical behavior, a simpler macroscopic mechanical characterization is sufficient. Such correlations between the chemical structure and macroscopic behavior provide the basis for the design of improved materials having desirable characteristics. In engineering, which is concerned with using materials in applications, a far more detailed characterization of the mechanical behavior is required for assuring adequate part stiffness and strength. As discussed in Chapter 5, prediction of part performance requires models for the response of solid plastics to thermal, mechanical, and electromagnetic stimuli. Plastics, especially thermoplastics, are viscoelastic materials – the phenomenology of, and idealized models for, which are discussed, respectively, in Chapters 14 and 7 – so that the response to mechanical stimuli is always time-dependent. However, in structural applications the stresses are normally maintained at sufficiently low levels at which the material behavior is approximately time-independent. Most procedures for mechanical design are based on time-independent models of material behavior. Chapter 4 discusses simple, highly idealized models for the mechanical behavior of plastics. This chapter describes various aspects of the phenomenology and the measurement of the thermomechanical behavior of plastics. While the underlying behavior of all materials is very complex – that of plastics being far more complex than of metals and ceramics – in engineering practice part performance is predicted by using very simple, time-independent conceptual models for material behavior. Most materials, especially plastics, exhibit time-dependent deformations under constants loads; this phenomenon is referred to as creep. Time-dependent reduction in stress under a constant deformation is referred to as stress relaxation. And the failure of a material under cyclic deformations – as the failure of a paper clip due to back-and-forth bending – is called fatigue. Existing methods of predicting part performance normally consider such behavior as secondary effects that need to be accounted for once part dimensions have been determined on the basis of simple time-independent models. Much of the work on characterizing the mechanical behavior of such materials – test types, interpretation of test data, constitutive equations, and yield criteria – and the use of mechanical data for predicting part performance are based on a metals mindset. This approach would be appropriate if the

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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mechanical behavior of plastics were qualitatively similar to that of metals, the only difference being in the magnitudes of parameters. But plastics are known to exhibit phenomena that metals do not: Plastics undergo stable necking, they can craze – a phenomenon unique to plastics, in which in local “failure” “separated” surfaces are connected through very large numbers of oriented fibrils that help maintain a level of material integrity – and their yield behavior is significantly affected by hydrostatic pressure. Also, they can fully recover from a yielded state on being heated to the glass transition temperature. Clearly, then, constitutive modeling must be based on an understanding of the different response phenomena that occur when such materials are subjected to thermomechanical loads. The simplest test for characterizing the mechanical properties of a material is the tensile test, in which a test specimen is pulled axially in a controlled manner. A measurement of the stress as a function of the strain (deformation) then characterizes the inherent stiffness of the material. Most tests for characterizing phenomenon such as creep, stress relaxation, and fatigue are modifications of the simple tensile test. Tests for characterizing biaxial deformations are more complex. The simplest biaxial test for plastics comprises stretching clamped sheets by lateral fluid pressure. In contrast to metals – which behave elastically till very small strains on the order of 0.002 – some plastics do not undergo permanent deformations till much larger strains on the order 0.06. While structural stiffness considerations would normally limit the allowable strains in plastics to much smaller strains, the large-deformation behavior is important for cold forming operations, such as sheet stamping, and for higher temperature operations such vacuum forming and blow molding. As such, the time-dependent, large-deformation behavior of plastics is also considered in the sequel.

15.2 Deformation Phenomenology of Polycarbonate The tensile test can be used to broadly classify materials into two groups: In some materials – such as stone, cast iron, and most ceramics – the stress-strain curve is sensibly linear and reversible and then, at a relatively low strain, the material abruptly fails. Such materials are said to be brittle. In other materials – such as aluminum, copper, and steel – the material continues to deform well beyond the small strain at which its stress-strain behavior is linear and reversible, and beyond which the material undergoes a permanent, unrecoverable deformation; in this region the material is said to have yielded. The deformation in this nonlinear range continues till the material necks and fails at relatively large strains. Such materials are referred to as being ductile. However, at low temperatures even a ductile material can fail in a brittle manner. As an example, a piece of soft rubber that behaves elastically at room temperature – causing it to bounce off a surface at which it is thrown – when cooled to liquid nitrogen temperature shatters like glass when hit with a hammer. And even the rate of deformation can change this behavior: Silly PuttyTM , a viscoelastic material sold as a play material, bounces off of a surface at which it is thrown; when a thin cylinder of this material is pulled slowly it behaves like a ductile material; when pulled at a fast rate it fails in brittle manner. Among plastics, thermosets have relatively high elastic moduli and strengths and tend to fail in a brittle mode. The room-temperature behavior of unfilled thermoplastics can be brittle, as in poly(methyl methacrylate) (PMMA), or ductile as in polyethylene (PE) and polycarbonate (PC). But, depending on the temperature, strain rate, and stress multiaxiality, these ductile materials can fail in brittle manner. Also, particle- and glass-fiber-filled thermoplastics tend to be brittle.

Mechanical Behavior of Plastics

This section first discusses the phenomenology of yield in bisphenol-A polycarbonate (PC) through tensile tests on thin, rectangular specimens cut from extruded sheets of a commercially available grade of PC. Because polymers yield in a tensile test by deformation localization, conducting such a test at a constant deformation rate is difficult, as is the generation true stress-strain curves. Then, biaxial yield phenomenon is explored through bulge tests in which clamped PC sheet is stretched by laterally applied fluid pressure. 15.2.1

Constant-Displacement-Rate Tensile Test

The current state of understanding of the yield of a material in a narrow, rectangular cross-sectioned specimen can be explained in terms of its load-displacement response in a tensile test. Since the load depends on the cross-sectional area, a more appropriate measure for it is the nominal stress, 𝜎 N , the load divided by the original cross-sectional area, also called the engineering stress. Figure 15.2.1 shows the variation of the nominal stress versus the displacement in polycarbonate (PC) in a tensile test conducted at a constant-displacement rate of 𝛿̇ = 0.25 mm s–1 (0.01 in s–1 ).

NOMINAL STRESS

σN (MPa)

150 TEMPERATURE = 22°C (72°F) •

δ = 0.25 mm s–1 (0.01 in s –1 ) 100

σ0 = σA A

D C

B 50

σd = σB PC O

0 0

50

100

DISPLACEMENT (mm) Figure 15.2.1 Stress displacement curve for a thin, rectangular polycarbonate (PC) specimen stretched in a tensile test at a constant-displacement rate. (Adapted with permission from V.K. Stokes and H.F. Nied, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 107 – 112, 1986.)

The specimen stretches homogeneously from O to A, where the stress attains a maximum, critical value 𝜎 0 = 𝜎 A . The stress then drops off precipitously to a lower value (point B), called the draw stress 𝜎 d = 𝜎 B , at which a neck is formed. Further increases in displacement (BC) first cause the neck to sensibly propagate at the draw stress 𝜎 d ; the nominal stress then starts to increase (CD) when the necked material reaches the shoulders of the specimen.

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Such changes in a PC tensile test specimen are illustrated by the photographs (Figure 15.2.2) of three specimens cut from a 1.5-mm thick PC extruded sheet, having 12.7-mm (0.5-in) wide rectangular cross-sections that were pulled to successively higher strains. These test specimens have the shape specified for ASTM D638 Type I bar. To better visualize the deformed states, these specimens were photographed between two transparent polarized sheets. Figure 15.2.2a shows the initiation of intersecting shear bands formed at the maximum load at point A in Figure 15.2.1. During the load drop-off along path AB these shear bands interact to eventually results in the formation of the stable neck shown in Figure 15.2.2b. Further extension results in this stable neck propagating (Figure 15.2.2c) along the specimen by the adjacent non-necked material undergoing necking, that is, in stable neck propagation at a constant load, during which the strain in the necked material remains constant. When the neck reaches the shoulders of the specimen, the load begins to increase, resulting in an increase in the strain in the neck.

(a)

(b)

(c) Figure 15.2.2 Photographs of PC tensile test specimens, placed between transparent polarizing sheets pulled to different strains, showing neck formation and propagation: (a) specimen pulled to point A in Figure 15.2.1, showing initiation of local yielding through intersecting shear bands, (b) neck formation during load drop-off along path AB in Figure 15.2.1, and (c) stable neck propagation.

Mechanical Behavior of Plastics

Because of the very large strains involved it is convenient to use the stretch, 𝜆 – defined as the ratio 𝜆 = ds∕ds of the deformed length ds of a small material line element originally of length d s – as a measure of strain. For small strains, the extension e = 𝜆 − 1 is the strain. A determination of the true stress, 𝜎 T , which is the load divided by the actual area, requires knowledge of the lateral strains. However, prior to the initiation of necking the deformations are essentially homogeneous. In this regime the true estimate can be obtained by assuming that the deformations are incompressible. This estimate for the true stress is then given by (Why?)

𝜎T = 𝜆𝜎N = (1 + e) 𝜎N

(15.2.1)

In PC, the stretch at yield is about 𝜆 = 1.06, so that the true stress can be as much as 6% higher than the engineering stress. Figure 15.2.3 shows the approximate true stress-stretch curve for PC, corresponding to the load-displacement curve in Figure 15.2.1. The stress and stretch increase homogeneously till point A at which the stretch is about 𝜆 = 1.06. At this point the specimen necks at some location; the stretch in the necked material is about 1.7. Because the shift from state A to B occurs almost instantaneously, the material undergoes an “instantaneous” jump in stretch from 𝜆 = 1.06 to 𝜆 = 1.7, resulting in a second homogeneous deformation field. Thus, the one-dimensional stress-stretch curve is only determined between O and A, and between B, C and D – which corresponds to an extension of the necked material that occurs only after the neck reaches the wider shoulders of the specimen resulting in a load increase.

150

TRUE STRESS (MPa)

D

100

B, C

A

1-D

? 50

0

1-D

O 1.0

2.0

1.5 STRETCH

λ

Figure 15.2.3 Stress-stretch curve for polycarbonate. The material deforms homogeneously from O to A, at which point the material yields and necking initiates. On yielding the stretch in the homogeneously deformed material jumps from 𝜆A ≈ 1.06 to 𝜆B ≈ 1.7 in the necked material. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

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What happens between A and B has not been characterized – hence the dashed line between A and B in Figure 15.2.3. While the homogeneous deformation portion OA of the stress strain is obtained from the strains measured by an extensometer, the portion BC is determined by measuring deformations of grids marked on the specimens – from a series of photographs taken during the test – and by superposing the stress-strain characteristics of the necked material measured in additional tests. At the molecular level, the polymer chains in the amorphous PC are randomly oriented in the first homogeneous phase (O to A in Figure 15.2.1), in which the material is sensibly isotropic. Beyond point A large deformations cause the chains to stretch and align along the stretch direction, resulting in a highly oriented non-isotropic material. Thus, necking may be looked upon as occurring due to a phase change in the material. This experiment raises several interesting questions: First, what happens when the specimen is subjected to a constant stress between the draw stress 𝜎 d and the threshold, or critical, stress 𝜎 0 ? Second, what occurs between points A and B during which the material yields to form a neck? Third, how are 𝜎 0 and 𝜎 d affected by deformation rate and temperature? And fourth, what is the phenomenology of yield for a multiaxial deformation field? Also of interest are the temperatures at which thermally induced recovery from yielded states occur. Answers to these questions are explored through two types of experiments: (i) Tensile tests on wide specimens show that the precipitous load drop-off occurs by the appearance of shear bands that coalesce to form a stable neck that then propagates along the specimen. The stretches (strains) in the shear bands and the necked material are very large. (ii) Bulge tests, in which circular sheets were subjected to biaxial deformation fields through lateral pressure-induced bulging. 15.2.2

*Considère Treatment of Yield

A construction due to Considère can be used for explaining the observed process of stable necking: Consider a tensile specimen with a rectangular cross section having original length, width, and thickness l0 , b0 , h0 , which on deformation become l, b, h, respectively, so that the axial stretch is 𝜆 = l∕l0 . On making the approximation that the deformation occurs at constant volume, so that l b h = l0 b0 h0 , the area of the deformed cross section is given by bh = b0 h0 ∕𝜆

(15.2.2)

For a given load P, the nominal, or engineering stress, 𝜎 N = P∕b0 h0 and the true stress 𝜎 T = P∕bh are then related through P = b0 h0 𝜎 N = bh 𝜎 T , so that

𝜎N =

𝜎T 𝜆

(15.2.3)

If yield is defined as the condition at which the stress does not change with a small increment of strain, or stretch, that is, by d𝜎 N ∕d𝜆 = 0, it follows that d𝜎N 1 d𝜎T 𝜎T = − =0 d𝜆 𝜆 d𝜆 𝜆2 that is, at yield d𝜎T 𝜎 = T (15.2.4) d𝜆 𝜆 This condition is satisfied by true stress-stretch curves in which the material “softens” at some point, that is, after continuing to increase with the stretch the rate of increase of stress with stretch begins to decline – as in Figure 15.2.4 – or after attaining a maximum, the stress actually declines – as in Figure 15.2.5. As shown in these two figures, this condition is initially satisfied at the yield point, Y,

Mechanical Behavior of Plastics

TRUE STRESS σ = σ (λ)

STRESS

Y

0

H

σY

σH

λY

λH

1

2

STRETCH λ Figure 15.2.4 Considère construction for a material in which the true stress-stretch softens, but in which the true stress increases monotonically with the stretch.

TRUE STRESS σ = σ (λ) H

σY

σH

λY

λH

STRESS

Y

0

1

2

STRETCH λ Figure 15.2.5 Considère construction for a material in which the true stress-stretch curve softens such that it exhibits both a maximum and a minimum.

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with coordinates (𝜆Y , 𝜎 Y ). But this condition is also satisfied at the “stress hardening” point, H, with coordinates (𝜆H , 𝜎 H ), at which the rate of increase of stress with stretch begins to increase. In materials with stress-stretch curves of the type shown in Figure 15.2.4, the stress increases monotonically with the stretch. However, in materials with stress-stretch curves as shown in Figure 15.2.5, like the one for PC in Figure 15.2.3, at point Y the stretch will instantaneously jump from 𝜆Y to about 𝜆H . (Why?) 15.2.3

*Uniaxial Extension of Wide PC Specimens

The tensile tests described in Section 15.2.1 were conducted on thin, ASTM D638 Type I specimens with a standard width of 12.7 mm (0.5 in) over the gauge length. In these narrow specimens the edges are not constrained, so that the stress is essentially one-dimensional, and the lateral strains are related to the longitudinal strain through 𝜀2 = 𝜀3 = − 𝜈𝜀1 . In very wide specimens, the state of stress close to the edges should essentially be one-dimensional. However, away from the edges, the material is constrained from moving in the width direction, resulting in a biaxial stress state in that region. In special, nonstandard tests, experiments PC specimens with widths up to 100 mm (4 in) were stretched in tension. Each specimen marked with a square grid, to help with visualizing deformations, was stretched at a constant-displacement rate, during which the deformation of the specimen was recorded by a video camera. Figure 15.2.6 shows representative frames from a video recording of a tensile test on a 100-mm (4-in) wide, 1.5-mm (0.06-in) thick, PC specimen, placed between two polarizing sheets to enhance the images, which was pulled at a displacement rate of 𝛿̇ = 0.25 mm s–1 (0.01 in s–1 ). Figure 15.2.6a shows the initial configuration of the undeformed specimen (𝜆 = 1). First, the material stretches homogeneously. Then, at a critical load, a shear band – material that has deformed by shear – forms; the formation of a shear band from the bottom right to the top left of the specimen can be seen in Figure 15.2.6b. On further extension the shear band widens (Figure 15.2.6c – g). At some stage a second shear band initiates (bottom left side in Figure 15.2.6h, and bottom left and top right in Figure 15.2.6j). The second shear band became progressively wider, leaving behind islands of non-yielded material (Figure 15.2.6k,l). Eventually, after these islands have yielded, the sheared (oriented) material forms the neck that propagates down the specimen. While the first shear band rotates the grid lines in a counterclockwise direction (Figure 15.2.6d – g, the second shear band causes these lines to rotate back (Figure 15.2.6j – l), until they are aligned along the original, undeformed directions. Throughout this yield history, no “plane stress” effects – reduction in the distance between the longitudinal lines – were observed, neither during the initial shearing (Figure 15.2.6b) nor during the reverse shearing (Figure 15.2.6c). Thus, the transition from the homogeneous, deformed state to the yielded state occurs via shear in which there are no edge effects. On unloading, the longitudinal stretch in the yielded material was on the order of 𝜆1 = 1.7, while the stretches in the width and thickness directions were about 𝜆2 ≈ 𝜆3 ≈ 0.7. The different regimes of the yielding process for PC (Figure 15.2.6), are schematically shown in Figure 15.2.7, in which the vertical lines are markers to indicate the onset and propagation of nonhomogeneous deformations. At a critical load, a shear band – material that has deformed by shear – appears across the otherwise homogeneously stretching material (Figure 15.2.7a). On further extension, the shear band widens (Figure 15.2.7b) – such widening of shear bands does not occur in metals. At some stage a second shear band initiates, as shown by the solid at the bottom left-hand side line in Figure 15.2.7b. In some cases, a corresponding, symmetric shear band also initiates, as indicated by the dashed line on the upper right-hand side of the figure. The second shear band became progressively wider, leaving behind an island of non-yielded material (Figure 15.2.7c). While the first shear band rotates the longitudinal grid

Mechanical Behavior of Plastics

lines in a counterclockwise direction (Figures 15.2.7b and 15.2.6e – g), the second shear band causes these lines to rotate back (Figure 15.2.6h – l), until all the lines are again aligned longitudinally.

(a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

(i)

(j)

(k)

(l)

Figure 15.2.6 Stills from a video recording of a tensile test on a 100-mm (4-in) wide PC specimen with a grid showing the evolution of a neck in a thin rectangular specimen with parallel grid lines. (a) Specimen with grid at zero stretch. (b) Shear band nucleates from bottom right to top left at an angle of about 35.25° with the horizontal. (c – g) The shear band widens; the grid lines in the yielded region undergo a leftward rotation. (h) A complementary (reverse) shear band nucleates (bottom left). (i) The second shear band widens. (j) A second reverse shear band nucleates (top right). (k, l) Both (reverse) shear bands begin to widen with the non-yielded “triangular” regions becoming progressively smaller. The interaction of these shear bands causes the initially sheared material to rotate back along the extension direction. After the shear bands have passed through all the material within the bands (not shown), the yielded region propagates as a stable neck. (Photos courtesy of SABIC.)

These experiments were conducted on specimens of different thicknesses (1.5, 3, and 6.35 mm; 0.06, 0.12, and 0.25 in) and widths (12.7, 25.4, 50.8, and 101.6 mm; 0.5, 1, 2, and 4 in). In each case, the transition from homogeneously deformed material at a stretch of 𝜆1 ≈ 1.06 to the oriented material with a stretch 𝜆1 = 1.7 occurred through shear bands that coalesced to form a neck that on further extension propagated along the specimen in a stable manner. However, the shear bands formed necks in different ways. For example, in some cases, two symmetric shear bands formed at the same instant, and neck formation occurred in a symmetric manner. These experiments on the tensile extension of thin rectangular specimens clearly show that the material can be homogeneously stretched only up to a limiting stretch of 𝜆1 ≈ 1.06. Further extension results

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Introduction to Plastics Engineering

θ

θ = 32.5°

(a)

(b)

(c)

(d)

(e)

Figure 15.2.7 Evolution of a neck in a thin rectangular specimen marked with parallel grid lines. The yielded material is indicated by the shaded regions. (a) A shear band nucleates and spreads at a constant angle across the specimen. (b) The shear band propagates, and a complementary (reverse) shear band nucleates (bottom left). (c) The second shear band widens and interacts with the first shear band causing the initially sheared material to rotate back along the extension direction. A second reverse shear band (top right) nucleates. (d) Both (reverse) shear bands begin to widen; the non-yielded “triangular” regions become progressively smaller. (e) After the shear bands have passed through all the material within the bands, the yielded region propagates as a stable neck. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

in strain localization in the form of shear bands in which the material has undergone very large shear. Although the material is still being pulled in the longitudinal direction, the state of stress in the shear bands is not uniaxial. The two sets of shear bands, each of which induces finite shear in the material, coalesce to move the sheared material back to a state in which it has undergone a stretch 𝜆1 = 1.7 in the longitudinal direction and stretches 𝜆2 ≈ 𝜆3 ≈ 0.7 in the lateral directions. This doubly sheared region constitutes the neck that is formed and consists of highly oriented material. On further extension of the specimen the necked material does not stretch any further. Rather, more of the homogeneously deformed non-necked material orients, and the neck propagates along the specimen in a stable manner under a constant uniaxial load. Finally, when the neck reaches the wider shoulder of the specimen, the load required to stretch the specimen at a constant-displacement rate increases, and the necked material is stretched beyond 𝜆1 ≈ 1.7. The deformation history of the material where the shear bands initiate is different from the deformation histories of the rest of the material: With reference to Figure 15.2.8, on loading along path OA, the shear bands are formed at point A at an initial nominal stress 𝜎 0 , after which the stress drops to a draw stress 𝜎 d at which the neck propagates along the specimen. Thus, in the rest of the material, the stretch first builds up to 𝜆1 ≈ 1.06. Then, a reduction of the stress from 𝜎 0 to 𝜎 d ≈ 0.75 𝜎 0 causes the stretch to decrease to 𝜆1 ≈ 1.05 along the path AB1 . The material is maintained at this reduced stretch until the

Mechanical Behavior of Plastics

advancing neck passes through, increasing the stretch to 𝜆1 ≈ 1.7 along path B1 B2 B3 . The effects of deformation rate and temperature on the critical stress 𝜎 0 , and the draw stress 𝜎 d are discussed in detail in Section 15.2.8.

A

σ0

STRESS

σd ≈ 0.75 σ0 B1

B2

B3

O 1.0

1.06 STRETCH λ

Figure 15.2.8 Shape of the stress-stretch curve. Neck formation at some point on the specimen is accompanied by a sudden reduction in the stretch from 𝜆A ≈ 1.06 to 𝜆B1 ≈ 1.05. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

15.2.4

*Definition and Measurement of Initial Yielding

Extensive tests were conducted to determine the yield point in terms of stress and strain for PC, polyetherimide (PEI), and poly(butylene terephthalate) (PBT). By following a schedule of loading and unloading on a uniaxial test specimen it was possible to determine the strain at which permanent deformation occurs. Figure 15.2.9 shows the response of a single PC specimen to successive loading and unloading for a stretch rate of 𝜆̇ = 10−2 s–1 . In these tests the yield was estimated to occur at an extension of e = 𝜆 − 1 = 0.06. Using this technique to find the initial yield point turns out to be somewhat of a problem in thermoplastics, since the behavior is noticeably viscoelastic and the recovery on unloading is a function of time. Nevertheless, by waiting a sufficient period of time it is possible to determine whether any further substantial recovery will occur, by observing the change in the rate of recovery. Here, the yield strain is defined as the strain beyond which the deformation does not return to zero within 15 minutes of unloading. Actually, in many of the tests described herein, the “recovery” strain upon unloading was monitored for hours or days to observe the rate of recovery and, from this information, it was concluded that the 15-minute stabilization period was a reasonable length of time to determine whether the strains were “permanent” or not. At room temperature, the strain just prior to the load drop, or drop in true stress (points A in Figures 15.2.1 and 15.2.3) was determined to be the strain beyond which additional deformations produce measurable permanent deformation. The permanent deformation is clearly visible in PC because shear band formation occurs during the drop in true stress.

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80 PC

STRESS (MPa)

342

50

•

λ = 10 –2 s–1 TEMPERATURE = 22°C (72°F)

0 0

0.05

0.08

EXTENSION e = λ – 1 Figure 15.2.9 Loading-unloading stress-extension curves for PC at T = 22°C and 𝜆̇ = 10−2 s−1 . (Adapted from V.K. Stokes and H.F. Nied, GE Corporate Research & Development Report No. 84CRD215, 1984, courtesy of SABIC.)

15.2.5

*Mechanical Behavior of Necked PC

To obtain a true stress versus stretch plot of a plastic over the entire stretch range that it can be subjected to, the necked material is stretched again to an ever-increasing stretch history in a strain-controlled test. This is accomplished by cutting out the necked region from a tensile test specimen, attaching extensometers, and retesting the oriented material in an extension-controlled test. Since lateral contractions are not measured by the extensometer, the stress is simply determined by dividing the load by the cross-sectional area prior to loading. A sample engineering stress versus extension plot of necked PC, which has been cyclically loaded twice at an extension rate 𝜆̇ = 10−3 s–1 is shown in Figure 15.2.10. The necked material used in this test was obtained by stretching a virgin specimen of PC at a displacement rate 𝛿̇ = 2.5 mm s–1 (0.1 in s–1 ). The final permanent deformation on unloading was a uniform stretch of approximately 𝜆f = 1.93. The extension shown in Figure 15.2.10 is calculated with respect to an “initial” length sf measured from the deformed plastic after unloading. This figure shows that substantial additional deformation is possible in the deformed PC, and that this deformation seems to have a distinct bilinear character. Extension on the order of e = 0.15 is possible without fracturing the material. Substantial viscoelastic recovery occurs on unloading of the oriented material, with the extension decreasing from e = 0.04 at D to e = 0.17 at E in a time span of 1 hour, during which the specimen carries no load. At the end of 1 hour, the specimen in Figure 15.2.10 was reloaded (EFG) at the same stretch rate (𝜆̇ = 10−3 s–1 ) to an additional extension of 0.1. Except for the shift due to the higher initial strain in the second test, the two curves are essentially identical.

Mechanical Behavior of Plastics

150 TEMPERATURE = 22°C (72°F) •

STRESS (MPa)

λ =10 –3 s–1

C

G

100

B F

50

1 HR A E

0 0

PC

1 HR D H 0.05

0.10

0.15

EXTENSION e = λ – 1 Figure 15.2.10 Stress-extension characteristics of necked (oriented) polycarbonate. (Adapted with permission from H.F. Nied and V.K. Stokes, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 113 – 118, 1986.)

15.2.6

*Composite Stress-Stretch Curve for PC

Tests on unloaded specimens showed that the amount of “springback” or strain recovery in the necked material was relatively small – a decrease in stretch on the order of 6%. For example, those material elements that in the displacement rate tests showed final stretches of 2.05 and 1.9 for displacement rates 𝛿̇ = 2.5 mm s–1 (0.1 in s–1 ) and 𝛿̇ = 0.25 mm s–1 (0.01 in s–1 ), respectively, had final stretches of 1.93 and 1.78 15 minutes after unloading. The recovery was definitely viscoelastic in nature, with approximately 4% recovery occurring within 15 seconds after unloading. Using the information obtained on stress and stretch, both before and after necking, together with data of the type given in Section 15.2.4, an approximate composite stress versus stretch plot of PC, which describes the material behavior for a continuously increasing stretch history, can be constructed. One such plot, which is a composite of true stress versus stretch data generated by using the photographic tests described in Section 15.2.3 and simple extension tests of the type described in Section 15.2.4, is shown in Figure 15.2.11. The dashed line (𝜆 > 1.7) represents the true stress versus stretch behavior of the oriented polymer, and is estimated from the engineering stress versus stretch curve (Figure 15.2.10) by assuming the deformations to be incompressible.

15.2.7

*Creep of PC at High Loads

In a constant-displacement-rate tensile test, a critical stress 𝜎 0 is required for initiating neck formation, after which the stress drops to 𝜎 d . This drop in stress raises an interesting question: What happens

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150

TRUE STRESS (MPa)

T = 22°C (72°F) •

λ e ≈ 0.01s–1 100

50 PC

0 1.0

2.0

1.5 STRETCH λ

Figure 15.2.11 Composite stress-stretch curve for polycarbonate. (Adapted with permission from H.F. Nied and V.K. Stokes, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 113 – 118, 1986.)

if the stress 𝜎 is maintained at a constant value 𝜎 c between 𝜎 d and 𝜎 0 , as schematically shown in Figure 15.2.12? (The curve for the constant-displacement-rate test is indicated by a dashed line.) To answer this question, tests were conducted in which the nominal stress 𝜎 N on the specimen was increased linearly with time to a prescribed value 𝜎 c (𝜎 d < 𝜎 c < 𝜎 0 ), at which it was held constant (Figure 15.2.12). The time tc to reach the stress 𝜎 c was adjusted to provide a nominally constant loading

σ0 NOMINAL STRESS

344

σc σd

1.0

tc

TIME

Figure 15.2.12 Loading history for a tensile specimen subjected to a constant nominal stress 𝜎 c between the critical stress 𝜎 0 for neck initiation and the draw stress 𝜎 d . The dashed line represents the stress (load)-time history for a constant-displacement rate test. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

Mechanical Behavior of Plastics

rate. These tests, in which only the longitudinal stretch 𝜆1 (longitudinal strain 𝜀1 = 𝜆1 − 1) was monitored as a function of time, show that the constant tensile stress causes the material to creep. When the stretch approaches 𝜆1 = 1.06, the material undergoes very rapid extension resulting in failure. What happens, of course, is that the cross-sectional area of the material decreases with increasing longitudinal stretch caused by creep, so that while the applied load is constant, the true stress continues to increase. It appears that the material necks at 𝜆1 ≈ 1.06, when the true critical stress needed to initiate necking is achieved. On necking, the stress required to propagate the neck decreases. However, because the test is done under load control, the machine increases the cross-head displacement rate in an effort to maintain the prescribed load; the very rapid extension of the specimen results in failure. The higher the value of 𝜎 c , the higher the creep rate, and the smaller the time at which the material undergoes stable necking. To confirm this hypothesis, the test procedure was modified to prevent the failure of the specimens caused by the uncontrolled increase in the cross-head displacement rate. Specimens were loaded at a constant-displacement rate to a predetermined nominal stress 𝜎 c , at which it was held constant (just as in the previous tests). To prevent uncontrolled displacement after load drop-off, the machine was programmed to change to displacement control once a prescribed displacement – chosen to ensure homogeneous stretches of about 1.1 – was attained. As in the previous tests, the stretch was monitored by means of a 12.7-mm (0.5-in) gauge-length extensometer. In two tests, standard 2.9-mm (0.114-in) thick ASTM D638 specimens were pulled in tension at a nominal stretch rate of 10−2 s−1 ; 𝜎 c was chosen such that (𝜎0 − 𝜎c )∕(𝜎0 − 𝜎d ) had values of 0.25 and 0.5. Both the specimens underwent creep at the constant loads. During the creep deformation, the stretch rate was approximately constant at about 3.5 × 10−4 and 10−5 s−1 for (𝜎0 − 𝜎c )∕(𝜎0 − 𝜎d ) = 0.25 and 0.5 and 0.5 (i.e. for the higher and lower constant loads), respectively, until a stretch of about 1.07, when the creep rate began to increase rapidly with the formation of shear bands. The strain history for the higher load case is shown in Figure 15.2.13; the strain rate began to increase at about 70 seconds when the stretch was about 1.075. The displacement cutoff was activated just before 80 seconds. By then a neck had formed outside the gauge length of the extensometer; the homogeneous deformation inside the gauge length had a stretch of 1.08.

1.10

STRETCH

(σ0 – σc )/(σ0 – σd ) = 0.25

1.05

PC 0 0

50 TIME (s)

100

Figure 15.2.13 Stretch history for creep deformation at (𝜎 0 − 𝜎 c )∕(𝜎 0 − 𝜎 d ) = 0.25. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

It took about an hour for the displacement cutoff to be activated at the lower load. Because of the low creep rate, it was possible to visually observe the formation of shear bands. Once shear bands appeared, more of them initiated with time as the stretch in the gauge length increased. One shear band began to

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widen; this appeared to coincide with an increase in the creep rate, resulting in the displacement cutoff being activated when the extensometer showed a homogeneous stretch of 1.09. At this point a wide shear band had formed. These exploratory tests show that creep deformation eventually results in strain localization (stable neck formation) at stretches in the range of 1.06 – 1.09. Tests were also done at stresses lower than 𝜎 d , for which the creep rate was very low. In many cases a crack initiates, which eventually results in brittle fracture caused by crack growth. Also, depending on the magnitude of the load, the material can craze (Section 15.6). 15.2.8

*Deformation-Rate and Temperature Effects

This section considers the effects of strain rate and temperature on the critical stress 𝜎 0 and the draw stress 𝜎 d . Standard 2.3-mm (0.09-in) thick ASTM D638 specimens were pulled in tension at ̇ constant-displacement rates corresponding to nominal stretch (strain) rates of 𝜆 = 10−4 , 10−3 , 10−2 , 10−1 , and 100 s–1 at temperatures of 22, 37.5, 51.5, and 65.5°C (72, 100, 125, and 150°F). In these tests, the stress-stretch, or stress-strain, curve has the typical shape earlier shown in Figure 15.2.8. The nominal stress increases up to the point A (critical stress 𝜎 0 ) at which necking causes the stress to suddenly drop to the draw stress 𝜎 d . On neck initiation, although the stretch in the necked material increases from 1.06 to 1.7, the stretch (strain) in the non-necked material actually decreases instantaneously, from A to B1 . For convenience, the portion AB1 of the stress-stretch curve will be simplified to AB2 . The stress-stretch curves at the five strain rates at 22°C (72°F) are shown in Figure 15.2.14, and the corresponding curves at 65.5°C (150°F) are shown in Figure 15.2.15. While the magnitudes of 𝜎 0 and 𝜎 d are affected by both the strain rate and the temperature – 𝜎 0 and 𝜎 d increase with increasing strain rates but

100 TEMPERATURE = 22°C (72°F)

NOMINAL STRESS (MPa)

346

50 •

λ = 10 0 s –1 = 10 –1 s –1 = 10 –2 s –1 = 10 –3 s –1 = 10 – 4 s –1

PC

0 0

1.05

1.1

STRETCH λ Figure 15.2.14 Stress-stretch curves at 22°C at nominal stretch rates of 10−4 , 10−3 , 10−2 , 10−1 , and 100 s−1 . (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

Mechanical Behavior of Plastics

100

NOMINAL STRESS (MPa)

TEMPERATURE = 65.5°C (150°F)

50

•

λ = 10 s –1 = 10 –1 s –1 = 10 –2 s –1 = 10 –3 s –1

PC

= 10 –4 s –1

0 0

1.05

1.1

STRETCH λ Figure 15.2.15 Stress-stretch curves at 65.5°C at nominal stretch rates of 10−4 , 10−3 , 10−2 , 10−1 , and 100 s−1 . (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

decrease with increasing temperatures – the ratio 𝜎 d ∕𝜎 0 appears to be insensitive to these two parameters, having a nominal value of about 0.75 for strain rates between 10−4 and 100 s−1 and temperatures between 22 and 65.5°C, as can be seen from Figure 15.2.16. It may appear from Figure 15.2.14 that, at room temperature, the stretch at which the load drops off suddenly increases from about 1.06 to 1.07 as the strain rate increase from 10−4 to 100 s−1 , and from Figure 15.2.15 that this change in the stretch is smaller at the higher temperature of 65.5°C. However, these tests were not sensitive enough to quantify the effect of strain rate on the stretch at which the load falls off. The actual (true) stress 𝜎 resulting from a tensile load will be larger than the nominal stress 𝜎 N based on the original cross-sectional area of the specimen. Estimates for 𝜎 can be obtained by assigning a suitable Poisson’s ratio. Consider a thin rectangular specimen of initial width b0 and thickness t0 that is stretched in the longitudinal direction to a stretch 𝜆1 , at which the width and the thickness are b and t, respectively. Let the stretches in the width and thickness directions be 𝜆2 and 𝜆3 , respectively. Then, the initial (A0 ) and final (A) cross-sectional areas of the specimen, and the initial (V0 ) and final (V ) volumes of material elements are related through A = 𝜆2 𝜆3 (15.2.5) A0 A V = 𝜆1 𝜆2 𝜆3 = 𝜆 V0 A0 1

(15.2.6)

For the small stretch at yield, 𝜆1 ≈ 1.06, the stretches are related to the normal (small) strains through 𝜆i = 1 + 𝜀i , i = 1, 2, 3, and the volume ratio is related to the (small) volumetric strain ev = 𝜀1 + 𝜀2 + 𝜀3

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1.5

DRAW STRESS / CRITICAL STRESS

348

22°C (72°F) 38°C (100°F) 51.5°C (125°F) 65°C (150°F)

1.0

0.5

0 10 – 4

10 – 3

10– 2

10– 1

10 0 –1

NOMINAL STRETCH RATE (s ) Figure 15.2.16 Ratio of the draw stress 𝜎 d to the critical stress 𝜎 c for four temperatures (between 22 and 65.5°C) and five

stretch rates (between 10−4 and 100 s−1 ). (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

through v∕v0 = 1 + ev . Assuming material isotropy and a Poisson’s ratio of v, ev = (1 − 2 𝜈 ) 𝜀1 = (1 − 2 𝜈 ) 𝜆1

(15.2.7)

so that

𝜎=

𝜆1 P P A0 P V0 𝜆 = 𝜎 = = A A0 A A0 V 1 1 + (1 − 2 𝜈 ) (𝜆1 − 1) N

(15.2.8)

which, for small strains, reduces to

𝜎 ≈ (1 + 2 𝜈 𝜀1 ) 𝜎N

(15.2.9)

Clearly, 𝜎 = 𝜆1 𝜎 N for an incompressible material (𝜈 = 0.5). The Poisson’s ratio for polymers is on the order of 0.4. For a stretch of 𝜆1 = 1.06, the values of 𝜎 ∕𝜎 N for Poisson’s ratios of 0.3, 0.35, 0.4, and 0.45 are 1.035, 1.041, 1.047, and 1.054, respectively. Thus, the true stress is on the order of 5% higher than the nominal stress. Figure 15.2.17 shows the room-temperature stress-extension curves for PC from a different series of tensile tests (compare with data in Figure 15.2.14) for characterizing the effects of stretch (strain) rate on the tensile modulus. This composite room-temperature stress-strain plot shows that increasing the strain rate at a fixed temperature increases the yield strain as well as the magnitude of the stress at which the material yields. Also, an increase in the stretch (strain) rate at a fixed temperature increases the extensional modulus for relatively large strains with little change in the small-strain modulus.

Mechanical Behavior of Plastics

80

STRESS (MPa)

PC

50

•

λ = 5×10 –1 s –1 = 5×10 –2 s –1 = 10 –2 s –1 = 10 – 4 s –1

TEMPERATURE = 22°C (72°F)

0 0.05

0

0.08

EXTENSION e = λ – 1 Figure 15.2.17 Room-temperature stress-extension curves for PC with the stretch rate as parameter. (Adapted with permission from V.K. Stokes and H.F. Nied, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 107 – 112, 1986.)

Additional information is provided in Figures 15.2.18 and 15.2.19, which depict the sensitivity of the yield stress and yield stretch (strain), respectively, to both strain rate and temperature. Over the temperatures range considered, the yield stress and strain essentially increase linearly with the logarithm of the strain and decrease linearly with increasing temperature. As expected, the yield stress and strain drop toward zero as the temperature approaches the glass transition temperature of PC (150°C). Note that the yield-extension data (Figure 15.2.19) has far more scatter than the yield-stress data (Figure 15.2.18). Using a least-squares fit, the data given in Figure 15.2.18 can be used for developing empirical expressions for relating the yield stress to the yield strain, strain rate, and temperature, resulting in the correlation

𝜎Y = C1 + C2 T + C3 log10 𝜆̇ + C4 T log10 𝜆̇

(15.2.10)

where C1 = 81.027 MPa,

C2 = − 0.28789 MPa∕°C

C3 = 3.3813 MPa,

C4 = 0.0094943 MPa∕°C

(15.2.11)

In this equation, 𝜎 Y is in MPa and T in °C. The straight lines in Figure 15.2.16 are calculated from Eq. 15.2.10. Note that the least squares fit was obtained by varying both T and 𝜆̇ . This expression is valid for PC over a wide temperature range which does not exceed the glass transition temperature. 15.2.9

*Biaxial Stretching of Clamped Circular PC Sheets by Fluid Pressure

A clear qualitative difference exists between the phenomenology of yield in uniaxial and biaxial extension: In a uniaxial tensile test, the transition from the non-yielded to the yielded material occurs through

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Introduction to Plastics Engineering

YIELD STRESS (MPa)

100

17.8°C (0°F) 22 (72) 65.5 (150)

50

93.3 (200)

PC

121.1°C (250°F)

20 10 – 4

10 – 3

10– 2

10– 1

10 0

101

•

–1 STRETCH (STRAIN) RATE λ (s )

Figure 15.2.18 Variation of the yield stress of PC with the stretch rate with the temperature as parameter. (Adapted with permission from V.K. Stokes and H.F. Nied, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 107 – 112, 1986.)

0.10

YIELD EXTENSION (STRAIN)

350

STRETCH (STRAIN) RATE •

λ = 10 s–1 • λ = 10 –1 s–1 • λ = 10 – 4 s–1

0.05

PC 0.02 –25

0

50

100

150

TEMPERATURE (oC) Figure 15.2.19 Variation of the yield-extension of PC with the temperature with the stretch rate as parameter. (Adapted with permission from V.K. Stokes and H.F. Nied, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 107 – 112, 1986.)

Mechanical Behavior of Plastics

strain localization in the form of shear bands – via a mechanism in which the strain field is no longer one-dimensional – through an almost discontinuous jump in the stretch from 𝜆1 ≈ 1.05 to 𝜆1 ≈ 1.7. In contrast, under biaxial tension, the material deforms continuously from an non-yielded to a yielded state; but strain localization can also occur. To map the phenomenology of biaxial deformation in polymers, clamped circular sheets of PC were stretched by a controlled laterally applied fluid pressure. In the tests described herein, 1.5-mm (0.06-in) thick circular sheet specimens of PC were clamped to provide circular test regions with radii of R = 101.5 mm (4 in). To facilitate measurement of the deformation, a grid formed by concentric circles at radial intervals of 6.35 mm (0.25 in) and eight radial straight lines at intervals of 45° were marked on each circular specimen. These specimens were loaded by oil that was pressurized by the controlled motion of a servohydraulically controlled piston. During each experiment, the applied pressure p and the displacement of the disk at its center (dome height) were monitored as functions of time. In this series of tests, sheet specimens were deformed to predetermined dome heights at which the specimens were unloaded by releasing the oil pressure. Five specimens were loaded to different final pressures at the same volumetric rate, for which the pressure-time and the pressure-dome-displacement histories are shown in Figures 15.2.20 and 15.2.21, respectively.

PRESSURE (MPa)

1.0

0.5

0 0

100

200

300

400

500

600

TIME (seconds) Figure 15.2.20 Pressure versus time histories for five tests, in which 1.5-mm thick circular disks of 101.5-mm radii were stretched by pressure generated by the injection of oil at a constant volumetric rate. Each specimen was stretched to a different pressure level after which no additional oil was injected; creep resulted in a decrease in the pressure. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

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1.0

PRESSURE (MPa)

352

0.5

0 0

0.5 1.0 NONDIMENSIONAL DOME DISPLACEMENT

Figure 15.2.21 Pressure versus nondimensional dome height for the five tests in Figure 15.2.20. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

The curves for the four lower pressures lie on the curve for the highest pressure. Three regimes can be identified: First, in the initial stage the pressure-displacement curve has a small slope; then the slope increases to an approximately constant value; finally, the slope decreases sharply to a lower (approximately) constant slope. In these tests, the piston was moved at a constant-displacement rate to force oil against the clamped sheet at a constant volumetric rate of 7.72 cm3 s−1 . The piston was programmed to stop when a prescribed pressure was attained, and the piston was maintained at this position for some time. Because of creep at these high pressures, the pressure in the constant volume of oil actually drops continually, first rapidly, and then more slowly, as can be seen from the pressure-time traces in Figure 15.2.20. The increase in the dome height caused by creep is rather small because of the low compressibility of the oil – a small increase in the volume under the dome causes a rapid decrease in the pressure, resulting in reduced creep. This small increase in the dome height appears as a slight rightward bulge in the unloading curves in Figure 15.2.21. The pressure was released after some time resulting in some elastic recovery, as shown by the decrease in the dome height along the unloading curves in Figure 15.2.21. The measured shapes of the domes after unloading are shown in Figure 15.2.22. The change in the curvature of the deflected shape near the clamped edge shows that bending effects are important at the clamped circular boundary.

Mechanical Behavior of Plastics

NONDIMENSIONAL DOME HEIGHT

1.0

0.5

0 0

0.5

1.0

RADIAL POSITION Figure 15.2.22 Unloaded shapes of the domes from the five tests in Figure 15.2.20. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

Under increasing pressure, the sheet deformed continually in the shape of a dome. However, at some stage in the deformation, strain localization occurred near the clamped edge but away from it. On further deformation, the localized region took on a V-shape aligned with the radial direction, with the vertex of the V pointing radially outward, as shown in Figure 15.2.23. Measurements of the deformation of a grid consisting of concentric circles and radial lines were used to calculate the radial stretch 𝜆1 and the hoop stretch 𝜆2 . Also the stretch in the thickness direction 𝜆3 was obtained by measuring the local deformed thickness. Because of the techniques used, the accuracy of stretch measurement was highest for 𝜆3 and lowest for 𝜆1 . The variations of the radial, hoop, and thickness stretches along the radius are shown, respectively, in Figures 15.2.24 – 15.2.26 for the specimens for which the pressure-time and pressure-displacement histories are shown, respectively, in Figures 15.2.20 and 15.2.21. Note that for larger radii, the curves for the thickness stretch 𝜆3 are based on thickness measurements on the thicker regions between the consecutive Vs. The thickness stretch inside each V is much smaller. The curves for 𝜆3 are the smoothest, while those for 𝜆1 are the least smooth. The dip in the thickness stretch near r∕R = 1 is caused by bending effects near the clamped edge. Because of symmetry, 𝜆1 should equal 𝜆2 at r∕R = 0, and the data in Figures 15.2.24 and 15.2.25 confirm this. For a spherical dome 𝜆1 = 𝜆2 and 𝜆3 would not vary over the specimen. The

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(a)

(b)

(c)

(d)

Figure 15.2.23 Strain localization in V-shaped region close to clamped edge. (a) Overall view of plane sheet formed into a dome. (b) View from the inside surface showing distribution of V-shaped regions. (c) Exterior view showing V-shaped regions. (d) Interior view showing V-shaped regions. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

approximately zero slopes of the 𝜆1 , 𝜆2 , and 𝜆3 curves in the neighborhood of r∕R = 0 show that the domes have spherical shapes only near r∕R = 0. The shapes of the 𝜆3 curves show that with increasing deformation (increasing dome heights), the equibiaxially stretched (spherical) region increases. This increase is important because a measurement of the curvature of this region, together with a simultaneous measurement of the pressure, can be used to obtain stress-stretch relations under biaxial deformations. For larger values of r∕R, the thickness stretch decreases (thickness reduction decreases) continuously. The measured variations of the radial and hoop stretches are compared in Figure 15.2.27. These data indicate that 𝜆1 < 𝜆2 near the clamped edge where the stretches are low. For higher stretches (smaller radii) 𝜆1 approaches 𝜆2 . Although the deformation increased monotonically with increases in pressure in most of the clamped sheet – right through yield into the post-yield regime – strain localization occurred near the clamped edges resulting in V-shaped depressions; these depressions were aligned radially, with the apex of the V pointing outward (Figure 15.2.23b – d). The deformations in the bulk of the deformed sheet merged continuously across the mouth of the V into its interior. However, the thickness jumped sharply across the sides of the V. For example, for the largest dome (solid squares in Figure 15.2.21), the thickness stretch at r∕R = 0.95 dropped from a value of 𝜆3 ≈ 1 in the homogeneously deformed region to 𝜆3 ≈ 0.75

Mechanical Behavior of Plastics

RADIAL STRETCH

2.0

1.5

1.0 0.9 0

0.5 RADIAL POSITION

1.0

Figure 15.2.24 Radial stretch 𝜆1 versus the radial position r∕R for the five unloaded domes from the tests in Figure 15.2.20. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

within the V. Clearly, the strain localization in these highly deformed regions is analogous to the strain localization that results in the formation of the stably propagating neck in a tensile test. The results of these biaxial stretch tests show a clear qualitative difference from the phenomenology of yield in a uniaxial tensile test: The transition from the non-yielded to the yielded material in a tensile test occurs through strain localization in the form of shear bands, via a mechanism in which the strain field is no longer one-dimensional, through an almost discontinuous jump in the stretch from 𝜆1 ≈ 1.05 to 𝜆1 ≈ 1.7. In contrast, under biaxial tension, the material deforms continuously from an non-yielded to a yielded state; both 𝜆1 and 𝜆2 increase monotonically, while 𝜆3 decreases monotonically. Interestingly, strain localization only occurs close to r∕R = 1. 15.2.10

Thermally Induced Recovery from a Mechanically Yielded State

Another unique feature of polymers is that they “remember” the original shape from which they have been deformed – they “recover” to the original undeformed state on subsequent heating to the glass transition temperature (Tg ≈ 150°C, 302°F, for PC). To study this effect, PC specimens were first stretched in tensile tests to a yielded state; the necked portions were then heated to different temperatures below Tg during which the recovery of the necked material was recorded.

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2.0

HOOP STRETCH

356

1.5

1.0 0.9 0

0.5 RADIAL POSITION

1.0

Figure 15.2.25 Hoop stretch 𝜆2 versus the radial position r∕R for the five unloaded domes from the tests in Figure 15.2.20. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

Several 1.5-mm-thick ASTM D638 bars with 152.5-mm nominal lengths were stretched in tensile tests to (unloaded) lengths of 197 mm (7.75 in). These specimens had approximately 108-mm (4.25-in) long yielded, stably necked regions. Necked material cut from these specimens was clamped in grips such that the material between the grips corresponded to an original (before the specimen was stretched) length of 50.8 mm (2 in); the clamped specimen and the grips were enclosed inside a temperature-controlled oven. To map the recovery of the specimens as a function of its temperature under no-load conditions, the specimen should ideally be heated in a load-controlled test with the load set to zero. But, because of limits on load resolution, the specimen will always be subjected to a small load. Since an increase in temperature causes the specimen to expand, to prevent any buckling the no-load setting for the load-control test was carefully set at 0+ . The temperature of the specimen was monitored by a thermocouple attached to its surface. The oven temperature was then increased and held at different temperatures (120, 125, 130, 138, 146, and 150°C) below the glass transition temperature. For these transient tests, Figure 15.2.28 shows the variations with time of the nondimensional specimen temperature, (T − Ta )∕(Tg − Ta ), where Ta is the ambient temperature, and the nondimensional recovery (li − l)∕(li − lf ) of the length, where li and lf are the initial and final lengths of the necked

Mechanical Behavior of Plastics

THICKNESS STRETCH

1.0

0.5

0 0

0.5 RADIAL POSITION

1.0

Figure 15.2.26 Thickness stretch 𝜆3 versus the radial position r∕R for the five unloaded domes from the tests in Figure 15.2.20. (Adapted with permission from V.K. Stokes and W C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

material. These tests show that the material recovers most of the “permanent” deformation at temperatures close to Tg . However, substantial recovery begins at temperatures well below Tg , as can be seen from the recovery curve for 120°C. (In these tests, heat transfer at the grips results in lower temperatures in the material close to the grips, so that the curves in this figure underestimate the final recovery.) While these tests show that the material begins to recover from “permanent” deformation below Tg , a one-to-one comparison of the recovery with the temperature would not be meaningful because of the transient temperature rise; such a correlation would not account for isothermal, time-dependent recovery. These preliminary results clearly show that the necked (oriented) material begins to recover from its deformed state at temperatures well below Tg . However, the total recovery in one hour at temperatures lower than about 140°C is less than about 20%. Also, the rate of recovery at these low temperatures is very small after a steady temperature has been attained. Close to the glass transition temperature, the material appears to rapidly recover the entire deformation. The results of these experiments raise interesting questions. Is there just one thermally activated recovery mechanism that accelerates the recovery process as the temperature approaches Tg , or are the recovery mechanisms different for high and low temperatures? At the lower temperatures (see the curves for 120 – 138°C)

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Introduction to Plastics Engineering

2.0

RADIAL STRETCH

358

1.5

1.0 0.9 1.0

1.5 HOOP STRETCH

2.0

Figure 15.2.27 Variation of the radial stretch 𝜆1 versus the hoop stretch 𝜆2 . (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

does the amount of recovery level off (zero recovery rate) or does the recovery continue at a slow rate? 15.2.11

Large-Deformation Applications

The ability of PC to recover from strains as large as 6% – the corresponding strain for metals is only about 0.2% – can be exploited in the design of “spring-like” structures for temporary energy absorption. The sequence of photos in Figure 15.2.29 shows the deformation and recovery of a PC box-beam prototype for an energy absorbing automotive bumper. Figure 15.2.29a shows the original beam shape made by vibration-welding a molded C-section to a back plate. Figure 15.2.29b shows the deformed shape resulting from the front face being pushed inwards. The 6-inch (15.25-cm) scale at the bottom of the profile shows that the box-beam has a width of about 5 inches (12.7 cm). And Figure 15.2.29c shows the recovery after the rather severe deformation the test beam was subjected to. The actual bumper capable of absorbing an 8 km h−1 (5 mph) frontal car impact into a barrier (Section 2.4.1) was made of a specially developed PC/PBT blend (Section 12.2.7). It was fabricated by vibration-welding an injection-molded C-section to a back plate (Section 21.7.2).

Mechanical Behavior of Plastics

NONDIMENSIONAL TEMPERATURE & RECOVERY

TEMPERATURE

1.0

150°C

146

138 130 125 120

150°C

0.5

146

RECOVERY

138 130

120

125

0 0

5

10

TIME (1000 s) Figure 15.2.28 Variations of the nondimensional recovery and the nondimensional specimen temperature versus the time, for

final temperatures of 120, 125, 130, 138, 146, and 150°C. The thin and thick lines correspond, respectively, to the temperature and the recovery. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

UNDEFORMED

LOADED BUMPER SECTION

UNLOADED

(a)

(b)

(c)

Figure 15.2.29 Recovery of a PC prototype bumper beam from large deformations. (a) Undeformed box beam. (b) Beam shape on the front face being pushed inwards. The 6-inch scale above the caption shows that the box-beam has a width of about 5 inches. (c) Recovered shape after unloading. (Photos courtesy of SABIC.)

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Introduction to Plastics Engineering

15.3 Tensile Characteristics of PEI Both polycarbonate (PC) and polyetherimide (PEI) are amorphous polymers with glass transition temperatures of 150 and 210°C, respectively. As such, they can be expected to exhibit qualitatively similar deformation phenomenology, with quantitative differences arising from their very different transition temperatures. 15.3.1

Constant-Displacement-Rate Tensile Test

A typical load-displacement response for PEI at room temperature is shown by the solid line in Figure 15.3.1. The dashed curve is the response of PC under the same conditions (see Figure 15.2.1).

150

NOMINAL STRESS (MPa)

360

TEMPERATURE = 22°C (72°F) •

δ = 0.25 mm s–1 (0.01 in s –1 )

A 100 B

PEI

C

50 PEI O

0 0

50

100

DISPLACEMENT (mm) Figure 15.3.1 Stress-displacement curve for a thin, rectangular polyetherimide (PEI) specimen stretched in a tensile test at a constant-displacement rate. The dashed line shows the corresponding curve for polycarbonate (PC). (Adapted with permission from V.K. Stokes and H.F. Nied, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 107 – 112, 1986.)

Except for the magnitude of the load in the extension test, the behavior of PEI (Tg ≈ 210°C) is very similar to that of PC (Tg ≈ 150°C). Initial yielding occurs at approximately 6% extension followed by a rather large load drop. Near the bottom of the load-drop a shear band forms in PEI followed by stable neck propagation at essentially constant load. Initially, as the load drops, there is little change in the cross-sectional area of the specimen, and strain softening occurs during load drop prior to the development of shear bands. Before the load decreases to its minimum value, two peninsulas of high strain join to form an intense shear band, resulting in the

Mechanical Behavior of Plastics

formation of a neck that propagates along the axial length of the specimen. Once the neck has propagated sufficiently, an average draw stretch of 𝜆1 = 1.8 is attained in the necked material. Figure 15.3.2 shows the true stress-stretch plot of PEI, which is very similar to PC. As with PC, the substantial region of uncertainty in the value of the stress is indicated by a dashed line. While the stretch quickly moves through this region of uncertainty, the thickness is not homogeneous over a sufficiently large gauge length to permit the estimation of the true stress as the load divided by the cross-sectional area. As the stretch approaches the limiting value of 𝜆1 = 1.8, the stress begins to increase due to an increase in load. A room-temperature tensile test on the necked material, as done for PC, was not possible for PEI because the specimen always fractured before a sufficiently long neck could propagate for preparing a specimen from the oriented-necked PEI. As such, a composite room-temperature stress-stretch curve by combining the data from the two sets of tests could not be constructed for PEI, as done for PC (Figure 15.2.8).

TRUE STRESS (MPa)

200

D

B, C

150

A

PEI

100 PC 50 TEMPERATURE = 22°C (72°F) •

δ = 0.25 mm s–1 (0.01 in s –1 ) 0

O 1.0

1.5

2.0

STRETCH λ Figure 15.3.2 Stress-stretch curve for polyetherimide. The material deforms homogeneously from O to A, at which point the material yields and necking initiates. On yielding the stretch in the homogeneously deformed material jumps from 𝜆A ≈ 1.06 to 𝜆B ≈ 1.8 in the necked material. The corresponding curve for PC (Figure 15.3.3) is also shown.

At room temperature, a photograph taken immediately after a PEI specimen fractured was used to determine the amount of “springback” after this sudden unloading. The springback in PEI varied from 5.6 to 7.8% of the final stretch prior to fracture, that is, the final stretches varied from 94.4 to 92.2% of

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Introduction to Plastics Engineering

the stretch prior to fracture. Higher springback was attained in regions where small cracks could not be detected by the naked eye, while smaller amounts of springback occurred in regions in which there was a substantial amount of crack activity. With regard to fracture of PEI, the development and opening of surface flaws in the necked material could be clearly observed from the photographs. 15.3.2

*Deformation-Rate and Temperature Effects

Figure 15.3.2 is a composite plot of the stress-strain behavior of PEI up to and including the yield point at room temperature 22.2°C (72°F). It shows that the behavior of PEI is analogous to that of PC (Figure 15.3.3). In PEI, not only do the yield stress and strain increase in magnitude with increasing strain rate, but apparently the “elastic” modulus increases dramatically above a stretch rate of 𝜆̇ = 10−3 s–1 .

150 PEI

STRESS (MPa)

362

100 •

λ = 10 s –1 = 10 –1 s –1 = 10 –2 s –1

50

= 10 –3 s –1

TEMPERATURE = 22°C (72°F)

0 0

0.05

0.08

EXTENSION e = λ – 1 Figure 15.3.3 Room-temperature stress-extension curves for PEI with the stretch rate as parameter.

Figure 15.3.4 shows that the same general relationship between yield stress, strain rate, and temperature exists for polyetherimide as for PC (Figure 15.2.16). Although there are wide variations in the yield stress that depend on the strain rate and temperature, a good empirical fit to the data in Figure 15.3.4 is the linear dependence of the yield stress on log-strain-rate and the temperature given in Eq. 15.2.10, with the constants for PEI being C1 = 140.76 MPa,

C2 = −0.55168 MPa∕°C

C3 = 6.8766 MPa,

C4 = 0.0090621 MPa∕°C

(15.3.1)

As with PC, the straight lines in Figure 15.3.4 are calculated from Eq. 15.2.10. Generally, the overall behavior up to yield of both PC and PEI is quite similar except for differences in the actual magnitudes of the stress and strain for these two amorphous materials.

Mechanical Behavior of Plastics

YIELD STRESS (MPa)

150 125 22°C (72°F)

100

65.5°C (150°F)

75 121.1°C (250°F)

50

PEI

25 10 – 4

10 – 3

10– 2

10– 1

STRETCH (STRAIN) RATE

10 0

101

•

λ (s–1)

Figure 15.3.4 Variation of the yield stress of PEI with the stretch rate with the temperature as parameter. (Adapted from V.K. Stokes and H.F. Nied, GE Corporate Research & Development Report No. 84CRD215, 1984, courtesy of SABIC.)

15.4 Deformation Phenomenology of PBT Previous sections have considered the deformation phenomenology of two amorphous polymers, PC and PEI, the temperature-dependent behavior of which are characterized by their glass transitions temperatures of 150 and 210°C, respectively. PBT, a semicrystalline polymers, has both a glass transition temperature (43.3°C, 110°F ) characterizing its amorphous phase and a melting temperature (232°C, 450°F ) characterizing its crystalline phase. The crystalline phase embedded in the amorphous phase constrains the deformation of the amorphous phase, thereby affecting the overall deformation characteristics of PBT. 15.4.1

Constant-Displacement-Rate Tensile Test

Following the tensile test procedures described in Section 15.2.1, Figure 15.4.1 shows the variation of the nominal stress versus the displacement in poly(butylene terephthalate) (PBT) in a tensile test conducted at a constant-displacement rate of 𝛿̇ = 0.25 mm s–1 (0.01 in s–1 ). For comparison, the corresponding curves for the amorphous polymers PC and PEI are also shown. Clearly, in contrast to the behavior of the amorphous plastics PC and PEI, the deformation history of semicrystalline PBT (20 – 30% crystallinity) along path A ′B ′C ′D ′ is qualitatively quite different: PBT exhibits two distinct load drops at room temperature. The amorphous matrix first yields along the first load drop-off (A ′B ′), resulting in gradual neck formation. The constraints from the crystalline phase then causes the material to deform relatively homogenously along “plateau” region B ′C ′, with very little additional necking that extends over the entire gauge section with a large radius of curvature involving

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Introduction to Plastics Engineering

150 TEMPERATURE = 22°C (72°F)

NOMINAL STRESS (MPa)

364

•

δ = 0.25 mm s–1 (0.01 in s –1 ) 100

PEI

PC A'

50

B'

C' D'

PBT

E'

O

0 0

50

100

DISPLACEMENT (mm) Figure 15.4.1 Stress-displacement curve (solid line) for a thin, rectangular poly(butylene terephthalate) (PBT) specimen stretched in a tensile test at a constant displacement rate. The corresponding curves for polycarbonate (PC) and polyetherimide (PEI) are shown by dashed lines. (Adapted with permission from V.K. Stokes and W.C. Bushko, Polymer Engineering and Science, Vol. 35, pp. 291 – 303, 1995.)

a very small decrease in the specimen thickness. Up to point C ′ there is no distinct boundary between necked material and material that has not necked. However, at the second load drop between points C ′ and D ′, a very intense shear band suddenly forms within the original necked region, with a corresponding dramatic decrease in thickness in the band region. Finally, the entire material orients, resulting in stable neck formation and propagation along D ′E ′, and a very distinct boundary is discernible between the necked and non-necked material. Since PBT exhibits severe necking at strains much larger than those at which necking occurs in PC or PEI, reliable extensometer measurements can be made only up to the second load drop at point C ′. Because the stretch at yield in PBT, is about 𝜆 = 1.15, it follows from Eq. 15.2.1 that the true stress can be as much as 15% higher than the engineering stress. The room-temperature, stress-stretch behavior of PBT is shown in Figure 15.4.2, in which the corresponding curves for PC and PEI are also shown. The PBT curve in this figure is labeled with letters that correspond to those for the same points on the stress-stretch curves given in Figure 15.4.1. Necking in PBT occurs very suddenly during the later portion of the load drop between C ′ and D ′. This neck propagates in a stable manner at a more or less constant load between D ′ and E ′. Even though the load remains constant during the entire post necking drawing process, the stress and stretch increase significantly during early neck propagation. Once the necking process has “stabilized,” no further increase in the stress or stretch is seen in the necked region until the neck reaches the shoulders of the test specimen. 15.4.2

*Definition and Measurement of Initial Yielding in PBT

Following the methodology described in Section 15.2.4 for establishing initial yielding in PC, Figure 15.4.3 shows the response of a single specimen to successive loading and unloading in PBT for a stretch rate of 𝜆̇ = 10−2 s–1 . In these tests the yield was estimated to occur at an extension of

Mechanical Behavior of Plastics

200 T = 22°C (72°F)

TRUE STRESS (MPa)

•

150

λ e ≈ 0.01s–1

PEI

100 PC

D'

A' B' C'

50

PBT

E'

O

0 0

2.0

1.5

2.5

3.0

3.5

STRETCH λ Figure 15.4.2 Stress-stretch curve for poly(butylene terephthalate). The material deforms homogeneously from O to A ′, at

which necking initiates. Neck formation occurs along A ′B ′C ′. Also shown are corresponding stress-stretch curves for PC and PEI. (Adapted from V.K. Stokes and H.F. Nied, GE Corporate Research & Development Report No. 84CRD215, 1984, courtesy of SABIC.)

80

STRESS (MPa)

PBT

50

•

λ =10 –2 s–1 TEMPERATURE = 22°C (72°F)

0 0

0.01

0.02

0.03

EXTENSION e = λ – 1 Figure 15.4.3 Loading-unloading stress-extension curves for PBT at T = 22°C and 𝜆̇ = 10−2 s−1 . (Adapted from V.K. Stokes and H.F. Nied, GE Corporate Research & Development Report No. 84CRD215, 1984, courtesy of SABIC.)

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Introduction to Plastics Engineering

e = 𝜆 − 1 = 0.03. As with PC and PEI, the yield strain for PBT is defined as the strain beyond which the deformation does not return to zero within 15 minutes of unloading. At room temperature, the strain just prior to the load drop, or drop in true stress in PC, PEI, and PBT (point A and A ′ in Figures 15.4.1 and 15.4.2), was determined to be the strain beyond which additional deformations produce measurable permanent deformation. The permanent deformation is clearly visible in PC and PEI, since shear band formation occurs during the drop in true stress. In PBT the evidence is subtler since the onset of deformation is not clearly visible. However, careful measurements with extensometers indicate that a certain amount of permanent deformation occurs during the initial load drop. Thus, for all three materials, the load drop during a uniaxial test is a clear indication of at least some permanent deformation. 15.4.3

*Mechanical Behavior of Necked PBT

A conventional stress-extension plot of necked PBT cyclically stretched twice at an extension rate of

𝜆̇ = 10−3 s–1 is shown in Figure 15.4.3. This oriented material was cut from the necked region of a PBT specimen previously stretched in the virgin state at a displacement rate 𝛿̇ = 2.5 mm s–1 (0.1 in s–1 ). The final permanent deformation in the necked region prior to cutting was a uniform stretch of 𝜆f = 3.6.

This final permanent deformation was measured 20 hours after unloading the specimen. The behavior of oriented PBT, isolated from the effects associated with non-necked PBT, is shown in Figure 15.4.4. Just as in the case of amorphous polymers, these stress-extension curves consist of two distinct regions. In the first region (approximately until e = 0.02), AB, the curve corresponds to elastic behavior wherein recovery is instantaneous upon loading. The second region, BC, in which the slope increases with the stretch, seems to be dominated by viscoelastic behavior with little permanent plastic deformation. For

150 TEMPERATURE = 22°C (72°F)

G

•

λ =10 –3 s–1

STRESS (MPa)

366

C

100

B 50

F PBT

0

A E 0

D

H 0.05

0.10

0.15

EXTENSION e = λ – 1 Figure 15.4.4 Stress-extension characteristics of necked (oriented) poly(butylene terephthalate). (Adapted from H.F. Nied and V.K. Stokes, GE Corporate Research & Development Report No. 84CRD235, 1984, courtesy of SABIC.)

Mechanical Behavior of Plastics

example, upon unloading the specimen at a rate of 𝜆̇ = 10−3 s–1 following the first loading cycle, and after a maximum additional extension in the PBT specimen of e = 0.097 (point C), the extension immediately after unloading was e = 0.035 (point D). Yet, after one hour at zero load (DE), the viscoelastic recovery resulted in a final extension of e = 0.0074 – a total “springback” of 92% from the maximum extension. Thus, after unloading the oriented PBT, an additional 28% recovery in strain occurs within the period of 1 hour. When the same PBT specimen is reloaded for a second cycle (EFG) the behavior is almost identical. The second curve, indicated by a dotted line, passes through the maximum extension point of the first loading cycle. Again, after unloading (GH), PBT experiences a 90% recovery of the maximum extension in one hour. As discussed in the section on PC, in the necked material the molecular chains are essentially aligned along the stress axis, resulting in a stronger and stiffer material. While such chain alignment also occurs in PBT, the aligned chains are connected through crystalline material, resulting in more complex morphology than in PC and PEI. Very high alignment levels in gel-spun Ultrahigh-molecular-weight polyethylene (UHMWPE) are used to produce fibers with strength-to-weight ratios about 8 – 15 times higher than for steel (Section 11.4.1.1). 15.4.4

*Composite Stress-Stretch Curve for PBT

Figure 15.4.5 shows the stress versus stretch data for PBT in the region where the plastic first shows signs of nonhomogeneous deformation and extensive drawing. Once yielding occurs the true stress remains more or less constant until a stretch of about 1.3, at which point the stretch increases significantly during the necking process. By the time the neck stabilizes to permit accurate cross-sectional area measurements, the stretch attains a magnitude of 𝜆 = 4. After that, during most of the drawing process, the stretch

TRUE STRESS (MPa)

T = 22°C (72°F)

100

•

λ e ≈ 0.01s–1 PBT

50

0 1.0

2.0

3.0

4.0

STRETCH λ Figure 15.4.5 Composite stress-stretch curve for poly(butylene terephthalate). (Adapted with permission from H.F. Nied and V.K. Stokes, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 113 – 118, 1986.)

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Introduction to Plastics Engineering

remains constant with slight variations in stress. As with PC, the post-yield data for PBT in Figure 15.4.4 was combined with photographic data to construct the composite stress-stretch relationship for PBT in Figure 15.4.5 beyond a stretch of 𝜆 = 4. In this figure the dashed line for 𝜆 > 3.5 represents results from extension tests on necked-oriented PBT. This composite plot does not represent material behavior at a constant extension rate 𝜆̇ and thus should not be interpreted in the sense of a conventional stress-strain curve. However, the overall stress versus stretch information generated in this manner is expected to be quite useful for modeling deformations during solid phase forming processes. 15.4.5

*Deformation-Rate and Temperature Effects

The type of pre-yield behavior seen in the amorphous polymers PC and PEI is not the same as that observed in the semicrystalline PBT. Figure 15.4.6 is a plot of the overall stress-strain behavior of PBT at room temperature, 22.2°C (72°F), for different strain rates prior to the material actually exhibiting nonhomogeneous deformation. As previously described, reliable extensometer measurements could be made to rather large strains for this reason. Unlike the amorphous polymers, the “elastic” modulus of PBT does not seem to change significantly with increasing strain rate, except close to the yield point. Figure 15.4.7 shows that the yield stress of PBT varies more or less linearly with increasing strain rate for temperatures at or below room temperature. As with PC and PEI, the expression in Eq. 15.2.10 for predicting the yield stress for the amorphous polymers as a function of strain rate and temperature seems to fit the PBT data quite well up to about 43.3°C (110°F), which is close to the glass transition temperature for PBT; the constants in this equation are C1 = 85.296 MPa,

C2 = −0.63509 MPa∕°C

C3 = 5.0513 MPa,

C4 = 0.011255 MPa∕°C

(15.4.1)

80

STRESS (MPa)

368

50 •

λ = 5×10 –1 s –1 PBT

= 10 –1 s –1 = 10 –2 s –1 = 10 – 3 s –1

TEMPERATURE = 22°C (72°F)

0 0

0.05

0.10

0.15

EXTENSION e = λ – 1 Figure 15.4.6 Room-temperature stress-extension curves for PBT with the stretch rate as parameter. (Adapted with permission from V.K. Stokes and H.F. Nied, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 107 – 112, 1986.)

Mechanical Behavior of Plastics

YIELD STRESS (MPa)

120

100 17.8°C (64°F)

80

22°C (72°F)

60

PBT 40 10 – 4

10 – 3

10– 2

10– 1

STRETCH (STRAIN) RATE

10 0

101

•

λ (s–1)

Figure 15.4.7 Variation of the PBT yield stress with the stretch rate at two different temperatures. (Adapted from V.K. Stokes and H.F. Nied, GE Corporate Research & Development Report No. 84CRD215, 1984, courtesy of SABIC.)

The characterization of the behavior of semicrystalline PBT between the glass transition temperature, 43.3°C (110°F), and the melting temperature, 232.2°C (450°F), is very important. The collective stress-stretch behavior of PBT at a strain rate of 𝜆̇ = 10−3 s–1 is shown in Figure 15.4.8 for several temperatures in the range of −17.8°C (0°F) to 65.6°C (150°F). It shows that below 37.8°C (100°F) there is very little change in the extensional modulus of PBT. Really interesting is the behavior of PBT between 37.8°C (100°F) and 51.7°C (125°F), where the sudden and dramatic drop in the extensional modulus occurs. Further increases in temperature greatly depress the modulus and eliminate any evidence of the load drop so clearly seen below the glass transition temperature. An important question arises: Does PBT exhibit any permanent plastic deformation in an isothermal test above the glass transition temperature? Also, if there is permanent deformation then, because at these elevated temperatures no load drop can be discerned, what is the strain level at which the material yields? Above the glass transition temperature, Tg , PBT is somewhat rubbery. That is, it may undergo very large strains and still recover much of its initial configuration upon unloading though this recovery is a time-dependent process. However, permanent deformations can be induced without severe necking in PBT above Tg . The yield point at these elevated temperatures was found by a systematic schedule of loading unloading and waiting for contractive recovery. The variations of the yield stress and the yield strain of PBT at a strain rate of 𝜆̇ = 10−3 s–1 , including points above Tg , are shown in Figure 15.4.9. Even though the yield stress declines with increasing temperature, the strain necessary for initial permanent deformation exhibits a dramatic jump above Tg . The amount of permanent deformation induced in an isothermal test in PBT is surprisingly small when compared with the large extensions necessary to achieve these final strains. That is, springback and subsequent time-dependent recovery are large. In any case, no amount of isothermal heat treatment above Tg , but below the melting point seems to cause the strains to return to zero. Furthermore, since necking

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Introduction to Plastics Engineering

80

STRESS (MPa)

PBT

−17.8°C (0°F ) 22°C (72°F )

50

37.8°C (100°F) 11.4°C (112°F) 65.6°C (150°F) •

λ = 10 –3 s–1

51.6°C (125°F ) 0 0

0.05

0.10

0.15

EXTENSION e = λ – 1

Figure 15.4.8 Stress-extension curves for PBT at a stretch rate of with the stretch rate of 𝜆̇ = 10−3 s–1 ,with the temperature as parameter. (Adapted with permission from V.K. Stokes and H.F. Nied, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 107 – 112, 1986.)

80

0.06 0.05

50

PBT •

λ = 10 – 3 s–1

YIELD EXTENSION e = λ – 1

YIELD STRESS (MPa)

370

0.01

0 – 40

0

50

100

TEMPERATURE ( o C)

Figure 15.4.9 Variations of the yield stress and the yield extension of PBT with the temperature at a stretch rate of 𝜆̇ = 10−3 s–1 , (Adapted with permission from V.K. Stokes and H.F. Nied, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 107 – 112, 1986.)

Mechanical Behavior of Plastics

occurs in PBT at about 20% extension, there is a maximum amount of permanent extension that can be obtained at a given temperature above Tg without necking. In PBT this maximum permanent extension is not large and is on the order of 2%. 15.4.6

*Post-Yield Behavior Prior to Necking

Since PBT does not exhibit intense nonhomogeneous deformation immediately after yielding, it is possible to monitor stress and strain in the post-yield regime using conventional techniques. When PBT does finally develop intense shear bands and localized necking, significant straining has already occurred well beyond the yield strain. From one-dimensional forming tests on PBT, indications are that the useful forming range in terms of strain will be limited to strain points between A ′ and C ′ shown in Figures 15.4.1 and 15.4.2. That is, useful permanent deformation for the purposes of forming can only occur by deforming PBT between the initial yield point (A ′) and necking at (C ′). However, a final judgment on this can only be made on the basis of tests that involve more complex deformations than the simple one-dimensional extension that has been considered thus far. Unlike polycarbonate, owing to the very severe reduction in cross-sectional area that occurs when the material necks, necked PBT exhibits a very strong propensity for immediate tearing. Furthermore, the esthetic appearance of PBT changes drastically with necking, primarily because of the thinning effect accompanied by greatly enhanced translucency. Thus, for PBT there is a lower and upper bound for strain, between which all forming deformation must occur. Clearly these bounds are themselves functions of both strain rate and temperature. Figure 15.4.10 shows the lower and upper forming bounds for PBT as a function of strain rate at 22.2°C (72°F) as an example of the available forming ranges below Tg . Note that while there is little scatter in the yield-extension data,

EXTENSION e = λ – 1

0.20

0.15

NECKING

0.10

PBT

0.05

YIELD

10 – 4

10 – 3

TEMPERATURE = 22°C (72°F)

10– 2

10– 1

10 0

101

•

–1 STRETCH (STRAIN) RATE λ (s )

Figure 15.4.10 Room-temperature upper and lower forming bounds for PBT as a function of the stretch (strain) rate. (Adapted with permission from V.K. Stokes and H.F. Nied, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 107 – 112, 1986.)

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Introduction to Plastics Engineering

considerable scatter occurs in the necking extension data. This scatter in the necking extension data may reflect the sensitivity of the material to any initial imperfections. 15.4.7

*Load History and Final Permanent Deformation

With the operational uniaxial forming range delineated by upper and lower bounds shown in Figure 15.4.10, the next step is to quantify the exact magnitude of axial permanent strain that can be accumulated within these bounds. For thermoplastics the final strain on unloading in the uniaxial test is sensitive to the entire temperature and loading history. In order to simulate the forming process and, at the same time, to isolate the effect of strain history on final deformation, isothermal tests were conducted in which the material was loaded at a given extension rate to an extension within the plastic range. This extension was held constant for various hold times after which the load was suddenly dropped to zero. Stress relaxation was measured during the hold period and, upon unloading, the extensional recovery was monitored as a function of time. These experiments simulated the actual forming process in a one-dimensional sense, except that the temperature was kept constant throughout the test. As an example, Figure 15.4.11 shows a dual record of the entire strain and stress history for a PBT specimen tested at 22.2°C (72°F) and initially strained at a constant extension rate of 𝜆̇ = 10−1 s–1 . At emax = 0.1 (e = 𝜆 − 1), the extension was held constant for 10 s, after which the load was suddenly dropped to zero in less than 0.01 s by cutting the specimen. In this test two regions are of primary interest: Region (b), where stress relaxation occurs during the hold in strain, and Region (c), representative of the strain recovery that occurs after sudden unloading. Note that beyond yield at e = 0.03, even though the strain continues to increase, the stress actually decreases before the “hold-time” fixed strain is reached. T = 22°C (72°F)

b

HOLD TIME = 10 s

0.10

•

λ = 10 – 1 s–1 50

b

a

PBT 0.05

a c

YIELD EXTENSION e = λ – 1

80

YIELD STRESS (MPa)

372

0

0 10

–1

10

0

10

1

10

2

10

3

10

4

TIME (seconds) Figure 15.4.11 Room-temperature stress and extension variations with time in a 10-second hold time test. (Adapted with permission from V.K. Stokes and H.F. Nied, ASME Journal of Engineering Materials and Technology, Vol. 108, pp. 107 – 112, 1986.)

Figure 15.4.12 compares the room-temperature (22.2°C) strain recovery from an initial maximum extension of emax = 0.1 at which the strain is held for times of 1, 10, 100, and 1,000 s before unloading,

Mechanical Behavior of Plastics

for initial extension rates of 𝜆̇ = 10−1 s−1 (solid lines) and 𝜆̇ = 10−3 s−1 (dashed lines). In this figure the strain-ratio is defined as the actual measured extension normalized by the fixed “hold-time” extension. Clearly, over six decades of time an enormous amount of strain recovery occurs in PBT, which strongly depends on the hold time. Discounting the essentially instantaneous contraction that occurs on unloading, the strain recovery after a short hold time decays very slightly when compared with the behavior after long hold times. The slow strain recovery following the initial “springback” in tests with long hold times is enormous, on the order of 60% or more. Specimens subjected to long hold times always exhibit markedly smaller initial strain contractions immediately after unloading in comparison to the large “instantaneous” contractions observed in specimens subjected to “short” hold times. Remarkably, after being subjected to strains of emax = 0.1, which for PBT is well into the plastic regime, the final permanent strain for initial extension rates of 𝜆̇ = 10−1 s−1 (solid lines) are only between 0.01 and 0.03. Decreasing the initial strain rate by two orders of magnitude to 𝜆̇ = 10−3 s−1 (dashed lines), has little effect on the form of the recovery strain and the final asymptotic strain values, except for the shortest hold times.

1.0 T = 22°C (72°F)

PBT

STRAIN RATIO (e / emax )

•

λ = 10 –1 s –1 emax = 0.01

0.5

HOLD TIME 1000 s 100 s 10 s 1s

•

λ = 10 –3 s –1

0 10– 2

10 – 1

10 0

10 1

10 2

10 3

10 4

10 5

TIME (seconds) Figure 15.4.12 Room-temperature strain recovery of PBT for four hold times, for emax = 0.01, 𝜆̇ = 10−1 s−1 (solid lines) and 𝜆̇ = 10−3 s−1 (dashed lines). (Adapted from V.K. Stokes and H.F. Nied, GE Corporate Research & Development Report No. 84CRD215, 1984, courtesy of SABIC.)

As Figure 15.4.13 shows, increasing the magnitude of the extension maintained during the hold time from emax = 0.1 (dashed lines) to emax = 0.15 (solid lines) does increase the asymptotic value of the permanent strain. However, the viscoelastic strain recovery is still enormous, amounting to approximately 70% of the total extension prior to unloading. It appears that permanent strains greater than e = 0.05 cannot be generated without undesirable necking, no matter what strain or load history is used. Generating large permanent strains becomes even more difficult with increasing temperatures, as can be seen from Figure 15.4.14, which shows data for the test temperatures of 37.8°C (solid lines) and 65.6°C (dashed lines). At the higher temperature (dashed lines), large viscoelastic recovery results in small permanent strains.

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Introduction to Plastics Engineering

STRAIN RATIO (e / emax )

1.0 PBT

emax = 0.015

HOLD TIME 1000 s

•

λ = 10 –3 s –1

100 s 10 s 1s

0.5

emax = 0.01 T = 22°C (72°F)

0 10– 2

10 – 1

100

10 1

10 2

10 3

10 4

10 5

TIME (seconds) Figure 15.4.13 Room-temperature strain recovery of PBT for four hold times, for 𝜆̇ = 10−1 s−1 , emax = 0.01, (solid lines) and emax = 0.015 (dashed lines). (Adapted from V.K. Stokes and H.F. Nied, GE Corporate Research & Development Report No. 84CRD215, 1984, courtesy of SABIC.)

1.0 T = 37.8°C (100°F)

STRAIN RATIO (e / emax )

374

emax = 0.1 •

λ = 10 –3 s

PBT HOLD TIME 1000 s 100 s 10 s 1s

0.5

T = 65.6°C (150°F)

0 10– 2

10 – 1

100

10 1

10 2

10 3

10 4

10 5

TIME (seconds) Figure 15.4.14 Strain recovery of PBT for four hold times, for emax = 0.01, 𝜆̇ = 10−1 s−1 , T = 37.8°C (solid lines)

and T = 65.6°C (dashed lines). (Adapted from V.K. Stokes and H.F. Nied, GE Corporate Research & Development Report No. 84CRD215, 1984, courtesy of SABIC.)

Mechanical Behavior of Plastics

Since the lower-temperature data in Figure 15.4.14 (solid lines) is near the glass transition temperature, the viscoelastic recovery can be expected to lie somewhere between the those shown in Figure 15.4.13 (below Tg ) and the dashed curves (65.6°C) in Figure 15.4.14 (above Tg ). The recovery immediately after unloading at 37.8°C shown in Figure 15.4.15 is similar to the recovery below Tg , while for longer times the mode of recovery is similar to behavior above Tg . Before unloading an extension of emax = 0.1 was maintained for 1,000 seconds for all three curves in Figure 15.4.15. Only a small percentage of the total extension emax can be attributed to “permanent” deformation.

1.0 HOLD TIME = 1000 s

STRAIN RATIO (e / emax )

•

emax = 0.1, λ = 10 –3 s TEMPERATURE

22°C (72°F) 37.8°C (100°F) 65.6°C (150°F)

0.5

PBT 0 10– 2

10 – 1

100

10 1

10 2

10 3

10 4

10 5

TIME (seconds) Figure 15.4.15 Strain recovery of PBT for a hold time of 1,000 seconds, for emax = 0.01, 𝜆̇ = 10−1 s−1 , and T = 22.2°C,

37.8°C, and 65.6°C. (Adapted from V.K. Stokes and H.F. Nied, GE Corporate Research & Development Report No. 84CRD215, 1984, courtesy of SABIC.)

The isothermal hold-time data show that recovery effects upon unloading are the dominant effects. With the range of available forming strains effectively restricted between e = 0.03 and e = 0.20, and with the resulting permanent deformation being small, severe constraints are placed on the isothermal formability of PBT. 15.4.8

Large-Deformation Applications

While in some applications, such as in energy absorbing bumper beams, full recovery from large deformations is desirable, other applications, such as in sheet forming, require permanent deformations. Because PC can recover from very large deformations at room temperature, it is not suited for cold sheet stamping as in steel. Figure 15.4.16 shows the results of cold stamping of thin PBT sheets between matched dies; the shape with three different “bump” shapes was chosen to simulate automotive wheel covers. The part on the left was formed at a faster rate than the one on the right. The difference in the part shapes made of the same material in the same dies, but at different forming rates, shows how the rate sensitivity of

375

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Introduction to Plastics Engineering

plastics (viscoelasticity) can affect formability. The part on the left exhibits “stress whitening” resulting from deformations in the highly stretched, oriented material.

Figure 15.4.16 Parts cold stamped in matching dies from thin PBT sheets. The part on the left was made at a faster forming rate; notice the stress whitening caused by highly stretched oriented material. (Photo courtesy of SABIC.)

15.5 Stress-Deformation Behavior of Several Plastics Thus far the focus has been on detailed descriptions of one grade of each of three high-performance plastics: Polycarbonate, an amorphous, high-toughness, high-stiffness, and high-strength plastic; polyetherimide, an amorphous, high-temperature, high-stiffness, and high-strength plastic; and poly(butylene terephthalate), a semicrystalline, high-stiffness, and high-strength plastic. Each of them is ductile, that is, they undergo large deformations before failure. However, not all thermoplastics are ductile. While these details illustrate the complexity of the mechanical response of amorphous and semicrystalline polymers, they do not address the substantial variations in different grades of the same material caused by factors such as differences in molecular weight distributions. Section 15.5.1 compares the mechanical behavior of several high-performance and relatively lower-performance plastics. In addition to stress-elongation curves for representative grades, tables provide ranges of property variations for each resin. The stress-elongation curves do not account for neck propagation, and really constitute stress-displacement plots of the type shown in Figures 15.2.1, 15.3.1, and 15.4.1. Thermoset plastics are more rigid, that is, they have larger elastic moduli, but are brittle – they fail without undergoing any plastic deformation. They are discussed in Section 15.5.2. 15.5.1

Thermoplastics

This section compares the stress-elongation behavior of several higher- and lower-performance amorphous and semicrystalline thermoplastics. For any resin, different grades can exhibit different elastic moduli and strength. In applications, data for the specific grade being considered have to be used. The comparisons in this section have been obtained from a compilation in a single source. The curves in the figures are only representative values. The range of variations of properties is listed in the accompanying tables.

Mechanical Behavior of Plastics

The stress-elongation curves do not account for neck propagation, and really constitute stressdisplacement plots. As such, the stress-elongation curves shown in this section are valid only till the onset of necking, beyond which they represent stress-displacement plots (Figures 15.2.1, 15.3.1, and 15.4.1.). Figure 15.5.1 shows the stress-elongation curves for several amorphous thermoplastics: Polyetheretherketone (PEEK), polyethersulfone (PES), polysulfone (PSU), modified poly(phenylene oxide) (M-PPO), and amorphous nylon (APA) – which absorbs moisture – is shown by two curves labeled APA-DRY and APA-SH, corresponding, respectively, to dry resin and resin at equilibrium at a standard humidity of 50%. Also shown for comparison are curves for PEI and PC. The curve for M-PPO is shown by a dashed curve to distinguish it from the curve for PC. All the plastics in this figure are ductile.

100 PEI

STRESS (MPa)

PES APA-DRY APA-SH

50

PC

PSU

M-PPO

AMORPHOUS RESINS

0 0

10

20

30

EXTENSION e = λ – 1 (%) Figure 15.5.1 Stress-elongation behavior of several high-performance amorphous thermoplastics. (Adapted with permission from “Polymeric Materials: Structure-Properties-Applications,” by G.W. Ehrenstein, Hanser Publishers, Munich, 2001.)

The variations in the key thermal and mechanical properties for different grades of the same resin are listed in Table 15.5.1. While the heading for the last column is this table is failure strain, the values should be interpreted with some care: As per the discussion on PC in Section 15.2.1, after neck formation while the non-necked necked material has an extension of e = 𝜆 − 1 = 0.06, the necked material has a constant extension of e = 0.7. Figure 15.5.2 shows the stress-elongation curves for several semicrystalline thermoplastics: nylon 6 (PA6) and nylon 6,6 (PA6,6) with curves for both dry resins and resins in equilibrium at a standard humidity of 50%, poly(ethylene terephthalate) (PET), and polyoxymethylene (POM). Also shown for comparison is the curve for and PBT. The curve for POM is shown by a dashed curve to distinguish it from the curve for PET. All the plastics in this figure are ductile. The variations in the thermal and mechanical properties for the resins in this figure are listed in Table 15.5.2.

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Introduction to Plastics Engineering

Table 15.5.1 Thermal and mechanical properties of high-performance amorphous resins.

T g (°C)

Plastic

Thermal limits (°C) short-time (long-time)

Density (g cm−3 )

Tensile modulus Yield stress Yield strain Failure strain (GPa) (MPa) (%) (%)

Amorphous PA (dry)

−152

120

1.12 – 1.15

2.8

90

7

20

Amorphous PA (50% H)

−114

80

1.12 – 1.15

2.7

85

6

>50

PC

145

135 (100)

1.2 – 1.24

2.2 – 2.4

55 – 65

6–7

100 – 130

PEI

215 – 230

190 (160)

1.27

2.9 – 3.0

85

6–7

>50

PES

225 – 230

210 (180)

1.37

2.6 – 2.8

80 – 90

5.5 – 6.5

20 – 80

M-PPO

140

120 (100)

1.04 – 1.06

2.3

50 – 55

3–5

36 – 45

PSU

185 – 190

170 (150)

1.24 – 1.25

2.5 – 2.7

70 – 80

5.5 – 6

20 to >50

(Adapted with permission from “Polymeric Materials: Structure-Properties-Applications,” by G.W. Ehrenstein, Hanser Publishers, Munich, 2001.)

100 PEEK PA6,6-DRY

STRESS (MPa)

378

PA6-DRY PET POM

50

PBT PA6,6-SH PA6-SH

SEMICRYSTALLINE RESINS

0 0

10

20

30

EXTENSION e = λ – 1 (%) Figure 15.5.2 Stress-elongation behavior of several high-performance semicrystalline thermoplastics. All these plastics are ductile. (Adapted with permission from “Polymeric Materials: Structure-Properties-Applications,” by G.W. Ehrenstein, Hanser Publishers, Munich, 2001.)

Table 15.5.2 Thermal and mechanical properties of high-performance semicrystalline resins.

Plastic

T g (°C)

T m (°C)

Thermal limits (°C) short-time (long-time)

Crystallinity (%)

Density (g cm−3 )

Tensile modulus (GPa)

Yield stress (MPa)

Yield strain (%)

Failure strain (%)

30

PA6 (dry)

−78

225 – 235

140 – 160 (80 – 100)

30 – 40

1.12 – 1.15

2.8

80

4

PA (50% H)

−28

225 – 235

140 – 160 (80 – 100)

30 – 40

1.12 – 1.15

1.0

45

25

>50

PA6,6 (dry)

−90

255 – 265

140 – 170 (80 – 100)

35 – 45

1.13 – 1.16

3

85

5

25

PA6,6 (50% H)

−39

225 – 265

140 – 170 (80 – 100)

35 – 45

1.13 – 1.16

1.6

60

20

>50

PBT

45 – 60

220 – 230

160 (100)

40 – 50

1.3 – 1.32

2.5 – 2.8

50 – 60

3.5 – 7

3.5 to >50 >50

PEEK

145

335

300 (250)

∼35

1.32

3.7

100

5

PET

70 – 80

250 – 260

170 (100)

30 – 40

1.33 – 1.40

2.1 – 3.1

55 – 80

4–7

>50

POM

−70

165 – 175

110 – 140 (90 – 100)

70 – 80

1.41 – 1.43

3 – 3.2

60 – 75

8 – 25

20 to >50

(Adapted with permission from “Polymeric Materials: Structure-Properties-Applications,” by G.W. Ehrenstein, Hanser Publishers, Munich, 2001.)

Introduction to Plastics Engineering

Figure 15.5.3 shows the stress-elongation curves for several lower-performance, or commodity, amorphous thermoplastics: poly(methyl methacrylate) (PMMA), poly(styrene-co-acrylonitrile) (SAN), poly(vinylidene fluoride) (PVDF), poly(vinyl chloride) (PVC) and plasticized PVC, polycarbonateacrylonitrile-butadiene-styrene (PC/ABS) blend, acrylonitrile-styrene-acrylate (ASA), acrylonitrilebutadiene-styrene (ABS), polystyrene (PS), high-impact polystyrene (HIPS), and plasticized PVC (80% PVC, 20% plasticizer). The curves for PC/ABS and ASA are shown by dashed curves to distinguish them from adjacent curves. Note that the three plastics PMMA, SAN, and PS in this figure are brittle. Also note by how much 20% plasticizer in PVC reduces its elastic modulus. The variations in the thermal and mechanical properties for the resins in this figure are listed in Table 15.5.3.

100 AMORPHOUS RESINS PMMA

STRESS (MPa)

380

SAN

PVC

50 PC/ABS ASA

PS

ABS

PLASTICIZED PVC

HIPS

0 0

10

20

30

EXTENSION e = λ – 1 (%) Figure 15.5.3 Stress-elongation behavior of several lower-performance amorphous thermoplastics. Note that the three plastics PMMA, SAN, and PS are brittle. Also note by how much 20% plasticizer in PVC reduces its elastic modulus. (Adapted with permission from “Polymeric Materials: Structure-Properties-Applications,” by G.W. Ehrenstein, Hanser Publishers, Munich, 2001.)

Figure 15.5.4 shows the stress-elongation curves for several lower-performance, or commodity, semicrystalline thermoplastics: syndiotactic polystyrene (PS-S), poly(vinylidene fluoride) (PVDF), polypropylene (PP), high-density polyethylene (HDPE), polytetrafluoroethylene (PTFE), and low-density polyethylene (LDPE). All the polymers are ductile, except for PS-S that is brittle. The variations in the thermal and mechanical properties for the resins in this figure are listed in Table 15.5.4. 15.5.2

Thermosets

In contrast to thermoplastics, most of which are ductile, thermosets are brittle. They are mostly used with reinforcing fibers. Figure 15.5.5 shows the stress-elongation curves for three thermosets: A polyester, an

Mechanical Behavior of Plastics

Table 15.5.3 Thermal and mechanical properties of lower-performance amorphous resins.

T g (°C)

Plastic

Thermal limits (°C) short-time (long-time)

Density (g cm−3 )

Tensile modulus (GPa)

Yield stress (MPa)

Yield strain (%)

Failure strain (%)

ABS

−85/95 – 105

85 – 95 (75 – 85)

1.03 – 1.07

2.2 – 3.0

45 – 65

2.5 – 3

15 – 20

ASA

−40/95

−85 to 95 (−75 to 85)

1.04 – 1.07

2.3 – 2.9

40 – 55

3.1 – 4.3

10 – 30

PC/ABS

−85/105/145

115 – 130 (105 – 115)

1.08 – 1.17

2 – 2.6

40 – 60

3 – 3.5

PMMA

105 – 120

85 – 95 (65 – 80)

1.15 – 1.19

3.1 – 3.3

60 – 80

PS

90 – 100

90 (80)

1.05

3.1 – 3.3

30 – 55

HIPS

−85/100

60 – 80 (50 – 70)

1.00 – 1.05

1.1 – 2.8

15 – 45

1.1 – 6

10 to >50

PVC

80

70 (60)

1.38 – 1.55

2.7 – 3.0

50 – 60

4–6

10 – 50

PVC-plasticized

−50 to 80

55 – 65 (50 – 55)

1.16 – 1.35

0.025 – 1.6

8 – 25

170 – 400

SAN

95 – 105

95 (85)

1.08

3.5 – 3.7

65 – 85

2.5 – 5

>50 2–6 1.5 – 3

(Adapted with permission from “Polymeric Materials: Structure-Properties-Applications,” by G.W. Ehrenstein, Hanser Publishers, Munich, 2001.)

100

STRESS (MPa)

SEMICRYSTALLINE RESINS

50 PVDF PS-S

PP

HDPE

PTFE

LDPE

0 0

10

20

30

EXTENSION e = λ – 1 (%) Figure 15.5.4 Stress-elongation behavior of several lower-performance semicrystalline thermoplastics. Except for PS-S, which is brittle, all the resins are ductile. (Adapted with permission from “Polymeric Materials: Structure-Properties-Applications,” by G.W. Ehrenstein, Hanser Publishers, Munich, 2001.)

381

Table 15.5.4 Thermal and mechanical properties of lower-performance semicrystalline resins.

Plastic

T g (°C)

T m (°C)

Thermal limits (°C) short-time (long-time)

Crystallinity (%)

Density (g cm−3 )

Tensile modulus (GPa)

Yield stress (MPa)

Yield strain (%)

Failure strain (%)

HDPE

50

PS-S

90 – 100

270

220 (150)

∼35

1.28

3.4 – 3.6

50 – 55

PTFE

125 – 130

325 – 330

280 (240)

55 – 90

2.13 – 2.23

0.4 – 0.75

20 – 40

PVDF

−35

170 – 175

– (150)

50 – 60

1.76 – 1.78

2 – 2.9

50 – 60

>50

1.5 – 2 >50 7 – 10

(Adapted with permission from “Polymeric Materials: Structure-Properties-Applications,” by G.W. Ehrenstein, Hanser Publishers, Munich, 2001.)

20 to >50

Mechanical Behavior of Plastics

100 THERMOSET RESINS

STRESS (MPa)

POLYESTER

EPOXY

50

RUBBER -TOUGHENED EPOXY

0 10

0

20

30

EXTENSION e = λ – 1 (%) Figure 15.5.5 Stress-elongation behavior of three thermoset resins. (Adapted with permission from “Polymeric Materials: Structure-Properties-Applications,” by G.W. Ehrenstein, Hanser Publishers, Munich, 2001.)

epoxy, and a rubber-toughened epoxy. The range of variations of properties is given in Table 15.5.5. Note how epoxy can be made ductile by embedding rubber particles, a process called rubber toughening. Table 15.5.5 Thermal and mechanical properties of lower-performance semicrystalline resins.

Plastic

T g (°C)

Thermal limits (°C) short-time (long-time)

Density (g cm−3 )

Tensile modulus (GPa)

Yield stress (MPa)

Failure strain (%)

Epoxy resin

70 – 200

(60 – 80)

1.17 – 1.25

Up to 4.2

Up to 100

1.5 – 2.5

Unsaturated polyester

70 – 150

180 (100)

1.2

3.2 – 3.5

50 – 77

1.2 – 2.5

(Adapted with permission from “Polymeric Materials: Structure-Properties-Applications,” by G.W. Ehrenstein, Hanser Publishers, Munich, 2001.)

15.5.3

Thermoplastic Elastomers

Thermoplastic elastomers (TPEs) are low-modulus, rubber-like materials that can be extruded and molded to form parts; they can be overmolded onto stiffer thermoplastics to provide a soft touch feel. Three commercially important TPEs – propylene-EPDM vulcanizate, copolyester, and urethane TPEs – are discussed in Section 11.9.1. This section provides basic stress-strain data for these three types of TPEs.

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One major difference between thermoplastics and TPEs is that in TPEs not all the strain is recovered on unloading, even for relatively low strains. The unrecovered strain is referred to as the tensile (or compressive) set. Figure 15.5.6 shows tensile and compressive stress-extension data for a thermoplastic vulcanizate (TPV); this data set is used as input for finite element analysis of parts made of this material. Two observations on this data set are noteworthy: First, compared to all the thermoplastics considered thus far, this material has a very low modulus – a tensile stress as low as 1.2 MPa causes and extension of 50%. This is an advantage in that it allows molding of relatively flexible parts. Second, the tensile and compressive stress-extensions are not symmetric – while a tensile stress as of about 1.2 MPa causes and extension of 50%, the compressive stress required to cause a compressive extension 50% (extension of −50%) is 3.5 MPa (stress of −3.5 MPa).

5 THERMOPLASTIC VULCANIZATE (Santoprene TM 101-55 TPV)

STRESS (MPa)

384

0

TENSION

COMPRESSION

–5 – 50

0

50

EXTENSION e = λ – 1 (%) Figure 15.5.6 Tensile and compressive stress-elongation behavior of TPV. (Adapted with permission from “SantopreneTM 101-55 TPV Finite Element Analysis Data Sheet,” courtesy of ExxonMobil.)

Mechanical Behavior of Plastics

Figure 15.5.7 shows stress-extension curves for a polyester TPE for large extensions over a wide temperature range. The curves for − 40 and −20°C end at the points where the specimens fail. For temperature between 0 and 150°C the tests were stopped before specimen failure. Notice that, at lower temperatures, the stress increase and sudden drop-off behavior, like in many thermoplastics, changes to a smooth transition at higher temperatures. Figure 15.5.8 shows these curves on an expanded extension scale to a lower extension of 50%. Clearly, this material has a higher modulus than the TPV considered previously.

STRESS (MPa)

POLYESTER TPE (Hytrel® 6356) – 40°C

50

NOTE: Extension tests for 0°C to 150°C stopped before specimen rupture.

– 20°C 0°C

120°C

150°C

0 0

100

23°C 40°C 60°C 90°C

200

EXTENSION e = λ – 1 (%) Figure 15.5.7 Tensile stress-elongation behavior of polyester TPE for large extensions at different temperatures. (Adapted with permission from “DUPONTTM HYTREL® Design Guide.” Courtesy of DuPont © 2016.)

As mentioned earlier, on unloading from a stressed state, even for relatively low loads TPEs are left with an unrecovered strain referred to as permanent set. Figure 15.5.9 shows the amount of tensile set as a function of the extension from which the part is unloaded. Figure 15.5.10 show the stress-extension curves for a thermoplastic urethane for large extensions for a temperature range of −20 to 100°C. Notice the very large extensions to which this material can be stretched without rupturing.

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STRESS (MPa)

POLYESTER TPE (Hytrel® 6356) – 40°C

50

NOTE: Extension tests for 0°C to 150°C stopped before specimen rupture.

– 20°C 0°C

150°C

0 0

20

10

120°C

30

23°C 40°C 60°C 90°C

40

50

EXTENSION e = λ – 1 (%) Figure 15.5.8 Tensile stress-elongation behavior of polyester TPE for lower extensions at different temperatures. (Adapted with permission from “DUPONT HYTREL Design Guide.” Courtesy of DuPont © 2016.)

2

TENSILE SET (%)

386

POLYESTER TPE (Hytrel ® 6356)

1

0 0

5

10

15

EXTENSION e = λ – 1 (%) Figure 15.5.9 Tensile set of polyester TPE as a function of extension. (Adapted with permission from “DUPONT HYTREL Design Guide.” Courtesy of DuPont © 2016.)

Mechanical Behavior of Plastics

100 THERMOPLASTIC URETHANE (Ellastollan® C95A)

STRESS (MPa)

– 20°C 23°C

50 60°C 100°C

0 500

0

1000

EXTENSION e = λ – 1 (%) Figure 15.5.10 Tensile stress-elongation behavior of thermoplastic urethane for large extensions at different temperatures. (Adapted with permission from “Ellastolan Material Properties Brochure” courtesy of BASF.)

Figure 15.5.11 shows loading-unloading curves for this material for extension up to 200%. The intercept of the unloading curve on the extension axis is the tensile set.

15.6 Phenomenon of Crazing Plastics exhibit the phenomenon of crazing – unique to plastics – in which an internal “failed” surface, which would in most materials result in separated surfaces, is connected by fine fibrils of highly oriented material that provide substantial residual strength to the material. Cracks, which are locally separated surfaces, initiate at imperfections in stressed materials. Figure 15.6.1a shows such a crack in a material stressed by a uniaxial stress 𝜎 y = 𝜎 0 ; in the initial stages the crack is confined to the nucleation site, and may not appear on the exterior surface. Under increasing stress the crack can grow and propagate, resulting in failure by a complete separation of the surfaces. In contrast to separated surfaces in a crack, in a craze these surfaces are tied by thin highly-drawn, high-stiffness, and high-strength material (Section 15.2.5) schematically shown in Figure 15.6.1b. As in a crack, the craze is confined to the nucleation site. All plastics do not craze before failing. For example, while thick polycarbonate specimens craze under impact loads, thin sheets do not craze before failing.

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THERMOPLASTIC URETHANE (Ellastollan ® C95A)

20

STRESS (MPa)

388

10

0 100

0

200

EXTENSION e = λ – 1 (%) Figure 15.5.11 Tensile loading-unloading stress-elongation behavior of thermoplastic urethane for large extensions. (Adapted with permission from “Ellastolan Material Properties Brochure” courtesy of BASF.)

σ0

σ0

y

•

x

σ0

σ0

(a)

(b)

Figure 15.6.1 (a) An internal crack in a material subjected to a tensile stress. (b) A craze in a material subjected to a tensile stress. The surfaces of the void, or “crack,” are bridged by very fine highly drawn material. The craze is confined to the nucleation site and does not extend to the part surface.

Mechanical Behavior of Plastics

In most applications of transparent plastics, crazes appear as reflective, fine bright white streaks in the interior. In nontransparent plastics the internally embedded crazes may not be visible. The term crazing is borrowed from its use in pottery wherein it refers to a network of lines or cracks in the fired glazed surface caused by glaze being under tension. This term was applied to amorphous polymers when approximately 3-mm thick bars subjected to tension were found to have numerous internal “crazes” that were made visible by the light reflected by them. Figure 15.6.2a shows internal crazes in a 4-mm (0.157-in) thick, 10 × 6.5 × 6 cm (4 × 2.5 × 2.25 in) open container made of PMMA kept in a room for about 10 years. Notice the extensive (internal) crazes on all walls and the bottom. Close-up views of the crazes in one side wall, and one end wall are shown,

(a)

(b)

(c) Figure 15.6.2 Internal crazes in a 4-mm-thick, 10 × 6.5 × 6 cm open container made of PMMA. (a) Overall view showing crazes in all walls and bottom. (b) Close-up view of crazes in side wall. (c) Close-up view of crazes in end wall.

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respectively, in Figure 15.6.2 parts b and c. The nucleation and growth of these crazes have been caused by tensile residual stresses in the part induced by the injection molding process. The structure of a craze is schematically shown in Figure 15.6.3. The fibrils of highly drawn material have diameters on the order of dc ≈ 5 nm. The spacing of the voids between the fibrils can be characterized by the distance sc ≈ 20 nm. And the craze thickness is on the order of tc ≈ 1 mm. Craze structure is closely related to the molecular architecture of the plastic. Fibril spacing and the lengths to which they grow before breaking depend on the degree of macromolecular chain entanglement; lower entanglement increases the propensity for crazing. BULK MATERIAL

HIGHLY DRAWN MATERIAL

VOID

y

•

tc

x

sc dc Figure 15.6.3 Schematic structure of a craze.

The crazing process involves the three phases of nucleation, craze formation, and craze extension. Crazes nucleate around defects in the material. Local voiding then results in the crack surfaces being connected by highly drawn, oriented material. The craze thickness tc can only increase by more adjacent material crazing in the lateral, x, direction. Figure 15.6.4a schematically shows the structure of a craze near its tip; the fibril diameter dc is larger near the tip where the material has not undergone as much drawing as in fibrils away from the tip. For the craze thickness tc to increase the craze tip has to advance laterally. One proposed mechanism for this involves an instability at the void-material interface – akin to the instability of the meniscus between two liquids of different densities that causes a “fingering,” or interpenetration of the lighter phase into the heavier phase – which causes the voids to “finger,” or penetrate into the bulk material. Figure 15.6.4 parts b – d show this mechanism at work; the dotted circles show the fibrils that are spawned as the interface moves to the right. The fibrillated structure of crazes, which appear as fine streaks at a macroscopic level, can only be visualized on a microscale. In the sequel, the structure of crazes is described by crazes in 800-nm thick polystyrene films: In these experiments, 800-nm thick dip-coated films of polystyrene placed on a ductile copper grid were exposed to toluene vapor to enable the films to adhere to the copper. The grid was then subjected to uniaxial stretch (Figure 15.6.5). The stretch was adjusted to ensure that crazes appeared in some of the grids, resulting mostly in through-thickness crazes. The craze structure was examined by using Field Emission Scanning Electron Microscopy (FESEM). The micrograph in Figure 15.6.6a shows a crack and a craze that developed in one of the grids, as visualized by FESEM. The close-up view in Figure 15.6.6b shows fully developed fibrils just ahead of the crack tip.

Mechanical Behavior of Plastics

y

•

x CRAZE TIP

•

(a)

(b)

(c)

(d)

x

z

Figure 15.6.4 Meniscus-instability model for craze growth. (a) Elongated fibrils near a craze tip; craze envelope indicated by a dashed line. Arrow indicates direction of craze tip advance. (b) Cross section through x = 0 plane showing the fingering into bulk material during craze advance; spawning of new fibrils indicated by dashed circles. (c, d) More fibrils being spawned by advancing crack tip. (Adapted from A.S. Argon and M.M. Salama, Philosophical Magazine, Vol. 36, pp. 1217 – 1234, 1977, courtesy of Taylor & Francis Ltd.)

63-µm DIAMETER COPPER WIRE GRID

~ 800-nm-THICK PS FILM ON COPPER WIRE GRID

1 mm

Figure 15.6.5 Arrangement for stretching 800-nm-thick films. The film mounted on a grid of 63-μm diameter copper wire was exposed to toluene vapor to enable the PS film to stick to the grid. The grid was then stretched with uniaxial tension. (Adapted from Kumar S. Arun, “Experimental Study of Crazing in Polystyrene,” MTech Thesis, IIT Kanpur, 2010; micrographs courtesy of Professor Sumit Basu, Indian Institute of Technology Kanpur.)

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1.75 µm

FIBRILS VOIDS TIE FIBRILS

10 µm

500 nm

(a)

(b)

Figure 15.6.6 Craze structure from FESEM micrographs. (a) Crack and a craze that developed in one of the grids. (b) Structure of fully developed fibrils just ahead of the crack tip. (Adapted from Kumar S. Arun, “Experimental Study of Crazing in Polystyrene,” MTech Thesis, IIT Kanpur, 2010; micrographs courtesy of Professor Sumit Basu, Indian Institute of Technology Kanpur.)

Figure 15.6.7 shows another fully developed craze with regularly spaced fibrils; the fibril diameter and spacing are notably uniform. 55 nm DIAMETER: 33 nm

REGULARLY SPACED FIBRILS

200 nm

Figure 15.6.7 Fully developed craze with regularly spaced fibrils; the fibril diameter and spacing are notably uniform. (Adapted from Kumar S. Arun, “Experimental Study of Crazing in Polystyrene,” MTech Thesis, IIT Kanpur, 2010; micrographs courtesy of Professor Sumit Basu, Indian Institute of Technology Kanpur.)

The collage in Figure 15.6.8 shows the structure in different parts of the craze. Inset (a) shows that the craze tip has irregularly spaced voids, and thick fibrils. Inset (b) shows regularly spaced thicker primitive fibrils develop just away from the tip. And inset (c) shows more uniform fibril spacing close to the crack tip. While these micrographs give an insight into the structure of crazes in very thin membranes, they show through-thickness crazes. Interior crazes in actual parts, such as those shown in Figure 15.6.3,

Mechanical Behavior of Plastics

500 nm

(c)

200 nm 200 nm

(b)

(a)

Figure 15.6.8 Collage showing different parts of the craze. (a) The tip of the craze has irregularly spaced voids, thick fibrils. (b) Slightly away from the tip regularly spaced thicker primitive fibrils develop. (c) Close to the crack tip, the fibril spacing is regular. (Adapted from Kumar S. Arun, “Experimental Study of Crazing in Polystyrene,” MTech Thesis, IIT Kanpur, 2010; collage courtesy of Professor Sumit Basu, Indian Institute of Technology Kanpur.)

nucleate in three-dimensional cavities. Another example of such a subsurface craze in a fatigued isotactic polypropylene (iPP) part is shown in Figure 15.9.6.

15.7 *Multiaxial Yield In engineering, the strength of a material is used to ensure that the stresses in a body do not cause component failure; this requires the characterization of failure criteria. Failure may broadly be classified into two categories: (i) Functional failure, in which the part fails to meet certain specifications without the material failing. For example, the deformation at a point may exceed specified limits or the part may buckle. Such failure can be avoided by a careful analysis followed by a suitable design. (ii) Material failure, in which the material may deform into the plastic region or may fracture; suitable criteria must first be established to prevent such material failure. Theories for such criteria are called theories of failure or yield criteria. Thus far, the bulk of the discussion on mechanical behavior of plastics has been based on data generated by tensile tests. The discussion in Section 15.2.9 shows that the yield behavior of polycarbonate in biaxial stretching is very different from that in the one-dimensional tensile test. In general, in a stressed part, all the stress components are present at any given point in a material.

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Because multiaxial tests are very difficult to perform, most information on failure is obtained by simple tests, such as the tensile and torsion tests, in which only a few of the stress components are nonzero. Theories of failure or yield criteria are required for predicting failure for the general stress field on the basis of information obtained in simple tensile or torsion tests. The phenomenon of yield or failure of plastics is very complex and is much less understood than the failure mechanisms of metals. A discussion of multiaxial yield of plastics will be preceded by a discussion of theories of failure originally developed for metals and ceramics. For this purpose, it is useful to use the three orthogonal principal stresses, 𝜎 1 , 𝜎 2 , and 𝜎 3 , to describe the state of stress in a body. In this space, the surface separating the safe states of stress from the yielded, or failed, states is called the yield surface. 15.7.1

Maximum Principal Stress Theory

According to this theory the material will occur, or the material will undergo elastic failure, when a principal stress reaches the yield point in tension for ductile materials or the ultimate strength for brittle materials; failure occurs when |𝜎 max | ≥ 𝜎 0 . This theory, attributed to Rankine, gives reasonably good results for brittle materials. A brittle material subjected to sufficient shear in a torsion test will fracture along a helix that makes an angle of about 𝜋 ∕4 with the axis of the applied torque, thereby indicating that the fracture is caused by the principal tensile stress. Let 𝜏 1 be the yield shear stress in a torsion test. Then, since pure shear stress 𝜏 1 is equivalent to a normal stress of the same magnitude, 𝜏 1 , on a plane at an angle 𝜋 ∕4 with the shear, failure should occur when 𝜏 1 ≥ 𝜎 0 if this theory is correct. Rankine’s theory does not give good results for ductile materials, which fail by plastic yield. Since the material will not fail as long as each of the three principal stresses is less than or equal to 𝜎 0 , the yield surface in stress space is a cube with sides of length 2𝜎 0 . Failure occurs for any state of stress outside this cube, while all the “safe” states are represented by points inside the cube. For a two-dimensional stress distribution, with 𝜎 3 = 0 say, the yield locus – the intersection of the yield surface with the plane 𝜎 3 = 0 – will be the square with sides 2𝜎 0 shown in Figure 15.7.1; the safe states of stress are represented by points inside this square. In this theory the yield strengths in tension and compression are assumed to be the same, which is known not to be the case for most materials. For example, brittle materials such as cast iron and cement are much stronger in compression than in tension. This phenomenon of the yield strengths in tension and compression not being the same is called the Bauschinger effect. 15.7.2

Maximum Shear Stress Theory

According to this theory, material failure occurs when the shear stress at a point reaches the maximum shear stress at failure, 𝜏 0 , in a simple tensile test. That is, failure occurs when 𝜏 max ≥ 𝜏 0 = 𝜎 0 ∕2. Since the maximum shear stress is the maximum of |(𝜎 1 − 𝜎 2 )∕2|, |(𝜎 2 − 𝜎 3 )∕2|, and |(𝜎 3 − 𝜎 1 )∕2|, it follows that failure occurs if |(𝜎 1 − 𝜎 2 )∕2| ≥ 𝜎 0 ∕2, |(𝜎 2 − 𝜎 3 )∕2| ≥ 𝜎 0 ∕2, or |(𝜎 3 − 𝜎 1 )∕2| ≥ 𝜎 0 ∕2. Thus the yield surface is the cylindrical surface formed by the intersection of the planes |(𝜎 1 − 𝜎 2 )| = 𝜎 0 , |(𝜎 2 − 𝜎 3 )| = 𝜎 0 , and |(𝜎 3 − 𝜎 1 )| = 𝜎 0 . Any point inside this surface, that is values of 𝜎 1 , 𝜎 2 , and 𝜎 3 such that |(𝜎 1 − 𝜎 2 )| < 𝜎 0 , |(𝜎 2 − 𝜎 3 )| < 𝜎 0 , and |(𝜎 3 − 𝜎 1 )| < 𝜎 0 , represents a state of stress for which failure will not occur. This theory is also referred to as the maximum stress difference theory, Coulomb’s theory, and Guest’s law. The maximum shear stress theory gives reasonably good results for ductile materials. The yield locus for a two-dimensional stress distribution, say with 𝜎 3 = 0, is given by the lines |(𝜎 1 − 𝜎 2 )| = 𝜎 0 , |𝜎 2 | = 𝜎 0 , and |𝜎 1 | = 𝜎 0 , which are shown in Figure 15.7.2. Any point within the shaded region is a safe point.

Mechanical Behavior of Plastics

σ2

A (σ0 , σ0)

σ1

O O

Figure 15.7.1 Yield locus in 𝜎 1 - 𝜎 2 space for the maximum principal stress failure theory for a two-dimensional stress distribution with 𝜎 3 ≡ 0.

σ2

A (σ 0 , σ 0)

O

σ1

Figure 15.7.2 Yield locus in 𝜎 1 - 𝜎 2 space for the maximum stress failure theory for a two-dimensional stress distribution with 𝜎 3 ≡ 0.

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In a torsion test, let the yield shear stress be 𝜏 1 . Then, if this theory were correct, failure would occur when 𝜏 1 = 𝜎 0 ∕2. However, tests have shown that on an average 𝜏 1 = 0.57 𝜎 0 . Thus the shear stress theory is safer by about (0.57 − 0.5)∕0.5, that is, by about 14%. Consider the case 𝜎 1 = 𝜎 2 = 𝜎 3 , which is the case of hydrostatic pressure or tension, for which the shear stress theory would not predict any failure, no matter how large this hydrostatic pressure is (Why?). Tests have shown that ductile materials can withstand very high hydrostatic pressures without undergoing failure. However, ultimately, for extremely high hydrostatic pressures the material, though ordinarily ductile, undergoes brittle fracture. This theory also does not account for the Bauschinger effect. 15.7.3

Maximum Principle Strain Theory

This theory, attributed to Saint Venant, predicts that failure occurs when the maximum normal strain exceeds the normal strain at failure in a simple tensile test. The normal strains are given by expressions of the form 𝜀1 = [𝜎 1 − v(𝜎 2 + 𝜎 3 )] ∕E; so that with 𝜀0 = 𝜎 0 ∕E it follows that the yield surface and the safe region are given by | 𝜎1 − 𝜈 (𝜎2 + 𝜎3 )| ≤ 𝜎0 | 𝜎2 − 𝜈 (𝜎3 + 𝜎1 )| ≤ 𝜎0 | 𝜎3 − 𝜈 (𝜎1 + 𝜎2 )| ≤ 𝜎0

(15.7.1)

so that the shape of the yield locus depends on the value of Poisson’s ratio. For a two-dimensional stress distribution with 𝜎 3 = 0, the yield locus and the safe region are given by | 𝜎 1 − 𝜈𝜎 2 | ≤ 𝜎 0 , | 𝜎 2 − 𝜈𝜎 1 | ≤ 𝜎 0 , and |𝜈 (𝜎 1 + 𝜎 2 )| ≤ 𝜎 0 . All the lines represented by these equations are shown in Figure 15.7.3 for 𝜈 = 0.25. The coordinates of A are (𝜎 0 ∕[1 − 𝜈 ], 𝜎 0 ∕[1 − 𝜈 ]) and those of A ′ are (𝜎 0 ∕2𝜈 , 𝜎 0 ∕2𝜈 ), so that OA∕OA ′ = 2𝜈∕(1 − 2𝜈 ). From this it follows that OA∕OA ′ ≤ 1 if 𝜈 ≤ 1∕3. For 𝜈 > 1∕3, OA∕OA ′ > 1, so that the lines 𝜎 2 = − 𝜎 1 ± 𝜎 0 ∕𝜈 will intersect the rhombus ABCD, which is the yield locus for 𝜈 ≤ 1∕3. Figure 15.7.4a shows the yield locus for 𝜈 = 1∕3, for which the lines EF and GH just touch the points A and B on the rhombus. For 𝜈 > 1∕3, the lines EF and GH intersect the rhombus, as shown for 𝜈 = 1∕2 in Figure 15.7.4b. For brittle materials this theory gives good results, which are better than those predicted by the maximum principal stress theory. However, this theory also does not account for the Bauschinger effect. 15.7.4

Strain Energy of Distortion Theory

This theory has grown out of the analytical work of Huber, von Mises, and Hencky, and the experimental work of Bridgman. According to this theory failure occurs when Wd ≥ Wd0 , that is when ( 𝜎1 − 𝜎2 ) 2 + ( 𝜎2 − 𝜎3 ) 2 + ( 𝜎3 − 𝜎1 ) 2 ≥ 2 𝜎0 2

(15.7.2)

It works well for ductile materials and gives better results than the maximum shear stress theory. For a two-dimensional stress distribution, with 𝜎 3 = 0 say, the yield locus and the safe region are given by

𝜎12 + 𝜎22 − 𝜎1 𝜎2 ≤ 𝜎0 2 This region is shown in Figure 15.7.5.

(15.7.3)

Mechanical Behavior of Plastics

E

σ2 = – σ1 + σ0 / ν

A'

σ2 σ2 = σ1 / ν + σ0

F

A

D

σ2 = ν σ1 + σ0 σ1

O

B

σ2 = ν σ1 – σ0

C

σ2 = σ1 / ν – σ0

G

σ2 = – σ1 – σ0 / ν H Figure 15.7.3 The rhombus ABCD is the yield locus in 𝜎 1 - 𝜎 2 space, for a two-dimensional stress distribution with 𝜎 3 ≡ 0, for the maximum principle strain failure theory for 𝜈 = 0.25. The lines EF and GH do not intersect the rhombus so long as 𝜈 < 1∕3.

An alternative equivalent theory is cast in terms of octahedral stresses, the stresses on the eight planes equally inclined to the principal stress axes. The normal stress, 𝜎 OCT , and the shear stress, 𝜏 OCT , on these planes are called the octahedral normal stress and the octahedral shear stress, respectively, and are given by 1 (15.7.4) ( 𝜎 + 𝜎2 + 𝜎3 ) 3 1 1 𝜏OCT = [(𝜎1 − 𝜎2 )2 + (𝜎2 − 𝜎3 )2 + (𝜎3 − 𝜎1 )2 ]1∕2 (15.7.5) 3 According to this theory failure occurs when 𝜏OCT ≥ 𝜏OCT0 , where 𝜏OCT0 = 0.47𝜎0 (Why?) is the octahedral shear stress at failure in a tensile test. A substitution of this failure criterion into Eq. 15.7.5 shows its equivalence to Eq. 15.7.4.

𝜎OCT =

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Introduction to Plastics Engineering

E

σ2

A

σ2

A'

A E

F

A'

D

F

D

σ1

O

O

B

G

G

σ1 B

H C

C

H (a)

(b)

Figure 15.7.4 (a) For 𝜈 = 1∕3 the lines EF and GH just touch the tips of the rhombus ABCD. (b) For 𝜈 < 1∕3 the lines EF and GH will intersect the rhombus ABCD, as shown in this figure for v = 0.25.

σ2

A (σ 0 , σ 0)

O

σ1

Figure 15.7.5 The yield locus in 𝜎 1 - 𝜎 2 space, for a two-dimensional stress distribution with 𝜎 3 ≡ 0, for the strain energy of distortion theory.

Mechanical Behavior of Plastics

15.7.5

Comparison of Failure Theories

In all the theories discussed so far, the yield strength of a material has been assumed to be the same in tension and compression. It is well known that this is not true. For most materials, the yield strength in compression is significantly higher than in tension. To this extent, all the theories described so far are in error. The yield loci for all the failure theories discussed here are shown together in Figure 15.7.6, in which the rhombus ABCD shows the failure locus for the maximum principal strain failure theory for 𝜈 = 0.25; the cross-hatched region corresponds to 𝜈 = 1∕2. This figure shows that for a two-dimensional stress distribution, the safe regions as predicted by the maximum shear stress theory and the strain energy of distortion theory are almost the same. Thus, even though the strain energy of distortion theory is closer to experimental results, the maximum shear stress theory is used for design purposes as it is simpler to use since the yield locus is defined by simple linear equations.

σ2

A

D

σ1

O

B

C Figure 15.7.6 Comparison of the yield loci for a two-dimensional stress distribution predicted by different theories of failure. The rhombus ABCD is the failure locus for the maximum principal strain failure theory for 𝜈 = 0.25; the shaded region corresponds to 𝜈 = 1∕2.

These theories of failure may also be assessed by comparing the strength in a torsion test predicted from the tensile strength with experimentally obtained values. Let the yield shear stress in a torsion and the yield tensile stress be 𝜏 1 and 𝜎 0 , respectively. The relation between 𝜏 1 and 𝜎 0 predicted by the

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different theories of failure is listed in Table 15.7.1, in which the second column gives the maximum utilizable strength as obtained from a tensile test. Thus for the maximum shear stress theory the entry in this column is 𝜎 0 ∕2 as this is the maximum shear stress 𝜏 0 in a tensile test for a tensile stress 𝜎 0 . Similarly, the maximum principal strain for such a test is 𝜀0 = 𝜎 0 ∕E, which is the entry in column 2. Table 15.7.1 Comparison of the predictions of four failure theories. Strength measured by tensile test

Strength measured by torsion test

Maximum principal strain theory

𝜎0 𝜎 0 ∕2 𝜀0 = 𝜎 0 ∕E

𝜏1 𝜏1 𝜀1 = (1 + 𝜈 ) 𝜏 1 ∕E

Strain energy of distortion theory

(1 + 𝜈 ) 𝜎 20 ∕3E

(1 + 𝜈 ) 𝜏 21 ∕E

Failure theory

Maximum principal stress theory Maximum shear stress theory

Relation between 𝜏 1 and 𝜎 0 predicted by failure theory

𝜏 1 = 𝜎0 𝜏 1 = 0.5 𝜎 0 𝜏 1 = 𝜎 0 ∕(1 + 𝜈 ) (𝜏 1 = 0.8 𝜎 0 for 𝜈 = 1∕4) 𝜏 1 = 0.577𝜎 0

The third column gives the maximum utilizable strength on the basis of the shear strength 𝜏 1 obtained in a torsion test. Thus, for example, in a torsion test with the shear stress being 𝜏 1 , the principal stresses are 𝜏 1 , −𝜏 1 , and 0, so that the maximum principal strain is 𝜀1 = [𝜏 1 − 𝜈 (−𝜏 1 + 0)]∕E = (1 + 𝜈 ) 𝜏 1 ∕E which is 1.25 𝜏 1 ∕E for v = 1∕4, the entry in column 3 corresponding to the maximum principal strain theory. For a failure theory to be correct, the corresponding entries in columns 2 and 3 should be equal. Obviously, if all the theories of failure were correct, then the relationship between 𝜏 1 and 𝜎 0 , given in the last column, would be the same for all the theories. The results of extensive experiments show that for ductile materials 𝜏 1 = (0.55 to 0.60) 𝜎 0 . Thus, for ductile materials, the maximum energy of distortion theory gives the best results. However, usually the maximum shear stress theory is used as it is simpler. The error that is introduced is small and leads to a safer design. For brittle materials, the maximum principal strain theory gives the best results. The energy of distortion theory also gives good results. Most materials exhibit the Bauschinger effect: the yield strengths for tension and compression are different. It is possible to construct theories of failure which account for this effect. One such theory is discussed in the next section. 15.7.6

Failure Theories for Plastics

In comparison to metals, the yield of plastics is strongly affected by hydrostatic pressure that for the triaxial stress state is given by p = − (𝜎 1 + 𝜎 2 + 𝜎 3 )∕3; the negative sign indicating that a positive p corresponds to compressive pressure. Several theories that account for such pressure effects have been proposed. The simplest of these assumes the von Mises yield criterion (Eq. 15.7.3) with the one-dimensional yield stress on the right-hand side of the equation being assumed to be a linear function of the pressure. For a two-dimensional stress distribution, the yield locus and the safe regions are assumed to have the form

𝜎12 + 𝜎22 − 𝜎1 𝜎2 ≤ [𝜎0 (1 + C0 p)]2 where the value of C0 is a characteristic of the material.

(15.7.6)

Mechanical Behavior of Plastics

Figure 15.7.7 shows the pressure-dependent elliptical shear yield locus, ABCD, for poly(methyl methacrylate), or PMMA, which accounts for failure, or yield, caused by shear effects. Notice that because of pressure effects, the ellipse is not centered at the origin, indicating that the material can withstand higher hydrostatic pressures than hydrostatic tension. However, because tensile stresses can cause failure due to crazing – the crazing locus is the line EAFCG in this figure – the safe region in which the material neither yields nor undergoes crazing is shown by the cross-hatched region AFCBA.

σ2 E D A CRAZING

F

SHEAR YIELDING

σ1

O C

G

B

PURE SHEAR σ1 = – σ2

Figure 15.7.7 Failure theory for plastics that accounts for the effects of hydrostatic pressure and crazing. (Adapted with permission from S.S. Sternstein and L. Ongchin, ACS Division of Polymer Chemistry Polymer Preprints, Volume 10, pp. 1117 – 1124, 1969.)

15.8 *Fracture All materials will have points of weakness at which stress can cause small cracks to nucleate. With increasing stress such cracks can grow resulting in catastrophic failure. The nucleation and growth of cracks fall under the discipline of fracture mechanics, the details of which are beyond the scope of this book. However, the sequel gives an elementary account of the rudiments of the subject.

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Section 5.11 discusses stress concentration – a predictable phenomenon in which the stress at a point can be much larger than the stress at nearby points. All materials have small internal flaws, stress concentrations at which can act as nucleation sites for cracks. Once a crack has formed, it does not necessarily advance, or propagate. The analysis of whether such a crack will grow forms the subject of linear elastic fracture mechanics. The simplest model problem analyzed is that of a crack being pulled apart by tensile stresses acting normal to the crack plane. Figure 15.8.1 shows a crack of length a, in a semi-infinite plate that, away from the crack, is subjected to a uniform tensile stress 𝜎 y = 𝜎 0 , normal to the crack plane.

σ0

y

r

θ O

a

P x

σ0 Figure 15.8.1 Geometry of an edge crack of length a in a semi-infinite plate subjected to a far-field uniform stress 𝜎 y = 𝜎 0 .

As shown in this figure, with the origin O of the coordinate system situated at the crack tip, the coordinates of a point P are (x, y) and (r, 𝜃 ) in rectangular and polar coordinates, respectively. It can be shown that the two-dimensional stresses 𝜎 x , 𝜎 y , and 𝜏 xy with respect to the rectangular, (x, y) coordinate system are given in terms of the polar coordinates, (r, 𝜃 ), by ( ) K 3𝜃 𝜃 𝜃 𝜎x = √ cos 1 − sin sin +··· 2 2 2 2𝜋 r ( ) K 3𝜃 𝜃 𝜃 𝜎y = √ cos 1 + sin sin +··· 2 2 2 2𝜋 r K

𝜏xy = √

𝜃

2

sin

𝜃

2

cos

3𝜃 +··· 2

(15.8.1) 2𝜋 r in which the higher order terms are negligible close to the crack tip, say for r < 0.1a. These expressions are only valid close to the crack tip. At large distances from the crack, r >> a, where the material does not experience the presence of the crack, the stress field is described by 𝜎 x = 0, 𝜎 y = 𝜎 0 , and 𝜏 xy = 0. In these equations the dependence of the stresses on r and 𝜃 is explicitly shown, and K is only a function of the far-field stress 𝜎 0 , the crack length a, and the specimen geometry (in the present discussion a cos

Mechanical Behavior of Plastics

semi-infinite half space with an edge crack). K is a measure of the overall stress intensity and is therefore called the stress intensity factor; it plays a critical role in fracture mechanics. In fracture mechanics, crack propagation analyses are based on the assumption that a crack will not grow until the stress intensity factor reaches a critical value, Kc , beyond which the crack will grow. The critical stress intensity factor Kc is a material property that is a measure for the fracture toughness of the material; it has dimensions of MN m−1.5 . For the edge crack shown in Figure 15.7.1, it can be shown that √ K = 1.12 𝜋 a 𝜎0 (15.8.2) The stress at failure, 𝜎 f , is then the stress at which K attains Kc , so that Kc 𝜎f = (15.8.3) √ 1.12 𝜋 a The older approach to design a part for strength, or failure, uses the types of yield criteria discussed in Section 15.7. The newer, fracture mechanics, fracture-toughness-based approaches use the type of failure stress shown in Eq. 15.8.3; in addition to requiring the fracture toughness of the material, this approach requires an estimate for the crack length a. This approach is particularly useful for understanding fatigue-crack growth (Section 15.9.2). This fracture-toughness model was developed for metals for which, for practical purposes, the material response is time-independent. Although this model is also applied to plastics, in which material behavior is both time and temperature dependent, the results of tests and analyses must be interpreted with care.

15.9 Fatigue Like most materials, plastics also undergo fatigue, a phenomenon in which a material fails under cyclic loads at load levels that would not cause failure under a constant load. A familiar example is that of a steel paper clip: straightening one of the legs at the curved end does not result in failure. But if the leg is straightened and bent back to its original shape several times, the paper clip fails. The heating of the bent material, a sure measure of the irreversible work done during this process, is large enough to be felt by finger touch. Thus, a part repeatedly loaded to stresses below the failure level can fail from fatigue. In a sense a load cycle induces a certain level of “damage” that accumulates with continuing load cycles. The nature of this damage is complex: in addition, the nature of the loading cycle – shape of the stress-time curve – the triaxial stress state affects fatigue life. Each cycle involves irreversible deformations that result in heat generation, which is particularly important for plastics as increase in temperature can result in rapid creep deformation. Fatigue of materials can be approached in two ways. In the older, traditional method the test specimen is subjected to a cyclic stress, or load, S, and the test is continued till the specimen fails, say at N cycles. Data at different stress levels is then used to construct the S-N, or stress versus cycles-to-failure, curve for the material. This simple, phenomenological approach to fatigue is subject to all the complexities described in the previous paragraph. In the second approach, fatigue is looked upon as damage accumulation resulting from cyclic crack propagation. It involves measuring of the crack growth per cycle as a function of the stress intensity factor, a concept defined in fracture mechanics (Section 1.8). This mechanistic approach provides an insight into the mechanism of fatigue. This approach, too, is subject to the complexities mentioned in the previous two paragraphs.

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15.9.1

The S-N Curve

In the simplest fatigue test, a smooth tensile test specimen is subjected to a load cycle in which either the stress or the deformation undergoes a sinusoidal variation between two levels. In a load-controlled (stress) test, the stress is cycled between maximum stress, 𝜎 max , and a minimum stress, 𝜎 min , in a sinusoidal manner at a frequency n. This results in a sinusoidal stress variation 𝜎 = 𝜎 mean + 𝜎 a sin [(2𝜋∕n)t], where 𝜎 mean = (𝜎 max + 𝜎 min )∕2 is the mean stress and 𝜎 a = (𝜎 max − 𝜎 min )∕2 is the stress amplitude. Other terms used in fatigue literature are the stress ratio R = 𝜎 min ∕𝜎 max , and the stress range Δ𝜎 = 𝜎 max − 𝜎 min . Positive values of R correspond to the stresses being positive throughout the fatigue cycle; R = 0 corresponds to 𝜎 min = 0. In fatigue tests 𝜎 max is always positive, so for negative values of R the minimum stress 𝜎 min is negative, or compressive. For 𝜎 min = − 𝜎 max , or R = −1, the stress cycles sinusoidally with a stress amplitude 𝜎 max about a zero mean stress. For a tensile fatigue test care has to be taken in choosing an appropriate value for a compressive minimum stress to prevent the specimen from buckling. But this restriction does not apply to flexural fatigue tests. Even for the one-dimensional stress fatigue tests considered here, mapping fatigue behavior involves a very large number of tests: For a fixed R, the number of cycles-to-failure has to be determined for a large number of values of 𝜎 max . And because of the inherent scatter in fatigue data, a large number of tests have to be conducted for each test condition. Also, although a test with sinusoidal stress variation is the easiest to do, it does not necessarily represent the stress variation that a part is subjected to. So, an important aspect of fatigue is the prediction of fatigue life of a part using data obtained from tensile or flexural (bending) fatigue tests. Based on experiments, the most important parameter affecting fatigue life is the mean stress. In its simplest form, fatigue data are obtained in the form of the number of cycles-to-failure at an applied mean stress, S. The data are then reported as plots of the mean stress versus the number of cycles-to-failure, N; since the cycles-to-failure are very large, on the order of 107 , for low stress levels, N is plotted on a logarithmic scale. Such plots are called S-N diagrams, which typically have the shape shown in Figure 15.9.1.

S-N CURVE

σY STRESS

404

ENDURANCE LIMIT

σE

0

5

10

FATIGUE LIFE (log N ) Figure 15.9.1 S-N fatigue-life curve showing definition of the endurance limit.

Mechanical Behavior of Plastics

At high stress levels a part fails at a low number of cycles. With decreasing stress levels, the part fails at increasingly larger number of cycles. For low enough stresses a plateau is reached at which the part will not fail, no matter how large the number of cycles. This stress level, 𝜎 E , is called the endurance limit of the material. For stresses below the endurance limit, fatigue will not cause the part to fail. Figure 15.9.2 shows the S-N curves for many high-performance, unfilled and filled thermoplastics. The data were obtained from 5-Hz, tension-tension fatigue tests a at a minimum stress of 𝜎 min = 0 at different values of the maximum stress 𝜎 max . Notice that the only plastics in this group that show some semblance of endurance-like behavior are PBT and 30-GF-PBT.

150 5 Hz 30-GF-PEI 40-GF-PC 30-GF-PBT 30-GF-M-PPO 20-GF-PC

STRESS (MPa)

100

50

PBT M-PPO PEI ABS PC

0 10 2

10 3

10 4

10 5

10 6

10 7

CYCLES TO FAILURE Figure 15.9.2 S-N fatigue curves for several unfilled and filled higher performance plastics. (Adapted with permission from Figure 8.1 in “Structural Analysis of Thermoplastic Components,” by G.G. Trantina and R.P. Nimmer, McGraw-Hill, Inc., 1994)

Figure 15.9.3 shows the S-N curves for many plastics. The data are from 30-Hz fatigue tests a at a mean stress of 𝜎 mean = 0 at different values of the stress amplitude 𝜎 a . Notice the endurance-like behavior of many plastics such as PTFE, PE, PP, PMMA, M-PPO, polystyrene, and epoxy. The fatigue lives of PET and dry nylon (PA) do not exhibit this type of behavior; instead, the life continues to decrease linearly with the cycles-to-failure on a logarithmic scale.

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25 EPOXY

30 Hz

STRESS AMPLITUDE (MPa)

406

20 PET

M-PPO

15

PMMA POLYSTYRENE

10 PP PA (Dry)

PE

5 PTFE

0 10 2

10 3

10 4

10 5

10 6

10 7

CYCLES TO FAILURE Figure 15.9.3 S-N fatigue curves for several unfilled plastics: Data obtained from 30-Hz fatigue tests at a mean stress of 𝜎 mean = 0 at different values of the stress amplitude 𝜎 a . (Adapted with permission from M.N. Riddell, Plastics Engineering, Vol. 30, No. 4, p. 71, 1974.)

While S-N curves provide a first level data for expected fatigue life in one-dimensional loading, scaling this information for fatigue-life expectancy requires mechanics models for fatigue failure.

15.9.2

*Fatigue-Crack Propagation

In specimens without flaws fatigue can nucleate cracks that then propagate in successive cycles, finally resulting in failure. But all parts have internal flaws – such as voids, for example – or external flaws – such as knit lines in molded parts (Section 17.3.1), which are vestiges of regions where flow fronts meet. During fatigue loading cracks propagate from such flaws. This process is simulated through fatigue tests on specimens that are carefully notched. Crack nucleation and propagation fall under the discipline of fracture mechanics, which is beyond the scope of this book. Once initiated, a crack does not necessarily advance, or propagate, in every fatigue cycle; instead it my jump after a series of cycle to leave striated, “chevron” structures on the crack surfaces. In plastics the crack tip may craze, so that crack propagation may be preceded by local crazing. The fatigue process may be divided into two parts. The number of cycles, Ni , to initiate and grow a crack to a size at which the threshold stress intensity is reached. And the number of cycles, Np , to grow the crack under subcritical conditions from the threshold size to the critical size at which failure is imminent.

Mechanical Behavior of Plastics

The total number of cycles-to-failure is then N = Ni + Np . In a component with an initial defect (crack), the number of cycles-to-failure is essentially Np . In properly molded parts made of high-performance resins, fatigue life is dominated by the initiation stage: Figure 15.9.4 shows the crack initiation site and the cyclic crack growth in a molded acetal ASTM D638 bar subjected to tensile fatigue. The initiation of the penny-shaped crack occurred at a void or other inhomogeneity and propagated subcritically under cyclic fatigue to a critical length prior to sudden failure. The specimen failed at about N = 106 cycles, of which Np was only about 186,000 cycles, so that failure was mainly dominated by the initiation stage with Ni ≃ 4.8 × 106 . Thus the initiation of the penny-shaped crack at the defect site governed the total life of the specimen.

CRITICAL RADIUS

INITIATION SITE

100 µm

Figure 15.9.4 SEM showing fracture surface in a polyacetal resin specimen subjected to tensile fatigue. The penny-shaped crack initiated at a void or other inhomogeneity at its center. (Adapted with permission from A.J. Lesser, “Fatigue Behavior of Polymers,” Encyclopedia of Polymer Science and Technology, John Wiley & Sons, 2002; original micrograph courtesy of Professor A.J. Lesser, University of Massachusetts, Amherst.)

Empirically it has been shown that over a large portion of fatigue-crack growth process the rate of crack growth, da∕dN, is governed by the Paris equation da = A ΔK n dN

(15.9.1)

da = AY n (Δ𝜎 ) n a n∕2 dN

(15.9.2)

√ in which, in terms of the cyclic stress level Δ𝜎 , ΔK = Y Δ𝜎 a is the cyclic range of the stress intensity factor, A and n are material constants, and Y is a factor that depends on the crack and structure geometry. Using the definition of ΔK, the Paris equation can be written as

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Assuming that the geometric factor Y does not change, an integration of this equation gives the number of cycles-to-failure, Nf , for a crack of initial length a0 to grow to the critical size af as [ ] 2 1 1 Nf = − , n≠2 (15.9.3) (n − 2) AY n (Δ𝜎 ) n a0 (n−2)∕2 af (n−2)∕2 The Paris equation was based on fatigue-crack growth data for metals, for which the linear fracture mechanics discussed in Section 15.8 is valid. But it is also used for fatigue-crack growth in plastics that, in contrast to metals, exhibit strong time- and temperature-dependent material behavior. Figure 15.9.5 shows a log-log plot of the crack growth rate, da∕dN, versus the cyclic the stress intensity range, ΔK, for several amorphous and semicrystalline plastics.

ΔK (103 psi in1/2 ) 0.5

1.0

5.0

2.0

10.0

10 0 10 2 ABS

10 1 PSO

10 2

10 3

PMMA

10 4 10 3

PS

10 5

PA 6,6

da /dN (in /cycle)

da /dN (mm/cycle)

408

10 4 M-PPO

10 6 PC

10 5 PVC

10 7 10 6 0.5

1.0

2.0

5.0

10.0

ΔK (MPa m1/2 ) Figure 15.9.5 Fatigue-crack propagation rates for amorphous and semicrystalline plastics. (Adapted with permission from R.W. Hertzberg, J.A. Manson, and M.D. Skibo, Polymer Engineering and Science, Vol. 15, p. 252 – 260, 1975.)

Mechanical Behavior of Plastics

Room-temperature fatigue tests conducted on isotactic polypropylene (iPP) bars cut from an iPP plaque – at a 2-Hz sinusoidal waveform, a constant peak- and mean-stress stress levels of approximately 70 and 35% of the yield stress, respectively – showed that an early manifestation of fatigue damage was the appearance of white lines with shear bands at an angle of 60 – 70° to the applied stress. This whitening, caused by light scattered by the evolving defects, first appeared between 1,000 and 10,000 cycles, suggesting that defects initiate well before 10,000 cycles. Scanning electron microscope (SEM) images of iPP samples fatigued to 10 6 cycles showed many defects with diameters in the 10 – 100 μm range and thicknesses up to about 1 μm. Many of the mature crazes appeared to have evolved from smaller crazes by coalescence. These SEMs showed that the crazes initiate and grow through both the bulk of the spherulites and the spherulite boundaries. Figure 15.9.6 shows one such craze with finely drawn fibrils between the defect surfaces, confirming that fatigue damage evolves through craze formation and growth. Note the arrows that show the direction of the imposed oscillatory tensile stress.

10 µm

Figure 15.9.6 SEM showing the detailed structure of a craze in an iPP sample fatigued to 10 6 cycles. The arrows show the direction of the imposed oscillatory tensile stress. (Adapted with permission from N.A. Jones and A.J. Lesser, Journal of Polymer Science: Part B: Polymer Physics, Vol. 36, 2751 – 2760, 1998; original micrograph courtesy of Professor A.J. Lesser, University of Massachusetts, Amherst.)

While transmission optical micrographs for specimens fatigued for less than 103 cycles show no crazes, those for greater than 10 5 cycles show many crazes with dimensions of about 6 μm. The series of micrographs in Figure 15.9.7 show the progression of fatigue damage from 10 5 to 10 6 cycles. As the number of fatigue cycles increases, the average length of the crazes increases, and the number of individual crazes decreases. There are also a number of distinctive features about the arrangement of these crazes. Figure 15.9.7f shows a process zone consisting of several smaller crazes surrounding a larger craze. A process zone consisting of yielded or damaged material surrounding the tip of a crack, commonly reported in the fracture of ductile metals, is also common in many polymeric materials. This micrograph shows that the crazes in iPP also evolve with an associated process zone, indicating that energy has to be expended to create the process zone in addition to the fibrillated material within the craze. Figure 15.9.7e shows that the crazes often form in cascades. The micromorphology of fatigue-crack fracture surfaces in plastics is quite complex. The SEM collage in Figure 15.9.8 shows the range of textures on the fracture surface of a fatigued aliphatic polyketone terpolymer. Near the initiation site, where crack nucleation subcritical growth occurs, the fracture surface exhibits greater texture. During subcritical growth the surface has a smoother, more uniform texture. The

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crack grows relatively slowly till it reaches the critical radius, after which the crack progresses very rapidly, resulting in failure. Beyond the critical radius the fracture surfaces is significantly rougher.

100,000

200,000

100 µm

100 µm (a)

(b)

400,000

600,000

100 µm

100 µm (c)

(d)

1,000,000

1,000,000 CRAZE

100 µm (e)

100 µm (f)

Figure 15.9.7 Optical micrographs of fatigue damage in iPP tensile bars showing the progression of fatigue damage with increasing numbers of fatigue cycles, shown on the upper left-hand side of each image. All the scale bars represent 100 μm. (a – d) Evolution of damage in the form of crazes a fatigue cycle of 105 , 2 × 105, 4 × 105 , and 4 × 105 , respectively. (e) Craze cascades. (d) Process zone consisting of several smaller crazes surrounding a larger craze. (Adapted with permission from N.A. Jones and A.J. Lesser, Journal of Polymer Science: Part B: Polymer Physics, Vol. 36, 2751 – 2760, 1998; original micrographs courtesy of Professor A.J. Lesser, University of Massachusetts, Amherst.)

Mechanical Behavior of Plastics

INITIATION SITE & SUBCRITICAL CRACK GROWTH

TEXTURE OF REMAINING LIGAMENTS

CRITICAL RADIUS

1 mm

Figure 15.9.8 SEM collage showing fatigue-induced fracture surface in an aliphatic polyketone terpolymer. Note the three phases of crack growth – crack initiation site and sub critical crack growth, the critical radius, and the surface texture associated with the remaining ligaments. (Adapted with permission from A.J. Lesser, “Fatigue Behavior of Polymers,” Encyclopedia of Polymer Science and Technology, John Wiley & Sons, 2002; original micrograph courtesy of Professor A.J. Lesser, University of Massachusetts, Amherst.)

High stresses in the vicinity of a crack tip cause significant damage below the crack surface. The energy absorbed in creating this damage helps in the high toughness of many plastics. The subsurface damage in the specimen with the fracture crack surface in Figure 15.9.8 is shown in the SEM in Figure 15.9.9. The damage is in the form of craze arrays having the highest density near the surface. FRACTURE SURFACE

DAMAGE BELOW FRACTURE SURFACE

10 µm Figure 15.9.9 SEM showing subsurface fatigue-induced damage in an aliphatic polyketone terpolymer. Note that the highest density of crazes occurs near the crack surface. (Adapted with permission from A.J. Lesser, “Fatigue Behavior of Polymers,” Encyclopedia of Polymer Science and Technology, John Wiley & Sons, 2002; original micrograph courtesy of Professor A.J. Lesser, University of Massachusetts, Amherst.)

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15.9.3

The Role of Hysteretic Heating

In contrast to metals and ceramics, the mechanical properties of plastics are very sensitive to the temperature in the normal temperature range in which plastic parts operate. Hysteretic heating during a fatigue test lowers the elastic modulus, which in turn results in larger and larger deformations in successive fatigue cycles. To avoid such hysteretic heating effects, fatigue tests for plastics therefore have to be conducted at relatively low frequencies. Because plastics have low thermal conductivities and diffusivities, for the same stress levels, or the same rate of hysteretic heating, a thinner part will run cooler than a thicker part. Also, for the same stress levels, higher test frequencies will result in higher hysteretic heat generation, resulting in lower fatigue lives.

15.10 Impact Loading Thermoplastics are rate sensitive materials, the qualitative nature of the failure of which depends on the strain rate – ductile failure at low strain rates, but brittle failure at high strain rates. The nature of failure also depends on the temperature: for a fixed strain rate, at which the material exhibits ductile failure at high temperatures, the failure mode changes to brittle failure at lower temperatures. Thus, the failure mode depends in a complex way on both the strain rate and the temperature. A rate-dependent material pulled in tension can fail in a ductile manner at high strains or abruptly in a brittle manner without undergoing plastic deformations. For instance, a thin rolled cylinder of Silly Putty – a play-dough-like material made of dimethyl siloxane, boric acid, silica, polydimethylsiloxane, decamethyl cyclopentasiloxane, glycerine, and titanium dioxide – will “flow,” deform continuously, when pulled apart slowly at the two ends, but will fail abruptly in a “brittle” manner when pulled rapidly. Rolled into a ball, this material bounces elastically just like a rubber ball. But left on a table it flows outward, albeit slowly. The rate sensitivity of plastics poses a major problem for mechanical design of thermoplastic parts because current mechanical design paradigms evolved for metals use, which for most applications are rate-independent materials. A rational approach to design would be based on analyses in which the constitutive equations for the material account both for rate and temperature effects. However, such constitutive equations are not available. As such, rate effects have to be treated in an ad hoc manner. The term “impact loading” is used in plastics for accounting for rate-dependent effects. The behavior of plastics under high loading rates is characterized by special impact tests – such as the Charpy and dart impact tests – which really do not characterize material behavior. Rather, they are component tests based on specific geometries. They do not determine fundamental material characteristics; they provide a qualitative understanding of how the material will behave in a specific geometry, at a given temperature, at a predetermined loading rate. 15.10.1

Instrumented Impact Test

One often used impact test in the plastics industry is an instrumented, high-speed multiaxial puncture test, called the Dynatup® impact test. In this test, schematically shown in Figure 15.10.1, an instrumented tup with a hemispherical head attached to a predetermined weight is allowed to drop from a predetermined height onto a horizontally mounted test specimen. Sensors establish the instant of impact and the impact velocity, from which the energy of impact can be calculated. Instrumentation on the tup continuously

Mechanical Behavior of Plastics

measures the tup displacement and force as a function of time. This load-displacement information can be used to calculate the maximum force, the energy at maximum force, and the total energy to break. HEMISPHERICAL INDENTER

METAL CLAMP

• R

R

PLASTIC DISK

r

Figure 15.10.1 Schematic drawing for an instrumented impact test. An instrumented tup is allowed to fall from a predetermined height onto a plastic disk held in place by an annular circular metal clamp.

Normally, either 102 × 102-mm (4 × 4-in) square plaques or 102-mm diameter disks are used for the tests. The specimens are clamped in place using pneumatic pressure. The chamber in which the specimen is clamped can be cooled to low temperatures in which a pre-cooled specimen can be placed to simulate impacts at low temperatures. Figure 15.10.2 shows the result of tests on 3-mm-thick, 60 × 60-mm polypropylene plaques that were impacted by a 12.7-mm diameter hemispherical tup. The specimen in Figure 15.10.2a failed in a ductile

(a)

(b)

Figure 15.10.2 Failure modes of two 3-mm-thick polypropylene plaques. (a) Ductile failure after substantial deformation. (b) Brittle failure with negligible plastic deformation. (Photos courtesy of Instron.)

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mode with substantial deformation prior to failure. The specimen in Figure 15.10.2b failed in a brittle manner with very little plastic deformation. Both tests were done at room temperature; the different failure modes reflect differences in the composition of the disk materials. The differences between ductile and brittle failure modes show up in the impact-load (force) versus time traces generated by the instrumented tup; such differences are shown by the two curves for ductile and brittle materials in Figure 15.10.3. Such data are used for estimating the impact energy absorbed by the impact.

60 50

BRITTLE FAILURE

40

FORCE (kN)

414

DUCTILE FAILURE

30 20 10 0 –10 0

0.5

1.0

1.5

TIME (ms) Figure 15.10.3 Variation of tup impact force with time. Full and dashed lines are traces for the tup impact force for ductile and brittle materials respectively. (Adapted from chart courtesy of Instron.)

Note that both the ductile and brittle failure modes in two samples of polypropylene, shown in Figure 15.10.2, occurred in room temperature tests. However, a material that fails in a ductile manner at room temperature can fail in a brittle mode at low temperatures. Depending on the test type, his ductile-brittle transition can occur over a range of temperatures. Again, this transition is not a material property; it depends on the part geometry, the stress state, and the temperature. As mentioned earlier, such impact tests do not measure fundamental properties. Rather they measure the performance of the material in specified geometries, and can provide guidance for choosing plastics for a specific application. A knowledge of the large-deformation behavior of a plastic – such as that shown in Figure 15.2.3 – can be used to simulate the type of behavior shown in Figure 15.10.2a. Figure 15.10.4 shows the result of one such numerical simulation that accurately captured the deformation during the Dynatup test of a plastic. 15.10.2

Ductile-Brittle Transition

Since the same plastic can fail in both ductile and brittle failure modes, an understanding of the mode in which a part will fail is important. This change in the failure mode from ductile to brittle is known to

Mechanical Behavior of Plastics

δ (a)

δ = 95. mm (0.375 in) (b)

δ = 159. mm (0.625 in) (c) Figure 15.10.4 Simulation of the deformations in an impact test. (a) Test geometry; 𝛿 is the vertical displacement of the lower plaque surface. (b, c) deformed shapes at two displacements. (Adapted with permission from R.P. Nimmer, Polymer Engineering and Science, Vol. 27, p. 263 – 270, 1987.)

depend on the both the temperature and the strain rate; lower temperatures and higher strain rates favor brittle failure. The temperature versus extension rate plot schematically shown in Figure 15.10.5 divides the failure behavior of PVC in tensile tests into four regimes: (i) In the rigid-brittle regime (lower right-hand corner) PVC fails in a brittle mode. The upper curved part of this region separates this behavior from the two ductile failure regimes of rigid ductile failure, so that this boundary defines the brittle-to-ductile transition. Clearly, in this regime, the higher the extension rate the higher the ductile-to-brittle transition temperature. (ii, iii) Above the brittle failure regime are two regimes in which the material fails in a ductile manner.

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This ductile regime is further divided into two regions by a vertical line. On the left the extension rates are so low that the heat generated by extension dissipates outward so that the test is essentially at isothermal conditions; the material properties do not change during the test. To the right the extension rates are so high that the heat generated during extension does not have enough time to dissipate out so that the extension occurs under adiabatic conditions; the resulting rise in temperature causes material changes during the test. (iv) Above the ductile failure regime, the material is in a soft state.

TEMPERATURE (°C)

416

200 150

PVC

SOFT

100 50 0

HARD DUCTILE (Isothermal)

HARD DUCTILE (Adiabatic)

HARD BRITTLE

– 50

10 – 5 10 – 4 10 – 3 10 – 2 10 – 1

1

10

100 1000

EXTENSION RATE (25.4 mm/s, 1 in/s) Figure 15.10.5 Schematic temperature-extension-rate diagram delineating ductile-brittle transitions in PVC. (Adapted from P.I. Vincent, Polymer, Vol. 1, p. 425 – 444, 1960, with permission from Elsevier.)

Much of the work on characterizing the regimes of ductile and brittle failure of plastics is done by tensile tests at different displacement rates on specimens placed in a cooled chamber. However, this procedure does not account for failure in a multi-stress component environment. Also, the yield of plastics is known to be sensitive to hydrostatic pressure (Section 15.7.6). This is important because many loaded components – such as the root regions in a ribbed plate, where the rib is joined to the plate – are subject to multiaxial stress states, in which failure can occur at lower stresses than those obtained from a tensile test. Some approximate correlations regarding the relation of the ductile-brittle transition to the stress state are made in terms of the “hydrostatic stress,” essentially the octahedral normal stress 𝜎 OCT defined in Eq. 15.7.4, and the “deviatoric,” or shear stress, which is the octahedral shear stress 𝜏 OCT defined in Eq. 15.7.5. The correlation is expressed in terms of the stress ratio rDB = 𝜎 OCT ∕𝜏 OCT . States of stress with large positive values of rDB , which requires the hydrostatic stress 𝜎 OCT to be positive, are associated with higher ductile-brittle temperatures TDB , than lower values of rDB that are dominated by shear. Such effects are illustrated by the results for polycarbonate for the four test geometries shown in Figure 15.10.6. Instead of a single transition temperature, TDB , the ductile-brittle transition occurs over a temperature range that is test-dependent: Above the highest temperature in this range PC always fails in a ductile mode. As the temperature is reduced from this maximum value, the failure mode is still ductile, but with the failure strain decreasing with the temperature. Below the minimum temperature of this range PC always fails in a brittle manner. Figure 15.10.6a shows a transition temperature range of

Mechanical Behavior of Plastics

TDB ≈ − 65 to − 90°C for PC in a tensile test, for which rDB ≈ 0.41, indicating that in a one-dimensional stress state PC will undergo ductile failure above temperatures of − 65°C, continue to fail in a ductile mode as the temperature is reduced but at decreasing strains, and will fail in a brittle manner at temperatures below − 90°C. For the tensile test of the plane-strain grooved specimen shown in Figure 15.10.6b, in which the stress state has the two stress components 𝜎 x and 𝜎 y with a larger rDB ≈ 0.8, TDB ≈ − 40 to −70°C, so that PC has a higher transition temperature of − 40°C. The notched-beam specimen shown in Figure 15.10.6c has all three stress components near the notch, resulting in higher values of the stress ratio rDB , which depends on the exact geometry of the notch; for this test, for PC TDB ≈ −20 to − 40°C,

σx

P

σy

σy

σx

x

σx

x y

•

σx

P

•

y

P

P

σz σx σy

TDB ≈ – 65 to – 90°C ≈ – 85 to – 130°F

TDB ≈ – 40 to – 70°C ≈ – 40 to – 95°F

(a)

(b)

σy σx

σz

σz z

σr

P

•

y x

σθ

σθ

P

σr σz

z •

r

TDB ≈ – 20 to – 40°C ≈ – 4 to – 40°F

TDB ≈ – 70 to – 95°C ≈ – 95 to – 140°F

(c)

(d)

Figure 15.10.6 Approximate ductile-to-brittle failure transition temperatures for PC for several geometries: (a) tensile bar, (b) grooved tensile bar, (c) notched beam, and (d) puncture disk. (Adapted with permission from Figure 5.48 in “Structural Analysis of Thermoplastic Components,” by G.G. Trantina and R.P. Nimmer, McGraw-Hill, Inc., 1994).

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having the highest limiting transition temperature of −20°C. In a clamped circular specimen subjected to a central load (Figure 15.10.6d), the geometry used in Dynatup impact tests, the region near the load is essentially subjected to biaxial tensile stresses, 𝜎 x and 𝜎 y ; the third component 𝜎 z is relatively small under the load and rapidly drops to zero at the lower plaque surface. For this case, the ductile-brittle transition temperature range is TDB ≈ −70 to − 95°C, indicating a limiting transition temperature of −70°C, which is the lowest among the four tests. Depending on their failure modes, plastics may be divided into three groups: (i) Those that are brittle even when unnotched, (ii) those that are brittle when notched, and (iii) those in which the specimens do not break completely, even when sharply notched. The previous discussion has shown that the failure mode is temperature-dependent. Table 15.10.1 classifies the failure modes of many commonly used plastics as a function of the ambient temperature. (Actually, this table presents a simplified, consolidated version of the data in the referenced table in which the failure modes are divided into four groups.) ABS exhibits the most complex failure modes; the asterisk on B indicates that the failure mode could be of the B or C type. Table 15.10.1 Failure characteristics of thermoplastics as a function of temperature. Temperature °C (°F) −20 (− 4)

−10 (14)

0 (32)

10 (50)

20 (68)

30 (86)

40 (104)

50 (122)

Polystyrene

A

A

A

A

A

A

A

A

Poly(methyl methacrylate)

A

A

A

A

A

A

A

A

Glass-filled nylon (dry)

A

A

A

A

A

A

A

B

Poly(4-methyl pentene-1)

A

A

A

A

A

A

A

B

Polypropylene

A

A

A

A

B

B

B

B

Craze-resistant acrylic

A

A

A

A

B

B

B

B

Plastic

Poly(ethylene terephthalate)

B

B

B

B

B

B

B

B

Acetal (Polyoxymethylene)

B

B

B

B

B

B

B

B

Cellulose acetate butyrate

B

B

B

B

B

B

B

B

Nylon (dry)

B

B

B

B

B

B

B

B

Polysulfone

B

B

B

B

B

B

B

B

High-density polyethylene

B

B

B

B

B

B

B

B

ABS

B

B*

B*

B*

B*

B*

B*

C

Rigid polyvinyl chloride

B

B

B

B

B

B

C

C

Polyphenylene oxide

B

B

B

B

B

B

C

C

High-impact polypropylene

B

B

B

B

C

C

C

C

Polycarbonate

B

B

B

B

C

C

C

C

Nylon (wet)

B

B

B

C

C

C

C

C

Polytetrafluoroethylene

B

C

C

C

C

C

C

C

Low-density polyethylene

C

C

C

C

C

C

C

C

Note: see text for definition of *. (Adapted from Table 21 in “Impact Tests and Service Performance of Thermoplastics,” by P.I. Vincent, The Plastics Institute Report (UDC reference number 678.073:531.66) 1971, © Institute of Materials, Minerals and Mining, with permission from Taylor & Francis Ltd.)

Mechanical Behavior of Plastics

Clearly, the ductile-brittle transitions discussed in this section are not material properties; in addition to the temperature, they depend on the geometry and deformation rate. As such, like the impact tests, they only provide broad guidelines for initial screening of materials for particular applications.

15.11 Creep Creep refers to the phenomenon in which the deformation continues to increase in a part subjected to a constant load. Creep does occur in ceramics and metals, but most of it is at elevated temperatures. In plastics creep can occur even at room temperatures; as discussed in Section 15.2.7, significant amount of room-temperature creep can occur in polycarbonate. Traditionally, an empirical approach for metals uses “isochronous curves” to interpolate load and time-temperature effects. This approach is not grounded in mechanics, and its blind application to plastics is questionable. Creep can be modeled as a complex, large-deformation temperature-dependent phenomenon, which is difficult to validate for complex loading histories even for one-dimensional loading. Modeling of creep is clearly not well understood, and further discussions on this topic are well beyond the scope of this book.

15.12 Stress-Deformation Behavior of Thermoset Elastomers Thermoset elastomers, or rubbers, are very elastic in that small loads can cause large deformations, almost all of which recovers on loading. Figure 15.12.1 shows the stress-elongation curve for vulcanized natural rubber. In contrast to thermoplastics, in which the stress levels are in Mega Pascals, the stress levels in rubber are in Pascals. Notice how even a low stress of 5 Pa can induce an elongation in excess of 6%.

TENSILE STRESS (Pa)

10 NATURAL RUBBER

5

0 0

5

10

EXTENSION e = λ – 1 (%) Figure 15.12.1 Load-extension curve for vulcanized natural rubber. (Adapted with permission from L.R.G. Treloar, Transactions of the Faraday Society, Vol. 40, pp. 59 – 70, 1944.)

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Loading-unloading curves for rubber exhibit hysteresis loops. And, depending on the load levels rubbers undergo permanent set. In particular, in many applications, compression set is an important consideration.

15.13 Concluding Remarks This chapter has provided a survey of the complex mechanical behavior of a broad range of plastics subjected to simple, thermomechanical loads. Clearly, within the normal range of temperature variations and loads, plastics differ from metals and ceramics in three important ways: First, the deformation of plastics even to time-independent loads is time-dependent. Second, plastics can recover from much larger strains on the order of 6% compared to only about 0.2% for metals. And, in comparison to metals, in which on unloading the material instantaneously recovers to unloaded state, the much slower recovery process is time-dependent. And third, whereas within the normal range of applications metal parts are insensitive to temperature changes, they have a large effect on plastic part performance. There are more differences between metals and plastics: Plastics exhibit stable necking in onedimensional tensile tests, in which permanent deformation only occurs after local necking. In two-dimensional testing, such as in a bulge test (Section 15.2.9) permanent set can occur without “necking” but some stress combinations can cause localized necking. And, “permanently” deformed parts can revert to the original state on heating. Plastics craze but metals do not. Besides affecting failure under static loads, crazing also affects fatigue damage. An understanding of how the thermomechanical response of plastics is both qualitatively and quantitatively very different from those of metals is important for not applying the metals-mind-set for rationally designing plastics parts. Structural design of metal parts using the process of first evaluating the stresses and deflections in parts caused by specified loads, and then using failure criteria to prevent failure is not directly applicable to plastics parts because of their time- and temperature-dependent properties and complex failure modes. New algorithms are needed for rational design of plastic parts. Just as an understanding of the material is important for structural part design, a similar in-depth understanding of thermomechanical behavior of plastics in the liquid state is important for understanding their flow in part processing conditions, such as in molds in injection molding and in extrusion through dies. While the complexity of the flow of liquid plastics has briefly been referred to in Section 6.7, this topic is the subject matter for rheology, which is not addressed in this book.

Further Reading The book, Structural Analysis of Thermoplastic Components, by G.G. Trantina and R.P. Nimmer, 1994, McGraw-Hill, Inc., has useful information on the subjects addressed in this chapter. In particular, the chapters on Failure, Designing for Impact with Plastic Materials, Time-Dependent Part Performance, and Fatigue: Cycle-Dependent Part Performance provide a more detailed discussion of these topics. It also discusses how the ad hoc, non-mechanics based empirical treatment of creep, developed for metals, can be applied to plastics.

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Part IV Part Processing and Assembly

The six chapters in this part provide a comprehensive, self-contained introduction to all aspects of part shaping (part processing) and assembly issues for all types of plastics – thermoplastics, thermosets, and elastomers. A large number of diagrams and photographs provide details of various processes; examples of actual parts made by such processes are given.

Chapter 16 Classification of Part Shaping Methods A rational classification of all part shaping (part processing) processes for all plastic material types is provided. How each process type affects the analysis and design of parts is also addressed.

Chapter 17 Injection Molding and its Variants Injection molding and all its important variants are discussed in detail in this long chapter. Examples of short-shot sequences are used to explain the physical processes that occur during molding without having to use complex fluid mechanics and heat-transfer tools. The effects of each process on moldable shapes are considered. Practical aspects of mold design are addressed.

Chapter 18 Dimensional Stability and Residual Stresses First, very simple “homogenous” models, are used introduce the complexities of processes that occur during mold filling and cooling, and how such processes affect part shrinkage and warpage. Then, successively more complex models, including those that account for temperature-dependent viscoelastic effects, are used to explain the processes that affect shrinkage and warpage.

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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Chapter 19 Alternatives to Injection Molding Several alternatives to injection molding of thermoplastics – extrusion, blow molding, rotational molding, thermoforming, and 3D printing – are described in detail. Photographs of actual parts illustrate the complexity possible in single parts.

Chapter 20 Fabrication Methods for Thermosets Major fabrication methods for thermosets – compression molding, reaction injection molding, pultrusion, filament winding, the use of prepregs, liquid resin transfer molding, and the fabrication of rubber parts including tires – are described in detail, with actual application examples.

Chapter 21 Joining of Plastics This chapter introduces all plastics joining and assembly methods – including adhesive bonding, mechanical fastening, and welding – with photographs of actual application examples.

423

16 Classification of Part Shaping Methods 16.1 Introduction Plastics are relatively expensive materials. Their usefulness stems from styling freedom in design – allowing for parts with complex surfaces – and from the complexity of parts that can be made – thereby cutting inventory and assembly costs. Together with part complexity, low-cost part manufacturing is what makes the use of otherwise expensive plastic materials competitive. In the plastics industry, the term processing has two connotations, as schematically shown in Figure 16.1.1. The first is resin manufacturing, involving scale-ups of chemistries for making resins. This includes compounding, in which additives and fibers are mixed to form different resin grades. The end products of thermoplastic resin manufacture, essentially a chemical engineering process, are cylindrical pellets with nominal lengths and diameters of 2- and 4-mm, respectively. For thermosets, the end products of resin manufacture are either powders or liquids; most elastomers, such as rubber, are also thermosetting materials. The second connotation of processing is conversion of resins (pellets) into parts, which can further be divided into the manufacture of bulk products – such as film, sheet, and profile extrusion for thermoplastics – and part shaping, the conversion of resins into diverse functional parts or components. The manufacture of bulk products is a specialized area requiring purpose-built machines. While this chapter mainly focuses on the classification, description, and evaluation of different part shaping, or part processing methods, bulk processing methods are also considered. The effect of each method on part design is also discussed. All part-shaping methods for thermoplastics involve heating of the material, sometimes to temperatures well above the melting point, and to cooling to room temperature after the part has been formed. Because plastics have very low thermal conductivities and thermal diffusivities, most plastic parts have thin walls to keep the cycle time (molding time) small. Typically, wall thicknesses are smaller than 6 mm (0.25 in). Smaller thicknesses on the order of 3 mm (0.125 in) are preferred. In many thin-wall applications, such as in cellular phones, the wall thicknesses are below 2 mm (0.08 in). Because of the very high viscosities of thermoplastic melts, molding thin-wall parts adds to processing complexity. In contrast to thermoplastics, all part shaping methods for thermosetting materials involve chemical reactions. Low part cost is achieved by making the part in one primary operation, in which the desired color is obtained by using colored pellets, and the desired surface finish is achieved by controlling the mold surface finish and the plastic melt temperature. Any subsequent operation, in which the part has to be painted or welded or adhesive-bonded, is called a secondary operation. The purpose of part and process design is to minimize the use of such secondary operations. Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

ELASTOMERS

THERMOSETS

ELASTOMERS

THERMOPLASTICS THERMOSETS THERMOPLASTICS

BULK PRODUCTS

RESIN MANUFACTURE – Pellets

RESIN CONVERSION INTO PRODUCTS

PART SHAPING

Introduction to Plastics Engineering

PLASTICS PROCESSING

424

Figure 16.1.1 Classification of plastics processing.

16.2 Part Fabrication (Processing) Methods for Thermoplastics In the plastics industry, part fabrication is called part shaping, or part processing, and even simply processing. As shown in Figure 16.2.1, part shaping methods fall into two broad categories. The first category comprises methods in which either molten resin is injected into a cavity formed by a two-sided mold – the shape of which determines the final nominal part shape, including the part thickness – or powder or heated sheet is placed in a heated mold cavity in which one mold half

Figure 16.2.1 Classification of part shaping methods for thermoplastics.

– Several variants

INJECTION MOLDING

COMPRESSION MOLDING

– Vacuum Forming

THERMOSTAMPING

ROTATIONAL MOLDING

– Mold surface only determines part surface geometry

SINGLE-SIDED MOLD

THERMOFORMING

INJECTION-COMPRESSION MOLDING

– Several variants

BLOW MOLDING

– Mold cavity determines both part surface geometry and thickness distribution

DOUBLE-SIDED MOLD

PART SHAPING FOR THERMOPLASTICS

Classification of Part Shaping Methods 425

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can be moved to compact the charge to a defined shape. Examples of processes in which a melt is injected into a fixed mold cavity include injection molding – the most commonly used method for making plastic parts – foam molding, and multiple live-feed injection molding. The second category comprises processes in which a one-sided mold is used to shape the outer surface of the part. Here the external shape is determined by the mold geometry, but the thickness distribution is not controlled by the mold; rather, it is determined by the process conditions. Examples of such processes include blow molding and thermoforming, rotational molding, and gas-assisted injection molding.

16.2.1

Processes Using Double-Sided Molds

As schematically shown in Figure 16.2.2, such processes can further be divided into two categories, the first of which comprises processes that make parts with solid walls. In some processes – such as injection molding, co-injection molding, and multiple live-feed injection molding – the molds do not move during the shaping process, so that the mold cavity shape remains unchanged during the process. In other processes – such as injection-compression molding, compression molding, and thermostamping – one-half of the mold pair moves during the shaping process. Injection-compression molding, compression molding, and thermostamping use the same kinds of molds and essentially are minor variants of the same process: In the compression molding an initial charge of molten material is placed an open mold, which is then closed to form the part; in injection-compression molding melt continues to be injected while the mold is closing. Instead of a plastic melt, in thermostamping a heated sheet is “compression molded.” The second category using double-sided molds comprises variants of the injection-molding process that make parts with nonsolid walls. Foam molded parts have thin solid skins surrounding cellular cores. Gas-assisted injection molding is used for making hollow parts with thin solid skins. Injection molding is by far the most important part shaping method. It has many variants such as co-injection molding – which can be used to mold parts with layers different of different plastics – and multiple live-feed injection molding, which improves the integrity of weld lines where two molten fronts meet, especially in fiber-filled materials. Injection molding and its variants are discussed in detail in Chapter 17. The important issues of shrinkage and warpage of injection molded parts are addressed in Chapter 18.

16.2.2

Processes Using Single-Sided Molds

As schematically shown in Figure 16.2.3, the three processes in this category are blow molding, thermoforming, and rotational molding. In blow molding a cylinder of molten material, called a parison, is pinched off at the two ends by two halves of a mold. Air is then injected to expand the material to the shape of the mold surface. The expansion of the parison results in nonuniform part thickness. This thickness variation can be controlled either by parison programming – in which the thickness of the parison along its length is varied during the extrusion process – or, more effectively, by injection molding a variable thickness preform. In vacuum forming a heated sheet is clamped on a mold, the surface of which has holes through which the air in the cavity between the sheet and the mold surface can be evacuated; hence the term vacuum forming. Atmospheric pressure then expands the sheet into the evacuated cavity to form a part matching the mold surface. In pressure forming, a higher pressure is used for forming the part, allowing for better surface control and reduced cycle time.

Figure 16.2.2 Classification of part shaping methods for thermoplastics using double-sided molds.

– Outer skin with different material core

COINJECTION MOLDING

– Several variants

INJECTION MOLDING

COMPRESSION MOLDING

– Better knit-line properties in fiber-filled materials

THERMOSTAMPING

– Hollow parts – Wall-thickness distribution not determined

GAS ASSISTED INJECTION MOLDING

– Variants of injection molding – Wall-thickness distribution may not be determined

PARTS WITH NONSOLID WALLS

MULTIPLE-LIVE-FEED INJECTION MOLDING

INJECTION-COMPRESSION MOLDING

– Cellular through-thickness structure – Part-thickness distribution determined

FOAM MOLDING

– Mold cavity determines both part surface geometry and wall-thickness distribution

PARTS WITH SOLID WALLS

– Mold cavity determines both part surface geometry and external wall-thickness distribution

DOUBLE-SIDED MOLD

Classification of Part Shaping Methods 427

INJECTION BLOW MOLDING – Molded, shaped preform

– Cylindrical extruded preform

– Pressure applied between two clamped sheets; materials need not be the same – Alternative for blow molding

* TWIN-SHEET FORMING

PRESSURE FORMING – Pressure used to force sheet against mold surface

VACUUM FORMING

– Powdered resin melted inside rotating, heated surface

– Vacuum on mold surface; atmospheric pressure applied

ROTATIONAL MOLDING

THERMOFORMING – Heated resin sheet forced by air against mold surface – Single or multilayer

EXTRUSION BLOW MOLDING

– Molten resin preform expanded by air pressure against mold surface – Single or multilayer

BLOW MOLDING

SINGLE-SIDED MOLD – Mold surface only determines part surface geometry

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Figure 16.2.3 Classification of part shaping methods for thermoplastics using single-sided molds. Note: The asterisk before twin-sheet forming indicates that this process uses a two-sided mold.

Classification of Part Shaping Methods

In a variant of pressure forming called twin-sheet forming, two heated sheets are clamped inside the cavity formed by two mold halves and, just as in blow molding, air is injected between the clamped sheets to form a hollow part in which the seam gets welded during clamping. For some applications, this process offers a low-cost alternative to blow molding. It has the added advantage that the two sheets used need not be of the same material. However, since this process uses a two-sided mold, it really does not fall under the one-sided mold category. But it is included here because the process steps and variables are the same as for pressure forming in a one-sided mold. Rotational molding uses a hollow, heated mold that can be rotated about two axes during the molding cycle. This process uses plastics in the form of a fine powder that is placed inside the mold. As the mold rotates the powder, heated by the mold surface, coats the surface, eventually forming a solid part. Circulating heated air inside the rotating cavity can provide additional heating. This process is suited for making large parts such as tanks for holding fluids, but is also used successfully for making small parts. These and other alternatives to injection molding are discussed in detail in Chapter 19.

16.3 Evolution of Part Shaping Methods Processing cost is the main driver for the evolution and selection of part processing methods. While not all shapes can be formed by every processing method, ingenious innovations in part design and processing capability now offer many competitive alternatives for part processing. The most versatile and productive part shaping method by far is injection molding; it is also the oldest and, perhaps, the most developed (Sections 17.2 – 17.7). In it, molten plastic well above the glass transition temperature is injected into a cooled, mold cavity. Cooling down the melt to form a solid part requires heat extraction from all the way to the part core. Because of the low thermal diffusivity of plastics this heat extraction is an inefficient process. Since the cooling time required varies approximately as the square of the part thickness, only relatively thin parts can be economically molded. Another limitation of this process is part size. Since the viscosities of molten thermoplastics are very high, very high pressures are required to force the melt through the mold cavity, resulting in very large clamp forces to prevent the two mold halves from opening. For example, the clamp force required for molding a 1 m2 panel is on the order of 10,000 tons. The large shrinkage of plastics on solidification causes two problems: shrink marks at the relatively thick sections at bosses and ribs, and part shrinkage and warpage. Several new types of part shaping methods were developed to overcome some limitations of the injection-molding process. The large part size issue has been addressed by the development of the vacuum forming, blow molding, and rotational molding processes (Sections 19.5.1, 19.3, and 19.4). Modifications to the basic injection-molding process have resulted in several variants that address shrink marks, shrinkage, molding thicker parts, and knit-line strength in fiber-filled materials. One early variant of injection molding is structural foam molding (Section 17.8.2). This class of processes was originally developed to avoid sink marks in injection molded parts. In one process – there are several variants – a gas charged melt is injected into a mold cavity, the volume of material injected being less than mold cavity volume. The hot melt surface coming in contact with the relatively cold mold walls solidifies, resulting in the formation of thin solid skins. The fall in pressure after injection causes the dissolved gas to come out of solution, forming bubbles that continue to grow in size. Because of the low thermal diffusivity of plastic melts, the core stays warmer than the skin regions, allowing more time for bubble growth. The nucleation and growth of the bubbles force the thin skin to be in contact with the mold walls, thereby avoiding sink marks, even in relatively thick parts. This process also results in lower residual stresses. The surface finish of parts is poor, however, requiring secondary finishing operations such as painting. As with most injection molded parts, foam components are typically thin-walled. Because foam molding is done at relatively low injection pressures, it requires less

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expensive molds than standard injection molding. Structural foam molding machines are very similar to regular injection molding machines. A second important variant of injection molding is gas-assisted injection molding – which too has several variants – that makes it possible to make large thick, hollow parts that cannot otherwise be molded (Section 17.8.5.2). In this class of processes, a slug of molten materials is first injected into a mold cavity, followed by injection of gas that pushes the molten slug while at the same time penetrating and coring it, resulting in a thin-walled hollow part. As with structural foam molding, this process results in lower residual stresses. This versatile process has been used for molding a variety of parts, such as integral hollow wheel chair tires and stress-free panels. This process requires special injection nozzles and controls for sequentially injecting molten resin and gas. More recently, in place of gas, liquid such as water is being used to push and core out molten slugs of plastic; the liquid cools the melt faster than gas resulting in reduced cycle times. Co-injection molding was developed to mold articles having different skin and core materials: The core can be of a cheaper material to reduce part cost (Section 17.8.4.1). Having a resin skin with a fiber-filled core allows for a relatively stiff and strong part that, at the same time, has a good surface finish. The skin can be of a material, such as a thermoplastic elastomer, that gives a soft feel on touching. Extensions of this technique include overmolding, in which a layer is molded onto an already molded substrate (Section 17.8.4.2). Injection-compression molding was developed to address the issue of reducing the pressures involved in injection molding (Section 17.8.1.2). Instead of forcing the melt through narrow passages in fixed molds, one-half of the mold is initially opened and a slug of molten material is placed on the second half. Then, additional material is injected as the top mold surface is lowered to form the mold cavity. This technique is used for molding optical media such as compact disks (CDs) and digital video disks (DVDs). During injection molding two molten fronts may meet and meld together – for example, when the flow front divides around a pin used for molding a hole and then meets again after the pin. The meld path on the surface is variously called meld line or a knit line; sometimes even a weld line. For resins and particulate-filled resins the main issue with meld lines is surface finish. However, in fiber-filled materials, improper mixing at meld surfaces results in fibers not properly bridging across such surfaces. As such the strength at meld lines is reduced. Multiple live-feed injection molding and push-pull molding were developed to overcome this limitation (Sections 17.8.6.1 and 17.8.6.2). In this process, the molten material is injected at two or more gates. Then, for a short time interval, while material is pushed in one gate, it is allowed to retract along the other gate, after which the process is reversed: material is pushed at the second gate and retracted at the first gate. In this way, the material is cycled back and forth; the complete mixing at the meld fronts results in complete bridging by fibers. Although molded part quality is improved, the additional equipment required and the molding cost limits this process to special applications requiring high performance of fiber-filled parts. In contrast to all the variants of injection molding, in all of which molten resin is injected into mold cavities, the starting point for vacuum forming is extruded sheet material that always has a good mirror-like surface. In vacuum forming sheets of plastic heated to temperatures below the glass transition temperature are clamped over molds and the cavity between the sheet and the mold is then evacuated. Atmospheric pressure forces the sheet to deform and take on the shape of the cavity. Because the sheet material used only has to be heated to below Tg , the surface finish of the part is controlled by the surface finish of the sheet, which is good. As such, in this is a low-pressure process the molds used are relatively inexpensive. Instead of only relying on vacuum to shape the part, in pressure forming additional pressure is used to force the sheet against the mold surface. In contrast to injection molding and its variants, while the shape of the part is determined by the mold geometry, the thickness distributions can vary greatly.

Classification of Part Shaping Methods

In a variant of vacuum/pressure forming, called twin-sheet forming, two heated sheets are clamped between two molds and air pressure is applied to force the two sheets against the two evacuated cavity surfaces (Section 19.5.4). The clamping force causes the clamped surfaces to weld, resulting in a hollow part. Two different but compatible materials can be used to form hollow parts with two different materials. There are two versions of the blow molding process. In the original process, called extrusion blow molding, a cylinder of molten material is first extruded. This cylinder of molten material, called a parison, is clamped between two mold halves, after which air is injected inside the clamped cylinder, forcing the parison to blow up like a balloon till it contacts the cold mold surfaces, and, on solidification, taking on its shape (Section 19.3.1). This process is similar to vacuum forming in the sense that a sheet is blown by air pressure to conform to the shape of a one-sided mold. But there is a major conceptual difference: Because in vacuum forming a solid sheet is heated to just below Tg , the surface finish of the part is determined by that of the sheet and to a much lesser extent by the surface finish of the mold. On the other hand, because the parison temperature in blow molding is well above Tg , the part surface finish is determined by that of the mold surface, requiring molds with better surface finish. As with vacuum forming, the thickness distribution across the part can vary greatly. This limitation can partially be overcome by varying the initial thickness of the molten parison by parison programming. However, the thickness distribution can be controlled far more accurately by the injection blow molding process, in which a preform is molded into a shape that, on blow molding, results in a desirable part thickness distribution (Section 19.3.2). Rotational molding, also called rotomolding (Section 19.4), was initially developed to make very large parts, such as water tanks, with relatively inexpensive molds. In this process a pre-measured powdered resin is placed inside a mold, which rotates about two axes, that is heated externally in an oven. As the powder tumbles over the mold surface it melts, taking on the shape of the mold surface. The wall thickness depends on the amount of resin powder used. Because the molding process occurs at atmospheric pressure, the molds are substantially less expensive.

16.4 Effects of Processing on Part Performance Several connotations of part performance have been discussed in Section 1.7, wherein Section 1.7.1 addresses the use of numerical procedures for predicting thermomechanical part performance. The basic procedure for using the finite element method for this is illustrated in Figure 16.4.1: A detailed knowledge of the part geometry – shape and thickness distribution – the local material properties such as stiffness and strength, and the applied loads and displacements are used for a finite element structural analysis of the part. This procedure is well suited for metals in which, in most cases, the part geometry and the local material properties are known a priori. For a given initial choice of geometry and material, a finite element structural analysis will predict part performance – its stiffness, strength, and other attributes such as energy absorption. If this does not meet the specified performance requirements, the geometry, and or the material are changed, as indicated by the chain-dashed lines, and the performance of the new configuration is analyzed. This iterative process is continued till a geometry-material combination satisfies performance requirements. In plastics, however, the complete part geometry and the local material properties may not be known in advance, and have to be determined from an analysis of the part fabrication process, which also determines the local material morphology. Then, a micromechanics analysis has to be used to determine the local mechanical properties. The resulting procedure for predicting part performance is far more complex with several feedback loops. Now that part shaping techniques have been

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CHANGE GEOMETRY

GEOMETRY • Shape • Thickness distribution

PART MATERIAL

STRUCTURAL MECHANICS • Finite Element Analysis

• Local stiffness • Local strength

PART PERFORMANCE • Stiffness • Strength • Energy absorption

CHANGE MATERIAL

Figure 16.4.1 Diagram illustrating the steps for a finite element structural analysis for predicting part performance.

classified, this section examines how part performance prediction is affected by different part shaping methods. The geometry of a functional part comprises two elements: The shape of its surfaces (surface geometry) and the thickness distribution. Consider the effect of processing on part geometry. For shaping processes using double-sided molds – such as injection molding – the mold cavity shape determines the final nominal part shape, including the part thickness. However, in processes in which a one-sided mold is used to shape the outer surface of the part, while the external part shape is determined by the mold geometry the thickness distribution is not controlled by the mold – rather, it is determined by the process conditions. Figure 16.4.2 shows most of the part shaping methods grouped according to the mold types used: In those in the lower half of the rectangular region, which use double-sided molds, the shaping method does not affect part geometry, which is determined by the mold geometry. In those in the other half, which use one-sided molds, while the part surface geometry is determined by the mold, the thickness distribution is not; it has to be determined by an analysis of the process. Next consider the effects of processing on local mechanical properties of the material that depend on the local morphology. These can be divided into three broad categories: (i) Amorphous resins, with or without particulate fillers, which behave like homogeneous materials on which processing has a small effect. The properties of these materials do not vary across the part and, therefore, as in the case of metals, can be defined through standard tests, such as standard ASTM D-638 for tensile tests, specified by ASTM International. (ii) Semicrystalline resins in which the local morphology, and hence the local mechanical properties, do depend on how the material is processed. And (iii), fiber- and flake-filled resins (both amorphous and semicrystalline), in which the local fiber and flake distribution and orientation, which are controlled by processing conditions, strongly affect properties. In the last two cases, standard test procedures do not measure meaningful material properties – rather, they determine “system (part-like) properties” that depend both on the specimen geometry and on the local material morphology. New procedures have to be developed for determining local properties of these materials as a function of processing. The effects of these two types of part shaping techniques on local material properties in the part are schematically shown in Figure 16.4.3.

• BLOW MOLDING • THERMOFORMING

PART GEOMETRY

• ROTATIONAL MOLDING • GAS-ASSISTED INJECTION

Part-thickness distribution affected by processing

MOLDING

UNAFFECTED BY PROCESSING

• INJECTION MOLDING

DETERMINED BY PROCESSING

Classification of Part Shaping Methods

• FOAM MOLDING • EXTRUSION • COMPRESSION MOLDING • MULTIPLE LIVE-FEED INJECTION MOLDING

Figure 16.4.2 Classification of part shaping methods according to whether or not they affect part geometry.

UNAFFECTED BY PROCESSING

•

AMORPHOUS RESINS – With and without particulates

DETERMINED BY PROCESSING

• SEMICRYSTALLINE RESINS

• ALL RESINS WITH FIBERS AND FLAKE FILLERS

•

ASTM TESTS PROVIDE MEANINGFUL DATA

• ASTM TESTS OF LITTLE VALUE

LOCAL MATERIAL PROPERTIES Figure 16.4.3 Classification of plastic materials by whether or not the local properties are affected by part shaping methods.

Combining the effects of processing on both geometry and material properties results in the four possible combinations schematically shown in Figure 16.4.4; note that each of these four cases corresponds to the number in the square in the lower right-hand side of each quadrant:

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• Case 1, in which the part geometry and local material properties are independent of processing; for example, in the injection molding of amorphous resins (lower first quadrant). • Case 2, in which the part geometry is independent of processing, but in which the local material morphology, and hence the local material properties, are determined by processing; for example, in the injection molding of semicrystalline or glass-fiber-filled resins (lower second quadrant). • Case 3, in which the part geometry is determined by processing, but the local material properties are independent of processing; for example, in the blow molding of amorphous resins (upper first quadrant). • Case 4, in which both the part geometry and the local morphology are determined by processing; for example, in the rotational molding of semicrystalline or glass-fiber-filled resins (upper second quadrant).

• • • •

• • • • •

BLOW MOLDING THERMOFORMING ROTATIONAL MOLDING GAS-ASSISTED INJECTION MOLDING

BLOW MOLDING THERMOFORMING ROTATIONAL MOLDING GAS-ASSISTED INJECTION MOLDING

3 AMORPHOUS RESINS WITH AND WITHOUT PARTICULATE FILLERS • • • • •

INJECTION MOLDING FOAM MOLDING EXTRUSION COMPRESSION MOLDING MULTIPLE LIVE FEED INJECTION MOLDING

INDEPENDENT OF PROCESSING

4 SEMICRYSTALLINE RESINS AND ALL RESINS WITH FIBERS AND FLAKE FILLERS • • • • •

1

INJECTION MOLDING FOAM MOLDING EXTRUSION COMPRESSION MOLDING MULTIPLE LIVE-FEED INJECTION MOLDING

LOCAL MATERIAL PROPERTIES

PART SHAPE UNAFFECTED BY PROCESSING

DEPENDS ON PROCESSING

PART GEOMETRY

SEMICRYSTALLINE RESINS AND ALL RESINS WITH FIBERS AND FLAKE FILLERS

AMORPHOUS RESINS WITH AND WITHOUT PARTICULATE FILLERS

PART SHAPE DETERMINED BY PROCESSING

LOCAL PROPERTIES DETERMINED BY PROCESSING

LOCAL PROPERTIES UNAFFECTED BY PROCESSING

INDEPENDENT OF PROCESSING

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2

DEPEND ON PROCESSING

Figure 16.4.4 Classification of the effects of processing on part geometry and local material properties into four categories. Each of these four cases corresponds to the number in the square in the lower right-hand-side of each quadrant.

Classification of Part Shaping Methods

In Case 1(lower first quadrant), as a first approximation structural performance can be predicted without a process analysis – just as in metals (Figure 16.4.1). However, in injection molding, the location of knit lines – surfaces where different flow fronts within a mold meet and meld together – along which the material is weaker, does require process analysis tools. Such tools have evolved over the past 50 years in the form of robust commercial mold filling and cooling software for injection molding that can determine filling patterns, knit-line locations, the effect of gating, and cooling line locations, and so on. Process analysis is also required for determining residual stresses, but methods for predicting them are not well developed. Just because molten plastic is injected into a mold does not guarantee that the mold will fill. So, an important role of process mechanics is to assess whether or not a mold will fill. If it does not, then process parameters – such as the melt temperature and the injection pressure – may have to be changed. If such changes still result in short shots (incomplete mold filling), the material may have to be changed, and or, the part thickness distribution may also have to be changed. Figure 16.4.5 shows how process mechanics fits into the part design process. Based on requirements an initial shape and a material are chosen for analysis. For a double-sided mold this will include the part thickness distribution; for a one-sided mold, only the surface shape will be specified. Then, process mechanics will determine whether a double-sided mold will fill or whether a sheet being forced against a one-sided mold will completely contact the mold surface. If the double-sided mold does not fill or the sheet does not completely contact the single-sided mold and the predicted thickness distribution is not acceptable, process parameters – such as the melt temperature and the injection pressure for a double-sided mold or the sheet temperature and the applied pressure for a single-sided mold – may have to be changed. If such changes do not produce the desired results, the material and or the part shape may have to be changed, as indicated by dashed lines.

GEOMETRY

SHAPE PART

PROCESS MATERIAL

PROCESS MECHANICS

• Thickness distribution

• Filling • Thickness • Morphology

Figure 16.4.5 The role of process mechanics in determining mold filling, part thickness distribution, and the local material morphology.

In addition to checking for mold filling and part thickness distribution, an important role of process mechanics is to predict the local part morphology – such as the distribution and orientation of fibers in fiber-filled materials or the local crystalline structure in semicrystalline materials. But knowledge of the local material morphology is not sufficient, as finite element analysis requires local material properties. These properties are determined by a micromechanics analysis that converts the morphological information into local mechanical properties. For an initial part shape and a material, the process for determining the full part shape – surface geometry and thickness distribution – and the

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local material property distribution throughout the part is then described by the schematic diagram in Figure 16.4.6. Iterations of this process finally determine the complete geometry and local mechanical properties of the part as inputs for finite element structural analysis.

GEOMETRY

SHAPE PART

PROCESS MATERIAL

PROCESS MECHANICS • Filling • Thickness • Morphology

• Thickness distribution

MICROMECHANICS • Local mechanical properties

Figure 16.4.6 Process for defining complete part shape and local material properties.

In the remaining three cases, a process analysis for predicting the part thickness distribution (Cases 3 and 4) and the material morphology (Cases 2 and 4) – together with a micromechanics analysis for predicting the local mechanical properties of the material – is a prerequisite for predicting the structural performance of parts. The processes for finite element structural analysis of parts for the four cases are shown, respectively, in Figure 16.4.7 parts a and b, and Figure 16.4.8 parts a and b. Note that the final iterations requiring changes in geometry and material when structural performance requirements are not met, indicated by chain-dashed lines in Figure 16.4.1, have not been shown in these figures. For each case, items that do not apply have either been left out or scored out. For example, for Case 1 (Figure 16.4.7a), the geometry – shape and thickness distribution – are known a priori, and is not affected by processing, but may have to be modified because of filling considerations (dashed lines and arrows). And micromechanics does not come into play because the material properties are also known in advance; the material may have to be changed because of filling considerations. In Case 2 (Figure 16.4.7b), process mechanics determines the local material morphology, which micromechanics converts into local properties. In Case 3 (Figure 16.4.7a), the material properties are known a priori but process mechanics determines the part thickness distribution. Case 4 (Figure 16.4.7b) is the most complex case in which, in addition to checking for mold filling and locations of knit lines, process mechanics determines the part thickness and the local material morphology, which a micromechanics analysis converts into local material properties. Clearly, then, plastic part design and performance prediction is substantially more complex than for metal parts. And this enormous complexity can only be handled by complex computer codes (software) that concurrently solve for the flow of highly temperature-dependent melts with fillers, the evolution of local material morphology, and the conversion of local morphology into local material properties. However, current codes are mainly used for filling and mold cooling analyses; evolution of local morphology, determination of local material properties, and characterization of shrinkage and warpage and residual stresses are beyond the capabilities of current codes.

PART

PART

MATERIAL

SHAPE

MATERIAL

SHAPE

PROCESS

PROCESS

• Filling • Thickness • Morphology

PROCESS MECHANICS

• Filling • Thickness • Morphology

PROCESS MECHANICS

• Thickness distribution

GEOMETRY

(Case 1). (b) Semicrystalline resins and all resins with fibers and flake fillers (Case 2).

(b)

• Local mechanical properties

MICROMECHANICS

(a)

• Local mechanical properties

MICROMECHANICS

• Thickness distribution

GEOMETRY

• Finite Element Analysis

STRUCTURAL MECHANICS

• Finite Element Analysis

STRUCTURAL MECHANICS

• Stiffness • Strength • Energy absorption

PART PERFORMANCE

• Stiffness • Strength • Energy absorption

PART PERFORMANCE

Classification of Part Shaping Methods

Figure 16.4.7 Procedure for finite element analysis of molded plastic parts. (a) Amorphous plastics in double-sided molds

437

PART

PART

MATERIAL

SHAPE

MATERIAL

SHAPE

PROCESS

PROCESS

• Filling • Thickness • Morphology

PROCESS MECHANICS

• Filling • Thickness • Morphology

PROCESS MECHANICS

(b)

• Local mechanical properties

• Thickness distribution

GEOMETRY

MICROMECHANICS

(a)

• Local mechanical properties

MICROMECHANICS

• Thickness distribution

GEOMETRY

• Finite Element Analysis

STRUCTURAL MECHANICS

• Finite Element Analysis

STRUCTURAL MECHANICS

• Stiffness • Strength • Energy absorption

PART PERFORMANCE

• Stiffness • Strength • Energy absorption

PART PERFORMANCE

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Figure 16.4.8 Procedure for finite element analysis of molded plastic parts. (a) Amorphous plastics in single-sided molds (Case 3). (b) Semicrystalline resins and all resins with fibers and flake fillers (Case 4).

Classification of Part Shaping Methods

16.5 Bulk Processing Methods for Thermoplastics The discussions in the previous sections have focused on methods for making discrete plastic parts. Some products, listed in Figure 16.5.1, are produced in bulk by continuous, high-volume processes, all of which involve the important process of extrusion. Almost all plastic forming techniques start with pellets that after drying, if necessary, are fed into a machine called an extruder, in which the pellets move inside a heated cylindrical barrel by means of a rotating screw (Figure 19.2.1). The relative motion between the screw and barrel surfaces causes frictional heating and mixing of the pellets, finally resulting in a uniform melt. This melt is extruded through dies – in a manner similar to toothpaste being extruded out of a tube – to form continuous parts. THERMOPLASTIC BULK PRODUCTS

FIBER

FILM

SHEET

PROFILE EXTRUSION

Figure 16.5.1 Thermoplastic bulk products.

16.5.1

Fiber Spinning

Synthetic fibers, a very large market for thermoplastics, are made by extruding either a polymer in a solvent solution or a molten resin through very fine holes having diameters on the order of 200 – 500 μm.The in-solution extruded fibers are either precipitated by passing through a bath or air dried to extract the solvent. Melt spinning refers to the case in which molten thermoplastic is extruded through holes and then stretched to further reduce the diameter. In practice, many fibers are simultaneously extruded through a spinneret, a die comprising from one to 10,000 holes (Section 19.2.1). 16.5.2

Film Blowing

The largest extruded products perhaps are thin films, widely used for packaging, made by a process called film blowing: A continuously extruded, thin-walled cylinder is pinched at the top and pulled continuously, while at the same time being expanded by pressurized air injected into the cavity to reduce the film thickness (Section 19.2.2). 16.5.3

Sheet Extrusion

In sheet extrusion, the molten plastic is extruded through flat dies. The flat extrudate then goes over rotating, sizing rolls that control the final thickness and surface finish. This process can also be used for making hollow, thin-walled sheets in which the two outer skins are connected by regularly spaced ribs. Extruded sheets are available in a variety of thickness ranging from very thin sheets for vacuum and thermoforming applications to thick sheets used for glazing (Section 19.2.3). 16.5.4

Profile Extrusion

Thin-walled, continuous, prismatic solid profiles, such as those used for vinyl (polyvinyl chloride: PVC) siding and plastic window frames in the construction industry, are produced by extruding melts through

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metal dies. The extruded profile is cooled by passage through a water bath. Extrusions with very complex cross sections, involving many closed and open cells can be formed by this process. It can also be used for making thick-walled profiles (Section 19.2.4). A special, but important, type of extrusion is that for making pipes, the diameters of which can range from fractions of centimeters to several meters. Multilayer parts, consisting of layers of different plastics, can be made by using co-extrusion, in which two or more molten plastics are simultaneously extruded through a die.

16.6 Part Processing Methods for Thermosets In thermosets, part fabrication involves chemical reactions resulting in solid parts that cannot be melted or reshaped other than by machining processes. The chemical reactions result in significantly longer molding cycles than for thermoplastics. These chemical reactions generate heat, and moisture that can result in voids. In contrast to thermoplastics – in which all part fabrication starts with plastic pellets – the starting material for thermoset part fabrication can have several different forms indicated in Figure 16.6.1: (i) Powders that react through exothermic reactions when heated. (ii) Chopped fibers impregnated with low-viscosity thermoset precursors, such as sheet molding (SMCs) and bulk molding compounds (BMCs), which can be injection molded in a manner similar to that for thermoplastics. (iii) Low-viscosity liquid resins that can either react within a mold or be injected into glass- or carbon-fiber woven preforms placed in a mold to produce highly-filled, high stiffness and high-strength parts. And (iv), sheets of aligned fibers in tacky precursors, called prepregs, which can be stacked and cured to produce lightweight, high-strength parts.

PART SHAPING FOR THERMOSETS

DRY POWDER RESIN

LIQUID RESIN

SHEET MOLDING COMPOUND

ADVANCED COMPOSITES

BULK MOLDING COMPOUND

Figure 16.6.1 Starting materials for thermoset part processing.

16.6.1

Processes Using Double-Sided Molds

As schematically shown in Figure 16.6.2, such processes can further be divided into three categories; one each for dry-powder, sheet molding and bulk molding compounds, and liquid-resin starting materials.

Classification of Part Shaping Methods

DOUBLE-SIDED MOLD – Mold cavity determines both part-surface geometry and thickness distribution

DRY POWDER RESIN

COMPRESSION MOLDING

SHEET & BULK MOLDING COMPOUND

LIQUID RESIN

TRANSFER MOLDING

INJECTION MOLDING

INJECTION-COMPRESSION MOLDING

REACTION INJECTION MOLDING

REINFORCED REACTION INJECTION MOLDING

STRUCTURAL REINFORCED REACTION INJECTION MOLDING

Figure 16.6.2 Part shaping methods using double-sided molds.

16.6.1.1 Processes Using Powder Resin

The compression molding process is similar to that for thermoplastics. A powder charge, placed on the lower half of a mold, is compressed by the moving upper half and the mold assembly is heated to start the thermosetting, curing reaction (Section 20.3). In transfer molding, a modification of compression molding, a measured, preheated molding compound charge is first placed in a cavity, and is then forced (transferred) by a hydraulic ram into a heated mold cavity through a sprue, where it is cured just as in compression molding (Section 20.4).

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16.6.1.2

Processes Using Sheet and Bulk Molding Compounds

Sheet and bulk molding compounds can be molded into parts by injection and injectioncompression molding, just as thermoplastics – except that the molds are heated for curing the injected material (Section 20.5). 16.6.1.3

Processes Using Liquid Resin

In the main process using liquid resin, reaction injection molding (RIM), two low-viscosity polymer precursors are mixed and injected at high pressure through an impinging mixer into a closed mold, where they react through an exothermic reaction to form the cured part (Section 20.6). In reinforced reaction injection molding (RRIM), fillers such as glass fibers or mica flakes are added to the liquids prior to mixing, resulting in stiffer parts than those made by RIM (Section 20.6.1). In structural reaction injection molding (SRIM) the low-viscosity of precursors are forced through a fiber mesh placed in a mold form rigid parts with very high levels of reinforcements (Section 20.6.2). 16.6.2

Processes Using Single-Sided Molds

For thermosets, the use of single-sided molds is referred as open form molding. In principle, the cosmetic surface is obtained by direct contact of the resin with the finished mold surface, after which the fiber-resin surface is applied behind this layer (Section 20.7).

16.7 Part Processing Methods Advanced Composites Advanced composites refer to high-performance polymer composites, mainly comprising highly-filled, continuous aligned-fiber composites used in the aircraft and aerospace industries (Chapter 25). Depending on part geometry, there are three main processes for fabricating advance composites: Pultrusion for straight structures with a uniform cross section, filament winding for cylindrical structures, and vacuum bag consolidation for relatively flat, laminated structures. 16.7.1

Pultrusion

In pultrusion – from pull and extrusion, in which material is pushed through a die – fibers impregnated with a low-viscosity thermoset resin are pulled through a heated (extrusion) shaping die, in which the resin cures and causes the fibers to set into the cross-sectional shape of the die (Section 20.8.1). By choosing an appropriate die, continuous composite parts, having fixed complex open- or closed-sections, can be made; the continuous profiles can then be cut to required sizes. 16.7.2

Filament Winding

Filament winding is an automated, continuous process in which fiber is wound around a mandrel (a long metal cylinder), which can be rotated back and forth in a controlled manner while the continuous-fiber dispensing head moves along the mandrel axis. In wet winding the fibers go through a resin bath before winding on the mandrel, which is then placed in an oven to cure the resin (Section 20.8.2).

Classification of Part Shaping Methods

16.7.3

Laminated Composites

Relatively flat high-performance structures are made by laminating plies comprising reinforcing unidirectional fibers or fabrics that, in most applications, are embedded in thermosetting resin matrices. Because such composites are used in critical applications – such as in aircraft – the forming processes require stringent controls for achieving defect-free uniform parts. 16.7.3.1 Prepregs

The main starting materials for advanced composites are prepregs (from pre-impregnated), reinforcing materials – unidirectional fibers or woven fabric – pre-impregnated with resin. A single prepreg layer is called a ply. A composite part may be built up from several plies with different orientations to achieve desired stiffness and strength; this combination of plies is called a lay-up. Manufacturing parts from prepregs and lay-ups requires pressures to consolidate the laminate and heat to initiate and complete the cure cycle for the resin. 16.7.3.2 Vacuum Bag Consolidation

Very large parts with complex two-dimensional curvatures are fabricated by using vacuum bags to surround the lay-up placed in a one-sided mold. In addition to help consolidate the part, the vacuum helps to remove air bubbles and the volatiles generated during the curing process. Autoclaves – large, heated pressure vessels – are used to apply higher pressures and high temperatures to the vacuum bagged assembly (Section 20.8.3.2). 16.7.3.3 Compression Molding

In this process the prepreg lay-up is placed in a matched mold and compression molded (Section 20.3).

16.8 Processing Methods for Rubber Parts Rubber part processing can be divided into to two categories: the first uses solid rubber and the second uses rubber in a liquid form. The process for curing rubber is called vulcanization. 16.8.1

Rubber Compounding

In contrast to plastics – in which the material input for part fabrication processes is in the form of pellets and sheets or liquid resins – natural and synthetic rubbers are supplied in the form of sheets folded into large bales. Like plastics, rubbers are compounded with many additives such as fillers for either reducing cost or for improving mechanical properties. Before mixing the additives, the rubber feedstock has to be cut into smaller pieces. And natural rubber may have to be masticated, or kneaded, in mixers to reduce its high viscosity by heat and shear. To obtain a uniform dispersion of the additives, during compounding the rubber and the additives are mixed in internal mixers, such as two-roll mills or Banbury mixers – in which the rubber is sheared between internal counter rotating rollers and the casing wall. Because of the high viscosities of base rubbers, the shearing and stretching in the mixers can raise the temperature to levels at which curing agents can cause premature vulcanization (curing). The compounding is therefore carried out in two stages: In Stage 1, only the non-vulcanizing additives are mixed with

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the rubber. The resulting mixture is called a masterbatch. In Stage 2 the masterbatch is mixed with the vulcanizing additives. Parts can then be made of this material by shaping and subsequent vulcanization. Several fabrication techniques for rubber parts are considered in the sequel. 16.8.2

Dry Rubber Compounding

In these processes, solid compounded rubber is used for making most load-bearing parts, including tires, conveyor belts, and pressure hoses. 16.8.2.1

Molding Processes

Just as other thermosets, compounded rubber can be molded by using compression, transfer, and injection molding. Of course, the pressures required for shaping the material in the mold and the cure (vulcanization) temperatures may be quite different. 16.8.2.2

Extrusion

Most rubber extrusion is done by using screw extrusion. The rubber in an extruded part is uncured, and has to be vulcanized in a separate operation. 16.8.2.3

Calendering

In calendering, used for converting compounded rubber into uncured sheet, the rubber is forced through a series of decreasing gaps between a series of rotating rollers. The last roll gap determines the sheet thickness (Section 20.9.2.3). 16.8.2.4

Reinforced and Coated Rubber Sheet

Several products are built up from rubber sheets reinforced with filaments or fabric; each such reinforced sheet is called a ply. Then, the fibers or fabric are fed into the roller gap, where pressure forces the rubber into intimate contact with the reinforcing material. 16.8.3

Wet Rubber Part Fabrication

Wet rubber forming techniques use liquid rubber for dip molding thin rubber parts and for dip coating parts with a layer of rubber or plastisol (Section 11.5.1.1). 16.8.3.1

Dip Molding

In dip molding a cleaned, heated metal or ceramic male mold (mandrel) is dipped into liquid rubber bath containing a vulcanizing agent. On withdrawing from the bath, the coated mandrel is placed in an oven where the rubber cures, taking on the shape of the mandrel surface. The part is stripped from the cooled mandrel (Section 20.9.3.1). 16.8.3.2

Dip Coating

Dip coating, a modification of dip molding, is used to coat portions of metal parts with rubber or PVC; instead using a mandrel, the actual part is used as the male “mold.” The coating can be for esthetic reasons, to provide a better gripping surface for hands – as in grips for hand tools – and for safety, as in coating metallic surfaces of toys (Section 20.9.3.2).

Classification of Part Shaping Methods

16.9 Concluding Remarks This chapter has focused on the evolution and comparison of part shaping methods for thermoplastics, thermosets, and elastomers. Clearly, the most important part fabrication process for thermoplastics is injection molding; Chapter 17 gives a comprehensive treatment of this important process and its variants. Chapter 18 addresses the important issues of part shrinkage and warpage associated with this process. And Chapter 19 gives a comprehensive description of alternatives to injection molding. While injection-molding type machines can be used for making thermoset parts, the molding process involves in situ chemical reactions; forming processes for thermosets are discussed in Chapter 20. It includes processes for making rubber parts such as tires. While the discussions in this chapter have focused on the methods used for part fabrication, and on how structural part performance is synergistically related to part fabrication methods, the important and difficult issues of polymer melt rheology, which play a critical role in the design of plastic part making machine and dies, have not been discussed. A vast amount of literature is, however, available on these important topics.

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17 Injection Molding and Its Variants 17.1 Introduction The most versatile and productive part shaping method for plastics by far is injection molding; it is also the oldest and, perhaps, the most developed. It is conceptually a very simple process: molten plastic, well above the glass-transition temperature, is injected into a cooled, mold cavity formed by a two-sided mold, the shape of which determines the final nominal part shape, including its thickness. The part is ejected on cooling. Several variants of injection molding have evolved, both to address some of its shortcomings as well as to enhance its capabilities.

17.2 Process Elements A typical injection molding machine comprises three main elements: (i) A screw and barrel system that converts solid plastic into a melt. (ii) A mold into which melt is injected through a nozzle at the end of the barrel. And (iii), a mold clamping system that keeps the mold closed against the high pressures at which melt is injected into the mold. These three elements are schematically shown in Figure 17.2.1. In this highly simplified diagram, Figure 17.2.1a shows the machine in the mold-open position. The motion of mold half attached to the movable plate is controlled by a hydraulically driven cylinder; this position is achieved by pumping hydraulic fluid through the port marked B, while at the same time allowing hydraulic fluid to exit from the port marked A. As shown in Figure 17.2.1b, mold closure is achieved by pumping in hydraulic fluid through port A, while allowing fluid to exit from port B. This not only closes the mold, but applies a clamping force to the closed mold through the hydraulic pressure acting on the cylinder. While this figure adequately explains the principle of the hydraulic clamping system, the actual design is far more complex. The raw, or input, material for injection molding consists of cylindrical plastics pellets with nominal diameters and lengths in the 3 – 4 mm and 3 – 6 mm ranges, respectively. Plastics pellets that absorb moisture, such as those of nylon for example, may have to be dried prior to being fed through a hopper into the barrel of the molding machine, wherein a mechanical screw moves the pellets forward. The resulting friction and the heat applied to the barrel melt the pellets. The melt accumulates at the end of the barrel from where a piston forces it through a nozzle into the mold. (In the plastics industry, the Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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CLAMPING UNIT

MOVING PLATE

TIE BAR

HOPPER

B A

MELT INJECTION NOZZLE

HEATED BARREL WITH SCREW

HYDRAULIC FLUID

FIXED MOLD HALF

MOLD FIXED TO MOVING PLATE

HYDRAULICALLY DRIVEN CYLINDER

(a)

B A

(b) Figure 17.2.1 Schematic diagram showing the three main elements of an injection molding machine: (i) A screw-barrel-nozzle system that converts solid pellets into a melt, (ii) a mold into which the melt is injected through a sprue-runner-gate system, and (iii) a mold clamping system for keeping the mold closed against the high injection pressures. (a) System in the mold-open position. (b) Mold in the closed position with applied clamping force.

process of producing a melt from solid pellets is referred to as plastication and the screw used for this purpose is called a plasticating screw.) The design of the screw and the barrel heating system requires special consideration to ensure (i) that the molten material mixes properly, and (ii) that molten material flows continuously without portions of it being trapped in recirculating flows – where a long residence time can degrade the material – to be shed off at longer intervals. (A schematic cross section of an extruder is shown in Figure 19.2.1.) When

Injection Molding and Its Variants

sufficient molten material has accumulated, it can be forced through a nozzle into the mold system either through a piston or, as in many machines, the screw reciprocates, acting as a piston. The nozzle of the barrel does not directly inject molten plastic into the mold cavity. Rather, the melt is injected into a sprue that feeds a runner system connected to the cavity through a gate. By a runner system simultaneously feeding several different cavities in a mold, several parts can be molded in each (inject-cool-eject) cycle. Figure 17.2.2 schematically shows the arrangement of sprues, runners, and gates for a three-plate, two-cavity mold. Figure 17.2.2a shows the two cavies filled with molten plastic forced from the nozzle through the sprue, runners, secondary sprues, and gates. When the mold is opened, the sprue-runner and secondary-sprue assembly separates out as do the two parts (Figure 17.2.2b). (The terms sprues, runners, and gates have been borrowed from terminology used for making cast iron parts.) Sprues, runners, and gates are discussed in detail in Sections 17.8.6 and 17.8.7. The clamping system is important because a low clamping force can result in thin “flash” oozing out from the mold parting surfaces. In mechanical clamping systems with toggle force magnifying mechanisms the required clamping force only develops when the mechanism locks in place. And changes in mold widths require readjustment of the mechanism. In contrast, hydraulic clamping systems do not require such adjustments, and controlled clamping forces can easily be applied. A typical injection molding cycle comprises four distinct sequential steps: 1. Mold closing. Mold halves are closed to form mold cavity having the shape of the desired part. 2. Mold filling. In this step molten plastic is injected into a closed mold cavity through the sprue-runner-gate system. During this process the pressure in the melt increases to force the melt into the cavity. 3. Packing. After the cavity is filled, the melt continues to be pressurized – this is referred to as the packing pressure phase. In addition to replicating the mold texture onto the plastic molded surface, the packing pressure allows more melt to be added to compensate for the large shrinkage that plastics undergo on cooling. Sometimes this phase is divided into an initial packing phase – to insure mold filling and surface replication – followed by a holding phase in which material is forced into the mold to compensate for cooling induced shrinkage. Because of its compressibility, the amount of resin forced into a mold cavity depends on the packing pressure – higher pressures will result in heavier parts with larger dimensions (Why?). Applying excessive packing pressure is referred to as overpacking and the resulting heavier parts are said to be overpacked. 4. Cooling phase. After the gate freezes off, during further cooling no more material can be added to compensate for continuing shrinkage. 5. Ejection. Once the part has cooled to a predetermined temperature, the mold is opened and the part is ejected. The time variation of pressure imposed on a mold during a molding cycle is schematically shown in the pressure-time diagram in Figure 17.2.3. The molding cycle starts with the closure of the mold (point marked A), after which melt injection from the nozzle into the sprue begins. The melt enters the mold at point B on the timeline, after which the injection pressure increases continually to force the melt deeper into the mold; the actual pressure-time profile depends on the nature of the flow front – for example radial versus straight. The mold is completely filled at point C. Then, the injection pressure is rapidly raised to a peak at point D to force the melt against the mold surfaces to replicate the surface texture and to eliminate any voids. This is followed by a holding pressure being applied along DEF to force more melt into the mold cavity to compensate for the shrinking volume of the melt as it solidifies. The gate freezes off – that

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STATIONARY PLATE

NOZZLE

MOVABLE CAVITY

GATE

MOVABLE CORE

(a)

PRIMARY SPRUE

RUNNER

SECONDARY SPRUE

MOLDED PARTS

(b) Figure 17.2.2 Layout of the sprue, runners, secondary sprues, and gates in a three-plate, two-cavity mold. (a) Melt from the nozzle is injected into the two mold cavities through the sprue, runners, secondary sprues, and two gates. (b) On opening the mold, the sprue-runner-secondary-sprue assembly and the two molded parts separate out.

is, the melt at the gate solidifies – at point F, after which no more material can be forced into the mold, so that the material shrinks continually resulting in the pressure decreasing along FG. Thereafter, the material continues to shrink in the mold under zero pressure. Finally, after demolding at point H, the part continues to shrink as it cools to the ambient temperature. This simple description describes the pressure

Injection Molding and Its Variants

history in a mold. But in an actual molding cycle, different points in a mold see different pressure-time histories; some aspects of such differences are discussed in Section 18.3. MOLD CLOSED MELT INJECTION INITIATED

MOLD PACKING HOLD PRESSURE

MOLD FILLING

IN-MOLD SHRINKAGE

D

MOLD CAVITY PRESSURE

E

F C MOLD-SURFACE CONTACT LOST

A

G

B

H

TIME Figure 17.2.3 The pressure inside a mold during a molding cycle.

The preceding discussion has described the generic mechanics of an injection molding cycle without considering the properties of the molding resin – such as its viscosity-temperature characteristics, how much it shrinks, and the temperature at which it degrades. First, at any given temperature the viscosities of different resins are very different as is their dependence on shear rates. The shear rates depend both on the speed of filling and the part thickness – thinner parts and faster filling rates cause higher shear rates. The simple schematic “molding diagram” in Figure 17.2.4 is useful for explaining which combinations of processing parameters result in well-formed molded parts. First consider mold filling. On entering the mold, the melt begins to freeze at the mold surfaces. Depending on the part thickness, if the injection pressure and the melt temperature are too low, the material close to the gate or at the gate will freeze off before the part is filled. This partially formed part is called a short shot (SS). Clearly, the SS will be longer if the melt temperature is increased. The locus of the pressure-temperature combinations at which the part just fully fills is indicated by the line AB in Figure 17.2.4; while processing states below this line will result in SSs, parts will be fully formed above this line. The actual shape of this curve will depend on the thermal properties of the resin, and its vertical position will depend on the part thickness.

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H

F

C

THERMAL DEGRADATION

K

EXCESSIVE SHRINKAGE

L

A

D

MOLDING WINDOW

FLASH

MELT INJECTION TEMPERATURE

452

I J E

B SHORT SHOT

G

• MOLD HOLDING PRESSURE Figure 17.2.4 Schematic pressure-temperature diagram showing processing window for injection molding.

Resins degrade at high temperatures. However, it is not just the temperature but also the duration of exposure to this high temperature that determines when a resin will degrade – for longer exposures degradation will occur at lower temperatures. At low molding pressures it takes longer to fill a part at a given high temperature; lower residence time at higher pressures will result in degradation occurring at higher temperature. The locus of the degradation points is indicated by line CD in Figure 17.2.4; the material will not degrade below this line. Note, however, that higher pressures cause higher shear rates – which also cause material degradation – so that when this effect is accounted for, the point D may move downward; thus, the curve CD could become near horizontal, or may even dip downward to the right (point D being lower than C). At low pressures, the amount of material packed into the mold may not be sufficient, resulting in higher part shrinkage. For any given pressure, the amount of packing will increase with the temperature (Why?), so that the excessive shrinkage boundary has the shape shown by the curve EF – lower shrinkage to the right of this curve. For a given mold clamping force, the tendency for the mold halves to slightly move apart under pressure can result in the melt oozing out at the parting line to form “flash.” For a given mold pressure, lower melt viscosities at higher temperatures will result in the boundary for flash formation having the shape shown by curve GH; flash being formed to the right of this boundary.

Injection Molding and Its Variants

Finally, integrating the four molding limiting cases results in the molding window, IJKL, within which acceptable parts can be molded. Note that the main utility of this forming diagram is to explain the complexities involved in part molding. The shape of the actual forming window has to be computed for each part geometry, the molding material, and the injection rate (speed). During the injection molding process, the plastic melt undergoes complex, transient flow and thermal effects that have a profound effect on part performance. This complexity cannot be analyzed by simple closed-form analyses. Only robust numerical codes are capable of computing the complex, nonlinear coupled flow, heat transfer, and freezing phenomena that occur during the molding of geometrically complex parts. The following sections discuss different aspects of this complex molding process. 17.2.1

Mold Filling

Depending on the material, temperature, and mold geometry, it is possible for the injected material to freeze before the mold cavity fills. Also, the flow in the mold can be influenced by walls and inserts. One technique for visualizing the part molding process is to mold SSs of varying time lengths: Successive “parts” are molded for longer times. Stacked composites of such SSs then shows how the flow fronts develop to eventually fill the part. 17.2.1.1 Filling of an Off-Center Gated Mold Cavity

When injected from one wall forming a constant-thickness mold cavity, the melt will flow uniformly in a radial direction; the flow front will then be a circle of increasing radius centered on the melt injection point (refer to Section 6.6.6). The circular flow front will continue to expand till it is blocked by a lateral mold wall. Figure 17.2.5 shows a rectangular mold cavity ABCD in which the melt is injected in an off-center gate located at G. Initially, the melt will flow radially till it touches the lateral wall AB at point H. Further melt injection will cause the contact point of the flow front at H to form a contact surface that moves along the two lateral directions indicated by the dashed arrows. The flow front away from the walls will continue to expand radially till it meets obstructions to flow. For the mold-gate geometry shown in Figure 17.2.5, the 12 stacked acrylonitrile-butadiene-styrene (ABS) short-shots (SSs) in Figure 17.2.6 illustrate the complexity of flow fronts resulting from their contact with mold walls. SS-1shows initial radial flow. In SS-2 the flow has gone beyond the limiting case for radial flow shown in Figure 17.2.5a with increased contact along AB. This contact area increases in the subsequent short-shots, SS-3 and SS-4, with the flow front almost touching the lateral mold wall DA. Notice the radial flow pattern in SS-1 to SS-4 away from wall AB. In SS-5 the flow front has progressed beyond just touching wall DA; the corner at A has filled with substantial contact with wall DA. In SS-6 to SS-9 the flow front has filled corner D with increased contact along the walls AB and DC. In SS-10 the flow front just touches wall BC close to corner B. In SS-11 the flow front has made substantial contact with wall BC. Finally, SS-12 shows the complete plaque with all the corners filled out. 17.2.1.2 Filling of a Double-Gated Cavity

If two or more gates are used, the flow fronts emanating from these gates will intersect. Because the contact points are not rigid, subsequent flow can convect, or move, the contact points. The interactions of the flow fronts in a two-gated cavity for semicrystalline polymer polypropylene (PP) are shown by the short-shot stacks in Figure 17.2.7. Notice that the left runner has a larger diameter than the right one, and that the melts are injected by small fan gates. The flow fronts from both gates radiate outwards with the

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C

D RADIAL MELT FLOW-FRONT

G

A

B

H

(a) G

MOLD CAVITY

(b) Figure 17.2.5 Geometry of an off-center gated plaque ABCD. (a) Plan view showing radial flow from gate located at G at the instant when the flow front touches the mold wall AB at H. Further melt injection will cause the flow front to form a contact surface with AB that moves in the two lateral directions indicated by dashed arrows. (b) Mold cross-section through gate G.

bottom portions spreading along the bottom lateral mold surface. The flow fronts from both gates then contact the upper mold surface and begin to spread across it. At some stage the spreading melt surfaces along the bottom surface come into contact. Further flow causes the contact surface to move upward, eventually filling up the space between the two flow fronts. The locus of the of the contact between these two flow fronts – variously referred to as a knit line, or a meld line, or a weld surface – plays an important role in surface esthetics, and can also affect strength; the formation of such surfaces are discussed in Section 17.3. 17.2.1.3

Effects of Material Differences on Flow in a Double-Gated Cavity

Differences in flow properties among different polymers can be expected to result in different flow patterns in the same mold and gating configurations. The interactions of the flow fronts in a double-gated cavity for the amorphous polymer ABS and the semicrystalline polymer PP are shown, respectively, by the short-shot stacks in Figure 17.2.8. In ABS, at the knit line – the locus of the flow front contact points – the contacting surface push, or extrude, fluid outward, thereby forming a complex knit area. The shape of the PP knit line is different with a well-defined knit line. The small “V” notch at the top of the last short-shot shows the difficulty in filling the last part of a knit line. Reflecting the differences in their flow properties, the detailed shapes of the flow fronts of ABS and PP are clearly different.

Injection Molding and Its Variants

C

D

12 11 10

G

9 B

A 1

2

3

4

5

6

7

8

Figure 17.2.6 Twelve stacked ABS short-shots of the plaque ABCD molded in the mold shown in Figure 17.2.3. Short-shot 1 shows the initial radial flow that continues till the radial flow front touches the mold wall AB, as shown in Figure 17.2.3a. (Adapted from Cornell University Injection Molding Program (CIMP) Progress Report No. 4, 1977, original photo courtesy of Professor K.K. Wang.)

G-1 RUNNER

G-2 SPRUE

RUNNER

Figure 17.2.7 Short-shot stacks for a double-gated cavity being filled by a PP melt. (Adapted from Cornell University Injection Molding Program (CIMP) Progress Report No. 5, 1978, original photo courtesy of Professor K.K. Wang.)

17.2.1.4 Effects of Slits in a Mold Cavity

Parts can have many features like slits, holes, and bosses, made by placing inserts inside the mold cavity around which flow fronts have to divide, flow, and reunite, resulting in complex flow fields. The previous section showed how flow fields from two gates meld together to fill a part. But the presence of obstructions in the mold cavity can cause such melding of flows emanating from a single gate. Figure 17.2.9 shows the short-shot patterns for ABS for an edge-gated rectangular cavity with two inserts to form slits. Notice the complexity of the flow around the slits.

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(a)

(b) Figure 17.2.8 Short-shot stacks for a double-gated cavity. (a) ABS. (b) Polypropylene. (Adapted from Cornell University Injection Molding Program (CIMP) Progress Report No. 4, 1977, original photos courtesy of Professor K.K. Wang.)

G Figure 17.2.9 Short-shot stacks for flow of ABS in an edge-gated cavity with two slits. (Adapted from Cornell University Injection Molding Program (CIMP) Progress Report No. 4, 1977, original photo courtesy of Professor K.K. Wang.)

To better visualize how the flow front separates and reunites around the slits, tracings of the first five SSs in this figure have separately been shown in Figure 17.2.10. In the first two figures, the flow front emanating from the single gate starts to wrap around the slits. In Figure 17.2.10c, after flowing around the vertical slit, the flow front recombines in a cusp at the slit surface. In Figure 17.2.10d, while the cusp on the vertical slit has formed into a flow front, the flow around the horizontal rectangular slot has yet to combine. Finally, as shown in Figure 17.2.10e, well-developed flow fronts are formed to the left and right of the inserts. As shown in Figure 17.2.9, these flow fronts eventually fill the entire mold cavity.

Injection Molding and Its Variants

(a)

(b)

(c)

(d)

(e) Figure 17.2.10 Progression of flow fronts around slits for the short-shot sequence shown in Figure 17.2.9.

17.2.1.5 Flow in a Double-Gated Cavity with Inserts

Figure 17.2.11 shows the short-shot sequence for the filling of a rectangular mold cavity, having three inserts, by two edge-gates for (a) ABS, and (b) PP. Notice the enormous complexity of the flow around the slits, and the flow of the melt around the hole on the right. As in Figure 17.2.8, the flow fronts for ABS and PP have different shapes. To better visualize the complexity of the advance, separation, and reunification of the flow fronts around the slits and the circular hole, tracings of all the nine SSs for polypropylene (Figure 17.2.11b) are shown in the sequence in Figure 17.2.12. At first, the flow fronts from each of the two gates progress

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(a)

(b) Figure 17.2.11 Short-shot stacks for a double-gated cavity with two slits and one circular inserts. (a) ABS. (b) Polypropylene. (Adapted from Cornell University Injection Molding Program (CIMP) Progress Report No. 4, 1977, original photos courtesy of Professor K.K. Wang.)

independently, wrapping around the two slits (Figure 17.2.12a – d). These two flow fronts then begin to coalesce (Figure 17.2.12e,f), finally forming a rightward moving flow front (Figure 17.2.12g – i) that separates and then recombines at the circular hole. 17.2.2

Part Thickness

The molten material begins to solidify as soon as it comes in contact with the cold mold walls. While mold filling and cooling of the melt at the mold surface occur simultaneously in a transient manner, an estimate for the time to cool a part may be obtained by assuming that, at time t = 0, the mold is filled with the melt at a uniform temperature T0 = Tmelt , at which time a mold temperature of T1 = Tmold is imposed on the melt. Let the part thickness be h. Then, from the results in Section 6.2.2, it follows that the time, tcool , for the mold temperature to be √ felt at the part mid-thickness – which is a good estimate for the cooling time – is given by h∕2 = 4 𝛼 tcool , so that tcool = h2 ∕(64 𝛼 ), where 𝛼 is the thermal diffusivity of the plastic melt, which typically is about 0.1 mm2 s-1 . Using this value of 𝛼 , the cooling times for part thicknesses of 0.5, 1, 2, 3, 4, 5, and 10 mm are 0.04, 0.16, 0.63, 1.4, 2.5, 3.9, and 15.6 seconds, respectively. Since machine productivity is very important for cost effective parts, most plastic parts have thin walls to keep the cycle time (molding time) small. Typically, wall thicknesses are smaller than 6 mm (0.25 in). Smaller thicknesses on the order of 3 mm (0.125 in) are preferred. In many thin-wall applications, such as in cellular phones, the wall thicknesses are below 2 mm (0.08 in). Because of the very high viscosities of thermoplastic melts, molding thin-wall parts adds to processing complexity.

Injection Molding and Its Variants

(a) G-1

G-2

G-1

G-2

G-1

G-2

G-1

G -2

G-1

G -2

G-1

G -2

G-1

G -2

G-1

G -2

G-1

G -2

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 17.2.12 Progression of PP flow fronts around two slits and a circular pin in a double-gated, rectangular cavity. These flow fronts correspond to the nine short-shots for PP shown in Figure 17.2.11b.

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The large shrinkage of plastics on solidification causes two problems: shrink marks at the relatively thick sections at bosses and ribs, and part shrinkage and warpage. 17.2.3

Mold Clamp Forces

A limitation of injection molding is part size. Because the viscosities of molten thermoplastics are very high, very high pressures are required to force the melt through the mold cavity, resulting in very large clamp forces to prevent the two mold halves from opening. For example, the clamp force required for molding a 1 m2 panel is on the order of 10,000 tons. The time variation of the pressure inside a mold during injection molding is schematically shown in Figure 17.2.3. The pressure profile in this simple figure is more representative of the pressure in the mold at the gate: During mold filling the highest pressure is at the gate, the pressure at the flow front being close to the atmospheric pressure. At the end of filling, the pressure inside the mold will vary from a high value at the gate to atmospheric pressure at the mold far end. At any instant, the force exerted by the melt on the mold halves will be given by (Why?) F=

∫A

p dA

so that the minimum force required to prevent the mold halves from opening will be the maximum value of this integral over the molding cycle. Once the mold has been filled, the pressure throughout the mold will be uniform at the holding pressure level. This is schematically shown in Figure 17.2.13. An estimate for the required clamp force, F, can be obtained by applying this pressure, p0 , over the entire projected mold area, A, giving F = p0 A. Injection molding covers a wide range of materials for which holding pressures can be as large as 100 MPa, for a 1 × 1 m (1 m2 ) part requires a clamping force 10,000 tons. For this high pressure, the clamping force for a 0.25 × 0.25 m (0.0625 m2 ) part would be about 650 tons. Even for a holding pressure of 50 MPa, the clamping forces for parts of these two sizes would be about 5,000 and 625 tons, respectively. So, parts in this size range require machines with large clamping force capabilities. 17.2.4

Mold Cooling

The heat from the melt extracted by the mold during the molding process has to be conducted away; else the mold temperature will increase with each molding cycle. Large-part molding can involve substantial amounts of heat extraction, requiring mold cooling by water circulating through pipes embedded in the mold. Let the plan area of the part in contact with the mold be A and let the part thickness be h. Then the heat Q to be extracted in each cycle is given by Q = 𝜌c Ah (Tmelt − Tmold ), where 𝜌 and c are, respectively, the density and the specific heat of the melt. The heat to be extracted per unit area of the part is then given by Q∕A = 𝜌c h (Tmelt − Tmold ). Now, the time during which the part is in the mold is given approximately by tcool = h2∕(64 𝛼 ). It then follows that the average heat flux – heat per unit area per unit time – into the mold is given by q = Q ∕(A tcool ) = 64 k (Tmelt − Tmold )∕h, where k = 𝜌c𝛼 is the thermal conductivity of the melt. This expression gives the total average heat flux, half of this amount going to each half of the mold.

Injection Molding and Its Variants

MOLD PROJECTED AREA

A

CLAMP FORCE MOLD NOZZLE

MELT

p0 MOLD

F = pA0 Figure 17.2.13 Estimate for mold clamp force.

This simple analysis shows that although the heat Q to be extracted from a part increases linearly with the part thickness, the rate of heat removal per unit area per unit time, q, which varies as the inverse of the part thickness, is smaller for thicker parts. This is a consequence of the part cooling time varying as the square of the part thickness.

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While the main function of the mold is to provide a cavity for molding parts, it also acts as a heat exchanger for removing heat from a cooling part.

17.3 Fountain Flow In the one-dimensional, pressure-driven steady flow of a Newtonian fluid in a rectangular channel discussed in Section 6.6.5, the velocity distribution normal to the flow direction has the parabolic distribution shown in Figure 6.6.2. Along the flow direction the pressure drops linearly with the distance, but the velocity remains unchanged. In the lateral, cross-flow direction the velocity has a parabolic distribution, but the pressure does not vary. The flow-pressure-drop equations for this flow form the basis for many engineering applications. However, they are not applicable to mold filling during which the flow is unsteady, or time-dependent. This time-dependent, developing flow during mold filling is called fountain flow. Elements affecting fountain flow are schematically shown in Figure 17.3.1. The melt injected into the mold at gate G first undergoes two-dimensional entrance flow at E; in addition to flow along the mold cavity, the material flows laterally to fill the corners at F and D. On coming in contact with the cold mold walls the melt begins to solidify, resulting in thin solidified layers over which the rest of the melt must flow with the parabolic-type velocity distribution shown, in which the velocity is zero at the surfaces of the solidified layers. At the flow front, ABC, where the melt meets air, the flow bifurcates resulting in lateral flows – similar to the flow on the top of a water fountain; hence the name – that carry the hot melt in the core to the cold walls. An understanding of this bifurcated flow at the flow front is important for understanding how polymer chains in the melt are deformed before being deposited on the mold walls. In conjunction with the cooling of the melt, this flow determines chain orientation in amorphous resins and the layered crystalline structure in parts molded from semicrystalline resins. These structures not only affect mechanical properties but are responsible for inducing optical birefringence. SOLIDIFIED LAYER MOLD CAVITY WALL

A

F

E

G

B

C

D MELT VELOCITY DISTRIBUTION

Figure 17.3.1 Melt injection into a mold cavity: Melt flow, solidification at mold walls, and flow bifurcation at the flow front.

Injection Molding and Its Variants

Clearly, such complex, time-dependent transient flows – which are accompanied by a time-varying channel size reduction, complex nonlinear temperature-dependent stress-strain-rate relations, and a melt-air interface – can only be analyzed through numerical simulation. However, even simple, highly idealized isothermal, Newtonian-flow analyses can provide insights into such complex flows. The results of a numerical simulation of an isothermal Newtonian fountain flow close to the flow front are shown in Figure 17.3.2, with the x-coordinate aligned along the cavity and the y-coordinate normal to the flow direction. For analytical convenience and ease of interpretation, the coordinates have been normalized as 𝜉 = x∕h and 𝜂 = y∕h. In terms of these normalized coordinates, the mold cavity has a height of 1 unit, and the flow profile is shown for a channel length of about 1.5 units. In the melt ABCDEF, the flow front ABC is moving to the right with a velocity V. The top half ABEF shows the velocity distribution; at the point of the arrow, the length and direction of the arrow indicates the magnitude and direction of the velocity, respectively. The bottom half BCDE shows the stream lines. MOLD CAVITY WALL

0.5

A

F

O

x

η = y/h

y

0

− 0.5

E B

V

C

D

0

0.5

1

1.5

ξ = x/h Figure 17.3.2 Melt flow front for an isothermal, Newtonian fluid in a rectangular cavity: Velocity vectors (top half) and stream lines (bottom half). (Adapted with permission from H. Mavridis, A.N. Hrymak, and J. Vlachopoulos, Polymer Engineering and Science, Vol. 26, pp. 449 – 454, 1986.)

Notice that the stream lines are straight till about 𝜉 = 0.75, and the velocity distribution across the mold cavity height has an essentially parabola-like shape associated with steady flow. But in the flow-front region ABC, which extends approximately from about 𝜉 = 0.75 to about 𝜉 = 1.75, the flow directions bend toward the mold walls; at point B, the flow front, which is moving to the right with velocity V, bifurcates at right angles to the flow direction. It appears that the two-dimensional fountain-flow effects

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are confined to a distance of only a few mold heights. However, the flow in this small region critically affects the microstructures in a molded part. Another way to look at these results is to move with the flow front, which will show the flow phenomenon as observed from a “stationary” flow front. This can be achieved by imposing a leftward velocity V to the mold cavity. The resulting velocity distribution and stream lines relative to the front are shown in Figure 17.3.3. This figure clearly shows the importance of the flow field at the flow front ABC; the change in the flow direction causes the hot melt in the central core around 𝜂 = 0 to be transported to the cold mold surface, where this hot melt solidifies.

0.5

V

A

F

y

O

x

η = y/h

464

0

− 0.5

E B

V

C

D

0

0.5

1

1.5

ξ = x/h Figure 17.3.3 Melt flow front for an isothermal, Newtonian fluid in a rectangular cavity: Velocity vectors (top half) and stream lines (bottom half) relative to a coordinate system moving with the flow front at B. (Adapted with permission from H. Mavridis, A.N. Hrymak, and J. Vlachopoulos, Polymer Engineering and Science, Vol. 26, pp. 449 – 454, 1986.)

While Figures 17.3.2 and 17.3.3 provide insights into the flow patterns near the flow front, they do not give a feel for the deformations that material elements undergo. This is addressed in Figure 17.3.4, which shows the result of the numerical simulation of the flow of a non-Newtonian, nonisothermal melt in a mold cavity. The numerical simulation tracks the shape of an initially rectangular material element, marked 1, as it moves with the flow, deforming into shapes 2 through 9. The element is stretched continually as it flows past the flow front ABC before being deposited on the cold mold wall. Finally, the original top end is pulled away from the wall resulting in the V-shaped geometries shown in shapes 8 and 9.

Injection Molding and Its Variants

0.5

A

F 9

y x

η = y/h

O

1

0

− 0.5

8 2

3

V

7 6 4 5

E B

C

D

0

0.5

1

V 1.5

ξ = x/h Figure 17.3.4 Melt flow front for a nonisothermal, non-Newtonian fluid in a rectangular cavity: Deformed shape of an initially rectangular material element (top half) and stream lines (bottom half) relative to a coordinate system moving with the flow front at B. (Adapted with permission from H. Mavridis, A.N. Hrymak, and J. Vlachopoulos, Journal of Rheology, Vol. 32, pp. 639 – 663, 1988.)

17.3.1

Meld Surfaces and Knit Lines

During injection molding two molten fronts may meet and coalesce together, either head-on – in which case the interface is called a knit, or weld surface – or flow together tangentially – this interface is referred to as a meld surface. The locus of the intersections of such surfaces with the part surface are referred to as the knit (or weld) and meld lines, respectively. The coalesced flow fronts can result in mechanically weak interfaces and in surface flaws that affect esthetics. 17.3.1.1 Head-on Welding of Two Flow Fronts

The simplest example of flow front coalescence is that of molding of a “picture frame,” schematically shown in Figure 17.3.5, in which the flow at the gate divides in two opposing directions, resulting in two flow fronts that meet head-on during the last stages of mold filling. The complexity of the melding of two opposing flow fronts meeting head-on, as in the previous “picture frame” example, can be gauged from Figure 17.3.6, which show the melt velocities and stream lines when the two flow fronts touch at point B. The left- and right-halves of this figure are from the numerically obtained flow pattern for isothermal, Newtonian fountain flow shown in Figure 17.3.3. Subsequent flow will result in the cavity AKB filling up by lateral flow. The subsequent flow surfaces are schematically

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K

A F-1

D

B

F-2

C

G

Figure 17.3.5 Division of melt flow on entrance at gate G in a “picture frame” mold ABCD. The flow paths of the two divided flows are indicated by dashed arrows. The two flow fronts, F-1 and F-2, indicated by solid arrows, approach each other and meld together at K.

shown (not calculated) by dashed lines. Clearly, even in this “head-on” coalescence of two flow fronts, after initial contact, the two flow surfaces meld together (dashed flow fronts from B to K). As the shrinking flow surface reaches the points on K the surface, trapped air makes it difficult for the melt to fully reach the mold surface, resulting in a tell-tale knit line. (Note the discussion regarding the V-shaped notch in Figure 17.2.3.) Reducing this effect would require unrealistically high pressures. Although this meld flow is very different from the flows that occur during butt welding, the meld surface KK is sometimes referred to as the weld surface and its trace on the part surface is called a weld line.

F

V

A

K

E B

D

C

V

K

Figure 17.3.6 Melt velocities and stream lines at the instant of two flow fronts touching (at point B). Flow fronts during subsequent filling of the cavity AKB are schematically indicated by dashed lines.

For resins and particulate-filled resins the main issue with meld lines is surface finish. However, in fiber-filled materials, improper mixing at meld surfaces results in fibers not properly bridging across such surfaces. As a result, material strength at meld lines in such materials can be much reduced.

Injection Molding and Its Variants

17.3.1.2 Melding of Flow Fronts Around a Pin

As shown by the short-shot sequences in Figures 17.2.9 and 17.2.10, the flow front in a single-gated mold can divide around obstacles, and the divided fronts can then meet again after flowing around the obstacles. The melding of such bifurcated flow fronts is more complex than the head-on melding discussed in the previous section. This section describes the simplest, non-head-on melding case, that of bifurcation and unification of flow around a pin used for molding a hole. Figure 17.3.7 schematically – not computed or through short-shots – shows how a flat flow front bifurcates and recombines after flowing around a pin. In this example a flat flow front approaches the bottom surface of the pin with a uniform velocity. On touching the lowermost pin surface at A, the flow is forced sideways, equally in both lateral directions, as indicated by the arrow below point A on the cylinder. Close to the cylinder surface the flow is forced outward, as indicated by the outward facing arrows (Figure 17.3.7c). Notice how slightly away from the cylinder the flow surface moves toward the cylinder surface as indicated by the inward-inclining arrows (Figure 17.3.7a – c). This trend continues till the flow front reaches the mid-height of the cylinder, after which the flow is driven toward the cylinder surface (Figure 17.3.7d – f). Eventually, the two bifurcated flows meet at the point B on the top surface of the cylinder (Figure 17.3.7f), and then meld together, resulting a very small unfilled region above point B, and dimpled surfaces (points D in Figure 17.3.7g,h) in the otherwise recombined flat flow front. Eventually, this flow front becomes flat. This flow field results in a surface blemish along BD. Some of these features can be seen from the last four short-shots in Figure 17.2.11b and from Figure 17.2.12g – i. B

B

B

B

A

A

A

A

(a)

(b)

(c)

(d)

D

D

B

B

B

B

A

A

A

A

(e)

(f)

(g)

(h)

Figure 17.3.7 Schematic description of flow bifurcation and recombination of flow around a circular pin. (a) A flat flow front approaching the pin from the bottom bifurcates at point B. (b – e) The bifurcated flow fronts wrap around the cylinder. (f) The bifurcated flow fronts touch. (g and h) The bifurcated flow fronts meld, leaving a very small unfilled region above point B and a dimpled, receding flat flow front.

The remainder of this section describes some results from experiments on the flow of molten polystyrene (PS) in a 3 × 80 × 100 mm cavity with a 20-mm diameter circular pin at its center. The

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geometry of the parts molded in this cavity is shown in Figure 17.3.8, in which the dashed lines indicate 8-mm wide specimens that were used to measure the characteristics of the surface flaw along the meld line AB. 3 mm 80 mm S-5 S-4

B

S-3 S-2 S-1

A

100 mm

468

20 mm

Figure 17.3.8 Geometry of molded part with a central hole. The dashed lines show the locations of five 8-mm wide specimens, S-1 through S-5, that were cut to measure the characteristics of the surface flaw along the meld line AB. (Adapted with permission from Koji Yamada, Kiyotaka Tomari, Umaru Semo Ishiaku, and Hiroyuki Hamada, Polymer Engineering and Science, Vol. 45, pp. 1180 – 1186, 2005.)

The flow around the pin was characterized by short-shots, some of which are shown in Figure 17.3.9. The bifurcated flow fronts approach each other at the top of the pin (Figure 17.3.9a), touch approximately 2 mm above the cylinder (Figure 17.3.9b) forming a knit surface, and then meld together (Figure 17.3.9c,d). In these figures the arrows indicate the flow directions and velocities at different points along the flow fronts. As the merged flow front expands, the material close to the cylinder stops moving – this stagnant region can clearly be seen in Figure 17.3.9d – eventually resulting in a very small, curvilinear-triangular-shaped unfilled region between the pin and solidified melt surfaces (Figure 17.3.9d). The cross sections of the V-notch at different points along the notch BC, as obtained by scanning electron microscopy (SEM) on polished surfaces of Specimens S-1 through S-5 are shown, respectively, in Figure 17.3.10a – e. Over 8-mm intervals from the pin surface at B, the V-notch depth drops from a maximum of 13.1 μm near B on S-1, to 9.8, 4.4, and 0.4 μm on S-2, S-3, and S-4, respectively. Specimen S-5 is essentially flat. As shown in this figure, the notch angles for these five specimens are, respectively, 40°, 63°, 132°, 165°, and 180°. Thus, the V-notch extends to a distance of about three times the pin radius. In addition to causing a surface blemish, the V-notch also affects the local strength. The schematic of a longitudinal cross section of a plaque along the plane ABC (Figure 17.3.11) shows three regions of the materials through the part thickness. The region marked V-notch is the outer surface that appears as a blemish. The region marked surface layer is a small region below the notch surface in which the bonding

Injection Molding and Its Variants

FLOW FLOW DIRECTION DIRECTION

1 mm mm INITIAL INITIAL FLOW FLOW FRONT FRONT CONTACT CONTACT

≈ 2mm 2 mm

UNFILLED REGION REGION

PIN

PIN

(a)

(b)

MERGED FLOW FLOW FRONTS

STAGNANT STAGNANT REGION REGION

PIN

PIN

(c)

(d)

Figure 17.3.9 Short-shots of molten polystyrene flow around a 20-mm diameter pin in a rectangular cavity. (a) The bifurcated fronts approach each other at the top of the pin. (b) The bifurcated flow fronts touch about 2 mm above the pin surface. (c) Motion of the merged flow fronts. (d) While the top flow front surface continues to move, the bottom surfaces stop moving resulting in a stagnant region that leaves a very small, curvilinear-triangular-shaped unfilled region next to the pin surface. (Adapted with permission from Koji Yamada, Kiyotaka Tomari, Umaru Semo Ishiaku, and Hiroyuki Hamada, Polymer Engineering and Science, Vol. 45, pp. 1180 –1186, 2005. Original photos courtesy of Dr. Koji Yamada.)

of the two melded surfaces is weak. Good bonding occurs in the core region, which can achieve the strength of the molded material. In addition to poor esthetics, the V-notch can act as a stress concentrator that can cause failure, especially under fatigue loading. 17.3.1.3 Effects of Gates, Part Geometries, and Materials on Knit Lines

In addition to the head-on coalescence discussed previously, flow fronts can come together in other ways: (i) The use of multiple gates for filling a cavity results in multiple flow fronts that eventually must coalesce to complete mold filling. (ii) In a single-gated mold the flow front can divide around an obstacle – such as a pin used for molding a hole – and the divided fronts can then meet again after flowing around the obstacle. And (iii), the use of multiple gates for parts with inserts results in even more complex coalescence patterns. Examples of knit-line formation for each of these mechanisms are discussed in the sequel.

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40 o

63 o

9.8 µm

13.1 µm

5 µm

5 µm (b)

(a)

165 o

132 o

4.4 µm

5 µm

0.4 µm

5 µm (d)

(c) 180 o

5 µm (e) Figure 17.3.10 SEMs of the surface notches on specimens cut from the plaques shown in Figure 17.3.8. (a – e) correspond, respectively, to specimens S-1through S-5. (Adapted with permission from Koji Yamada, Kiyotaka Tomari, Umaru Semo Ishiaku, and Hiroyuki Hamada, Polymer Engineering and Science, Vol. 45, pp. 1180 – 1186, 2005. Original photos courtesy of Dr. Koji Yamada.)

Figure 17.3.12 shows the short-shots (SSs) for the filling of a rectangular cavity with polypropylene using two gates, G-1 and G-2. The resulting knit line, the locus of the contact points of the two fronts during the filling process is shown by the white dashed line. Figure 17.3.13 shows the SSs for the single-gated flow of ABS melt in a rectangular cavity having two slits. The evolving flow fronts (Figure 17.2.6) show how the flow wraps around the slits to coalesce,

Injection Molding and Its Variants

C

B CORE REGION SURFACE LAYER V-NOTCH

A

Figure 17.3.11 Central cross section of plaque in Figure 17.3.8 through plane ABC showing details of three regions in the meld zone.

G-1 RUNNER

G-2 SPRUE

RUNNER

Figure 17.3.12 Knit lines superposed on short-shot stacks. (a) A double-gated cavity being filled by a PP melt. (b) A single-gated cavity with two slits and one circular insert being filled by an ABS melt. (c) A two-gated cavity with two slits and one circular insert being filled by a PP melt. (Adapted from Cornell University Injection Molding Program (CIMP) Progress Report No. 5, 1978, original photo courtesy of Professor K.K. Wang.)

forming the two knit lines, indicated by the two separate, black dashed lines in the order they were formed. Figure 17.3.14 shows the SSs for the double-gated flow of a PP melt in a rectangular cavity having two slits and a circular pin. The evolving flow fronts (Figure 17.2.7) show how the flow wraps around the slits and pin to coalesce, forming the five knit lines, indicated by the five, white dashed lines in the

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2 1

G Figure 17.3.13 Two knit lines, shown by black dashed lines, superposed on short-shot stacks for a single-gated cavity with two slits being filled by an ABS melt. (Adapted from Cornell University Injection Molding Program (CIMP) Progress Report No. 4, 1977, original photo courtesy of Professor K.K. Wang.)

2

3

4

5

1

Figure 17.3.14 Five knit lines superposed on short-shot stacks for a double-gated cavity with two slits and a circular insert being filled by a PP melt. (Adapted from Cornell University Injection Molding Program (CIMP) Progress Report No. 4, 1977, original photo courtesy of Professor K.K. Wang.)

order they were formed. Notice how the first two knit lines, marked 1 and 2, intersect to form the third knit line, 3. 17.3.2

The Role of Numerical Simulation

While short-shots do provide insights into the flow fields during mold filling, using this technique for visualizing flow fields in widely different part geometries and mold and gate configurations is impractical. Instead, numerical simulation of the flow in a particular geometry provides a practical means for analyzing and troubleshooting moldabilty issues. For the flow fronts for PP in a two-gated cavity visualized by the short-shot stacks in Figure 17.2.7, the knit line is shown by the white dashed line in Figure 17.3.15a. The corresponding flow fronts predicted by a numerical simulation of the flow are shown in Figure 17.3.15b. Such simulations provide means of assessing gating configurations for complex geometries: based on simulations, the gate configurations can be modified to move knit lines to regions where esthetics are less important or to regions of lower stresses to compensate for the lower strengths of knit lines.

Injection Molding and Its Variants

G-1

G-2

(a) KNIT LINE

G-1

G-2 (b)

Figure 17.3.15 (a) Knit line (dashed curve) visualized from short-shot stacks for a double-gated cavity being filled by a PP melt. (b) Flow front simulation for the double-gated cavity showing the position of the knit line. (Adapted from Cornell University Injection Molding Program (CIMP) Progress Report No. 5, 1978, original photo courtesy of Professor K.K. Wang.)

17.4 Part Morphology The discussion on fountain flow (Section 17.3) highlighted the complexity of flow during injection molding. Not only is material from the central, hotter core continually being deposited on the colder mold wall (Figure 17.3.3) – where it is quenched (rapidly solidified) – material elements undergo significant amounts of stretching and alignment (Figure 17.3.4). The stretching and alignment change the local properties and causes varying birefringence. It also affects residual stresses. In a clear amorphous thermoplastic, this alignment can be visualized by placing the part between to polarizing sheets. Fountain flow has a more dramatic effect on the morphologies of parts molded from semicrystalline materials, for which faster cooling rates result in smaller crystalline structures. Figure 17.4.1 shows the morphology of an approximately 160 by 170 μm transverse section of a polyoxymethylene (POM) (acetal) part molded in a flat mold. This transverse morphology has five distinct zones: Starting from the top surface, there is a 25 μm-thick skin; then a 10 μm-thick zone with twisted lamella; followed by a 15 μm-thick zone of fine asymmetric spherulites; a 50 μm-thick zone of oblate spherulites; finally followed by a core of spherulites. Thus, below the 25 μm-thick skin, there is a 75 μm-thick region in

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SKIN

OBLATE SPHERULITES

TWISTED LAMELLA

FINE ASYMMETRIC SPERULITES

SPERULITIC CORE

100 µm Figure 17.4.1 Polarized light micrograph showing five-layer morphology in injection-molded polyoxymethylene (acetal). (Original photo courtesy of Professor Musa Kamal.)

SKIN INTERMEDIATE LAYER

SPERULITIC CORE

100 µm Figure 17.4.2 Polarized light micrograph showing three/four-layer morphology in injection-molded high-density polyethylene (HDPE). (Original photo courtesy of Professor Musa Kamal.)

Injection Molding and Its Variants

which the crystalline structure changes from twisted lamella, to fine asymmetric spherulites to oblate spherulites, after which the morphology in the core region is fully spherulitic. This shows that even in a pure resin the internal structure (morphology) of an injection-molded part is far from homogeneous. Furthermore, the morphology in the same part made of a different material can be very different. As an example, Figure 17.4.2 shows the morphology of an high-density polyethylene (HDPE) part molded in the same mold used for the POM part. The morphology is quite different – the HDPE part only has three to four layers in comparison to the five in POM.

17.5 Part Design Each part fabrication, or processing, technique imposes limitations on the shapes that can be made, so that part shapes have to be modified to conform both to formability and function needs. This section addresses some of these practical considerations affecting the shapes of injection-molded parts. 17.5.1

Part Stiffening Mechanisms

In Chapter 5 it was shown that bending stiffness in thin-walled beams can be obtained by using cross sections in which the walls are away from the neutral axis. It is possible to achieve desired part stiffness by using closed sections – such as a D-shaped closed section – or by an open channel section, such as a C-shaped section. Also, appropriate molded-in ribbing can enhance the bending stiffness of a beam. However, as discussed in Chapter 5, the torsional stiffness of open sections is dramatically smaller than that of a corresponding closed section. This difference is important because it is not practical to mold long beams having closed cross sections, especially if the cross section varies along the length or if the part has features such as local indents/bosses or if the walls curve along the length. As such, special attention has to be paid to achieving desired levels of torsional stiffness. Figure 17.5.1 shows two long beams, each having a width b, a web depth d, and a uniform wall thickness t. From the discussion is Section 5.2, for bending in the x-y plane, that is bending by a moment Mz , the D-section is stiffer than the C-section. However, these stiffnesses can be equalized by doubling the thickness of the C-section web (Why?). But the torsional stiffnesses of the two sections are very different: It follows form Sections 5.6 and 5.5 that the torsional stiffness of the “D” section (Figure 17.5.1a) and the “C” section (Figure 17.5.1b) are given, respectively, by JD = 2(bd) 2 t∕(b + d) JC = (2b + d) t 3∕3 so that their ratio is given by JD 6 = (bd∕t) 2 JC (b + d)(2b + d) Since b, d ≫ t, this ratio is very large. As an example, with b = d, this ratio is JD ∕JC = (b∕t)2 , a very large number. One way of achieving higher torsional stiffness in such open-sectioned beams is to mold in one or more cross ribs of the type shown in Figure 17.5.2. An alternate means of providing additional torsional stiffness would be to mold in one or more ribs that span the channel opening, as indicated in Figure 17.5.3.

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l

b d

y O··

x z (a) b d

(b) Figure 17.5.1 Geometry of (a) a closed D-section beam, and (b) an open C-section beam.

Figure 17.5.2 Torsional stiffening of a molded C-section beam by molded-in cross ribs.

17.5.2

Molding-Driven Features

Besides the restrictions imposed on how an injection-molded part can be stiffened, additional design considerations include cycle-time reduction, sink-mark reduction, and ease of part ejection from a mold. The sequel addresses such issues. 17.5.2.1

Part Thickness Distribution

As discussed in Section 17.2.2, cycle-time reduction requires that plastic parts be thin-walled. So, while in a metal-based part adequate bending stiffness may be provided by the shape shown in Figure 17.5.4a, this geometry is not suitable for plastics. Instead, an equivalent geometrical stiffness can be obtained

Injection Molding and Its Variants

(a)

(b) Figure 17.5.3 Torsional stiffening of a molded C-section beam by molded ribs.

by the thin-walled design shown in Figure 17.5.4b. Figure 17.5.4a,b shows equivalent geometries for an even thicker part. The same principle applies for the recessed part shown in Figure 17.5.5.

(a)

(b)

(c)

(d)

Figure 17.5.4 Original and uniform thickness part geometries with equivalent bending stiffnesses.

At times it may be necessary to have a surface, such as AB shown in Figure 17.5.6a, offset from the adjoining surface (CD), as for example, for mounting a device. Since better processability requires more uniform, gradually thickening parts, the required surface offset can be obtained by the shape shown in Figure 17.5.6b.

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(a)

(b)

Figure 17.5.5 Original and preferred uniform thickness part geometries for better processability.

A

B

C

D (a)

(b)

Figure 17.5.6 Original and preferred uniform thickness part geometries for better processability.

17.5.2.2

Part Shrinkage

Plastics undergo very large-volume reductions – on the order of 3%, which translates into linear part reductions on the order of 1% – that result in the molded part being smaller than the mold dimensions. If this shrinkage were uniformly distributed through the part, it could be compensated for by uniformly increasing the mold cavity dimensions. However, the shrinkage is not uniformly distributed; in addition to inducing residual stresses, such differential shrinkage can cause the part to warp. The mechanisms underlying shrinkage and warpage are addressed in Chapter 18. Shrinkage during molding can result in unacceptable surface unevenness, the most common of which are “sink marks,” which are small local surface depressions. The sequence of diagrams in Figure 17.5.7 shows the evolution of a sink mark in a uniform thickness T-section. The first figure (Figure 17.5.7a) shows the mold filled with a melt at the instant at which the gate has frozen off, so that more material cannot be injected to compensate for the subsequent shrinkage; for simplicity, it is assumed that surface layers have not frozen off. Figure 17.5.7b shows the part an instant later when a thin layer of the melt in contact with the mold surfaces has solidified. On straight sections the solidification front progresses uniformly. At the corners marked A, melt included in a 90° region is cooled on two sides, so that it cools more rapidly resulting in a faster progression of the solidification front at these convex corners. However, at the concave corners marked B, the two sides have to cool a 270° region, resulting in a slower solidification at such reentrant corners. These trends continue as more solidification occurs, as shown in Figure 17.5.7c,d. During the solidification progression shown in Figure 17.5.7a – d the entire outer solidified skin shrinks around the molten core. At some stage (Figure 17.5.7e), the bulk of the part solidifies leaving a molten core in the center; instead of the entire solidified material shrinking further, it is easier for the shrinkage to be taken up by the skin being pulled inwards, resulting in a depression of the surface, indicated by S – this depression is called a sink mark. Further into the solidification process (Figure 17.5.7f) the sink wall thickness becomes so large that it is no longer able to accommodate additional shrinkage. Then, because the pressure in the residual melt is so low, dissolved gases are released resulting in the formation

Injection Molding and Its Variants

FLANGE

MELT

A

B

RIB

A (a)

(b)

(c)

S

S

HOLE SOLID

(d)

(e)

(f)

Figure 17.5.7 Schematic diagrams showing the mechanism of sink-mark formation in a T-sectioned part.

of porous voids. Finally, a cavity is formed. The formation of sink marks above ribs can be avoided by reducing the residual melt in the final phase; as a rule of thumb, sink marks can be avoided by limiting the thickness of the rib to less than 60% of the flange thickness. In addition to reduced thickness, processing considerations normally limit the height of ribs to five times the flange thickness. Sink marks appear whenever a part cross section has a large enough area in which a small pool of melt is left near the end of the solidification process. As, an example, Figure 17.5.8 shows a solidification sequence at a rectangular bend in a part.

S

A

S

B

HOLE

A (a)

(b)

Figure 17.5.8 Evolution of a sink mark in an L-sectioned part.

(c)

(d)

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Unless rectangular corners are considered absolutely necessary, such sink marks can be avoided by rounding the corners marked A and B. The more uniform the thickness at the corner, the more uniform the solidification will be, so that in addition to no sink marks, residual stresses in the part will be reduced. Progressively better designs for a corner are shown in Figure 17.5.9.

(a)

(b)

(c)

(d)

Figure 17.5.9 Progressively better designs for a corner.

17.5.2.3

Part Warpage

Differences in shrinkage between the outer surfaces of a part can result in the out of plane distortion – even in a nominally flat panel, as shown in Figure 17.5.10. Such distortions are referred to as warpage. LARGER SHRINKAGE

SMALLER SHRINKAGE

Figure 17.5.10 Curvature induced in a flat panel by differential shrinkage of surfaces.

Differential shrinkage within a part can be caused by different cooling rates on the two molding surfaces, by material nonhomogeneities, and by internal constraints within the mold that may allow some regions to relax more during the solidification process. Cooling rates can affect local shrinkage by the amount of viscoelastic relaxation in amorphous resins that occurs during solidification; such effects are discussed in Sections 18.7 and 18.8. In semicrystalline materials cooling rates affect local crystal size and orientation, and crystallinity levels; the resulting material nonhomogeneity causes differential shrinkage, and, thereby warpage. Changes in mold surface temperatures by controlling mold cooling and gate location can be used to reduce shrinkage induced by nonhomogeneous cooling rates. Material nonhomogeneities can be caused by fillers, especially anisotropic fillers such as flakes and chopped fibers, the local orientations of which are affected by the local flow during filling. The rigidity of such fillers affects local shrinkage and thereby warpage. Internal constraints in a complex part, such as ribs and bosses, which prevent local material movement within a mold, affect viscoelastic relaxation processes. The resulting effects on local shrinkage then also affect warpage.

Injection Molding and Its Variants

The simple example in Figure 17.5.10 illustrates the mechanism of warpage in one plane. Point-to-point shrinkage differences can cause actual parts to twist into undesirable shapes. Examples of warped parts are shown in Figure 17.5.11. The detailed description of the parts includes information on the types of molds and gates used – topics that are addressed in Section 17.7.

A B C

A

250 mm

100 mm

(a)

(b)

B

C

100 mm

100 mm

(c)

(d)

Figure 17.5.11 Warped injection-molded parts. (a) Warped HDPE protective end cap for pipe. (b) Warped PBT ribbed test part. (c) Warped PP baby-wipe container base. (d) Warped PC circuit board carriage. (Photographs courtesy of Beaumont Technologies, Inc.)

Figure 17.5.11a shows a highly warped HDPE end cap, having a diameter of about 460 mm and a wall thickness of 2 mm, used for protecting the ends of metal pipes. The warpage is caused by a combination of wall-thickness variations and poor mold cooling near the center. Figure 17.5.11b shows a warped poly(butylene terephthalate) (PBT) T-section test part, in which the flange (bottom) dimensions are about 320 × 50 × 2.5 mm, and the rib (web) is about 15 × 1.25 mm; the curvature in the surface marked ABC results from warpage. This part was molded in a test mold designed to evaluate the influence of rib thickness on warpage. In order to create as linear as possible flow along the part length, a 50-mm wide fan gate was used along the near part edge. Figure 17.5.11c shows a warped PP baby-wipe container base with overall dimensions of about 100 × 205 × 75 mm, and a wall thickness of 1.5 mm; the curvature in the edge marked ABC is caused by warpage. The part was center-gated at the top, using a hot runner. Warpage is caused by thermal variations in the mold at the corners, which is s problem for molding plastic boxes. And Figure 17.5.11d shows a warped polycarbonate (PC) circuit board carriage (holder) with base dimensions of about 305 × 240 mm, which has six snap fits for holding the circuit board; the rib in the

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foreground should have been straight but is bent due to warpage. The perimeter rib wall thickness is about 2.5 mm, while that in the central region is about 1.3 mm with 2.5-mm thick flow leaders radiating out from the center to the edges. The part was molded using a three-plate, cold-runner mold with eight cold drops feeding into each of the eight flow leaders. Warpage in this part was mainly caused by part thickness variation. This relatively complex part illustrates how much thought has to go into part, mold, and runner design to prevent warpage. Of the four parts shown in Figure 17.5.11, the first three are of semicrystalline materials – for which warpage is always an issue – only the last is made of an amorphous material. This shows that for all molded plastics parts, including amorphous ones, warpage can be an issue. 17.5.2.4

Draft Angles

Expansion from packing pressure and solidification induced shrinkage can make it difficult to eject a cooled part from a mold consisting of two halves – the half with a concave surface, which forms the external part surface, is called the cavity; the half with a convex surface which forms the internal part surface, is called the core. Figure 17.5.12a shows a filled mold cavity for an open box-shaped part with straight rectangular sides. During opening of the mold (Figure 17.5.12b), shrinkage at the outer surface of the part will tend to shrink away from the mold surface, so that gripping the core due to shrinkage will cause the part to move with the core; full demolding requires an ejector pin to force the part away from the core (Figure 17.5.12c). During the initial mold opening phase (Figure 17.5.12b), the outer surface of the part slides against the lateral cavity surfaces against which it is in forced contact (Why?). This dragging motion results in scruff marks on the lateral part surfaces (Figure 17.5.12c). To facilitate part release, the side walls are slightly tapered, as shown in Figure 17.5.12e – h. This slope relative to a right-angled surface is referred to as the draft angle. Resins manufacturers’ Design Guides give recommendation for the draft angles to be used; typically, draft angles on the order of 1° are used. The same principle applies to the shapes of ribs (Figure 17.5.13), which have draft angles on the order of 1°. For simplicity, the shapes of most parts in this chapter have been shown with sharp corners. However, most corners in actual parts have rounded corners and lateral sides with suitable draft angles. As examples, the shapes of molded parts schematically shown in Figures 17.5.13 and 17.5.5b would have the shapes shown, respectively, in Figure 17.5.14. 17.5.2.5

Boss Geometries

Bosses are round, solid or hollow projections from flat walls. They are primarily used for assembly with other components using self-tapping screws and force-fitted plugs. They can also be used for force-fitted pins as drive shafts and to act as positioning guides. Figure 17.5.15a,b schematically show the generic geometry of a boss on a plate. Note the reduced thickness at the bottom of the boss to prevent to prevent sink marks and voids. The true shape requires appropriate tapers (draft angles) and rounded corners. Figure 17.5.15c,d shows a more realistic shape with a taper on the external surface; the internal surfaces can be tapered, but part function may require cylindrical surfaces. The wall thickness of a boss, which is limited by molding requirements, may not provide sufficient stiffness and strength, and can result in the boss failing at the base. Additional stiffness and strength can be provided through gussets, a form of ribs, which attach the boss sidewall to the base. Figure 17.5.16a,b shows a boss in the middle of a plate stiffened by four triangular gussets. Since bosses directly attached to sidewalls result in thick regions prone to sink marks and voids, bosses are located at away from such walls (Figure 17.5.16c,d) and corners (Figure 17.5.16e,f).

Injection Molding and Its Variants

STATIONARY CAVITY

DRAFT ANGLE

NOZZLE CONTACT

(e)

(a)

MOVABLE CORE EJECTION PIN

(f)

(b)

SCRUFF MARKS

(c)

NO SCRUFF MARKS

(g)

DRAFT ANGLE

(d)

(h)

Figure 17.5.12 Molding of part. (a) Without draft. (b) With draft angle.

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DRAFT ANGLE

RIB FLANGE

Figure 17.5.13 Taper on molded ribs. DRAFT ANGLE DRAFT ANGLE

(a)

(b)

Figure 17.5.14 Actual shapes of parts with appropriate tapers and radiused corners. (a) Ribbed flange. (b) Part from Figure 17.5.5b.

Since meld/knit lines weaken the structural performance of a boss, special care has to be taken to design the gating system to ensure that meld/knit lines do not form around it. 17.5.2.6

Molded-In Inserts

Metal inserts can be molded into a part to provide internal threaded surfaces to which parts can be attached by means of machine screws. This makes it possible to open and close an assembly many times, which may be required for part servicing and replacement. A metal insert can also be used to provide a bearing surface for metal shafts. Both types of these molded-in inserts are schematically shown in Figure 17.5.17. Prior to mold closure, the metal insert is locked into place in the mold. On cooling, the injected plastic shrinks around the metal insert, resulting in a tight fit. Because viscoelastic stress relaxation can result in a loosening of the interfacial bond, the outer surfaces of the inserts are knurled; additional mechanical interlocking is provided by building-in protrusions on the outer surfaces.

Injection Molding and Its Variants

(a)

(c)

REDUCED THICKNESS TO PREVENT VOIDS

(b)

(d)

Figure 17.5.15 Generic shape of a boss integrally molded onto a plate. (a) Schematic diagram. (b) Actual part shape with radiused corners and external taper.

17.5.3

Plastic Hinges

Plastic hinges, also referred to as living hinges, are thin pieces integrally molded with two parts that allow substantial rotary motion between them. The example of a single-piece molding of a pair of polypropylene forceps shown in Figure 1.5.17 was made possible by using four plastic hinges, the details of which are shown in Figure 17.5.18, that correspond to the forceps in open and closed positions, respectively. In this application, each opening and closing cycle reverses the stresses in the hinges, so that failure due to fatigue can be a concern. However, polypropylene has excellent fatigue resistance, so that a properly designed part can withstand several hundred thousand cycles, which is more than sufficient to cover the number of opening-closing cycles expected in this application. Plastic hinges are extensively being used in packaging applications in which a lid on a bottle has to be opened and closed many times. The sequel describes the types of hinges used in such applications.

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GUSSET

(a)

(c)

(e)

(d)

(f)

GUSSET

(b)

Figure 17.5.16 Bosses stiffened by gussets. (a and b) Four gussets in a centrally located boss. (c and d) Boss near a sidewall stiffened by two triangular and one rectangular gusset. (e and f) Boss near a corner stiffened by two rectangular gussets.

THREADED METAL INSERT KNURLED SURFACE

(a)

KNURLED SURFACE

(b) Figure 17.5.17 Molded-in metal inserts. (a) Threaded insert. (b) Insert for bearing.

Injection Molding and Its Variants

HINGE

HINGE

(a)

(b) Figure 17.5.18 Plastic forceps. (a) Open position showing four integrally molded plastic hinges. (b) Hinges in closed position.

First consider a hinged box (Figure 17.5.19) in which the lid can easily be opened and closed any number of times without having to worry about fatigue. This box is made of three parts – the top and bottom and the hinge pin – which have to be assembled to form it. This box can be molded in one piece by using an integrally molded hinge, as shown in Figure 17.5.20. The detail in Figure 17.5.20a shows typical dimensions of such plastic hinges. Because the lids are required to fully open, the hinge has to open by at least 180°. That much flexing requires the hinges to be very thin to control the stresses. Such hinges were the first applications to plastic closures. Recent designs have added thin webs that act as springs during lid opening and closing, making it possible for the lid to easily pop open when slightly opened, and to snap closed when slightly closed. How this bistable mechanism works is best explained by means of a mechanical model in which a hinged box has two springs attached next to the hinge, as shown in Figure 17.5.21a,b. As the lid is lifted from its horizontal position the springs stretch and resist the upward motion of the lid. Similarly, the springs resist an opening of the lid. The bistable nature of this design can be understood from the enlarged lid-opening sequence shown in Figure 17.5.21c – h. The chain-dashed line in Figure 17.5.21c, at an angle of 𝜃 0 with the horizontal, shows the position at which the spring axis passes through the pin center; the spring sees the maximum extension at this position.

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HINGE

HINGE

PIN

(a)

(b)

(c)

Figure 17.5.19 Hinged box. (a) Top and side views of open box. (b) Partially closed. (c) Fully closed.

Increasing 𝜃 results in increasing tension in the spring, which resists this lid-closing operation. For 𝜃 < 𝜃 0 , the lid will fall back to its horizontal position when the closing force is released. This will continue till 𝜃 = 𝜃 0 , the point of maximum spring extension. For 𝜃 > 𝜃 0 (Figure 17.5.21g), the spring tension will tend to close the lid, so that it will snap closed (𝜃 = 90°) when the closing force is released. Alternatively, the lid-closing process could be described as a lid-opening process by the opening angle 𝜙, in terms of which the critical angle is 𝜙0 , as shown in Figure 17.5.21h. The amount of force required to open the lid can be controlled by the stiffness of the spring. And a desired neutrally stable position, 𝜃 0 , can be obtained by changing the horizontal and vertical offsets h and v, respectively (Figure 17.5.21f). The offset, ho , of the spring attachment point from the box surface provides an additional means for varying 𝜃 0 beyond 90°. An implementation of this bistable mechanism with a plastic hinge for a square box with a lid is shown in Figure 17.5.22. In addition to a plastic hinge, the box and lid are connected by two thin, integrally molded webs on both sides of the hinge. As the lid is closed, these webs stretch and act as tension springs, thereby providing a bistable lid in the same way that the springs do in Figure 17.5.21. Because of the way the mold has to be cut for molding such features, it is now more common for the hinge and webs to form one connected feature. One implementation of this design is shown in

Injection Molding and Its Variants

THIN "HINGE"

(b) 0.1 - 0.3 mm

1.5 - 2.0 mm

(a) Figure 17.5.20 Plastic box with integrally molded hinge. (a) Top and side views of open box. Detail shows approximate hinge dimension. (b) Partially closed. (c) Fully closed.

Figure 17.5.23. This results in the external shape of the hinge-web feature having the shape of a bow-tie knot, as shown in the side view of Figure 17.5.23c. Variations of this integrated hinge-web design are now extensively used for plastic bottle caps, which allow material to be squeezed out of bottles by flipping the cap open without having to unscrew the cap. The photographs of one such polyethylene tomato-sauce bottle cap in Figure 17.5.24 show the hinge-web in open and closed positions. The molding of such assemblies having a combination of relatively thick and very thin portions poses a challenge for gating design. Having the melt flow into the main body through the thin web-hinge can result in early freeze-off without the part filling properly. The gate needs to be placed away from the hinge-web in the thicker portions of the part in such a way that the hinge-web fills last. And for uniformity, the flow front should be parallel to the hinge-web axis.

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SPRING

HINGE

HINGE

PIN

(a)

(b)

θ0 θ

θ < θ0 (d)

(c)

(e)

ϕ0 tan θ0 = v/h

θ = θ0

θ > θ0

θ = 90°

v h (f)

h0 (g)

(h)

Figure 17.5.21 Hinged box with attached springs (a) Top and side views of open box. (b) Top and side views of closed box. (c) Fully closed. (c) Detail of hinge in fully open position. (d and e) Lid in the process of being closed. Notice extension of springs. (f) Lid in position with maximum spring extension. (g) Lid closed beyond maximum spring extension. (h) Lid in fully closed position.

Injection Molding and Its Variants

WEB

Z

X

X Y

Y

SECTION ON ZZ

HINGE

Z

OUTWARD CURVED WEB

(c)

(a)

HINGE

HINGE

WEB

WEB

SECTION ON XX

SECTION ON YY (b)

Figure 17.5.22 Plastic box with integrally molded hinge and separate thin web springs. (a) Top and side views of open box, and a cross-sectional side view showing details of the plastic hinge in between two web springs. (b) Sections on XX and YY show details of the plastic hinge and the thin web that acts as a tensile spring. (c) Top, front, and side views of the closed box. In the closed position, the web being longer than the attachment distance causes the web to pop outwards.

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WEB

Z

HINGE

X

X

Y

Y

SECTION ON ZZ

492

Z

OUTWARD CURVED WEB

(c)

(a)

HINGE

HINGE

WEB

WEB

SECTION ON XX

SECTION ON YY (b)

Figure 17.5.23 Plastic box with integrally molded hinge and integral web springs. (a) Top and side views of open box, and a cross-sectional side view showing details of the plastic hinge with two attached web springs. (b) Sections on XX and YY show details of the plastic hinge and the inclined thin web that acts as a tensile spring. (c) Top, front, and side views of the closed box. In the closed position, the web being longer than the attachment distance causes the web to pop outwards. Notice the external bow-tie shape of the integrated hinge and web.

Injection Molding and Its Variants

HINGE WEB

(a)

(b)

Figure 17.5.24 Plastic bottle cap with integrally molded hinge and web springs. (a) Integral lid and base in open position showing hinge and thin webs. (b) Back view with lid in almost closed position. Notice the “bow-tie” shape of the integrally molded hinge-web structure.

17.6 Large- Versus Small-Part Molding Injection molding covers a wide range of materials for which holding pressures can vary in the range of 5 – 50 MPa. For a high-side holding pressure of 50 MPa, a 1 × 1 m (1 m2 ) part requires a clamping force 5,100 tons; for a 0.25 × 0.25 m (0.0625 m2 ) part the clamping force is 320 tons. For a low-side holding pressure of 5 MPa, the corresponding clamp forces required are 50 and 32 tons, respectively. So, parts in this size range require machines with large clamping force capabilities. Another way to assess part size is through the ratio of the amount of material in the sprue-runner gate system relative to the amount of material in the part. This ratio increases with a decrease in the part size. At some part size this ratio becomes too large and alternative means have to be found to reduce this ratio. Parts can also be characterized through their thickness. Most standard parts have thicknesses in the 3 – 5 mm range, and most mold designs and injection molding machines and molding parameters have been developed for this standard part-thickness range. For hand-held devices, such as cell-phone casings, part thicknesses are in the 0.5 – 1.5 mm range, referred to as thin-wall molding, mold design and molding parameters fall outside normal experience, and require special attention. In yet another important regime, parts have thicknesses and features in the submillimeter regime, and weights in the milligram range. 17.6.1

Thin-Wall Molding

In early applications of injection molding parts had thicknesses in the 2 – 5-mm range, for which molding parameters were empirically optimized for different materials and part sizes. These parameters did not work with newer applications, such as cell phones and plastic containers that had reduced part thicknesses in the range of 1 – 2 mm. Problems with molding such parts resulted in the concept of thin-wall molding. In mold filling, hot melt from the central core in the advancing flow front is continually deposited on the mold walls by the mechanism called fountain flow. As soon as filling begins, the melt in contact with

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the colder mold surface begins to solidify, thereby continually reducing the flow channel height. As the part thickness decreases, the solidifying layer can interfere with the filling process, which can result in short-shots. Because of this effect the molding window for conventional parts shown in Figure 17.2.3 does not work for parts in the sub 2-mm range. Molding parameters for such applications have to be reworked. To understand how part thickness affects processing, consider the model problem of molding a flat plaque of length l and thickness h. Figure 17.6.1 shows the relevant geometry at the instant at which thin layers of thickness 𝛿 have solidified at the two mold surfaces (indicated by the hatched regions); uav is the average melt filling velocity. Two important time scales are the mold-fill time, tf , and the melt-solidification time, ts . Clearly, for proper molding tf ≪ ts (Why?). HOT MELT

SOLIDIFIED LAYER

MOLD SURFACE

δ

h

uav

l Figure 17.6.1 Model mold geometry for analyzing the interactions of melt filling and freezing during injection molding.

Assuming that the pressures at the entrance and the end of the mold are p and 0, respectively, it follows from Eq. 6.6.13 that uav scales as (Why?) uav = −

p h2 h2 dp h2 p = = 12𝜇 dx 12𝜇 l 12𝜇 l

(17.6.1)

where 𝜇 is the melt viscosity. Then tf =

12𝜇 l = (l∕h)2 uav p

(17.6.2)

From Eq. 6.2.8, the thickness of the time t for a solidified layer of thickness 𝛿 to develop is t=

𝛿2 4𝛼

(17.6.3)

so that the time, ts , for a total solidification of the molten layer, that is, for 𝛿 = h∕2, is given by ts =

h2 16 𝛼

(17.6.4)

The condition tf ≪ ts for proper mold filling then results in 192

𝛼𝜇 p

(l∕h2 )2 ≪ 1

(17.6.5)

Injection Molding and Its Variants

Alternatively, the time, t0 , required for the solidification of a layer of thickness 𝛿 0 is given by t0 =

𝛿02 4𝛼

(17.6.6)

Then, the condition for the frozen layer to have a thickness 𝛿 0 , that is, for tf = t0 , is 48

𝛼𝜇 p

(l∕h2 )2 = 𝛽 2 ,

𝛽 = 𝛿0 ∕h

(17.6.7)

in which, to facilitate easy mold filling, the nondimensional frozen skin thickness, 𝛽 = 𝛿 0 ∕h, can be assigned a low value, such as 𝛽 = 0.1. From the two ways of arriving at the condition for proper mold filling given in Eqs. 17.6.5 and 17.6.7, the nondimensional group describing the filling and freezing processes is (𝛼𝜇 ∕p) (l∕h2 )2

(17.6.8)

For a constant value of this group, the filling-freezing process would be the same for different values of the variables. In this group, the material properties are grouped in (𝛼𝜇 ∕p), having the dimensions of length-square, in which the thermal diffusivity of thermoplastics falls in the range of 0.1 – 0.25 mm2 s−1 . So, the material properties affect the filling process mainly through the ratio of the viscosity to the pressure, 𝜇 ∕p, which has the dimensions of time (seconds). The part geometry, here represented by the length l and the thickness h, affects the filling-freezing effect through the nondimensional ratio (l∕h2 ) 2 . These analyses clearly indicate that part size effects are measured by the ratio (l∕h2 ) 2 , so that for a fixed value of this ratio, the same appropriate value of 𝜇 ∕p should result in the same quality of moldings in parts 2 2 of different lengths and thicknesses so long √ as the ratio (l∕h ) remains the same. For example, parts with √ (l, h) pairs (L, H), (L∕2, H∕ 2 ), (2L, 2 H ), are geometrically equivalent for the part filling-freezing process. So, a large part with a large thickness can be equivalent to a small part with a much smaller thickness. For a fixed (𝜇 ∕p) (l∕h2 ) 2 , doubling the length while keeping the thickness constant would result in a four-fold reduction in (𝜇 ∕p), which, for a constant viscosity, would require a four-fold increase in the injection pressure. Halving the length would require a four-fold increase in (𝜇 ∕p), which translates into a fourfold decrease in the pressure. Thickness variation has a much larger effect: doubling the thickness while keeping the length constant would result in a 16-fold increase in (𝜇 ∕p), or a 16-fold reduction in pressure. And halving the thickness would require a 16-fold decrease in (𝜇 ∕p), corresponding to a 16-fold increase in the pressure. From this discussion it should be clear that the use of thin-wall molding to only describe small thin-walled parts, such as cell-phone casings, is incorrect. Thicker, but larger parts exhibit the same processing issues. A better description of relative part thickness, as affecting molding, is described by the ratio (l∕h2 ) 2 . For the same molding results, the processing variables scale as ( p∕𝜇 ) ∼ ( l∕h2 )2 , so that thinner parts, corresponding higher values of (l∕h2 ) 2 require higher values of the ratio p∕𝜇 , which can be achieved by higher injection pressures and lower melt viscosities. Lower melt viscosities can be obtained by using lower viscosity grade materials and by increasing the melt temperature. The (l∕h2 ) 2 for thin-walled cellular phone parts was much larger than for conventional parts for which the ratio p∕𝜇 had been optimized. Such parts required higher injection pressures and melt temperatures, and higher mold surface temperatures. Figure 17.6.2 schematically shows the conventional and thin-wall molding windows ABCDE and EFGH, respectively.

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THERMAL DEGRADATION

G

H

C

D

THIN-WALL MOLDING WINDOW

E CONVENTIONAL MOLDING WINDOW

F FLASH

EXCESSIVE SHRINKAGE

MELT INJECTION TEMPERATURE

496

A B SHORT SHOT

MOLD HOLDING PRESSURE Figure 17.6.2 Schematic pressure-temperature diagram showing shift in processing window for thin-wall injection molding.

17.6.2

Micromolding

Micromolding, also called microinjection molding, is used for very small parts having submillimeter dimensions, or for larger parts with functional features on this length scale. While molding thin-wall parts only required increasing the pressure and temperature capabilities of existing molding machines, because of the very small part volumes – much smaller than the volumes of sprues and runners from existing machines – micromolding required the development of morphologically different machines. The brief review of the evolution of new micromolding machines given next is instructive. The first machines for molding very small parts were downsized versions of two existing types of injection molding machines. In the downscaled single-step system, schematically shown in Figure 17.6.3, a single screw with a check valve (ring) at the end was used to both melt (plasticate) the resin and to force the melt through a nozzle. These machines used screws with diameters in the 12 – 18-mm range, with clamping forces in the 50 – 550 kN (11,250 – 123,500 lb) range. The large melt cushion – the volume of melt in the barrel prior to injection – deviations from the check valve (check ring), and long flow lengths made process control difficult for small melt shots: especially since a 1-mg melt shot only required a 0.0056-mm (0.22-mils) stroke on a 14-mm diameter screw. Also, in each molding cycle, the thermal separation of the sprue from the melt cushion resulted in a cold material slug at the nozzle tip. The large sprue and the cold slug resulted in excessive material waste.

Injection Molding and Its Variants

SPRUE

MELT RECIPROCATING SCREW WITH CHECK VALVE

MICROMOLDED PART

COLD MATERIAL SLUG

Figure 17.6.3 Schematic diagram of a downscaled, single-step microinjection molding machine. (Adapted with permission from a Wittmann Battenfeld presentation.)

In addition to all the limitations of the one-step machines, an alternative downscaled two-step system, schematically shown in Figure 17.6.4, which uses a screw for melting resin and a separate injection plunger (ram) for pushing the melt thorough the nozzle, requires a check valve that can leak. The molding cycle starts with the screw forcing melt into the main barrel, after which the check valve is closed and the plunger moves forward to force the melt through the nozzle into the sprue. Machines in this category used screws (extruders) with diameters in the 12 – 18-mm range, plungers with diameters in the 3 – 12-mm range, and clamping forces in the 10 – 500 kN range. SCREW

CHECK VALVE SPRUE

INJECTION PLUNGER

MELT

MICROMOLDED PART

COLD MATERIAL SLUG

Figure 17.6.4 Schematic diagram of a downscaled, two-step microinjection molding machine. (Adapted with permission from a Wittmann Battenfeld Inc. presentation.)

The first successful, true micromolding machine used the three-step configuration schematically shown in Figure 17.6.5, in which the motion of a dosing piston is used to accurately inject the right amount of melt into the injection barrel, after which the check valve is closed. The injection plunger then pushes this charge all the way into the sprue, so that on part ejection no cold material slug is left in the barrel.

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DOSING PISTON

MELT CUSHION

SCREW SPRUE HEATER

MICROMOLDED PART

MELT

INJECTION PLUNGER

Figure 17.6.5 Schematic diagram of a two-step microinjection molding machine. (Adapted with permission from a Wittmann Battenfeld Inc. presentation.)

The efficiency of this machine is best explained by comparing the material use parameters for the same part with those for the downsized machine. For the standard, downsized machine shown in Figure 17.6.3, the melt cushion, the sprue, and the cold-slug volumes were 5.2, 0.9, and 0.15 cm3 , respectively, so that the combined volume of the melt cushion, the cold slug and the sprue was 6.25 cm3 . In contrast, for the micromolding machine in Figure 17.6.5, there really is no melt cushion, and since the sprue volume is only 0.13 cm3 , the combined volume of the material in the cushion and sprue is 0.13 cm3 ; the ratio 6.25∕0.13 of the melt cushion of the standard machine to that of the micromolding machine is then 48 : 1. A newer version of the micromolding machine, schematically shown in Figure 17.6.6, uses a two-step process in which a screw without an integrated check ring is used both to melt the material and to inject the material. The check valve and the dosing piston are eliminated. Instead, a back pressure sensor is used to control the forward and backward motion of the screw to accurately inject the right amount of melt into the injection barrel.

Injection Molding and Its Variants

SCREW

BACK PRESSURE SENSOR

MELT CUSHION SPRUE

HEATER

MICROMOLDED PART

MELT

INJECTION PLUNGER

Figure 17.6.6 Schematic diagram of a two-step microinjection molding machine: In the first step the screw melts the plastic; in the second step the screw acts as a plunger to inject the plastic into the mold. (Adapted with permission from a Wittmann Battenfeld Inc. presentation.)

The molding cycle for this machine is summarized in Figure 17.6.7. Figure 17.6.7a shows the part-ejected configuration in which the plunger is in its most extended position, and in which the rotating screw is moving outwards as it continues to melt and push more resin into the reservoir. Once sufficient resin has been melted, the screw stops rotating and the plunger is withdrawn to open the port connecting the barrel to the screw reservoir, and the screw moves inwards to push a metered amount of resin into the barrel (Figure 17.6.7b). The injection pressure is monitored throughout this process; this information is used to control the switchover point, thereby precisely guaranteeing part-to-part consistency. The plunger then pushes this melt forward, while the screw starts to close and begins to melt plastic for the next shot while the plunger is still injecting material into the mold (Figure 17.6.7c). Finally (Figure 17.6.7d), the plunger pushes the metered melt through the sprue into mold. Demolding the part brings the system back into the initial position shown in Figure 17.6.7a. Another comparison of downsized conventional molding technology and micromolding technology is through the sizes of sprues and parts. Figure 17.6.8a shows the sprue and parts sizes for a conventional machine configuration. For sprue weights in the 1,000 – 2,000 mg range, a 10-mg part is about 1% of the molding shot, so that the molding machine is unable to accurately control part weight. The sprue

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∙

(a)

∙

(b)

∙

(c)

(d)

Figure 17.6.7 Molding cycle for a two-step microinjection molding machine. (a) With injection plunger closed, screw rotation melts resin and the screw moves outwards as reservoir fills. (b) Plunger moves leftward to open access to molten reservoir, and forward motion of the screw fills the barrel. The exact amount is metered by feedback to the screw controller by a signal from the back pressure sensor determining the exact position of the metering screw. (c) The plunger moves forward, closes the access to the melt pool, and pushes the metered melt into the barrel, while always measuring the injection pressure, which is used to control the switch over point to guarantee part-to-part consistency. The screw begins to melt a new charge. (d) The plunger pushes the melt through the sprue into the mold, thereby forming the part. During this portion of the cycle the screw continues to melt more resin. Demolding the part takes the machine back to the configuration shown in (a). (Adapted with permission from a Wittmann Battenfeld Inc. presentation.)

and part sizes for a micromolding machine are shown in Figure 17.6.8b, for which, for sprue weights in the 50 – 200 mg range, a 10-mg part is about 20% of the shot weight, so that the molding machine is able to better control part weight. For the same part size, the ratio of the standard sprue weight to the micromolded sprue weight is about 20 : 1. The smaller sprues and runners in micromolding machines, a collection of which are shown in Figure 17.6.9, result in faster cycle times, smaller material wastage, and better and more consistent part quality.

Injection Molding and Its Variants

MOLDED PART

MICRO SPRUE

STANDARD SPRUE

(a)

MOLDED PART

(b)

Figure 17.6.8 Sprue and part sizes. Note, for part size estimation, the matchsticks have a 1.8 mm × 1.8 mm cross-section. (a) Part molded on downsized standard machine. (b) Part molded on micromolding machine (Aesculap AG & Co. KG.). (Adapted with permission from a Wittmann Battenfeld Inc. presentation.)

Figure 17.6.9 Sprue, runners, and gates for micromolded parts. The matchstick has a 1.8 mm × 1.8 mm cross-section. (Photo courtesy of Wittmann Battenfeld Inc.)

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Newer micromolding machines are clean-room compatible, have automated vision inspection systems, and can be used for a wide range of precision molding operations such as molding of elastomers, insert molding, multicomponent molding, and combining several different processes. This technology is used across a whole spectrum of industrial segments: Hearing aids, instruments, and implants in the medical industry; connectors and mobile phone parts in the telecommunication industry; gears, latches, motors, actuators, and valves for micro-mechanical applications; micro switches, sensors, valves, and security system parts in the automotive industry; lenses, opto-couplers, sensors, glass fiber conductors in optics applications; micro parts for cameras, laptops, and displays for the electronics segment; and lab-on-a-chip and data carriers for diagnostics applications. The following series of photographs show examples of diverse micromolded applications. Figure 17.6.10 shows the large variations possible in the dimensions of micromolded parts. While the polyoxymethylene (POM) paper clip (Figure 17.6.10a) is about 14-mm wide by 0.7-mm thick, the sensor (Figure 17.6.10b) has feature on a 20-μm scale. Figure 17.6.11 shows four examples of micromolded parts for medical applications. The first two (Figure 17.6.11a,b) are examples of implantable resorbable parts. The third (Figure 17.6.11c) is a 40-μm polyoxymethylene microfilter used in medical acoustics. And the fourth is a polyoxymethylene 28-pin hearing aid micro sub connector. Figure 17.6.12 shows a microswitch locking lever and a micronozzle. Figure 17.6.13 shows polyetheretherketone (PEEK) microgears for high-temperature applications. Figure 17.6.14 shows several types of micromolding applications involving two materials: Figure 17.6.14a shows a POM insert molded plug weighing 17 mg. Figure 17.6.14b shows a metal band overmolded with a 30% glass-filled liquid crystal polymer (LCP). And Figure 17.6.14c shows a part made by overmolding a soft thermoplastic elastomer (TPE) on a hard PA66 core. (The overmolding process is described in Section 17.8.4.2.)

(a)

(b)

Figure 17.6.10 Large length scales of micromolded parts. (a) 14-mm wide by 0.7-mm thick POM paper clip. (b) 20-μm polycarbonate sensor. (Photos courtesy of Wittmann Battenfeld Inc.)

Injection Molding and Its Variants

(a)

(b)

(c)

(d)

Figure 17.6.11 Medical applications of micromolded parts. Note, for part size estimation, the matchsticks have a 1.8 mm × 1.8 mm cross-section. (a) 2-mg, bioresorbable clip. (b) 11-mm3 , implantable bioresorbable clip (Aesculap AG & Co. KG.). (c) POM medical acoustics 40-μm microfilter; part volume 0.56 mm3 . The “line” on the part is a human hair with a diameter in the 0.5 – 0.07 mm range. (d) POM hearing aid 28-pin micro sub connector; part volume 3 mm3 . (Photos courtesy of Wittmann Battenfeld Inc.)

(a)

(b)

Figure 17.6.12 (a) POM microswitch locking lever; part volume 0.7 mm3 . (b) Micronozzle made of Catamold® TZP-A (Lechler GmbH). (Photos courtesy of Wittmann Battenfeld Inc.)

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Figure 17.6.13 PEEK microgears for high-temperature applications. (Photo courtesy of Kleis Gears Inc.)

(a)

(b)

(c)

Figure 17.6.14 (a) POM insert molded plug; part weight 17 mg. (b) Metal band overmolded with 30% glass-filled LCP; weight of plastic 0.38 g. (c) Two-component soft part; soft TPE (16 mg) overmolded on a hard PA66 (1 mg) core. (Photos courtesy of Wittmann Battenfeld Inc.)

17.7 Molding Practice While major advances in injection-molding machines, including the use of sensors and real-time computer control, have made injection molding a very productive part fabrication (processing) method, it is novel innovations in mold design that have made possible the molding of very complex shapes. This section introduces the basic mold types; the principles of operation are described using simple schematic diagrams. These simple descriptions hide the mechanical complexity of even the simplest mold – such as guide pins on one mold half that mate with holes in the other half – which can only be shown by using engineering drawings for molds.

Injection Molding and Its Variants

Basically, a mold consists of two halves, the cavity – which forms the external shape of the part – and the core – which forms the internal shape of the part. When closed, the surfaces of these two mold halves form the cavity into which molten plastic is injected to form the part. The two halves mate along an interface – called the parting line – which can be a plane normal to the mold separation direction, or stepped, or angled to accommodate the part shape. Vestiges of the parting line on molded parts can affect part esthetics. In cold-runner molds, the molding process results in plastics wastage in the form of sprues, runners, and gates that can be ground and reused. There are two types of cold-runner molds: In two-plate molds the solidified sprue-runner is attached to the part and has to be removed after molding; in three-plate molds the runner system separates from the part at the part ejection stage. In hot-runner molds heated runners keep the plastic in a molten state, and this material is not ejected with the part, so that the material in the runner system is not wasted. Because of demanding esthetics for plastic parts, molding technology has to address several issues: vestiges of the sprue-part or gate-part interfaces, vestiges of parting lines where the inner surfaces of the two mold halves meet, meld lines, and surface blemishes caused by flow patterns. And improper gating can result in warping of even well-designed parts. 17.7.1

Two-Plate Cold-Runner Mold

With reference to Figure 17.7.1a, a two-plate mold comprises a stationary cavity plate and a movable core. When in full contact, these two parts form the cavity into which molten plastic is injected to form the part. Molten plastic is injected by the injection-molding-machine nozzle at the nozzle contact on the mold. The molding sequence for a single-cavity, or single part, mold is schematically shown in Figure 17.7.1. The core is first moved (Figure 17.7.1a) to close the mold (Figure 17.7.1b). Then molten plastic is injected into the resulting cavity (Figure 17.7.1c). After solidification the core is retracted (Figure 17.7.1d), during which the solidified sprue-part remains attached to the core because of forces generated by shrinkage. Then the ejection pin pushes the part away from the core (Figure 17.7.1e), resulting in the ejected assembly shown in Figure 17.7.1f. Finally, the sprue is removed resulting in the part (Figure 17.7.1g). Such molds have a single parting surface (Figure 17.7.1c) where the two mold halves meet, which can leave a vestigial parting-surface mark on the part. Also, the removal of the sprue results in a vestigial mark on the outer surface of the part. If the outer surface is required to be blemish free, reverse injection, as shown in Figure 17.7.2, can be used to move the sprue to the inside surface. Notice the relocation of the injection pin as the solidified part will tend to stick to the (upper) core. A two-part mold can also be used for molding several parts in each cycle. Figure 17.7.3 schematically shows the cavity and core geometry for simultaneously molding two parts. Note the central notch on the core surface; on molding it forms the sprue puller to facilitate the removal of the sprue. A central sprue feeds two runners that feed the part cavities through two gates. The core moves to contact the stationary cavity to generate the runner, gate, and part cavities (Figure 17.7.3a), into which molten plastic is injected (Figure 17.7.3b). The core opens with the part attached to it (Figure 17.7.3c). The notched sprue puller at the bottom of the sprue forces the sprue to move with the part. Then the molded part, with the sprue-runner-gate attached to it, is forced out by the ejection pin (Figure 17.7.3d), resulting in the assembly shown in Figure 17.7.3e. Finally, two parts are obtained by removing sprue-runner-gate assembly.

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SPRUE GATE STATIONARY CAVITY

NOZZLE CONTACT

(d) (a)

SPRUE GATE MOVABLE CORE EJECTION PIN

(e) (b)

SPRUE

(f) (c) (g)

Figure 17.7.1 Schematic diagrams showing the molding sequence for a part in a single-cavity, two-plate cold-runner mold.

(b) (a) (c)

Figure 17.7.2 Relocation of sprue-part interface through reverse injection molding. The top half is the fixed mold half.

Injection Molding and Its Variants

(a)

(b)

(c)

(d)

SPRUE

(e) GATE

RUNNER

SPRUE PULLER

Figure 17.7.3 Two parts simultaneously molded in a two-plate cold-runner mold. Note the notched sprue puller at the top central surface of the core.

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Two-plate, cold-runner molds are the most commonly used relatively inexpensive molds. Because the sprue-runner system is ejected in each mold cycle, color changes are easy to accommodate. One limitation is that in multi-cavity molding – several parts molded in the same mold – the gate has to be located at the part periphery (Figure 17.7.3). The main disadvantage of such molds, especially for large production volumes, is that the sprue-runner system has to be separated from the part and reground for reuse. 17.7.2

Three-Plate Cold-Runner Mold

A three-plate mold has an intermediate movable plate having the cavity (Figure 17.7.4a). The stationary plate has the primary sprue and a protruding reverse edge that acts as a sprue puller (see detail at the top left of Figure 17.7.4d). The second movable cavity plate has a runner and a secondary sprue. The movable core has the same function as in the two-plate mold. The molding sequence for a single-cavity mold is schematically shown in Figure 17.7.4. After the mold is closed (Figure 17.7.4a), molten plastic is injected into the resulting cavity (Figure 17.7.4b). On solidification the movable cavity and core are retracted (Figure 17.7.4c). The sprue puller keeps the sprue-runner-sprue system attached to the stationary plate. And forces generated by shrinkage keep the solidified sprue-part attached to the core. In the final step (Figure 17.7.4d) ejection pins are used to eject the sprue-runner-sprue assembly and the part. One advantage of the three-part mold is that the sprue-runner-sprue assembly is automatically separated from the part. Another advantage is that in a multi-cavity mold, the gate does not have to be at the part periphery as in the two-plate mold case (Figure 17.7.3e). Figure 17.7.5 shows a two-cavity mold in which each part is gated on the top surface. Note the pin gate used to facilitate separation of the part from the secondary sprue (detail on right-hand side of Figure 17.7.5a). 17.7.3

Molds for Parts with Undercuts

Parts having undercuts that prevent straight ejection by retracting the mold core require more complex molds with additional retractable cores. The principle of how a sliding core can be used for molding undercuts is illustrated by the molding sequence shown in Figure 17.7.6. In the mold-closed position (Figure 17.7.6a) a laterally movable slider provides the core required for forming the undercut. After the part has cooled, the retraction of the core causes the secondary core to slide sideways (Figure 17.7.6b,c), till it clears the outermost surface of the undercut part. The part can then be ejected by using an ejection pin (Figure 17.7.6d). In this simple example in which the slider is guided by a pin (Figure 17.7.6c), the slider is automatically pulled outward as the main core is retracted. Hydraulic or pneumatic actuated slides are used when secondary core retraction requires more complex relative motion. Clearly, molding parts with undercuts result in more complex, expensive molds. A sliding mechanism for molding internal undercuts is schematically shown in Figure 17.7.7. After the part has been molded (Figure 17.7.7a), sliders are used to retract the movable core (Figure 17.7.7b,c), thereby making it possible to eject the molded part (Figure 17.7.7d). While these schematic figures illustrate the underlying principles, the actual mechanisms required can be quite complex. 17.7.4

Molds with Collapsible Cores

Internal threads in plastic bottle lids can be molded by a threaded metal core that, after molding, can be unscrewed – for example, by using a rack-and-pinion mechanism. While this works for internal threads, it cannot be used for molding internal dimples, grooves, and slots. In such cases collapsible cores have to be used.

Injection Molding and Its Variants

SPRUE EJECTION PIN STATIONARY PLATE

SPRUE PULLER

NOZZLE CONTACT

MOVABLE CAVITY

MOVABLE CORE PART EJECTION PIN

(b)

(a) SPRUE PULLER

(d)

(c) PRIMARY SPRUE RUNNER

SECONDARY SPRUE

(e)

(f)

Figure 17.7.4 Schematic diagrams showing the molding sequence for a part in a single-cavity, three-plate cold-runner mold.

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PIN GATE

(a)

(b)

PRIMARY SPRUE RUNNER

SECONDARY SPRUE

(c)

(d)

Figure 17.7.5 Schematic diagrams showing the molding sequence for simultaneously molding two parts in a double-cavity, three-plate cold-runner mold. (a) Closed and filled mold assembly. The right-hand side of this figure shows the detailed geometry of the pin gate tip. (b) Mold opening, and part and sprue-runner ejection. Note the notched sprue puller at the top left surface of the core.

Injection Molding and Its Variants

(a)

(b)

PIN SLIDER

(d)

(c)

UNDERCUT

(e)

(f)

Figure 17.7.6 Schematic diagrams illustrating the use of a pin-and-slider secondary core for molding a part with an external undercut. (Adapted with permission from “Part and Mold Design Guide” courtesy of Covestro.)

The sequence of schematic diagrams in Figure 17.7.8 illustrates how a collapsible core can be used for molding internal threads in a bottle cap. Figure 17.7.8a shows the mold in the closed position with molten plastic injected into the cavity formed by the cavity half and the core, which is held in the expanded position by the core pin and the two sliders, marked S. After melt solidification the core pin is retracted (Figure 17.7.8b). Then the core is collapsed by moving the sliders inward (Figure 17.7.8c), after which the entire core assembly is retracted (Figure 17.7.8d). The molded cap in the cavity mold half can then be ejected by an ejection pin.

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(a)

(b)

(c)

(d) Figure 17.7.7 Schematic diagrams illustrating the use of a pin-and-slider secondary core for molding a part with an external undercut. (Adapted with permission from “Part and Mold Design Guide” courtesy of Covestro.)

Injection Molding and Its Variants

COLLAPSIBLE CORE IN MOLDING POSITION

S

CORE IN COLLAPSED MODE

S

CORE PIN IN MOLDING POSITION

(a)

(b)

(c)

(d)

Figure 17.7.8 Schematic diagrams illustrating the use of a collapsible core for molding internal threads in a bottle cap. (Adapted with permission from “Part and Mold Design Guide” courtesy of Covestro.)

17.7.5

Hot-Runner Molds

The large amounts of plastics generated by the sprue-gate-runner system in a cold-runner mold, which have to be reground, can be avoided by using molds with hot runners. In such molds the sprue-runner-gate system is continually heated to keep the plastic in a molten state, so that the solidified mold is ejected without the sprue-runner-gate that is retained in the mold. As a result, the amount of molten plastic used per cycle is limited to the part volume. The overall cycle time is reduced – shorter melt injection and shorter cooling times because the sprue-gate-runner does not have to be cooled – and the smaller shot sizes require relatively smaller molding machines. Also, injecting molten plastic directly at the gates allows for better part quality control. However, in comparison to cold-runner systems, color changes require more time because the molten plastic in the sprue-runner-gate system has to be purged with each color change. Figure 17.7.9 is a highly simplified diagram illustrating the essential features of a hot-runner, two-cavity mold. The two main functional units are (i) the hot manifold, which contains the larger diameter runner that conveys the melt from the molding-machine nozzle to selected points behind the cavity plate, and (ii) hot drops, which feed the melt directly into the mold cavity. Because the plastic in the runner and the drops is retained in a molten state in the manifold-cavity-plate assembly – thereby not generating scrap as in cold-runner molds – the runner can be of larger diameter, resulting in lower pressure drops and mold clamping forces. Note that many details have been glossed over in this simplified diagram. For example, a drop is not a monolithic component. Rather it has a screwed-in tip insert that controls the final flow channel shape and diameter of the injection tip. Because

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HOT MANIFOLD CARTRIDGE HEATER

HEATER BAND

SPACER

AIR GAP

DROP CAVITY

(a) HOT RUNNER HEATER BAND

MOLTEN PLASTIC

CAVITY PLATE DROP

HEATER BAND

CORE PLATE

(b) Figure 17.7.9 Schematic diagrams illustrating the essential features and working of a hot-runner mold.

Injection Molding and Its Variants

a frozen sprue-runner-gate assembly is not ejected, from an operational standpoint a hot runner system can be looked upon as a direct extension of the molding-machine nozzle into the cavity mold surface. The runner and the drops can be heated in several ways. Figure 17.7.9 shows an externally heated runner manifold and externally heated drops. In this case the hot runners are heated by external cartridge heaters and each drop by an individual heating band; the manifold and drop heaters have separate temperature controls. Air gaps are used to insulate the hot manifold and drops to minimize heat transfer to the cavity plate. Also, a hot-runner system can be used for cold-gating of cold-runner molds, resulting in less scrap than in a conventional cold-runner mold. While eternally heated manifolds and drops have the lowest pressure drops, leakage of molten plastic can destroy the heating systems. The temperatures of hot runners and the drops can be controlled in several ways: external heating; internal heating, in which the heater is inside the runner channel or the drop; and the use of insulation. In addition to the scheme shown in Figure 17.7.9, several combinations of these methods are in use, such as externally heated manifold with internally heated drops, internally heated manifold and drops, insulated manifold with internally heated drops, and insulated manifolds and drops. 17.7.6

Sprues, Runners, and Gates

The function of the sprue-runner-gate system in injection molding is to convey molten plastic from the molding-machine nozzle tip to the top of the mold. These three terms have been borrowed from the terminology for molding cast iron and cast steel parts, wherein molten metal is poured into a mold through a sprue, from where the melt is conveyed by runners to several points on the casting surface, to enter the gating cavity through gates. In metal castings the sprue-runner-gate system has to be knocked off from the main casting, as in cold-runner molding of plastics. In some simple cases (Figures 17.7.1 and 17.7.2) the sprue feeds directly into the gate but, in general, the sprue feeds the gate(s) through a runner (Figures 17.7.3 – 17.7.5). 17.7.6.1 Runner Configurations

For a required runner length, cold runners must have sufficiently large diameters to reduce pressure drops to acceptable levels. But larger diameters can result in more regrind. Multi-cavity molds, especially for large-volume production of small parts, can have a very large number of cavities, requiring complex runner systems with branches having different diameters. Such diameter changes can result in different filling rates with time lags among the flows reaching the gates. As an example, Figure 17.7.10 shows the short-shot sequences for the sprue-runner-gate system corresponding to the double-gated filling of ABS in the mold shown in Figure 17.2.7a. Notice that, for the same time (same short-shot), the larger diameter runner on the left (Runner-1) fills further than the smaller diameter runner on the right (Runner-2). However, because the right runner is shorter, its flow front reaches its gate (Gate-2) marginally earlier than the arrival of the left flow front at Gate-1. But because the left gate is bigger than the right gate, mold filling starts at about the same time at both gates. In this balanced, well-designed runner-gate system, the mold fills in a relatively even, balanced fashion. Clearly, in a multi-cavity mold, parts molded in different cavities will be the same if they see the same processing histories. This requires that the temperatures and flow rates of the melt fronts arriving at each gate be the same. Consider the 16-cavity mold schematically shown in Figure 17.7.11, in which the melt is injected at sprue S at which it divides into two large runners, SL and SR. The 16 cavities are then fed through smaller lateral runners as shown. After feeding the cavities marked 1, the flow in the main runners, SL and SR, feeding the cavities marked 2 will be reduced by the amount fed to cavities 1. It follows that in this runner system, the flow volumes feeding cavities decreases with the distance from the sprue. With this scheme, cavities having the same number see the same flow histories and can be expected to produce equivalent parts. As a result, each molding cycle will produce four groups of parts, each group having four equivalent parts that are different from those in the other three groups. For this reason, this

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type of system is referred to as a geometrically unbalanced runner system. To improve parts equivalency, such unbalanced runners have to be artificially balanced by continually increasing the diameters of the side feeder runners with their distance from the sprue.

GATE-1

GATE-2

SPRUE RUNNER-1

RUNNER-2

Figure 17.7.10 Short-shot sequence the sprue-runner-gate system for the filling of the double-gated cavity with two slits and one circular insert with ABS shown in Figure 17.2.7a. (Adapted from Cornell University Injection Molding Program (CIMP) Progress Report No. 4, 1977, original photo courtesy of Professor K.K. Wang.)

4

3

2

1

L

1

2

3

4

R

S

4

3

2

1

1

2

3

4

Figure 17.7.11 Geometrically unbalanced runner system for a 16-cavity mold. All cavities fed from runners connected to main central runner.

Injection Molding and Its Variants

Figure 17.7.12 shows a 12 cavity mold in which the runners are artificially balanced by increasing the diameters of the side runners (part feeders) with their distance from the sprue.

Figure 17.7.12 Artificially balanced runner system for 12-cavity mold. The diameters of the side feeders increase with the distance from the sprue. (Photo courtesy of Beaumont Technologies, Inc.)

An alternative runner system for the 16-cavity mold is shown in Figure 17.7.13. Clearly, in this system of cascaded runners, each cavity is fed by melts that have seen identical flow histories, so that all 16 molded parts will be equivalent. Such systems are therefore referred to as geometrically balanced runner systems.

S

Figure 17.7.13 Geometrically balanced runner system for a 16-cavity mold. All cavities fed from runners connected to main central runner.

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17.7.6.2

Imbalances from Flow Asymmetry

In steady flow in a channel the highest shear rates – and therefore the hottest regions – are highest at the boundaries and zero at the center (see Figure 6.6.2 in Section 6.6.5), which will be cooler. When the flow splits in two directions at a rectangular junction into a secondary runner, the sharp turn in the flow causes the hottest highly sheared material close to the primary runner walls to flow along the walls of the secondary runner closest to the junction, and the less-sheared flow from the central core flows along the opposite wall of the secondary runner. Thus, on splitting into the secondary runner, the originally symmetric flow around the centerline of the primary runner becomes asymmetrical, as indicated in Figure 17.7.14. HOTTER MELT

COOLER MELT

HIGHER-SHEAR, HOTTER, LOWER-VISCOSITY REGION

PRIMARY RUNNER

LOWER-SHEAR, COOLER, HIGHER-VISCOSITY REGION

SECONDARY RUNNER

Figure 17.7.14 Asymmetry in flow through a runner caused by a flow-split at a rectangular junction into a secondary runner. (Adapted with permission from Figure 6.21 in “Mold Design,” by J. Beaumont, in “Injection Molding Handbook,” T.A. Osswald, L-S. Turng, and P.J. Gramann, (Eds.), Hanser Publishers, Munich, 2002.)

This process of causing flow asymmetries at rectangular junctions, at which a flow splits into two flows, at successive junctions is schematically shown in Figure 17.7.15, in which the successive junctions are marked by A, B, C, and D, with the subscripts L and R referring, respectively, to the left and right branches of each bifurcated flow. Because of such flow asymmetries, even parts molded in balanced-flow runner systems of the type shown in Figure 17.7.13 may see different processing histories, thereby affecting part geometry and surface finish. Figure 17.7.16 shows parts molded in an eight-cavity mold fed by a geometrically balanced runner system, with four parts marked A, B, C, and D on both the upper (with subscript U) and lower (with subscript L) parts of this figure. Note that the runner and part configurations of the upper parts can be obtained by rotating the lower parts by 180° about the sprue. As such, the parts A U and A L should be identical, which they are. The same holds true for the upper and lower versions of the parts B, C, and D. However, while the parts A and D are similar, they are very different from the similar parts C and D. These molded parts show that, as a result of flow asymmetries caused at flow splits in runners, even the use of a geometrically balanced runner system does not provide balanced filling and packing in all the cavities, thereby resulting in nonuniform parts.

Injection Molding and Its Variants

DL

EL

COOLER

COOLER

G

FL CL

DR

FR

BL

CR

BR

HOTTER HOTTER MELT

ER

HOTTER

AR

AL

COOLER MELT

Figure 17.7.15 Schematic representation of continuing complex asymmetries in flow caused by flow splits at rectangular junctions into successive secondary runners.

AU

BU

CU

DU

DL

CL

BL

AL

Figure 17.7.16 Parts molded using a geometrically balanced runner system for an eight-cavity mold. The differences in part shapes are caused by flow asymmetries resulting from flow splits at rectangular runner junctions. (Photo courtesy of Beaumont Technologies, Inc.)

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The imbalances caused by flow asymmetries can be alleviated by using MeltFlipper® technology in which flow flippers introduced in the runner flow paths reduce flow asymmetries by rotating the flow. As an example, Figure 17.7.17 shows the dramatic improvement in part uniformity by using MeltFlipper runners.

C B

B

A B

B

FLOW ROTATORS

Figure 17.7.17 Dramatic improvement in part uniformity obtained by using MeltFlipper runners, a patented method and design. In this eight-cavity mold melt flippers are used at runner junctions such as at A, B, and C. (Photo courtesy of Beaumont Technologies, Inc.)

17.7.7

Gate Types

A gate is the interface at which melt transported by a runner is injected into the mold. No matter how carefully designed, the gate-part interface always leaves an undesirable vestigial mark. Gate selection and design involve several considerations: (i) Assuring smooth flow fronts within the mold, (ii) control of knit line locations, (iii) minimization of gate vestigial marks, and (iv) ease of separating sprue-runner-gate systems from parts. How gate location, number of gates, and inserts affect filling patterns have been illustrated in earlier figures: the interaction of flow fronts with cavity walls for a single gate in Figure 17.2.6, the interaction of two flow fronts emanating from two gates in Figures 17.2.7 and 17.2.8, the interaction of the flow front from a single gate with slits in Figure 17.2.9, and the interaction of flow fronts from a two gates with slits in Figure 17.2.11. The complexity of knit line formation can be gauged from Figure 17.3.12 (single-gated cavity), Figure 17.3.13 (single-gated cavity with inserts), and Figure 17.3.14 (double-gated cavity with inserts). Vestigial marks are best handled by using gates to move them to locations where they will be least noticed. But proper gate geometry can also help. Ease of separating the sprue-runner-gate from the part requires a small gate area, which must be balanced to ensure that injection velocities and shear rates do not become too high. The sequel describes a few of the many gate types in use.

Injection Molding and Its Variants

17.7.7.1 Sprue Gate

The simplest gate type is the sprue gate in which the molding-machine nozzle directly injects melt into the mold as shown in Figures 17.7.1 and 17.7.2. To facilitate part ejection in a cold-runner mold the sprue is tapered. Material regrind is reduced as there are no runners. It is mainly used for symmetric or cylindrical parts with the gate located centrally. In a three-plate cold-runner mold the sprue-runner-gate assembly is automatically stripped from the part. To facilitate separation of the part from the secondary sprue, a gate with a restriction, called a pin gate, is used. (Details of such a restricted gate are shown on right-hand side of Figure 17.7.5a.) 17.7.7.2 Edge Gate

In edge gating the gate is located at the parting line along and edge. How this type of gate is used in a multi-cavity gate is shown in Figure 17.7.3. Figure 17.7.18 shows details of two variants of the part-gate interface for an edge-gated rectangular part. Figure 17.7.18a shows a lapped edge gate, which will result in a vestigial mark on the main flat surface. The enlarged detail shows the geometry for easy removal of the gate-runner-sprue assembly. Figure 17.7.18b shows a notched edge gate, in which the vestigial mark will appear on the thin, lateral surface.

(a)

(b)

Figure 17.7.18 (a) Lapped edge gate. (b) Notched edge gate.

17.7.7.3 Fan Gate

One disadvantage of an edge gate is that the initial flow will be radial (see Figure 17.2.6). A flatter flow front can be achieved by using a fan gate, schematically shown in Figure 17.7.19a, in which the runner

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A B

A B C

C

(b)

(a)

(c) Figure 17.7.19 (a and b) Fan gates. (c) Film gate.

feeds the triangular piece ABC, in which radial flow injects melt all along the edge AC. To improve the distribution of the melt along AC, in one variant (Figure 17.7.19b), the runner extends into two branches BA and BC, resulting in better distribution of the melt along AC. Another way to improve the distribution along AC is to use the film gate shown in Figure 17.7.19c. 17.7.7.4

Diaphragm Gate

Figure 17.7.20 shows a diaphragm gate, which is very useful for molding cylindrical parts. The symmetric flow field results no weld lines and reduced part distortion. This type of a gate is essentially an axisymmetric version of the gates shown in Figure 17.7.18. 17.7.8

Jetting

Jetting refers to the phenomenon in which a stream of the melt from the gate enters the mold in the form of a cylindrical jet; in his case the mold does not initially begin to fill by the progression of a flow front. The short-shot in Figure 17.7.21a shows a jet attached to an evolving flow front. In this case a jet was first extruded at the gate before the material spread laterally to form a flow front. In the short-shot in

Injection Molding and Its Variants

PART

DISK-SHAPED RUNNER

DIAPHRAGM GATE

Figure 17.7.20 Diaphragm gate.

Figure 17.7.21b, the initial material extruded at the gate shot out in the form of a high-velocity jet that traveled all the way to the far gate surface where it began to pile up. Then, after a regular flow front formed, it pushed the folded jet forward. Once a regular flow front develops it pushes the jetted material forward, eventually filling the mold to form a part. However, even after packing, the initial jetting will form multiple meld lines, resulting in poor surface finish and mechanical flaws. Jetting is caused when, on entering the mold, the initial high-velocity extruded material does not impinge on any surface; the lack of any resistance allows the high-velocity jet to stream across the mold cavity. Jetting can be avoided by proper gate design to ensure that the material entering a mold at a gate immediately impinges on a mold surface. Of the two gate designs shown in Figure 17.7.22, the center-gated design in Figure 17.7.22a is more prone to jetting. Because the lap gate shown in Figure 17.7.22b forces the initial flow to contact the side and bottom mold walls, the chances for jetting are very small. 17.7.9

Mold Venting

During molding, the advancing melt flow front pushes the ambient air causing it to compress and create an undesirable back pressure. Molds therefore have vents through which this air can escape. They must be so placed that air is not trapped in pockets. This requires a careful consideration of the mold filling pattern to evaluate where the compressed air will tend to accumulate. A complex mold may require several vents. Vents are most easily placed at parting line surfaces, where very shallow recesses are ground into the surfaces to allow air to escape. The depths of such recesses have to be large enough to allow air to escape

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RUNNER PART SHORT-SHOT

JET

(a)

(b) Figure 17.7.21 Short-shots illustrating the phenomenon of jetting. (a) The initial jet is being pushed forward as the mold begins to fill. (b) The jet contacted the rear mold surface where it began to fold as the mold fills. (Photos courtesy of Beaumont Technologies, Inc.)

PART

CENTER GATE

LAP GATE RUNNER

(a)

(b)

Figure 17.7.22 Two types of gates. (a) The center-gated runner geometry is more prone to result in jetting. (b) The lap-gated geometry greatly reduces the chances for jetting.

Injection Molding and Its Variants

but not so large as to allow molten plastic to enter the recess. Typically, recess depths are in the range of 0.015 – 0.03 mm, and their widths are about 2 mm. Air from these vents exits into about 0.5-mm deep relief areas from where it escapes into the atmosphere. Parting-line vents have the advantage that they are easy to clean. Other options include using clearances at sliding surfaces such as those at ejector pins and retracting cores. When molding ribs, a split cavity mold with a parting line centered on the rib can be used to provide vents. In special cases, in which vents cannot be conveniently located, vacuum is used to evacuate a mold. This requires the mold cavity to be sealed by gaskets. As with many practical mold design issues, special venting schemes have to be tailored to suit particular applications. 17.7.10

Mold Cooling

Mold cooling is required to extract the heat released from the solidification of molded parts (Section 17.2.4). While the mold temperature will change during a molding cycle, proper mold cooling can be used to obtain a quasi-steady-state temperature in the sense that each point in the mold undergoes a fixed cyclic temperature change. Mold cooling is achieved by pumping water through cooling channels in the mold. A network of cooling holes is produced by drilling intersecting holes in the mold block. Ends of such holes are closed off to form a network of series or parallel circuits through which the cooling water is pumped. Cooling circuit design is a very important aspect of mold design. Special techniques have to be used to cool larger cores and corners of box-shaped parts. 17.7.11

Summary Comments

Section 17.7 has focused on molding practice, innovations in which have immensely contributed to the success of end-use injection molding technology. More than the development of injection molding machines, it is novel innovations in mold design that have made possible the molding of incredibly complex shapes. The simple descriptions in Section 17.7 neither showcase the ingenious innovations in mold design that make possible the molding of very complex parts, nor do they give an idea of the mechanical complexity of molds. A mold is much more than just a part shaping device – it also functions as a heat exchanger, undergoing temperature cycles during each part molding cycle. The pressure-temperature cycle during solidification has a marked effect on residual stresses in a part and on part warping – the subject matter for Chapter 18. Given the cyclic nature of thermal events during molding and the discrete distribution of cooling lines, only computer simulation can characterize the actual thermal history seen by a molded part. Part filling is a complex, transient coupled flow and heat transfer event (Sections 17.3 and 17.4) in which the temperature of the mold surfaces also changes cyclically. Thus, a complete characterization of the pressure-temperature history seen by a part requires an integrated analysis of the melt and mold systems – again, a complex problem that can only be solved through computer simulation.

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17.8 Variants of Injection Molding Although injection molding is by far the most important and versatile process for making plastic parts, it does have shortcomings. First, the high injection pressures required to force melts through mold cavities of large parts require more expensive molds and higher capacity molding machines. Second, because of the large volumetric shrinkage that plastic melts undergo on solidification, is the tendency to form sink marks caused by delayed solidification in thicker regions. Third, is the inability of molding thin-walled hollow parts. And, fourth, is the inability of molding parts with layers of different materials. 17.8.1

Methods for Reducing Injection Pressure

The high injection pressures required for forcing molten plastics through long paths in molds for large parts is a shortcoming of standard injection molding that can be overcome. Large pressures require sturdier and more expensive molds, result in higher residual stresses, and increase the chances of flash at mold parting lines. Also, larger pressures require higher capacity, more expensive molding machines. Two methods for reducing the injection pressure are discussed in the sequel; the first uses innovative gating schemes, and the second requires a movable mold cavity. Note that both these methods use standard injection molding machines. 17.8.1.1

Sequential Gating

The longer the distance of a flow front in a mold from the gate is, the higher the pressures required to move the melt forward will be. One strategy for reducing the injection pressure is to shorten the flow distance by using a hot-runner manifold to simultaneously inject molten resin through multiple gates (see Figures 17.2.7 and 17.2.8). This limits the flow distance to that required for flow the fronts to meet. By balancing the gate sizes the positions of the resulting knit lines can be changed. The trend toward integrating many features into a single large part – such as integrating automotive grills, light openings, fascias, and the bumper into a single molding – results in complex cavity geometries with increased flow-length to wall-thickness ratios. In such complex parts it is difficult to predict the actual flow pattern during molding, even by using numerical codes. And it is even more difficult to determine the optimum processing conditions, gate locations, and hot runner diameters to locate knit lines in desired locations. Moreover, because the gate tips have to be cooled, small amounts of solidified material at the gate opening from the previous part can result in poor surface finish. These difficulties have been overcome with the development of sequential valve gating systems in which the timing and amount of injection at a gate can be accurately controlled by actuating computer-controlled valves. With such systems the hot runner does not directly connect the gate to the cavity. Instead, the gate opening is sealed by a retractable pin that can be moved by a programmed signal; also, the rate at which this pin is retracted and the distance to which it is retracted determines when and how much material is injected at the gate (Figure 17.8.1). How programmed sequential gating works can be understood from the filling sequence for three programmed gates schematically shown in Figure 17.8.2. Each of the three gates marked A, B, and C is connected to the same hot-runner manifold (not shown). The pins in these gates can be actuated independently; they can open or close the gates, the rate of opening and closing can be controlled, as can the area of the gate opening. (Note that the thickness and mechanical details of the gate tip have not been shown.) In this example, filling starts at Gate A (pin retracted) and Gates B and C closed (Figure 17.8.2a).

Injection Molding and Its Variants

HOT RUNNER

MOVABLE PIN

PIN MOTION

GATE

GATE

PART

(a)

PART

(b)

Figure 17.8.1 (a) Regular gate for a hot-runner system. (b) Gate with a programmable pin.

The melt spreads as a circular disk in the mold cavity (see Figures 17.2.1 and 17.2.2) till, in this case, it contacts the left mold wall (Figure 17.8.2b,c), after which the material flows to the right – during which the pressure required to maintain the flow continues to increase – till it just crosses the second gate (Gate B). At this time Gate A is closed and Gate B is opened while Gate C remains closed (Figure 17.8.2d). From then onwards, Gate B forces flow to the right, the pressure increasing continually from a low value till the flow front reaches Gate C (Figure 17.8.2e). Then Gate B is closed and Gate C is opened to fill the remaining portion of the cavity (Figure 17.8.2f,g). In this way the long cavity is filled sequentially so that the effective flow length for filling is the distance between the gates rather than the entire cavity length. This process results in much lower filling and packing pressures, requiring less expensive molds; it also reduces the molding cycle time. Also, the lower pressures prevent overpacking, reduce the tendency of flash generation at mold parting lines, lower residual (molded-in) stresses, and result in better surface finish. It is a very effective method for molding thin-wall parts. In conventional hot-runner molding, after each molding cycle some molten material can drool from the gate opening and cause stringing which can affect the quality of subsequent parts. This problem is eliminated in programmed sequential molding because each of the gates can be closed just before demolding. 17.8.1.2 Injection-Compression Molding

The injection-compression molding process is designed to reduce the high molding pressures required in injection molding. This is achieved by injecting a melt volume equal to the part volume into a partially closed mold. The injection pressures are lower because the melt initially flows through a larger mold cross section. The mold cavity thickness is reduced to the design value by closing the mold, during which process the entire cavity is filled and packed. The compression from mold closure causes a more uniform pressure distribution throughout the mold cavity, which in turn results in more uniform packing, and lower shrinkage and residual stresses in the part.

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A

B

C

A

B

C

A

B

C

A

B

C

A

B

C

(a)

(b)

(c)

(d)

(e)

Figure 17.8.2 Mold filling sequence in a mold with three gates fitted with programmable gate opening/closing pins.

The sequential steps are schematically shown in Figure 17.8.3. In this process layout, the movable core starts at an intermediate position in which the open cavity height is larger than for the fully closed mold (Figure 17.8.3a). The melt volume required to fill and pack the part is then injected into this partially

Injection Molding and Its Variants

A

B

C

A

B

C

(f)

(g) Figure 17.8.2 (Continued)

closed mold (Figure 17.8.3b); the larger cavity height allows this material to flow at lower injection pressures. Closure of the mold then causes this material to flow into the mold (Figure 17.8.3c), and eventually to fill the fully closed cavity (Figure 17.8.3d). The solidified part is then demolded (Figure 17.8.3e – h), just as in conventional injection molding. In this two-stage process, the change in the flow condition when transitioning from the injection to the compression phase can leave vestigial marks referred to as “hesitation” or witness marks. One method for overcoming this shortcoming is to initiate the compression phase before the filling phase ends; the resulting continuous flow front yields parts with much better surface finish. In yet another variation, melt is injected into a closed mold in which the movable core is held in place at a relatively low closing force. As the injected melt pressure builds up, it forces the movable core to open the mold cavity, thereby reducing the required injection pressure. At some stage a controlled closure of the movable core imposes compression and an eventual filling of the part. Any injection molding machine with accurate shot-volume control can be used for this process. However, in comparison to standard injection molding, an additional pressure-controlled module is required to control the position of the movable core. Injection-compression molding is the process technology for molding audio, video, and Blu-ray discs. Because it results in lower residual stresses and birefringence, it is also used for molding optical lenses. 17.8.2

Structural Foam Molding

Structural foam molding processes, which are modifications of the injection molding process, were originally developed to avoid sink marks in injection-molded parts. Of the several variants, in one process – often referred to as the low-pressure process – a pressurized, gas-charged melt is injected into a mold cavity, with the volume of material injected being less than mold cavity volume. The hot melt surface coming in contact with the relatively cold mold walls solidifies, resulting in the formation of thin solid skins. The fall in pressure after injection causes the dissolved gas to come out of solution, forming bubbles that continue to grow in size. The low thermal diffusivity of plastic melts causes the core to stay warmer than the skin regions, allowing more time for bubble growth, resulting in larger bubbles in

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STATIONARY CAVITY

NOZZLE CONTACT

(a)

(e) MOVABLE CORE EJECTION PIN

(b) (f)

(c) (g)

(h)

(d)

Figure 17.8.3 Mold filling sequence in injection-compression molding.

Injection Molding and Its Variants

the core than near the skins. The nucleation and growth of the bubbles force the thin skin to contact the mold walls, thereby avoiding sink marks, even in relatively thick parts. However, the surface finish of parts is poor, requiring secondary finishing operations such as painting. The most commonly used materials for structural foams are low- and high-density polyethylene. Polystyrene is used for molding furniture. Polypropylene is used for higher stiffness and strength applications. Other materials used include ABS, nylon, modified poly(phenylene oxide) (M-PPO), polycarbonate, and polyetherimide (PEI). The use of a gas requires an auxiliary system with an accumulator to supply pressurized gas for injection into the melt; the gas used most often is nitrogen. Alternatively, the pellets used for molding are compounded with a blowing agent that generates a gas on heating; in this case an auxiliary system is not required. The expanding gas generates a pressure in the range of 2 – 3.5 MPa (300 – 500 psi) that is sufficient for driving the initial charge to fill the mold extremities. As with most injection-molded parts, foam components are typically thin-walled. The trend is toward reducing the part wall thickness, typically from 6.35 mm (0.25 in) to 4 mm (0.157 in). Because foam molding uses relatively low injection pressures – about one-tenth of the pressures for molding resins – it requires less expensive molds than standard injection molding. Structural foams are typically described in terms of a general density reduction resulting from the foaming process. The level of nominal density reduction is controlled by process parameters and directly relates to the amount of material injected into the mold cavity. With reference to Figure 17.8.4, let the volume of the material injected into a mold cavity having volume v0 be v, and the density of the (unfoamed) resin be 𝜌0 . Then, since the same mass having initial volume v ends up filling the mold cavity volume v0 , by neglecting the mass of the dissolved gas or foaming agent, the average density 𝜌 of the foamed material can be defined by 𝜌0 v = 𝜌 v0 . The percent density reduction is then defined as 100 (𝜌0 − 𝜌)∕𝜌0 = 100 (v0 − v)∕v0 . Structural foam parts are normally designed for nominal density reduction levels in the range of 5 – 25%, although higher density reductions are possible, especially in thicker parts. Figure 17.8.5 shows the cross-sectional morphologies of 4-mm thick, 12.7-mm wide modified poly(phenylene oxide) structural foam bars for nominal density reductions of 5 and 15%. Clearly, for the same thickness part, the morphologies for different density reduction foams can be quite different. Actually, the nominal density reduction used for filling a part does not describe the local density reduction in a part. This is illustrated by the densities of 18, 19 × 152.5-mm (0.75 × 6-in) rectangular bars cut from a 152.5 × 457-mm (6 × 18-in), M-PPO-SF edge-gated, molded plaque (Figure 17.8.6). The average density of each bar was determined by dividing its weight by its volume. The average densities of the 18 bars along the plaque, for two plaque thicknesses of 4 and 6.35 mm, and two nominal density reductions of 5 and 15%, are shown in Figure 17.8.7. First, the highest density occurs near the gated end and drops off along the plaque length. Second, the density drop-off along the plaque length is more pronounced for the 15% nominal density reduction plaques – both for the 4- and 6.35-mm thick plaques. And third, the density differences between the 4- and 6.35-mm thick bars are relatively small for the 5% density reduction foam, but much larger for the 15% density reduction foam. Clearly, then, the nominal part density reduction is a poor indicator of the actual local density reduction. The change in the local density reduction is also reflected in the local skin-core morphology. Figure 17.8.8 shows the skin-core morphologies of 3, 19 × 152.5-mm specimens cut from 6.35-mm thick edge-gated plaques (Figure 17.8.3) for three plaque nominal density reductions of 5, 15, and 25%. The three specimens for each of the three nominal density reductions have been cut from the gated end, from the middle, and the end of the plaques – the actual positions have been indicated by the specimen numbers.

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CAVITY VOLUME

v0

GATE

VENTED MOLD

(a) INJECTED MELT VOLUME

v

(b)

(c)

CORE

SKIN

(d) Figure 17.8.4 Schematic diagram illustrating basic aspects of foam molding. (a) Geometry of a vented mold having a cavity volume of v0 . (b) Melt volume v injected into the mold. (c) Gas evolution in the melt fills the entire mold cavity. (d) Ejected foam part; detail shows the typical skin-core morphology.

Injection Molding and Its Variants

(b)

(a)

Figure 17.8.5 Cross-sectional morphologies of 12.7-mm wide M-PPO-SF bars. (a) 4-mm thick bar cut from 5% density reduction injection-molded plaque. (b) 4-mm thick bar cut from 15% density reduction injection-molded plaque. (Adapted with permission from V.K. Stokes, R.P. Nimmer and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1491 – 1500, 1988.)

457 mm

152.5 mm

SPECIMEN 1

SPECIMEN 18

Figure 17.8.6 Layout of 18 19 × 152.5-mm (0.75 × 6-in) test specimens cut from 152.5 × 457-mm (6 × 18-in), M-PPO-SF edge-gated, molded plaque.

A comparison of Figure 17.8.8a with Figure 17.8.8d shows that specimens having different nominal density reductions – here 5 and 15% density reductions – can have very similar morphologies. Similarly, a comparison of Figure 17.8.8f with Figure 17.8.8i again shows that specimens having different nominal density reductions – here, 15 and 25% density reductions – can have very similar morphologies. Clearly, structural foams are not homogeneous materials and their morphologies, and hence mechanical properties, depend on the processing conditions and part geometry. As such, the methods used for characterizing the mechanical properties of resins are not appropriate for structural foams, which really are material systems rather than materials. The mechanical characterization of structural foams is discussed in detail in Chapter 23. 17.8.2.1 Alternative Foam Molding Processes

Some of the shortcomings of the low-pressure process – such as poor surface finish, nonuniform cellular core, and relatively porous skins – can be avoided by using the gas counterpressure process, in which the mold is initially filled with an inert gas to maintain a pressure of 1.5 – 3.5 MPa (200 – 500 psi) during filling. Maintaining this pressure requires the mold and sliding surfaces, such as those of ejection pins, to be sealed by O-rings. After the short-shot has been filled into the cavity, the pressurized gas is vented in a controlled manner to allow the material to expand into the cavity. This process gives a more uniform bubble distribution in the core and the more controlled expansion results in a better surface finish. However, in this process the nominal density reduction is limited to about 10%, in comparison to up to 25% for the low-pressure process.

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NOMINAL DENSITY REDUCTION 15%

5%

1.2

4 mm

6.35 mm

4 mm

6.35 mm

1.1

DENSITY (g cm –3 )

534

1.0 0.9 0.8 0.7 0

5

10

15

20

SPECIMEN NUMBER Figure 17.8.7 Variations of the average local density of rectangular specimens cut from 4- and 6.35-mm thick, 5 and 15% density reduction, M-PPO-SF edge-gated, molded plaques. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

Less often used is the high-pressure foam molding process in which the cavity is first filled with the gas-charged or blowing-agent-filled melt. After skins have formed at the mold surface, the material is expanded to form a cellular core by creating more space by retracting a specially designed core; this feature limits the complexity of the parts that can be molded. Alternatively, after the mold has been filled, the pressure generated by gas released by the blowing agent is used to push material back into the runner system. Although this process is more controllable, resulting in better surface finish and a more uniform core, it requires more expensive molds because (i) the injection pressures are equivalent to those in the regular injection molding process and, (ii) the cost of a movable core. 17.8.2.2

Advantages, Disadvantages, and Applications

The main advantages of structural foams are (i) low pressures allowing molding of large parts with thicker ribs and bosses, (ii) elimination of sink marks by the pressure on the solidifying skin imposed by the gas pressure in the slowly solidifying core, (iii) molding thicker parts, (iv) lower residual stresses resulting in better dimensional control and less part warpage, (v) better acoustic and thermal insulation resulting from the skin-core morphology, and (vi) better stiffness to weight ratios in relatively flat parts. The main disadvantage of structural foams is poor surface finish from the surface swirl pattern caused by the molding process.

NOMINAL DENSITY REDUCTION

Injection Molding and Its Variants

5% (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

15%

25%

Figure 17.8.8 Variations of the skin-core morphologies of specimens cut from 6.35-mm thick, 5, 15, and 25% nominal density reduction, M-PPO-SF edge-gated, molded plaques. (a) 5%, specimen 1, (b) 5%, specimen 7, (c) 5%, specimen 18, (d) 15%, specimen 7, (e) 15%, specimen 10, (f) 15%, specimen 16, (g) 25%, specimen 2, (h) 25%, specimen 6, and (i) 25%, specimen 16. (Adapted with permission from V.K. Stokes, Journal of Materials Science, Vol. 27, pp. 5073 – 5083, 1987.)

Typical, diverse applications of structural foams include enclosures for audio speakers, automobile load floors and glove box doors, battery trays, material handling pallets, playground slides, refuse bins, satellite dishes, and water skis. Photos of many structural foam applications are shown in Reference 1 (pp. 448 – 449) and Reference 2 (pp. 75 – 79).

17.8.3

Microcellular Foam Molding

Microcellular foams are rigid thermoplastics with gas bubbles of diameters on the order of 10 μm. Also in development are nanocellular foams with diameters on the order of 0.1 μm. In contrast, conventional extrusion processes used for making polymer foams – such as for foam mattresses or building insulation – result in bubbles with diameters larger than 0.5 mm. In its earliest form, making microcellular foams was a two-stage process, in which a PS specimen from a plastic sheet is first allowed to absorb nitrogen in vessel at a pressure of about 100 bar (1,500 psi) till an equilibrium concentration of nitrogen has been reached, after which the specimen is removed. This gas-laden specimen, containing about 100 times more gas than is soluble in PS at 1 atm, is thermodynamically unstable. This specimen is then heated to near the glass-transition temperature of PS, resulting in an almost instantaneous nucleation of a large number of bubbles; the interaction of light with the small bubbles causes the specimen to turn from clear to white. This process, now referred to as the “batch solid-state microcellular process,” is still used today to explore new polymers and polymer-nano-composite systems;

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the term “solid state” is used to emphasize that the entire foaming process occurs in the solid state near the glass-transition temperature, well below the melting point of the polymer. From the limited property data available for microcellular foams, the strength is proportional to the solid-fraction of the foam – a 25% density reduction foam has 75% of the strength of the solid polymer. The impact strength depends on the gas-polymer system. In poly(vinyl chloride) (PVC), the impact strength decreases with increasing void-fraction, but in CPET, a crystallizable poly(ethylene terephthalate) with a nucleating agent, the impact strength remains nearly constant for void fractions down to 40%. In a variant, referred to as the “semi-continuous process,” a roll of plastic film is converted into to a roll of microcellular film. This process has been used to mass produce poly(ethylene terephthalate) microcellular sheet that has been used in packaging for food items, such as coffee cups, bowls, and trays. Up to 50% weight reduction in food packaging has been demonstrated by a switch from solid plastics. Figure 17.8.9 shows SEMs of microcellular and nanocellular structures in specimens of poly(methyl methacrylate) (PMMA) and PC. Figure 17.8.9 parts a and b show, respectively, the cellular structures

PC

PMMA

200 µm

50 µm

(b)

(a) PC

PMMA

1 µm

1 µm

(c)

(d)

Figure 17.8.9 SEMs of solid-phase generated microcellular and nanocellular foams PMMA and PC. (a) Structure of PMMA microcellular foam. (b) Structure of PC microcellular foam. (c) Structure of PMMA nanocellular foam. (d) Structure of PC nanocellular foam. (Photos courtesy of Professor Vipin Kumar, University of Washington.)

Injection Molding and Its Variants

of PMMA and PC microcellular foams having about 109 cells cm−3 . Because of the lower viscosity of the PMMA melts the cells are much larger in PMMA (Note the differences in the scales.). Clearly, the cells are much smaller and of more uniform size than in structural foams. The corresponding structures in nanocellular foams are shown, respectively, in Figure 17.8.9c and d. Clearly, at about 1015 cells per cm−3 , the cells are much smaller. Here again, the cells in PMMA are larger. In contrast to the solid-state processing of microcellular and nanocellular structures discussed previously, such structures can also be generated by using single-phase, gas-charged supercritical polymer melts. In continuous microcellular foam injection molding processes, nitrogen or carbon dioxide is injected at high pressures into the melt in the screw of a conventional injection molding machine to form a single-phase, supercritical solution of the gas in the melt. This melt is injected into molds where the reduced pressure results in the nucleation of very small bubbles. Just as in structural foam molding, the initial contact of the supercritical melt with the relatively cold mold walls forms a solid skin. However, in contrast to structural foam molding, the very high dissolved gas content results in much smaller, relatively more uniform microcellular structures. And, in contrast to the solid-state processes discussed before, coalescence of the bubbles results in somewhat larger bubbles and in bubble diameter gradients in the part. As with structural foam molding, the formation of bubbles continually expands the melt volume, thereby filling out potential sink marks and reducing warpage. This mechanism replaces the packing phase in conventional injection molding, thereby substantially reducing cycle time. Figure 17.8.10 shows a 2012 Ford Escape SUV and Kuga CUV instrument panel, the first microcellular foam instrument panel. In comparison to conventional solid injection molding, this long-glass and talc-filled PP foam panel, made by using MuCell® microcellular foam injection molding technology, reduced the weight by 0.45 kg (1 lb), the cycle time by 15%, the mold clamp tonnage by 45%, and saved about $3 per vehicle.

Figure 17.8.10 Large instrument panel made by injection molding microcellular parts made of long-glass filled and talc-filled polypropylene. (Photo courtesy of SPE Automotive Division.)

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17.8.4

Multimaterial Molding

In many applications optimal part performance requires the use of two or more materials in the same part. Generic examples include cost reduction by encapsulating a core made of a cheaper material, such as regrind, by a skin of a more expensive material; using a high-quality surface skin to encase materials with poor surface finish, such as structural foams and highly filled resins; using a fiber-filled core to obtain EMI shielding and using a resin skin for esthetics; and encasing a core to provide the required part stiffness and strength by an elastomeric layer to obtain a soft-touch surface. Two such processes are discussed in the sequel. In the first type, generically referred to a as coinjection molding, two different materials are sequentially or simultaneously injected into the same mold to form a layered part with a skin of one material encasing a core of the second. In the second, called overmolding, a subcomponent molded of one material is placed in a larger mold, and a second material is injected into the space between the mold cavity and the subcomponent that essentially acts as a core. 17.8.4.1

Coinjection Molding

In coinjection molding, also called sandwich molding, two separate injection units are used to sequentially inject different materials into a mold through one gate. This is made possible by using a rotatable switching valve to direct flow to the gate from one injection unit at a time. The principle underlying this process can be understood from the highly simplified schematic representation shown in Figure 17.8.11. First, a short-shot of the skin material is injected into the mold (Figure 17.8.11a). (The path of the skin resin from the injection unit to the gate is more clearly shown in the enlarged view of the switching valve assembly on the top of this figure.) During filling of the short-shot, because of fountain flow, material from the hotter core moves close to the mold surfaces where the cooled mold surfaces cause a thin layer of the resin to solidify. Then, after rotating the switching valve, core resin is injected into the mold, where it begins to displace the core of the short-shot deeper into the mold (Figure 17.8.11b). This process is continued till the mold is almost full (Figure 17.8.11c). Injection is then switched to the skin material to encapsulate any vestiges of the core material at the sprue-gate junction, and also to fill and pack the part (Figure 17.8.11d). Demolding results in a part (Figure 17.8.11e) in which the core is totally encapsulated by the skin material, even at the sprue-part interface, so that a totally encapsulated part is obtained on sprue removal. For this process to work, the different injection phases have to be precisely timed. For example, if the short-shot is not sufficient for continuing to form skin to contain the core material being injected, it could directly contact the mold surfaces – referred to as “core breakthrough,” or “core surfacing” – resulting in an undesirable surface finish. Several variants of this process are now available that allow more flexibility in terms how and when the materials are injected into a mold. In one version, special nozzles, into which two or three materials are fed through concentric channels, are used to simultaneously inject the skin and core materials through a gate without having to use a switching valve. The skin-core structure is then controlled by the thickness distribution of the resins in the concentric flow through the nozzle. Coinjection molding is used in a wide variety of automotive applications, such as headlamp reflectors, door handles, and distributor caps; insulted beverage mugs; housings for copiers, printers, and televisions; microwave dishes; outdoor furniture; and toilet seats. 17.8.4.2

Overmolding

Overmolding, also called multicomponent or two-shot molding, is a two-step process in which (i) first, conventional injection molding is used to mold a subcomponent, which (ii) is then placed in a larger mold cavity into which a second material is injected to partially or fully encapsulate the subcomponent

Injection Molding and Its Variants

SKIN RESIN

SKIN RESIN

CORE RESIN

CORE RESIN

FLOW-SWITCHING VALVE DETAIL

DETAIL

(a)

(b)

(c)

(d)

(e)

Figure 17.8.11 The coinjection molding process. (a) Short-shot of skin resin injected into mold cavity. (b) After rotation of the flow-switching valve molten core material is injected into the short-shot. (c) The mold almost filled. (d) After another rotation of the flow-switching valve, more skin material is injected to pack the part, and completely encapsulate the core. (e) Cross section of molded coinjected part.

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to form the part. The subcomponent essentially acts like an insert in the second molding step. As such, overmolding essentially consists of two separate injection molding processes, one for each material. One way to reduce tooling and cycle-time costs is to have the subcomponent molded into the core of the mold, after which the subcomponent-core assembly is rotated to fit inside a larger cavity into which the second material is molded. Alternatively, the mold can be made with a slide the retraction of which creates the desired larger cavity. Molding the subcomponent and the final part can then be further facilitated by using a flow-switching valve of the type shown in Figure 17.8.11a. An early important application of this process was for molding typewriter keys with permanent characters; this type of application is still important for molding keys for computer desktops and laptops and hand-held devices such as telephones. One high-volume automotive application of this process is in two-color taillight lenses; in one case, a special three-station mold and three materials have been used to mold a five-color taillight assembly. Applications requiring soft-touch surfaces made of TPEs include tooth brush handles (Figures 2.2.9 and 2.2.10) and mobile phone housings. Other applications include portable tool and electric shaver housings. Examples of overmolded micromolded parts are shown in Figure 17.6.14. 17.8.5

Hollow Parts

One major shortcoming of injection molding is the inability to mold complex, thin-walled hollow parts the geometries of which do not permit the use of retractable cores. Two processes for molding such parts are discussed in the sequel. The first of these uses a fusible core that can be melted out after the part has been molded. While this process adds to the cost, it produces parts with tightly controlled dimensions with smooth inner surfaces. In the second process, a mold is first filled with a short-shot of resin and gas is then injected into the core to produce a hollow part; in this less expensive process the part thickness cannot be controlled accurately. 17.8.5.1

Fusible-Core Molding

In fusible-core molding, a core in the shape of the internal cavity of a desired hollow part is first cast or molded from a low-melting point material, such as an eutectic bismuth-tin alloy. This core is placed as an insert inside a hollow cavity having the external shape of the part, and resin is injected into the resulting cavity. Ejection of the cooled molding essentially results in the desired part shape but with the embedded core. In the last step the core is melted out by heating; the core material is recycled for making more cores. For this process to work, the melting point of the core alloy must be lower than that of the molten overmolding plastic. But the core does not melt during molding because its high thermal diffusivity and thermal inertia keeps the core surface temperature below its melting point. The molded assembly is then immersed in a heated bath where the core melts out; the hollow part is ready after being cleaned. The core material is recycled with very little loss. Because the fusible core has to be cast or molded, and then melted after demolding, this is an energy intensive process. But it does produce dimensionally controlled parts with smooth inner and outer surfaces. The core can be made of other materials requiring less energy, such as wax or salt, which can be melted or dissolved out after the molding operation; recycling of such core materials can involve significant material loss. While the surface finish of the interior may not be a consideration in many applications, surface smoothness can be important; for example, in automotive air-intake manifolds where surface roughness can impede the flow of air. With the push for weight reduction, when plastics were considered for automotive air-intake manifolds, fusible-core molding rapidly became the technology of choice, and dedicated systems were developed for mass producing plastic manifolds. Initially, fusible-core molding

Injection Molding and Its Variants

was used for making bulk molding compound (BMC) manifolds, but the costs associated with longer cycle times for thermosets quickly led to glass-filled PA 66 becoming the material of choice; glass-filled PA 6 was also used. For some time, this application represented the largest applications of this technique, a photo sequences detailing different steps for which is shown in Reference 1 (pp. 372 – 373). Fusible-core technology is no longer used for making plastic air-intake manifolds. Instead, they are now made by vibration-welding two injection-molded halves (Figure 21.7.6). Although fusible core technology has been used for other applications such as turbocharger housings, high-performance bicycle wheels, and plumbing fixtures, very few companies now use this technology for making precision, low-volume parts with very complex geometries from high-temperature thermoplastics, such as PEEK and PEI, for the aerospace, medical, industrial, and military sectors. Figure 17.8.12 shows two views of a complex, aerospace filter housing made of carbon-fiber-filled PEEK using fusible core technology. The plastic two-part housing shown in this figure has the same shape as the aluminum housing it replaced, but is 53% lighter.

Figure 17.8.12 Two-part aerospace filter housing made of carbon-fiber-filled PEEK using fusible core technology. The two-part housing shown in this figure is 53% lighter than the aluminum housing it replaced. (Photos courtesy of Egmond Plastic BV.)

17.8.5.2 Gas-Assisted Injection Molding

The gas-assisted injection molding (GAIM) process starts off with the injection of a short-shot of resin into a regular mold – just as in coinjection molding (Figure 17.8.9a). Then, a pressurized inert gas, such as nitrogen, is injected at the same gate where it tunnels through the short-shot and pushes its boundary forward. Although at first glance this appears to be similar to the coinjection process, there is a big difference: Whereas in coinjection molding the viscosities of the skin and core materials are of the same order of magnitude (on the order of 10 3 Pa s), in GAIM the viscosity of the skin material melt (on the order of 10 3 Pa s) is about eight orders of magnitude larger than the viscosity of the gas (on the order of 2 × 10 −5 Pa s). As a result, the flow behaviors of these two processes are qualitatively very different. First, the low pressures required to push the gas into the material requires less expensive molds. And second, because the low-viscosity gas tends to tunnel through the path of least resistance, when the flow path is not properly designed, the gas can sometimes penetrate in lateral directions resulting in undesirable effects called fingering. Also, the gas does not require a separate injection head, and it can be injected through the same nozzle assembly. The main sequential steps in GAIM of a hollow part – a handle – are schematically shown in Figure 17.8.13. The mold assembly comprises an upper mold half with a sprue, a lower half with two pins, and a nozzle assembly with a nitrogen injection line (Figure 17.8.13a). The nitrogen line has two

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MELT NITROGEN OUT

NITROGEN IN NITROGEN LOWER MOLD HALF WITH TWO PINS NOZZLE SPRUE

UPPER MOLD HALF

(d)

NITROGEN

(a) MELT

(e) NITROGEN

(b)

NITROGEN

(f) 250 mm

(c)

(g)

Injection Molding and Its Variants

Figure 17.8.13 Steps in the gas-assisted injection molding of a hollow part. (a) Mold and injection head components. (b) Short-shot injection of melt into the mold cavity. (c) Gas injection into short-shot. (d) Intermediate stage in gas and melt flow front advance. (e) Complete filling and packing stage. (f) Recycling of gas. (g) Cross section of molded hollow part. (Adapted with permission from Figure 1.27, p. 23, in “Innovation in Polymer Processing: Molding,” J.F. Stevenson (Ed.), Hanser Publishers, Munich, 1996.)

ports, one each for injection and recovery. After closing the mold, the nitrogen ports are closed, and a short-shot of the molten resin is injected into the mold (Figure 17.8.13b). Then the melt inlet and nitrogen exit ports are closed and nitrogen gas is injected into the short-shot; the nitrogen begins to tunnel through the melt following the path of least resistance (Figure 17.8.13c). With sustained injection pressure the nitrogen continues to tunnel through the advancing short-shot (Figure 17.8.13d), until the part is filled with a skin of the resin; the pressure is maintained to keep the skin material pressed against the mold surface till the melt solidifies (Figure 17.8.13e) – this helps to eliminate sink marks. After the part has solidified, the nitrogen-in port is closed and the nitrogen-out port is opened; the gas released from the part can be recycled (Figure 17.8.13f). Finally, demolding results in the hollow part shown in Figure 17.8.13g. This part, a handle with a nominal length of 250 mm (10 in), was chosen to highlight how thick the walls of the hollow part are. Note several features: (i) The use of pins to form cavities used for inserting screws for attaching the handle to a wall. (ii) The part has a hole on the top left-hand side surface. And (iii), because of the way the melt flows, the thickness distribution is neither uniform nor symmetric. This thickness distribution cannot be made symmetric even by injecting the melt at the part center because, in a gas-containing liquid, flow instabilities at a T-junction can cause asymmetries in the bifurcated flows. In normal molding, the flow front is nominally at atmospheric pressure. From there the pressure in the melt increases all the way up to the sprue. As such, the longer the part is, the higher the required injection pressure will be, requiring sturdier, more expensive molds. In contrast to this, in gas-assisted molding the injection pressure only increases till the short-shot has been injected, after which the gas is injected at a sensibly much lower pressure (on the order of 0.5 – 25 MPa). This results in significantly lower mold clamp forces in comparison to regular injection molding of a part with the same geometry and, therefore, requires less expensive molds. In the example in Figure 17.8.13 the melt and gas are injected through one nozzle into one gate. But in variants of this process the resin and melt can be injected through different gates. This separation allows for multiple points for gas injection. The size of the short-shot has to be carefully chosen. If it is too short, the gas tunnels through the flow front, a condition called “blow out.” The larger the short-shot the smaller will be the distance through which the gas penetrates. This can result in large portions of the part having very thick sections, requiring very large cooling times, and having pronounced sink marks. Because the gas tends to follow the path of least resistance in the short-shot melt, the local part thickness cannot be controlled. For example, around a bend in the part geometry, the gas will tend to flow closer to the inner surface, as shown in Figure 17.8.14a; this effect is exacerbated by the higher temperature

(a)

(b)

Figure 17.8.14 Asymmetry in hollow part geometry due to gas flow at a bend.

(c)

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at the inner corner surface from poorer cooling resulting in a lower melt viscosity. The large differences in the thickness between the inner and outer surfaces of the bend can be reduced by rounding the bend, as indicated in Figure 17.8.14 parts b and c. The thickness distribution along a part can be made to approach a desired shape by proper gating and gas-injection points. Since determining the part stiffness and strength requires a prior knowledge of the part thickness distribution, once the process parameters (including location of gates) have been chosen, the thickness distribution has to be obtained by a computer simulation of the process. The top photograph in Figure 17.8.15 shows an approximately 355-mm (14-in) long polycarbonate GAIM foot-rest. The two lower photographs are of the part with a portion of the “handle” surface cut away, showing the shape of the gas channel. The part thickness in this oval-sectioned part is fairly uniform.

Figure 17.8.15 Photographs of a 355-mm long polycarbonate GAIM foot-rest. Top picture is an external view of the part. In the bottom two pictures a portion of the surface has been cut away to show the shape of the internal gas channel. Notice how uniform the thickness of the hollow portion is. (Photo courtesy of Sajar Plastics, LLC.)

Figure 17.8.16 shows the evolution of the channel shape in an approximately 510-mm (20-in) long ABS GAIM chair arm that has a rectangular cross section. An external view of the chair arm, together with views of three cut-away sections, is shown in Figure 17.8.16a. The channel shapes at the cut outs are better visualized in the enlarged views of the three cut outs from a different perspective in Figure 17.8.16b – d. Notice the nonuniform thickness of the hollow portions. In addition to molding hollow parts, gas-assisted molding is used for relatively flat parts that are stiffened by “hollow” ribs of the type shown in Figure 17.8.17. The sink marks normally associated with molded ribs (Section 17.5.2.2) are eliminated by the packing pressure exerted by the gas. And the wider rib geometry near the flat surface also contributes to the part stiffness.

Injection Molding and Its Variants

(a)

(b)

(c)

(d)

Figure 17.8.16 Photographs of a 510-mm long, ABS GAIM chair arm. (a) Photographs of external views of the chair arm and three cut away sections, showing the evolution of the gas channel shape. (b – d) Enlarged views of the three cut outs from a different perspective. Notice the nonuniform thickness of the hollow portions. (Photo courtesy of Sajar Plastics, LLC.)

The design of such gas stiffening channels must account for several effects resulting from the gas tending to flow along the path of least resistance: (i) Instead of forming a desired uniform channel, the gas can flow, or leak out, laterally into thinner sections in an unstable way, forming irregular finger-like gas pockets. This undesirable process is referred to as fingering. (ii) In contrast to the flow of resins, which at a T-junction bifurcates equally into two branches, in GAIM the gas bubble becomes unstable

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(a)

(b)

(c)

(d)

(e)

(f)

Figure 17.8.17 Rib and channel shapes for stiffening of panels by hollow ribs. (Adapted with permission from Figure 1.40, p. 37, in “Innovation in Polymer Processing: Molding,” J.F. Stevenson, (Ed.), Hanser Publishers, Munich, 1996.)

at a junction so that the two branches do not have the same shape. And (iii), instead of continually penetrating the short-shot melt, the bubble can punch through, resulting in one or more bubbles forming slugs of material separated by gas bubbles, thereby adversely affecting the part thickness distribution. The GAIM process has many applications. Automotive applications include door hardware modules, door and passenger assist handles, bumper fascias. Other diverse applications include copier and computer panels, chair bases, lawn and garden furniture, paint brush handles, material handling pallets, washing machine agitators, window frames, and toilet seats. The following series of photographs illustrate the complexity of the parts that the GAIM process is capable molding. Figure 17.8.18 shows the interior of an ABS medical equipment base, in which gas channels are used to feed ribs and bosses. In addition to two molded-in vents for cooling fans, the portion of the part shown has five molded-in metal inserts that act as nuts for metal screws. Figure 17.8.19 shows the underside of a laboratory equipment front cover of a PC/ABS medical equipment base, in which gas channels are used to feed ribs and bosses. The portion of the part shown has two molded-in metal pins that act as male inserts for snap fits onto the base. Figure 17.8.20 shows the ribbed supporting structure of a 30-GF-PBT medical equipment tray, in which the locations of the gas channels are indicated by arrows. Figure 17.8.21 shows the ribbed supporting structure of an ABS medical tray with handles, in which the locations of the gas channels are indicated by arrows. In addition to gas channels (indicated by white arrows) that feed the ribbed structure, the hollow handles are formed by gas channels (black arrows). Figure 17.8.22 shows the back surface of a PC/ABS ATM fascia, in which the locations of the gas channels are indicated by black arrows. Notice the large sizes of the channel that act as stiffening ribs.

Injection Molding and Its Variants

METAL INSERT

COOLING FAN VENT

50 mm BOTTOM

GAS CHANNELS

Figure 17.8.18 Location of gas channels in the interior of an ABS medical equipment base. The distance between the two vertical ribs is about 60 mm (2.25 in). (Photo courtesy of Sajar Plastics, LLC.)

METAL PIN

100 mm

Figure 17.8.19 PC/ABS Laboratory Equipment Front Cover. The top left edge shown is about 265-mm (10.5-in) long. The arrows show the locations of the gas channels. (Photo courtesy of Sajar Plastics, LLC.)

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100 mm

Figure 17.8.20 Ribbed supporting structure in an approximately 460-mm long by 305-mm wide (18 × 12-in), 30-GF-PBT medical equipment tray. The arrows show the locations of the gas channels. (Photo courtesy of Sajar Plastics, LLC.)

250 mm

Figure 17.8.21 Ribbed supporting structure in an approximately 635-mm wide by 560-mm high (25 × 22-in), ABS medical tray with handles. The arrows show the locations of the gas channels. (Photo courtesy of Sajar Plastics, LLC.)

17.8.5.3

Summary Comments

From the discussions in the previous two subsections it should be clear that while the fusible-core molding process produces parts with predefined geometries with tightly controlled dimensions, it requires significantly more investments. It essentially consists of two molding operations for each part, one for molding the fusible core, and the second for molding the part, so that the overall cycle time per part is

Injection Molding and Its Variants

100 mm Figure 17.8.22 Back surface of a PC/ABS ATM Fascia: the arrows show the location of the gas channels. The opening is for a 65 by 65 mm (2.5 × 2.5-in) window. The arrows show the locations of the gas channels. (Photo courtesy of Sajar Plastics, LLC.)

large. As such, it is used in high-volume applications requiring complex hollow plastic parts, such as automotive air inlet manifolds. In contrast, the GAIM process, for which only a gas-injection system and associated controls have to be added, requires much less additional investment in comparison to standard injection molding. As a result, this process is widely used for a very large number of applications. However, the GAIM process cannot produce parts with specified wall thicknesses. The capabilities of process simulation tools, required for designing part geometry, and selecting process conditions for obtaining desirable part thickness distributions, are limited by effects such as fingering. 17.8.6

Knit and Meld Line Esthetics and Integrity

For resins and particulate-filled resins the main issue with knit- and meld lines is surface finish. However, because of improper mixing, fibers not properly bridging across knit and meld surfaces in fiber-filled materials results in reduced strengths at such interfaces. Two modified versions of the injection molding process have been developed to address these issues: multiple-live-feed injection molding (MLFIM) and push-pull molding. In both these methods, described in the sequel, after mold filling a to-and-fro motion is imparted to the melt to mix the interface and to remove vestigial knit and meld lines. 17.8.6.1 Multiple-Live-Feed Injection Molding

In multi-live-feed injection molding MLFIM – also called shear-controlled orientation injection molding (SCORIM) – melt from an injection molding nozzle bifurcated into two streams is used to fill a mold

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through two gates in such a way that the filled material can be oscillated back and forth to achieve better mixing at knit and meld surfaces. The main machine and process features for this process are shown in the simplified schematic diagram in Figure 17.8.23: Flow from the main nozzle is injected into a flow-bifurcation module where it is divided into two streams, each of which feeds melt into two different gates in the part mold. Each of the two flow paths in the bifurcation module has a servohydraulically driven piston, marked A and B, that can be used to control the motion of the melt in the mold. First, the mold is filled either through one or both gates. Then, for short period of time, the pistons A and B are oscillated 180° out of phase, so that while the melt is pushed in through one gate, an equivalent amount of melt is pushed back through the second gate. This small cyclic, oscillatory flow induces a reversing macroscopic shearing motion in the melt; the resulting mixing at the flow at knit and meld surfaces improves both the knit/meld-surface integrity and the surface finish. After mixing, both the pistons act in phase to apply packing pressure till the part solidifies. The scheme in Figure 17.8.23 shows the use of a hot-runner feed system in which pin gates have been used for minimizing melt drool at the gate between parts. Standard cold-runner molds can also be used. In addition to a normal injection molding machine, the MLFIM process requires a special flow-bifurcating module with two servohydraulically actuated pistons. The resulting capital cost increase can only be justified in special applications, with fiber or flake fillers, in which knit/meld-surface integrity and surface finish are important. Figures 9.45 and 9.46 in Reference 1 (pp. 425 – 426) show the improvements in fiber orientation made possible by this process. 17.8.6.2

Push-Pull Injection Molding

Push-pull injection molding is similar to MLFIM molding in that melt in the mold is oscillated to-and-fro before the packing phase. However, it differs in that two computer-controlled injection heads are used to independently feed two gates in place of the flow-bifurcation module used in MLFIM. The mold cavity is first filled through both nozzles (Figure 17.8.24). After a weld surface is formed, the first (main) nozzle forces more material at the gate, thereby causing melt to flow back in the second (secondary) nozzle; this flow is accommodated by a retraction of the screw in the second injection unit. Then the second unit forces material through the second gate back into the first injection head. This to-and-fro motion is obtained through synchronized computer control of the of the two injection heads. After a predetermined number of cycles, both the nozzles act in unison to apply packing pressure till the part solidifies. Each to-and-fro motion deforms the meld surface to cause mixing. This motion, and the subsequent packing pressure, eliminates voids. The back and forth motion generates molecular orientation in the melt that is desirable in LCPs. This process uses two injection heads as in coinjection molding of two plastics in a two-injection head molding machine. So, although it is more expensive than a standard injection molding machine, this process uses conventional machines. As with MLFIM, this process can only be justified in special applications, with fiber or flake fillers, in which knit/meld-surface integrity and surface finish are important, such as in demanding automotive and aerospace applications. Examples of parts molded by this process are shown in Figures 9.48, 9.49, and 9.50 in Reference 1 (pp. 429 – 430).

Injection Molding and Its Variants

MELT

INJECTION MOLDING BARREL NOZZLE

FLOW BIFURCATION MODULE PISTON MOTION

A

B

SERVOHYDRAULICALLY DRIVEN PISTON

MOLTEN PLASTIC

CARTRIDGE HEATER

HEATED MANIFOLD BLOCK STATIONARY MOLD CAVITY MOVABLE CORE

POST-FILL SHEARING

PIN GATE

Figure 17.8.23 Schematic diagram showing essential elements of the multi-live-feed injection molding system. The main difference from standard injection molding is the insertion of a flow-bifurcating module between the injection-molding nozzle and the mold, which feeds two gates in (heated) manifold block. Servohydraulically controlled pistons A and B in this block are used to impart post-fill shearing motion to the melt in the mold.

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NOZZLE-1

NOZZLE-2

STATIONARY CORE

MOVABLE CAVITY

POST-FILL SHEARING

Figure 17.8.24 Schematic diagram showing essential elements of the push-pull injection molding system. The main difference from multiple-live-feed molding is that oscillatory flow is obtained by using two synchronized computer-controlled injection heads, one each for two gates.

17.8.7

In-Mold Decoration and Lamination

Applying an external design on a molded part by painting or printing is a secondary operation that adds to the part cost. In-mold decoration (IMD) processes have been developed to produce decorated parts in a one-step process. They may be classified into two broad categories: IMD and in-mold lamination (IML). In IMD, the decoration is laminated onto a film of the same material as the part resin. For large-volume applications the designs may be prepared in the form of film rolls from which the decorations can be fed into a molding machine. The IMD process starts with a design on a roll being clamped between two mold halves with the design facing the mold half corresponding to the exterior part surface. Then melt is injected into the mold where it contacts the film and pushes it against the mold and, at the same time, the film fuses with the molten plastic. On cooling the molded part has the desired design. This description only explains the principle underlying the IMD process. There are many innovative modifications that make it possible to produce parts with high-quality surfaces with effects not possible by painting and printing on a molded part. Examples of IMD applications include automotive lenses, bumper fascias, and exterior mirror housings. In the IML process, instead of using a thin film as in the IMD process, a multilayered textile laminate is placed in the parting surface of a mold facing the mold half corresponding to the exterior part surface. It is then overmolded on the inner side with a resin. There are several modifications of this process that make it possible to use a variety of fabric types, and to layer the fabric with an intermediate layer of

Injection Molding and Its Variants

elastomeric foam to produce a soft, giving feel to the external textile surface, which can be important for seat applications. Examples of IML include automotive interior panels, and molded chair seats with laminated textile fabric exteriors. Examples of in-mold decorated and in-mold laminated parts molded are shown in Figures 9.20, 9.21, and 9.22 in Reference 1 (pp. 393 – 394).

17.9 Concluding Remarks The process of injection molding and its variants can be looked at from three different perspectives: (i) The design and fabrication of molding machines. (ii) The design and fabrication of molds. And (iii), the use of computer simulation tools to better design plastic melt flow paths in molds, layout of cooling lines in molds, and to predict residual stresses, shrinkage, and warpage in parts. The performance of injection molding machines has benefited enormously from computer control that, besides providing real-time monitoring of temperatures and flow rates, makes possible the delivery of the melt at the right place at the right time. A critical element of molding machines is the extruder – the main function of which is to melt pellets and deliver a uniform melt at the nozzle – the improvements in the design of which have significantly contributed to the quality of molded parts. Several variants of injection molding have evolved, both to address some of its shortcomings as well as to enhance its capabilities. But far more than machine improvements, it is novel innovations in mold design and complexity that have made possible the molding of very complex parts. A mold is much more than just a part shaping device – it also functions as a heat exchanger, undergoing temperature cycles during each part molding cycle. The pressure-temperature cycle during solidification has a marked effect on residual stresses in a part and on part warping. Given the cyclic nature of thermal events during molding and the discrete distribution of cooling lines, only computer simulation can characterize the actual thermal history seen by a molded part. Part filling is a complex, transient coupled flow and heat transfer event in which the temperature of the mold surfaces also changes cyclically. Thus, a complete characterization of the pressure-temperature history seen by a part requires an integrated analysis of the melt and mold systems – again, a complex problem that can only be solved through computer simulation.

References The following two books have useful practical information on the subjects discussed in this chapter. They are listed here mainly because of photos of relevant applications. 1. T.A. Osswald, L-S. Turng, and P.J. Gramann, (Eds), Injection Molding Handbook, Hanser Publishers, Munich, 2002. 2. Jack Avery, Injection Molding Alternatives, a Guide for Designers and Product Engineers, Hanser Publishers, Munich, 1998.

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18 Dimensional Stability and Residual Stresses 18.1 Introduction Most plastic part shaping processes – in particular injection molding and its variants – involve solidification of molten resins, during which they undergo very large decreases in volume, so that the final dimensions of a part can be significantly smaller than that of the cavity in which it was molded. This change in dimensions is generically referred to as shrinkage. As an example, on demolding, the length of a meter-long part can shrink by as much as 1 cm. Differences in shrinkage between the two surfaces of a part can result in the out of plane distortion – even in a nominally flat panel. Such distortions are referred to as warpage. Dimensional stability is used as a generic term for issues relating to shrinkage and warpage – both for the immediate, short term effects observed after demolding and for the smaller, longer term effects due to relaxation in the material. Another important, and mostly undesirable, consequence of molding processes are self-equilibrating stresses induced in a part as a result of nonuniform cooling during solidification. Besides adding to the stresses in the part caused by externally applied loads, these thermally induced stresses – called residual stresses – can act as nuclei for chemically induced cracking, also called stress corrosion cracking. Residual stresses can be relieved by annealing, in which the part is cooled slowly after being heated to a temperature at which the stresses can relax. However, such annealing processes also result in dimensional changes that, in addition to changes in linear part dimensions, can also affect warpage. Thus, issues of dimensional stability and residual stresses are coupled and need to be considered simultaneously. From a part design and processing standpoint, mold design requires an a priori knowledge of the dimensional changes that occur on molding. For, then, the mold dimensions can be adjusted to achieve the desired final part dimensions. This strategy is used in making metal castings, for which patterns are purposely made larger to compensate for shrinkage on solidification. Plastic mold design attempts to compensate for shrinkage while sizing molds to achieve specified part dimensions. The increased mold dimensions are based on shrinkage data for the material being molded; the tacit assumption being that shrinkage is a material property that is unaffected by part geometry and processing conditions. This strategy assumes that the shrinkage is affine, or uniform, so that every part dimension shrinks by the same ratio, thereby assuring a part in which each dimension is a scaled down version of the mold dimension. However, local shrinkage is far from uniform and varies across the part depending on the part geometry and processing conditions. As a result, the uniform shrinkage strategy

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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cannot be used for obtaining parts in which all the dimensions meet specifications. Instead, only critical dimensions can be controlled, and that too by an iterative trial-and-error process. While critical dimensions may be achieved, this strategy can result in higher than necessary residual stresses. Prediction and control of shrinkage, warpage, and residual stresses are, perhaps, the most difficult generic problems associated with the use of plastics. This chapter presents three simple models of increasing complexity to explain the mechanics of the underlying processes.

18.2 Problem Complexity The injection molding process is not as simple as it might seem at first glance. An injection-molded part is obtained at the end of a series of coupled fluid and heat-transfer transient processes. First, as molten resin is injected into the mold, complex transient flow processes, in which hot material is continually moved to the mold surface, occur at the moving resin-air interface, or flow front, that is at ambient pressure; this flow is referred to as fountain flow (see Section 17.3). As the part fills, the pressure at the gate is increased to provide the necessary pressure drop to drive the flow. Second, the molten material begins to solidify on contact with the mold surface, so that the local flow-channel height changes continuously during the mold filling process. Third, temperature changes, caused by heat transfer and viscous dissipation, result in significant changes in the local melt viscosity during filling. These changes in the local viscosity, together with changes in the local pressure gradient, result in continually changing local shear rates that affect the local orientation in the melt. The highest pressure in the injection molding cycle normally occurs at the gates at the end of the filling phase. Fourth, on filling, the part continually shrinks owing to a decrease in the specific volume with decreasing temperature. To maintain part dimensions and shape after filling, pressurization is used to add molten resin to make up for the shrinkage resulting from volumetric changes. This pressure, which is maintained until the material at the gate freezes, is called packing pressure. Fifth, layers of solidifying molten material begin to support tensile and shear stresses even in the absence of motion. The sequential solidification of the layers results in self-equilibrating residual stresses. Because thermoplastics exhibit time-dependent (viscoelastic) thermomechanical behavior at near-glass-transition temperatures, the local stresses “relax” continually until the local temperature falls substantially below the glass transition temperature. This relaxation causes the stresses to redistribute continually. Finally, on demolding, the material can “spring back” from the mold constraints and, depending on the mold temperature at which it is ejected from the mold, the part can undergo further relaxation at ambient conditions. Although the local strains caused by volumetric changes during solidification of molten plastics, and the consequent overall changes in part dimensions relative to the part size, are quite small – on the order of 10−2 – they can result in residual stresses; also, differential shrinkage can adversely affect part shape. This chapter only addresses dimensional stability and residual stresses in amorphous materials. The treatment of these issues for semicrystalline materials, which in general shrink far more than amorphous materials, and for which additional analysis is required for the evolution of crystallinity, is well beyond the scope of this book.

18.3 Shrinkage Phenomenology As a first step toward understanding the phenomenology of shrinkage, consider the injection molding of a simple, edge-gated 177.8 × 355.6 × 3-mm (7 × 14 × 0.118-in.) rectangular plaque schematically shown

Dimensional Stability and Residual Stresses

in Figure 18.3.1. The mold surface had thin lines scribed at 25.4-mm (1-in) intervals that show up as rectangular grids on molded plaques. Measurements of the distances between these grid lines can be used to calculate the shrinkage at each grid location, both in the flow, or fill, direction and the cross-flow, or width, direction. The mold had four pressure transducers, making it possible to measure the local pressure at the nozzle and at the three locations in the plaque marked G (Gate), C (Center), and E (End) in the figure.

Cʹ

C

Bʹ

B

G

C A

177.8

E Aʹ

y

25.4

x 25.4 355.6

Figure 18.3.1 Plaque geometry. All dimensions in millimeters. (Adapted from “Dimensional Stability of Thermoplastic Parts: Model Experiments,” by A.J. Poslinski, W.C. Bushko, V.K. Stokes, and L.R. Cosma, SPE Technical Papers, Vol. XII, ANTEC 96 pp. 486 – 490, 1996, courtesy of Taylor & Francis Ltd.)

The large dimensions of the plaques made shrinkage measurements more accurate. A generous, thicker than the plaque gate assured gate solidification occurring only after complete solidification of the plaque, thereby assuring a continuous liquid path into the cavity during the packing phase. The data discussed in the sequel were obtained from molding experiments on a low-viscosity polycarbonate. The nominal molding conditions were a mold temperature of 93°C (200°F), a melt temperature of 316°C (600°F), an injection pressure of 117.2 MPa (17,000 psi), and a hold time of 20 seconds. The effect of the cavity pressure on the distribution of local shrinkage was investigated for five nominal hold pressures of 27.6 (4,000), 41.4 (6,000), 55.2 (8,000), 68.9 (10,000), and 82.7 MPa (12,000 psi). Five plaques were molded for each processing condition. Because the pressure histories for the same nominal processing condition showed very little variation within each batch of five plaques, the averages of five traces for the gate, center, and the cavity end were used to represent the cavity-pressure histories for that batch. Figure 18.3.2 shows the four pressure-time histories measured by the four pressure transducers during the molding of a 3-mm (0.118-in)-thick low-viscosity polycarbonate plaque for a nominal hold pressure of 55.2 MPa (8,000 psi). The first curve (NOZZLE), for the pressure at the nozzle, is a measure of the pressure just ahead of the gate, which increases as more and more molten resin is forced into the mold. After the mold fills up – in about 0.5 seconds in this case – the nozzle pressure is dropped and maintained at a constant lower level to “pack” the part. The role of this “packing pressure” is to continually force material into the mold to compensate for the shrinkage resulting from the volume reductions caused by

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the falling temperatures during solidification. The small “undershoot” in the pressure just before packing starts is a transient from the controller changing the pressure level. At about nine seconds, the gate freezes off and the pressure at the nozzle begins to decrease.

100

PRESSURE (MPa)

558

NOZZLE GATE

50 CENTER

END

0 0

50

100

TIME (s) Figure 18.3.2 Pressure traces recorded for a nominal hold pressure of 55.2 MPa (8,000 psi). (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

The second curve (GATE) in Figure 18.3.2 is the pressure trace from transducer 1 (at G in Figure 18.3.1), just after the gate. The pressure starts to increase at transducer 1 sometime after the pressure begins to increase at the nozzle. This offset represents the time for the resin flow front to flow from the nozzle to transducer 1. The pressure at this location then builds up to a peak, which is lower than that at the nozzle when the mold is completely filled. When the pressure at the nozzle is switched to the packing level, the pressure at transducer 1 first dips well below the packing level, and then increases to just below the packing pressure. After the gate solidifies, the high pressure within the plaque gradually decreases as a result of decreasing volume of the material within the plaque. The third curve (CENTER) in Figure 18.3.2 is the pressure history at the middle of the mold cavity, as measured by transducer 2 (at C in Figure 18.3.1). This curve is qualitatively similar to that obtained at transducer 1. However, the time offset at which the pressure begins to increase is larger, the peak pressure achieved is lower, and the undershoot in the pressure is greater. The fourth curve (END) in this figure is the pressure history at transducer 3 (at E in Figure 18.3.1), located near the end of the mold cavity. Here the time offset is even larger and there is no peak pressure. Figure 18.3.2 shows that the large (300 mm) length of the mold cavity and its small thickness (3 mm) resulted in a significant pressure drop along the mold, and that different parts of the plaque see very different pressure histories during the molding process. The end farthest from the gate sees the lowest maximum pressure. Now the regions close to the gate have hot material flowing through the core till the mold fills and therefore see relatively lower cooling rates. In contrast to this, the regions farthest from the gate, which do not see this continuing heat input in the core, see larger cooling rates. This end therefore freezes off first, as indicated by the earliest decline in pressure – after no more material can be added, the pressure falls off as the material shrinks on cooling. The mid-plaque region sees a higher pressure and

Dimensional Stability and Residual Stresses

for a longer time, indicating that this region solidifies after the far end. And the region nearest the gate is the last to freeze. For a nominal hold pressure of 55.2 MPa (8,000 psi), the distributions of flow and cross-flow direction shrinkages along polycarbonate plaques are shown, respectively, in Figures 18.3.3 and 18.3.4. The standard deviations of the measured values are indicated by error bars. Notice the larger scatter for the data from five plaques near the plaque ends. The flow direction shrinkage has a smaller variation among longitudinal lines AA ′, BB ′, and CC ′. Both the flow and cross-flow shrinkages increase from about 0.5% at the gate to about 0.7% at the plaque end, an increase of about by about 40%. This increase correlates with the local packing pressure being systematically lower from the gate to the far end. The shrinkage along centerline BB ′ exhibits a noticeable decrease at about x = 200 m. This anomalous decrease in flow shrinkage was observed at this location at all other hold pressures not shown here. It is attributed to the central pressure transducer projecting deeper into the plaque causing significantly deeper imprints on all the plaques, thereby introducing an undesirable constraint resulting in the anomalous flow shrinkage in its vicinity.

0.8

SHRINKAGE (%)

y = 50.8 mm 0.7

y = 101.6 mm

0.6

0.5 y = 25.4 mm 0.4 0

100

200

300

POSITION x (mm) Figure 18.3.3 Flow direction shrinkage along lines AA ′, BB ′, and CC ′ (see Figure 18.3.1) for a nominal hold pressure of 55.2 MPa (8,000 psi). (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

Figure 18.3.5 shows the distribution of the averaged combined flow and cross-flow shrinkage along the plaque length: For each longitudinal point along the plaque, the flow and cross-flow shrinkages at the corresponding points along the three lines AA ′, BB ′, and CC ′ were combined and averaged. The rationale for this averaging process is the assumption that flow and cross-flow shrinkages and shrinkages at lines AA ′, BB ′, and CC ′ are statistically similar and represent the same physical quantity. This assumption is justified for molding amorphous materials, such as polycarbonate, when a uniform melt flow front is achieved along the plaque during molding.

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0.8

SHRINKAGE (%)

y = 101.6 mm 0.7

y = 25.4 mm

0.6

0.5 y = 50.8 mm 0.4 0

100

200

300

POSITION x (mm) Figure 18.3.4 Cross-flow direction shrinkage along lines AA ′, BB ′, and CC ′ (see Figure 18.3.1) for a nominal hold pressure of 55.2 MPa (8,000 psi). (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

0.8

SHRINKAGE (%)

560

0.7

0.6

0.5

0.4 0

100

200

300

POSITION x (mm) Figure 18.3.5 Combined flow and cross-flow direction shrinkage along the plaque (see Figure 18.3.1) for a nominal hold pressure of 55.2 MPa (8,000 psi). (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

Dimensional Stability and Residual Stresses

Figures 18.3.6 – 18.3.8 show, respectively, the distributions of the flow, cross-flow, and combined shrinkages for four packing pressures of about 28, 48, 56, 70, and 82 MPa. Clearly, high packing pressures result in lower in-plane shrinkage. The differences in the shrinkage along the plaque – shrinkage increasing with distance from the gate – again correlates with the local packing pressure and the time for which it is effective (see Figure 18.3.2).

27.6 MPa 41.5 MPa

0.8

SHRINKAGE (%)

55.2 MPa 0.7

0.6

68.9 MPa

0.5

82.7 MPa 0.4 0

100

200

300

POSITION x (mm) Figure 18.3.6 Combined (from lines AA ′, BB ′, and CC ′) flow direction shrinkage along the plaque (see Figure 18.3.1) for five hold pressures. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

All the traces consistently show decreasing shrinkage with increasing cavity pressure. The combined flow and cross-flow shrinkage (Figure 18.3.8) varies from 0.44% at 82.7 MPa (12,000 psi) in the gate region to 0.77% at 27.6 MPa (4,000 psi) at the end of cavity. The experimental data presented in this section clearly shows that shrinkage in a part is not uniform. Observed differences in local flow and cross-flow shrinkage are normally caused by fillers and by differences in-mold constraints resulting from the geometry of the part being molded. Also, anisotropic crystallinity gradients in semicrystalline materials can cause anisotropic in-plane shrinkage. Shrinkage can be reduced by applying higher packing pressures. Clearly, the “shrinkage” of interest to a user is not a material property. Rather, it is a part, or a system, property that depends on the material and the processing conditions – as shown previously by the data for different packing pressures. This raises many interesting questions: What material characteristics contribute to shrinkage? Do the mold and melt temperatures affect shrinkage? How does part geometry, such as thickness and the constraining effects of ribs and bosses, affect shrinkage? What about the effects of filling rates? And what affects do all these variables have on part warpage and residual stresses? There are no simple answers to these practical questions. The most important material characteristic affecting dimensional stability is the volumetric change in the material as it shrinks on cooling. This property is characterized by the so-called PVT diagram, which is the subject of the next section.

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27.6 MPa

0.8

41.5 MPa

SHRINKAGE (%)

55.2 MPa

0.7

0.6

68.9 MPa

0.5

82.7 MPa

0.4 0

100

200

300

POSITION x (mm) Figure 18.3.7 Combined (from lines AA ′, BB ′, and CC ′) cross-flow direction shrinkage along the plaque (see Figure 18.3.1) for five hold pressures. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

27.6 MPa 41.5 MPa

0.8

55.2 MPa

SHRINKAGE (%)

562

0.7

0.6

68.9 MPa 82.7 MPa

0.5

0.4 0

100

200

300

POSITION x (mm) Figure 18.3.8 Combined (from lines AA ′, BB ′, and CC ′) flow and cross-flow direction shrinkage along the plaque (see Figure 18.3.1) for five hold pressures. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

Dimensional Stability and Residual Stresses

18.4 Pressure-Temperature Volumetric Data As discussed in Section 10.3.1, the variation of the specific volume – the volume of a unit mass of the material, which is the reciprocal of the density of the material – with pressure and temperature is normally described by a PVT diagram for the material. Figure 10.3.1 shows the specific-volume temperature, (v, T), curve for an amorphous melt slowly cooled along the path ABC. While the transition from the essentially linear v-T variations along most of the molten region AB and the solidified region BC occurs over a broad temperature range temperature, referred to as the glass transition, the glass transition temperature Tg , is defined by the intersection at G of the asymptotes of the essentially linear variations on either side of this transition. As shown in Figure 10.3.3, this glass transition temperature depends on the pressure p, so that it should be understood that Tg = Tg ( p). For amorphous materials, such as polycarbonate, the specific-volume temperature curve for each constant pressure exhibits a sharp change in slope over a very small temperature range that, for practical applications, is idealized to a point representing Tg . In the idealized PVT diagram for a low-viscosity polycarbonate (Lexan 101) in Figure 18.4.1, which is typical for amorphous materials, lines corresponding to

1.00

SPECIFIC VOLUME (cm3 g–1)

A0 0.95

p = 0 MPa

0.90 S0

100

•

0.85

E0

200

•

•S 0.80

0.75 0

100

200

300

TEMPERATURE (°C) Figure 18.4.1 PVT diagram for polycarbonate (Lexan 101). (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

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Introduction to Plastics Engineering

constant pressures, p, show the variation of the specific volume, v, with temperature, T. The v-T curve is continuous but has a discontinuous slope at Tg = Tg ( p). Normally, the symbol Tg is used for the temperature at the solid-liquid transition temperature at p = 0, which is point S0 in Figure 18.4.1. More generally, the glass transition temperature Tg = Tg ( p) at pressure p is the location of the point on the liquid-to-solid transition line S0 S corresponding to this pressure. The normally defined glass transition temperature is then Tg (0), which will be abbreviated as Tg (0) = Tg . Here, point S lies on the 250 MPa isobar. 0 This representation presumes a one-to-one equation of state of the form v = v ( p, T) that uniquely determines the specific volume v for each pressure and temperature combination. However, this equilibrium relationship is only valid when the temperature and pressure are varied sufficiently slowly to maintain thermodynamic equilibrium. For example, for each constant pressure, the isobars (constant pressure lines) on a PVT diagram determine the specific volume as a function of temperature only for very slow cooling rates. For higher cooling rates, the shapes of the v-T curves, especially the position of the sharp slope change representing Tg , change with the cooling rate (Figure 10.3.2). Thus, for given initial and final temperatures and pressures, and an initial specific volume, the final specific volume is path-dependent, that is, it depends on how the pressure and temperature varied from the initial to the final state. PVT data do not describe this nonequilibrium behavior. The cooling rates during injection molding are very high – clearly, change of state is not through an equilibrium process – so that the specific volume at any temperature and pressure obtained from equilibrium PVT data would, at best, be approximate. However, since nonequilibrium data are difficult to obtain and are normally not available, and since such processes are difficult to analyze, the analysis of thermal processes during molding is based on specific-volume data obtained from equilibrium PVT data. The PVT data show that polymer melts undergo very large volumetric shrinkage on solidification, or freezing. For example, from Figure 18.4.1, the specific volumes of polycarbonate at atmospheric pressure, p = 0, at temperatures of 300 and 20°C are 0.958 and 0.836 cm3 gm−1 , respectively. This translates into a volumetric shrinkage of 12.7%, which corresponds to a linear shrinkage of 4.2%. This would be the volumetric shrinkage in a mass of molten polycarbonate that fills the mold cavity at atmospheric pressure, and is then allowed to cool down to room temperature without the addition of more molten material. The curves on a PVT diagram represent the variation of the specific volume of the same, fixed mass of material as the pressure and temperature are changed. In injection molding, molten material is continually forced into the mold cavity to compensate for the decreasing volume of the solidifying. How the PVT diagram can be used to assess the effects of packing, during which the mass of material changes, is discussed in Section 18.5. 18.4.1

Quantification of PVT Data

Using this idealized representation for PVT data, the specific volume for any given pressure and temperature can be characterized by the double-domain modified Tait equation of state (DDTE) { [ ]} p v( p, T ) = v0 (T ) 1 − C ln 1 + B(T) v0 (T ) = b1x + b2x (T − b5 ) B (T ) = b3x exp [−b4x (T − b5 )]

(18.4.1)

in which the subscript x indicates the solid phase when the temperature T is less than the glass transition temperature Tg ( p) or the melt otherwise. The transition temperature, Tg , which determines the boundary

Dimensional Stability and Residual Stresses

between the solid and melt domains, is assumed to have a linear dependence on the pressure p Tg ( p) = b5 + b6 p

(18.4.2)

For the PVT curves for low-viscosity polycarbonate shown in Figure 18.4.1, the constant C = 0.0894, and the coefficients bi x (i = 1, 2, 3, 4), b5 , and b6 have the values listed in Table 18.4.1. Table 18.4.1 Double-domain modified Tait equation constants for polycarbonate. Constant

Solid

Melt

b1

854.16

854.16

mm3 g−1

b2

0.159

0.562

mm3 (g °C)−1

b3

299.95

182.88

b4

1.7051 × 10−3

3.9864 × 10−3

b5

144.01

b6

0.34877

Units

MPa

°C−1 °C °C (MPa)−1

On substituting the temperature on the liquid-solid transition line from Eq. 18.4.2 in Eq. 18.4.1, the specific volume on the solid-liquid transition line is given by { [ ]} p vS ( p) = (b1x + b2x b6 p) 1 − C ln 1 + exp (b4x b6 p) (18.4.3) b3x where p denotes the pressure at the time of solidification. Thus, Eq. 18.4.3 gives specific volume of the melt that solidified under pressure p. Equation 18.4.3 does not uniquely define vS ( p), because the parameters bi x , i = 1, 2, 3, 4 can be chosen either from the solid domain or from the liquid domain. The solidification line must be uniquely defined by the equation of state, since it is a common border between both domains. This uniqueness requires bi (solid) = bi (melt), i = 1, 2, 3, 4. The imposition of these conditions would eliminate the double-domain character of the Tait equation that would no longer discriminate between the solid and melt phases. As normally understood, the DDTE is continuous at the solidification point only on the p = 0 isobar. For p > 0, the isobars are discontinuous at the solidification point, that is, the specific volume undergoes a discontinuity. This discontinuity is normally overlooked because the DDTE, which is an empirical fit to data, is not considered accurate in the solidification regime where nonequilibrium effects are important. However, this regime is very important for estimating shrinkage; even a small discontinuity in the specific volume can cause large errors in the shrinkage estimates. This discontinuity in the specific volume along the solidification line can be resolved by requiring that vS ( p) be uniquely defined to first order in the solidification pressure. An expansion of vS ( p) in terms of p gives ( ) b1x vS ( p) = b1x + b2x b6 − C p+··· (18.4.4) b3x

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This modified uniqueness condition then requires that b1 (solid) = b1 (melt) b2 (solid) b6 − C

b1 (solid) b (melt) = b2 (melt) b6 − C 1 b3 (solid) b3 (melt)

(18.4.5)

The first part of Eq. 18.4.5 guarantees the continuity of v(0) = v(0, T) on the PVT diagram. While imposing a constraint on coefficients b2 x and b3 x , the second equation does not overconstrain the Tait equation; it can still discriminate between the solid and melt phases. The function vS ( p) (Eq. 18.4.3) and its linear approximation (Eq. 18.4.4) are shown in Figure 18.4.2.

0.86

SPECIFIC VOLUME (cm3 g–1)

566

0.85

TRANSITION LINE

0.84 LINEAR APPROXIMATION

0.83 0

50

100

PRESSURE (MPa) Figure 18.4.2 Linear approximation for the transition line defining the transition temperature. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

In terms of the specific volumes vS and vE0 at the solidification and final states, respectively, the volumetric shrinkage SV defined by vS − vE 0 SV = (18.4.6) vS can then be expressed as a function of the cavity pressure at solidification as SV =

vS + 𝛽 p + · · · − vE 0

vS0 + 𝛽 p + · · ·

0

≈

vS − vE 0

vS

0

0

−

vE 𝛽 0

v2S 0

p

(18.4.7)

Dimensional Stability and Residual Stresses

where 𝛽 = C b1 ∕b3 − b2 b6 and vS0 = b1 . The last step in obtaining Eq. 18.4.7 involves an expansion in 𝛽 p – which is small compared to vS0 , even for pressures up to 100 MPa – in which only linear terms in p are retained. The ratio (vS − vE )∕vS can also be approximated as 𝛼V (TS − TE ), where 𝛼 V is a 0 0 0 0 0 volumetric thermal expansion coefficient for the solid phase at ambient pressure, and TS0 − TE0 is the temperature difference between points A and F, so that SV = 𝛼V (TS − TE ) − 0

0

vF 𝛽 v2S

p

(18.4.8)

0

where TS = Tg (0) = Tg = b5 is the glass transition temperature. 0 0 If every portion of the part solidifies at the same time and the same pressure, the volumetric shrinkage SV will be equally distributed in all directions, so that the linear shrinkage will approximately equal 1∕3 of the volumetric shrinkage SV . Thus, this linear shrinkage – which would be the linear shrinkage of a part that has been cooled infinitely slowly – referred to as the isotropic shrinkage SI , is given by SI = 𝛼L (Tg − TE ) − 𝛾 p 0

0

(18.4.9)

where 𝛼 L = 𝛼 V ∕3 is a linear coefficient of thermal expansion for the solid phase and 𝛾 = vE0 𝛽 ∕3v2S . 0

18.5 Simple Model for How Processing Affects Shrinkage A highly idealized model for the molding process, in which the cooling rates are assumed to be so slow that the melt cools homogeneously – so that there are no temperature gradients in the solidifying melt – can be used to understand how packing pressure affects shrinkage in amorphous resins. The homogeneous temperature assumption rules out cooling rate and fill-time, or cycle time, effects. Also, the pressure will be assumed to be uniform throughout the material, ruling out the possibility of any pressure gradients. The injection molding process has the three phases of mold filling, packing, and solidification and demolding, for which the material in the mold sees the idealized pressure-time history shown in Figure 18.5.1. For easy visualization of this molding process, Figure 18.5.2 schematically shows these three phases for the filling of a cubic mold cavity of volume VM ; the arguments in this section are valid for a mold of arbitrary shape, however. In both Figures 18.5.1 and 18.5.2, like letters represent like events during molding. During the filling phase, AB (Figure 18.5.1), the temperature of the melt is assumed to be constant. As material is injected into the mold cavity, the pressure rises linearly from zero (atmospheric pressure) to a value pB . At the end of mold filling, the mass of material in the mold has the mold cavity volume, VM (Figure 18.5.2a), but the original volume V0 of this mass at the atmospheric pressure was larger, as schematically shown by the square (top of Figure 18.5.2a) at A being shrunk down to the mold shape at B. During the packing phase, BC (Figure 18.5.1), while the injected material cools homogeneously to the solidification temperature, the constant packing pressure continually forces molten material into the mold cavity to compensate for material shrinkage. Again, while the volume of material in the mold remains constant at VM during the packing phase, the mass of material increases continually from B to C. At the end of the packing phase (point C), all the material in the mold has solidified, so that more material cannot be forced into the mold cavity (Figure 18.5.2b). This “packed” material is in a compressed state at the packing pressure pB . Further cooling results in a fall in the pressure in the solidified material till,

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PACKING

•

C

•

MATERIAL IN CONTACT WITH MOLD

Cfo

MATERIAL NOT IN CONTACT WITH MOLD

FILLING

•B

50

COOLING

80

PRESSURE (MPa)

568

0

•

Dfo

A

0

•

D

•

10

E

• 20

TIME (s) Figure 18.5.1 Idealized pressure history seen by a material during molding.

at point D, the pressure falls to zero (atmospheric pressure). At this point, the part loses contact with the mold surface. Additional cooling no longer changes the pressure but the material continues to shrink till demolding at point E, at which point the part has a final volume VF (Figure 18.5.2c). On demolding, the molten part originally occupying the mold cavity volume VM finally has the volume VF . The volumetric part shrinkage SV is then defined by SV =

VM − VF VM

(18.5.1)

While shrinkage as defined here is less than unity, it is more commonly expressed in terms of percent shrinkage. For the range of processing conditions normally used in injection molding, volumetric shrinkage is on the order of 0.01 or 1%. Because of homogeneous cooling and the absence of any flow effects, the cube of molten material shrinks equally in all directions. As such, for the relatively small shrinkage that plastics undergo, the linear shrinkage SL is given by SL =

1 S 3 V

(18.5.2)

Dimensional Stability and Residual Stresses

VF

V0

VM

VM

p = pA → pB

p = pB → pC

p = pC → pD

Mold Filling

Packing

In-Mold Cooling

(a)

(b)

(c)

Figure 18.5.2 Schematic showing three phases for the filling of a cubic mold cavity of volume VM .

The control of linear shrinkage in molded parts is of great importance. The subsections that follow use the PVT diagram to assess the effects of packing-pressure history on shrinkage. 18.5.1

Constant Packing-Pressure History

Consider the shrinkage of the cube of material discussed before in terms of the PVT diagram for a low-viscosity polycarbonate shown in Figure 18.5.3. As before, on this figure the vertical segment AB corresponds to mold filling during which while the melt temperature remains constant, the pressure increases from pA = 0 (atmospheric pressure) at A to pB = 50 MPa at B. During the packing phase, BC, the pressure remains constant at pB = 50 MPa. Note, however, that material is constantly being added to the mold along the path ABC on the PVT diagram. For this reason, this path has been shown by a dashed line. The packing phase ends at point C on the freezing line, at which point the material in the mold has solidified, and after which no more material can be forced into the mold. Let the mass of the packed material at point C be m C . From point C to point D, while the volume of this fixed mass m C of solidified material remains constant at the mold volume VM , the pressure falls as the material cools from a temperature TC at C to TD at D. Thus, cooling in the mold occurs at constant specific volume vC = VM ∕m C , the specific volume at C. At point D the pressure on the solidified material becomes the atmospheric

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1.00 A

SPECIFIC VOLUME (cm3 g–1)

570

•

0.95

•B

p = 0 MPa

0.90 S0

D E

0.85

•

•

•

100

C

•

200

•S

0.80

0.75 0

100

200

300

TEMPERATURE (°C) Figure 18.5.3 Filling-packing path for a constant packing-pressure history of 50 MPa. (Note that TE = 20°C.)

pressure pD = pA = 0. Then, along DE, the solidified material cools at atmospheric pressure to the final temperature TE = 20°C. (Note that TE = 20°C throughout Sections 18.5.1 – 18.5.3.) During this cooling phase the material is no longer in contact with the mold, and the specific volume of the solidified material decreases from vD = vC = VM ∕m C to vE = VF ∕m C , where VF is the final volume of the material. It follows that the volumetric and linear shrinkage are given, respectively, by m v − m C vE v − vE VM − VF = C C = C VM m C vC vC 1 SL = SV (18.5.3) 3 Clearly, in this model the initial temperature TA has no effect on the shrinkage. For example, as shown in Figure 18.5.4, starting at a lower temperature TA ′ at A ′, and filling to the same packing pressure pB ′ = pB = 50 MPa results in the molding path A ′B ′CDE, in which the final shrinkage is again determined only by the specific volumes vC and vE at points C and E, respectively, on the PVT diagram. Thus, mold filling starting at any temperature along the isobar p = 0 to the right of the solidification point S0 will result in the same final shrinkage as long as the packing pressure is the same. From the PVT diagram for this low-viscosity polycarbonate, for pB = 50 MPa the specific volume along the solidification line S0 S is vC = 0.8529 cm3 gm−1 and vE = 0.8363 cm3 gm−1 , so that from Eq. 18.5.3 the linear shrinkage is SL ( p = 50) = 0.65 %. SV =

Dimensional Stability and Residual Stresses

SPECIFIC VOLUME (cm3 g–1)

1.00

0.95

Aʹ

0.90

p = 0 MPa

E

0.85

•

D1

•

S0

•

D

•

C

•

B'

•

•

C1

•

B

•

B1

•

200

•

100

A

•S

0.80

0.75 0

100

200

300

TEMPERATURE (°C) Figure 18.5.4 Two filling-packing paths, ABCDE and AB1 C1 D1 E, for a constant packing-pressure history of 50 MPa show that the starting initial temperature does not affect final shrinkage. Shrinkage paths ABCDE and AB1 C1 D1 E for constant packing-pressure histories 50 and 100 MPa, respectively, show that packing pressure level does affect shrinkage. (Note that TE = 20°C.)

However, the pressure at which the material is packed does affect shrinkage. For example, packing to a higher pressure pB along the molding path AB1 C1 D1 E results in a lower shrinkage 1 (vC − vE )∕vC , which is smaller than that for the molding path ABCDE shown in Figure 18.5.4. 1 1 Again, the initial temperature does not affect the final shrinkage, only the packing pressure does. For pB1 = 100 MPa, vC1 = 0.8429 cm3 gm−1 , so that with vE = 0.8363 cm3 gm−1 the linear shrinkage is SL ( p = 100) = 0.26 %, which is smaller than SL ( p = 50). By following the filling-packing pathAB0 C0 D0 E shown in Figure 18.5.5, in which the point D0 (not explicitly shown in the figure) coincides with point E, so that vC0 = vE , the final shrinkage is exactly zero at this critical packing pressure p0 , which for this material is 135 MPa; the solidified material attains the ambient atmospheric pressure at E. For higher packing pressures, such as along AB2 C2 shown in Figure 18.5.6, the point D2 (not shown in the figure) on the p = 0 line, at which vD2 = vC2 , occurs to the left of the demolding point E, so that the molding cycle occurs along AB2 C2 D2′ E, where the point D2′ on C2 D2 corresponds to the demolding temperature. At this temperature, the material is still under pressure with vD ′ = vC2 < vE . On 2 demolding along D2′ E the compressed material expands with the pressure dropping to zero. Clearly, with vD ′ = vC < vE , the volumetric shrinkage SV is negative, indicating an expansion in the dimensions of 2 2 the part. For example, for pB = 200 MPa, vC = 0.8262 cm3 gm−1 , so that SL ( p = 200) = − 0.41 %. 2

2

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Introduction to Plastics Engineering

1.00

SPECIFIC VOLUME (cm3 g–1)

572

A

•

B

•

B0

•

0.95 p = 0 MPa

0.90 S0

•

D

•

E

0.85

•

C

•

C0

•

0.80

135 250

•S

0.75 0

100

200

300

TEMPERATURE (°C) Figure 18.5.5 Shrinkage paths AB0 C0 E that results in zero shrinkage occurs for a packing pressure of 135 MPa. (Note that TE = 20°C.)

In this model, the effect of a constant packing pressure history on shrinkage is then clearly given by v ( p) − vE SV ( p) = C vC ( p) where vC = vC ( p) is the specific volume at the point where the isobar for packing pressure p intersects the solidification line S0 S. Figure 18.5.7 shows the variation of the linear shrinkage with the packing pressure. 18.5.2

Effect of Gate Freeze-Off

Sometimes the melt in the gate can freeze-off before the material in the mold has solidified; this phenomenon is called gate freeze-off. Once the gate has frozen off, applying packing pressure behind the frozen material does not cause additional material to be forced into the mold. Gate freeze-off is schematically shown in Figure 18.5.8, in which, instead of packing till the solidification point C, gate freeze-off occurs at point Cf , after which the melt in the mold cools till the pressure drops to zero at point Df , thereafter the material is no longer in contact with the mold surface. With reference to Figure 18.5.8, the molding cycle occurs along the path ABCf Df S0 E; material is packed into the mold only along the dashed path ABCf . Clearly, vDf = vCf , and 1 1 vCf − vE SL = SV = 3 3 vC f

Dimensional Stability and Residual Stresses

SPECIFIC VOLUME (cm3 g–1)

1.00 A

•

B

•

0.95 p = 0 MPa

0.90 S0

•

D

E

0.85

•• Dʹ

•

2

0.80

B0

C

•

C0

• •

C2

• B • 2

•S

135 250

0.75 0

100

200

300

TEMPERATURE (°C) Figure 18.5.6 Over packed part. Shrinkage paths AB0 C0 E that results in zero shrinkage. Molding path AB2 C2 D ′2 E results in an increase in the part size. (Note that TE = 20°C.)

As shown in Figure 18.5.9, the earliest freeze-off occurs at the end of filling (point B), for which the molding cycle is along the path ABD0 S0 E. As freeze-off occurs at later and later times, the freeze-off point Cf on the PVT isobar moves to points such as C ′, for which vD ′ = vC ′ , and finally to point C2 on the solidification line, corresponding to no freeze-off. Now in a real injection molding cycle, the freeze-off point would be indicated by a freeze-off time. But because time is not a parameter in the highly idealized molding cycle being discussed in this section, different freeze-off points can be indicated by the temperature on the isobar at the instant of freeze-off. For a fixed injection temperature of TInjection = 300°C, Figure 18.5.10 shows the variation of the linear shrinkage as a function of the freeze-off temperature for different packing pressures. In this figure, the rightmost points corresponding to T = 300°C, represent the linear shrinkage for no packing, that is, for the cases in which the gate is closed at the end of injection (point B). Clearly, although shrinkage decreases with increasing injection pressures, it is very large in the absence of packing. As the freeze-off point moves to the left on the isobars, the shrinkage decreases continuously. The left-most points correspond to freeze-off occurring along the solidification line (dashed line); this corresponds to full packing. Thus, premature freeze-off results in increased shrinkage.

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Introduction to Plastics Engineering

LINEAR SHRINKAGE (%)

1.0

0.5

0.0

– 0.5

–1.0 0

50

100 150 200 PACKING PRESSURE (MPa)

250

Figure 18.5.7 Variation of the linear shrinkage with the packing pressure.

1.00

SPECIFIC VOLUME (cm3 g–1)

574

A

•

B

•

0.95

Df

•

0.90 S0 E

0.85

•

•

50

Cf

•

p = 150 MPa C

0.80

•S

250

0.75 0

100 200 TEMPERATURE (°C)

300

Figure 18.5.8 Gate freeze-off. After filling along path AB, and packing at constant pressure along path BCf , the gate freezes off at Cf , after which no more material is added to the part. (Note that TE = 20°C.)

Dimensional Stability and Residual Stresses

SPECIFIC VOLUME (cm3 g–1)

1.00 A

•

B

•

0.95 D0

•

• •C • • Dʹ

0.90

S0 E

0.85

•

Cʹ

•

p = 150 MPa

2

50

250

•S

0.80

0.75 0

100 200 TEMPERATURE (°C)

300

Figure 18.5.9 Variations in gate freeze-off. (Note that TE = 20°C.)

5

TInjection = 300oC

LINEAR SHRINKAGE (%)

4

p = 0 MPa

3

50

2

100

•

1

0

–1 100

135

•

•

150

••

•

•

200 250

150 200 250 300 FREEZE-OFF TEMPERATURE (°C)

Figure 18.5.10 Variation of the linear shrinkage with freeze-off temperature.

350

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Introduction to Plastics Engineering

18.5.3

Effect of Packing Duration

This section addresses questions of the type: Is it necessary to apply the packing pressure till the part has solidified? During which phase of the molding cycle is the packing pressure most effective in reducing shrinkage? Figure 18.5.11 corresponds to the case in which, after injecting molten material into the mold along path AB, packing pressure is applied till point C, at which point the packing pressure is reduced to zero along path CD. In this cycle while new material is added to the mold along the dashed path BC, material is removed from the mold along path CD as the packing pressure drops to zero. Clearly (Why?), in this case the linear shrinkage is given by SL = (vD − vE )∕3vD , and this shrinkage is the same as that obtained via the zero-pressure path along DS0 E. Thus, if the packing pressure is removed prior to solidification at point C, the entire effect of packing at a higher pressure is lost, and the much higher shrinkage corresponds to the filling process starting at point D.

1.00 A

SPECIFIC VOLUME (cm3 g–1)

576

•

0.95

•B

D

• • • •C

0.90 D0

E

0.85

•

S0

•

p = 150 MPa

C0

50

•S

0.80

250

0.75 0

100

200

300

TEMPERATURE (°C) Figure 18.5.11 Packing pressure applied only during path BC. Material is added along path AB. (Note that TE = 20°C.)

Next consider the case (Figure 18.5.12) in which after injection at zero pressure at A, no more material is added to the mold along path AB. If the pressure is then increased along path BC, requiring the addition of more material into the mold, and this level of packing pressure is maintained to solidification along the isobar CC0 , then the linear shrinkage will be SL = (vC0 − vE )∕3vC0 . But this is the same as the linear shrinkage that would result from the packing pressure being applied all along the path AB1 C0 . Thus, according to this simplified model, the final shrinkage really depends on the packing pressure applied just prior to solidification. That is, packing pressure need not be applied until just before solidification. In summary, according to this model, no matter how high the packing pressure applied during the molding cycle, if this pressure is removed prior to solidification, then the linear shrinkage is the same

Dimensional Stability and Residual Stresses

1.00

SPECIFIC VOLUME (cm3 g–1)

A

•

0.95

•

B

0.90 D0

E

0.85

•

S0

50

•

•

• • •C

B1 p = 150 MPa

C0

•S

0.80

250

0.75 0

100

200

300

TEMPERATURE (°C) Figure 18.5.12 Variation of the linear shrinkage with freeze-off temperature. (Note that TE = 20°C.)

as if no packing pressure was applied. Also, if no packing pressure is applied during the early part of the molding cycle, and packing pressure is applied just prior to freezing, then the linear shrinkage is the same as if this packing pressure was applied along B1 C0 . 18.5.4

Summary Comments

The highly idealized model for the molding process, in which there are no temperature gradients in the solidifying melt, does not account for cooling rate and fill-time, or cycle time, effects. And the uniform pressure assumed throughout the material rules out the possibility of any pressure gradients. While this model addresses shrinkage, it cannot account for residual stresses and warpage. Also, this model does not address shrinkage in semicrystalline resins in which the PVT data very strongly depend on cooling rates that affect both the size of crystals and the amount of crystallinity developed. For a constant packing pressure, this model predicts that the initial melt temperature does not affect shrinkage. Mold filling starting at any temperature along the isobar p = 0 to the right of the solidification point will result in the same final shrinkage so long as the packing pressure remains the same. However, the pressure at which the material is packed does affect shrinkage. In a real injection molding cycle, the freeze-off point is indicated by a freeze-off time. But because time is not a parameter in the highly idealized molding cycle discussed in this section, different freeze-off points are indicated by the temperature on the isobar at the instant of freeze-off. Although shrinkage decreases with increasing injection pressures, it is very large in the absence of packing. As the freeze-off point moves to the left on the isobars, the shrinkage decreases continuously. Thus, premature freeze-off results in increased shrinkage.

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Introduction to Plastics Engineering

According to this model, no matter how high a packing pressure is applied during the molding cycle, if this pressure is removed prior to solidification then the linear shrinkage is the same as if no packing pressure was applied. Also, the final shrinkage really depends on the packing pressure applied just prior to solidification. That is, packing pressure need not be applied until just before solidification.

18.6 *Solidification of a Molten Layer Although the simplistic, highly idealized model for the molding process discussed in Section 18.5 provides a first level understanding of how of packing pressure effects shrinkage, its shortcomings limit the usefulness of its conclusions: This model does not account for temperature gradients in the solidifying melt during molding; rules out cooling rate and fill-time, or cycle time, effects; and cannot account for pressure gradients and variable pressure histories. As such, it cannot predict residual stresses and part warpage; the following two sections address these shortcomings. 18.6.1

*Freezing of a Molten Layer

To assess how temperature gradients affect shrinkage, the next level of complexity considers the solidification of a layer of melt in which the temperature and pressure are allowed to vary across the layer thickness but not along it. Figure 18.6.1 shows the geometry for this simple model for a freezing molten layer, initially at temperature Tmelt (x, y, z, 0) ≡ T0 , which is cooled between two mold surfaces, the upper and lower ones being maintained at fixed temperatures of TU and TL , respectively. Although this model allows for temperature and pressure gradients, the temperature and pressure can only vary across the layer thickness in the x-direction, but not in the lateral y- and z-directions.

Tmold (h/2, y, z, t) ≡ TU A

B x

Tmelt (x, y, z, 0) ≡ T0

•

O

578

x = h/2

y x = – h/2

C

D Tmold ( – h/2, y, z, t) ≡ TL

Figure 18.6.1 Geometry for the freezing of a molten layer.

Because each layer is constrained in the same manner in the y- and z-directions, in this model the stresses and the strains induced by them must have the form (Why?)

𝜎x = 𝜎x ( x, t), 𝜀x = 𝜀x ( x, t),

𝜎y = 𝜎z = 𝜎L ( x, t) 𝜀y = 𝜀z = 𝜀L ( x, t)

(18.6.1)

Dimensional Stability and Residual Stresses

in which the subscript L indicates the longitudinal direction (y- and z-directions). The shear stresses and shear strains are identically zero (Why?). Since this model does not have melt flow, additional assumptions have to be made for “injecting” molten fluid into the melt to model packing-pressure effects. This is achieved by specifying the imposed packing pressure and assuming that the material for “packing” is supplied by a source term. Also, this model does not account for fountain flow effects which cause the centrally hot layers to be transported to the cold-mold walls. As such, this model is still a rather crude representation of the actual conditions during molding. However, it does provide insights into the effects of varying temperature- and pressure histories. This model can be used to evaluate overall shrinkage by tracking the solidification resulting from heat transfer from the melt to the mold surfaces, both of which are assumed to be at the same temperature TU = TL = Tmold . The analysis proceeds by first numerically solving the one-dimensional transient heat conduction equation (Eq. 6.2.3) subject to the temperature boundary conditions specified at the mold surfaces. For any time t, this solution then defines the location of the point xsolid (t) at which the melt has reached the solidification temperature. At this time, all locations |x| ≥ xsolid ≤ h∕2 have solidified and the stress distribution in this region can be determined by integrating the equilibrium equations for stress subject to the lateral constraint conditions. The built-up stresses result in a net longitudinal force that is required to prevent the material from moving laterally. When the layer is “demolded,” this net force reduces to zero, resulting in shrinkage and a residual stress distribution in the solidified material. The integration of the equilibrium state requires a stress-strain model for the solidified material. In the simplest case, this material is assumed to be an elastic solid, the constitutive equations (stress-strain relations) for which are discussed in Section 4.3. The resulting analysis of the freezing process is discussed in Section 18.6.2. A more realistic model assumes that the solidifying material is viscoelastic material, the constitutive equations for which are addressed in Sections 7.3 and 7.7. Numerical solutions for the shrinkage of viscoelastic melts are discussed in Section 18.7. Numerical examples for shrinkage discussed in Sections 18.6.2 and 18.7 will all be for the freezing of a 3-mm thick layer at an initial temperature of T0 = 300°C, and a mold temperature of TU = TL = Tmold = 80°C. The simulations can then be compared with the experimental results for the molding of 3-mm thick plaques molded at comparable the mold and melt temperature conditions discussed in Section 18.3. Asymmetrical residual stresses resulting from different upper and lower mold-surface temperatures TU and TL can cause part warpage. Numerical simulations for such warpage in a 3-mm-thick solidifying viscoelastic melt, for T0 = 300°C, and the upper and lower mold temperatures varied such that T (0, t) = (TL + TU )∕2 = 80°C, is discussed in Section 18.8. 18.6.2

*Fluid to Elastic-Solid Freezing Model

In the simple freezing model discussed in the sequel, solidified layers are assumed to behave elastically. With reference to the discussion in Section 18.6.1, consider an elastic layer at a distance x from the center of the cavity thickness that is subjected to stresses 𝜎 x (x, t), 𝜎 y = 𝜎 z = 𝜎 L (x, t), the corresponding strains being 𝜀x (x, t), and 𝜀y = 𝜀z = 𝜀L (x, t). To these strains generated by the stresses must be added the isotropic shrinkage SI (x, t) – just as thermal strains in an elastic solid are added to those generated by stresses (Section 4.4). The through-thickness strain 𝜀x (x, t), and the in-plane strains 𝜀y and 𝜀z are then

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Introduction to Plastics Engineering

related to stresses 𝜎 x , 𝜎 y , and 𝜎 z , and the isotropic shrinkage SI (x, t) through 1 (𝜎 − 2𝜈𝜎L ) − SI E x 1 𝜀y = 𝜀y = 𝜀L = [(1 − 𝜈 ) 𝜎L − 𝜈𝜎x ] − SI (18.6.2) E where E and 𝜈 are, respectively, the Young’s modulus and the Poisson’s ratio of the material, which is assumed to be elastic from the instant it turns solid (Why does SI = SI (x, t) term have a negative sign?). The isotropic shrinkage at time t is

𝜀x =

SI (x, t) = 𝛼L [Tg − T(x, t)] − 𝛾 p(x)

(18.6.3)

in which the pressure at the time of solidification, p, is only a function of the layer x. While in the mold, the solidified layer is subject to the same time-dependent cavity pressure. Equilibrium in the thickness direction requires 𝜎 x = − p(t), so that 𝜎 x is time-dependent. After demolding and cooling to the final constant, uniform temperature, 𝜎 x = 0, 𝜎 L (x) is the residual stress, and 𝜀L (x) is the final in-plane strain in the layer. All the layers located between x = ± h∕2 will have the same final strain 𝜀L (x) = 𝜀L , which equals the negative of the final in-plane part shrinkage SL , so that from the second equation in Eq. 18.6.2 1−𝜈 𝜎L (x) − SI [ p(x)] (18.6.4) E A substitution for SI from Eq. 18.6.8, an integration of Eq. 18.6.4 across the wall thickness gives − SL =

− h [SL − 𝛼L (Tg − TF )] − 𝛾

h∕2

∫−h∕2

1−𝜈 𝜎 dx E ∫−h∕2 L h∕2

p(x) dx =

(18.6.5)

The right-hand side of this equation, essentially the total in-plane force, must equal zero for an unconstrained, demolded part, so that SL = 𝛼L (Tg − TF ) − 𝛾

h∕2

1 p(x) dx h ∫−h∕2

(18.6.6)

or SL = 𝛼L (Tg − TF ) − 𝛾 peff

(18.6.7)

where peff is the thickness averaged cavity pressure defined by h∕2

peff =

1 p(x) dx h ∫−h∕2

(18.6.8)

The parameter peff , referred to as the effective pressure, measures how a time-varying cavity pressure affects in-plane shrinkage. While two different cavity-pressure histories can result in the same effective pressure, each effective pressure corresponds to a unique shrinkage. An elimination of SL from Eqs. 18.6.4 and 18.6.6 gives the residual stress 𝜎 L (x) E (18.6.9) 𝛾 [ peff − p(z)] 1−𝜈 While this model does give a formal expression for the residual stress, the estimate in Eq. 18.6.9 is rather crude. For, consider the case of a constant cavity pressure p(t) ≡ p0 . In this case p(t) = p0 gives

𝜎L (z) =

Dimensional Stability and Residual Stresses

peff = p0 , so that the model predicts zero residual stresses. As discussed in Section 18.7, for this case a viscoelastic material model predicts substantial residual stresses that depend on the applied pressure – a result that is supported by experiments. As such, the residual stresses (Eq. 18.6.9) predicted by the simple model in this paper are not reliable, and must be interpreted with care.

18.6.3

*Numerical Example for a 3-mm-Thick Plaque

Figure 18.6.2 shows a typical cavity-pressure history p(t) for the end of cavity region of a mold cavity. The initial rapid increase in the cavity pressure during injection is followed by a slowly decaying pressure during the packing phase. Variations in the cavity pressure during solidification cause changes in the transition temperature Tg ( p) = b5 + b6 p (Eq. 18.6.2).

30

PRESSURE (MPa)

PRESSURE AT PLAQUE END

20

EFFECTIVE PRESSURE

10

0 0

5

10

14

TIME (s) Figure 18.6.2 Cavity pressure at cavity end as a function of time for a hold pressure of 82.7 MPa (12,000 psi). (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

For this example, the temperature history was obtained from a numerical solution to the heat-transfer equation (Eq. 6.2.3) using a constant thermal diffusivity of 0.135 mm2 s−1 . A uniform initial temperature of 300°C was assumed over the domain −1.5 ≤ x ≤ 1.5, and a constant mold temperature of 80°C was imposed at time zero on the surfaces at x = ±1.5. The position of the solid/melt interface x = x (t) is then determined from this temperature history T (x, t) by using T (x, t) = b5 + b6 p(t) ⇒ x = x (t). Figure 18.6.3 shows x = x (t) for the packing-pressure history in Figure 18.6.2. To illustrate the effect of elevated cavity

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pressure on the solidification front, Figure 18.6.3 also shows the location of the solid/melt interface for p = 0.

1.5 SOLID-LIQUID INTERFACE (mm)

582

FOR PRESSURE HISTORY AT PLAQUE END

1.0

FOR IDENTICALLY ZERO PRESSURE

0.5

0 0

5

10

12

TIME (s) Figure 18.6.3 Position of the solid-liquid interface at the cavity end as a function of time for a hold pressure of 82.7 MPa (12,000 psi). (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

Figure 18.6.4 shows the cavity pressure as a function of solid/melt interface position. The effective pressure peff , the thickness average of p(z), is 15.2 MPa, as indicated by the dashed horizontal line. The definition of the effective pressure requires the integration of cavity pressure from the initiation of solidification at the external surfaces x = ∓ h∕2 till, for symmetric cooling, the central plane x = 0 solidifies. For analyzing the development of in-plane shrinkage during solidification, it is convenient to define the partial effective pressure p̃ eff (t) ⎧ 2 h∕2 p(z) dz for t < tsol ⎪ p̃ eff (t) = ⎨ h ∫z (t) ⎪p for t ≥ tsol ⎩ eff

(18.6.10)

where tsol is the solidification time. Figure 18.6.5 shows the time history of the partial effective pressure for the case of symmetric cooling considered in this example. Clearly, the cavity pressure at the end of the solidification contributes more to the final part shrinkage than does the initial packing pressure. For t > 11.1 s, the cavity pressure does not affect shrinkage because all the material has solidified. It can be shown that even for a constant cavity pressure the curve in Figure 18.6.5 is not a straight line. The nonlinearity comes primarily from the nonlinear progression of the solidification front.

Dimensional Stability and Residual Stresses

PRESSURE AT Tg (°C)

30

ACTUAL PRESSURE AT PLAQUE END

20

EFFECTIVE PRESSURE

10

0 0

0.5 1.0 SOLID-LIQUID INTERFACE (mm)

1.5

Figure 18.6.4 Solidification front cavity pressure at the cavity end as a function of the through-thickness position for a hold pressure of 82.7 MPa (12,000 psi). The effective pressure peff , the thickness average of p(x), is 15.2 MPa, is indicated by the dashed horizontal line. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

18.6.4

*Effective Pressure as an Independent Variable

The effective pressure defined in Eq. 18.6.8 is a through-thickness average of the pressure- and temperature-dependent solidification pressure. It combines all geometric and processing parameters into one number, which can be calculated for any point in the cavity when the local pressure history is known from a direct measurement or a melt flow analysis. The only material property indirectly affecting the effective pressure is the material thermal diffusivity required for calculating the temperature distribution. Most unfilled amorphous thermoplastics have a very similar thermal diffusivity. Thus, for this class of materials, the effective pressure depends entirely on non-material process parameters. Consequently, the effective pressure is, perhaps, the most appropriate independent processing-geometry parameter for presenting shrinkage data. Figure 18.6.6 shows shrinkage data from Figure 18.3.8 as a function of the effective pressure calculated for each point from the measured cavity-pressure history. The simple solidification model, on which the effective pressure is based, can be used to predict local shrinkage from the PVT diagram for the material, described by Eq. 8.4.1 with the required parameters defined as in Table 18.4.1 for polycarbonate. The predicted shrinkage does agree with the measured values. The model captures the material contraction that occurs during the abrupt cooling in the mold. Eq. 18.6.7 indicates a linear dependence of the in-plane shrinkage on effective pressure, which accounts for all non-material parameters. The intercept and the slope in Eq. 18.6.7 are two parameters of practical interest because they can be considered as two material parameters that determine material shrinkage. The intercept b = 𝛼L (Tg − TF )

(18.6.11)

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EFFECTIVE PRESSURE (MPa)

30

20

10

0 5 TIME (s)

0

10

12

Figure 18.6.5 Cavity end partial effective pressure as a function of time for a hold pressure of 82.7 MPa (12,000 psi). (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

0.9 0.8 SHRINKAGE (%)

584

0.7 0.6 0.5 0.4 0.3 0

10

20 30 40 50 EFFECTIVE PRESSURE (MPa)

60

Figure 18.6.6 Shrinkage as a function of effective pressure. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 59, pp. 1648 – 1656, 2019.)

Dimensional Stability and Residual Stresses

depends on the linear coefficient of thermal expansion 𝛼 L and the glass transition temperature Tg , both of which are material properties, and on the final temperature TF , which can be assumed to be the room temperature of 20°C. Coefficient b gives an upper limit for the in-plane shrinkage of the material. The slope v 𝛽 a = − F2 (18.6.12) 3vS 0

depends on the final and glass transition specific volumes, and the coefficient 𝛽 = Cb1 ∕b3 − b2 b6 , which can be obtained from the PVT diagram for the material. Slope a determines the sensitivity of material shrinkage to the variation in the local cavity pressure. 18.6.5

*Summary Comments

The simple solidification model developed for predicting the effect of time-varying temperature and pressure histories on part shrinkage predicts a linear dependence of shrinkage on an “effective pressure,” which combines the thermal diffusivity of the material, the wall thickness, and the time-varying cavity pressure into a single parameter that is uniquely related to the shrinkage. The effective pressure is shown to effectively correlate in-plane shrinkage data obtained from injection molding experiments, in which shrinkage was measured at 25-mm intervals along the length and width of rectangular plaques molded in an instrumented mold. The solidification model characterizes two material parameters, which can be estimated from the PVT diagram for the material, that describe the sensitivity of the shrinkage to the local cavity-pressure history. While this model also gives an expression for the residual stresses, the estimates are rather crude, and must be interpreted with care. For example, contrary to well-established phenomenology, this model predicts zero residual stresses for a constant cavity pressure.

18.7 **Viscoelastic Solidification Model While the discussion in Sections 18.6.2 – 18.6.4 on the solidification of a molten layer, in which the solidified layers are assumed to be elastic, does account for varying pressures and through-thickness temperature variations, it does not account for viscoelastic relaxation effects that affect residual stresses. Although this model provides insights into how the packing-pressure history affects shrinkage, residual stresses are not well modeled. This section extends the analysis of the model described in Section 18.6.1 to viscoelastic melts. It addresses questions of the type: Is it necessary to apply the packing pressure till the part has solidified? During which phase of the molding cycle is the packing pressure most effective in reducing shrinkage? With reference to Section 18.6.1, this section discusses the freezing of a 3-mm thick layer of a viscoelastic melt at an initial temperature of T0 = 300°C, and a mold temperature of TU = TL = Tmold = 80°C. 18.7.1

Viscoelastic Material Model

During cooling the viscoelastic layer undergoes relaxation at a varying temperature, requiring a temperature-dependent viscoelastic material model. The simplest model for this is the thermorheologically simple material described in Section 7.7.1, the experimental basis for which is discussed

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in Section 14.3. This model requires two relaxation moduli – which can be the shear and bulk moduli – and a model for the shift function. In addition, the PVT relationship is required for modeling volumetric changes. All the simulations in Sections 18.7 and 18.8 will be for a low-viscosity polycarbonate for which the shear and bulk relaxation moduli are shown in Figure 18.7.1, and the corresponding shift function is shown in Figure 18.7.2.

4 3 LOG MODULUS (MPa)

586

BULK MODULUS

2

1 0 SHEAR MODULUS

–1 –2 –3 –2 –1

0

1 2 3 4 LOG TIME (s)

5

6

7

Figure 18.7.1 Shear and bulk relaxation moduli for a low-viscosity polycarbonate. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol.35, pp. 365 – 383, 1995.)

PVT data for this material for three pressure levels is shown in Figure 18.7.3. The DDTE model for this data are discussed in Section 18.4, in which Figure 18.4.1 shows the modeled variation of this data. 18.7.2

Temperature Distribution in a Solidifying Melt

For all the examples that follow, the temperature history was obtained from a numerical solution to the heat-transfer equation (Eq. 6.2.3) using a constant thermal diffusivity of 0.135 mm2 s−1 . A uniform initial temperature of 300°C is assumed over the domain −1.5 ≤ x ≤ 1.5, and a constant mold temperature of 80°C is imposed at time zero on the surfaces at x = ± 1.5. On demolding, the plaque surfaces are subject to an ambient temperature of 20°C. In the resulting transient temperature distribution shown in Figure 18.7.4, the melt cools rapidly because of effective heat transfer across the cold-mold walls maintained at 80°C. At 6 seconds (dashed curve), the entire thickness has solidified, that is, T(0, 6) = Tsolid = Tg . The solid “plaque” continues to cool till 30 seconds, at which it attains a uniform temperature and is demolded. Because thereafter the plaque

Dimensional Stability and Residual Stresses

7

LOG SHIFT FUNCTION

5

0

–5 –6 140

150

160

170

180

TEMPERATURE (°C) Figure 18.7.2 Shift function for a low-viscosity polycarbonate.

SPECIFIC VOLUME (cm3 g–1)

0.95

0 MPa 0.90 50 MPa

0.85

100 MPa

0.80 20

100

200

TEMPERATURE (°C) Figure 18.7.3 PVT diagram for a low-viscosity polycarbonate.

300

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300

0 0.1

250 TEMPERATURE (°C)

588

0.5 2 4

200

6

150

8 10

100

15 30 40 50 t=∞

50 0

– 1.5

– 1.0

0 0.5 – 0.5 POSITION (mm)

1.0

1.5

Figure 18.7.4 Temperature distributions across the solidifying melt thickness. At about t = 6 s all the material has essentially solidified. At about t = 30 s, at which all the material is essentially at the mold wall temperature of 80°C, the solidified slab is “demolded.” It then continues to cool more slowly. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

surfaces are in contact with ambient at a temperature of 20°C, the much less efficient heat transfer to the air the plaque continues to cool but at a significantly lower rate. The central plane, x = 0, eventually attains the ambient temperature at about 90 seconds. Because of the poor heat-transfer characteristics of the melt, there is a significant lag in the central-plane temperature, as shown in Figure 18.7.5. 18.7.3

Evolution of Shrinkage and Residual Stresses

A specific example for a constant packing pressure of 20 MPa will be used to explain the evolution of shrinkage and residual stresses in a solidifying melt. Figure 18.7.6 shows the cavity-pressure history for this simple case. Till the centerline solidifies –attains the glass transition temperature– at t = 6 s, the model keeps injecting molten material to maintain the cavity pressure at 20 MPa. Thereafter, no more material can be added, resulting in the cavity pressure falling off; the actual variation is obtained from the freezing model for the material. As discussed in Section 18.6.1, while between the mold surfaces, the solidifying melt cannot have any lateral strains. To compensate for the shrinking volume during cooling, this constraint induces lateral tensile stresses in the layers, which add up to a restraining tensile force on the solidifying melt. The in-plane and through-thickness strains for this example are shown in Figure 18.7.7. After all the material has solidified (point A) at about t = 6 s, while the through-thickness strain begins to decrease along the

Dimensional Stability and Residual Stresses

300

TEMPERATURE (°C)

250 200 CENTRAL PLANE

150 SURFACE

100 50 0 0

50 TIME (s)

100

Figure 18.7.5 Temperature lag between surface and central-plane temperatures. Notice the significantly lower cooling rates on demolding into the ambient air. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

dashed curve AB ′, the in-plane strain remains zero till “demolding” (point B) at t = 30 s. On demolding, when the lateral constraining force becomes zero, the in-plane strain jumps instantaneously from point B to point C elastically, and the through-thickness strain jumps from point B ′ to point C ′; these strains then continue to relax viscoelastically along the curves BDE and B ′ D ′ E ′, respectively. The buildup of the in-plane (lateral) stresses, 𝜎 y = 𝜎 z = 𝜎 L ( x, t), during cooling in the mold is shown in Figure 18.7.8. Initially, when the packing pressure is applied at t = 0+ s, the entire melt will be under a compressive stress 𝜎 x ( x, 0+ ) = 𝜎 y ( x, 0+ ) = 𝜎 z ( x, 0+ ) = − 20 MPa. As the material cools, strain constraints imposed in the solidified layers near the cold surfaces induces progressively larger tensile in-plane stresses. But the molten material in the core will continue to have a compressive stress of 20 MPa. Once material injection stops on solidification at t = 6+ s, all the layers will have progressively larger tensile stresses. The curve for t = 30− s shows the stresses just before demolding. Notice that after material injection stops after t > 6+ s, all the stresses exhibit a “dip” at near about x = ± 0.75 mm. This corresponds to the location on the t = 6 s temperature distribution curve at which the lateral stress equals the applied packing pressure of 20 MPa. On demolding at t = 30+ s, to satisfy the no lateral force condition these stresses instantaneously redistribute elastically into the stress distribution for t = 30 s shown in Figure 18.7.9. The final stress distribution is the residual stress distribution, which is tensile in thin outer layers and in a large middle core. Notice, again, the shape change at about x = ± 0.75 mm. Since the peak tensile and compressive residual stresses of about 6 and 5 MPa, respectively, are comparable, the layers having tensile and compressive stresses are also comparable. A central core, of about −0.5 ≤ x ≤ 0.5 mm, is in

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20 CAVITY PRESSURE (MPa)

590

15

10

5

0 0

5 TIME (s)

10

Figure 18.7.6 Cavity-pressure history. The pressure is maintained at 20 MPa by injecting more material to compensate for material shrinkage. After all the material solidifies at about t = 6 s, no more material can be added and the pressure falls off. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

tension and, except for very thin layers at the surface that are in tension, the remaining areas are in compression. An integration of the lateral stresses across the thickness gives the lateral restraining force per unit length in the lateral direction. A division of the lateral force per unit length by the part thickness (3 mm) gives the average in-plane stress, the evolution of which is shown in Figure 18.7.10. Initially, from point A till point B ′ at which the entire melt has solidified, the through-thickness restraining average stress is compressive and equals the packing pressure of 20 MPa. After freeze-off at B ′ at about t = 6 s, the stresses, and therefore the average stress, begin to decrease. On continued cooling the solidified plaque detaches from the mold surface (point C ′), at which point the through-thickness restraining force and the average stress become zero (path C ′D ′G ′). Initially, the in-plane reaction restraining average stress also equals the packing pressure. The constraints on the solidifying outer layers results in this compressive restraining force and average stresses decreasing. After freeze-off at B, this compressive restraining average stress decreases more rapidly, becomes zero at point C, and then tensile along path CDE. On demolding at E, the reaction average stress becomes zero (path FG). 18.7.4

Effects of Packing-Pressure Level

The parametric results in this section extend the results of Section 18.7.3 by evaluating the effect of packing pressure levels on shrinkage and residual stresses. Figure 18.7.11 shows the cavity-pressure

Dimensional Stability and Residual Stresses

A

0

B 20 MPa

STRAIN (%)

Cʹ THROUGH-THICKNESS

C

– 0.5

Dʹ

Eʹ

D

E

Bʹ IN-PLANE

– 1.0 0

50 TIME (s)

100

Figure 18.7.7 Variations of the in-plane and through-thickness strains with time. Note that shrinkage is the negative of the strain. After all the material has solidified (point A) at about t = 6 s, while the through-thickness strain begins to decrease along the dashed curve AB ′, the in-plane strain remains zero till “demolding” (point B) at t = 30 s. On demolding, when the lateral constraining force becomes zero, the in-plane strain jumps instantaneously from point B to point C, and the through-thickness strain jumps from point B ′ to point C ′; these strains then continue to relax along the curves BDE and B ′D ′E ′, respectively. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

histories for several packing pressures ranging from 0 to 100 MPa. As for all the examples in Section 18.7, all the material freezes off at t = 6 s, after which the pressure begins to drop off, and demolding occurs at t = 30 s, which is followed by elastic spring back and relaxation. Figure 18.7.11 shows that for packing pressures below a little less than 60 MPa the cavity pressure drops to zero prior to demolding. But for pressures above this value, the solidified material remains under pressure prior to demolding. This difference causes a qualitative change in the evolution of in-plane and through-thickness strains that, for a packing-pressure level of 20 MPa, is described in detail in Section 18.7.3 (Figure 18.7.7). Figure 18.7.12 shows the effect of the packing-pressure level on the evolution of in-plane strain. For pressures above about 60 MPa, the part actually springs back to a large dimension before relaxation starts. And for sufficiently large pressures, the final in-plane dimension can be larger than the cavity size. The final in-plane shrinkage varies from about 1% for a zero packing pressure to about zero for a packing pressure of 100 MPa. The effect of the packing-pressure level on the evolution of the through-thickness strain is shown in Figure 18.7.13. For packing pressures just below about 60 MPa the cavity pressure decays to zero before demolding (Figure 18.7.11), at which point the plaque detaches from the mold surfaces, resulting in

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20

t = 30 s 20

15 10

STRESS (MPa)

10

8

0 0.1 2 4

– 10

6 20 MPa

– 20 –1.5

0 0.5 –0.5 POSITION (mm)

–1.0

1.0

1.5

Figure 18.7.8 Build up of lateral stresses in the melt. Note the qualitative change in the stress after complete solidification at t = 6 s. The curve for t = 30 s shows the stresses just before demolding. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

8 t = 30 s

RESIDUAL STRESS

50 s 30 s

5

STRESS (MPa)

592

0

20 MPa

–5 – 1.5

– 1.0

0 0.5 – 0.5 POSITION (mm)

1.0

1.5

Figure 18.7.9 Lateral stress distributions after demolding at t = 30 s. Note the slow changes after demolding. The final stress distribution is the residual stress distribution. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

Dimensional Stability and Residual Stresses

20 E

RESTRAINING FORCE (MPa)

D

10

IN-PLANE

0

C

F Dʹ

Cʹ

G Gʹ

THROUGH-THICKNESS

– 10 B 20 MPa

A

– 20 0

Bʹ

10

20

30

TIME (s) Figure 18.7.10 Lateral in-plane and through-thickness restraining forces per unit lateral length. The letters on the curves correspond to like events shown in Figure 18.7.7. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

a negative strain well before demolding. The jump in strain on demolding for pressures below about 50 MPa results from the Poisson effect involved in the elastic redistribution of stresses to affect a zero net in-plane force. Above this pressure, while there still will be a jump in strain, the entire plaque expands on demolding. The variations of the final in-plane and through-thickness shrinkages with the packing pressure are shown in Figure 18.7.14. The through-thickness shrinkage is systematically smaller than the in-plane shrinkage. The effect of the packing-pressure level on the in-plane residual stresses is shown in Figures 18.7.15 and 18.7.16. For packing pressures below 25 MPa (Figure 18.7.15) the magnitude of the residual stresses increases with the packing pressure – higher tensile stresses in the core and subsurface, and higher compressive stresses elsewhere. For packing pressures below 20 MPa the core tensile stresses exhibit a flat plateau that can be attributed a rubbery solid region in the core during the decay of the pressure to zero. While this figure only shows residual stresses for pressures up to 25 MPa, the discussions in this paragraph apply to pressures up to 35 MPa. As shown in Figure 18.7.16, above a packing pressure of 35 MPa the residual stresses decrease with increasing packing pressures. 18.7.5

Effect of Packing-Pressure Duration

This section considers the effect of cutting off the packing pressure prior to gate freeze-off. The dashed curve in Figure 18.7.17 shows cavity-pressure history for a packing pressure of 40 MPa, in which the

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100

100 CAVITY PRESSURE (MPa)

594

80

60

50

50 40 30 20 0

0 0

20

10

30

TIME (s) Figure 18.7.11 Cavity-pressure histories. For each packing pressure level, the material solidifies at about t = 6 s, and the part is demolded at 30 s. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

entire plaque would solidify at time tsolid = 6 s. For the results in this section, the packing pressure is applied for a shorter time tstep ≤ tsolid . Because the entire plaque has not solidified, material can actually flow out of the cavity to maintain a zero cavity pressure during tstep ≤ t ≤ tsolid . The time of this packing pressure cutoff is described by the packing index 𝜆 = tstep ∕tsolid , with 𝜆 = 0 and 𝜆 = 1 corresponding, respectively, to zero packing pressure and the packing pressure being applied till freeze-off at tsolid = 6 s. The variations of the in-plane and through-thickness shrinkages with the packing index are shown in Figure 18.7.18. Clearly, the duration for which the packing pressure is applied has a significant effect on the shrinkage. The in-plane (through-thickness) shrinkage decreases (increases) with the duration for which the pressure is applied. Application of the pressure till complete freeze-off can reduce the in-plane shrinkage by about 50%; however, the through-thickness shrinkage increases by about 40%. Figure 18.7.19 shows the effect of the packing index 𝜆 = tstep ∕tsolid on the in-plane residual stress distribution. The very sharp transition from the compressive stresses in the subsurface layers to the flat, tensile core results from the instantaneous drop off in the packing pressure. Note that the higher residual stresses in this example result from the higher packing pressure of 40 MPa in comparison to the lower pressure of 20 MPa used in earlier examples. 18.7.6

Effect of Gate Freeze-Off Time

This section considers the effect of gate freeze-off at time. The dashed curve in Figure 18.7.20 shows cavity-pressure history for a packing pressure of 20 MPa, in which the entire plaque would solidify at time

Dimensional Stability and Residual Stresses

IN-PLANE STRAIN (%)

0.4

0

100 MPa 80 60

–0.5

40

20 –1.0

0 0

50 TIME (s)

100

Figure 18.7.12 Effects of the packing-pressure level on the evolution of the in-plane strain (negative value of shrinkage). Notice that for larger pressures the plaque can actually spring back to a larger dimension on demolding before relaxation starts. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

tsolid = 6 s. For the results in this section, the packing pressure is applied for a shorter time tfreeze ≤ tsolid , at which the gate freezes off, and after which no more material can be added to the melt. The time of this early freeze-off is described by the gate freeze-off index 𝜉 = tfreeze ∕tsolid , with 𝜉 = 0 and 𝜉 = 1 corresponding, respectively, to gate freeze-off occurring just after filling at t = 0 and at t = 6 s. The variations of the in-plane and through-thickness shrinkages with the gate freeze-off index are shown in Figure 18.7.21. Clearly, early gate freeze-off has a very large effect on the through-thickness shrinkage, which decreases from about 8.2% for freeze-off occurring just after filling at t = 0 to just about 0.62 % if freeze-off occurs at t = 6 s – a reduction of shrinkage by about 92%! This sensitivity of the through-thickness shrinkage to early freeze-off can affect surface quality, or appearance, of the part. Early freeze-off has very little effect on the in-plane shrinkage. Figure 18.7.22 shows the effect of the gate freeze-off index 𝜉 = tfreeze ∕tsolid on the in-plane residual stress distribution. Except for the case of 𝜉 = 1, which corresponds to “freeze-off” occurring only after the entire “plaque” has solidified, for 𝜉 = < 1 the core solidifies under a zero packing pressure, resulting in the residual stress exhibiting a flat plateau in the core. The transition from the outer skin to the core is smooth because the cavity pressure drops off slowly after the gate freezes off. The tensile stresses in the core and the subsurface compressive stresses increase with freeze-off occurring later (increasing 𝜉 ). These residual stresses are qualitatively similar to those for the packing-pressure-duration case shown in Figure 18.7.19 for a higher packing pressure level of 40 MPa, which partly accounts for the difference in the magnitudes of the stresses for between the two cases.

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THROUGH-THICKNESS STRAIN (%)

0.4

100 MPa 0

80 40

60 40 20

– 0.5 20 – 1.0

0

0

– 1.5 0

50 TIME (s)

100

Figure 18.7.13 Effects of the packing-pressure level on the evolution of the through-thickness shrinkage. Notice that for higher pressures the final plaque thickness can actually be larger than the cavity height. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

1.0

SHRINKAGE (%)

596

IN-PLANE

0.5

0

THROUGH-THICKNESS

– 0.2 0

50 PACKING PRESSURE (MPa)

100

Figure 18.7.14 Variations of the in-plane and through-thickness shrinkages with packing pressure. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

Dimensional Stability and Residual Stresses

8 25 MPa

RESIDUAL STRESS (MPa)

20 MPa

5 15 10 5

0

0 5 10 15

–5

20 25

–1.5

– 1.0

– 0.5

0

0.5

1.0

1.5

POSITION (mm) Figure 18.7.15 Residual stresses for packing pressures between 0 and 25 MPa. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

8 30 MPa 40

RESIDUAL STRESS (MPa)

5

60 80 100

0 100 80 60 40

–5 – 1.5

– 1.0

0 0.5 – 0.5 POSITION (mm)

30

1.0

1.5

Figure 18.7.16 Residual stresses for packing pressures between 30 and 100 MPa. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

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CAVITY PRESSURE (MPa)

70

λ=

tsolid

50

tstep tsolid

tstep

0 5

0

10

14

TIME (s) Figure 18.7.17 Step packing-pressure history. The constant packing pressure of 40 MPa is reduced to zero at t = tstep . The packing index is defined by 𝜆 = tstep ∕tsolid . (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

1.5

THROUGH-THICKNESS

SHRINKAGE (%)

598

1.0

IN-PLANE

0.5

40 MPa

0 0

0.5 PACKING INDEX

1.0

Figure 18.7.18 Variations of the in-plane and through-thickness shrinkages with the packing index 𝜆 = tstep ∕tsolid . (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

Dimensional Stability and Residual Stresses

RESIDUAL STRESS (MPa)

10

0 0 1.0 40 MPa

– 10

λ = 0.8 0.6

– 20 – 1.5

λ=

0.2 0.4

– 1.0

tstep tsolid

0 0.5 – 0.5 POSITION (mm)

1.0

1.5

Figure 18.7.19 Variations of the in-plane residual stress with the packing index 𝜆 = tstep ∕tsolid . (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

18.7.7

Summary Comments

The example of the evolution of shrinkage and residual stresses (Section 18.7.3) for a constant packing pressure shows that the final part shrinkage and residual stresses are the end results of very complex processes during solidification in the mold – during which the part is constrained to prevent any shrinkage, resulting in the buildup of stresses – and after demolding when the restraining forces in the mold become zero and the part shrinks to satisfy this condition. Furthermore, the shrinkage and stresses continue to change after demolding owing to stress relaxation. The final shrinkage and residual stresses are then the equilibrium values of these variables. The residual stress distribution is tensile in thin outer layers and in a large middle core. A central core is in tension and, except for very thin layers at the surface that are in tension, the remaining areas are in compression. The effect of packing pressure levels on shrinkage and residual stresses have been evaluated for several packing pressures ranging from 0 to 100 MPa. For the specific geometry and mold temperature conditions considered, for packing pressures below a little less than 60 MPa the cavity pressure drops to zero prior to demolding. But for pressures above this value, the solidified material remains under pressure prior to demolding. This difference causes a qualitative change in the evolution of in-plane and through-thickness strains. For pressures above about 60 MPa, the part actually springs back to a large dimension before relaxation starts. And for sufficiently large pressures, the final in-plane dimension can be larger than the cavity size. The final in-plane shrinkage varies from about 1% for

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30

CAVITY PRESSURE (MPa)

600

tsolid tfreeze 20

10 tfreeze

ξ= t solid

0 5

0

8

TIME (s) Figure 18.7.20 Packing-pressure history with early gate freeze-off. The gate freezes off at t = tfreeze ≤ tsolid . The freeze-off index is defined by 𝜉 = tfreeze ∕tsolid . (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

a zero packing pressure to about zero for a packing pressure of 100 MPa. For packing pressures below 35 MPa the magnitude of the residual stresses increases with the packing pressure – higher tensile stresses in the core and subsurface, and higher compressive stresses elsewhere. For packing pressures below 20 MPa the core tensile stresses exhibit a flat plateau that can be attributed a rubbery solid region in the core during the decay of the pressure to zero. In contrast, for packing pressures above 35 MPa the residual stresses decrease with increasing packing pressures, and do not exhibit a central flat plateau. The effects of gate freeze-off time have been evaluated for a fixed packing pressure of 20 MPa by applying the packing pressure for shorter and shorter times than time for plaque solidification. Early gate freeze-off has very little effect on in-plane shrinkage but a very large effect on through-thickness shrinkage. This sensitivity of the through-thickness shrinkage to early freeze-off can affect part appearance. Except for the case in which “freeze-off” occurs only after the entire “plaque” has solidified, for earlier freeze-off the core solidifies under a zero packing pressure, resulting in the residual stress exhibiting a flat plateau in the core. The transition from the outer skin to the core is smooth because the cavity pressure drops off slowly after the gate freezes off. The tensile stresses in the core and the subsurface compressive stresses increase with freeze-off occurring later.

Dimensional Stability and Residual Stresses

9

SHRINKAGE (%)

20 MPa

THROUGH-THICKNESS

5

IN-PLANE

0 0

0.5 GATE FREEZE-OFF INDEX

1.0

Figure 18.7.21 Variations of the in-plane and through-thickness shrinkages with the gate freeze-off index 𝜉 = tfreeze ∕tsolid . Notice the large effect on the through-thickness shrinkage. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

20 MPa

RESIDUAL STRESS (MPa)

5

0 0 1.0

–5

ξ = 0.8 0.6 0.2 0.4

– 10 – 1.5

– 1.0

tfreeze

ξ= t solid

0 0.5 – 0.5 POSITION (mm)

1.0

1.5

Figure 18.7.22 Variations of the in-plane residual stress with the gate freeze-off index 𝜉 = tfreeze ∕tsolid as parameter. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 35, pp. 365 – 383, 1995.)

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18.8 **Warpage Induced by Differential Mold-Surface Temperatures This section extends the results of Section 18.7 for shrinkage to warpage induced by unequal mold-surface temperatures. The plaque geometry will be the same as in all the examples in Section 18.7, as will be the viscoelastic material model described in Section 18.7.1.The only change will be that the upper and lower mold surfaces will be at different temperatures. Examples similar to those in Section 18.7.3 – 18.7.6 will explore the effects of various molding parameters on the warpage caused by the asymmetric stress distributions resulting from differences in the mold-surface temperatures. 18.8.1

Temperature Distribution in a Solidifying Melt

For the examples that follow, the temperature history was obtained from a numerical solution to the heat-transfer equation (Eq. 6.2.3) using a constant thermal diffusivity of 0.135 mm2 s−1 . A uniform initial temperature of 300°C is assumed over the domain −1.5 ≤ x ≤ 1.5, and the upper and lower mold temperatures, TU and TL , respectively , are kept constant with TL > TU . The effects of the mold-surface temperature differential ΔTmold = TL − TU is assessed by varying the upper and lower mold temperatures as TU = 80 − (ΔTmold )∕2 and TL = 80 + (ΔTmold )∕2, respectively, so that the mid-plane temperature is maintained at T(0, t) = (TU + TL )∕2 = 80°C. As in Section 18.7, on demolding the plaque surfaces are subject to an ambient temperature of 20°C. Figure 18.8.1 shows the asymmetric, transient temperature distribution in the melt for the case with Δ tmold = 20°C, which corresponds to TL = 90°C and TU = 70°C. The melt cools off rapidly because of effective heat transfer across the cold-mold walls. At 6 s (dashed curve), the entire thickness has solidified, that is, T(0, 6) = Tsolid = Tg . The solid “plaque” continues to cool till 30 s, when it has essentially attained a linear temperature between TL = 90°C and TU = 70°C, at which time it is demolded. Because the plaque surfaces are then in contact with ambient at a temperature of 20°C, with the much less efficient heat transfer to the air the plaque continues to cool at a significantly lower rate. The central plane, x = 0, eventually attains the ambient temperature at about 90 seconds. Because of the poor heat-transfer characteristics of the melt, there is a significant lag in the central-plane temperature, as shown in Figure 18.8.2. 18.8.2

Constant Packing-Pressure Level

This section discusses the warpage resulting from two differential mold-surface temperatures of ΔTmold = 20°C and 40°C for the case of a constant packing-pressure of 40 MPa. The dashed curve in Figure 18.7.17 shows how for ΔTmold = 0 the pressure falls off after the entire “plaque” solidifies at t = 6 s. It turns out that the corresponding pressure curves for ΔTmold = 20°C and 40°C are almost indistinguishable from the curve for ΔTmold = 0. As shown in Figure 18.8.3, the in-plane and through-thickness shrinkages are insensitive to differences in the mold-surface temperature. Figure 18.8.4 shows the in-plane residual stress distributions for ΔTmold = 0, 20, and 40°C. Clearly, although the residual stresses for ΔTmold = 20 and 40°C are asymmetric, they are not significantly different from the symmetric distribution for ΔTmold = 0°C. This is in consonance with the very small variations for the shrinkages shown in Figure 18.8.3. The peak residual stress near x = 0 tends to undergo a slight

Dimensional Stability and Residual Stresses

300

0 2

TEMPERATURE (°C)

250 200

6

150 10

100

Δtmold

30 40 50 t=∞

50 0 – 1.5

– 1.0

0 0.5 – 0.5 POSITION (mm)

1.0

1.5

Figure 18.8.1 Temperature distributions across the solidifying melt thickness for Δ tmold = 20°C. At about t = 6 s all the

material has essentially solidified. At about t = 30 s, at which all the material has attained a steady-state linear variation from TL = 90°C at the lower mold surface to TU = 70°C at the upper mold surface, the solidified slab is “demolded.” It then continues to cool more slowly. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

shift toward the lower, colder mold surface. And the highest compressive stresses in the neighborhood of x = 1 are slightly lower than those in the neighborhood of x = −1. Figure 18.8.5 shows the time evolution of the warpage, as measured by the curvature of the mid-plane, for ΔTmold = 0, 20, and 40°C. As expected, there is no warpage for the symmetric case of equal mold-surface temperatures (ΔTmold = 0°C). While in the mold, the “plaque” is constrained to remain flat. Because of the asymmetric cooling the in-plane stresses also evolve in an asymmetric manner. As a result, in addition to in-plane restraining forces, bending moments are required to hold the “plaque” flat. On demolding, at 30 seconds in this case, both the restraining forces and bending moments become zero. In addition to shrinkage, this causes the plaque to initially warp with the side corresponding to the cooler surface becoming concave. During the continued post-demolded cooling this curvature first continues to decrease – the plaque becoming flatter – becomes zero, and then reverses its curvature so that the cooler side becomes convex. In its final shape the side corresponding to the cooler mold surface becomes convex. Figure 18.8.6 shows that the final curvature, and therefore the warpage, increases with the differential mold-surface temperature. For this temperature range the curvature is essentially proportional to ΔTmold . A feel for the relationship between warpage and curvature follows from the result that, when the radius of curvature (reciprocal of the curvature) is significantly larger than the plaque length, the curvature is proportional to the central height of a uniformly warped plaque placed on a flat surface.

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Introduction to Plastics Engineering

300

250 TEMPERATURE (°C)

604

200 CENTRAL PLANE

150

UPPER SURFACE LOWER SURFACE

100 50

0

50 TIME (s)

100

Figure 18.8.2 Temperature lag between surface and central-plane temperatures. Notice the significantly lower cooling rates on demolding into the ambient air. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

18.8.3

Effect of Packing-Pressure Level

The effect of the packing-pressure level is evaluated by varying it from 0 to 10 MPa for a fixed differential mold-surface temperature of ΔTmold = 20°C. The cavity-pressure histories are essentially the same as those for the case of ΔTmold = 0°C shown in Figure 18.7.11. The time evolution of the in-plane and through-thickness strains is essentially the same as for the ΔTmold = 0°C case shown, respectively, in Figures 18.7.12 and 18.7.13. The variation of the shrinkage – the negative of the final equilibrium strains – with the packing pressure are shown in Figure 18.7.14. Figure 18.8.4 showed that although for a constant packing pressure of 40 MPa the residual stresses for ΔTmold = 20 and 40°C are asymmetric, they are not significantly different from the symmetric distribution for ΔTmold = 0°C. And the peak residual stress near x = 0 tends to undergo a slight shift toward the lower, colder mold surface. It turns out that for the case of ΔTmold = 20°C being considered in this section, the residual stresses are very similar to those for the case of ΔTmold = 0°C shown in Figures 18.7.15 and 18.7.16 for packing pressures varying from 0 to 25 MPa and from 30 to 100 MPa, respectively. However, asymmetric cooling causes the residual stress curves to undergo a slight leftward shift, and the highest compressive stresses in the neighborhood of x = 1 are slightly lower than those in the neighborhood of x = −1. Figure 18.8.7 shows the variation of the final (equilibrium) curvature with the nominal packing-pressure level. For zero packing pressure the cold-mold surface side warps to a concave shape, the curvature of which – and hence the warpage – decreases with increasing packing pressure. This curvature becomes

Dimensional Stability and Residual Stresses

1.0 40 MPa

SHRINKAGE (%)

IN-PLANE

0.5

THROUGH-THICKNESS

0 0

20 30 10 DIFFERENTIAL MOLD-SURFACE TEMPARATURE Δ tmold (°C)

40

Figure 18.8.3 In-plane and through-thickness shrinkages are insensitive to the differential mold-surface temperature. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

zero at a packing pressure of about 12.5 MPa. Further increases in the packing pressure reverse the curvature with the cold-mold surface side becoming convex, and the curvature continues to increase till a packing pressure of about 40 MPa. Further increases in the packing pressure result in a small monotonic decrease in the curvature. 18.8.4

Effect of Gate Freeze-Off

This section discusses the warpage at different packing pressures resulting from a fixed gate freeze-off occurring at 𝜉 = tfreeze ∕tsolid = 0.83, and fixed differential mold-surface temperature of ΔTmold = 20°C. The dashed curves in Figure 18.8.8 – which are the same as in Figure 18.7.11 in Section 18.7.4 – show how for ΔTmold = 0 the pressure falls off after the entire “plaque” solidifies at t = 6 s. It turns out that the corresponding pressure curves for ΔTmold = 20 and 40°C are almost indistinguishable from the curve for ΔTmold = 0. The solid curves in this figure show the cavity pressures for gate freeze-off occurring at t = 5 s, which corresponds to a gate freeze-off index of 𝜉 = 0.83. As in Figure 18.7.11, the cavity pressures drop off to zero before the melt solidifies (attains the glass transition temperature) for packing pressures lower than about 60 MPa. For higher pressures, the cavity pressures are well above zero when demolding occurs at t = 30 s. Figure 18.8.9 shows that the in-plane and through-thickness shrinkages drop off monotonically for the packing pressure levels considered. A comparison with Figure 18.7.14, which corresponds to the case

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8 Δ tmold = 0ºC

40 MPa

RESIDUAL STRESS (MPa)

606

20

5

4

0

–5 – 1.5

– 1.0

0 0.5 – 0.5 POSITION (mm)

1.0

1.5

Figure 18.8.4 In-plane residual stresses residual stresses for ΔTmold = 0, 20, and 40°C for a fixed packing pressure of 40 MPa.

(Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

of no freeze-off, shows that packing pressure is less effective in reducing shrinkage with gate freeze-off. Also, gate freeze-off has a larger effect on through-thickness shrinkage. The effect of the packing-pressure level on the in-plane residual stresses is shown in Figures 18.8.10 and 18.8.11. For packing pressures below 30 MPa (Figure 18.8.10) the magnitude of the residual stresses increases with the packing pressure – higher tensile stresses in the core and subsurface higher compressive stresses. The core tensile stresses exhibit flat plateaus that are inclined as a result of the asymmetric cooling of the mold surfaces. As shown in Figure 18.8.11, above a packing pressure of 40 MPa, the residual stresses decrease with increasing packing pressures. The variation of the final curvature, or warping, with the nominal packing pressure is shown in Figure 18.8.11. For zero packing pressure the cold-mold surface side warps to a concave shape, the curvature of which decreases with increasing packing pressure. This curvature becomes zero at a packing pressure of about 10 MPa. Further increases in the packing pressure reverse the curvature, and the curvature continues to increase till a packing pressure of about 45 MPa. More increases in the packing pressure result in a monotonic decrease in the curvature. A comparison with Figure 18.8.7 shows that gate freeze-off induces much higher warpage. 18.8.5

Summary Comments

The examples of the evolution of warpage (Section 18.8.2) for a constant packing-pressure of 40 MPa and two differential mold-surface temperatures of ΔTmold = 20 and 40°C show (i) that the pressure curves for

Dimensional Stability and Residual Stresses

0.8 COOLER

40 MPa

WARMER

CURVATURE (1/m)

0.5 COOLER

WARMER

DEMOLDING

Δ tmold = 0ºC

0

20 40

– 0.2 0

50

80

TIME (s) Figure 18.8.5 Evolution of plaque curvature for a fixed packing pressure of 40 MPa. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

ΔTmold = 20 and 40°C are almost indistinguishable from the curve for ΔTmold = 0, and (ii) that although the residual stresses for ΔTmold = 20 and 40°C are asymmetric, they are not significantly different from the symmetric distribution for ΔTmold = 0°C. The peak residual stress near the mid-plane tends to undergo a slight shift toward the lower, colder mold surface. The time evolution of the warpage, as measured by the curvature of the mid-plane, is quite complex: While in the mold, the “plaque” is constrained to remain flat. Because of the asymmetric cooling the in-plane stresses also evolve in an asymmetric manner. As a result, in addition to in-plane restraining forces, bending moments are required to hold the “plaque” flat. On demolding both the restraining forces and bending moments become zero. In addition to shrinkage, this causes the plaque to initially warp with the side corresponding to the cooler surface becoming concave. During the continued post-demolded cooling this curvature first continues to decrease – the plaque becoming flatter – becomes zero, and then reverses its curvature so that the cooler side becomes convex. In the final shape the side corresponding to the cooler mold surface becomes convex. The final curvature, and therefore the warpage, increases with the differential mold-surface temperature. The effect of the packing-pressure level, as evaluated by varying it from 0 to 10 MPa for a fixed differential mold-surface temperature of ΔTmold = 20°C, show (i) that the cavity-pressure histories are essentially the same as those for the case of ΔTmold = 0°C, and (ii) that the time evolution of the in-plane and

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Introduction to Plastics Engineering

0 COOLER

CURVATURE (1/m)

608

WARMER

– 0.05

40 MPa

– 0.10 0

10 20 30 DIFFERENTIAL MOLD-SURFACE TEMPARATURE Δ tmold (°C)

40

Figure 18.8.6 Variation of the final curvature with the differential mold-surface temperature ΔTmold for a fixed packing pressure of 40 MPa. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

through-thickness strains is essentially the same as that for the ΔTmold = 0°C. For zero packing pressure the cold-mold surface side warps to a concave shape, the curvature of which decreases with increasing packing pressure. This curvature becomes zero at a packing pressure of about 12.5 MPa. Further increases in the packing pressure reverse the curvature with the cold-mold surface side becoming convex, and the curvature continues to increase till a packing pressure of about 40 MPa. Further increases in the packing pressure result in a small monotonic decrease in the curvature. The effects of gate freeze-off time have been evaluated for packing pressures in the range of 0 to 100 MPa, for a fixed gate freeze-off occurring at 83% of the time before solidification, and a fixed differential mold-surface temperature of ΔTmold = 20°C. The in-plane and through-thickness shrinkages drop off monotonically for the packing pressure levels considered. The packing pressure is less effective in reducing shrinkage with gate freeze-off. Also, gate freeze-off has a larger effect on through-thickness shrinkage. For packing pressures below 30 MPa the magnitude of the residual stresses increases with the packing pressure – higher tensile stresses in the core and subsurface higher compressive stresses. The core tensile stresses exhibit flat plateaus that are inclined as a result of the asymmetric cooling of the mold surfaces. Above a packing pressure of 40 MPa, the residual stresses decrease with increasing packing pressures. For zero packing pressure, the final shape of the plaque warps with the cold-mold surface side having a concave upward shape, with its curvature decreasing with increasing packing pressure. This curvature becomes zero at a packing pressure of about 10 MPa. Further increases in the

Dimensional Stability and Residual Stresses

0.05

FINAL CURVATURE (1/m)

COOLER

WARMER

0

– 0.05 COOLER

WARMER

– 0.10

Δ tmold = 20ºC – 0.15 0

50 PACKING PRESSURE (MPa)

100

Figure 18.8.7 Variation of the final curvature with the packing pressure for a fixed differential mold-surface temperature of

ΔTmold = 20°C. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

packing pressure reverse the curvature, and the curvature continues to increase till a packing pressure of about 45 MPa. More increases in the packing pressure result in a monotonic decrease in the curvature. In this chapter, only the warpage induced by the cooling of a homogeneous melt in a mold having different surfaces temperatures has been considered. However, in a nonhomogeneous material warpage will occur even when the two mold surfaces are equal. One important example of a nonhomogeneous material is a fiber-filled resin.

18.9 Concluding Remarks In this chapter, the processes that result in residual stresses and shrinkage and warpage in parts molded from amorphous resins have been explained at three increasingly complex levels, in all of which the material is assumed to be homogeneous: 1. At the simplest level, a highly idealized model, in which there are no temperature and pressure gradients in the solidifying melt, is used to trace the history of a material in an idealized injection molding cycle. While this model predicts shrinkage, it cannot account for residual stresses and warpage. And it does not account for cooling-rate, fill-time, and cycle-time effects. For a constant packing pressure, this model predicts that the initial melt temperature does not affect shrinkage. Mold filling starting at any temperature along the p = 0 isobar to the right of the solidification point will result in the same final shrinkage so long as the packing pressure remains the same. However, the pressure at which the material is packed does affect shrinkage.

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Introduction to Plastics Engineering

100 CAVITY PRESSURE (MPa)

610

100

tsolid = 6 s

80

tfreeze = 5 s

ξ = 0.83

60 50 40

20

0 0

10

20

30

TIME (s) Figure 18.8.8 Cavity-pressure histories for gate freeze-off occurring at t = 5 s shown by solid lines. Full freeze-off occurs at t = 6 s, shown by dashed lines. The gate freeze-off index for each pressure level is 𝜉 = tfreeze ∕tsolid = 0.83. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

In a real injection molding cycle, the freeze-off point is indicated by a freeze-off time. But because time is not a parameter in the highly idealized molding cycle discussed in this section, different freeze-off points are indicated by the temperature on the isobar at the instant of freeze-off. Although shrinkage decreases with increasing injection pressures, it is very large in the absence of packing. As the freeze-off point moves to the left on the isobars, the shrinkage decreases continuously. Thus, premature freeze-off results in increased shrinkage. According to this model, no matter how high a packing pressure is applied during the molding cycle, if this pressure is removed prior to solidification, then the linear shrinkage is the same as if no packing pressure had been applied; the final shrinkage is determined by the packing pressure just prior to solidification. 2. At the next level, the temperature and the pressure are allowed to vary in the thickness direction, and the solidified layers are assumed to behave like elastic solids. This simple solidification model predicts that the effect of time-varying temperature and pressure histories on part shrinkage is measured by a linear dependence of shrinkage on an “effective pressure,” which combines the thermal diffusivity of the material, the part wall thickness, and the time-varying cavity pressure into a single parameter. The effective pressure is shown to correlate in-plane shrinkage data obtained from injection molding experiments. The solidification model characterizes two material parameters, which can be estimated from the PVT diagram for the material, that describe the sensitivity of the shrinkage to the local cavity-pressure history.

Dimensional Stability and Residual Stresses

1.8

ξ = 0.83

SHRINKAGE (%)

1.5

THROUGH-THICKNESS

1.0

0.5 IN-PLANE

Δtmold = 20ºC

0.0 – 0.2 0

50 PACKING PRESSURE (MPa)

100

Figure 18.8.9 Variations of the in-plane and through-thickness shrinkages with packing pressure for a gate freeze-off index 𝜉 = tfreeze ∕tsolid = 0.83, and fixed differential mold-surface temperature of ΔTmold = 20°C. (Adapted with permission from

W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

While this model also gives an expression for the residual stresses, the estimates are rather crude, and must be interpreted with care. For example, contrary to well-established phenomenology, this model predicts zero residual stresses for a constant cavity pressure. 3. At the third, most sophisticated level, the second level treatment of the freezing of a layer of between two mold surfaces is extended by treating the molten and solidifying amorphous resin as a viscoelastic material. The example of the evolution of shrinkage and residual stresses for a constant packing pressure shows that the final part shrinkage and residual stresses result from two very complex processes: (i) Solidification in the mold, during which the part is constrained from shrinking, resulting in the buildup of stresses. And (ii), redistribution of the stresses after demolding to satisfy the condition of the restraining mold forces becoming zero. Furthermore, the shrinkage and stresses continue to change after demolding owing to stress relaxation. The final shrinkage and residual stresses are then the equilibrium values of these variables. The residual stress distribution is tensile in thin outer layers and in a large middle core; the remaining areas are in compression. For different packing pressure levels, below a certain packing pressure, the cavity pressure drops to zero prior to demolding. But above this pressure, the solidified material remains under pressure prior to demolding. This difference causes a qualitative change in the evolution of in-plane and through-thickness strains. For the higher packing pressures, the part actually springs back to a large dimension before

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Introduction to Plastics Engineering

8 10 RESIDUAL STRESS (MPa)

612

ξ = 0.83

5

30 MPa

20 10

0

0 10

–5

20

Δ tmold = 20ºC

30

– 10 – 1.5

– 1.0

0 0.5 – 0.5 POSITION (mm)

1.0

1.5

Figure 18.8.10 Residual stresses for packing pressures between 0 and 30 MPa for a gate freeze-off index 𝜉 = tfreeze ∕tsolid = 0.83,

and fixed differential mold-surface temperature of ΔTmold = 20°C. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

relaxation starts. And for sufficiently large pressures, the final in-plane dimension can be larger than the cavity size. For low packing pressures the magnitude of the residual stresses increases with the packing pressure – higher tensile stresses in the core and subsurface, and higher compressive stresses elsewhere. For still lower packing pressures, the core tensile stresses exhibit a flat plateau. In contrast, for packing pressures above this value the residual stresses decrease with increasing packing pressures, and do not exhibit a central flat plateau. Early gate freeze-off has very little effect on in-plane shrinkage but a very large effect on through-thickness shrinkage. This sensitivity of the through-thickness shrinkage to early freeze-off can affect part appearance. Except for the case in which “freeze-off” occurs only after the entire “plaque” has solidified, for earlier freeze-off the core solidifies under a zero packing pressure, resulting in the residual stress exhibiting a flat plateau in the core. The transition from the outer skin to the core is smooth because the cavity pressure drops off slowly after the gate freezes off. The tensile stresses in the core and the subsurface compressive stresses increase with freeze-off occurring later. The time evolution of the warpage, as measured by the curvature of the mid-plane, is quite complex: On demolding both the in-mold restraining forces and bending moments become zero. In addition to shrinkage, this causes the plaque to initially warp with the side corresponding to the cooler surface becoming concave. During the continued post-demolded cooling this curvature first continues to decrease – the plaque becoming flatter – becomes zero, and then reverses its curvature so that the cooler

Dimensional Stability and Residual Stresses

8

RESIDUAL STRESS (MPa)

10

40 MPa

60

ξ = 0.83

80 100

5

0 100

–5 Δ tmold = 20ºC – 10 – 1.5

– 1.0

0 0.5 – 0.5 POSITION (mm)

80 60 40

1.0

1.5

Figure 18.8.11 Residual stresses for packing pressures between 40 and 100 MPa for a gate freeze-off index 𝜉 = tfreeze ∕tsolid

= 0.83, and fixed differential mold-surface temperature of ΔTmold = 20°C. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

side becomes convex. In the final shape the side corresponding to the cooler mold surface becomes convex. The final curvature, and therefore the warpage, increases with the differential mold-surface temperature. Effects of packing pressure levels: For zero packing pressure the cold-mold surface side warps to a concave shape, the curvature of which decreases with increasing packing pressure. This curvature becomes zero at a certain packing pressure. Further increases in the packing pressure reverse the curvature with the cold-mold surface side becoming convex, and the curvature continues to increase till a packing pressure to a certain packing-pressure level, above which increases in the packing pressure result in a small monotonic decrease in the curvature. The packing pressure is less effective in reducing shrinkage with gate freeze-off. Also, gate freeze-off has a larger effect on through-thickness shrinkage. The evolution and prediction of residual stresses and shrinkage and warpage in injection-molded parts is perhaps the most complex of all processes in plastics engineering. Even in the most complex, level-3 analysis in this chapter, only a very simple flat mold geometry has been considered. In a real part, different regions will be cooling and freezing at different rates and lateral movements in the freezing material will occur. As a result, the processes during the freezing of the part have to be solved as an integrated whole.

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0.05 COOLER

FINAL CURVATURE (1/m)

614

WARMER

0

Δ tmold = 20ºC – 0.05 COOLER

WARMER

– 0.10

ξ = 0.83 – 0.15 0

50 PACKING PRESSURE (MPa)

100

Figure 18.8.12 Variation of the final curvature with the packing pressure for a gate freeze-off index 𝜉 = tfreeze ∕tsolid = 0.83, and fixed differential mold-surface temperature of ΔTmold = 20°C. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Engineering and Science, Vol. 36, pp. 322 – 335, 1996.)

The analysis for the processes during molding of semicrystalline materials is even more complex because it must account for the evolution of crystallization. In this chapter, warpage is induced by the cooling of a homogeneous melt in a mold having different surfaces temperatures. However, in a nonhomogeneous material warpage will occur even when the temperatures of the two mold surfaces are equal. One important example of a nonhomogeneous material is a fiber-filled resin.

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19 Alternatives to Injection Molding 19.1 Introduction The previous two chapters have focused on the injection molding process and its variants. While injection molding is the most versatile of all plastics fabrication processes used for making precision, submillimeter- to meter-size parts, it is not suited for very large parts for which the very high molding pressures require prohibitively expensive molding machines and molds. Although variants such as structural foam molding and gas injection molding overcome some of the limitations of the standard injection molding process, they too have limitations. Several alternative processes have been developed for reducing the cost of plastics parts. They may be classified into three broad categories: (i) Those in which the starting material is in the form of plastic pellets, as in injection molding, in which parts are formed from molten pellets. They include processes for making continuous fibers, thin films, profiles, and sheet – which are produced in bulk by continuous, high-volume processes – and blow molding. (ii) Processes in which the starting plastic material is in powder form; the only commercially important process for this group is rotational molding. And (iii), processes in which the starting material is plastic sheet. The commercially important processes in this category are different versions of thermoforming. All plastic forming techniques in the first category use an extruder to convert solid thermoplastic pellets into a melt. For this reason, the design and performance of extruders are important aspects of part conversion processes.

19.2 Extrusion Almost all plastic forming techniques start with pellets that, after drying if necessary, are fed into a machine called an extruder, in which the pellets move inside a heated cylindrical barrel by means of a rotating screw. The relative motion between the screw and barrel surfaces results in frictional heating and mixing of the pellets resulting in a uniform melt. In its simplest form an extruder consists of a barrel, parts of which are externally heated, surrounding a coaxial rotating screw which moves the pellets axially (Figure 19.2.1). The processes in an extruder that result in solid particles being converted into a uniform melt involve several steps along the extruder screw: (i) First, the solid pellets are fed through a hopper and conveyed along the extruder barrel. In this zone the solid pellets take up more space than their total volume because of voids between individual solid pellets; conveying compacts the pellets. (ii) In a second zone, also called the transition zone, the Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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pellets melt from frictional heat generation and externally applied heat, resulting in a decreasing volume along the screw, the diameter of which is increased to adjust for this decreased volume. (iii) In the third zone, also referred to as the metering zone, the screw acts as a pump to build up pressure in the melt. HOPPER

BAND HEATER BARREL

FILTER SCREW

DIE CONVEYING

MELTING

PUMPING

Figure 19.2.1 Schematic diagram of a single screw extruder.

The hot melt produced in an extruder is used as the input to many plastics processes. In intermittent part forming processes – such as in injection molding in which the melt is used to fill mold cavities, and in blow molding where an extruded molten cylinder is forced against a mold cavity by air pressure – the extruder output is not continuous, but is interrupted to provide the melt for each part molding cycle. In continuous part forming processes – such as in fiber spinning, film blowing, wire coating, and profile extrusion – the melt is extruded through dies – in a manner similar to toothpaste being extruded out of a tube – to form continuous parts. An understanding of the material motion in these complex processes is critical for the proper design of a screw to obtain optimal extruder performance. The motion of both the solid and molten phases through the extruder involves complex, recirculating (secondary) paths which can result in unnecessary increased residence time that can degrade the molten resin. Also, if not properly designed, the recirculating motions can trap material for long periods of time; when this trapped material is intermittently ejected, it can affect part performance. Extruder design – a highly specialized subject, requiring complex analyses of the motion of the solid and melt phases through the extruder channel, and the judicious use of experimental data – is well beyond the scope of this book. 19.2.1

Fiber Spinning

Synthetic fibers are made by extruding either a polymer in a solvent solution or a molten resin through very fine holes having diameters on the order of 200 – 500 μm. The in-solution extruded fibers are either precipitated by passing through a bath or air dried to extract the solvent. Melt spinning, the focus of this section, refers to the case in which molten thermoplastic is extruded through holes and then stretched to further reduce the diameter. Actually, many fibers are simultaneously extruded through a spinneret, a die comprising from 1 to 10,000 holes.

Alternatives to Injection Molding

Figure 19.2.2 illustrates the basic aspects of melt spinning. On extrusion through a fine hole, die swell (Section 6.7.1) first causes the diameter to increases before decreasing (Figure 19.2.2a). The extrudate is pulled at a high velocity, typically greater than 50 ms−1 . If the change in density between the melt and solid phases is ignored, conservation of mass requires that the hole radius R, and average extrusion velocity V, are related (Why?) to the final fiber radius r, and velocity v, through r∕R = (V∕ v)1/2 . With the average spinneret velocity being about two orders of magnitude smaller than the take up velocity, it

V

SPINNERET PLATE

POLYMER MELT

R

MELT FILTER

SPINNERET COOLING AIR

DIE SWELL

EXTRUSION DIE

GUIDE ROLLS

v

(a)

r

FILAMENT TAKE-UP ROLL

FILAMENT WIND-UP BOBBIN

(b)

Figure 19.2.2 Schematic diagram illustrating fiber melt spinning. (a) Extrusion of a melt through a small-diameter hole. Note the die swell in the material exiting the hole. (b) Setup for the simultaneous melt spinning of multiple fibers.

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follows that r = R ∕10. Because of the fine holes in the spinneret, the melt is first extruded through a filter (screen pack) to eliminate small particles (Figure 19.2.2b). After passing over guide rolls the fibers are pulled by high-speed filament take-up rolls that stretch the fibers, which are finally wound onto bobbins. Textile yarn consists of multiple fibers and is described by the number and diameter of the fibers. For example, a 70/40 yarn consists of 40, 70-denier filaments (extruded fibers). (A denier or den is a unit for the linear mass density of fibers, defined as the mass in grams of 9,000 m of the fiber. A single strand of silk is approximately one denier.) The mechanics of fiber spinning is quite complex. The extrudate first undergoes extensional flow till cooling air causes it to solidify, after which it undergoes mechanical stretching. This process is subject to many time varying disturbances that can cause instabilities, resulting in nonuniformities in the fiber diameter, and even in brittle fracture. Because of their importance to fiber production, such instabilities have been studied both analytically and experimentally. Technical aspects of these important issues are well beyond the scope of this book. The first artificial fibers, generically known as Rayon, were made from cellulose. The first successful and still used fibers are of nylon. Perhaps the largest application of melt spinning is in the production of poly(ethylene terephthalate) (PET) yarn for textile applications. A host of different synthetic fibers, including hollow ones, are now available to serve many different textile and carpet markets. Microfibers are the newest class of important synthetic textile fibers. 19.2.2

Film Blowing

Extensively used for packaging, thin films account for some of the largest applications of extrusion. They are made by the process of film blowing, schematically shown in Figure 19.2.3, in which a thin-walled cylinder, continuously extruded through an annular die and pinched at the top, is pulled upwards. Simultaneously, the cylindrical shape is expanded by air pressure injected into the cavity. On exiting from the die, the molten cylinder is cooled by air blown from rings. Expansion caused by the air pressure, and the vertical pull results in a thinning of the cylinder. At some stage the film freezes off, after which its thickness changes little. The solidified film is guided by rolls and flattened by nip rolls; the flattened tube is then wound on take-up rolls. If the application requires bags, in a second operation the flattened tube can be heat-sealed at regular intervals. Or, if only film is required, the flattened sheet can be slit at the sides. The biaxial stretching of the film in the circumferential and axial directions improves its mechanical properties and makes it less permeable. As with fiber spinning, if the change in density between the melt and solid phases is ignored, conservation of mass requires (Why?) that the initial radius R, the initial thickness H, and average extrusion velocity V, of the extruded cylinder are related to the final film radius r, its final thickness, h, and its final velocity v, through h ∕H = (RV∕r v). 19.2.3

Sheet Extrusion

Plastic sheet is made either by extruding a flat melt onto a chilling roll – referred to as cast film extrusion – or by forcing extruded melt through gaps in counter rotating rolls – referred to as calendering. Calendering produces better quality sheet. Cast film extrusion is better suited to making films, which refers to thicknesses smaller than 250 μm; larger thicknesses are referred to as sheet material.

Alternatives to Injection Molding

FLATTENED TUBE TO TAKE-UP AND WIND-UP ROLLS

NIP ROLLS GUIDE ROLLS FILM THICKNESS

h

v

V

EXTRUDATE THICKNESS

H

MELT FREEZE ZONE

r

R

COOLING AIR

EXTRUSION DIE AIR PRESSURE

Figure 19.2.3 Schematic diagram illustrating film blowing.

19.2.3.1 Cast Film Extrusion

In cast film extrusion a thin, flat melt is extruded onto a highly polished, cooled (chilled) rotating roll (Figure 19.2.4), where one side is rapidly cooled (quenched). The second side is quenched by a second roll that also helps to strip the almost solid sheet off the chilling roll. The film/sheet then passes through a series of idler rolls – which are very important for maintaining tension in thin films – and pull rolls that, through speed control, provide additional tension. The finished product is then wound onto rolls. The film/sheet thickness is mainly controlled by the surface speed of the chill roll and to a smaller extent by the tension imposed by the pull rolls. Since there are no lateral controls on the solidifying extrudate, accurate thickness control is difficult. Differential cooling of the two sides can result in differing surface textures. The mechanics of the film casting process is subject to instability issues similar to those occurring in fiber spinning.

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EXTRUDER HEAD CAST SHEET DIE STRIPPING ROLL

• MOLTEN EXTRUDATE

•

TO IDLER, PULL, AND WIND-UP ROLLS

ROTATING CASTING ROLL

Figure 19.2.4 Schematic diagram illustrating cast film extrusion.

19.2.3.2

Calendered Sheet Extrusion

In sheet calendering, molten plastic extruded through flat dies passes through precision gaps in pairs of counter rotating rolls, called calendering rolls (Figure 19.2.5). The flat extrudate then goes over rotating, sizing rolls that control the final thickness and surface finish. The gap (A) in the first pair of rolls and their speeds control the initial sheet thickness and speed. Slightly thinner gaps (B, C) in the subsequent roll pairs result in film/sheet with very accurate thickness control. Highly polished roll surfaces and their squeezing action result in high-quality surfaces. Depending on the thickness of the final product, the film/sheet is either wound onto rolls or goes through a cutting station to produce rectangular sheets. The sheet may also pass through another set of rolls to texture the surface through an embossing operation. Extruded sheets are available in a variety of thicknesses ranging from very thin sheets for vacuum and thermoforming applications to thick sheets used for glazing. 19.2.4

Profile Extrusion

Thin-walled, continuous, prismatic solid profiles, such as those used for vinyl (PVC) siding and plastic window frames in the construction industry, are produced by extruding melts through metal dies. The extruded profile may pass through a second die – called a calibrator – that better sizes the part shape; it is then cooled by passage through a water bath. Because of the shrinkage that molten plastics undergo on solidification, the die shape has to be larger by as much as 10 %.

Alternatives to Injection Molding

EXTRUDER HEAD MOLTEN EXTRUDATE

•

• A

B

•

ROTATING CALENDERING ROLLS TO PULL AND WIND-UP ROLLS OR SHEET CUTTING STATION

C

•

Figure 19.2.5 Schematic diagram illustrating calendered sheet extrusion.

19.2.4.1 Open Profiles

Figure 19.2.6 shows a simplified version of a profile extrusion die for extruding a channel section; how this die is attached to an extruder has not been shown. The flow from an extruder first enters a tapering portion of the die that then channels the flow into the L-shaped portion that has a uniform section, referred to as the land. In this section the flow develops a steady profile that then is extruded out of the die head, to form the channel extrusion shown in Figure 19.2.6b. Even for this simple case, the mechanics of extrusion is quite complex. First, die swell (Section 6.7.1) causes the extruded profile to first expand and then narrow down. And post-extrusion differential cooling can cause the part shape to warp. To obtain desired shape and dimensions the extrudate has to be drawn through another die, called a calibrator, and then through a water bath for cool down. Second, and more importantly, depending on the geometry and thickness of the profile, the die shape may have to be changed to obtain the desired cross section. As an example, to obtain a square cross section (Figure 19.2.7a), the die exit would have the shape shown in Figure 19.2.7b.

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CONSTANT LAND LENGTH

(a)

(b)

Figure 19.2.6 Die shape for extruding L-shaped profile. (a) Simplified die shape. (b) Cross section of extruded profile.

(a)

(b)

Figure 19.2.7 (a) Extruded square profile. (b) Die exit shape for extruding square profile.

Detailed die design involves many materials related considerations. For example, because of elastic recovery effects in viscoelastic melts, shorter die land lengths result in higher die swell. Also, the die must be designed to ensure a uniform exit velocity over the cross section. Depending on the shape of the profile, this may require varying land lengths in the die. An example with varying die land lengths is shown in Figure 19.2.8. Coextrusion can be used to produce profiles with two different compatible materials, such as hard and soft combinations of the same material. The second material may be injected through an auxiliary extruder, such as by using the side-fed arrangement shown in Figure 19.2.9, in which the two materials are joined under pressure within the die body to produce a durable bond. For noncompatible materials adequate bonding can be achieved by using adhesives.

Alternatives to Injection Molding

VARYING LAND LENGTH

FIXED LAND LENGTH LARGER LAND LENGTH

(a)

(b)

Figure 19.2.8 Die shape with varying land length. (a) Simplified die shape showing essential features. (b) Cross section of extruded profile. (Adapted from Guidelines for the Extrusion of Sarlink courtesy of Teknor Apex Company.)

MAIN FLOW FLEXIBLE MATERIAL

RIGID MATERIAL

SIDE FEED

(a)

(b)

Figure 19.2.9 Variable land-length die with side feed for second material. (a) Simplified die shape showing essential features. (b) Cross section of extruded profile. (Adapted from Guidelines for the Extrusion of Sarlink courtesy of Teknor Apex Company.)

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By using several extruders, such coextrusion can be extended to produce multimaterial profile extrusions of the type shown in Figure 19.2.10.

(a)

(b)

(c)

Figure 19.2.10 Multimaterial profiles. (a) Two materials. (b) Two materials profile with split flow. (c) Profile with three materials. (Adapted from Guidelines for the Extrusion of Sarlink courtesy of Teknor Apex Company.)

19.2.4.2

Closed Profiles

Tube and pipe manufacturing constitute a special but important application of extrusion technology, a highly simplified die design for which is shown in Figure 19.2.11. The desired annular extruded shape is produced by a bush-mandrel assembly in which the mandrel is kept in place by spider legs that rigidly attach the mandrel to the bush. (In an actual die, the mandrel-bush assembly is made of several pieces, both to facilitate assembly and mandrel alignment.) The central air hole is to pressurize the cylindrical extrudate to prevent its collapse. AIR HOLE

x

SPIDER LEG

x (a)

SPIDER LEG

MELT

MANDREL

BUSH

(b)

Figure 19.2.11 Die for extruding pipes. (a) End view of die; this face is attached to an extruder. The mandrel is attached to the die bush through three spider legs. (b) Section through XX. (Adapted from Guidelines for the Extrusion of Sarlink courtesy of Teknor Apex Company.)

In free extrusion, mainly used for small-diameter flexible tubes, say having diameters from a few mm to a cm, the extrudate is directly quenched in a water bath, and internal air pressure is used to control the tube shape and diameter. In rigid pipes, the diameters of which can range from fractions of centimeters to hundreds of centimeters, the extrudate is passed through a calibrator immersed in cooling water that uses vacuum to control

Alternatives to Injection Molding

the outside pipe diameter (Figure 19.2.12). The quenched pipe is pulled by caterpillar tracks and then cut to desired lengths. VACUUM

EVACUATED WATER-FILLED TANK EXTRUDED PIPE TO PIPE PULLING CATERPILLAR

COOLING WATER

PERFORATED CALIBRATOR

Figure 19.2.12 Schematic diagram showing a perforated calibrator in a water-filled tank. Vacuum applied to the water pulls the cooling outer pipe wall to the calibrator inner surface.

Pipe extrusion requires some of the largest extruders. Figure 19.2.13 shows the extrusion line for a 2400-mm diameter high-density polyethylene (HDPE) pipe with a wall of about 90 mm. The cutout pipe cross section on the left, with a forklift in the background, gives an idea of how large this pipe system is.

Figure 19.2.13 Extrusion line (right) for a 2400-mm diameter HDPE pipe with a wall of about 90 mm. The cut section on the left gives an idea of how large this pipe is. (Photo courtesy of P.E.S Co.)

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This process is not limited to circular tubes and pipes. Extraordinarily complex shapes (Figure 21.6.2) with multiple open and closed sections are routinely extruded for thermoplastic window and door frame applications – for which the material of choice normally is PVC. And some applications require coextrusion for making parts that have rigid and soft surfaces. This process is also used for making multiwall sheet (Figure 19.2.14) for glazing applications (Figure 2.5.1a) – for which the material of choice is polycarbonate. This closed-cell structure not only stiffens the sheet with less material, it provides extra thermal insulation.

Figure 19.2.14 Cross section of a multiwall extruded sheet.

19.2.5

Coating

Coating, which refers to adhering a layer of plastic to a substrate, covers a broad range of applications, such as coating electric wires, plastic-film coated paper, audio and video tapes, and adhesive tapes. Equally broad are coating processes. This section describes the principles underlying two such processes. In wire coating, schematically shown in Figure 19.2.15, a metal wire is pulled through an extrusion die. Molten plastic gets deposited onto the wire by the drag flow (cylindrical version of flow discussed in Section 6.6.1) created by the moving wire surface. If necessary, pressure applied by the extruder can be used to enhance the flow through the die. EXTRUSION DIE

COATING

WIRE

TO COOLING LINE AND WIND-UP ROLL

MOLTEN PLASTIC

Figure 19.2.15 Schematic diagram of the wire-coating process.

In the knife-coating process (Figure 19.2.16), a knife is used to control the thickness of the coating dragged onto a substrate from a recirculating pool of molten material behind the knife.

Alternatives to Injection Molding

KNIFE

COATING

SUBSTRATE

TO COOLING LINE AND WIND-UP ROLL

MOLTEN PLASTIC

Figure 19.2.16 Schematic diagram of the knife-coating process.

19.3 Blow Molding Blow molding, a process for making hollow parts by blowing a thin, cylindrical molten shell into a mold, is a low-pressure process that requires relatively inexpensive molds, and the formed parts are relatively stress-free. Several variations of the original extrusion blow molding process have created a versatile suite of processes that are used for making a very wide variety of products. As with the description of other forming methods, only the basic principles underlying the process have been highlighted through simple schematic diagrams that gloss over important engineering details. Although all real plastic parts have rounded corners, for simplicity, parts in the sequel are mostly shown with sharp corners. 19.3.1

Extrusion Blow Molding

In extrusion blow molding, parts are formed by extruding molten plastic into a thin cylinder, called a parison, which is expanded by air pressure into a mold. The principle of this process is illustrated in Figure 19.3.1. The key parts for extrusion blow molding (Figure 19.3.1a) are the extrusion head that produces the molten cylindrical parison, the pinch plates that are used to seal off the bottom of the parison, the blow pin through which pressurized air is injected to inflate the sealed parison, and the mold halves that form the cavity into which the parison is inflated. The mold halves have notches on the top inside surfaces to accommodate the blow pin when the molds close. The molding cycle starts with the extrusion of the parison, which is pinched off at the bottom and slightly pressurized to prevent its collapse (Figure 19.3.1b). The mold halves next close around the parison, which is then pressurized to force the parison against the interior mold walls (Figure 19.3.1c) to form the part shape. During this phase flash is generated at the mold parting line. On cooling the part with flash is released (Figure 19.3.1d). The final part is obtained (Figure 19.3.1e) on removing the flash. Note that actual parts have rounded corners, not the sharp corners shown in this illustrative example. One shortcoming of extrusion blow molding is that the part wall thickness is not uniform – during inflation, the farther a portion of the parison has to travel before contacting a mold surface, the thinner the local part thickness will be. In analyzing part thickness variations, a good assumption is that the local thickness of the parison gets locked in when the parison touches a wall. In subsequent

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DIE HEAD

BLOW PIN MOLD

PARISON

PINCH PLATE

PINCHED PARISON

(a)

(b)

FLASH

FLASH

(c)

(d)

(e)

Figure 19.3.1 Schematic diagram illustrating different phases of the extrusion blow molding process. (a) Initial layout showing two mold halves, two pinch plates, die head with an extruded molten parison, and the blow pin. Note the notches on the top inside surfaces of the molds to accommodate the blow pin. (b) The pinch plates pinch and seal off the lower end of the parison. (c) The mold halves close and air injected through the blow pin forces the parison against the interior mold surfaces. (d) On cooling the mold halves and the pinch plates open, and the formed part is released. Notice the flash generated on the parting line of the molds. (e) Final part after flash removal.

inflation, it is only the free, expanding parison that can stretch, resulting in local thinning. This thinning is schematically illustrated in Figure 19.3.2, which shows a parison expanding into a deep cavity. The intermediate shapes of the parison shapes, marked 1 through 4, show continuing thinning of the parison. Clearly, the deeper the cavity is, the thinner the portions of the parison that expand the most will be.

Alternatives to Injection Molding

PARISON

AIR PRESSURE

1 2 3

MOLD CAVITY

4

EVOLVING PARISON SHAPE

Figure 19.3.2 Schematic diagram illustrating the evolution of part thickness variation in extrusion blow molding. The sequence of intermediate shapes, 1 through 4, shows the thinning that occurs with continuing expansion of a parison into a deep cavity.

After the parison touches the bottom mold surface, the parison starts expanding into the corners. Figure 19.3.3 shows a portion of the parison in contact with the surface near a reentrant corner; the portions in contact with the mold surfaces have frozen-in thicknesses. In subsequent inflation steps only the unsupported parison portions stretch. This process results in a continual thinning of the part as it contacts the mold surface, so that in this example the thinnest portion of the part is at the reentrant corner (Figure 19.3.3). There are two strategies for improving the thickness distribution in blow-molded parts. In the first, called parison programming, the parison is not extruded as a uniform-thickness cylinder. In the second, a plug is used to expand the parison. These two processes are described in the following two subsections. 19.3.1.1 Parison Programming

Because different parts of a parison undergo different amounts of stretch, a uniform-thickness parison results in a nonuniform-thickness part. In some cases, a uniform thickness may not be the right choice from a part performance standpoint. For example, the thickness distribution may be adjusted to improve the part stiffness or strength. In parison programming the thickness distribution along an extruding parison is controlled to obtain a desired thickness distribution in a part.

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PARISON

AIR PRESSURE

MOLD

(a)

(b)

THINNEST POINT

(c)

(d)

Figure 19.3.3 Schematic diagram illustrating the evolution of part thickness variation in extrusion blow molding process. (a) Part of a parison close to a corner. (b) As the parison inflates under air pressure, the portions of the parison not in contact with the mold become thinner. (c) Further thinning during parison inflation. (d) The thinnest part wall thickness occurs at the corner.

The principle of parison programming is schematically illustrated in Figure 19.3.4. In standard extrusion blow molding, the parison is extruded through an annular slit formed by a circular opening in an extrusion die bushing and the outer circumference of a fixed mandrel (Figure 19.3.4a). The thickness distribution in an extruded container formed from such a uniform-thickness parison is shown in Figure 19.3.4b. In parison programming the axial position of the mandrel is changed in a controlled manner so as to obtain a desired parison thickness distribution (Figure 19.3.4c). The more uniform thickness distribution so obtained is shown in Figure 19.3.4d. Clearly, parison programming requires more expensive blow-molding machines. However, this additional cost is acceptable in large-volume part production in which plastic material savings from more optimized designs can result in substantial overall savings. Parison programming is facilitated by computer simulation codes that predict the wall thickness distribution in a part made from a parison with a prescribed axial wall thickness. Such codes are also capable of solving the inverse problem – determine the parison thickness distribution to achieve a desired part wall thickness distribution.

Alternatives to Injection Molding

FIXED MANDREL

PROGRAMMABLE MANDREL

EXTRUSION DIE BUSHING

PARISON

(a)

(c)

0.020 in (0.51 mm)

0.033 in (0.84 mm)

0.021 in (0.53 mm)

0.033 in (0.84 mm)

0.023 in (0.58 mm)

0.025 in (0.64 mm)

0.025 in (0.64 mm)

0.023 in (0.58 mm)

0.029 in (0.74 mm)

0.025 in (0.64 mm)

0.031 in (0.79 mm)

0.033 in (0.84 mm)

0.033 in (0.84 mm)

0.033 in (0.84 mm)

(b)

(d)

Figure 19.3.4 Effect of parison thickness on container wall thickness. (a) Uniform-thickness parison extruded from a fixed mandrel extrusion die. (b) Container wall thickness distribution for fixed mandrel die. (c) Nonuniform-thickness parison produced by a programmed mandrel. (d) Container wall thickness distribution for programmed mandrel extrusion die. (Adapted with permission from Figure 7 – 17 in “Tool and Manufacturing Engineers Handbook: Volume VIII, Plastic Part Manufacturing,” edited by Philip E. Mitchell, Society of Manufacturing Engineers, 1996.)

19.3.1.2 Deep-Draw Blow Molding

The local thickness of a parison freezes-in on contact with a mold wall. This property is used to prevent thinning of a parison expanded by air by using a plug to push the parison into a deep cavity; during this process the thickness of the parison in contact with the mold will not change. This process is illustrated (Figure 19.3.5) by the deep-draw blow molding of a double-walled container, which otherwise would be assembled from two parts. The overall arrangement, schematically shown in Figure 19.3.5a, consists of three parts: (i) The extrusion head and the pinch plates used, respectively, for extruding and pinching off the parison. (ii) The core part of the mold that has a plug (movable, sliding core) which can slide relative to the core plate. And (iii), the cavity assembly comprising a fixed cavity plate and two folding cavity plates that can be rotated to form the final cavity shape. The molding process starts with the extrusion of a parison that is pinched off by pinch plates. This allows the parison (bag) to inflate evenly (Figure 19.3.5a). The molds then begin to close around the parison while the movable core pushes a portion of the parison. During this process, the thickness of

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BLOW PIN DIE HEAD

PINCH PLATE ACTUATOR

MOVABLE CORE

CAVITY PLATE

CORE PLATE INFLATED PARISON PINCHED PARISON

FOLDING CAVITY PLATE

(a)

(b)

HOLLOW DEFLASHED PART

(c)

(d)

Figure 19.3.5 Schematic diagram illustrating different phases of the deep-draw blow-molding of a hollow container. (a) Immediately after extrusion the parison is pinched-off and kept inflated as the extrusion continues. (b) The sliding core begins to push the parison as the core and cavity begin to close. (c) The plug advances to its maximum position, pushing the parison along. The folding cavity plates rotate to a horizontal position; at the same time the movements of the core and cavity close the mold halves to pinch off the parison, which is torn from the parison by a spurt of air from the blow pin. (d) The mold begins to open – the folding plates rotate to the open position, and the core retracts. The part is removed and deflashed. Note that this figure only shows the deflashed part. (Adapted with permission from Figures 7-38, 7-39, and 7-40 in “Tool and Manufacturing Engineers Handbook: Volume VIII, Plastic Part Manufacturing,” edited by Philip E. Mitchell, Society of Manufacturing Engineers, 1996.)

Alternatives to Injection Molding

the parison in contact with core plug surface, which will eventually form the bottom of the deep-drawn part, will not change (Figure 19.3.5b). The plug advances to its maximum position, pushing the parison along. The folding cavity plates rotate to a horizontal position; at the same time the movements of the core and cavity close the mold halves, pinching off the parison, which is torn from the parison by a spurt of air from the blow pin, resulting in the configuration shown in Figure 19.3.5c. The mold begins to open – the folding plates open, and the core retracts (Figure 19.3.5d). The part is removed and deflashed. This completes the molding cycle. This process can be used to make many parts that previously could not be blow-molded. The folding cavity allows the molding of outside features that conventionally would require undercuts. This process has been used for making flower planters that have realistic brick patterns on all four outside walls. It has also been used for making one-piece hollow water coolers that traditionally were made from blow-molded shells that had to be bonded together. It has also been used to make round containers. 19.3.1.3 Flashless Blow Molding of Tubular Parts

For forming long, curved tubular structures standard blow molding would result in unacceptable levels of material waste in the form of flash. Several processes, generically referred to as 3D (three-dimensional) blow molding, have been developed to efficiently mold such parts. In one version of flashless blow molding, a tubular parison is vertically extruded from a fixed extrusion head onto a 3D mold cavity-half that is moved by a computer-controlled table to position the cavity to follow the mold cavity path (Figure 19.3.6). During this process the parison is partially inflated to prevent its collapse. A mating mold half then closes the mold and the parison is blown to form the part, which has very little scrap, mainly at each end. The main disadvantage of this process is the relatively long time during which the parison is in contact with only one-half of the mold, which can result in premature freezing of the parison surface. This can be overcome by using higher mold temperatures. FIXED EXTRUSION HEAD

3-DIMENSIONAL MOLD CAVITY 3-DIMENSIONAL PARISON

PARISON

z y

MOVABLE MOLD CAVITY

(a)

(b)

•

x

MOLD MOVES IN ALL THREE DIRECTIONS

Figure 19.3.6 Schematic diagram illustrating fixed extrusion head, 3D blow-molding of a non-planar tube. (a) A tubular parison is extruded onto a 3D movable mold cavity. (b) A computer-controlled table moves the mold cavity to facilitate the filling of the cavity. (Adapted from Blow Moulding Processing Manual. Courtesy of DuPont © 1999.)

In a modified version of this scheme the mold cavity remains fixed but the die head is moved by computer control to fill the 3D cavity. Figure 19.3.7 shows photos of two types of flashless, blow-molded TPE parts: (a) Constant velocity joints (CVJ) made of Hytrel®, a copolyester TPE (Section 11.9.2). (b) Hytrel propeller shaft boot.

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(a)

(b)

Figure 19.3.7 Parts made by flashless blow-molding process. (a) CVJ boot seals. (b) Hytrel propeller shaft boot. (Photos courtesy of ABC Group Inc.)

An alternative innovative concept, referred to as the 3D suction process, uses a specially designed mold through which air can be sucked to facilitate the movement of a cylindrical parison through a circuitous mold. Figure 19.3.8a shows the elements of the hardware for this process: In this example two mold halves are shown in the closed position forming a circuitous, cylindrical mold cavity. Besides the die head with a blow pin, there are four pinch-off plates that, on closing, also form a part of the mold cavity. The bottom of the mold has a suction cup through which air is sucked out via a vacuum line. Figure 19.3.8b shows a small-diameter parison that is drawn through the circuitous cavity by the imposed suction at through the suction cap; the air flow around the parison helps to pull and guide the parison till the lower end of the mold. The shutter plates are then closed to pinch off the parison, and air is blown through the blow pin to expand the parison into the mold cavity (Figure 19.3.8c). The mold is then opened and the part is deflashed (Figure 19.3.8d). This process works well for smaller diameter air ducts and pipes, especially when the part cross section does not change much along its length. Figure 19.3.9 shows two parts made by the 3D suction blow molding process: (a) The automotive integrated turbocharger duct combines the air-intake duct with charge air cooler and integrates both into the intake manifold. This new design reduced the air-intake loop volume by up to 50% (for better engine response), at the same time lowering package space 40% and part-count, weight, and costs by 20%. Made from a high-performance recyclable, soft TPV – PA6 and acrylic rubber (ACM) – this design eliminated the need for bellows. Part (b) The automotive turbo resonator was made from polyamides in a two-step process. First, the 20-GF-PA 6,6 insert is injection molded; while increasing structural stability, this insert also eliminated a metal ring used in previous designs. The insert is then overmolded with 35-GF-PA 6,6 using a 2D suction blow molding process to form the outer resonator. Particular attention was required for the parison to enter concentrically over the previously injection molded insert and to remain centered on cooling after blow molding. This design resulted in a 30 wt% saving over the original stainless steel. 19.3.1.4

Multilayer Extrusion Blow Molding

In multilayer extrusion blow molding, special extrusion heads, in which different materials can be internally coextruded through separate dies, are used to simultaneously coextrude several different plastics into a multilayer annular parison. This multilayer parison can then be blow molded by any one

Alternatives to Injection Molding

DIE HEAD WITH BLOW PIN

SHUTTER PINCH-OFF PLATE

CLOSED MOLD HALVES

CYLINDRICAL MOLD CAVITY

CYLINDRICAL PARISON

SUCTION CAP

SUCTION

(a)

(b) CLOSED SHUTTER PINCH-OFF PLATES

FLASH

MOLD HALVES OPEN

PRESSURIZED EXPANDED PARISON

FLASH

(c)

DEFLASHED PART

(d)

Figure 19.3.8 Schematic diagram illustrating the 3D suction blow-molding process. (a) Setup comprising a closed mold with a circuitous cylindrical cavity, pinch-off plates, a suction cap, and an extrusion head with a blow pin. (b) A small-diameter parison is drawn through the circuitous cavity by suction imposed at the suction cap. (c) Closed shutter plates pinch-off the parison, and air pressure expands the parison into the mold cavity. (d) The mold is opened and the part deflashed. (Adapted from figures courtesy of Kautex Machinenbau GmbH.)

of the blow molding methods described in the previous sections. The two main reasons for using this more expensive process are (i) to use a cheaper inner layer to reduce cost, and (ii) to improve barrier properties – reduce gas diffusion through the part thickness – in food packaging applications. Adhesive tie layers have to be used when some of the functional layers are of incompatible materials.

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(a)

(b)

Figure 19.3.9 Parts made by 3D suction blow-molding process. (a) Integrated turbocharger duct. (b) Plastic Turbo resonator (Photos courtesy of SPE Automotive Division.)

This technique makes it possible to use regrind material as a core sandwich layer to provide the bulk of stiffness and strength inside thin outer layers of virgin plastic to provide appropriate appearance surfaces. Industrial applications include plastic drums and jerry cans. One of the largest applications of multilayer blow molding is in making plastic automotive gasoline fuel tanks that, unlike metal fuel tanks, increase crashworthiness safety because the plastic tanks can bend and flatten, rather than tear, thereby preventing gasoline spillage. Because of the complex shapes into which plastics can be molded, plastic fuel tanks can use available space far more efficiently, and internal parts can be inserted during the molding process. And plastic fuel tanks are corrosion resistant. Besides functional requirements of stiffness and strength, plastic fuel tanks must adhere to strict hydrocarbon emission standards through the wall of the tank. The two sets of requirements can be met by a six-layer design in which the inner and outer layers are made of virgin HDPE, a large thickness of reground HDPE, two layers of compatibilized polyethylene as adhesive layers, and a layer of ethylene vinyl alcohol (EVOH) as a barrier layer for gasoline – the gas barrier properties EVOH are 4,400 times better than those of HDPE. While the thicknesses of the different layers may vary from design to design, the nominal thickness distributions, starting from the tank inside, are on the order of 25% virgin HDPE inner layer, 2% adhesive tie layer, 2 – 3% EVOH barrier layer, two adhesive tie layers, 40% regrind (recycled) HDPE layer, and a15% HDPE outer layer. The total thickness of the HDPE layers is on the order of 2.5 – 5 mm (0.1 – 0.2 in), and that of the EVOH layer is in the range of 0.1 – 0.3 mm (0.004 – 0.012 in). Multilayer blow-molded bottles and containers are widely being used for packaging. Because an oxygen barrier layer is required for food items, such as tomato ketchup, plastic squeezable bottles may be made with a six-layer configuration in which the polyethylene may be replaced by polypropylene. When gas permeation is not an issue fewer layers may be used.

Alternatives to Injection Molding

19.3.1.5 Blow Molding with Encased Modules

The first major advance in making automotive plastic gas tanks was the development of a six-layer coextrusion blow molding machine, which made it possible to include a barriers layer to prevent gas permeation through the walls, thereby drastically reducing emissions. In the early blow molded gas tanks the internal modules, such as fuel-delivery modules, baffles, and level sensors, had to be inserted through a hole cut in the skin. This limited the size and complexity of the modules that could be used. And subsequent welds at the hole increased the cost and were prone to leaking gas. The next advance was the development of a “ship-in-a-bottle” (SIB) technique in which a spreader is used to widen the bottom of a parison sufficiently to enable a robotic arm to insert a rigid carrier with the fuel-delivery system inside the parison. Mold closure and subsequent blowing formed the part. The SIB process was used to produce the first plastic saddle-shaped fuel tank for a partial zero emissions vehicle (PZEV) – automobile with zero evaporative emissions from its fuel system. However, the placement of fuel system components was again limited by the 25-cm (10-in) plastic parison opening, and internal welds were difficult to make. In a subsequent refinement of the process, a preformed blow-molded parison is first separated into two halves, and a robot then inserts the fuel system components into designated areas of the tank assembly. This method does not have any size or location limitations. As a measure of innovation in blow molding, multilayer plastic automotive tanks can now be blowmolded with as many as 15 components – such as rollover valves, fuel-delivery modules, and level sensors – inside the tank. The molded tank then requires only one hole to insert the fuel-delivery module. Figure 19.3.10a shows a CAD rendering of the GM T172 Equinox “saddle-shaped” blow-molded gas tank with many molded-in components, including baffles to prevent noise due to sloshing of fuel in the tank, especially during vehicle start up and stopping, a feature considered especially important for hybrid vehicles. Note the large number of internal components that include two baffles. The actual modules used in this gas tank had (Figure 19.3.10b) two baffles, electrical saddles, fuel-delivery modules, grade vent valve, and vapor line.

(a)

(b)

Figure 19.3.10 Automotive gas tank with encased fuel-delivery module, (a) CAD model showing poisoning of internal module. (b) Actual fuel-delivery module used. (Photos courtesy of Kautex Machinenbau GmbH.)

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For this fuel tank, the fuel-delivery parts are molded and assembled into the fuel-delivery module shown in Figure 10.3.10b. Then a novel process is used to encase this module inside the tank during the blow-molding step. In addition to a two-part mold, this novel process uses a preformed module to encase the fuel-delivery module in a six-layer vertical parison. The principle of this process is explained by using a very simple set of schematic diagrams. Figure 19.3.11a shows the two mold halves together with the intermediate preformed module. As the six-layer parison comes out of the extrusion die head, it is slit up the side with a saw, making a sheet of plastic (Figure 19.3.11b). A robot takes the sheet off the head and wraps the sheet around the preform module that sits in between the mold halves, resulting in the assembly shown in Figure 19.3.11c. PARISON WITH VERTICAL SLIT PREFORMED MODULE

MOLD

(a)

(b)

(c)

Figure 19.3.11 (a) Two mold halves with the preformed module. (b) Vertical parison with a vertical slit made by a saw. (c) Preformed module encased in hot plastic sheet. (Adapted from figures courtesy of Kautex Machinenbau GmbH.)

The next steps in the molding operation are shown in Figure 19.3.12: The mold closes on the preform module encased in plastic, and the parison is inflated into the mold halves (Figure 19.3.12b). The mold opens and the perform module comes out, leaving each mold half with a hot plastic tank shell (Figure 19.3.12c). A robot arm with all the internal components places the module between the mold halves and welds the components to the inner surface of the tank shell (Figure 19.3.12d). The robot arm comes out and the mold fully closes (Figure 19.3.12e). The partially expanded shell is blown into the mold cavity, encapsulating the components inside the six-layer tank (Figure 19.3.12f). Knives from the right mold half cut the shell weld surface (Figure 19.3.12g). The tank is cooled and then extracted from the mold by a robot arm (Figure 19.3.12h). A cutaway view of a molded tank with the top surface and some parts removed is shown in Figure 19.3.13. As a result, not shown is the internal weld of the fuel line to the wall, which prevents gas leakage normally associated with older methods in which the internal modules were inserted by large holes cut in the blow-mold. The welds are made by parts being pressed into hot plastic.

Alternatives to Injection Molding

(a)

(b)

(c)

(d)

(e)

(f)

Figure 19.3.12 Steps in encapsulating the fuel-delivery module during the blow molding process. (a) The mold closes on the preform module encased in plastic. (b) The parison is inflated into the mold halves. (c) The mold opens, the perform module is removed, leaving each mold half with a hot plastic tank shell. (d) A robot arm places the fuel-delivery module between the mold halves and welds the components to the inner surface of the tank shell. (e) The robot arm exits and the mold fully closes. (f) The partially expanded shell is blown into the mold cavity, encapsulating the components inside tank. (g) Knives inside the mold half cut the shell weld surface. (h) After cooling the tank is extracted from the mold by a robot arm. (Adapted from figures courtesy of Kautex Machinenbau GmbH.)

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(g)

(h)

Figure 19.3.12 (Continued)

SUB-SIDE BAFFLE

MAIN-SIDE BAFFLE

SUB-SIDE FUEL DELIVERY MODULE

GRADE VENT VALVE

INTERNAL TRANSFER LINE (ELECTRICAL SADDLE)

Figure 19.3.13 Cutaway view of a molded tank with the top surface and some parts removed. (Photo courtesy of Kautex Machinenbau GmbH.)

19.3.2

Injection Blow Molding

In injection blow-molding a preform is injection molded to replace the parison used in extrusion blow-molding. This preform can have threads molded at the head and the thickness distribution along the preform can be tailored to produce the desired thickness distribution in the part. While still hot the preform can be used as a parison for molding the part. Alternatively, cold performs can be heated and then blow molded.

Alternatives to Injection Molding

While the injection blow molding process is widely used for large-volume production using multistation machines, the basic underlying principles are illustrated in the schematics in Figure 19.3.14. First a preform with a prescribed thickness distribution along its length is injection molded with a core comprising a circular pin that has a central axial hole. As shown in Figure 19.3.14a, high-quality threads required in the part can be molded integrally with the preform. The hot preform and core assembly is then placed in a mold that grips the threaded portion (Figure 19.3.14b), and air is blown through the pin to force the inflated preform surface against the mold wall. In a variation called stretch blow molding, before blowing, the preform is mechanically stretched by pushing the pin axially, as shown in Figure 19.3.10c. While blow molding does produce tangential stretching, the mechanically induced axial stretching in stretch blow molding results in a biaxial stretching of the blown surface – resulting higher optical clarity, better mechanical properties, and lower permeability. HOLE IN CORE PIN

CORE PIN

STRETCHED PREFORM

INJECTION-MOLDED PREFORM MOLD

(a)

(b)

(c)

Figure 19.3.14 Schematic diagrams illustrating different aspects of the injection blow-molding process. (a) Injection-molded preform with the core pin used during molding. (b) The preform-pin assembly is placed in a mold and the preform is expanded by pressurized air pushed through the hole in the pin. (c) In stretch blow-molding the core pin first stretches the preform before pressurized air is used to shape the part.

Alternatively, instead of directly blowing the hot injection-molded preform, the preforms are cooled as parts; they can be made by a molder from where the preforms can be shipped to a blow molding facility. In this case the cold preforms have to be heated prior to the blow molding operation. Clearly, this process adds to the initial machine cost because it requires both injection and blow molding machines. But it offers several advantages in large-volume production: Injection molding allows (i) a more precise control of the thickness distribution of the preform, (ii) superior dimensional control of the neck and screw portions of screw-top bottles and containers, and (iii) better surface appearance. And (iv), since there are no pinch-offs, material wastage and trimming are eliminated.

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19.3.3

Part Stiffening

Like many hollow parts made by several fabrication techniques, stiffening ribs of the type used in injection-molded parts are not possible, so that alternative means have to be used to stiffen hollow parts having large unsupported panel-like structures. Lateral load transfer between adjacent panels (walls) of blow-molded parts is facilitated by connecting them through rib-like structures. Another stiffening technique is to fill the hollow spaces with foam. Figure 19.3.15a,b show a conical tack-off formed by an appropriate design of the mold that pushes a conical protrusion in one wall against the opposing wall. The clearances are designed to provide sufficient pressure to affect a pressure weld at the mating hot surfaces. Large panels can be stiffened by using multiple in-line tack-offs shown in Figure 19.3.15c; other, staggered patterns are also used.

X

X

(a)

COMPRESSION WELD OVER SMALL AREA

CONICAL TACK-OFF

(c)

(b) Figure 19.3.15 Schematic diagrams illustrating the use of a conical tack-off. (a) Front view of panel with a single conical tack-off. (b) Cross section through XX showing compression weld at contacting surfaces. (c) Front view of panels showing arrays of conical stiffening tack-offs.

Alternatively, continuous rib tack-offs can be used (Figure 19.3.16). Shorter, in-line or staggered ribs are also used. 19.3.4

Summary Comments

While this section has introduced several types of blow-molding methods, they now comprise a large set of versatile forming technologies that can be used to mold simple to very complex parts. Parts can also be molded with metal inserts. The discussion in Section 19.3 does not list the numerous products that can be made by blow molding. A partial list of applications: containers with handles for water, milk, juices, and oils; PET bottles for water

Alternatives to Injection Molding

COMPRESSION WELD

(a)

RIB TACK-OFF

(b) Figure 19.3.16 Schematic diagrams illustrating the use of continuous rib tack-offs. (a) Cross section of panel showing compression welds at contacting surfaces. (b) Front view of panels with an array of continuous rib tack-offs.

and carbonated beverages; multilayer bottles and containers with gas barriers for food; trash containers; tool cases; patio furniture; stadium seats; toys; mannequins for the fashion industry; canoes and kayaks; material handling pallets; automotive bumpers; and automotive gas tanks. As discussed in Section 19.3.1.4, multilayer plastic automotive tanks can be blow molded with as many as 15 components – such as rollover valves, fuel-delivery modules, and level sensors – inside the tank. The molded tank then requires only one hole to insert the fuel-delivery module.

19.4 Rotational Molding In rotational molding – also called rotomolding, rotoforming, and rotocasting – a partially plasticpowder-filled, heated mold rotating about two axes is used for making hollow plastic parts. The principle underlying this process is shown in Figure 19.4.1. (a) The mold is closed after a pre-measured amount of powdered resin is placed in the lower half of an opened mold. (b) The closed mold is then rotated about two axes in a heated chamber. As the powder tumbles over the mold surface it melts, taking on the shape of the mold surface. (c) The mold is then placed in a cooling chamber in which the biaxial rotary motion is continued till the molten plastic solidifies. (d) The mold is then opened and the part removed.

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UPPER MOLD HALF

MOLD HEATING

LOWER MOLD HALF

ARM

POWDER

MOLD ATTACHMENT PLATE

(a)

(b)

MOLDED PART

MOLD COOLING

(c)

(d)

Figure 19.4.1 Schematic diagram illustrating the four phases of a rotomolding cycle. (a) Measured amount of plastic powder placed in the lower half of an opened mold. (b) Closed mold is rotated about two axes in a heated chamber. (c) Mold placed in a cooling chamber continues to be rotated about two axes. (d) After solidification the part is removed from the mold.

Alternatives to Injection Molding

Because of the slow rotational speeds used, the plastic powder in the rotating mold is not pushed against the walls by centrifugal forces as, for example, in the spin casting method used for metal pipes. Rather, as schematically shown for a cylindrical mold in Figure 19.4.2, at slow rotational speeds the powder charge in the bottom of the stationary mold begins to tumble under the action of gravity. Figure 19.4.2a shows the powder charge, with surface AB, at the bottom of the cylinder. As the cylinder begins to rotate, friction causes the charge to also rotate with the cylinder (Figure 19.4.2b). But when the surface AB attains a critical angle, the material at B begins to slide in the direction BA (Figure 19.4.2c). This tumbling action also causes the powder particles to mix.

•

• POWDER

A

•

B

B

B A

(a)

A

(b)

(c)

Figure 19.4.2 Motion of powder charge in a cylindrical mold. (a) Powder with surface AB placed in the lower half of a cylindrical mold. (b) As the cylinder is rotated, the powder charge initially rotates with the cylinder. (c) When the charge surface attains a critical angle, the particles at B begin to slide and tumble in the direction BA.

The powder in contact with the mold heats up by conduction from the heated mold surface. At some stage the melting of this powder results in a thin layer of molten plastic coating the mold surface. The continually tumbling powder particles stick to the molten layer and, on melting, cause the molten layer to thicken. This thickening continues till all the powder has melted, eventually resulting in a layer of uniform thickness covering the entire mold surface. Clearly, the molten film thickness, and therefore the part thickness, will depend on the amount of resin powder used in the charge; the larger this charge is, the thicker the part will be. Because of the rotation of the mold about two axes, the tumbling, melting, and molten film formation is actually a complex process the analysis of which is made more difficult by the intermittent nature of tumbling and by the way in which powder particles are captured by the molten plastic film. Instead of complete melting some particles may sinter together, especially in the inner surfaces, resulting in small voids in the finished part. Also, the thickness may have small variations across the part. The molds can be configured with trap-doors through which additional material can successively be added without having to open and cool the mold. This makes possible multilayer solid parts with enhanced barrier properties or parts with interior layers of a cheaper material. Sandwich structures with foam cores can be made by adding a second charge of plastic mixed with a foaming agent. A third charge can then provide a solid skin. Rotomolding is the technology of choice for molding very large parts – such as septic, oil, and chemical storage tanks – that can be molded to relatively uniform thicknesses up to about 13 mm (0.5 in). It has now evolved into a versatile technology for making complex, hollow parts with relatively uniform-thickness

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walls, and is used for molding a wide variety of parts in many different sectors, such as in the automotive (instrument panels), toy (outdoor furniture, ride-on toys, and playhouses), materials handling (trash cans and pallets) industries; the examples cited within parentheses are just representatives of the very large number of products for which rotomolding can be used. The following examples illustrate the complexity of parts that can be molded-in in a wide variety of applications. Figure 19.4.3a shows a 183-cm (72-in) long rotationally molded HDPE spineboard used for providing rigid support during transport of patients suspected of having spinal injuries. The

(a)

CAVITY FILLED WITH PUR FOAM

STRAPPING PINS

(b)

(c) Figure 19.4.3 Spineboard for transporting injured patients. (a) Outer rotationally molded HDPE shell. (b) Lateral cross section showing hollow interior that is later filled with PUR foam. (c) View showing insert-molded LDPE injection-molded pin. (Photos courtesy of Formed Plastics, Inc.)

Alternatives to Injection Molding

cross-sectional view in Figure 19.4.3b shows the uniform-walled hollow interior and the two strapping pins at each end. These injection-molded pins are insert-molded during the rotational molding of the spineboard shell (see detail in Figure 19.4.3c). This requires that the pin portions visible in the final product be shielded during the molding process; it is embedded in the solid mold portions that help shape “handles” on the two sides. Other than filling the hollow internal cavity after molding, the shell-pin assembly is molded in a one-step process. Clearly, such parts, which can easily be molded by this process, cannot be injection molded or by variants of that process. Figure 19.4.4a shows a rotationally molded, LLDPE laboratory reservoir with an integrated site glass. The photos of the two halves of the longitudinal sections in Figure 19.4.4 parts b and c show the complex hollow structure of the interior. Notice the uniform wall thickness throughout the part. This part has several metal inserts (Figure 19.4.4d,e). In particular, the detail of the metal insert shown in Figure 19.4.4d – along a cross section perpendicular to the view in Figure 19.4.4b – shows an insert that connects the interior to the outside.

(b)

(c)

(d)

(e)

(a)

Figure 19.4.4 Rotationally molded, LLDPE laboratory reservoir with an integrated site glass. (a) External view. (b, c) Complex hollow structure of the interior. (d, e) Metal inserts. (Photos courtesy of Formed Plastics, Inc.)

Figure 19.4.5 shows the 51-cm (20-in) high, LLDPE housing of a 20-gallon, wet-dry vacuum cleaner rotationally molded with metal inserts. Portions of this housing are hollow. Figure 19.4.6a shows a three-piece rotationally molded ground penetrating radar cart assembly, the main components of which are, (i) a 51-cm wide by 129-cm long (20-in × 32-in) LLDPE hollow chassis (Figure 19.4.6b), (ii) an 89-cm (35-in) long LLDPE handle, and (iii) a cross-linked PE instrument enclosure.

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Figure 19.4.5 Rotationally molded LLDPE housing for a 51-cm high, 20-gallon wet-dry vacuum cleaner. (Photo courtesy of Formed Plastics, Inc.)

LLDPE HANDLE CROSS-LINKED PE ENCLOSURE

HOLLOW CHASSIS

LLDPE CHASSIS

(a)

(b)

Figure 19.4.6 Three-piece rotationally molded ground penetrating radar assembly. (a) The handle and chassis are made of LLDPE and the instrument enclosure is made of cross-linked PE. The handle is 89 cm long. (b) View of the 51-cm wide by 129-cm long hollow chassis. (Photos courtesy of Formed Plastics, Inc.)

Alternatives to Injection Molding

Figure 19.4.7 shows a three-piece rotationally molded tent heater assembly. The top and bottom 135-cm (53-in) long hollow structures are made of LLDPE. The 94-cm (37-in) long fuel tank, which nests in the lower structure, is made of cross-linked PE. The heater fits in the space between the upper and lower hollow structures.

Figure 19.4.7 Three-piece rotationally molded tent heater assembly. The top and bottom 135-cm long hollow structures are made of LLDPE. The 94-cm long fuel tank, which nests in the lower structure, is made of cross-linked PE. The heater fits in the space between the upper and lower hollow structures. (Photo courtesy of Formed Plastics, Inc.)

Figure 19.4.8 shows a rotationally molded LLDPE 2-part recycling bin, which is 81-cm long by 58-cm wide by 81-cm high (32 × 23 × 32 in).

Figure 19.4.8 Rotationally molded 81-cm long by 58-cm wide by 81-cm high LLDPE two-part recycling bin. (Photo courtesy of Formed Plastics, Inc.)

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19.4.1

Rock-and-Roll Rotational Molding

In rock-and-roll rotational molding, a variant of the process schematically shown in Figure 19.4.9, the part is rotated (rolled) about the major (long) axis, but only rocked to-and-fro about the minor axis. The combination of this rotary and rocking motion is adequate for distributing the powdered plastic all over the heated mold surface. In comparison to standard rotational molding machines with biaxially rotating molds, such rock-and-roll machines significantly reduce machine size and the complexity of making long aspect-ratio parts such as large tanks, kayaks, and lampposts. The rotary motion about the major axis is faster than the rocking motion about the minor axis. The rocking motion can be as large as 𝜃 = ± 45°. ROTATING MOLD

ROCKING MOTION

MAJOR AXIS

± θ°

Figure 19.4.9 Schematic diagram showing motion modes in a rock-and-roll rotational molding machine for making elongated parts. The mold rotates about the major axis, and rocks at a lower speed about the minor axis.

Figure 19.4.10 shows a very large rock-and-roll rotational molding machine. Its size can be gauged from the height of the man standing next to the ladder (lower left part of photo). In this machine, the rock-and-roll drum is 10,668-mm (35-ft) long and has a diameter of 6,096 mm (20 ft). Figure 19.4.11a shows a very large rotationally molded HDPE Protein Fractionator Tank. It has a diameter of 2,134 mm (84 in) and a height of 3,658 mm (144 in), and weighs 272 kg (600 lbs). Its size can further be gauged by the view showing inserts being placed inside the mold (Figure 18.4.11b). Notice the graphics appliqué on the upper right used for the in-mold decoration on the outer surface of the tank (Figure 19.4.11a). The mold for this part is augmented by 22 inserts; a detailed view of some of them in place in the mold is shown in Figure 19.4.11c. Figure 19.4.12 shows the metal insert for molding the 38-mm (1.5-in) diameter threaded hole in an HDPE part. Figure 19.4.13a shows a large, 363 kg (800 lb), 4, 877 × 2, 438 × 1524-mm (16 × 8 × 5-f t) HDPE rotationally molded swim spa just prior to demolding. The cover of the mold, the underside of which generates

Alternatives to Injection Molding

Figure 19.4.10 Large rock-and-roll rotational molding machine. Its size can be gauged from the height of the man standing next to the ladder (see lower left part of photo). (Photo courtesy of Innovative Rotational Molding.)

the inner surface of the mold, is shown in Figure 19.4.13b. Notice how this mold cover has been fabricated by welding sheet-metal components, thereby reducing mold cost.

19.4.2

Advantages and Limitations

In addition to being able to mold very large hollow parts, rotational molding has many advantages: (i) Because the pressure inside the mold is low, the molds – made of sheet metal or cast aluminum – are relatively inexpensive and easy to make; lead times for molds are smaller. (ii) Parts have a uniform thickness. (iii) Parts do not have sprue marks and knit lines. For esthetic parts, molds can be designed to incorporate parting lines into part features. And there is no waste because of the absence of runners and flash. (iv) When cooled properly, the parts are relatively stress-free. (v) Parts with complex shapes can be made with specified wall thicknesses in the range of 1.5 – 13 mm. (vi) Stiffening features and insert can be integrally molded-in. (vii) Little or no draft angles are required. (viii) Many types of undercuts can be molded-in. (ix) The material and its color can be changed easily. (x) Multilayer parts can be molded. (xi) High-quality graphics can be molded-in.

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(a)

(b)

(c)

Figure 19.4.11 Large rotationally molded HDPE Protein Fractionator Tank. (a) External view of 2,134-mm diameter, 3,658-mm high tank. (b) Internal view showing inserts being placed inside mold. Notice the in-mold graphics appliqué (upper right). (c) Detail showing inserts in place inside the mold. (Photo courtesy of Innovative Rotational Molding.)

Alternatives to Injection Molding

(a)

(b)

Figure 19.4.12 Internal threads in rotationally molded part (a) Threaded hole in HDPE part. (b) Steel insert for molding threads. (Photo courtesy of Innovative Rotational Molding.)

(a)

(b)

Figure 19.4.13 Rotationally molded large swim spa. (a) Molded tub in mold just prior to demolding. (b) Outer view of sheet-metal fabricated mold cover; the inner surface of the tub is generated by the bottom surface. (Photo courtesy of Innovative Rotational Molding.)

The main limitation of this process is the long cycle times resulting from the external heating and cooling times for the mold; the thicker the wall, the longer the cycle time. Another major disadvantage is the limited availability of the materials in powder form; converting resin pellets into powder adds to the cost. The main materials used in rotomolding are different grades of low-, medium-, and high-density polyethylenes. Other thermoplastic materials used in growing applications include polypropylene, polyvinyl chloride, polyvinylidene fluoride, ABS, polyesters, polycarbonate, several nylons, polyoxymethylene (acetal), modified polyphenylene oxide, and polyphenylene sulfide.

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19.4.3

Part Morphology

The complex morphology of injection-molded semicrystalline parts (Figures 17.4.1 and 17.4.2) results from the high shear rates and high differential cooling caused by the melt contacting the cold mold during part molding, resulting in high residual stresses in the part. In contrast, in rotational molding the melt is not subjected to shear and, because during molding the interior surface is subjected to a high uniform temperature, and cooling by air is more gradual, rotationally molded parts have a relatively homogeneous morphology. The maximum air temperature attained inside the mold is an important process variable that is monitored and used for process control. Too low an air temperature can result in poor fusion among the molten particles and in air bubbles being trapped within the part wall. The decrease in melt viscosity at the right temperature, and an increase of the air pressure inside the mold, allow the bubbles to be pushed to the mold surface. Too high a temperature may cause the polymer to degrade at the inner surface; this can prevent the formation of spherulites at that surface, thereby degrading mechanical properties. The morphology of rotationally molded semicrystalline polymers correlates well with the maximum air temperature inside the mold; this temperature is a valuable tool for quality control. The occurrence of air bubbles or degradation of the polymer at the inner surface are easily detected by polarized light microscopy. The relatively homogeneous morphology of an approximately 4-mm thick, rotationally molded polyethylene part is shown in Figure 19.4.14. (Note the differences in the scales among the six micrographs.) The first three figures correspond to a part for which the internal air temperature during

1mm

(a)

1mm

(d)

1mm

1mm

(b)

(c)

0.1 mm

0.1 mm

(e)

(f)

Figure 19.4.14 Polarized light micrographs showing morphology of rotationally molded polyethylene parts. (a) Relatively homogeneous spherulitic morphology across the thickness of an approximately 4-mm thick part. The left surface was in contact with the mold; the right, wavy internal surface was in contact with air that reached a maximum temperature of 195°C. (b) Morphology of the left half of the part at a higher magnification. (c) Morphology of the right half of the part at a higher magnification. (d) Bubble-free morphology of the middle portion of a part molded at a maximum internal air temperature of 241°C. (e) Relatively homogeneous morphology of middle portion of the same part at a higher magnification. (f) Relatively homogeneous morphology of the inner portion with right surface in contact with air. The spherulitic structure at this inner does not indicate any degradation. (Photos courtesy of Professor J. Oliveira, University of Minho.)

Alternatives to Injection Molding

molding reached 195°C; the holes in these three figures are trapped air bubbles. The maximum air temperature for the part used in the last three figures was 241°C. Notice the relatively uniform morphology through the part thickness and the absence of air bubbles. The higher magnification micrographs in Figure 19.4.14 parts e and f confirm the relatively homogeneous part morphology. Also, at this molding temperature, the spherulitic structure at the inner (right) surface (Figure 19.4.14f) does not indicate any material degradation. 19.4.4

Part Design

This process can be used to produce parts with little or no draft angles. And parts with a fairly large range of undercuts can also be made. Innovations have made it possible to mold very complex parts with inserts, screw threads, and in-mold decoration. However, how this can be achieved is beyond the scope of this book. 19.4.4.1 Approaches to Part Stiffening

Like many hollow parts made by several fabrication techniques, stiffening ribs of the type used in injection-molded parts are not possible, so that alternate means have to be used to stiffen hollow parts having large unsupported panel-like structures. As with blow molding, one approach is mold rib-like structures of the type shown in Figure 19.4.15.

Figure 19.4.15 Top and front views of a ribbing scheme for stiffening a rotomolded box structure.

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This approach can be extended to mold-in an extended network of rib-like stiffening structures. As an example, Figure 19.4.16 shows a scheme used for stiffening box-like structures. In the side, top, and front views the non-hashed regions provide the rib-like stiffening features. Figure 19.4.16d (Section on XX) shows a cross section through one of the lateral ribs; notice the inside surfaces of the side, rear, and lateral inclined surfaces, and the cross-shaped ribs on the front view. Figure 19.4.16e (Section on YY) shows the cross-sectional shapes of the lateral, depressed, and rib surfaces. Note that in real parts all the corners in the molded structure will be rounded – sharp corners in this figure have been shown for simplicity.

X

X

DEPRESSED SURFACE

Y

(a)

(d)

Y

(b) INCLINED EXTERNAL SURFACE

(e)

INCLINED SURFACES DEPRESSED SURFACES

(c)

(f)

Figure 19.4.16 Scheme for stiffening a large rotationally molded box structure. In (a) side view, (b) top view, and (c) front view lateral inclined surfaces are indicated by lightly dot-filled areas and the depressed surfaces are indicated by more densely-filled areas. The white, unfilled regions provide the stiffening features. (d) The Section on XX shows across section through one of the raised lateral ribs. (e) The Section on YY shows the cross-sectional shapes of the lateral, depressed, and rib surfaces. (f) Shading indicating inclined and depressed surfaces.

An alternate approach to stiffening mimics the tack-offs used for stiffening blow-molded structures. Figure 19.4.17 schematically shows continuous stiffening tack-offs that in rotomolding practice are referred to as kiss-offs.

Alternatives to Injection Molding

MOLD

RIB KISS-OFF

POWDER

(a)

(b) RIB KISS-OFF

(c)

(d) Figure 19.4.17 Schematic diagram illustrating the use of continuous kiss-offs for panel stiffening. (a) Closed mold with molding powder. (b) Part inside mold. (c) Top view of panel with kiss-offs. (d) Front view of panel with kiss-offs.

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Y

Y

X

X

(a)

(b)

(c)

(d)

(e) Figure 19.4.18 Schematic diagram illustrating panel stiffening through local kiss-offs (tack-offs). (a) Front view of panel with local circular kiss-offs. (b) Part shape inside mold. (c) Top view of panel at Section XX. (d) Top view of panel at Section YY. (e) Cross section of mold at XX. (f) Cross section of mold at YY.

Alternatives to Injection Molding

Instead of continuous kiss-offs (tack-offs), local kiss-offs, as schematically shown in Figure 19.4.18, can be used for panel stiffening. Figure 19.4.18a shows the front view of a panel with local kiss-offs. Figure 19.4.18 parts b and c show, respectively, the top views of the Sections on XX and YY. And Figure 19.4.18 parts d and e show the cross sections of the mold corresponding, respectively, to the cross sections on XX and YY.

19.5 Thermoforming In thermoforming a film or sheet of thermoplastic heated to a rubbery state is used to mold parts at very low pressures. This requires amorphous materials to be heated above their glass transition temperatures; the temperature has to be high enough to soften the material yet not so high as to result in sheet sag. For semicrystalline thermoplastics the forming temperatures are below the melting temperatures as these materials rapidly lose strength above this temperature. In a typical thermoforming operation, the heated film or sheet is formed in a mold wherein, on cooling below the glass transition temperature, the formed shape is set, and the part can be demolded. Because of the low forming pressures, the molds used in thermoforming are relatively inexpensive. Several variants of thermoforming are discussed in the sequel. 19.5.1

Vacuum Forming

In vacuum forming, the oldest of thermoforming techniques, a heated sheet is clamped over a mold after which the mold is evacuated. Atmospheric pressure then pushes the sheet against the mold to form the part. The different stages of vacuum forming with a male mold – also called drape molding – are shown in the schematic diagram in Figure 19.5.1. A vented mold (CDEFG) is mounted on an attachment plate (GH) that is connected to a vacuum line (Figure 19.5.1a), and a clamped, heated sheet is lowered onto the mold surface. The clamped, heated sheet first comes in contact with the upper surface (CD) of the mold (Figure 19.5.1b) to which it adheres. As the clamps are further lowered, while the portion CD of the sheet does not stretch, the portions AD and CB continue to stretch till the sheet touches the mold attachment plate at G and H, respectively (Figure 19.5.1c). Then, on evacuation of the mold, atmospheric pressure forces the sheet to contact the mold surface (CDEF) (Figure 19.5.1d). During this process the portions DE and EF of the sheet undergo further stretching. The molded sheet is removed after cooling (Figure 19.5.1e). The final part is obtained after trimming the excess portions (Figure 19.5.1f). Since stretching causes the sheet to become thinner, the part will be thickest at the bottom and thinnest at the open end. A simple measure for the depth of draw d is the draw ratio d∕D. Clearly, in this simple case, the larger the draw ratio the thinner the more stretched portions of the part will be. However, for more complex parts this simple definition does not provide a good measure for the maximum thinning in a part. Even definitions based on area increases are of limited value. Actual part thinning has to be obtained from computer simulations of the process. Figure 19.5.2 shows different stages of vacuum forming of the part in Figure 19.5.1 with a female mold. A clamped, heated sheet is lowered onto the mold surface of a vented mold (CDEFGH) mounted on a sealed box attached to a plate (GH) that is connected to a vacuum line (Figure 19.5.1a). The clamped, heated sheet first comes in contact with the upper surface (EF and DC) of the mold (Figure 19.5.1b) to which it adheres. The clamps are further lowered to seal the sheet against the upper mold surface EF and DC (Figure 19.5.1c). During this sealing process while the portions EF and DC of the sheet do

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CLAMP

VACUUM VENT

G

HEATED SHEET

D

A

C MOLD

F E

H

VACUUM LINE

G

D

H

MOLD ATTACHMENT PLATE

(b) D

C

H

G

B

F E

(a) D

C

G

C

F E

H

VACUUM

(c)

(d) THICK

THIN

d

660

D (e)

(f)

Figure 19.5.1 Schematic diagram illustrating vacuum forming with male mold. (a) Clamped, heated sheet being lowered onto a male mold with vents (CDEF) that is mounted on mold attachment plate (GH). (b) Clamped, heated sheet in contact with the upper surface (CD) of the mold. (c) The clamps are lowered till the portions AD and CB of the sheet touch the mold attachment plate at G and H, respectively. (d) On evacuation of the mold, atmospheric pressure forces the sheet to contact the mold surface (CDEF). (e) Molded sheet removed after cooling. (f) Final part obtained after trimming excess portions.

not stretch, the portion AE and CB undergo stretching. Then, on evacuation of the mold, atmospheric pressure forces the sheet to contact the mold surface (EFGHDC) (Figure 19.5.1d). During this process while the sheet portions EF and DC do not stretch, the portion FGHD undergoes stretching. Note that this stretching is not uniform: potions of the sheet contacting the mold surface do not stretch further – the

Alternatives to Injection Molding

HEATED SHEET

A E F

B AE F

D C G

SEALED MOLD SUPPORT BOX

H

D C B G

(a) E F

(b) E

D C

A

H

B

A

(c)

F

D C G

H

B

(d)

THICK

(e)

THIN

d

D

(f)

Figure 19.5.2 Schematic diagram illustrating vacuum forming with a female mold. (a) Clamped, heated sheet being lowered onto a male female mold with vents (CDEFGH) that is mounted a sealed box. (b) Clamped, heated sheet in contact with the upper surface (CD and EF) of the mold. (c) The clamps are further lowered to seal the sheet against the surfaces EF and DC. (d) On evacuation of the mold, atmospheric pressure forces the sheet to contact the mold surface (CDEFGH). (e) Molded sheet removed after cooling. (f) Final part obtained after trimming excess portions.

last portions to contact the mold surface are stretched the most and are therefore the thinnest. The molded sheet is removed after cooling (Figure 19.5.1e). The final part is obtained by trimming the excess portions (Figure 19.5.1f). Note that because stretching causes the sheet to thin, in contrast to the part formed by using a male mold (Figure 19.5.1f) the part formed by a female mold will be thickest at its open end and thinnest at its bottom. Applications of vacuum forming include boat hulls, kayaks, and tuck instruments panels.

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19.5.2

Pressure Forming

In vacuum forming the sheet is forced against the mold by atmospheric pressure, which may not be sufficient for reproducing fine part surface details and texture. Such details and finer surface texture can be captured by applying higher forming pressures. Figure 19.5.3 schematically shows stages of pressure forming in a female mold. The mold assembly has two components – a vented mold mounted on a sealed box attached to a plate that is connected to a vacuum line, and a pressure box (Figure 19.5.3a). A clamped, heated sheet is lowered onto the mold surface, where it first comes in contact with the upper surface of the mold to which it adheres. The clamps are further lowered to seal the sheet against the upper mold surface (Figure 19.5.3b). Then the pressure box is lowered onto the sheet to form an airtight seal (Figure 19.5.3c). After which, a simultaneous evacuation of the mold and an application of air pressure on the top sheet surface forces the sheet against the female mold (Figure 19.5.1d). The molded sheet is removed after cooling (Figure 19.5.1e). The final part is obtained by trimming the excess portions (Figure 19.5.1f). Pressure forming can also be done on a male mold. Pressure forming is used in a wide variety of applications such as refrigerator liners, pallets, luggage, tool boxes, and business equipment housings. 19.5.3

Plug-Assisted Thermoforming

The discussions in the previous section point out one limitation of the thermoforming process: nonuniform part thickness. And the thickness distribution in a part depends on the type of mold used (compare Figures 19.5.1f and 19.5.2f). Several techniques have been developed to reduce extreme differences in part thickness variations. One such method is to use a plug to limit the thickness reduction; the principle for plug-assisted forming in a female mold is illustrated in Figure 19.5.4. A clamped, heated sheet is lowered onto the mold surface of a vented mold mounted on a sealed box attached to a plate that is connected to a vacuum line (Figure 19.5.4a). The clamped, heated sheet first comes in contact with the upper surface of the mold to which it adheres, and the plug is lowered to contact the upper sheet surface along AB (Figure 19.5.4b). Then the male plug pushes the sheet into the female mold cavity (Figure 19.5.4c). During this process the portion AB of the sheet adhering to the plug surface does not stretch. Then, a simultaneous evacuation of the mold, and an application of air pressure on the top sheet surface atmospheric pressure forces the sheet to contact the mold surface (Figure 19.5.4d). The molded sheet is removed after cooling (Figure 19.5.1e). The final part is obtained by trimming the excess portions (Figure 19.5.4f). Because the sheet portion AB in contact with the plug does not stretch during plug insertion, the sheet retains its original thickness at the end of insertion, so the only stretching in this portion of the sheet occurs only during the phase shown in Figure 19.5.4d. This is why plug-assisted forming results in less part thickness variation than in standard forming in a female mold. Figure 19.5.4 shows the steps in plug-assisted pressure forming. In plug-assisted vacuum forming external pressure is not applied and the part is formed by atmospheric pressure forcing the sheet against the evacuated mold surface.

Alternatives to Injection Molding

PRESSURE BOX HEATED SHEET

SEALED MOLD BOX

(a)

(b) PRESSURISED AIR

VACUUM

(c)

(d)

(e)

(f )

Figure 19.5.3 Schematic diagram illustrating pressure forming with a female mold. (a) Clamped, heated sheet being lowered onto a male female mold with vents that is mounted a sealed box. (b) Clamped, heated sheet sealed against the mold surfaces. (c) Pressure box seated on top sheet surface. (d) Sheet formed by simultaneous evacuation of the mold and application of air pressure on the top sheet surface. (e) Molded sheet removed after cooling. (f) Final part obtained after trimming excess portions.

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PLUG

A

FEMALE MOLD

B

(a)

(b) PRESSURISED AIR

A

B

VACUUM

(c)

(d)

(e)

(f)

Figure 19.5.4 Schematic diagram illustrating plug-assisted forming into a female mold. (a) Clamped, heated sheet and plug assembly being lowered onto a male female mold with vents that is mounted a sealed box. (b) Clamped, heated sheet sealed against the mold surfaces. (c) Descending plug forces sheet into female mold. (d) Sheet formed by simultaneous evacuation of the mold and application of air pressure on the top sheet surface. (e) Molded sheet removed after cooling. (f) Final part obtained after trimming excess portions.

Alternatives to Injection Molding

Figure 19.5.5 shows a large calf hutch made by plug-assisted thermoforming of 9.5-mm (0.375-in) thick polyethylene sheet. This large part has approximate dimensions of 2, 502 × 1, 537 × 1, 511-mm (98.5 × 60.5 × 59.5-in), and weighs about 42.7 kg (94 lbs). The figure shows (a) a photograph of the plug (top) and the thermoformed part (bottom of photo), and (b) a close-up view of the thermoformed part. Notice the use of fans to cool the plug and part.

(a)

(b)

Figure 19.5.5 Calf hutch made by plug-assisted thermoforming of 9.5-mm (0.375-in) thick polyethylene sheet. This large part has approximate dimensions of 2, 502 × 1, 537 × 1, 511 mm (98.5 × 60.5 × 59.5in), and weighs about 42.7 kg (94 lbs). (a) Photo shows the plug (top) and the thermoformed part (bottom of photo). (b) A close-up view of the thermoformed part. (Photos courtesy of Hampel.)

19.5.4

Twin-Sheet Forming

In twin-sheet forming hollow parts are made by simultaneously thermoforming two separately heated sheets in two mold halves; the sheets are joined together by the squeezing action at the junction of the two molds. The principle of twin-sheet molding is shown in the schematic diagram in Figure 19.5.6. The starting arrangement is shown in Figure 19.5.6a. Two separately heated sheets are clamped together with a spacer at one (right) end and a space with a hole – through which air can be injected between the sheets – at the other (left) end. Note that one (left) end of both the mold halves have recesses cut in to prevent the air hole from being closed off during the molding process. The sheets and mold-half assemblies are brought into contact (Figure 19.5.6b) and, after injecting low-pressure air to slightly inflate the sheets, the mold halves are forced together; the consequent pinching of the heated sheets causes them to bond together (Figure 19.5.6c). Notice that the (left-hand) recesses on both halves of the molds allow space to keep the air-injection hole intact. Then, simultaneous pressurization of the interior and evacuation of the exterior surfaces (Figure 19.5.6d) results in the part shown in Figure 19.5.6e. The final part (Figure 19.5.6f) is obtained after a trimming operation. Instead of forming both part halves simultaneously, each half can be formed separately and bonded together in a final pinching operation. While the illustration used in this section shows a large-volume part, twin-sheet forming can be used for forming relatively flat parts, the stiffness of which can be enhanced by molding in pinch-offs similar to tack-offs in blow molding and kiss-offs in rotational molding. The hollow parts produced by this process can be further stiffened by injecting polyurethane foam.

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SPACE FOR AIR CHANNEL

SPACER HEATED SHEET

HEATED SHEET AIR INLET

CLAMP

(a)

(b) VACUUM

PRESSURISED AIR

AIR

(c)

(d) VESTIGE OF AIR-INJECTION HOLE

(e)

(f)

Figure 19.5.6 Schematic diagram illustrating twin-sheet forming in female molds. (a) Two clamped, heated sheets and upper female mold assembly being lowered onto lower mold assembly. (b) Clamped, heated sheets sealed contact both mold surfaces. (c) Sheets sealed by pinching action of mold surfaces and sheets pushed apart by low air pressure. (d) Sheets formed by simultaneous evacuation of the mold and application of air pressure on the interior sheet surfaces. (e) Molded sheet removed after cooling. (f) Final part obtained after trimming excess portions.

Alternatives to Injection Molding

Applications of twin-sheet forming include bed liners for pickup trucks, roofs for agricultural tractor cabs, pallets, garage door panels, and walls and doors for sheds. Figure 19.5.7 shows a saddle bag – having approximate dimensions of 405 × 202 × 255 mm (16 × 8 × 10 in), with a weight of 1.25 kg (2.75 lbs) – made by twin-sheet forming 6.35-mm (0.25-in) thick HMWPE sheets. Notice how twin-sheet forming makes it possible to stiffen the hollow top and the hollow saddle body. Also observe how the top is connected to the saddle by an integral, molded-in plastic hinge.

(a)

(b)

Figure 19.5.7 Saddle bag made by twin-sheet forming 0.635-mm thick HMWPE sheets. The bag has approximate dimensions of 405 × 202 × 255 mm, and weighs 1.25 kg. (a) External view shows the closed saddle bag. (b) Photo with top open shows the hollow structure of the lid (top) and the bag. Note how the top is connected to the saddle by a molded-in hinge. Also note the smooth finish of all the surfaces. (Photos courtesy of Hampel.)

19.5.5

Advantages and Limitations

The main advantage of thermoforming processes is that they can be used for making large parts – such as boat hulls and tops for camper trucks – at relatively low costs. The low forming pressures result in low-cost molds and relatively stress-free parts. Thermoforming has two main disadvantages. First, it uses sheet materials – so that thermoforming is the second step in making parts – which adds to the material cost. And second, the part trimming required after part molding results in a significant wastage of the original sheet. 19.5.6

Part Stiffening

Thermoformed parts, especially those with panel-like sides, can be stiffened by molding in rib-like depressions. Two ways of stiffening box-like structures are schematically shown in Figure 19.5.8.

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X

X

(a)

(b) Y

Y

(c)

(d)

Figure 19.5.8 Schematic diagram illustrating stiffening of box-like thermoformed containers. (a) Bottom view of a box-like container stiffened by a depressed rib (hashed region). (b) Section on XX showing the ribs on the bottom surface. (c) Bottom view of a box-like container stiffened by a depressed flat surface (hashed region). (d) Section on YY showing the bottom panel stiffened by a depressed bottom surface (hashed region).

19.5.7

Mechanical Forming

Early in sheet applications, following steel sheet-stamping practice, attempts were made to stamp thermoplastic sheet using matched dies (see Figure 15.4.16). However, because of permanent deformation of thermoplastics requires very large strains, most of which are recovered on unloading, cold sheet stamping is not a viable process for these materials. However, heated thermoplastic sheets can be compression molded between matched molds. The principle of this process is schematically illustrated in Figure 19.5.9.

Alternatives to Injection Molding

CLAMPED, HEATED SHEET

(a)

(b)

(c)

(d)

(e)

(f)

Figure 19.5.9 Schematic diagram illustrating mechanical forming of heated sheet in matched molds. (a) Clamped, heated sheet and upper mold half being lowered onto lower mold. (b) Clamped, heated sheet sealed contacts both mold surfaces. (c) Sheet compression molded between matched molds. (d) Molds opened. (e) Molded sheet removed. (f) Final part obtained after trimming operation.

19.6 Expanded Bead and Extruded Foam There are two classes of thermoplastics foams that are widely used for nonstructural applications: Because of their low densities, expanded bead foams are widely used in packaging applications and for disposable food containers. They are also being used for filling shells of load-bearing parts such as automotive bumpers. The second class comprises extruded foam that is widely used as an insulating material in building and construction applications. 19.6.1

Expanded Bead Foam Molding

The expanded bead foam molding process is used for making light, low-density parts. This process starts with submillimeter sized thermoplastic beads (pellets), charged with dissolved hydrocarbons that act as

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blowing agents, which are shipped to part processors in sealed bags. These beads are first pre-expanded by heating by steam or hot air to obtain the desired reduced density. Parts are formed by filling pre-expanded beads into a clamped mold and then heating them – either through conduction or by directly flowing steam through the beads – resulting in the beads fusing together. Almost all the density reduction occurs during pre-expansion, thereby determining the density of the part; the density changes very little during the molding process. The first and most often used thermoplastic for expanded bead foam molding is polystyrene, for which the commonly used blowing agent used is i-pentane or n-pentane or c-pentane, the vaporization temperatures for which are 28, 35, and 49°C, respectively; of the three the most often used is n-pentane. This material is called expandable polystyrene (EPS). On heating for pre-expansion, the EPS beads begin to soften at the glass transition temperature (Tg = 85°C) at which the blowing agent begins to expand the beads; the bulk of the expansion occurs on further heating at temperatures in the range of 100 – 110°C. The end product of this process is a relatively rigid and tough, closed-cell foam. Because larger beads have relatively lower surface areas from which the blowing agent can permeate through, larger beads can be used to obtain lower pre-expanded densities. EPS is widely used for food cups, bowls, plates, and disposable trays. It is also used for carry-out food packaging, which includes hinged lid clam-shell containers. Custom-shaped EPS moldings are used for cushioning products inside boxes. And ubiquitous “peanuts” are used as fillers for cushioning items inside shipping containers. Rigid molded 2 × 8 f t or 4 × 8 f t bead-board (bead sheet panels) is used in building insulation applications. 19.6.2

Extruded Foam

Extruded polystyrene foam (XPS) is obtained by extruding molten polystyrene charged with a blowing agent through an extruder. The resulting sheet material has a more homogeneous closed-cell structure. Also, in contrast to bead-board, the outer surfaces are smoother. The largest application of XPS sheet is for building insulation. It is also used in crafts and for making models, especially architectural models.

19.7 3D Printing Plastic components tend to have complex shapes that are not easy to visualize. While computer generated models and advances in computer graphics made it possible to visualize the shape and looks of a design with increasing realism, making prototypes was a slow, tedious process. Then, a host of technologies to address this problem were developed. The first such prototyping techniques could only make parts suitable for evaluating the look, feel, and fit of a part. Next, techniques were developed for making parts that were strong enough for functional testing. The earliest of these processes was stereolithography (SLA), in which liquid photopolymers in a tank are cured, layer-by-layer, by a guided, computer-controlled, ultraviolet (UV) laser beam. In an alternative, selective laser sintering (SLS) process, a UV laser beam is used for a layer-by-layer sintering of a thermoplastic powder bed to obtain the desired part. Another form of 3D printing involves the deposition of photopolymer droplets from small-diameter nozzles, similar to inkjet printers for 2D printing on paper, which are then hardened (cured) by UV light. In the fused deposition modeling (FDM) method,

Alternatives to Injection Molding

thin beads of molten thermoplastic extruded in layers through fine nozzles harden immediately in air to form the part. These represent a few of the fast-emerging new technologies for using digital technology to produce parts. These developments have evolved into the new discipline of 3D printing in which a digital model of the object is used to build or “print” 3D solid objects in a layer-by-layer process. It involves several steps: First, a complete digital model of the 3D part is constructed using CAD software. This model is then digitally sliced into thin, flat layers. Finally, a 3D printer is used to sequentially lay down these thin flat layers of material to build the desired solid shape, based on the CAD model. These rapid prototyping techniques can be used to make prototype assemblies, or models, that can be used to evaluate the look and feel of parts, which is particularly important for plastic parts for which esthetics are important. They have now matured to a level at which they can be used to rapidly make actual parts; they are especially competitive for making smaller part runs. In summary, 3D printing technology has made it possible to quickly make models to firm up conceptual designs, prepare functional prototypes, craft jigs and fixtures for manufacturing, and make custom end-use parts. Figure 19.7.1a shows an aluminum fixture assembled from several parts that was replaced by a single, elegant and ergonomic, 3D printed part made of ABS plastic (Figure 19.7.1b). In addition to being cheaper, the turnaround time from concept to part is much shorter for the printed part.

(a)

(b)

Figure 19.7.1 (a) Metal fixture. (b) Fixture replaced by a single elegant, ergonomic 3D printed plastic part. (Photos courtesy of Stratasys Ltd.)

These technologies can also be used for making plastic molds that are suitable for injection and blow molding. Figure 19.7.2a shows two halves of a 3D printed plastic mold of Digital ABSTM with the molded part in place in the left half of the mold. The hand-held molds give an idea of their size. Figure 19.7.2b shows two halves of a Digital ABS 3D printed plastic mold and the resulting intricate polyethylene part. Of course, both the molded parts in these figures could have been printed, but producing them in printed molds makes it possible to rapidly make more parts.

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(a)

(b)

Figure 19.7.2 (a) 3D printed plastic mold with molded part in left mold half. (b) 3D printed plastic mold and intricate polyethylene molded part. (Photos courtesy of Stratasys Ltd.)

Figure 19.7.3 shows a one-step 3D printed human foot model in which different photopolymer resins were used for the bones and the encasing clear tissue. Such models are made from 3D images created from medical CAT scan or MR digital files.

Figure 19.7.3 Photo of a human foot model, produced from a CAT or MRI digital file. (Photo courtesy of Stratasys Ltd.)

19.8 Concluding Remarks This chapter has provided a high-level overview of part fabrication alternatives to injection molding. Simple schematic diagrams have been used to highlight basic principles; important engineering details have been glossed over. Even then, not all methods in this category have been covered. While computer simulation of deformation and flow has now evolved into a useful tool for mold and die design, the many advances that have made manufacture of incredibly complex plastic parts possible

Alternatives to Injection Molding

are based on major innovations in process and equipment design, and even more on innovations in mold and tool design. For example, multilayer plastic automotive tanks can now be blow molded with as many as 15 components – such as rollover valves, fuel-delivery modules, and level sensors – inside the tank. The molded tank then requires only one hole to insert the fuel-delivery module! Each of the processing methods in this chapter evolved from simple machines designed to make simple parts that could not easily be made by other methods. With the growing use of plastics, each of these technologies now has sophisticated computer-controlled machines that, in conjunction with innovative tooling, are used to efficiently make multimaterial parts – if necessary, with metal inserts – that can be decorated during the part forming process. The evolution of these technologies have made possible an unending number of everyday use items such as water bottles, food packaging, ketchup bottles with multifunctional lids, containers with handles, sophisticated toys, baby strollers, canoes and kayaks, many automotive components – such as foam-filled automotive bumpers, fenders, integrated bumper-grill assemblies, and complex lighting systems – and a host of medical devices, such as sophisticated one-time use syringes that not only accurately meter the amount of medicine injected, but also provide protection against accidental pricking by the injecting needle. The most recent of the technologies discussed in this chapter, 3D printing, has enormous potential. It is a “disruptive” technology in the sense that it will change the way small-run parts are made: Fast turnaround times from concept to CAD models to rapidly made printed prototypes that, after functional testing, can be made in larger runs in 3D printed plastic molds. This reduces both cost and time to market. It is part of the more general additive manufacturing revolution, in which 3D objects are made by adding materials, such as plastics, metals, ceramics, and even concrete, in layers.

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20 Fabrication Methods for Thermosets 20.1 Introduction The previous three chapters have discussed fabrication, or part processing, methods for thermoplastics, which have very high viscosities in the molten state; thermoplastics can be remelted and shaped into new parts, and are therefore easy to recycle. In contrast, thermosets have very low viscosities, and part fabrication involves chemical reactions resulting in solid parts that cannot be melted or reshaped other than by machining processes; they are difficult to recycle. Because of the chemical reactions, molding cycles for thermoset are significantly longer than for thermoplastics. These chemical reactions generate heat, and moisture that can result in voids. In contrast to thermoplastics – in which the only way the chemical structure of the plastics affects a part forming process is through temperature-dependent physical properties such as viscosity – in thermosets the very varied exothermic reaction chemistries – which may involve just one, or more reactive components – can have a much larger effect on forming processes that have to be tailored for each material class. A review of Chapter 13 will aid in understanding this chapter. Especially important for thermosets is an understanding of the gel point, beyond which the low-viscosity reactants rapidly react to form highly viscous liquids or solids. Thermoset part fabrication has several distinct forms. The oldest involves compression molding of filled powdered material. Chopped fibers impregnated with low-viscosity thermoset precursors, such as bulk molding compounds (BMCs), can be injection molded in a manner similar to that for thermoplastics. Chopped fibers impregnated with low-viscosity thermoset precursors can be sprayed onto molds and then cured. Low-viscosity resin can be injected into glass- or carbon-fiber woven preforms placed in a mold to produce highly filled, high-stiffness, and high-strength parts. And sheets of aligned fibers in tacky precursors, called prepregs, can be stacked and cured to produce lightweight, high-strength parts. The exothermic reactions during the curing process of thermosets produce gases and water vapor that can result in undesirable porosity in products. Such voids can be suppressed by maintaining high pressures during the curing process.

20.2 Gel Point and Curing During part forming, resins undergo exothermic chemical reactions resulting in continually growing crosslinked polymer networks; this phase is called curing. The gel point marks the conversion of low-viscosity reactants into a highly viscous liquid or solid. Although at this stage the growing polymer Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

Introduction to Plastics Engineering

networks in the reacting material have a low number-average molecular weight, the weight-average molecular weight is very high. Continuing polymerization during curing results in essentially all the material transforming into a collection of rigid, crosslinked networks. The differences in the chemistries of thermoset systems result in different curing rates. However, the curing processes are qualitatively similar; they will be illustrated by means of schematic viscosity-versus-time and mechanical-properties-versus-time curves for thermoset systems in which the initial reacting materials are low-viscosity liquids. Figure 20.2.1 schematically shows the change in viscosity of a thermosetting system, maintained at a constant temperature, as cross-linking reactions result in continuingly growing polymer networks. While the viscosity should increase monotonically with the increasing network size, this figure shows a decrease in the viscosity from point A to B. This decrease is caused by the increase in temperature from the heat released by the exothermic reaction; from point A to B the decrease in viscosity due to a temperature increase masks the viscosity increase caused by the increasing network size. From point B onwards, the effect of network size begins to dominate the viscosity, which at point C attains the original viscosity of the reactants at A. From here onwards the viscosity increases steadily till the gel point at time tGel (point D), after which the viscosity increases very rapidly along DE, resulting in a solid with “infinite viscosity.” E

INCREASING VISCOSITY

VISCOSITY

676

D

A

SOLID

C B

tGel TIME Figure 20.2.1 Isothermal viscosity variation with time for a thermosetting liquid. (Adapted with permission from “Handbook of Thermoset Plastics,” by H. Dodiuk, S.H. Goodman, 2013, Elsevier.)

The effect of the temperature of the reactants on the gel time is schematically shown in Figure 20.2.2. At lower temperatures the reaction rates slow down. Because of very slow reaction rates, gelation is yet to occur at the lowest temperature, T1 . This is also true at the higher temperature T2 . At still higher temperatures, higher reaction rates releasing larger amounts of heat cause faster curing, resulting in lower gel times. The dashed line in this figure shows that the viscosity does not change at temperature T0 ; at this temperature the thermoset components do not react and therefore can be stored at this or lower temperatures. Initiation of cross linking requires raising the temperature to a level appropriate for the system.

VISCOSITY

Fabrication Methods for Thermosets

INCREASING TEMPERATURE

T4 T3

T2 T1

T0

tGel3

tGel4

TIME Figure 20.2.2 Effect of temperature on the gel time for a thermosetting liquid. (Adapted with permission from “Handbook of Thermoset Plastics,” by H. Dodiuk , S.H. Goodman, 2013, Elsevier.)

VISCOSITY

The mass of the reactants also affects the gel time: Because of the poor thermal diffusivity of plastics, heat generated by the exothermic reactions is conducted away more slowly through larger masses. The consequent increase in local temperature then results in shorter gel times for larger masses. This effect is schematically shown in Figure 20.2.3, in which the masses m1 and m2 have not reached their gel points. And, as shown in this figure, a mass m0 or lower do not react at all.

m4

m3

INCREASING MASS

m2 m1 m0

tGel 4

tGel3 TIME

Figure 20.2.3 Effect of resin mass on the gel time for a thermosetting liquid. (Adapted with permission from “Handbook of Thermoset Plastics,” by H. Dodiuk, S.H. Goodman, 2013, Elsevier.)

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Figure 20.2.4 schematically shows the evolution of mechanical properties with the cure time. Before reaching the gel stage at time tGel the mechanical properties are very low – those corresponding to the low-viscosity reactants. Beyond the gel point the mechanical properties asymptotically increase with the cure time to the values for fully completed reactions. About 90% of the asymptotic value is obtained at the early part of the asymptotic plateau, which may be referred to as the cure time tCure .

MECHANICAL PROPERTIES

678

INCREASING MECHANICAL PROPERTIES

tGel

SOLID

tCure TIME

Figure 20.2.4 Variation of mechanical properties with time for a thermosetting liquid. (Adapted with permission from “Handbook of Thermoset Plastics,” by H. Dodiuk, S.H. Goodman, 2013, Elsevier.)

20.2.1

Shelf Life of Precursors

Although the previous section has discussed liquid thermosetting precursors, that discussion also applies to other reactants including solids. Figure 20.2.2 shows that at temperatures below T0 the thermosetting precursors do not react. This temperature sensitivity of thermosetting precursors is used for preparing and storing premixed components that can be stored in a single container. While theoretically this material can be stored indefinitely, loss of moisture or absorption of gases can have a significant effect on shelf life. The change in viscosity versus cure time variation shown in Figure 20.2.1 results from the ongoing polymerization of the precursors; besides depending on the concentrations of the reacting precursors, the shape of this curve also depends on the temperature and cure time. By allowing the cure to proceed to an intermediate level, well before the gel point, and then dropping the temperature to T0 , a storable material with a higher viscosity can be obtained. This process is called B-staging. Stored B-staged materials result in shorter cure times for part fabrication.

20.3 Compression Molding In the compression molding process for thermosets, schematically shown in Figure 20.3.1, a measured amount of granular molding compound, called the charge, which could be in a partially cured

Fabrication Methods for Thermosets

state, is first placed in the bottom half of a mold (Figure 20.3.1a). The top half of the mold is then lowered, during which the applied high pressure and temperature causes the granular material to soften and flow and contact all parts of the mold cavity, eventually resulting in complete mold closure; the material then cures under high pressure and temperature (Figure 20.3.1b) in the heated mold. The part is then demolded (Figure 20.3.1c), resulting in the clean part (Figure 20.3.1d) without a sprue. MOVABLE CORE

MOLDING COMPOUND

STATIONARY CAVITY

EJECTION PIN

(a)

(b)

(d)

(c) Figure 20.3.1 Steps in compression molding of thermosetting solids.

A sufficiently high-temperature has to be used to initiate and sustain the curing reaction. The initial molding compound charge can be preheated to reduce the cycle time. High pressures in the mold have to be maintained to prevent the vapor and gases generated from the curing reaction from forming bubbles and cavities. Typical granular materials used in compression molding include phenolics, melamine, urea-formaldehyde, and epoxy-based precursors (Sections 13.2.1 – 13.2.2.3). Thermosetting elastomers can also be compression molded.

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Because of the loose powder-like initial charge, the molds have to be in the vertical configurations shown in Figure 20.3.1. Instead of lowering the top half of the mold, the mold can be closed by raising the lower half. The absence of runners and gate systems in compression molding results in less material wastage, and the mechanical properties are more uniform than in parts made by injection molding. Also, the molds are simpler and less expensive. The mechanical properties of compression-molded parts are better than of identical parts made by transfer molding (Section 20.4) or thermoset injection molding (Section 20.5). The main disadvantage of compression molding is the longer cycle times required for resin curing. In addition to powders and pellets, the charge could be in the form of putty-like materials such as bulk- and sheet-molding compounds (Section 13.2.7) that are precursors of long-glass (>25 mm) filled polyester, vinyl ester, or epoxy resins. Among other applications, Sheet-molding compound (SMC) has been used for making large, relatively flat automotive parts such as hoods, fenders, and spoilers; in these large applications the SMC precursor is cut to conform to the mold geometry. Figure 20.3.2 shows a 137-cm (54-in) long, 154-kg (70-lb), compression-molded vinyl-ester-based SMC pickup box liner developed for the 2001 Ford Explorer Sport Trac. This first use of a one-piece composite liner for a pickup cargo box resulted in a weight saving of 110-kg (50-lb) over a steel liner that had to be assembled by welding over 20 pieces of steel stampings. Besides superior wear and scratch resistance, compression molding allows for better styling and esthetics. This technology is now used in many truck applications.

Figure 20.3.2 Compression-molded SMC composite pickup box. (Photo courtesy of SPE Automotive Division.)

20.3.1

Compression Molding of Thermoplastics

Compression molding is also used for molding parts from glass- or carbon-fiber filled thermoplastic sheets. Depending upon the types of fibers used, several types of such thermoplastic sheet material are available. They may be classified into those that use chopped fibers and those that use continuous fibers. Long, chopped fiber-filled based thermoplastic composite sheet (LFT) is made by using a wet slurry process adapted from paper making technology. Chopped fibers and polymer powder with suitable additives are mixed in water to form a slurry, which is pumped onto a vacuum filter belt that removes most

Fabrication Methods for Thermosets

of the water. The resulting intimately mixed fibers and polymer material is passed through a drier and then consolidated in a continuous double-belt press. This slurry process can be used with high-viscosity polymers and high fiber loadings (60 – 70% by weight). In another version, the thermoplastic impregnates a random, continuous-fiber glass mat; this material is called a glass mat thermoplastic (GMT). Randomly oriented long-glass-fiber mat impregnated with a molten thermoplastic – most commonly polypropylene – is inserted between continuously moving heated metal belts constrained to maintain a constant thickness. The pressure from the lateral constraint on the moving belt forces the molten thermoplastic – the viscosity of which is lowered by the heat transfer from the heated belt – into the interstitial space in the fiber mat. In the last part of the system the continuously moving belt is cooled resulting in a continuous thermoplastic impregnated sheet. This process produces sheets with thicknesses of about 4 mm, with 20 – 40 wt% of fibers. Appropriately cut blanks of such material placed in a compression mold flow laterally to fill the mold shape. The flow of the material in the mold causes the fibers in the glass mat to undergo preferential orientation. Plaques of different shapes can be stacked in a mold to produce parts of varying thickness. Compression molding of LFT or GMT can be used to produce large parts at relatively low cost. However, parts made in this way can have highly anisotropic properties, and can exhibit wide variations in local mechanical properties. Such wide variations in mechanical properties of GMTs are discussed in Chapter 24.

20.4 Transfer Molding Transfer molding is a modification of compression molding in which the measured, preheated molding compound charge is first placed in a cavity called the pot (Figure 20.4.1a). The charge in the pot is then forced (transferred) by a hydraulically driven ram into the heated mold cavity through a sprue (Figure 20.4.1b), where it is cured just as in compression molding. The part is then demolded by opening the mold and using an ejection pin to push the part out (Figures 20.4.1c – e). Because the mold is closed when the charge is forced into the mold cavity, transfer molding produces more precise parts with less flash along the parting line. And this process is better suited for parts with metal inserts that are easier to retain in a closed mold. The overall cycle times for transfer molding are shorter than for compression molding. However, because of the cull sprue (Figure 20.4.1c) and the sprue (Figure 20.4.1d) this process produces more scrap material that cannot be reused as in thermoplastics.

20.5 Injection Molding Aged BMC (Section 13.2.7) comes in the form of bulky short-glass-filled chunks, having a dough-like consistency that can be injection molded. This requires modified injection molding machines in which the metered BMC charge is either forced into the mold by a ram or by a modified screw. In contrast to thermoplastics – in which a very hot melt is injected into a mold – the BMC charge is rapidly injected at room temperature into a heated mold, in which the BMC cures to form a solid glass-filled part. Fiber breakage can be reduced by proper nozzle design. Figure 20.5.1 shows an automotive electronic throttle control module made by injection molding thermoset polyester-based BMC. The motor housing, gear box, and the gear pins for the plastic gears are

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TRANSFER RAM MOLDING COMPOUND

TRANSFER POT

CULL SPRUE

MOLD CAVITY

MOVABLE CORE EJECTION PIN

(a)

(c)

SPRUE

(d)

(e) PARTING SURFACE

(b) Figure 20.4.1 Steps in transfer molding of thermosets.

Fabrication Methods for Thermosets

Figure 20.5.1 Electronic throttle control module body made of injection-molded thermoset polyester-based BMC. (Photo courtesy of SPE Automotive Division.)

molded into the unit. A snap-on PBT cover eliminated the need for fasteners. First introduced in 2007, this precision molded housing resulted in a 28% weight reduction and an 18% cost reduction compared to the aluminum housing it replaced. The use of plastics also reduced ice freeze-up. Molding made it possible to incorporate a stop feature to zero the blade and to handle multiple spring-loaded impacts. 20.5.1

Injection-Compression Molding

In injection-compression molding the BMC charge is first injected into a partially closed mold, after which the part is compression molded into its final shape. The main advantage of this process is that, in contrast to compression molding in which the charge has to be cut from SMC sheets and placed in an open mold, the charge is directly injected into the mold, thereby reducing cycle time.

20.6 Reaction Injection Molding (RIM) In reaction injection molding, known as RIM, two low-viscosity polymer precursors are mixed together and then injected at high pressure through an impinging mixture into a closed mold, where the precursors react through an exothermic reaction, finally resulting in a cured thermoset material. Polyurethane reaction injection molding (PU-RIM) (Section 13.2.8) is the most widely used material in RIM processing. Depending on the chemistry of the precursors, this process can be used for molding flexible foam parts, sandwich structures with solid outer skins with foam cores, and even solid parts. This is one of the processes used for making automotive bumpers, air spoilers, and fenders. Figure 20.6.1a shows a Siemens Edge CT Scanner, in which most of the external, appearance parts are made of plastic. The large, annular fascia, shown in Figure 20.6.1b, is PU-RIM part that has a diameter of about 180 cm (6 ft) and weighs about 20 kg (45 lb).

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(a)

(b)

Figure 20.6.1 (a) External view of Siemens Edge CT Scanner. Most of the external casing is made of plastic. (b) Large front fascia made or PUR RIM. (Photos courtesy of Siemens.)

20.6.1

Reinforced Reaction Injection Molding (RRIM)

RIM parts can be stiffened by using fillers such as glass fibers or mica flakes, in which case this process is referred to as RRIM; one use for this process is for molding rigid-foam automotive panels. In comparison to the injection molding of thermoplastics – which have very high melt viscosities – the very low viscosity of the thermosetting precursors results in low pressures inside the mold, which therefore require much lower clamping forces, resulting in lower tooling costs. The low molding pressures make it possible to mold very large low-cost parts. The possibility of producing large, paintable parts having high-density skins surrounding low-density cores make RIM and RRIM attractive for many automotive and office equipment applications. But, as with all thermoset processing methods, the molding cycle times are longer than for injection molding of thermoplastics. A good example of a RRIM application is the replacement of a cast aluminum housing in a wastewater pump by PUR RRIM. The cutaway view of the original cast aluminum housing, together with the direction of waste flow, is shown in Figure 20.6.2a. The multi-part aluminum assembly and the pump and check valve were replaced by 11 PUR RRIM parts (Figure 20.6.2b). Another component, the lift core, was replaced by a low-density integral hard foam part. The smallest and largest PUR parts were 74 g and 13.5 kg, respectively. At 28.5 kg, the total weight of the 12 parts was about half that of the original aluminum assembly. And, at about 42 kg, the weight of the entire PUR-based system including the pump motor was much less than about 78 kg for the original aluminum-based system. Assembly cost reduction was achieved by molding in metal threaded inserts for components that needed to be opened,

Fabrication Methods for Thermosets

and cores were molded-in for self-threading screws for permanent connections; adhesives were used to seal chambers.

(a)

(b)

Figure 20.6.2 STRATE AWALIFT wastewater pump. (a) Cutaway diagram showing complexity of the original aluminum housing and the direction of waste flow. (b) External view of the PUR RRIM assembly. (Photos courtesy of Talis UK.)

20.6.2

Structural Reaction Injection Molding (SRIM)

SRIM makes use of the low viscosities of thermosetting precursors to mold rigid parts with very high levels of reinforcements. In this process the mixed precursors are forced through a fiber mesh placed in the mold. Depending on the performance requirements, prior to placing in the mold, the fiber preform could be shaped to provide high levels of reinforcements in three-dimensional parts.

20.7 Open Mold Forming In open mold forming a one-sided mold surface is used to form fiber-filled thermoset parts requiring only one finished surface. In principle, the cosmetic surface is formed by direct contact of the resin with the finished mold surface, after which the fiber-resin surface is applied behind this layer. This low-cost, labor-intensive process can be used for making large fiber-reinforced parts such as boat hulls and decks, truck cabs and fenders, bathtubs, and shower stalls. The actual forming process proceeds outward from the cosmetic surface. For facilitating release of the cured part, the mold surface is coated with a mold release agent. For obtaining a high-quality surface finish on the part surface, a gel coat, comprising a pigmented resin is then sprayed on. This is followed by the application of a barrier coating for reducing cracking in the gel coat. A thin layer of the reinforcing fibers is then applied. Finally, the bulk of the fiber reinforcement – which can be fibers or fiber mats – is applied in layers intermixed with the resin. The applied material is then compacted by means of rollers and the part cured. The different steps in the process are illustrated schematically in Figure 20.7.1. The simplest version of this process uses a manual hand lay-up of the resin and fibers. In the semi-automatic version of the process, the resin and the chopped fibers are sprayed on. In both versions the resin-mat surface has to be compacted with squeegees and rollers to reduce voids.

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MOLD RELEASE

GEL COAT

MOLD

(b)

(a) FIBER-RESIN MAT

FINISHED SURFACE

(c)

(d) Figure 20.7.1 Steps in open mold molding forming of concave fiber-filled thermoset parts.

Figure 20.7.1 shows the fabrication of a part having a concave finished surface. A convex finished surface can be formed by using a concave open mold, as schematically shown in Figure 20.7.2. Prior to curing, a foam or balsa-wood core can be attached to the outer fiber surfaces, after which the outer surfaces of the core are covered with layers of fiber and resin. Curing then results in thick parts, having thin fiber skins imbedding cores parts with thin fiber skins, with high bending and torsional stiffnesses.

20.8 Fabrication of Advanced Composites Advanced composites refer to high-performance polymer composites, mainly comprising highly filled, continuous aligned-fiber composites used in the aircraft and aerospace industries (Chapter 25). High-performance can be obtained by using glass fibers imbedded in epoxy matrices (Section 13.2.6). Still higher stiffness and strength are obtained by using carbon fibers embedded in thermoset matrices that for high-temperature applications can be bismaleimides (Section 13.3.2).

Fabrication Methods for Thermosets

GEL COAT

MOLD RELEASE

MOLD

MOLD

(a)

(b) FINISHED SURFACE

FIBER-RESIN MAT

MOLD

(c) MOLD

(d) Figure 20.7.2 Steps in open mold molding forming of convex fiber-filled thermoset parts.

Depending on part geometry, there are four main processes for fabricating advance composites: Pultrusion for straight structures with a uniform cross section; filament winding for cylindrical structures; vacuum bag consolidation for relatively flat, laminated structures; and vacuum-assisted liquid-resin transfer molding for very large, complex structures. In each of these processes, the reinforcing material starts as a single glass or carbon fiber, or as a polymeric filament. Because of their fragility, fibers are supplied in bundles called strands. These fibers are used in composites as untwisted collections of parallel strands (assembled roving) or parallel continuous filaments (direct roving); they are generically referred to as rovings. A large number of filaments gathered together without twist is called a tow. Rovings and tows are either used directly or in the form of woven or engineered stitched fabric.

20.8.1

Pultrusion

In pultrusion – from pull and extrusion, in which material is pulled through a die – fibers impregnated with a low-viscosity thermoset resin are pulled through a heated (extrusion) shaping die, where the resin cures, thereby causing the fibers to set into the cross-sectional shape of the die. By choosing an appropriate die, continuous composite parts with fixed complex open- or closed-sections can be made; the continuous profiles can then be cut to required sizes.

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This process is schematically shown in Figure 20.8.1. Fibers from a creel – fibers wound on bobbins to facilitate unwinding and arranging fiber tows – are fed into guides that arrange them for downstream use in the pultrusion system. While these fibers provide the tensile modulus and strength of the pultruded part, the transverse, or lateral, properties are essentially limited by those of the resin. To substantially improve transverse properties, the upper and lower surfaces of continuous tow collection are capped by continuous strand mats, in which randomly oriented continuous fibers result in near isotropic properties. The fiber and strand-mat combination arranged in the guides then goes through a resin bath for impregnating the fibers with resin; excess resin is removed as the coated system exits the resin bath. For improving the surface texture, the coated system can be covered by thin surface veils – thin, synthetic non-woven monofilaments mats – that result in smooth resin-rich surface. The veil, impregnated-fiber, and mat combination is fed through a preformer that removes excess resin from the veil surface. The material is then pulled by a caterpillar-like system through a heated forming die in which the thermoset resin cures and sets the system into its final form. The continuous profile exiting the heated die is cooled and then cut to required lengths. FIBER CREEL CONTINUOUS STRAND MAT SURFACING VEIL

PULLING SYSTEM

GUIDES PREFORMER

RESIN BATH

CUT-OFF SAW

HEATED FORMING AND CURING DIE

CONTINUOUS GLASS FIBER

Figure 20.8.1 Schematic diagram showing the generic layout for a pultrusion line.

The photo in Figure 20.8.2 shows the function and complexity of the guides used for organizing fiber tows and continuous strand mats. 20.8.2

Filament Winding

Filament winding is an automated, continuous process in which fiber is wound around a piece of tooling, called a mandrel, which can be rotated back and forth in a controlled manner; the fiber is fed from a continuous-fiber dispensing head that moves along the mandrel axis (Figure 20.8.3). The rotary motion of the mandrel and the axial motion of the fiber-dispensing head are computer controlled to obtain desired

Fabrication Methods for Thermosets

GLASS MAT ROLL GLASS MAT ROLL

GUIDE

GLASS ROVING

GLASS MAT

VEIL STITCHED TO E-GLASS MAT

Figure 20.8.2 Glass rovings and glass mat moving through guides. (Photo courtesy of Creative Pultrusions, Inc.)

fiber orientation for particular applications. In wet winding the fibers go through a resin bath before winding on the mandrel. The wound mandrel is then placed in an oven where the resin is cured. After cooling the mandrel is removed. This process is often aided by using release agents, extraction equipment, or simply by designing the mandrel to collapse on itself. Such filament wound parts exhibit very high hoop strength. Computer control allows the filaments to be wound at controlled angles to the mandrel axis, with different layers wound at different angles: High winding angles – close to circular winding – result in high hoop stiffness and strength. Low angles – which align the fibers closer to the mandrel axis – will result in parts with higher axial stiffness and strength. Also, control of fiber tension during the winding process can be used to obtain desired part attributes: lower tension results in more flexible parts; higher tension results in more rigid, higher strength parts. A variation of this process uses towpreg or prepreg – continuous fiber pre-impregnated with partially cured resin – thereby eliminating the need for a resin bath. The use of prepreg tape enhances the productivity and versatility of this process. In still another variation, called dry winding, the fibers are wound without resin; the dry fiber structure is then used as a preform for the RTM process. While circular-sectioned mandrels are used for making cylindrical parts such as tubes, this process can be used for making symmetrical parts with rectangular cross sections by using mandrels with appropriate cross sections. Even more complex asymmetrical shaped parts can be made by using mandrels in which the cross section changes along the mandrel axis. The final part parameters (geometry) of the part are used by special software, which controls the machine winding process, to generate the winding pattern. Winding simulations can be used ensure that the machine can wind the part.

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INCOMING FIBERS

RESIN BATH GUIDE RAIL

GUIDE RING

ROTATING MANDREL

(a) FIBER CREEL

GUIDES (COMBS)

MOVING CARRIAGE

RESIN BATH

NIP ROLLERS

FIBER TOWS

GUIDE RING

ROTATING MANDREL

(b) Figure 20.8.3 Schematic diagram for the filament winding process. (a) Isometric view showing the motions of the rotating mandrel and the fiber-dispensing head. (b) Top view showing more details of the fiber-dispensing head mounted on a reciprocating moving carriage.

Fabrication Methods for Thermosets

The photos in Figure 20.8.4 show different stages in the winding of a tube with a length of 1,219 mm (48 in) and a diameter 203 mm (8 in).

(a)

(c)

(b)

(d)

Figure 20.8.4 Four stages in the filament winding of a 1219-mm (48-in) long tube having a diameter of 203 mm (8 in). (Photos courtesy of McClean Anderson).

The photos in Figure 20.8.5 show different stages in the winding of a fiberglass bottle/tank for use in LPG tanks or pressurized water tanks. It has a length of 46 cm (18 in) and a diameter of about 15 cm (10 in). The process starts with a blow-molded plastic tank mounted on a mandrel attached to a lathe chuck (Figure 20.8.5a); a center attached to a second chuck then secures the free end. The winding process is more complex than for a cylinder (Figure 20.8.4) because the fibers have to be wound around the hemispherical ends. Successive steps of the winding process are shown in Figure 20.8.5 parts b – h. Note that the photos in Figure 20.8.5 parts e and h show views from the side opposite to that in Figure 20.8.5a – notice how the blow-molded tank is attached to the headstock chuck through a stub shaft. The other photos show progress of the winding as observed from the mandrel corresponding to the free side shown in Figure 20.8.5a. Figure 20.8.5i shows the wound tank ready to be cured. In this example, the blow-molded liner, which served as the mandrel, becomes part of the finished product.

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(a)

(b)

(d)

(g)

(e)

(h)

(c)

(f)

(i)

Figure 20.8.5 Stages in the filament winding of a 46-cm (18-in) long bottle having a diameter of 25 mm (10 in). (Photos courtesy of McClean Anderson).

20.8.3

Laminated Composites

Relatively flat high-performance structures are made by laminating plies comprising reinforcing unidirectional fibers or fabrics that, in most applications, are embedded in thermosetting resin matrices. Because many advanced composites operate at high temperatures, the processing methods have to be capable of handling high-temperature resins. And because such composites are used in critical applications – such as in aircraft – the forming processes have to have stringent controls for achieving defect-free uniform parts.

Fabrication Methods for Thermosets

20.8.3.1 Prepregs

While the final performance of an advanced composite depends on the fiber and resin used, the main starting materials for advanced composites are prepregs (from pre-impregnated), reinforcing materials – unidirectional fibers or woven fabric – pre-impregnated with resin. The prepregs can be B-staged (Section 20.2.1) and stored at reduced temperature. Prepregs are mainly produced in two ways. In the hot-melt process – which is applicable to both unidirectional fiber and fabric prepregs – a paper substrate is first coated with a thin layer of heated resin. After placing the reinforcement on this surface resin impregnation is affected in an impregnation machine that applies heat and pressure from nip rollers. Then, after applying a release film on the surface, the prepreg is wound onto a core. In the solvent dip process – that only works with fabric prepregs – the fabric is dipped in a bath of resin dissolved in a solvent. The final prepreg is obtained by drying the solvent in a drying oven. A single prepreg layer is called a ply. A composite part may be built up from several plies with different orientations to achieve desired stiffness and strength; this combination of plies is called a lay-up. Manufacturing parts from prepregs and lay-ups requires pressures to consolidate the laminate and heat to initiate and complete the cure cycle for the resin. 20.8.3.2 Vacuum Bag Consolidation

Very large parts with complex two-dimensional curvatures are fabricated by using vacuum bags to surround the lay-up placed in a one-sided mold: This process uses a flexible bag, the evacuation of which imposes a one-atmosphere pressure on the lay-up in the mold. In addition to helping to consolidate the part – by acting like a matched mold – the vacuum helps in removing air bubbles and the volatiles generated during the curing process. Autoclaves – large, heated pressure vessels – are used to apply higher pressures (0.3 – 0.7 MPa, 45 – 100 psi) and high temperatures to the vacuum bagged assembly. The higher pressures result in better quality laminated parts with fewer, smaller voids. The mold surface on which the prepreg lay-up is placed is first coated with a release agent, and the open surface of the lay-up is covered by a release film that can be solid or perforated to allow for removal of gases and small amounts of resin. This film is covered by a breather layer that, by allowing air flow under the vacuum bag, facilitates a uniform vacuum over the release film. In some cases, the vacuum bag is formed by applying a bagging film over the assembly and this film is then sealed at the mold surface by means of sealant tape. 20.8.3.3 Compression Molding

In this process the prepreg lay-up is placed in a matched mold and compression molded (Section 20.3). 20.8.3.4 Pressure Bag Molding

In this process, which is suitable for simple hollow sections, such as tubes, the lay-up with a flexible bag inside it is placed inside a hollow mold. During the cure cycle the flexible bag is inflated to apply consolidation pressure on the lay-up.

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20.8.3.5

Liquid-Resin Transfer Molding

One way of making composite parts with very high reinforcing levels is to place a reinforcing preform inside a mold, and to then inject a low-viscosity liquid resin under pressure to help it fill out the interstitial pores in the preform. The resin saturated preform can then be cured at room temperature or in a high-temperature autoclave. This process has two drawbacks: large parts require very high pressures for proper infusion of the resin into the pores, and residual bubbles of air result in porosity that affects the mechanical performance of the part. A host of processes for mitigating these limitations of the resin infusion process have been developed. In one such resin-transfer molding process, SCRIMPTM (Seemann Composites Resin Infusion Molding Process), originally developed for naval applications, resin infusion is assisted by applying vacuum to preforms placed in a sealed mold. In essence, the idea is to help resin infusion by evacuating the preform in the mold by sealing it with a flexible bag that can be evacuated, and by then allowing the vacuum to suck in the resin through the reinforcing fibers. The distance that the resin has to cover during infusion can be reduced by having multiple resin injection ports. The basic principle underlying this process is illustrated in Figure 20.8.6. The process starts with a one-sided mold, like in Figure 20.7.2, in which the mold surface corresponds to the finished appearance surface of the part (Figure 20.8.6a), the surface of which is treated with an anti-stick release agent. The fiber reinforcement is placed on the mold surface, with different parts of the mold being covered different number of layers to provide the desired part thickness distribution (Figure 20.8.6b). The mold is then covered with a flexible bag and the ends are sealed at the mold-bag surface with tapes (Figure 20.8.6c), and the space between the inner mold surface and the bag, which has the fiber reinforcing preform, is the evacuated by means of a vacuum pump (Figure 20.8.6d). This not only evacuates the air but also helps in consolidating the reinforcing fibers. After vacuum has been achieved, liquid resin containing a catalyst is sucked in through a port in the bag surface (Figure 20.8.6 parts e and f); the resin flows radially within the evacuated reinforcing fibers due to the one-atmosphere difference between the free surface of the open resin reservoir and the fiber reinforcement. The circular flow fronts corresponding to Figure 20.8.6 parts e and f are shown, respectively, in the plan views in Figure 20.8.6g and h. For large parts circular flow fronts unnecessarily increase the resin flow distance for infusing the entire part. Instead, a perforated tube below the bag surface can be used to create straight flow fronts, as shown in Figure 20.8.7a,b. Multiple resin injection lines can be used create several flow fronts to shorten the flow distance for the resin during infusion (Figure 20.8.7c,d). In the simple schematics in Figure 20.8.6 the fiber core is shown as single entity. In actual practice, as shown in Figure 20.8.8, the resin infusion process works with complex reinforcement systems that can include foam or balsa-wood cores. A porous fabric layer is placed just below the vacuum bag to facilitate even flow of the resin over the lay-up. Although all the layers shown are of uniform thickness, the fiber/fabric and core layers can vary in thickness to suit the geometric and mechanical requirements for the component. The original SCRIMP process aimed at naval applications has spawned several similar processes optimized for different types of products. Such processes can produce very large parts – such as hulls for boats/ships, thick decks, sailboats, railcar bodies, large shipping containers, bridge decks, and wind turbine blades – in thicknesses varying from a 3.2 mm (0.125 in) to 150 mm (6 in) with high fiber loadings of about 60 – 75 wt% (50 – 65 vol%).

Fabrication Methods for Thermosets

MOLD

FIBER REINFORCEMENT

SURFACE COATED WITH RELEASE AGENT

(b)

(a) VACUUM VACUUM BAG

(c) VACUUM

(d) CATALYZED LIQUID RESIN

VACUUM

(e)

RESIN

(f) RESIN FLOW FRONT VACUUM BAG

VACUUM LINE

(g)

(h)

Figure 20.8.6 Principle of vacuum-assisted liquid-resin-transfer molding process. (a) Open mold surface treated with an anti-stick release agent. (b) Reinforcing fiber system preformed on mold surface. (c) Fiber preform covered with flexible bag with edges sealed on the mold surface. (d) Space between sealing bag and mold surface evacuated by means of vacuum pump. (e, f) After evacuation, liquid resin with premixed catalyst sucked into the reinforcing fibers. (g, h) Plan view showing radially advancing resin flow front inside fiber reinforcement.

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CATALYZED LIQUID RESIN

PERFORATED RESIN INFUSION TUBE

RESIN

RESIN FLOW FRONT

VACUUM BAG VACUUM LINE

(a)

(b)

(c)

(d)

Figure 20.8.7 Straight flow fronts. (a, b) Linear flow front obtained by using a perforated linear resin injection line. (c, d) Two linear flow front obtained by using two perforated linear resin injection lines. VACUUM BAG CORE

LAYER TO FACILITATE RESIN FLOW FIBER/FABRIC LAYUPS MOLD

Figure 20.8.8 Schematic diagram showing different layers in the liquid-resin infusion process for fabricating a composite.

Although the prepreg-autoclave route produces the highest quality wind turbine blades, significant cost reduction has been achieved by using vacuum-assisted resin infusion molding methods. A good example is 88.4-m (290-ft) long composite blade shown in Figure 20.8.9a – currently the world’s longest wind turbine blade – three of which are used in each 8-MW wind turbine. Its size can be gauged from the heights of the men standing next to the central support. This blade is made from glass and

Fabrication Methods for Thermosets

(a)

(b)

(c)

(d)

Figure 20.8.9 Fabrication of very large composite wind turbine blades. (a) Photo of the world’s largest wind turbine blade inside manufacturing bay. (b) Fabrics being placed in a mold. (c) Machine laying fabric in the mold. (d) Two-sided mold for gluing halves together. (Photos courtesy of LM Wind Power.)

polyester fiber fabrics infused with polyester resins. It is molded in two halves using vacuum-assisted resin infusion. Figure 20.8.9b shows fabrics being placed in a very large mold, and Figure 20.8.9c shows a machine laying fabric in the mold. The two molded halves are glued together in a two-sided mold (Figure 20.8.9d). 20.8.3.6 Sandwich Structures with Prepreg Skins

Light, stiff structures can be made by bonding high-stiffness laminates – such as those made with carbon fiber – to lightweight honeycomb cores (Figure 20.8.10), which for high-temperature applications is made of Nomex® (Section 11.7.7). For less demanding applications cores made of plastic foam or balsa wood can be used. The process for making a honeycomb sandwich structure is schematically shown in Figure 20.8.11. An adhesive film is generally used to bond cured laminates to cores. For very large parts, a paste that is cure-compatible with the laminate is used to splice core sections and to fill tight corners and edges.

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Figure 20.8.10 Top and front view of a honeycomb core. PREPREG SKIN ADHESIVE FILM

HONEYCOMB CORE

ADHESIVE FILM PREPREG SKIN

(a)

(b) Figure 20.8.11 Process for making a honeycomb sandwich structure.

20.9 Fabrication of Rubber Parts Several types of thermoset elastomers, also called rubbers, are described in Section 13.4. The largest application of rubbers is for making tires and tire inner tubes, for which the two most widely used are carbon-black filled polyisoprene (natural rubber) (Section 13.4.1.1) and poly(styrene-co-butadiene) (SBR rubber) (Section 13.4.1.5). Sulfur is the main chemical used for the cross-linking process, called vulcanization in the rubber industry. Activators (zinc oxide and stearic acid) and accelerators (mercaptobenzthiazole) are used to

Fabrication Methods for Thermosets

reduce the curing time, which with the sole use of sulfur can be on the order of 8 hours. The use of these agents shortens the sulfur cross links to a few atoms, resulting in better thermal stability. Rubber part processing can be divided into to two categories: The first uses solid rubber and the second uses rubber in a liquid form. 20.9.1

Rubber Compounding

In contrast to plastics – in which the material input for part fabrication processes is in the form of pellets and sheets or liquid resins – natural rubber has traditionally been supplied in the form of sheets folded into large bales. Because rubber processing lines were designed for such large bales, synthetic rubbers also tend to be supplied in bales. Like plastics, rubbers are compounded with many additives such as fillers for either reducing cost or for improving mechanical properties, antioxidants, antiozonants, thermal and ultraviolet (UV), stabilizers, plasticizers, and mold release agents. By far, the most commonly used filler is a colloidal form carbon, called carbon black that, in addition increasing the tensile strength and abrasion and tear resistance, also provides UV protection; it also accounts for the characteristic black color of most rubbers. China clays are used with pigments when colors other than black are required. Before mixing the additives, the rubber feedstock is cut into smaller pieces. And natural rubber may have to be masticated, or kneaded, in mixers to reduce its high viscosity by heat and shear. To obtain a uniform dispersion of the additives, during compounding the rubber and the additives are mixed in internal mixers, such as two-roll mills or Banbury mixers – in which the rubber is sheared between internal counter rotating rollers and the casing wall. Because of the high viscosities of the base rubbers, the shearing and stretching in the mixers can raise the temperature to levels at which curing agents could result in premature vulcanization. The compounding is therefore carried out in two stages: In Stage 1, only the non-vulcanizing additives are mixed with the rubber. The resulting mixture is called a masterbatch. In Stage 2 the masterbatch is mixed with the vulcanizing additives. Parts can then be made of this material by shaping and subsequent vulcanization. Several fabrication techniques for rubber parts are considered in the sequel. 20.9.2

Dry Rubber Part Fabrication

In these processes, solid compounded rubber is used for making most load-bearing parts, including tires, conveyor belts, and pressure hoses. 20.9.2.1 Molding Processes

Just as other thermosets, compounded rubber can be molded by using compression, transfer, and injection molding. Of course, the pressures required for shaping the material in the mold and the cure (vulcanization) temperatures may be quite different. The elasticity of rubbers makes demolding easier than for the more rigid thermoplastics and thermosets: Rubber parts do not require drafts and undercuts can be more easily molded without slides. 20.9.2.2 Extrusion

Most rubber extrusion is done by using screw extrusion. To prevent vulcanization in the extruder, they tend to be shorter than those used for thermoplastics. Die design has to account for the die swell that rubber exhibits during extrusion. The rubber in an extruded part is uncured, and has to be vulcanized in a separate operation.

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20.9.2.3

Calendering

Calendering, schematically shown in Figure 20.9.1, is used for converting compounded rubber into uncured sheet. In this process the rubber is forced through a series of decreasing gaps between a series of rotating rollers. The last roll gap determines the sheet thickness. Better sheet thickness control can be achieved by finishing an extruded sheet by a calendering roll. COMPOUNDED RUBBER

•

•

ROTATING CALENDERING ROLLS

•

UNCURED SHEET

• • Figure 20.9.1 Calendering process for making uncured rubber sheet.

20.9.2.4

Reinforced and Coated Rubber Sheet

In several demanding applications, such as tires and conveyor belts, the products are built up from rubber sheets reinforced with filaments or fabric; each such reinforced sheet is called a ply. Such plies are made by a calendering process, schematically shown in Figure 20.9.2, in which rubber is first calendered into a sheet. Then, in the last calendering step, the fibers or fabric are fed into the roller gap, where pressure forces the rubber into intimate contact with the reinforcing material. This process is also used for making waterproof cloth for outdoor applications. 20.9.3

Wet Rubber Part Fabrication

Wet rubber forming techniques use liquid rubber for dip-molding thin rubber parts and for dip-coating parts with a layer of rubber or plastisol (Section 11.5.1.1).

Fabrication Methods for Thermosets

COMPOUNDED RUBBER

•

•

FIBERS OR FABRIC

•

•

ROTATING CALENDERING ROLLS

RUBBER IMPREGNATED FIBERS OR FABRIC

• •

Figure 20.9.2 Calendering process for making uncured rubber sheet reinforced with fibers or fabric.

20.9.3.1 Dip Molding

The dip-molding process for rubber is schematically shown in Figure 20.9.3. A cleaned metal or ceramic male mold, called a mandrel, heated in an oven (Figure 20.9.3a), is dipped into liquid rubber bath containing a vulcanizing agent (Figure 20.9.3b), where rubber in contact with the hot mandrel surface begins to dry. On withdrawing from the bath, the mandrel, which has a thin layer of rubber coating on its surface, is placed in an oven where the rubber cures, taking on the shape of the mandrel surface (Figure 20.9.3c). The mandrel is then dipped in a cold water bath to cool the cured part (Figure 20.9.1d), after which the mandrel is withdrawn from the bath, and the part is stripped from the mandrel (Figure 20.9.3e). The elasticity of the rubber allows parts to easily be stripped (removed), even from complex-shaped mandrels. The thickness of the part is controlled by the mandrel temperature and the dip time: The hotter the mandrel and the longer the dip time, the thicker the coating will be. This process is used with a large variety of thermosetting elastomers such as natural latex rubber, neoprene, nitrile rubber, and silicones, in many colors, textures, and hardness. This process is also used for molding thermoplastic poly(vinyl chloride) (PVC) parts using plastisol (Section 11.5.1.1). Examples of dip molded products are plastic bags, and bicycle handlebar grips. Two very large applications of this process are for making natural latex condoms and surgical gloves, the following detailed description of the process of making which gives an insight into the complexities of the process. Just in the medical field alone, the annual world-wide consumption of rubber gloves is about 100 billion. They are made in large automated factories that produce about 50,000 latex gloves per hour. A

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OVEN

LIQUID RUBBER

(a)

(b) COLD WATER

(d)

MANDREL

(c)

PART STRIPPED FROM MANDREL

702

(e)

(f)

Figure 20.9.3 Dip-molding process for liquid molding of rubbers. (a) Mandrel heated in oven. (b) Heated mandrel dipped in liquid rubber. (c) Mandrel with rubber film heated in oven; rubber vulcanizes. (d) Mandrel with vulcanized rubber part cooled in water bath. (e) Rubber part stripped from mandrel. (f) Finished rubber part.

very large number of aluminum or ceramic hand-shaped mandrels are attached to conveyor belts that cycle them through different cleaning, dipping, curing, and stripping stations. First, the mandrels pass through aqueous soap solution, followed by a dip through a bleach station to remove any material on the mandrels left from the previous run. Then, after passing through a hot-water tank, the mandrels pass through a chemical bath to be coated with a layer to which the liquid rubber will stick. This is followed by dipping in the latex rubber solution compounded with vulcanizing agents. After dipping the mandrels

Fabrication Methods for Thermosets

are spun to remove drops, and then moved to the baking oven in which the latex is vulcanized into solid rubber. After the baking step, the hands are washed and dried, and then spun through brushes to roll up cuffs on the gloves to facilitate the extraction of the gloves from the mandrel. Latex gloves can easily be sucked off the mandrels with the aid of low-pressure air assist. However, synthetic rubber gloves are stickier and have to be removed manually. Finally, the gloves are tested for holes by air inflation; some gloves are filled with water to check for leaks. This process has several advantages: (i) It has short lead times, and can be used for making both small and very large number of parts. (ii) Both glossy and matt surface finishes can be made. (iii) Color changes are easily made. (iv) Various levels of elasticity and surface hardness can be obtained by proper selection and compounding of rubber. And (v), the (male) molds are inexpensive and last long because they are not subject to stresses. The main disadvantage is that the thickness distribution over the part cannot be controlled. It is essentially a uniform part thickness process. And even this thickness cannot be accurately controlled as, for example, in injection molding. Also, the cycle times for this process are relatively large. 20.9.3.2 Dip Coating

Dip coating is used to coat portions of metal parts with rubber or PVC. The process is a modification of dip molding in which, instead using a mandrel, the actual part is used as the male “mold.” The coating can be for esthetic reasons, to provide a better gripping surface for hands – as in grips for hand tools – and for safety, as in coating metallic surfaces of toys. Multiple layers added in different dipping operations can be used to provide different characteristics in the coating. This process can be used to apply coating having different characteristics and textures: Smooth and shiny, with open- or closed-cell foam, and textured to look and feel of suede or leather. 20.9.4

Manufacture of Reinforced Rubber Parts

Fiber-reinforced rubber parts constitute a class of flexible advanced composites (Section 25.4). The largest application of such advanced composites is in automotive tires – and to a much lesser extent in tires for agricultural, construction, and mining equipment. Other applications include V-belts for machines, conveyor belts for conveying raw material, and pressure hoses – high-pressure hoses can have very complex layered morphologies (Section 25.4.3). This section outlines fabrication methods for tires, conveyor belts, and pressure hoses. 20.9.4.1 Tires

Tires are complex multilayer, flexible annular structures for supporting the loads of moving vehicles. The functional parts of a tire, shown in Figure 25.4.1, are the carcass, the structural element that transmits the load from the tread to the axle; the tread, the traction and wear surface that contacts the road; the beads, metal-wire rings at the internal diameter to facilitate seating and load transfer at the tire-rim interface; the inner liner, which provides a smooth surface for inflatable inner tubes or an air seal surface for tubeless tires; and the outer liner that protects the carcass and provides an esthetic surface. The carcass is assembled from fiber-reinforced rubber plies. To facilitate the flexing of the carcass, the inner and outer linings are made from a more elastic, softer rubber. The tread is made of a harder rubber that, while providing sufficient friction at the road-tread interface, resists wear. The carcass is built up from several plies of unidirectional fibers embedded in a rubber matrix. Older, biased-ply tires (Figure 25.4.2) have several diagonal plies, each making an angle of 35 – 45° with the

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tire cross section, in which the fiber directions alternate at ± (35 – 45)°. The bead ring is made of rubber coated steel wire rings. The innermost carcass ply wraps around these rings and around a filler, and the end is bonded back onto the ply. The inner rubber liner covers the inside and outside surfaces of the carcass. Newer tires have bias-ply belts that provide additional support to the carcass (Figure 25.4.3a). Most modern tires now are radial tires (Figure 25.4.3b), in which the fibers in the plies lie in radial planes; the belts can be reinforced with synthetic fibers or with steel wires – in which case they are referred to as steel-belted radial tires. Tire manufacture starts with compounding several types of rubbers for the inner liner, the carcass, the outer liner, and the tread. Calendering is used for making several types of rubber fabric- and fiber-embedded plies by using the calendering process shown in Figures 20.9.1 and 20.9.2. Since steel wires do not adhere to the rubber, they are coated with brass. In these sheets, the fibers/steel wires are aligned along the length. Pieces are cut from the calendered sheets to obtain plies with the right bias orientation. In a separate operation, the bead coils are made by winding steel wire into rings and impregnating and coating them with rubber. While tires are built up in several different ways, they all involve wrapping different layers of calendered rubber, fiber-filled rubber, and tread on a rotating cylindrical drum – the tire building drum – of a tire building machine. A newer process for making radial tires uses a compound tire building drum (Figure 20.9.4) comprising two segmented steel cylinders connected to a hollow rubber cylinder. This drum assembly rotates on a shaft that allows axial motion of the metallic end pieces, thereby making it possible to use air pressure to inflate the rubber shell outwards. RUBBER HOLLOW CYLINDER

(a)

METAL CYLINDER MADE OF RADIALLY MOVABLE PIECES

(b)

Figure 20.9.4 Tire building drum. (a) Two steel cylinders made of segmented, radially moveable steel pieces connected to a central inflatable hollow rubber cylinder. (b) Axial movement of the end piece caused by inflation of the central rubber cylinder.

Using this type of tire building drum, the steps in one process for making radial tires are schematically shown in Figure 20.9.5: (a) A wide strip of synthetic rubber that is impervious to air is tightly wound on the drum; this will end up as the inner tire surface of the tubeless tire, replacing the separate inner tubes used in older tires. (b) A wider strip with the fibers aligned along the drum axis – along the tire radial

Fabrication Methods for Thermosets

direction – is tightly laid on the first rubber layer; this casing ply is the main structural, load transmitting component of the tire carcass. (c) Two profiled rubber strips with protrusions are attached at both ends as shown; the wire bead rings are placed at the ends as shown in (d). (e) The casing ply is folded over the rings. (f) Then rubber strips A are tightly placed over the fold, and rubber strips, B, are wound toward the middle – they are used for attaching the sidewalls and belts. (g) Wide rubber strips, spanning from the edges of the strips B to the tire bead are attached as shown; they will form the tire sidewalls. Note that in order to clearly show the different steps, in the schematic diagrams in Figure 20.9.5a – g the different layers are shown with gaps. In actual practice, each layer is tightly wound on the layer below and pressed (rolled) in place; in corner and stepped regions a rotating tool can be used to eliminate air gaps; at the end of the process shown in (g), the actual part will have the shape shown in Figure 20.9.5h, with no air gaps between layers. The layers are kept in place by the tack (stickiness) in the uncured rubber layers.

(a)

PROFILED RUBBER

RADIAL PLY

(b) (c)

WIRE BEAD RING

(d) (e) A A (f)

B

B

RUBBER STRIP

RUBBER SIDEWALL

(g) (h) Figure 20.9.5 Buildup of radial tires. (a) Wide strip of impervious synthetic rubber tightly wound on tire building drum. (b) Wider strip with the fibers aligned along drum axis tightly laid on first rubber layer. (c) Two profiled rubber strips with protrusions attached at both ends. (d) Wire bead rings placed at the ends. (e) Casing ply folded over the rings. (f) Rubber strips A tightly placed over the fold and rubber strips B wound towards the middle. (g) Wide rubber strips spanning from the edges of the strips B to the tire bead attached. (h) Actual assembly shape shown with no air gaps between layers; layers kept in place by uncured rubber tack.

The first step ends with all but the belt and tread layers in place (Figure 20.9.5h); these layers are shown laid out on the tire building drum in Figure 20.9.6a. In the second step the rubber shell is inflated with air pressure, forcing this assembly into the toroidal shape shown in Figure 20.9.6b; this process is made possible by the axial movements of the two serrated steel drums. Two belts (Figure 20.9.6c) – with

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the fibers in the layers in symmetrically opposite directions about the circumferential tire section – are placed on the central zone, partially covering the strips B (Figure 20.9.5f). Then, a thicker, profiled layer of rubber is laid on top of the belt (Figure 20.9.6d); this layer covers the top edges of the sidewall rubber layers. This tire, called a green tire because the rubber in the various layers is in the tacky, uncured state, is removed and ready to be cured to the final tire shape.

(b)

(a) TWO BELT PLIES

TREAD

(c)

(d)

Figure 20.9.6 Buildup of green tire. (a) Rubber layers on tire buildup drum from first step (Figure 20.9.6 h). (b) Rubber shell inflated with air pressure. (c) Two belts with fibers in symmetrically opposite directions placed on circumferential tire surface. (d) Thicker, profiled layer of rubber laid on top of the belt.

Fabrication Methods for Thermosets

In the third, final curing step, the green tire is cured (vulcanized) in a heated tire mold, the inside of which is shaped to form the tread and the rest of the external shape of the tire. The curing process is schematically shown in Figure 20.9.7: The green tire is placed in the lower half of a tire mold, which is placed inside an outer casing (Figure 20.9.7a). Then the top halves of the tire mold and the outer casing are used to close the mold and the casing, after which steam is injected in the space between the casing and the mold and pressurized boiling water is injected into the inflatable bladder inside the mold (Figure 20.9.7b). The steam heats the tire mold, and the hot water expands the bladder into the inner surface of the green tire. The fully expanded bladder forces the outer green tire surface against the mold. The heat from the steam and the hot water cures the tire material into its final shape (Figure 20.9.7d). The finished tire is obtained on demolding. OUTER CASING LOWER HALF TIRE MOLD LOWER HALF

INFLATABLE BLADDER

GREEN TIRE ELASTIC BLADDER

(b)

(a)

CURED TIRE

(c)

PRESSURISED BOILING WATER

STEAM

(d)

Figure 20.9.7 Schematic diagram showing the tire curing (vulcanization) process. (a) The green tire is placed in the lower half of a tire mold, which is placed inside an outer casing. (b) The top halves of the tire mold and the outer casing are used to close the mold and the casing. Steam is injected in the space between the casing and the mold, and pressurized boiling water is injected into the inflatable bladder inside the mold. (c) The steam heats the tire mold, and the hot water expands the bladder into the inner surface of the green tire. (d) The fully expanded bladder forces the outer green tire surface against the mold. The heat from the steam and the hot water cures the tire material into its final shape. The finished tire is obtained on demolding.

This section has described the principle underlying one of many processes for making tires. In some of the older processes the carcass was not inflated prior to the addition of the tread – they did not have

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the relatively inextensible belts – so that the green tire was essentially a cylinder with a diameter of the bead ring. This cylinder was stretched to the tire shape inside a tire mold during the curing cycle. 20.9.4.2

Conveyor Belts

Belts used in industrial conveyor systems, which are designed for conveying large amounts of raw material such as coal to power plants, are made of several layers of reinforced elastomers, the complex structures of which are described in Section 25.4.2. Conveyor belts are made by curing multiple layers of elastomers and reinforced elastomers by the calendering process for reinforced and coated rubber discussed in Section 20.9.2.4. 20.9.4.3

Pressure Hoses

A hose is a flexible, hollow tube for transporting fluids between two locations; smaller diameter ones are also called tubes. Flexible tubes of polyethylene (PE) and PVC are made by extrusion; small diameter PVC tubing used in medical drip systems is a good example of a low-pressure application. The term hose includes tubular structures used in many diverse applications – from the very flexible common garden hose to reinforced tubes for higher pressure applications – with very different pressure, temperature, and chemical environmental requirements. Hoses for suction applications must be reinforced to prevent them from collapsing during use. High-performance hoses – such as those operating at high pressures and temperatures, and hostile chemical environments – are complex, multilayered structures made of several layers of elastomers and reinforcing fibers, including steel. The structure of different high-performance hoses and their structures are described in Section 25.4.3. The simplest unreinforced rubber (elastomeric) hoses can be made by vulcanizing extruded tubes. Hoses with more complex layered structures can be assembled on a mandrel prior to vulcanization: By using the types of machine motion shown in Figure 20.8.3, a tape of the elastomer for the inner hose wall is first wound on a steel mandrel in an overlapping helical pattern. A reinforcing layer of elastomer impregnated fabric strip is then helically wound in the same way. In the simplest hose structure, a layer of elastomeric tape of the material, which will form the outer surface of the hose, is wound in a helical pattern. To provide compression during vulcanization, the assembly is covered by a tightly wound layer of nylon tape. The mandrel is then placed inside a steam heated autoclave where the elastomeric layers consolidate and cure into a void free structure. After cooling, the nylon compression is peeled off and the mandrel is removed by using special techniques. Instead of using helically wound tape, a braiding machine can be used for covering an extruded tube with fibers in different patterns; the fibers can also be interlocked by a knitting process. Braiding can also be used for laying steel wire layers. In relatively small diameter tubes for very high-pressure applications the outer surface can be braided with narrow, thin tapes of stainless steel. This section has given an introduction to how some hoses are made. The actual manufacturing techniques are very sophisticated. High-performance hoses can have very complex architectures in which stainless steel interlock slip winding can be used to protect the elastomeric layers (Figures 25.4.21 and 25.4.22).

20.10 Concluding Remarks This chapter has addressed part fabrication technologies for thermosetting resins and elastomers. Fabrication of resin parts, advanced composites, rubber parts, and reinforced rubber parts – normally considered disparate subjects – has been addressed in a coherent systematic manner.

Fabrication Methods for Thermosets

Some of the molding methods for thermosetting resins, such as injection molding, are similar to those for thermoplastics. But because of curing reactions, thermoset resin molding has much longer cycle times. Because of their flexibility injection-molded rubber parts are easier to mold with undercuts. One major difference between the two types of resins is in the extensive use of thermosets as matrices for advanced composites that have very large amounts of continuous reinforcements, for which special fabrication techniques, such as pultrusion, filament winding, and the use of prepregs, have been developed. And new applications for very large parts, such as hulls and decks for boats and ships, blades for very large wind turbines, have resulted in the invention and refinement of vacuum-assisted resin infusion systems. Another difference is a how a single application type – all type of tires, for bicycles, to motor cycles, to automobiles, to aircraft, to mining equipment – has resulted in the creation of a standalone industry.

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21 Joining of Plastics 21.1 Introduction Because plastic parts of sufficient complexity could be molded, joining of plastics did not receive much attention in the past, and joining methods were considered secondary operations to be avoided if possible. Important exceptions were high-performance composites made of thermoset resin-based unidirectional laminates, mainly used by the aerospace industry, for which adhesive bonding and mechanical fastening technologies were developed. Now joining of plastics and plastic composites has become important for several reasons. First, such materials are increasingly being used in complex structural assemblies, in which joining considerations and cost are important. Second, the emerging structural (load-bearing) applications of polymeric materials require structural joints that must withstand static and fatigue loads. Third, weldable filled and unfilled thermoplastic resins are being used in many structural applications. A good example is the first all-plastic 8-km h−1 (5-mph) bumper, made of an unfilled thermoplastic polycarbonate (PC)/poly(butylene terephthalate) (PBT) blend (Section 12.2.7), which was fabricated by vibration welding two injection-molded parts (Figures 2.4.1 and 21.7.5). Welding is also an important consideration in high-performance thermoplastic composites. For optimizing part performance, the efficient use of plastics and plastic composites requires different parts of a structure to be made of different materials, which may include metals. For example, thin-walled, blow-molded structural parts can be stiffened by filling the cavity with low-density foam. Plastic parts, such as automobile doors, now satisfy design requirements through the use of several different plastics and composites in combination with metals. Such innovative ways of using plastics need new insights and technologies for multimaterial part design, fabrication, and assembly. This need has driven development of a new generation of multiprocessing machines that can execute two or more plastics processes simultaneously or serially, in the same mold, to fabricate multimaterial parts. Such technologies are placing increasing demands on joining methods for attaching or bonding different types of plastics to other plastics or metals. Joint design depends on the materials being used, on the required performance, and on the joining technology used; joint performance will depend on the characteristics of the process used. In structural applications, the important issues for joint performance are: • • • •

How strong is the joint under static loads? How resistant is it to impact loads? What is the effect of residual stresses? How does the joint perform under fatigue loading?

Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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• • • •

How does creep of the resin(s) affect long-term joint performance? What effect does resin aging have on joint performance? What is the effect of fillers such as particulates and fibers? How do environmental factors influence joint performance?

The overview of joining methods for plastics and composites in this chapter has been grouped into three broad categories: mechanical fastening, adhesive bonding, and welding. These categories have then been further classified into established technology groups, each of which is then briefly discussed.

21.2 Classification of Joining Methods Joining of plastic materials and their composites can broadly be divided into mechanical fastening and bonding (Figure 21.2.1). Bonding can be further divided into adhesive bonding, solvent bonding – which may be considered as a special type of adhesive bonding – and welding. Mechanical fastening and adhesive bonding can be used for joining all materials, including metals. And the parts being joined need not be of the same material. On the other hand, welding, which requires the materials at the joint interface to melt, is only applicable to thermoplastics. One advantage of mechanical fastening is that, except for hot staking, the joint can be opened and closed many times, thereby making it possible to remove defective parts. Adhesive bonding and welding result in permanent joints; one exception is induction welding that allows a welded joint to be opened.

JOINING METHODS FOR PLASTICS

MECHANICAL FASTENING • • • • •

Rivets, screws, ... Spring clips Snap fits Metal inserts Hot staking

BONDING

ADHESIVE BONDING • Single-component adhesives • Two-component adhesives • Solvent bonding

Figure 21.2.1 Classification of joining methods for plastics.

WELDING • Requires melting & soldification at interface • Only applicable to thermoplastics

Joining of Plastics

21.3 Mechanical Fastening Mechanical fastening can either be permanent, or consist of joints that can be opened and closed. Snap fits, spring clips, screws, and metal inserts are used to provide openable joints. Rivets and hot staking are examples of methods used for achieving permanent joints. 21.3.1

Snap Fits

Snap fits provide a very versatile and widely used fastening mechanism that is ideally suited to the ease with which geometric complexity can be molded into plastic parts. A snap-fit consists of a snap-on lug that can lock into a recess. The lug is kept in place by a plastic spring, which is the plastic cantilevered arm that attaches the lug to the body of the part. The number of times that such a joint can be opened and closed can be increased by designing the plastic spring for fatigue. While snap-fit joints are convenient for rapid assembly, they have not been used in demanding load-bearing applications. Recent analyses have led to improved snap-fit designs with improved load-carrying capacities and life. The principle of snap fits can be understood from the cantilever-hook system, shown in Figure 21.3.1, in which one of the parts (left part in Figure 21.3.1a) has a cantilever with a hook, ABCDE, at the end. This hook is inserted into a cavity, with surface HI, in the second part (at the right). During insertion, the wedge surface DE on the hook first comes in contact with the right part, as shown in Figure 21.3.1b. A further insertion on the hook requires an axial force that, in combination with the reaction and friction force at H, will cause the cantilever to deflect as shown in Figure 21.3.1c. Finally, when the point C on the finger goes past point I on the second part, the cantilever snaps back, resulting in the locked configuration shown in Figure 21.3.1d. In this permanent locking technique, the left part cannot be retracted by a leftward pull. A retractable cantilever-hook locking system is shown in Figure 21.3.2. In this design, in addition to the wedge surface DE, the finger has a wedge surface BC on its back face and a matching wedge surface IJ on the second part. In this case, when the first part is pulled leftward, the point I on the second part slides on the back surface BC of the finger, causing the cantilever to deflect, and thereby allowing the part to retract. Thus far the focus has been on the principles on which snap fits are based. Actual parts are so designed that in the locked position the parts match in such a way that the cantilever cannot move anymore to the right. Clearly, even in a retractable joint, the opening (retraction) force should be larger than the closing (insertion) force. This can be controlled by the slopes of the surfaces BC and DE – the larger the slope is, the larger the force will be. The variation of the axial force versus the axial displacement of the cantilever in Figure 21.3.2 is schematically shown in Figure 21.3.3: Point P corresponds to a displacement at which the surface DE just touches the corner H (see Figure 21.3.1b). Then, as the point of contact moves along ED on the wedge, that is, as the cantilever begins to deflect, the axial force on the cantilever varies along PQ. Point Q corresponds to D just touching H. After that H moves along surface DE that, because of the deflected shape of the cantilever now has a gentler slope; the corresponding force varies along QR. In the design shown, the lengths of CR and HI are the same, so that after the point C reaches H, the point of contact moves along IJ, and the corresponding force decreases along RS. (Actually, this unloading along RS will remain the same as long as the length of segment HI is less than or equal to that of segment CD. But segment HI is larger than CD; then, instead of unloading along RS, the force will remain constant at the level at R until point C reaches point I, after which it will follow the path RS.)

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A

C B

D E

G

F

H

I

(a)

(b)

(c)

(d)

Figure 21.3.1 Assembly of two parts using a cantilever-hook locking system. (a) First part (left) with a finger at the end of a cantilever. (b) Wedge surface on finger contacts second part (right). (c) Continued insertion causes cantilever to bend. (d) Locked finger prevents opening of joint.

A

C B

D E F

G

J H

I

(a) Figure 21.3.2 Assembly with a retractable cantilever-hook locking system.

(b)

Joining of Plastics

T

FORCE

RETRACTION FORCE

INSERTION

RETRACTION

INSERTION FORCE

U

R

Q

S

P

V W

O POSITION

Figure 21.3.3 Force-displacement curve for a retractable cantilever-hook locking system.

For opening the joint the part at the left is pulled to the left, which causes the flank BC to rub against I, and the cantilever to deflect. Because of the high slope of BC, the retraction force is high. As BC moves past point I, the axial retraction force follows the path ST, which peaks at point T when C reaches point I. The load then immediately falls off, along TU, to the level required for maintaining the deflection of point C to the height corresponding to the surface HI. Then, as point C is dragged across the surface IH, the force decreases along UV, and then along VW as the surface DE moves past point H. Note that in Figure 21.3.3, zero displacement of the parts is measured from the locked position shown in Figure 21.3.2b. So, while insertion is along PO, with the cantilever moving from left to right, retraction is along OW with the cantilever moving from right to left. Snap fits come in many different forms and are very widely used assembling plastic parts industrywide – from toys to consumer and automotive applications. An example of a cylindrical snap-fit assembly is shown in Figure 2.2.14a. 21.3.2

Use of Screws

Besides offering a means for joining different components, screw joints make it possible to open an assembly for maintaining or replacing internal components. While the strongest screw joints in plastics can be made by using machine screws mating with metal inserts in the plastic part, it is easier to use self-threading metal screws specially designed for plastics.

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The use of self-threading screws requires a plastic boss with a molded or drilled pilot hole, in which the screw forms the thread during assembly. This reduces molding and assembly costs by eliminating molded threads or secondary tapping operations for threaded inserts. The geometry and defining dimensions of a self-threading screw and plastic boss are shown in Figure 21.3.4. The main parameters for the screw are its root diameter dr , outside thread diameter dT , thread pitch p, and thread angle 𝛼 ; other parameters are its overall length l, the thread engagement length lT , and the head diameter dH (Figure 21.3.4a). In comparison to metal screws, screws for plastics have larger pitches, larger values of the ratio dT ∕dr , and lower thread angles – typically for plastic screws 𝛼 = 30°, compared to 𝛼 = 60° for metal screws. The actual values of these parameters are specified by screw manufacturers.

dB

dH

dT

hL

dP

α l

lT

lE

h

p dr dT

t 0.6t

(a)

(b)

Figure 21.3.4 Parameters for defining the geometries of plastic screws and bosses. (Adapted with permission from F. Dratschmidt and G.W. Ehrenstein, Polymer Engineering and Science, 37, 744, 1997.)

Figure 21.3.4b shows the nomenclature for the dimensions a plastic boss: wall thickness t, boss height h, pilot-hole diameter dP , boss diameter dB , depth of lead-in cavity hL – its diameter is slightly larger than the thread diameter dT – and the thread engagement length lE – the final threaded length inside the boss hole. The pilot-hole diameter lies between the screw root and thread diameters, that is, dr < dP < dT . A screw joint is assembled in the three phases schematically shown in Figure 21.3.5. First, the screw is inserted into the pilot hole. Then the screw is driven into the pilot hole by means of a torque applied to the head. During this process the screw forms threads in the boss wall by displacing plastic, some of which flows toward the screw root. At the end of this phase the screw head seats on the assembly surface (Figure 21.3.5b). In the final, third phase the screw is tightened by applying additional torque that turns the screw by a small amount, which induces a tensile load on the screw shank and substantial stresses in the boss (see inset in Figure 21.3.5c). Some level of these stresses helps in generating the desired clamping force, but excessive stresses can cause the joint to fail in one of several ways.

Joining of Plastics

(a)

(b)

(c)

Figure 21.3.5 Three phases of assembly of a screw joint. (a) Initiation of screw insertion. (b) Completion of crew insertion; screw head seated on assembly surface. (c) Screw tightening resulting is stressed boss. (Adapted with permission from EJOT Verbindungstechnik GmbH & Co. KG).

The variations with insertion time in the applied torque, T, and the resulting axial force, F, generated in a screw are schematically shown in Figure 21.3.6. The torque first increases along path OA to the forming torque T1 required to form the first thread in the boss. Then, as the screw penetrates along the boss pilot hole, the torque increases along path AB to TD at point B at which the screw head contacts the part surface (Figure 21.3.5b); at this point the screw threads have penetrated along the full engagement length lE . Up to this point there is no axial force, or preload, on the screw. Any further twisting of the screw results in a rapidly increasing torque along path BCE and the preload force along path KLM (Figure 21.3.5c). The screw is tightened to a torque TT that is smaller than the maximum torque TS , called the stripping torque, at which the material fails and the preload force drops off. The axial force FT , at tightening torque TT , is then the axial tensile preload force on the joint; this tensile force on the screw keeps the joint “tightened.” The preload force FT on a screw joint can cause it to fail in one of four ways illustrated in Figure 21.3.7. This force is transmitted to the plastic at the outer surface of the screw and, for a screw engagement length of lE , results in a shear stress 𝜏 P = FT ∕(𝜋 dT lE ) acting on a cylindrical surface of diameter dT and length lE (Why?). When this shear stress reaches the ultimate shear stress 𝜏 Pu of the boss (plastic, hence the subscript P) material, the screw strips out from this cylinder, carrying with it plastic embedded in the

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TS

E

STRIPPING TORQUE

TT

C

TIGHTENING TORQUE

TD

DRIVING TORQUE

FS

M L

B

FT

A

T1

O

K

SCREW PRELOAD FORCE

SCREW INSERTION TORQUE

718

N

TIME(s) Figure 21.3.6 Schematic variations of the screw insertion torque, T, and the axial preload force generated in a screw with the insertion time. (Adapted with permission from F. Dratschmidt and G.W. Ehrenstein, Polymer Engineering and Science, Vol. 37, pp. 744 – 755, 1997.)

threads. This stripping mode is shown in Figure 21.3.7a, and occurs when the axial preload exceeds FS = (𝜋 dT lE ) 𝜏Pu Clearly, for a given screw, the stripping load increases linearly with the engagement length, so that stripping can be avoided by increasing lE . While stripping can be avoided by increasing the engagement length, the preload force can be large enough to cause a tensile failure in the screw shank, as shown in Figure 21.3.7b. In terms of the ultimate tensile strength, 𝜎 Mu , of the screw (metal, hence the subscript M) material, the screw will fail in this mode when the axial force reaches FT = (𝜋 dT 2 / 4) 𝜎Mu Figure 21.3.7c shows boss fracture with near axial cracks caused by high hoop stresses generated by a small pilot-hole diameter, a small wall thickness, or large thread angles. In the fourth failure mode shown in Figure 21.3.7d, for small wall thicknesses the boss can undergo axial tensile failure at the last (final) thread in the boss or at the boss ground plane marked BB. In terms of the ultimate tensile strength, 𝜎 Pu , of the plastic, the boss will fail in this mode when axial preload force reaches

𝜋

( d 2 − dp 2 ) 𝜎Pu 4 B An important measure of the performance of a threaded joint is the screw preload (clamp force), FT , which corresponds to the tightening torque TT . The stress relaxation rate in the viscoelastic plastic depends on the preload. Joints tightened with high torques have high initial preloads that quickly decay in few hours. Therefore, the preload should only be high enough to guarantee joint integrity over time. FT =

Joining of Plastics

(a)

(b)

B

(c)

B

(d)

Figure 21.3.7 Failure modes self-threading screw joints. (a) Screw pull our due to stripping of boss material. (b) Failure of screw shank. (c) Axial cracking of boss wall. (d) Tensile failure of boss. (Adapted with permission from F. Dratschmidt and G.W. Ehrenstein, Polymer Engineering and Science, Vol. 37, pp. 744 – 755, 1997.)

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Special screw shapes have been developed for self-threading screws. Figure 21.3.8 shows three screws with symmetric threads with nominal diameter dT = 5 mm, root or core diameter dr = 3 mm, and pitch p = 2.2 mm. The thread tip angle in each of them is 𝛼 = 30°. However, in the screw thread shown in Figure 21.3.8a has a varying thread angle that steadily increases toward the core from 30° at the tip. In the screw in Figure 21.3.8b the thread has a constant angle of 30°, but the root, or core, is has a modified shape. And, in contrast to the previous two screws in which the core cross sections are circular, the core in the screw shown in Figure 21.3.8c has a trilobal cross section. This design helps to prevent the screw from loosening by rotation, even under intense vibrations.

2.2 mm

3 mm

5 mm

(a)

90° 60° 40° 30°

30° 2.2 mm

3 mm

5 mm

(b)

30°

Figure 21.3.8 Examples of three types of screw thread geometries. (Adapted with permission from F. Dratschmidt and G.W. Ehrenstein, Polymer Engineering and Science, Vol. 37, pp. 744 – 755, 1997.)

Joining of Plastics

2.2 mm

3 mm 5 mm

(c)

30° Figure 21.3.8 (Continued)

21.4 Adhesive Bonding Adhesive bonding offers the potential for joining any two materials. It is particularly important for joining thermoset plastics, which cannot be welded. In the past, all high-performance plastic composites had thermoset matrices. Joining needs for demanding aerospace applications of such composites have led to the development of a whole class of structural adhesives. Adhesives have also been developed, and are extensively used, for bonding metals to metals and nonmetals in the aerospace and automotive industries. Structural adhesives are those that can be used in the load-bearing applications. Adhesives can be classified into two broad categories: (i) Two-component systems, in which the two components are mixed just prior to use. The two components react chemically during the cure cycle, resulting in the desired bond. (ii) Single-component systems. The most commonly used structural adhesives are epoxies, urethanes, and acrylics – which are two-component systems – and the single-component class comprising cyanoacrylates, anaerobics, and hot melts. Epoxies represent the most widely used class of versatile adhesives. The two components are mixed, in equal measure, prior to use. Epoxies require long cures, or cycle times, that can be reduced by applying heat. Epoxy bonds tend to be rigid and fail in a brittle manner. In contrast to epoxies, the two components of acrylic adhesives are not mixed before use. Rather, one component is applied to one surface, and the second component is applied to the second surface. The curing reaction occurs when the surfaces are mated. Cure times are shorter than for epoxies, and result in more flexible bonds. Anaerobics are single-component adhesives that cure by oxygen deprivation. Cyanoacrylates cure in seconds at room temperature. Special formulations have been developed for medical applications, such as for sealing wounds. Many variants, such as Krazy Glue®, are available for everyday bonding applications. In adhesive bonding, the (parts) surfaces to be joined are called adherents. Surface preparation of adherents prior to the application of adhesives, including a degreasing step, is crucial for joint strength and reliability.

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Adhesive bonding is a relatively mature technology, in which chemistry has played an important role. Special tests have been developed for characterizing the strengths of adhesive joints. Mechanics principles have been applied to understand the mechanisms that affect joint strength. 21.4.1

Solvent Bonding

In solvent bonding, or solvent welding, the surfaces of the parts to be joined are treated with a solvent that swells and softens the surface. Pressure applied to the joint allows polymer chains to diffuse across the joint interface. The evaporation of the solvent from the joint interface then results in bonding of the two surfaces. Solvent bonding only works for thermoplastics. Cycle time is governed by the rate of solvent evaporation, and can be shortened by the use of heat. Instead of using pure solvents, solutions containing the plastic to be joined can be used. This makes it easier to fill mismatching surfaces. Solutions containing the monomer of the plastic together with a polymerizing catalyst are also used. The solution can be tailored to effect polymerization at the joint interface at room temperature or at higher temperatures. Example of solvents used in this process include formic acid for bonding PA6 and PA6,6, and methylene chloride and ethylene dichloride for PC and PC/ABS. Solvent bonding is a slow process, not suited to large-scale manufacture. It is mainly used in specialty applications. It can also be useful in a one-of-a-kind application.

21.5 Welding Because welding requires melting and subsequent freezing of the materials to be joined at the joint interface, it is only applicable to thermoplastic materials. Depending on how the heat for melting is supplied, as indicated in Figure 21.5.1, welding can broadly be classified as: (i) thermal bonding, comprising hot-gas welding, extrusion welding, hot-tool (hot-plate) welding, and infrared heating, (ii) friction

WELDING OF THERMOPLASTICS

THERMAL • • • • •

Hot gas welding Extrusion welding Hot-tool welding Infrared heating Laser welding

FRICTION (MECHANICAL) • Spin Welding • Vibration Welding (100–250 Hz) • Orbital welding • Ultrasonic Welding (20–40 kHz)

ELECTROMAGNETIC • • • •

Figure 21.5.1 Classification of welding methods for thermoplastics.

Resistance (implant) welding Induction welding (5–25 MHz) Dielectric heating (1–100 MHz) Microwave heating (1–100 GHz)

Joining of Plastics

(mechanical) welding, comprising spin welding, angular vibration welding, orbital welding, vibration welding, and ultrasonic welding, and (iii) electromagnetic bonding, which includes resistance (implant) welding, induction welding, dielectric heating, and microwave heating. Since welding was considered a secondary operation – in contrast to primary fabrication methods, such as injection molding – most work on welding was done by companies that developed different welding methods. As a result, companies making plastic parts requiring welding normally work with a welding machine vendor to design welding equipment, including specialized tooling, often on a turnkey basis. Much of the work on understanding welding methods has been driven by joint performance requirements in structural applications, such as in the automotive sector. In addition to other physical properties, resin manufacturers are now also concerned with the weld strength of their resins. They tend to consider lower weld strengths as a disadvantage. But from an engineering, or applications, standpoint, inherent lower weld strength is not important, because it is the joint strength that matters. Very strong joints can be designed even with materials with lower weld strengths.

21.6 Thermal Bonding In thermal welding the heat for melting the weld interface is provided by directly applying heat by contact conduction, radiative infrared heating, or through laser radiation. These methods cover a broad range of techniques from one-of-a-kind applications, to batch production, to large-scale production. Equipment and tooling costs are relatively low. 21.6.1

Hot-Gas Welding

In the hot-gas welding method a thermoplastic filler rod and the parts to be joined are heated by a hot-gas stream. In this respect it is similar to gas welding of metals, but in contrast to metal welding, the filler rod is not melted. Rather, the rod is pushed into the joint and heated until it softens sufficiently to fuse with the workpiece material. The process can be automated. The main advantage of hot-gas welding is its flexibility. Simple, portable equipment can be used for fabricating large, complex one-of-a-kind parts, or for carrying out repairs of thermoplastic parts. It is a slow process that is difficult to control, especially in the commonly used manual mode, and therefore is not suited for mass production. 21.6.2

Extrusion Welding

Extrusion welding is similar to hot-gas welding except, in place of using a filler rod, molten filler material is extruded into the joint. Hot gas is still needed to heat the joint region in the workpiece. This process is preferred for the automatic welding of large assemblies. It is the technology of choice for welding large plastic sheet liners for landfills. This process is suited for welding assemblies, such as for attaching plastic pipes to plastic tanks (Figure 2.5.6). 21.6.3

Hot-Tool (Hot-Plate) Welding

In the hot-tool welding process, the surfaces to be joined are brought to the “melting” temperature by direct contact with the matching surfaces of a heated metallic tool. The molten surfaces are then brought

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together, and the interface is allowed to cool and solidify under controlled pressure, resulting in a weld. In many applications, such as the joining of plastic pipes, the surfaces to be joined are flat, so that the hot tool is a hot plate. This was true of most early applications and is the reason this process is also called hot-plate welding. However, in many applications, such as automotive headlamps and rear lights, the doubly-curved joint-interface surface requires complex tools that allow the hot surfaces to match the contours of the two halves to be joined. The applicability of this process to complex joint geometries is one of the major advantages of this process. The hot tool element is made of metal and is heated by internal tubular heaters or heating cartridges. Temperature sensors makes it possible to accurately control the surface temperature of the tool. To prevent the molten layer of plastic from adhering to the tool, the tool surface can be coated with a layer of polytetrafluoroethylene (PTFE). Flat hot plates can be covered by PTFE-coated glass cloth. However, the use of such coatings limits the maximum tool surface temperature to about 260°C (500°F). In some materials, the molten film has a tendency to adhere more to the hot-tool surface than to the film itself. This can cause a large portion of the molten film to adhere to the tool when the part being heated is separated from the tool. In such cases, the polymer surfaces can be heated by radiation and convection. However, this process is difficult to control, and the uneven heating can result in nonuniform weld quality. The hot-tool welding process currently in use can be described in terms of the four phases schematically shown by the pressure-time graph in Figure 21.6.1: (i) In phase 1, the part surface is brought in contact with the hot-tool surface (Figure 21.6.1a). A relatively high pressure, p1 , is used to ensure that the part

p(t) II

I

III

IV

p1 p2

•

t

HOT TOOL

p(t)

WELD BEAD

p(t)

(a)

p0

(b)

p0

(c)

Figure 21.6.1 Four phases of the hot-tool welding process. (a) Matching surfaces brought in contact with hot tool. (b) Parts separated and hot tool retracted. (c) Molten films in contact solidify under pressure.

Joining of Plastics

surface matches the tool surface. This pressure is maintained until the molten plastic begins to flow out laterally. (ii) In phase 2 (see pressure-time graph on top of Figure 21.6.1a), the inter-part tool pressure is reduced to a much lower value, p2 , to allow the molten film to thicken. The rate of film thickness growth is controlled by heat conduction through the film. (iii) When a sufficient molten film thickness has been achieved, the part and tool are separated. This is phase 3 (Figure 21.6.1b). (iv) The molten interfaces of the parts to be joined are then brought in contact and held under pressure until the interface solidifies (Figure 21.6.1c). During this fourth phase the molten film flows laterally while cooling. The duration of phase 3 should be kept to a minimum to prevent cooling of the molten-film surfaces. The important welding parameters for this process are then: • • • • •

Hot-tool temperature Pressure and duration of phase 1 Melt pressure and duration of phase 2 Change-over time Weld pressure and duration of phase 4.

Feedback control can be used to control the amount of material melting and flow during the melting and matching phases. Alternatively, as in displacement-controlled hot-tool welding, hard stops can be used to control the amount of initial melt outflow during the melt phase and the melt displacement during the matching phase. In this process, besides the platen-temperature used for heating the weld surfaces, the three important control variables are the melt and weld penetrations, and the melt time. The weld penetration is the distance by which the part surface protrudes beyond the matching point of the stops on the hot platen and the part holder; after the part contacts the hot platen this is the thickness of the material that is allowed to melt and flow laterally. The specimen then remains in contact with the platen; this time is called the melt time. The longer the melt time is, the more the thickness of the molten layers will be just before the molten surfaces are separated from the platen and brought together. Stops control the distance by which the molten layers come together with consequent lateral outflow; this distance is called weld penetration. And the seal time is the duration for which the joining surfaces are kept in contact to affect the weld. An understanding and quantification of the process phenomenology is important for several reasons. First, if the right tool temperature is not used, the interface material can degrade. Second, the cooling rate in phase 4 – which is controlled by the initial film thickness, the initial film temperature, and the weld pressure – determines the rate of crystallization in semicrystalline polymers. Third, the amount of lateral flow in phase 4 will influence the amount of orientation induced in the weld zone. In principle, any polymer that melts on heating can be welded by hot-wool welding. By using different surface temperatures for two halves of an assembly it is also possible to weld dissimilar materials, an area that has not been explored systematically. Tail-light automotive assemblies represent important applications of the welding of dissimilar plastics. Hot-tool welding is a relatively simple process. Its main advantages are: • It is forgiving: surface inaccuracies can be taken into account during phases 1 and 2. • It can handle joints with complex geometry. • By controlling the hot-tool surface temperature, the temperature of the molten film in phase 2 can be accurately controlled. This is especially important in materials that begin to degrade at temperatures slightly above those required to obtain a molten film.

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• Any plastic that melts on heating can be welded. • Selected dissimilar thermoplastics can be welded. The main disadvantage of the process is that it is relatively slow. Weld times range from 10 to 20 seconds for small items to 30 minutes for large pipes. This process could also become important for assembling large thermoplastic panel structures for construction and housing applications. In such applications, the development of small, portable hot-tool devices could aid the on-site welding of additional components onto thermoplastic panels. It is already the technology of choice for welding thermoplastic window frames, mainly assembled by hot-tool welding ABS and poly(vinyl chloride) (PVC) extruded profiles. Figure 21.6.2 shows how extruded sections, in this case of ABS, are mitered and hot-tool welded; the weld flash has to be machined off with a knife or an automated deflasher.

Figure 21.6.2 Hot-tool welding of an ABS window frame: Extruded sections are mitered and hot-tool welded. Profiles with two different colors have been used in the frame for easier visualization of the joint. (Photo reproduced with permission from V.K. Stokes, Polymer Engineering and Science, Vol. 37, pp. 692 – 701, 1997.)

Figure 21.6.3 shows two automotive applications. In Figure 21.6.3a the two parts on the left are a dual material red and clear poly(methyl methacrylate) (PMMA) lens and the ABS base to which it is hot-tool welded to form the tail-light assembly shown in the right. Figure 21.6.3b shows a glove box cover, made by welding the upper polypropylene (PP) part on the left to the lower polypropylene molded base, designed to absorb a knee impact by crushing; two versions with different top colors are shown on the right. Figure 21.6.4 shows a blow-molded gas tank made from multilayer sheet in which both the outer surfaces are of high-density polyethylene (HDPE). A blow-molded shell having the desired tank shape is placed in a special hot-tool welding and finishing machine in which (i) appropriate holes are made and the nozzles shown in Figure 21.6.4a are welded on, and (ii) cable clips and retainers are welded to the outer shell for clamping electrical wires and for attaching fuel pipes and heat shields, and for attaching the fuel tank to the car body. The finished gas tank is shown in Figure 21.6.4b.

Joining of Plastics

(a)

(b)

Figure 21.6.3 Hot-tool welding of automotive parts: (a) Dual material PMMA lens welded to a molded ABS base to form a tail-light assembly. (b) Polypropylene glove box cover. (Photos courtesy of bielomatik Leuze GmbH + Co. KG.)

(a)

(b)

Figure 21.6.4 Hot-tool welding of a polyethylene blow-molded gas tank. (a) Molded components to be welded blowmolded shell. (b) Finished gas tank with welded components. (Photos courtesy of bielomatik Leuze GmbH + Co. KG.)

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Hot-tool welding is the technology of choice for joining large-diameter plastic pipes, for which special machines have been developed for on-site handling and welding large-diameter pipes. The sequence of photos in Figure 21.6.5 shows the on-site hot-tool welding of 1600-mm diameter HDPE pipes having a wall thickness of 61.2 mm. On the left of Figure 21.6.5a is one end of a pipe being readied for welding to a section of pipe shown on the lower right. The pipes come in 12-m lengths that after welding can form km-length continuous pipelines. On the far right of this figure a just welded section can be seen; the end is covered to prevent contamination and cooling of the weld bead during welding by the air draft induced by temperature differences between the two pipe ends. Figure 21.6.5b shows the hot-tool assembly on a carrier. Figure 21.6.5c shows the hot-tool being lowered into position between two pipe ends to be joined. Figure 21.6.5d shows the hot-tool in place with hydraulics pushing the two clamped pipe ends together. Figure 21.6.5e shows the weld bead (lower right). The welded pipe is then pushed into the holding pond to enable another pipe section to be joined to the open end.

(a)

(d)

(b)

(c)

(e)

Figure 21.6.5 Hot-tool welding of 1600-mm diameter HDPE pipes: (a) One end of a pipe being readied for welding to a section of pipe shown on the lower right. (b) Hot-tool assembly on a carrier. (c) Hot-tool being lowered into welding position. (d) Hot-tool in place with hydraulics pushing the two clamped pipe ends together. (e) Weld bead on the lower right. (Photos courtesy of SKZ – German Plastic Center.)

Joining of Plastics

21.6.3.1 Weld Morphology

Because of melting, flow, and solidification, the morphology of the material in the weld zone is very different from that of the surrounding material. These differences can affect the short- and long-term behavior of the welded zone. This section describes the morphology of hot-tool welds of polycarbonate (PC) and poly(butylene terephthalate) (PBT); the corresponding morphologies of vibration welds of these two materials are discussed in Section 21.7.2.1. For the weld tests, the test specimens for PC were cut from 3.0-, 5.8-, and 12.0-mm thick extruded sheet material. Test specimens for PBT were obtained from specimens cut from 153 × 203-mm edge-gated injection-molded 3.2- and 6.1-mm thick plaques of PBT. The edges of each specimen were machined to obtain rectangular blocks of size 76.2 × 25.4-mm × thickness for ensuring accurate alignment of the surfaces during butt welding. Displacement-controlled hot-tool welding was used for preparing the weld specimens. Welded specimens for the morphologies discussed in this section were made at melt and weld penetrations of 0.13 and 0.66 mm, respectively, and a seal time of 10 seconds. PC welds were made at a hot-tool temperature of 245°C; two hot-tool temperatures of 305 and 320°C were used for PBT. Each welded specimen was cut normal to the 2.54-mm face to expose the cross-thickness weld zone. Optical microscopy was used to examine the morphology of the weld zone in two ways. First, optical macrographs were used to visualize the entire weld zone. Then, transmission electron microscopy (TEM) was used to study the detailed morphology of the weld zone in PBT welds. Hot-tool welds of 3.0- and 5.8-mm thick PC specimens were made at a hot-tool temperature of 245°C, melt and weld penetrations of 0.13 and 0.66 mm, respectively, at three melt times of 10, 15, and 20 seconds. Macrographs of the morphologies of hot-tool welds of 3.0-mm thick PC specimens made at a weld temperature of 245°C melt and weld penetrations of 0.13 and 0.66 mm, respectively, and melt times of 10, 15, and 20 seconds are shown, respectively, in Figure 21.6.6a – c. Clearly, higher melt times result in more bubbles in the weld zone. For the same weld conditions as for the 3.0-mm thick PC specimens, macrographs of the weld morphologies of 5.8-mm thick PC specimens are shown in Figure 21.6.6d – f. The case with a melt time of 10 seconds has the least amount of outflow. The interface where the two molten surfaces contacted can clearly be seen (Figure 21.6.6d); this interface has bubbles and debris picked up from the hot-tool surfaces. For a melt time of 15 seconds (Figure 21.6.6e) the weld interface is not visible except near the edges where the presence of bubbles is evident. In each of these two thicknesses, bubbles are present throughout the weld interface. The weld bead is formed by the material at the melt interface being forced out by the squeeze flow. At any lateral cross section, the molten material at the interface has the highest velocity, which increases with the distance from the center. On exiting from the edge, the melt bifurcates, thereby forming a bead having a mushroom-shaped cross section with a dimple (Figure 21.6.6a – f). Clearly, the hot-tool welds are nonuniform with the thickness increasing from the center to the edges. This heat-affected-zone (HAZ) shape results from radiative heating of the specimen lateral surfaces during the heating phase, which causes the molten layer at the outer edges to be thicker at the end of the heating phase. This HAZ thickness variation can therefore be expected to increase with increasing hot-tool temperatures. Bubbles are present in all hot-tool welds; these bubbles are concentrated around the central weld plane. They are elongated by the squeeze flow and tend to be larger near the edges. For each thickness, the weld zone thickness increases with the lateral flow. Furthermore, the sudden reduction in the pressure in the melt as it exits to the atmosphere causes the moisture to come out of solution, resulting in a “foaming” of the flash. The weld beads in hot-tool welding are thicker, have fewer bubbles,

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and tend to form the characteristic mushroom shape that results from the bulk flow from the central core pushing out at higher velocity and then dividing laterally. Because of their thickness, hot-tool weld beads can affect weld strength.

1 mm

1 mm

(a)

(d)

1 mm

1 mm

(b)

(e)

1 mm

1 mm

(c)

(f)

Figure 21.6.6 Macrographs of the weld morphology of displacement-controlled hot-tool welds of PC specimens made at a weld temperature of 245°C (473°F), and melt and weld penetrations of 0.13 mm (0.005 in) and 0.66 mm (0.026 in). (a) – (c) show the morphologies for 3.0-mm thick specimens for melt times of 10, 15, and 20 seconds, respectively. (d) – (f) show the morphologies for 5.8-mm thick specimens for melt times of 10, 15, and 20 seconds, respectively. (Adapted with permission from V.K. Stokes, Polymer Engineering and Science, Vol. 43, pp. 1576 – 1602, 2003.)

Macrographs of the morphologies of displacement-controlled hot-tool welds of 3.2-mm thick PBT specimens made at a weld temperature of 305°C, melt and weld penetrations of 0.13 and 0.66 mm, respectively, and melt time of 15 and 20 seconds are shown, respectively, in Figure 21.6.7 parts a and b. And macrographs of the morphologies of hot-tool welds of 6.1-mm thick PBT specimens made at a weld

Joining of Plastics

temperature of 320°C, melt and weld penetrations of 0.13 and 0.66 mm, respectively, and melt times of 10, 15, and 20 seconds, are shown, respectively, in Figure 21.6.7 parts c – e. Figure 21.6.7 shows that hot-tool weld beads in PBT have a bulbous, bubble-free shape, which is very different from the mushroom-shaped, bubble-filled beads in PC hot-tool welds.

1 mm

1 mm

(a)

(c)

1 mm

1 mm

(b)

(d)

1 mm (e) Figure 21.6.7 Macrographs of the weld morphology of displacement-controlled hot-tool welds of PBT specimens made at a

weld temperature of 305°C (473°F), and melt and weld penetrations of 0.13 mm (0.005 in) and 0.66 mm (0.026 in). (a) – (c) show the morphologies for 3.0-mm thick specimens for melt times of 10, 15, and 20 seconds, respectively. (d) – (f) show the morphologies for 5.8-mm thick specimens for melt times of 10, 15, and 20 seconds, respectively. (Adapted with permission from V.K. Stokes, Polymer Engineering and Science, Vol. 43, pp. 1576 – 1602, 2003.)

Figure 21.6.8 shows the morphology of hot-tool weld of 3.2-mm thick PBT specimens made at a weld temperature of 305°C, melt and weld penetrations of 0.13 and 0.66 mm, respectively, and a melt time of 15 seconds. Clearly, the molten and resolidified zone is not of uniform thickness: the thickness of

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the resolidified zone increases significantly near the edge, at the right extreme of the figure. Notice the sandwich structure of the resolidified zone.

250 µm

Figure 21.6.8 Micrograph showing the weld morphology of a hot-tool weld of 3.2-mm thick PBT specimens made at a weld temperature of 305°C, melt and weld penetrations of 0.13 and 0.66 mm, respectively, and a melt time of 15 seconds. The resolidified zone is not of uniform thickness, and the thickness of this zone increases significantly near the edge, at the right extreme of the figure. Notice the sandwich structure of the resolidified zone. (Adapted with permission from V.K. Stokes, Polymer Engineering and Science, Vol. 43, pp. 1576 – 1602, 2003.)

The morphology of a hot-tool weld of 6.1-mm thick PBT specimens made at a weld temperature of 320°C, melt and weld penetrations of 0.13 and 0.66 mm, respectively, and a melt time of 10 seconds is shown in Figure 21.6.9. The molten and resolidified zone is not of uniform thickness (Figure 21.6.9a). The high-magnification TEM in Figure 21.6.9b, from a region where the resolidified zone is relatively thin, shows that the weld zone has a sandwich structure comprising two outer regions, with very little if any crystallinity, surrounding a central core with substantial crystallinity. The top and bottom outer layers are about 70 and 50 μm thick, respectively. The crystalline core, which is about 35 μm thick, has crystallinity gradients with very small spherulites at the outer edges; the spherulites in the middle are almost as large as in the bulk material. The TEM in Figure 21.6.9c is from a relatively thick resolidified zone. In this case, while the outer layers are about 35 – 40 μm thick, the central core is about 105 μm thick. Crystallinity gradients across this core are clearly visible. Figure 21.6.9 parts b and c show evidence of flow-induced orientation in the outer layers, especially at the two interfaces between the resolidified regions and the bulk material. 21.6.3.2

Weld Strength

The weld morphologies reported in the previous section show that the weld zone is not homogeneous; it consists of complex, nonhomogeneous layered structures. As such, any weld strength measurements are average system-like properties in the sense explored in Chapters 22 – 24, so that measured weld strengths will depend on the specimen geometry, such as the specimen thickness. In glass-filled materials the strength will also depend on the region of the molded plaque from which the test specimen is obtained. For the weld strengths reported in this section, test specimens for PC were cut from 3.0-, 5.8-, and 12.0-mm thick extruded sheet material. Test specimens for other resins were obtained from specimens cut from 153 × 203-mm edge-gated injection-molded 3.2- and 6.1-mm thick plaques. The edges of each specimen were machined to obtain rectangular blocks of size 76.2 × 25.4-mm × thickness for ensuring accurate alignment of the surfaces during butt welding. Two specimens with machined lateral edges were hot-tool welded along the thickness direction of the 25.4-mm × thickness edges, resulting in a 152.4 × 25.4-mm × thickness (6 × 1-in × thickness) bar that has a continuous “flash,” and hence a continuous weld, except at the ends of the 25.4-mm (1-in) side (Figure 21.6.10a). To obtain a homogeneous

Joining of Plastics

250 µm

100 µm

(a)

(b)

(c)

Figure 21.6.9 Micrographs showing the weld morphology of a hot-tool weld of 6.1-mm thick PBT specimens made at a weld

temperature of 320°C, melt and weld penetrations of 0.13 and 0.66 mm, respectively, and a melt time of 10 seconds. (a) The resolidified zone is not of uniform thickness. (b) High-magnification TEM from a relatively thin resolidified zone, showing two 70 μm (top) and 50 μm (bottom) outer layers surrounding a 35-μm thick crystalline core. (c) High-magnification TEM from a relatively thicker resolidified zone: a 105-μm-thick crystalline core is surrounded by two 35 – 40 μm-thick outer lagers that have very little crystallinity. Notice indications of flow-induced orientation in these outer layers. (Adapted with permission from V.K. Stokes, Polymer Engineering and Science, Vol. 43, pp. 1576 – 1602, 2003.)

weld specimen, 3.17-mm (1/8-in) layers are routed out from each end, resulting in a 152.4 × 19-mm × thickness (6 × 0.75-in × thickness) rectangular specimen (Figure 21.6.10b), which is then routed down to a standard ASTM D638 tensile test specimen with the welded butt joint in its center, as shown in Figure 21.6.10c. These bars are then subjected to a constant displacement rate tensile test corresponding

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to a nominal strain rate of 10 −1 s−1 . During each strength test, the average strain across the weld interface is measured with a 25.4-mm (1-in) gauge-length extensometer. Because of the local nature of failure this extensometer only establishes the lower limit of the strain at failure; the actual strain can be much higher in many cases. WELD FLASH

152.4 mm (6 in)

25.4 mm (1 in)

WELD FLASH

(a)

19 mm (0.75 in)

(b)

12.7 mm (0.5 in) 50.8 mm (2 in)

(c) Figure 21.6.10 Geometry of specimens for determining the strength of butt welds.

Table 21.6.1 lists data on achievable hot-tool weld strengths and strains-to-failure for welds of the 10 thermoplastics: ABS, modified poly(phenylene oxide) (M-PPO), modified polyphenylene oxide/polyamide blends (M-PPO/PA), PA-6,6 (dry-as-molded, DAM), and PA-6,6 (50% relative humidity, RH), polycarbonate (PC), polybutylene terephthalate (PBT), polycarbonate/acrylonitrile-butadienestyrene blends (PC/ABS), polycarbonate/poly(butylene terephthalate) blends (PC/PBT), polyetherimide (PEI), and polyvinyl chloride (PVC). Since the weldability polyamides (nylons) is affected by the moisture content, the data for PA-6,6 are reported both for DAM parts and for parts equilibrated in a room-temperature 50% RH environment.

Table 21.6.1 Achievable strengths of 10 hot-tool welded thermoplastics. a) PA-6,6

ABS 37.9 (2.25 %)

M-PPO 31.7 (2.37 %)

M-PPO/PA 54 11.1 %)

(DAM) 72.6 (19.3)

a) PA-6,6 (50 % RH) 63.5 (21.4)

PC 66.5 (7 %)

PBT 59.5 (3.53 %)

PC/ABS 63 (4.42 %)

PC/PBT 54.8 (4.12 %)

PEI 111.6 (7.1 %)

PVC 63.2 (3.46 %)

1.0

0.8

1.0

0.54

0.58

1.0

0.99

0.91

0.98

0.95

0.98

2.2 %

2%

6.28 %

1.51 %

1.01 %

6.2 %

4.15 %

2.61 %

4.2 %

2.36 %

3.38 %

a) Welds on 3.2-mm thick specimens.

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In this table, the resins in the first row are arranged in alphabetical order. The first number below each resin gives the tensile strength of the resin in MPa, and the second number gives the percentage strain at failure. For example, the grade of ABS used had a tensile strength of 44 MPa and a strain at failure of 2.2%. Clearly, except for PA-6,6, very high hot-tool weld strengths are achievable in both amorphous and semicrystalline resins. Except for PC – for which test specimens were cut from 5.8-mm thick extruded PC sheet – the weld specimens for these materials were cut from injection-molded plaques of different thicknesses: 3.2-mm thick PA-6,6; 5.08-mm thick ABS and M-PPO; 6.35-mm thick ABS; M-PPO/PA; PBT; PC/ABS; and PEI. The welds for these materials were made at hot-tool temperatures of 245°C (475°F), 305°C (580°F), 350°C (660°F), 335°C (635°F), 350°C (660°F), 274°C (525°F), 335°C (635°F), 290°C (555°F), 320°C (610°F), 410°C (770°F), 274°C (525°F). For describing the strengths of welds between dissimilar resins, they are arranged in alphabetical order in the first row and first column, as shown in Table 21.6.2. The first number below each resin gives the tensile strength of the resin in MPa, and the second number gives the percentage strain at failure. For example, the grade of M-PPO used had a tensile strength of 31.7 MPa and a strain at failure of 2.37%. The first of the two numbers along the diagonal corresponding to the same resin gives the relative weld strength achievable (0.8) based on the strength of the resin, and the second number is the nominal strain at failure (2%) across the weld. Table 21.6.2 Achievable strengths among hot-tool welds of some dissimilar thermoplastics. M-PPO 31.7 (2.37%)

M-PPO

0.8

31.7 (2.37%)

2%

PC 66.5 (7%)

PBT 59.5 (3.53%)

PC/ABS 63 (4.42%)

PEI 111.6 (7.1%)

a) 0.4

0.58%

PC

1.0

b) 1.0

66.5 (7%)

6.2%

4.96%

2.54%

PBT

b) 1.0

0.99

d) 0.9

59.5 (3.53%)

4.96%

4.15%

2.84%

PC/ABS

a) 0.4

0.91

63 (4.42%)

0.58%

2.61%

PEI 111.6 (7.1%) a) b) c) d)

c) 0.84

c) 0.84

d) 0.9

0.95

2.54%

2.84%

2.36%

Hot-tool temperature of 245°C (475°F). Dual hot-tool temperatures: PC 246°C (475°F), PBT 302°C (575°F). Dual hot-tool temperatures: PC 329°C (62,475°F), PEI 399°C (750°F). Dual hot-tool temperatures: PEI 427°C (800°F), PBT 274°C (525°F).

The off-diagonal entries list available weld-strength data for welds between dissimilar resins. The first of the two numbers represents the ratio of the weld strength to the strength of the weaker of the two resins, and the second number gives the nominal strain at failure. For example, the numbers 0.4%, 0.58% in the entry corresponding to the M-PPO row and the PC/ABS column indicates that welds between M-PPO and PC/ABS can attain 0.4 of the strength of ABS M-PPO – which is the weaker of the two materials – and that the nominal strain to failure is 0.58%. This weld strength was obtained at a hot-tool temperature of 245°C (475°F).

Joining of Plastics

High weld strengths among the dissimilar resins PC, PBT, and PEI could only be obtained by using different hot-tool temperatures for the pairs of materials: For PC to PBT welds, PC 246°C (475°F), PBT 302°C (575°F); for PC to PEI welds, PC 329°C (62,475°F), PEI 399°C (750°F); and for PEI to PBT welds, PEI 427°C (800°F), PBT 274°C (525°F). Table 21.6.3 lists data on achievable hot-tool weld strengths and strains-to-failure for welds 6.1-mm thick specimens of PBT and 30-GF-PBT, both of which were welded at hot-tool temperatures of 335°C (635°F). The low relative strength, 0.55, of the glass-filled material corresponds to an actual strength of 53.7 MPa, which is 82.6% of the strength of the base resin (654 MPa). This table also has data for the welding of 3.2-mm thick specimens of M-PPO and 30-GF-M-PPO, which were welded at hot-tool temperatures of 335°C (635°F) and 290°C (555°F), respectively. Here, the relative weld strength of 0.77 of the glass-filled M-PPO, which corresponds to 61.7 MPa, which is more than the strength of the base resin (43.9 MPa). Table 21.6.3 Achievable strengths of hot-tool welds of glass-filled grades of PBT and M-PPO. a) PBT

a) 30-GF-PBT

b) M-PPO

c) 30-GF-M-PPO

65 (3.5%)

97.6 (3.15%)

43.9 (2.5%)

80.1 (1.78%)

a) PBT

0.99

0.55

0.8

0.77

65 (3.5%)

4.15%

%

2%

1.33%

a) 6.1-mm thick specimens welded at hot-tool temperature of 335°C (635°F). b) 3.2-mm thick specimens welded at hot-tool temperature of 305°C (580°F). c) 3.2-mm thick specimens welded at hot-tool temperature of 290°C (555°F).

21.6.4

Infrared Welding

Infrared welding is a variant of the hot-tool welding process in which the surfaces to be joined do not contact the matching surfaces of a heated metallic tool. Rather, the parts are brought close to its high-temperature surfaces, and melting of the part surfaces occurs by radiant heating; since the part surface does not contact the tool, stringing associated with hot-tool welding does not occur. The molten surfaces are then brought together, and the interface is allowed solidify under pressures. The three phases of the process are schematically shown in Figure 21.6.11: Phase 1 corresponds to the heating phases 1 and 2 of hot-tool welding. And phases 2 and 3 of this process correspond, respectively, to phases 3 and 4 of hot-tool welding. Thus, other than how the mating surfaces are melted, the achievable weld is the same as for hot-tool welding. Initially, the radiative surface heating was achieved by using a red-hot metal piece, as shown in Figure 21.6.11. An important application was for welding flexible, complex thermoplastic elastomeric extrusions into rectangular gaskets for refrigerator doors. Then, the metallic heaters were replaced by ceramic heaters with reflectors to focus infrared radiation on the surface to be heated. With substantial progress in the design of infrared radiative heat emitters, which can be embedded on 3D surfaces that match complex surfaces with 3D welding contours, this process is now used for assembling instrument panels, door trims and center consoles with complex, two- or three-dimensional welding contours.

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p(t) II

I

III

t

• HOT TOOL

WELD BEAD

p0

(a)

(b)

p0

(c)

Figure 21.6.11 Three phases of the infrared welding process. (a) Matching surfaces brought in proximity of the heated hot tool without touching its surface. (b) Parts separated and hot tool retracted. (c) Molten films in contact solidify under pressure.

The rate of heating and the depth of the material penetrated by infrared radiation depend on the thermal absorptive characteristics of the material. Cycle times can be reduced by matching this to the frequency of infrared radiation, emitters for which are available in the short-wave (0.78 – 2 μm) and medium-wave (2 – 4-μm) bands. Shorter waves are absorbed in deeper layers, while medium waves are more effective in surface layers. Power-level and time control on emitters are used for optimizing the welding processes. Figure 21.6.12 shows two molded polypropylene parts that are infrared welded to form the base for a hot-water pressure washer. Both the parts facing upward in Figure 21.6.12a, and in the assembly position in Figure 21.6.12b, show the complexity of the parts. In both these figures the white line above the lower part shows the location of the infrared heater element. The image in Figure 21.6.12c shows the welded assembly. Figure 21.6.13a shows the process for infrared welding of the air ducts for the window defrost (top part) to the inside of the instrument panel (bottom part). The insert for the instrument cluster (lower part of Figure 21.6.13b) is inserted in an intermediate step. 21.6.5

Laser Welding

Laser welding uses a laser beam to heat the surface to be welded. In contrast to wide areas that can be heated by infrared welding, lasers provide sharp, high energy-density beams that can heat precisely defined areas. Computer-controlled optics can be used to heat wider areas by rapidly scanning the beam

Joining of Plastics

(a)

(b)

(c) Figure 21.6.12 Images of an infrared-welded assembly. (a) Both parts facing upwards. (b) Parts in welding position. (c) Welded part. In all three images the white line shows the position of the infrared heating element. (Photos courtesy of bielomatik Leuze GmbH + Co. KG.)

across the surface or to heat a complex seam. This way, instead of using a complex, contoured array of infrared emitters to heat a complex 3D surface, a single or a bank of laser beams can be used to heat complex seams by controlling the beam position. This eliminates the need for a different complex heating tool for each different application: in laser welding, only the beam path needs to be reprogrammed.

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(a)

(b)

Figure 21.6.13 Glass-filled polypropylene, infrared-welded instrument panel assembly. (a) Air ducts for the window defrost (top part) welded to the inside of the instrument panel (bottom part). (b) The insert for the instrument cluster (lower part) is inserted in an intermediate step. (Photos courtesy of bielomatik Leuze GmbH + Co. KG.)

Laser welding is done in one of the two ways schematically shown in Figure 21.6.14: (i) In the first mode the laser beam directly illuminates and heats the surfaces to be welded (Figure 21.6.14a). Mirrors can be used to split the beam to simultaneously scan both surfaces. Once the surfaces have melted, the mirror assembly is withdrawn, and the parts are brought together, just as in hot-tool welding. The beam is directed alternately to each of the secondary mirrors by the main, rapidly oscillating mirror in such a

OSCILLATING MIRROR

LASER BEAM

LASER BEAM ABSORBING MATERIAL

TRANSPARENT MATERIAL

B

A BEAM SPLITTING MIRROR

(a)

MOLTEN MATERIAL

(b)

Figure 21.6.14 Schematic layouts of laser welding processes. (a) Beam splitting mirrors used to heat surfaces through beam scanning. (b) Through-transmission infrared heating of weld interface.

Joining of Plastics

way that the entire area to be welded is scanned by the beam. The laser scan has to be fast enough to uniformly heat the surface without some parts cooling off. (ii) In the second mode (Figure 21.6.14b), called through-transmission infrared welding, the two parts to be welded – one of which has to be transparent to the laser beam – are first clamped in their assembled (welded) mode. Then the joints surface is illuminated by a laser beam passing through the transparent part to heat the surface of the second, radiation absorbing part. Because of the relatively low cost of lasers, laser welding is becoming an important welding technique for applications such as those requiring clean, debris-free joints. Figure 21.6.15a shows a PA6 (nylon 6) steering oil container assembled by laser welding two parts. This assembly was switched to laser welding from vibration welding to avoid contamination by debris. Figure 21.6.15b shows a quality-level-temperature (QLT) sensor for predicting oil quality and detecting oil level, made of PA6,6 (nylon 66). Laser welding is ideal for this application because of the sensitive, embedded electronics – ultrasonic welding could damage the electronics, and IR welding could damage the chips and wire bonds from peripheral radiation heating. An interesting feature of this application is the black lid, made by using a special colorant that is transparent to infrared laser radiation but looks black in visible light.

(a)

(b)

Figure 21.6.15 Laser welding examples. (a) PA6 steering oil container. (b) PA6,6 QLT sensor for predicting oil quality and detecting oil level. (Photos courtesy of bielomatik Leuze GmbH + Co. KG.)

21.7 Friction Welding In friction welding the heat for melting the weld interface is provided by frictional heating of the interface through relative motion. Of the four commercially important methods under this category, in spin, vibration, and orbital welding, frictional heat is generated through in-plane rubbing of the interface. Spin welding works for tubes and rods. Vibration welding and orbital welding – which is a variant of

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vibration welding – are suited for structural welds in both small and large parts. But they can only be used for welding plane or near-plane interfaces. In the fourth commercial category, ultrasonic welding, the relative motion at the weld interface in most applications is normal to the interface. Although in some applications the relative motion can be along the interface. 21.7.1

Spin Welding

Spin welding, also called friction welding, can be used for welding components along plane, circular mating surfaces. In this process, the parts to be joined are rotationally rubbed relative to each other, under pressure, about an axis normal to the plane surfaces to be joined (Figure 21.7.1). The resulting frictional heating at the plane interface causes the plastic to heat and melt. When the relative motion is stopped, the molten film solidifies under pressure, resulting in a weld. The main process parameters for spin welding are the weld (relative rubbing) speed, the weld pressure, and the weld time. This process can produce high-quality welds. The process phenomenology for spin welding is similar to that for vibration welding, so it should be possible to estimate the effects of process conditions on spin welded joints from similar information on vibration-welded joints. The main advantage of this process is its simplicity and speed. Drill presses and lathes can easily be modified to make prototype parts. The main disadvantage of the process is that it can only be used for bonding circular cross-sectioned parts that do not require angular alignment. Variants of this process can be used when angular alignment is important, or when the parts do not have a circular cross section. Instead of unidirectional circular motion, the parts can be rubbed through an oscillatory rotary motion.

WELD INTERFACE

•

•

•

p0

ω = 2πn

Figure 21.7.1 The spin welding process.

21.7.2

Vibration Welding

In vibration welding, also called linear welding and linear friction welding, frictional work done by vibrating two parts, under pressure, along their common interface is used to generate heat to effect a weld (Figure 21.7.2). This process is ideally suited to the welding of thermoplastic parts along relatively flat seams. The process can also accommodate seams whose out-of-plane curvature is small.

Joining of Plastics

p0 VIBRATORY MOTION z = a sin(2πnt) WELD INTERFACE y

O•

x z

Figure 21.7.2 The vibration welding process.

In this process, the parts to be joined are placed in fixtures and then brought together under pressure. The parts are then vibrated in the plane of the interface until the interfacial material melts. The vibratory motion is stopped, and the molten film is allowed to solidify, under pressure, resulting in a weld. The main process parameters are the weld frequency of the oscillatory motion, its amplitude, and the weld time. Most industrial machines operate at a fixed weld frequency of 120-Hz, although 240-Hz machines are also available. The amplitude of vibration is normally less than 5 mm (0.2 in) and the weld time varies from 1 to 10 seconds. In most industrial machines the vibratory motion is obtained by exciting a tuned spring-mass system, either by means of electromagnets or hydraulically. As a result, the amplitude of vibration cannot conveniently be independently controlled and requires a change in the mass of the oscillating components. A typical vibration weld for a thermoplastic has four phases. In the first phase, Coulomb friction generates heat at the interface, raising its temperature to the point at which the polymer can undergo viscous flow. In the second phase, the interface begins to melt and the mechanism of heat generation changes from solid Coulomb friction to viscous dissipation in the molten polymer. The molten polymer begins to flow in a lateral direction, resulting in an increase in the weld penetration – the distance by which the parts approach each other as a result of lateral flow. In the third phase, the melting and flow are at a steady state, and the weld penetration increases linearly with time. When the machine is shut off, the weld penetration continues to increase because the weld pressure causes the molten film to flow until it solidifies. This is phase 4. These four phases, which are schematically shown in the penetration-time plot in Figure 21.7.3, are also typical of spin welding. Of course, in contrast to vibration welding, heat is generated in the first three phases of spin welding by a continuous, unidirectional rotational motion. Tests on a range of polymers have shown that the most important parameter affecting the strengths of welds of the same material is the weld penetration. Very high weld strengths, equal to those of the resin, can be achieved for penetrations greater than a threshold value. This threshold is most likely the penetration, 𝜂 T , at the end of the transient phase at which a steady state is achieved.

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η

PENETRATION

744

•

• II

I

ηT

III

IV

t3

t4

• •

t1

•

t2

t

tc TIME Figure 21.7.3 Four phases of the vibration welding process.

As indicated in Figure 21.7.3, the total cycle time tc , is the sum of the cycle times for the four individual phases. Larger weld pressures on the weld interface, and higher weld frequencies, result in shorter phases 1 and 2. Once a steady state has been achieved, the vibratory can be cut off, thereby reducing the overall cycle time. This process has several potential advantages for joining large, flat-seamed, thermoplastic parts: (i) relatively short cycle times, (ii) simple equipment, and (iii) insensitivity of the process to surface preparation. In contrast to hot-tool welding, in which the interfaces to be welded are heated conductively, the heating is localized. This process is very controllable and is much less likely to cause material degradation at the interface because of overheating. The main disadvantage of this process is that it is limited to flat-seamed parts. Also, this process is not suited to welding low-modulus thermoplastics, such as some thermoplastic elastomers (TPEs). The first phase of vibration welding normally produces solid debris, which can be eliminated by preheating the surfaces by infrared radiation. This modified process has been used for making particle-free joints in applications such as air-duct pipes, fluid containers, tank systems, and filter housings. Figure 21.7.4 shows a glass-filled polypropylene dishwasher-pump housing made by vibration welding two molded parts. The first, innovative, path-breaking all-plastic bumper, directly attached to the car frame, capable of an 8-kph (5-mph) barrier impact, was used in the 1984 Ford Escort shown in Figure 2.4.1; the car on the right has a conventional steel bumper beam. Except at the two ends and the mounting points, the bumper beam has the D-shaped cross section made by vibration welding a molded vertical back plate to a ribbed C-shaped molded section shown in Figure 21.7.5a. Figure 21.7.5b shows the fracture surfaces of such a beam tested to failure; the integrity of the cross section of the failed surfaces illustrates how strong vibration welds can be. This application likely represents one of the most demanding performance requirements for vibration welds.

Joining of Plastics

(a)

(b)

Figure 21.7.4 Glass-filled polypropylene dishwasher-pump housing. (a) Two glass-filled polypropylene injection-molded components. (b) Welded assembly. (Photos courtesy of GE Appliances.)

(a)

(b)

Figure 21.7.5 Vibration welded PC-PBT bumper. (a) D-shaped bumper cross section (b) Welded tested to failure. Notice that the welds did not fail. (Photos courtesy of Ford Motor Company.)

Of all later automotive applications, this bumper may be regarded as the first automotive full-sized structural application of an unreinforced thermoplastic. A blend of polycarbonate and poly(butylene terephthalate) – with rubber modifiers to provide ductility even at low temperatures – was specially

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developed for this application. This blend is now routinely used in many applications, such as lawnmower decks, which require impact resistance. A relatively more recent application is that of automotive inlet manifolds made by welding 30 wt% glass-filled PA6,6 (nylon 66) injection-molded parts by a two-step vibration welding process. Figure 21.7.6a shows upper and lower halves of ducts that were vibration welded. Figure 21.7.6b shows the before and after photos of a second assembly that sits on top of the engine, which was welded in a separate operation. One end of the welded duct is then attached to the top part (Figure 21.7.6b) by means of screws; the second end attaches to the engine.

(a)

(b)

Figure 21.7.6 Welded, injection-molded automotive manifolds made of 30 wt% glass-filled PA6,6 (nylon 66). (a) Upper and lower halves of injection-molded manifold parts welded to form one assembly. (b) Before and after photos of a second manifold assembly vibration-welded in a second operation. (Photos courtesy of bielomatik Leuze GmbH + Co. KG.)

This type of a complete vibration-welded and bolted plastic manifold assembly for a Volkswagen engine is shown in Figure 2.4.4. Notice the complexity in this assembly that vibration welding made possible. 21.7.2.1

Weld Morphology

The preparation of PC and PBT weld specimens is described in Section 21.6.3.1. For studying the morphology of the HAZ, 5.8-mm thick undried PC specimens were welded at a weld frequency of 120 Hz, a weld amplitude of 1.59 mm, a penetration setting of 0.51 mm, and three weld pressures of 0.52, 1.72, and 6.89 MPa. Macrographs of the weld cross sections of the PC welds at three weld pressures are shown in Figure 21.7.7a – c. At the lowest weld pressure of 0.52 MPa, Figure 21.7.7a shows gas bubbles in the neighborhood of the central weld plane. The density of these bubbles increases from the center to the edges. It is known that during welding the pressure acting on the molten zone is the highest at the mid thickness of the specimen, and decreases continuously toward the edges where the pressure is atmospheric. A Newtonian-viscosity based flow model predicts a parabolic pressure distribution. The differences in the bubble distribution across the weld zone can be explained by nonuniform pressure

Joining of Plastics

1 mm

1 mm

(a)

(d)

1 mm

1 mm

(b)

(e)

1 mm

1 mm

(c)

(f)

Figure 21.7.7 Macrograph of morphologies of 120 Hz vibration welds of PC and PBT. Welds of 5.8-m thick PC specimens at weld pressures of (a) 0.52, (b) 1.72, and (c) 6.89 MPa. Welds of 6.1-m thick PBT specimens at weld pressures of (d) 0.52, (e) 1.72, and (f) 6.89 MPa. (Adapted with permission from V.K. Stokes, Polymer Engineering and Science, Vol. 43, pp. 1576 – 1602, 2003.)

distribution in the melt. The increase in temperature in the weld zone results in the absorbed moisture coming out of solution. However, the vapor pressure has to be higher than the local pressure in the melt for bubbles to form. Higher weld pressure can then be expected to result in fewer bubbles, as is the case. At a weld pressure of 1.72 MPa (Figure 21.7.7b), bubbles are only visible at the outermost edges. At the highest pressures 6.89 MPa no bubbles are visible in the weld zone (Figure 21.7.7c). The welds have a fairly uniform thickness except near the edges. As expected, the thickness of the weld zone decreases with increase in weld pressure. A comparison of the morphologies in Figure 21.7.7 with those in Figure 21.7.6 shows that vibration and hot-tool welds have very different morphologies. First, while in vibration welds the HAZs are almost

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of uniform thickness across the weld except near the edges, those in hot-tool welds are nonuniform with the thickness increasing from the center to the edges. The structures of the bead, or “flash,” are also quite different. In vibration welding, the intermittent vibratory motion results in the flash being extruded out in a non-monotonic, time-dependent manner. The flash tends to divide into two thin “sheets” the nonsmoothness in which is caused by the pulsatile nature of the lateral flow. Furthermore, the sudden reduction in the pressure in the melt as it exits to the atmosphere causes the moisture to come out of solution, resulting in a “foaming” of the flash. In contrast to vibration welding, the weld beads in hot-tool welding are thicker, have fewer bubbles, and tend to form the characteristic mushroom shape that results from the bulk flow from the central core pushing out at higher velocities and then dividing laterally. Because of their thickness, hot-tool weld beads can affect weld strength far more than vibration weld beads. Macrographs of the morphology of 120 Hz vibration welds of 6.1-mm thick PBT specimens made at weld pressures of 0.52, 1.72, and 6.89 MPa are shown, respectively, in Figure 21.7.7d – f: In contrast to PC, the macrographs do not show any bubbles in the PBT welds. The weld zones are of uniform thickness, except near the edges. The weld-zone thickness decreases with increasing weld pressure. Figure 21.7.8 shows the morphology of a 120 Hz vibration weld of 6.1-mm thick PBT specimens made at a weld pressure of 6.89 MPa. The macrograph in Figure 21.7.8a shows that the molten and resolidified zone has a constant thickness of about 150 μm. Near the edge this zone has a “winged” structure. The high-magnification TEM in Figure 21.7.8b from the “winged” region closer to the edge shows that the approximately 130-μm thick resolidified region has an interesting layered structure with many features. The high-magnification TEM in Figure 21.7.8c shows the morphology of a 120 Hz vibration weld of 6.1-mm thick PBT specimens made at a weld pressure of 3.45 MPa. Note the absence of any significant crystallinity in the resolidified zone, which is about 140 μm thick. The high-magnification TEMs in Figure 21.7.9 show the morphology of a 120 Hz vibration weld of 6.1-mm thick PBT specimens made at a weld pressure of 1.72 MPa. Figure 21.7.9a shows that the molten and resolidified zone away from the edges has a constant thickness of about 140 μm; this region has very little crystallinity. Figure 21.7.9b shows evidence of small crystallites in the 165-μm thick resolidified zone closer to the edge. Notice the large crystallinity gradients across the weld zone and the large, squeeze-flow induced distortion of the small spherulites, especially at the interfaces between the resolidified material and the bulk, parent material. A comparison of the morphologies in Figures 21.7.7d – f to 21.7.9 with those in Figures 21.6.7 – 21.6.9 shows that vibration and hot-tool welds of PBT have very different morphologies. First, while in vibration welds the weld zones almost of uniform thickness across the weld except near the edges, the weld zones in hot-tool welds are nonuniform with the thickness increasing from the center to the edges. The most striking difference in the morphologies of PBT vibration and hot-tool welds is in the structures of the HAZ. In vibration welding the melt thickness is small, and the temperature of the solid material adjacent to the melt is low, resulting in relatively high cooling rates during the final joining phase. As a result, the molten material does not have sufficient time to form crystals of any significant size. In contrast to this, in hot-tool welding the melt layer is relatively thicker and, because of heat conduction, the adjacent solid is warmer, resulting in relatively slower cooling rates. This difference is sufficient to allow a layer of material in the middle of the melt to crystallize. The structures of the bead, or “flash,” are also quite different. In vibration welding of PBT, the flash has a wing-like shape similar to that of PC, except that the PBT flash does not have bubbles. In contrast to vibration welding, the weld beads in hot-tool have a bulbous shape. Again, bubbles are absent, except for the occasional bubble resulting from the air trapped at the weld interface during the joining phase

Joining of Plastics

that is transported out by the lateral motion of the melt. This transport is made easier in PBT because its melt viscosity is a much lower than that of PC.

250 µm

100 µm

(a)

(b)

(c)

Figure 21.7.8 Micrographs showing the weld morphology of a 120 Hz vibration weld of 6.1-mm thick PBT specimens. Weld pressure of 6.89 MPa: (a) The molten and resolidified zone has a constant thickness of about 150 μm. Near the edge this zone has a “winged” structure. (b) A high-magnification TEM from the “winged” region closer to the edge. (c) High-magnification TEM of the morphology of a weld made at a weld pressure of 3.45 MPa. Note the absence of any significant crystallinity in the resolidified zone, which is about 140-μm thick. (Adapted with permission from V.K. Stokes, Polymer Engineering and Science, Vol. 43, pp. 1576 – 1602, 2003.)

21.7.2.2 Weld Strength

As with the hot-tool weld morphologies discussed in Section 21.6.3.2, the morphologies of vibration welds reported in the previous section show that the weld zone is not homogeneous. It consists of complex, nonhomogeneous layered structures, so that any weld strength measurements are average

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100 µm

750

(a)

(b)

Figure 21.7.9 High-magnification TEMs showing the morphology of a 120 Hz vibration weld of 6.1-mm thick PBT specimens made at a weld pressure of 1.72 MPa. (a) The molten and resolidified zone away from the edges has a constant thickness of about 150 μm and has very little crystallinity. (b) Small crystallites can be seen in the resolidified zone closer to the edge. (Adapted with permission from V.K. Stokes, Polymer Engineering and Science, Vol. 43, pp. 1576 – 1602, 2003.)

system-like properties in the sense explored in Chapters 22 – 24, and the measured weld strengths will depend on the specimen geometry, such as the specimen thickness, used. In glass-filled materials the strength will also depend on the region of a molded plaque from where the test specimen is obtained. As for hot-tool welds, for the weld strengths reported in this section, test specimens for PC were cut from 3.0-, 5.8-, and 12.0-mm thick extruded sheet material. Test specimens for other resins were obtained from specimens cut from 153 × 203-mm edge-gated injection-molded 3.2- and 6.1-mm thick plaques. The edges of each specimen were machined to obtain rectangular blocks of size 76.2 × 25.4-mm × thickness for ensuring accurate alignment of the surfaces during butt welding. Two specimens with machined lateral edges were vibration welded along the thickness direction of the 25.4-mm × thickness edges, resulting in a 152.4 × 25.4-mm × thickness (6 × 1-in × thickness) bar that has a continuous “flash,” and hence a continuous weld, except at the ends of the 25.4-mm (1-in) side (Figure 21.6.10a). Welded specimens were then routed down to the standard ASTM D638 tensile test specimen with the welded butt joint in its center shown in Figure 21.6.10c. These bars are then subjected to a constant displacement rate tensile test corresponding to a nominal strain rate of 10 −1 s−1 . During each strength test, the average strain across the weld interface is measured with a 25.4-mm (1-in) gauge-length extensometer. Because of the local nature of failure this extensometer only establishes the lower limit of the strain at failure; the actual strain can be much higher in many cases. Table 21.7.1 lists data on achievable vibration weld strengths and strains-to-failure for welds among the nine thermoplastics: ABS, modified poly(phenylene oxide) (M-PPO), modified poly(phenylene oxide)/polyamide blends (M-PPO/PA), polycarbonate (PC), poly(butylene terephthalate) (PBT),

Table 21.7.1 Achievable strengths of some vibration welded thermoplastic resins. ABS 44 (2.2 %)

M-PPO 45 (2.5 %)

M-PPO/PA 58 (>18 %)

PC 68 (6 %)

PBT 65 (3.5 %)

ABS

0.9

0.76

0.83

0.8

44 (2.2 %)

2.1 %

2.1 %

1.7 %

1.6 % 0

M-PPO

0.76

1.0

0.22

0.24

45 (2.5 %)

2.1 %

2.4 %

0.35 %

0.4 %

M-PPO/PA

0.22

1.0

0.29

58 (>18 %)

0.35 %

>10 %

0.75 %

0

0.24

0.29

1.0

1.0

68 (6 %)

1.7 %

0.4 %

0.75 %

6%

1.7 %

PBT

0.8

0

0

65 (3.5 %)

1.6 % b)

b)

0.96 3.5 %

b)

b)

0.20

b)

b)

b)

50 (>10 %)

1.51 %

0.47 %

4.9 %

2.5 %

PMMA

62.6 (2.26 %)

1.0

0.95

0.99

4.9 %

2.75 %

3.06 % 0.21

0.98

a)

0.18 0.47 %

0.36 %

1.0

0.95

0.94 1.15 %

0.95

0.18

4.1 %

0

4.1 %

0.73

0.95

0.73 1.51 %

0.98

PC/PBT

2.75 %

0.65

2.5 %

0.85

0

62.6 (2.26 %)

0.45

2.3 %

0.65

a) PMMA

1.05 %

0.45

1.14 %

PEI 119 (6 %)

1.14 %

0.7

1.8 %

1.05 %

PEI

0.7

0.76

1.8 %

119 (6 %)

0.20

b)

0.41 %

1.8 %

0.76

0.85

1.36 %

60 (4.5 %)

0.85

1.0 1.7 %

PC/PBT 50 (>10 %)

1.8 %

0.41 %

0.83

b)

b)

1.36 %

PC

PC/ABS

PC/ABS 60 (4.5 %)

b)

0.26

0.53 %

b)

0.26

0.53 % 1.0 >15 % c)

1.0

6%

0.94

0.99

0.21

1.0

1.15 %

3.06 %

0.36 %

2.24 %

a) Welds of PPMA to all materials on 5.8-mm thick specimens. b) Data obtained through tests on 3.2-mm thick specimens. c) High strength was only achieved through high-frequency welds (250 and 400 Hz).

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Introduction to Plastics Engineering

polycarbonate/acrylonitrile-butadiene-styrene blends (PC/ABS), polycarbonate/poly(butylene terephthalate) blends (PC/PBT), polyetherimide (PEI), and poly(methyl methacrylate) (PMMA). Except for PC – for which test specimens were cut from 5.8-mm thick extruded sheet of PC – the weld specimens for these materials were cut from injection-molded plaques of different thicknesses: 5.08-mm thick ABS, M-PPO; 6.35-mm thick ABS, M-PPO/PA, PBT, PC/ABS, and PEI. Table 21.7.2 lists data on achievable vibration weld strengths and strains-to-failure for welds of the six thermoplastics: acrylonitrile-styrene-acrylic (ASA), both DAM and equilibrated at 50% relative humidity (50%RH) amorphous nylon (PA-A (DAM), PA-A (50%RH)), DAM and 50%RH equilibrated nylon 6 (PA-6 (DAM) and PA-6 (50%RH)), DAM nylon 6,6 (PA-6,6 (DAM)), polyphenylene oxide/modified poly(phenylene sulfide) blend (PPO/PPS), and polyvinyl chloride (PVC). Except for PPO/polyphenylene sulfide (PPS) – for which test specimens were cut from 3.175-mm thick injection-molded plaques – the weld specimens for these materials were cut from 6.1- to 9.35-mm thick injection-molded plaques. Table 21.7.2 Achievable strengths of vibration welded ASA, nylons, and PVC. ASA 32.5 (2.9%)

PA-A (DAM) 120 (6.34)

PA-A (50% RH) 120 (6.34)

PA-6 (DAM) 67.7 (2.61)

PA-6 (50% RH) 62.0 (2.26)

PA-6,6 (DAM) 78.5 (4.5)

PPO/PPS 52.6 (4.58%)

PVC 63.2 (3.46%)

0.46

0.97

1.0

1.0

1.0

1.0

0.89

0.97

0.9%

6.76%

6.91%

3.42%

4.15%

4.5%

2.56%

3.30%

Table 21.7.3 lists data on achievable vibration weld strengths and strains-to-failure for welds of particulate- and glass-filled grades of PA-6,6, PBT, PEB, PEI, and SMA. Table 21.7.3 Achievable strengths of vibration welds of particulate- and glass-filled grades of PA-6,6, PBT, PEB, PEI, and SMA. 33-GF-PA-6,6 PBT 15-GF-PBT 30-GF-PBT 10-PF-PBT 30-PF-PBT 65-PF-PEB 30-GF-PEI 16-GF-SMA 163.8 (3.06%) 65 (3.5%) 86.7 (2.4%) 90.6 (2.9%) 63.2 (3.9%) 52.7 (2.6%) 52.7 (0.74%) 124.4 (1.68%) 84.3 (2.5%)

0.57

0.96

0.67

0.53

0.89

0.62

0.87

0.7

0.35

1.2%

3.5%

1.45%

1%

2.3%

0.9%

0.61%

1.3%

1.27%

Table 21.7.4 lists data on achievable vibration weld strengths and strains-to-failure for welds of 20-GF-M-PPO to itself and to M-PPO. Table 21.7.4 Achievable strengths of vibration welds of 20-GF-M-PPO to itself and to M-PPO. M-PPO

20-GF-M-PPO

45 (2.5%)

82 (3.2%)

M-PPO

1.0

a)

45 (2.5%)

2.4%

1.95%

20-GF-M-PPO

a) 1.0

0.71

82 (3.2%)

1.95%

1.5%

a)

Data obtained through tests on 3.2-mm thick specimens.

1.0

Joining of Plastics

21.7.3

Orbital Welding

In conventional orbital welding, each point on the moving component executes a small-radius circular motion, such that all the points move in phase. The mechanisms of melting and flow are the same as for vibration welding. By varying the amplitudes in two directions, linear vibration welding can be programmed along different directions in a planar weld. This can simplify tooling for off-axis welds. 21.7.4

Ultrasonic Welding

Ultrasonic welding uses high-frequency (15 – 70 kHz), low-amplitude (15 – 60 μm, 0.5 – 2.5 × 10−3 in) mechanical vibrations to locally heat and melt thermoplastics at a joint interface to effect welds. The parts to be joined are held together and subjected to ultrasonic vibrations under constant interfacial pressure. Although the precise mechanisms causing localized heating are not well understood, heat generation is believed to occur by a combination of surface and intermolecular friction. Two types of joints are made by this process (Figure 21.7.10). In the first and main type (Figure 21.7.10a), the ultrasonic vibrations occur in a direction normal to the surfaces to be joined. This process does not work for absolutely flat contact surfaces. Instead, one of the surfaces to be joined has a triangular protrusion, called an energy director, the apex of which contacts the second flat surface. In the second joint type, called a shear joint, the principle for which is shown in Figure 21.7.10b, a major component of the ultrasonic vibrations is parallel to the mating surfaces. Energy directors can be used to augment this process, as in the “mash” joint design shown in Figure 21.7.10c. Heat generation can be assumed to occur mainly by frictional (shear) forces at the interface.

VIBRATORY MOTION

y = a sin(2πnt) p0

p0

BUTT JOINT

SHEAR JOINT

MASH JOINT

(a)

(b)

(c)

p0

ENERGY DIRECTOR

Figure 21.7.10 Types of joints made by the ultrasonic welding process. (a) Butt joint. (b) Shear joint. (c) Mash joint.

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Ultrasonic vibrations required by this process are generated by electrically exciting disk-shaped piezoelectric or magnetostrictive transducers that are clamped under pressure between metal blocks. The transducer (converter) assembly is designed to resonate at the desired frequency. The converter is coupled to an additional component called a booster, in which a reduced cross section results in an amplification of the amplitude of the mechanical vibration. The tool (attached to the booster) that transmits this booster motion to the workpiece is called a horn, or a sonotrode. In addition to transmitting the mechanical vibration and the contact force, the sonotrode can be designed to effect an additional amplification of the amplitude. Further, the working end of the sonotrode is designed to mate with the workpiece geometry. For forming operations, such as staking and swaging, the sonotrode shape determines the end part geometry. Proper sonotrode design and tuning for the plastic material volume in a part is crucial for this process. In ultrasonic welding, the acoustic stack, comprising the converter-booster-sonotrode assembly, is lowered under pressure onto the workpiece held in an anvil, or fixture. While maintaining this pressure at the sonotrode-workpiece interface, the ultrasonic signal is turned on. The energy director in the workpiece weld interface heats up, melts, and flows out laterally. Of course, during this process, the sonotrode moves downwards with the workpiece, thereby maintaining the desired pressure at the interface, while keeping the amplitude constant. The ultrasonic vibration is then turned off, and the thin, molten plastic film is allowed to cool under pressure, resulting in a weld. The distance between the sonotrode-workpiece interface and the joint interface affects the weld process and weld quality. Ultrasonic joints are therefore further classified into two categories (Figure 21.7.11):

SONOTRODE

NEAR-FIELD WELDING

FAR-FIELD WELDING

(a)

(b)

Figure 21.7.11 Two regimes of the ultrasonic welding process. (a) Near-Field Welding. (b) Far-Field Welding. (Adapted from figure courtesy of Herrmann Ultrasonics, Inc.)

Joining of Plastics

In near-field, or direct ultrasonic, welding the sonotrode-workpiece interface is “close” to the joint interface; say within 6 mm (0.25 in). In far-field, or indirect ultrasonic, welding this distance is “large;” say, larger than 6 mm. One reason for this distinction is that the larger this distance is, the greater will be the attenuation of the ultrasonic motion caused by damping, resulting in a reduction of the energy arriving at the joint interface. More important is the effect of part resonance on the amplitude of motion at the energy director, and the consequent effect on heating and melting. The relative sizes of the converter-booster-sonotrode assembly for three welding frequencies are shown in Figure 21.7.12. Lower weld frequencies result in higher amplitudes and larger sonotrode widths, thereby making possible longer weld seams. The higher amplitudes help in providing more energy for melting plastic, which can be important for welding high-performance plastics with higher glass-transition temperatures and melting points. One advantage of the 20-kHz system is that this frequency is at the upper end of human hearing and, therefore, does not always require sound abatement. Still lower-frequency systems operating at 15 kHz, with bigger sonotrodes and larger amplitudes can be used when the 20-kHz systems give marginal results; they are more effective for welding semicrystalline plastics. And, because there is lower attenuation through the material, these low-frequency systems are more effective for welding softer, lower-modulus plastics, and in far-field welding. Because of their lower sonotrode amplitudes, high-frequency systems are more appropriate for welding smaller, more delicate parts.

CONVERTER

BOOSTER

SONOTRODE

20 kHz

30 kHz

35 kHz

(a)

(b)

(c)

Figure 21.7.12 Relatives sizes of the converter-booster-sonotrode assembly for (a) 20 kHz, (b) 30 kHz, and (c) 35 kHz. (Adapted from figure courtesy of Herrmann Ultrasonics, Inc.)

During welding, the workpiece undergoes very complex motions. The transmission of ultrasonic vibrations through the workpiece sets up standing wave patterns in which there are nodes with zero displacements, and other regions with high-amplitude motion. The current view is that the joint interface must be placed in regions of maximum stress, which are not in the regions of maximum motion. Thus, in ultrasonic welding, part geometry plays a crucial role in the transmission of energy to the joint and in how it affects the heating/melting phenomenon at the joint interface.

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Currently, ultrasonic welding is the most widely used welding method for thermoplastics. Cycle times are short, typically on the order of a second. The equipment is compact and lends itself to automation. Its major shortcoming is that current technology does not permit the welding of large seams. Also, it is not suited to welding low-modulus thermoplastics, such as some TPEs. Variants of ultrasonic welding, such as staking, spot welding, swaging, insertion, and embedding are addressed in a separate subsection. Figure 21.7.13 shows how a pair of injection-molded ABS parts are ultrasonically welded together to make a garden hose nozzle. Notice the layouts of the energy directors. The weld must be such that the garden hose does not leak at the seams and at the two inserts. Because of its size this part was welded by a 20-kHz machine. The sequence of photos in Figure 21.7.14 shows how an innovative horn (sonotrode) design can be used to assemble a gear box with PA6 gears in a 33-GF-PA6,6 housing with a long, curved seem. The housing parts with the gears inside are placed in welding fixture and then welded at the eight-shaped seam by using the multi-headed tool to form a hermetic energy-director weld joint. These two examples show before-and-and after views of ultrasonically welded parts. The versatility of this technique is illustrated next by several ultrasonically assembled parts. Figure 21.7.15a shows daytime running lights assembled by ultrasonically welding a clear plastic PMMA lens to a black plastic ABS body along a 3D seam. Besides a hermetic seal the appearance surface had to have an unblemished (Class A) surface. A composite welding head made of three sonotrodes at different angles was used for making this weld. A number of individually screwed-in tips made it possible to quickly adjust the sonotrode assembly for welding different light assemblies. Figure 21.7.15b shows the back view of a side-view-mirror housing made by ultrasonically welding the top to the bottom. Notice the narrow clear plastic lens for blinking turn signal light. Figure 21.7.15c shows two plastic polyester (PET) end caps bonded to polyester caps in which a pleated paper filter is embedded. And Figure 21.7.15d shows a 40-GF-PA6,6 cap ultrasonically inserted into an aluminum die-cast housing. 21.7.4.1

Ultrasonic Staking, Spot Welding, Swaging, Insertion, and Embedding

This section addresses variants of ultrasonic welding, such as staking, spot welding, swaging, insertion, and embedding. Ultrasonic Staking Ultrasonic staking provides means for joining thermoplastic components with metals

and other nonweldable materials. Figure 21.7.16 shows the staking of a thin part to a thicker thermoplastic part with a molded hollow protrusion. The descending sonotrode (Figure 21.7.16a) melts the protrusion and shapes its surface to the form shown in Figure 21.7.16b. The protrusion does not need to be hollow; a solid protrusion with an appropriately shaped sonotrode head can be used to form a rivet-like head. After melting the protrusion and forming the head the vibratory motion of the sonotrode is turned off and is kept in place till the newly formed head has solidified under static pressure to assure a tight joint. Figure 21.7.17 shows the staking of a small, approximately 10 × 20 mm high-speed connector chip, commonly used in PCs, onto a thermoplastic substrate. Earlier this part would have been heat staked, but the process was too slow. Besides being much faster, ultrasonic staking is more accurate and repeatable. Staking is done on each of the eight molded round staking posts (pins) with an approximate diameter of 0.5 mm, and on the two the approximately 0.25 × 0.5 mm rectangular pins on the right side. In this figure, the pictures on the left and right sides show, respectively, the assemblies before and after the staking operation.

Joining of Plastics

(a)

(b)

(c)

(d)

Figure 21.7.13 Ultrasonic welding of ABS garden hose nozzle. (a) External view of parts. (b) Internal view of parts. (c) Parts assembled for welding. (d) Welded part with nozzle and hose nut. (Photos courtesy of Herrmann Ultrasonics, Inc.)

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(a)

(b)

(d)

(c)

(e)

(g)

(f)

(h)

Figure 21.7.14 Ultrasonic welding of gear box with PA6 gears in a 33-GF-PA6,6 housing. (a, b) Internal views of two housing halves. (c, d) Views of how housing halves fit. (e) PA6 gears with shafts and bearings. (f) Parts assembled for welding. (g) Welding fixture. (h) Welding head with multiple tools. (Photos courtesy of Herrmann Ultrasonics, Inc.)

Joining of Plastics

(a)

(b)

(c)

(d)

Figure 21.7.15 (a) Daytime running lights assembled by ultrasonically welding clear plastic PMMA lens to black ABS body. (b) Rear view of ultrasonically assembled rear-view-mirror housing. (c) Fabric filter with plastic end caps. (d) Plastic cap ultrasonically inserted into an aluminum die-cast housing. (Photos courtesy of Herrmann Ultrasonics, Inc.)

HORN

(a)

(b)

Figure 21.7.16 Ultrasonic staking. (a) Parts to be joined with descending sonotrode. (b) Top thin part staked to bottom thermoplastic part. (Adapted from figures courtesy of Herrmann Ultrasonics, Inc.)

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BEFORE

AFTER

(a)

(b)

(c)

(d)

Figure 21.7.17 Before staking (a, c) and after staking (b, d) photos of staking posts (pins) in a high-speed connector chip. (Adapted from photos courtesy of Herrmann Ultrasonics, Inc.)

Ultrasonic Spot Welding Two flat parts can be spot welded without having to mold-in special joint features

and energy directors. As shown in Figure 21.7.18, a sharp sonotrode head penetrates the upper plate into the lower plate, pushing molten material laterally at the joint interface to form a spot weld – the molten and resolidified material at the joint interface and under the sonotrode at the top surface are shown as cross-hatched regions. Ultrasonic Swaging Swaging offers an alternative to when staking pins cannot easily be molded into a

part. As shown in Figure 21.7.19, a shaped sonotrode head is used to melt and reshape a protrusion in one part over a second part. Swaging along an entire periphery ensures locking of the second part within the first part. Ultrasonic Insertion Threaded metal inserts and metal bearings can be ultrasonically inserted into a molded pilot hole in a molded thermoplastic part. The process of insertion is schematically shown in Figure 21.7.20. To increase pull-out strength the metal surface has grooves into which molten resin

Joining of Plastics

HORN

Figure 21.7.18 Ultrasonic spot welding. The molten and resolidified material at the joint interface (spot weld) and under the sonotrode at the top surface are shown as cross-hatched regions. (Adapted from figures courtesy of Herrmann Ultrasonics, Inc.)

HORN

(a)

(b)

Figure 21.7.19 Ultrasonic swaging. (a) Parts to be swaged with descending sonotrode. (b) Left part swaged onto the second thermoplastic part. (Adapted from figures courtesy of Herrmann Ultrasonics, Inc.)

HORN KNURLED SURFACES

(a)

(b)

Figure 21.7.20 Ultrasonic insertion of metal insert. Note the use of grooves to increase pull-out strength, and knurling of surface to increase rotational resistance during screw tightening. (Adapted from figures courtesy of Herrmann Ultrasonics, Inc.)

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will flow and solidify during insertion. Also, to increase rotational resistance during screw tightening portions of the insert surface are knurled. Ultrasonic Embedding The process of embedding uses an energy director to force molten material into

a second component as shown in Figure 21.7.21. This process is useful for attaching a thermoplastic component to a nonwoven fabric or to an incompatible material having an open or porous structure. The cross-hatched region in Figure 21.7.21b shows the melted and resolidified material.

HORN

(a)

(b)

Figure 21.7.21 Before (a) and after (b) schematic figures showing ultrasonic embedding. The cross-hatched region shows the melted and resolidified material. (Adapted from figures courtesy of Herrmann Ultrasonics, Inc.)

21.8 Electromagnetic Bonding In electromagnetic welding the heat for melting the weld interface is provided by electrical energy in the form of resistive heating or high-frequency radiation. Of the several available processes, induction welding is the most commonly used. 21.8.1

Resistance (Implant) Welding

In this process an electrically conductive wire or braid is placed within the joint interface and resistively heated by the passage of an electrical current. The heat causes the surrounding plastic to melt, and a weld is effected by subsequent cooling. The metal wire or braid remains within the joint, and can therefore affect its strength. The use of such inserts also increases the overall cost. The main advantage of this process is its simplicity and its applicability to complex joints in large parts. Weld times are short, less than 30 seconds even for the largest parts. A variant of resistance welding, called electrofusion, is being used for joining small diameter ( ET (EBL < ET ) when E = E (y) increases (decreases) with increasing y. Also, EBL not equaling ET clearly indicates that neither of them are true material properties. As such, EBL and ET will be called, respectively, the bending modulus and the tensile modulus for historical reasons only. The ratio EB ∕ET is a measure of how nonhomogeneous the through-thickness variation E = E (y) is. 22.5.4

Bending of Nonhomogeneous Bar in the Higher Stiffness Mode

Next consider the bending of the bar in the higher stiffness mode due to a bending moment My , in the configuration shown in Figure 22.5.3 (why is the moment labeled − My ?).

x

x

y

z

b

– My

– My

d (a)

(b)

Figure 22.5.3 Bending of a nonhomogeneous bar of breadth b and depth d in the higher stiffness mode due to a bending moment −My . (a) Longitudinal arcuate shape. (b) Lateral cross section.

Then, following the same arguments as for the bending case discussed previously, the bar will bend into an arcuate shape such that 𝜀z = x∕R and 𝜎 z = E 𝜀z = E (y) x∕R. As before, the absence of a longitudinal force implies ∫

𝜎z dA = 0

⇒

∫

E (y) x dA = 0

so that x = 0 passes through the E-weighted centroid. And since the stress distribution must equal the moment My , − My =

∫

x (𝜎z dA) =

1 E (y) x2 dA R∫

⇒

− My Jxx

=

1 , R

or

My Jxx

=

1 −R

(22.5.6)

where Jxx = ∫ E (y) x2 dA. (The negative sign on R implies that a positive bending moment My will cause the beam to bend concave upwards instead of concave downwards as shown in Figure 22.5.3.) Then, just as for the bending of a bar due to Mx , an effective “bending modulus,” EBH , for the higher

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stiffness mode can be defined as the “modulus” of a homogeneous bar with the same stiffness, that is, My My 1 = ≡ R EBH Ixx Ixx

⇒

EBH =

d∕2

b∕2

∫ E (y) x2 dA Jxx = Ixx ∫ x2 dA

Now Jxx = =

∫ (

2

E (y) x dA = b∕2

∫−b∕2

( 2

E (y) x dx dy =

b∕2

(22.5.7) ) x dx

d∕2

2

E (y) dy ∫−d∕2 ∫−b∕2 ∫−b∕2 ∫−d∕2 ) ( b∕2 ) d∕2 d∕2 1 1 2 2 x (d dz) E(y) (b dy) = x dA E (y) dA ∫−b∕2 bd ∫−d∕2 A ∫−d∕2

= Ixx ET It then follows that EBH = ET ! In retrospect, this result should not be so surprising. For, any thin slice of the beam between y and y + dy has a uniform modulus E (y), so that the beam is made up of thin uniform slices each having a different but constant modulus. The fact that the stiffness of a beam depends linearly on the width allows for an addition of the stiffnesses of the individual slices, resulting in the higher stiffness mode “bending modulus” being the same as the “tensile modulus.” Thus, the effective bending “modulus” depends on the axis about which the bending occurs; again, a clear indication of the fact that the bending “moduli” are not material properties. Rather, such averages determine the stiffnesses of specific geometrical structures. Such moduli would be more useful if they can be used to determine the stiffnesses of other structures.

22.6 Short-Fiber-Filled Systems This section discusses the distribution of mechanical properties of in short-fiber filled injection-molded parts. The levels of anisotropy and spatial variation of the mechanical properties in such parts are characterized by data from molded plaques of a commercial, 30 wt% short-glass-filled poly(butylene terephthalate), VALOX® 420, which will be referred to as 30-GF-PBT. The manufacturers’ data sheets show this material having a flexural modulus of 7.6 GPa, a flexural strength of 190 MPa, and a tensile strength of 119 MPa. These data-sheet mechanical properties were determined by tensile and flexural tests on injection-molded bars, in which the fibers are preferentially aligned in the length direction. Data-sheet properties of three short-glass-filled plastics are compared in Table 22.6.1. Table 22.6.1 Data-sheet properties from material suppliers’ tests on injection-molded tensile bars.

Material

Flexural modulus (GPa)

Tensile strength (MPa)

Flexural strength (MPa)

30-GF-PBT (VALOX 420)

7.6

119

190

30-GF-PC (LEXAN 3413)

6.6

100

153

33-GF-PA-6,6 (ZYTEL 70G33)

8.7

186

262

Fiber-Filled Material Materials – Materials with Microstructure

Data from injection-molded plaques are used to address the following questions: How anisotropic is the local tensile modulus in different regions of molded parts? How nonhomogeneously are these moduli distributed across parts? How repeatable are these modulus distributions among different parts molded under the same conditions? How does part thickness affect local properties? How do processing conditions affect part properties? And what is the effect of fiber length on material properties? 22.6.1

Tensile Modulus

This subsection addresses the variation of the flow- and cross-flow-direction tensile moduli in molded plaques of three different thicknesses. Data from tests on several plaques assess the repeatability of data. Details of how the tests were carried out are important for understanding the complexity of material property variations across plaques. 22.6.1.1 Test Procedures

The data described herein were obtained from two sets of molded plaques. The first set, consisting of 152 × 203-mm (6 × 8-in) injection-molded, fan-gated plaques were molded in nominal thickness of 1.9, 3.2, and 6.35 mm (0.07, 0.125, and 0.25 in) in a machine with a free-flow check ring to reduce fiber breakage. The plaques were molded at an injection speed corresponding to a fill time of 5 seconds; the other processing conditions are listed in Table 22.6.2. These plaques were used to determine the distribution and variation of mechanical properties across the plaques in the flow and cross-flow directions. The second set, consisting of 76 × 280-mm (3 × 11-in), 3.05-mm (0.12-in) thick edge-gated plaques, were molded at three injection speeds corresponding to fill times of 1, 2, and 5.6 seconds; the molding conditions are listed in Table 22.6.3. These plaques were primarily used to evaluate the effects of processing conditions on tensile properties. All the mechanical tests were done under ambient conditions (22°C and 50% relative humidity) on a servo-hydraulic test machine. The modulus and strength of the materials were determined through tensile tests on specimens cut from these plaques. For mapping the modulus variations over the plaques, Table 22.6.2 Processing conditions for injection-molded short-fiber-reinforced 152 × 203-mm (6 × 8-in) plaques having three thicknesses. Material

30-GF-PBT

Plaque thickness (mm)

1.9

3.2

6.3

Mold temperature (°C)

79

79

79

238

238

249

246

246

252

238

238

243

227

227

238

Hold time (s)

19

19

37

Screw speed (RPM)

90

90

95

Nozzle temperature (°C)

Front barrel temperature (°C)

Middle barrel temperature (°C) Rear barrel temperature (°C)

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

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Table 22.6.3 Processing conditions for 76 × 280-mm (3 × 11-in) injection-molded plaques. Fill time (s)

1

Mold temperature (°C)

2

5.6

66

66

66

Barrel temperature (°C)

260

260

260

Injection speed (mm s−1 )

101.6

50.8

17.8

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

12.7-mm (0.5-in) wide rectangular strips were pulled to an approximate strain of 0.2%, under force control, at a nominal strain rate of 10−2 s−1 . The strain over a 12.7-mm gauge length was monitored by an extensometer, and the tensile modulus was calculated from the initial portion of the stress-strain curve between 0.1 and 0.5% strain. To characterize variations, the tensile modulus of the plaque material was determined at 12.7-mm intervals – both along the flow and cross-flow directions – over approximately 19.35 × 103 mm2 (30 in2 ) and 15.48 × 103 mm2 (24 in2 ) portions of plaques. For this, the plaques were cut into 12.7-mm wide specimens: Lines were first drawn on each plaque at 12.7-mm intervals, to mark 12.7-mm segments for subsequent tensile tests. The 12.7 mm wide strips were then cut, at right angles to these lines, starting from the centerline of the plaque and working outward. In this way, the plaques were cut into numbered, rectangular 12.7 mm wide strips with marks at 12.7-mm long intervals. As schematically shown in Figures 22.6.1 and 22.6.2, such strips were cut in both the flow and cross-flow directions – one direction per plaque: the flow-direction modulus was determined at 12 segments on each of 10 strips; the cross-flow modulus was determined at eight segments on each of 12 strips. The thickness and the width of each 12.7-mm wide segment on each strip were measured and recorded. 203 mm 10 8 7 6 5 4 3

12.7-mm STRIPS

9

152 mm

786

2 1

1 2 3 4 5 6 7 8 9 10 11 12 12.7-mm SEGMENTS

Figure 22.6.1 Layout of twelve 12.7-mm segments on ten 12.7-mm wide rectangular strips cut in the flow direction from 152 × 203-mm (6 × 8-in) fan/edge-gated plaque. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

11

12.7-mm SEGMENTS

1 2 3 4 5 6 7 8

12

9

10

8

7

6

5

3

4

1

2

Fiber-Filled Material Materials – Materials with Microstructure

12.7-mm STRIPS

Figure 22.6.2 Layout of eight 12.7-mm segments on twelve 12.7-mm wide rectangular strips cut in the cross-flow direction from 152 × 203-mm (6 × 8-in) fan/edge-gated plaque. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

22.6.1.2 Directional and Spatial Modulus Variation

The flow and cross-flow direction tensile moduli over 12.7 × 12.7-mm (0.5 × 0.5-in) regions were determined across central portions of two plaques (plaques I and III, respectively). The specimen layouts shown in Figures 22.6.1 and 22.6.2 make possible 120 and 96 measurements to map, respectively, the distributions of flow- and cross-flow direction tensile moduli on these rectangular plaques. The 120 flow-direction tensile moduli for the 12 segments on each of the 10 strips cut from plaque I along the flow direction in Figure 22.6.1 are plotted between two consecutive vertical grid lines in Figure 22.6.3a, in which the data points are indicated by circles. The 12 data points from each flow-direction specimen are arranged in the 1 – 12 order numbers shown in Figure 22.6.1 (from the gated to the opposite end), with data for increasing strip numbers being added sequentially. Thus, in Figure 22.6.3a, the numbers 1 – 12 on the abscissa correspond, respectively, to the 12 segments on strip 1; numbers 13 – 24 correspond, respectively, to the 12 segments on strip 2; and so on, till the numbers 109 – 120 on the abscissa correspond, respectively, to segments 1 – 12 on strip 10. These 120 flow-direction tensile moduli along the 10 strips, shown in Figure 22.6.3a, are re-plotted in Figure 22.6.3b to show how the flow-direction modulus varies across the plaque width. The 10 measurements between any two neighboring vertical gridlines in Figure 22.6.3b show the variation of the flow-direction modulus at a fixed segment location on different specimens across the width. Again, the numbers on the abscissa are organized in the sequence of strips 1 – 10 and from segments 1 to 12 based on the numbering scheme shown in Figure 22.6.1. Thus, in Figure 22.6.3b, the numbers 1 – 10 on the abscissa correspond, respectively, to the 10 segments at location 1 on each of the 10 strips; numbers 11 – 20 correspond, respectively, to the 10 segments at location 2 on each of the 10 strips; and so on, till the numbers 111 – 120 on the abscissa correspond, respectively, to the 10 segments at location 12 on each of the 10 strips. Figure 22.6.3a shows that the central region of the plaque (data points 13 – 108) has lower flow-direction tensile moduli than the two outer strips near the plaque sides (data points 1 – 12 and

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109 – 120 respectively). At each distance from the gate, these outer strips have the highest flow-direction tensile modulus, which increases with the flow distance from the gated end (data 1 and 109) and peaks near the end where material flow stopped (data 12 and 120). Notice that in each strip of the middle region, the flow-direction modulus increases with the distance from the gate, peaks somewhere in the plaque center, and then decreases to the local minimum value for the strip at the last segment (segment 12 of each strip or data 24, 36, 48, …, 108.).

FLOW-DIRECTION MODULUS (GPa)

9

8

7 0

24

48

72

96

120

FLOW-DIRECTION STRIPS (a) 9

FLOW-DIRECTION MODULUS (GPa)

788

8

7 0

50

100

120

CROSS-FLOW DIRECTION SEGMENTS (b) Figure 22.6.3 Flow-direction tensile moduli of two 152 × 203 × 6.1-mm, 30-GF-PBT plaques (plaques I and II), (a) for flow-direction strips from side to side, and (b) across strips along cross-flow direction from gate end to the end away from the gate. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

Fiber-Filled Material Materials – Materials with Microstructure

CROSS-FLOW DIRECTION MODULUS (GPa)

Between each successive pair of vertical gridlines, Figure 22.6.4a shows the cross-flow direction tensile moduli measured on eight segments on each of the 12 strips cut from plaque III along the cross-flow direction (Figure 22.6.2). The eight data points from each cross-flow direction specimen are arranged in the 1 – 8 order numbers shown in Figure 22.6.2 (from one 203-mm edge to the other), with data for increasing strip numbers being added sequentially. Thus, in Figure 22.6.4a, the numbers 1 – 8 on the abscissa correspond, respectively, to the eight segments on strip 1; numbers 9 – 16 correspond, respectively, to the eight segments on strip 2; and so on, till the numbers 89 – 96 on the abscissa correspond, respectively, to segments 1 – 8 on strip 12.

5.6 5.4 5.2 5.0 4.8 4.6 0

16

32

48

64

80

96

CROSS-FLOW DIRECTION STRIPS

CROSS-FLOW DIRECTION MODULUS (GPa)

(a) 5.6 5.4 5.2 5.0 4.8 4.6 0

24

48

72

96

FLOW-DIRECTION SEGMENTS (b) Figure 22.6.4 Cross-flow direction tensile moduli of two 152 × 203 × 6.1-mm, 30-GF-PBT plaques (plaques III and IV), (a) for cross-flow direction strips from gated end to the end away from the gate, and (b) across strips along flow direction, from side to side. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

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These 96 cross-flow direction tensile moduli along the 12 strips, shown in Figure 22.6.4a, are re-plotted in Figure 22.6.4b to show how the cross-flow direction modulus varies across the plaque length. The 12 measurements between any two neighboring vertical gridlines in Figure 22.6.4b show the variation of the cross-flow direction modulus at a fixed segment location on different specimens across the 203-mm length. As before, numbers on the abscissa are organized in the sequence of strips 1 – 12 and from segments 1 – 8 based on the numbering scheme shown in Figure 22.6.2. Thus, in Figure 22.6.4a, the numbers 1 – 12 on the abscissa correspond, respectively, to the 12 segments at location 1 on each of the 12 strips; numbers 13 – 24 correspond, respectively, to the 12 segments at location 2 on each of the 10 strips; and so on, till the numbers 85 – 96 on the abscissa correspond, respectively, to the 12 segments at location 8 on each of the 12 strips. The flow and cross-flow tensile moduli appear correlated – high values of one appear to be accompanied by low values of the other. This correlation can, again, be explained by the dependence of the tensile stiffness on fiber orientation, since material that is stiffer in the fiber-aligned direction is less stiff in the transverse direction. The maximum variation across plaques appears to occur over similar regions in these plaques. As with the flow-direction tensile modulus variation in each flow-direction strip (Figure 22.6.3a), the difference between the local maximum and minimum cross-flow direction tensile modulus within each band is the largest at the plaque center (between data 37 and 48 in, see Figure 22.6.4b). The large variation of the elastic moduli within each plaque most likely results from nonhomogeneous distributions of the fiber content and the orientation, caused by material flow during injection molding. The data for the distribution of the elastic moduli in molded plaques discussed in this section clearly show that data-sheet properties are inadequate for describing the actual distribution of elastic moduli in parts. For example, they fail to account for the large differences for the properties in the flow- and cross-flow directions. This will become even more apparent form the discussions in the succeeding sections. 22.6.1.3

Repeatability of Modulus Data

To test for the repeatability of the flow and cross-flow direction tensile modulus distributions, the tests described before were repeated on specimens cut from two additional 6.1-mm thick, 30-GF-PBT plaques (plaques II and IV, respectively). The average flow-direction tensile moduli were 7.98 and 7.99 GPa for plaques I and II, respectively. The average cross-flow direction tensile moduli for plaques III and IV were 5.21 and 5.16 GPa, respectively. The maximum and minimum flow-direction tensile modulus for the plaques (Figure 22.6.3) were 8.57 and 7.21 GPa for plaque I, and 8.44 and 7.38 GPa for plaque II, respectively, with corresponding standard deviations (𝜎 s) of 0.22 GPa (2.8% of the plaque mean) and 0.21 GPa (2.6% of the plaque mean). The maximum and minimum cross-flow direction tensile moduli (Figure 22.6.4) were 5.50 and 5.00 GPa for plaque III, and 5.53 and 4.97 GPa for plaque IV, with a standard deviation of 0.10 GPa (2.0% and 1.9% of respective means) for both plaques. The variation of tensile moduli over these injection-molded 30-GF-PBT plaques appears to be higher for the flow than for the cross-flow direction. The upper and lower limits of the flow-direction tensile modulus are separated, respectively, by 0.59 GPa (7.3% of the mean, or 2.6 𝜎 ) and −0.77 GPa (−9.7% of the mean, or −3.5 𝜎 ) from the sample means in plaque I, and by 0.45 GPa (5.7% of the mean, or 2.2 𝜎 ) and −0.61 GPa (−7.6% of the mean, or −2.9 𝜎 ) in plaque II. The upper and lower limits of the cross-flow tensile modulus are separated from the sample means, respectively, by 0.29 GPa (5.5% of the mean, or 2.8 𝜎 ) and −0.21 GPa (−4.0% of the mean, or −2.0 𝜎 ) in plaque III, and 0.37 GPa (7.2% of the mean, or 3.6 𝜎 ) and −0.19 GPa (−3.7% of

Fiber-Filled Material Materials – Materials with Microstructure

the mean, or −1.9 𝜎 ) in plaque IV. The distributions of flow-direction tensile moduli for plaques I and II have longer tails at the lower values; in contrast, the cross-flow tensile modulus distribution exhibits longer tails at the higher values. When the data from plaques II and IV are superposed on Figures 22.6.3 and 22.6.4, respectively, they show that the sets of measurements from the two pairs of plaques (I, II and III, IV) are quite repeatable, both for the flow- and cross-flow direction tensile moduli. Figures 22.6.3 and 22.6.4 show that two sets of measurements from the two pairs of plaques (I, II and III, IV) are quite repeatable, both for the flow- and cross-flow direction tensile moduli. The repeatability of the flow-direction data appears to be somewhat better than that of the cross-flow data. The association between two data sets x and y can be measured by the Pearson product-moment correlation coefficient n ∑

r=

(xi − x)(yi − y)

i =1

(n − 1) 𝜎x 𝜎y

(22.6.1)

where x and 𝜎 x are, respectively, the sample mean and standard deviation of data set x; y and 𝜎 y are, respectively, the sample mean and standard deviation of data set y, and n is the number of samples for both data sets. The data pair are uncorrelated, strongly correlated, and negatively correlated, respectively, when r = 0, r → 1, and r → −1. The correlation coefficient for the 120 flow-direction tensile moduli measured from plaques I and II is 0.81. The corresponding correlation coefficient for the 98 cross-flow direction tensile moduli measured from plaques III and IV is 0.78. The spatial repeatability of the data from two different plaques can be visualized by comparing modulus contours for the plaques. Using the same data for the 6.1-mm 30-GF-PBT plaques as in Figures 22.6.3 and 22.6.5a,b show the contours of the flow-direction moduli in plaques I and II, respectively. Clearly, even though the local modulus exhibits spatial variations, the plaque-to-plaque data are quite repeatable. Similarly, the cross-flow modulus contours for plaques III and IV shown, respectively, in Figure 22.6.5 part c and d, indicate that plaque-to-plaque data are repeatable. Test results in the next section show that the local flow and cross-flow tensile moduli data also exhibit plaque-to-plaque repeatability for much thinner, 1.9-mm thick plaques. These sets of results provide the basis for assuming that short-fiber filled injection-molded parts will have very small part-to-part property variations. 22.6.1.4 Effects of Plaque Thickness on the Tensile Modulus

During injection molding, the shear stress on the molten material increases with reductions in the mold cavity gap height. As a result, fiber alignment in thin and thick molded plaques can be different. To investigate the effects of plaque thickness on the distribution and variation of the tensile modulus, additional flow direction and cross-flow direction specimens were cut from 1.9-mm thick, 152 × 203-mm, 30-GF-PBT plaques, as per the layouts shown in Figures 22.6.1 and 22.6.2. The flow-direction and cross-flow direction tensile moduli were determined on two plaques for each direction: Plaques V and VI for the flow-direction and Plaques VI and VIII for the cross-flow direction (a total of four plaques). The average flow-direction tensile moduli for the 1.9-mm thick plaques were 8.61 and 8.52 GPa for plaques V and VI, respectively. The average cross-flow direction tensile moduli were 4.67 and 4.64 GPa for plaques VII and VIII, respectively. The maximum and minimum flow-direction tensile moduli were,

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FLOW-DIRECTION MODULUS (GPa) PLAQUE I

8.1–8.6

PLAQUE II

7.6–8.1 7.1–7.6

FLOW DIRECTION

FLOW DIRECTION

(a)

(b)

CROSS-FLOW DIRECTION MODULUS (GPa) PLAQUE III

5.3–5.6

PLAQUE IV

5.0–5.3

FLOW DIRECTION

FLOW DIRECTION

(c)

(d)

Figure 22.6.5 Contours showing variations of tensile moduli in 152 × 203 × 6.1-mm, 30-GF-PBT plaques. (a) and (b) Flow-direction moduli in plaques I and II, respectively. (c) and (d) Cross-flow direction moduli in plaques III and IV, respectively. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

respectively, 9.00 and 7.85 GPa for plaque V, and 8.93 and 7.83 GPa for plaque VI, with the standard deviations being 0.24 GPa (2.8% of the plaque mean) and 0.26 GPa (3.1% of the plaque mean). The maximum and minimum cross-flow direction tensile moduli for the same plaque thickness were, respectively, 4.96 and 4.40 GPa for plaque VII, and 4.83 and 4.41 GPa for plaque VIII, with standard deviations of 0.12 and 0.09 GPa (or 2.5% and 2.0% of the respective plaque means) for both plaques. As with the 6.1-mm thick plaques, the 1.9-mm thick plaques have higher tensile moduli variations in the flow than in the cross-flow direction. However, in comparison to the moduli for the 6.1-mm thick plaques, the average tensile modulus for the 1.9-mm thick plaque increases in the flow direction by about 7% but decreases in the cross-flow direction by about 11.5%. That is, the ratio of the flow to the cross-flow direction tensile modulus increases with decreasing plaque thickness. One explanation for these large differences among the moduli variations in the 1.9- and 6.1-mm thick plaques is possible differences in the thicknesses of the skin and core regions.

Fiber-Filled Material Materials – Materials with Microstructure

The modulus contours for the 1.9-mm 30GF-PBT plaques in Figure 22.6.6a,b show the distribution of the flow-direction moduli in plaques V and VI, respectively. Similarly, Figure 22.6.6c and d show the cross-flow modulus contours for plaques VII and VIII, respectively. Clearly, the plaque-to-plaque data are quite repeatable.

FLOW-DIRECTION MODULUS (GPa) PLAQUE V

8.6–9.0

PLAQUE VI

8.2–8.6 7.8–8.2

FLOW DIRECTION

FLOW DIRECTION

(a)

(b)

CROSS-FLOW DIRECTION MODULUS (GPa) PLAQUE VII

4.7–5.0

PLAQUE VIII

4.4–4.7

FLOW DIRECTION

FLOW DIRECTION

(c)

(d)

Figure 22.6.6 Contours showing variations of tensile moduli in 152 × 203 × 1.9-mm 30-GF-PBT plaques. (a) and (b) Flow-direction moduli in plaques V and VI, respectively. (c) and (d) Cross-flow direction moduli in plaques VII and VIII, respectively. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

The repeatability of plaque-to-plaque data obtained from 1.9-mm and 6.1-mm thick plaques provide the basis for assuming that short-fiber filled injection-molded parts will have very small part-to-part property variations. The upper and lower limits for the flow-direction tensile moduli of 1.9-mm thick plaques are separated from the sample means by, respectively, 0.39 GPa (4.5% of the mean, or 1.6 𝜎 ) and−0.76 GPa (−8.8%, or −3.1𝜎 ) for plaque V, and by 0.41 GPa (4.8%, or 1.6 𝜎 ) and−0.69 GPa (−8.1%, or −2.6 𝜎 ) for plaque VI. The upper and lower limits for the flow-direction tensile modulus for the 3.0-mm thick plaque IX are

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separated from the sample mean by, respectively, 0.37 GPa (4.4% of the mean or 2.3 𝜎 ) and−0.27 GPa (−3.2% or −1.8 𝜎 ). As with the 6.1-mm thick plaques, the distributions of the flow-direction tensile moduli from two 1.9-mm thick plaques also exhibit longer tails at the lower values. But, the distribution of the flow-direction tensile moduli measured on the only 3.0-mm thick plaque tested does not exhibit this tail. The upper and lower limits for the cross-flow direction tensile moduli of the 1.9-mm thick plaques are separated from the sample means by, respectively, 0.29 GPa (6.2% of the mean, or 2.5 𝜎 ) and−0.27 GPa (−5.8% or −2.3 𝜎 ) in plaque VII, and by 0.19 GPa (4.1% or 2.1𝜎 ) and−0.23 GPa (−4.9% or −2.4 𝜎 ) in plaque VIII. In comparison to the cross-flow direction tensile modulus distributions for the 6.1-mm thick plaques, the distributions for the two 1.9-mm thick plaques seem to have more symmetric tails. The variations of the flow- and cross-flow-direction tensile moduli of 1.9-mm and 6.1-mm thick plaques are compared in Figure 22.6.7. The two images on the left, Figure 22.6.7a and b, are for the 1.9-mm thick plaques; those on the right, Figure 22.6.7c and d, are for the 6.1-mm thick plaques. The figures on the top (Figure 22.6.7a and c) and bottom (Figure 22.6.7b and d) correspond, respectively, to the flow- and cross-flow direction moduli, respectively. (The flow and cross-flow data for the 1.9-mm thick plaques are from plaques V and VII, respectively, and the corresponding data for the 6.1-mm thick plaques are for plaques I and III, respectively. The slight differences in the contours from those shown in Figures 22.6.6a,c and 22.6.5a,c are caused by the different scales used for the tensile moduli.) The two arrows on the sides of the contour level bars indicate the nominal minimum and maximum moduli in the plaque. The actual maximum and minimum values indicated by the arrows are, respectively: 9.0 and 7.85 for the flow direction, and 4.96 and 4.40 GPa for the cross-flow direction of the 1.9-mm thick plaques; and 8.57 and 7.21 for the flow direction, and 5.50 and 5.0 GPa for the cross-flow direction of the 6.1-mm thick plaques. Using the average of the maximum and minimum values as an approximate measure for the differences between the flow- and cross-flow-direction properties, for the 1.9-mm thick plaque the averages of 8.43 and 4.68 result in a value of 1.8 for the ratio of the flow to cross-flow properties. With averages of 7.89 and 5.25, this ratio is 1.5 for the 6.1-mm thick plaque. To characterize the effects of plaque thickness on the flow-direction and cross-flow direction tensile moduli, the values of these moduli were averaged in one direction (say, the cross-flow direction) and plotted along the other direction (flow direction). Figure 22.6.8 shows the variation of the averages for the flow-direction modulus for three plaque thicknesses. The variation of these averages for the cross-flow direction modulus distributions for two plaque thicknesses is shown Figure 22.6.9. For the 12 segments on the 10 flow-direction strips per plaque, the flow-direction tensile modulus averaged over 20 values for two 1.9- and 6.0-mm thick plaques, and 10 values for the only 3.0-mm thick plaques is plotted at each of the 12 locations in Figure 22.6.8a. The horizontal bars above and below each symbol represent the maximum and minimum for the same set of 20 values for the 1.9and 6.1-mm thick plaques and 10 values for the 3.0-mm thick plaque. Because tests were done on only one 3.0-mm thick plaque, in contrast to two each for the 1.9- and 6.1-mm thick plaques, the spread between the maximum and minimum at each location in Figure 22.6.8a is expected to be smaller for the 3.0-mm thick plaque than for the 1.9- or 6.1-mm thick plaques. Similarly, for each of the 10 strips, each of which has 12 segments, each location on Figure 22.6.8b shows the average, maximum, and minimum flow-direction tensile modulus from 24 values for two each 1.9- and 6.0- mm thick plaques. The horizontal bars above and below each symbol again indicate the maximum and minimum of the 24 values from the two 1.9- and 6.1-mm thick plaques used to obtain the average at each location in Figure 22.6.8b.

Fiber-Filled Material Materials – Materials with Microstructure

FLOW-DIRECTION MODULUS (GPa) Tensile Test Direction

10

10 8.5–9.0 8.0–8.5 7.5–8.0

5 GPa

5

7.0–7.5

(a)

(c)

1.9-mm-THICK PLAQUES

6.1-mm-THICK PLAQUES

10

GPa

10 5.5–6.0 5.0–5.5 4.5–5.0

5 GPa

5

4.0–4.5

(d)

(b)

GPa

Tensile Test Direction

CROSS-FLOW DIRECTION MODULUS (GPa) Figure 22.6.7 Contours of the flow- (Figure 22.6.7a,c) and cross-flow (Figure 22.6.7b,d) direction tensile moduli for 1.9-mm (Figure 22.6.7a,b) and 6.1-mm (Figure 22.6.7c,d) thick 30-GF-PBT plaques. The flow and cross-flow data for the 1.9-mm thick plaques are from plaques V and VII, respectively, and the corresponding data for the 6.1-mm thick plaques are for plaques I and III, respectively. The slight differences in the contours from those shown in Figures 22.6.6a,c and 22.6.5a,c are caused by the different scales used for the tensile moduli.

For the data for 1.9-, 3.0-, and 6.1-mm thick plaques, each of the three curves in Figure 22.6.8a,b clearly show that the flow-direction tensile modulus increases with decreasing thickness. This increase in modulus with decreasing plaque thickness can be attributed to fiber orientation differences in the shell-core-shell microstructure among plaques of different thickness. The higher fiber alignment in the shell region of the thicker plaque results in a higher flow-direction tensile modulus. For the 1.9- and 6.1-mm thick plaques, the average flow-direction tensile modulus increases along the flow direction from the gate to a maximum and then decreases to a minimum at the plaque end. The maximum average modulus occurs at a distance of about 60% of the plaque length from the gate for the 1.9-mm thick plaques and at a distance of about 40% for 6.1-mm thick plaques. The locations of these

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FLOW-DIRECTION MODULUS (GPa)

9

8

1.9 mm

3 mm

6.1 mm

7 0

2

4

6

8

10

12

FLOW-DIRECTION DIVISIONS (a) 9

FLOW-DIRECTION MODULUS (GPa)

796

8

1.9

3

6.1

7 0

2

4

6

8

10

CROSS-FLOW DIRECTION DIVISIONS (b) Figure 22.6.8 Summary of flow-direction tensile moduli of 1.9-, 3-, and 6.1-mm thick, 152 × 203-mm 30-GF-PBT plaques. (a) Maximum, minimum, and average of each flow-direction strip from the gate end to the end opposite to the gate, and (b) maximum, minimum, and average of the segments across all flow-direction strips plotted from side to side. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

maxima correlate with the core being thinner and the core fiber orientation being higher at the middle of the plaques. In contrast to this, the flow-direction tensile modulus is almost constant throughout the 3.0-mm thick plaque (Figure 22.6.8a). As shown in Figure 22.6.8b, in these plaques the distributions of average flow-direction tensile modulus along the cross-flow direction are “W” shaped, having local maxima at the center and at the edges. Again, the high flow-direction moduli at the edges can be attributed to the thinner core along the edges of the plaques.

CROSS-FLOW DIRECTION MODULUS (GPa)

Fiber-Filled Material Materials – Materials with Microstructure

5.6 1.9 mm

6.1 mm

5.4 5.2 5.0 4.8 4.6 4.4 0

2

4

6

8

10

12

FLOW-DIRECTION DIVISIONS

CROSS-FLOW DIRECTION MODULUS (GPa)

(a) 5.6 1.9

6.1

5.4 5.2 5.0 4.8 4.6 4.4 0

2

4

6

8

CROSS-FLOW DIRECTION DIVISIONS (b) Figure 22.6.9 Summary of cross-flow direction tensile moduli of 1.9- and 6.1-mm thick, 152 × 203-mm 30-GF-PBT plaques. (a) Maximum, minimum, and average of each cross-flow direction strip from the gate end to the end opposite to the gate, and (b) maximum, minimum, and average of the segments across all cross-flow direction strips plotted from side to side. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

In contrast to the flow-direction moduli (Figure 22.6.8a), the curves in Figure 22.6.9a and b for the cross-flow direction tensile moduli for the 1.9-mm thick plaques clearly have lower values than do those for the 6.1-mm thick plaques. As shown in Figure 22.6.9a, the average cross-flow modulus for both thicknesses first decreases slightly from the gate, along the flow direction, to a minimum and then rises to the maximum at the end of the plaques. A comparison of Figure 22.6.8a with Figure 22.6.9a indicates that the flow and cross-flow moduli appear to be correlated – high values of one appear to be

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accompanied by low values of the other. Figure 22.6.9b shows distributions of the average cross-flow tensile moduli along the cross-flow direction for plaques with these two thicknesses. For the 6.1-mm thick plaques (Figures 22.6.8b and 22.6.9b), only the cross-flow and the flow-direction moduli are correlated. The corresponding moduli for the 1.9-mm thick plaques are uncorrelated; most likely because, while in Figure 22.6.8b the flow-direction tensile modulus along the cross-flow direction was measured on 10 strips, in Figure 22.6.9b the cross-flow direction tensile modulus was measured on only eight segments. The pattern of flow-direction modulus variation within plaques, or the qualitative modulus distribution, is similar for all the plaques: high and low moduli are distributed over similar regions in these plaques. The average, maximum, minimum, and standard deviation of the flow and cross-flow moduli for injection-molded, 30-GF-PBT plaques, for the two thicknesses, are summarized in Tables 22.6.4 and 22.6.5. The flow and cross-flow moduli appear to be correlated – high values of one appear to be accompanied by low values of the other. And the difference between the flow and cross-flow moduli becomes smaller with increasing thickness. The modulus variation within each plaque (standard deviation) appears to decrease with plaque thickness. Plaque-to-plaque repeatability appears to be independent of plaque Table 22.6.4 Flow-direction tensile moduli of 152 × 203-mm (6 × 8-in) 30-GF-PBT injection-molded plaques. Flow-direction tensile modulus (GPa) Thickness (mm)

Plaque

Average

Standard deviation (%)

Maximum (𝜎 )

Minimum (𝜎 )

1.9

V

8.61

0.24 (2.80)

9.00 (1.6)

7.85 (3.1)

VI

8.52

0.26 (3.10)

8.93 (1.6)

7.83 (2.6)

3.0

IX

8.36

0.16 (1.88)

8.73 (2.3)

8.09 (1.8)

6.1

I

7.98

0.22 (2.80)

8.57 (2.6)

7.21 (3.5)

II

7.99

0.21 (2.61)

8.44 (2.2)

7.38 (2.9)

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

Table 22.6.5 Cross-flow direction tensile moduli of 152 × 203-mm (6 × 8-in) 30-GF-PBT injection-molded plaques. Cross-flow direction tensile modulus (GPa) Thickness (mm)

Plaque

Average

Standard deviation

Maximum

1.9

VII

4.67

0.12 (2.46 %)

4.96 (2.5𝜎 )

4.40 (2.3𝜎 )

VIII

4.64

0.09 (2.01 %)

4.83 (2.1𝜎 )

4.41 (2.4𝜎 )

III

5.21

0.10 (1.98 %)

5.50 (2.8𝜎 )

5.00 (2.0𝜎 )

IV

5.16

0.10 (1.93 %)

5.53 (3.6𝜎 )

4.97 (1.9𝜎 )

6.1

Minimum

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

Fiber-Filled Material Materials – Materials with Microstructure

thickness, in the sense that the correlation coefficients (defined in Eq. 22.6.1) for different plaques of the same thickness are comparable, regardless of the plaque thickness (Tables 22.6.6). However, the flow-direction modulus has slightly better repeatability than the cross-flow direction modulus. Table 22.6.6 The repeatability of flow- and cross-flow-direction tensile moduli of 152 × 203-mm (6 × 8-in), 1.9- and 6.1-mm thick, injection-molded plaques of 30-GF-PBT. Thickness (mm)

Plaques

Correlation coefficient

Flow

V, VI

0.83

Cross-flow

VII, VIII

0.77

Flow

I, II

0.81

Cross-flow

III, IV

0.78

Direction

1.9 6.1

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

22.6.1.5 Effects of Injection Speed on the Tensile Modulus

Flow-induced fiber distribution and orientation can be affected not only by the thickness and geometry of mold cavity, but also by processing conditions such as injection speed, and melt and mold temperatures. Consequently, the mechanical properties of injection-molded, short-fiber reinforced plastics are influenced by such process variables. To examine the effects of injection speed on the tensile modulus, a set of 76 × 279 × 3.05-mm (3 × × 0.12-in), 30-GF-PBT plaques were injection-molded, under controlled conditions (Table 22.6.3), at injection speeds corresponding to nominal fill times of 1.0, 1.9, and 5.6 seconds. By using the technique described in the preceding sections, the tensile modulus was determined at 12.7-mm intervals over a 63.5 × 229-mm (2.5 × 9-in) region of the plaques, as shown in Figure 22.6.10. However, only flow-direction tensile specimens are feasible for these narrow plaques. For the same fill time of 2 seconds, the effective processing conditions for these 76 × 279-mm plaques is different from that for the 152 × 203-mm plaques because of differences in part volume and tooling. The 152 × 203-mm plaques

GATE

279.4 mm 5 4 3 2 1

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

12.7-mm SEGMENTS

76.2 mm

Figure 22.6.10 Layout of 18 12.7-mm segments on five 12.7-mm wide rectangular strips cut in the flow direction from 76 × 279-mm fan/edge-gated plaques. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

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having the closest part volume to the 76 × ×3.05-mm (3 × 11 × 0.12-in) plaques are the 1.9-mm thick plaques (64.7 versus 58.6 cm3 ). For each of the three filling times, the distributions of the flow-direction tensile moduli were determined for three plaques, and the plaque-to-plaque variations were evaluated using the testing method described in the preceding sections. Each symbol at a segment location in Figure 22.6.11a represents the flow-direction tensile modulus averaged over the five segments at that location on the five strips, and

FLOW-DIRECTION MODULUS (GPa)

9

8

1 sec

2 sec

5 sec

7 1

5

10

15

18

FLOW-DIRECTION SEGMENTS (a) 9

FLOW-DIRECTION MODULUS (GPa)

800

8

1 sec

2 sec

5 sec

7 1

2

3

4

5

CROSS-FLOW DIRECTION DIVISIONS (b) Figure 22.6.11 Summary of flow-direction tensile moduli of 76 × 279 × 3.05-mm 30-GF-PBT plaques molded at fill times of 1.0, 1.9 and 5.6 seconds. (a) Maximum, minimum, and average of segments across the flow-direction strips plotted from the gate end to the end opposite to the gate, and (b) maximum, minimum, and average of each flow-direction strip plotted from side to side along the cross-flow direction. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

Fiber-Filled Material Materials – Materials with Microstructure

then again averaged over three plaques; one symbol for each of the 18 locations along the strips cut from a 76 × 279-mm plaque. Thus, each data point is the average of 15 values – five segments from the five specimens on a plaque times three plaques. The horizontal bars above and below each symbol represent the maximum and minimum of the same group of 15 values. Similarly, Figure 22.6.11b shows the average flow-direction tensile modulus for each of the five flow-direction specimens in the plaque. Each data point is the average of 54 values – 18 segments on each specimen on a plaque times three plaques. The horizontal bars above and below each symbol again indicate the maximum and minimum of the same set of 54 values. Corresponding to the data for 1.0, 1.9, and 5.6 seconds fill times, the three sets of curves in Figure 22.6.11a,b clearly show that the flow-direction tensile moduli increase with fill time – or decrease with injection speed – although the difference between the corresponding averages is less than 0.5 GPa. This fill-time dependence can be explained by differences in the shell-core-shell microstructure in plaques filled in different fill times. Fiber orientation measurements show that the average shell orientation in the flow direction is greater for longer fill times than for shorter fill times; the difference in the shell orientation increases slightly with the distance from the gate. The cores for shorter fill times have more random in-plane fiber orientation in comparison to the higher overall flow-direction fiber alignment in cores for longer fill times. Thus, it appears that slower filling (longer fill times) gives a thicker shell and a higher fiber flow-direction alignment in the core, resulting in a higher flow-direction tensile modulus. For all injection speeds, in these plaques the flow-direction tensile modulus increases along the flow direction from the gate to a maximum between segments 4 and 8, that is, between 35 and 45% of the plaque length, and then decreases to a minimum at the plaque end (Figure 22.6.11a). This trend correlates well with the thinnest core occurring at 133 mm from the gate and thicker cores at 45 and 225 mm from the gate (14). In these plaques, the distribution of the flow-direction tensile modulus along the cross-flow direction has a “V-like” shape, having a minimum in the middle and peaks at the edges (Figure 22.6.11b). This shape can, again, be attributed to the core being thinner along the edges than in the middle of the plaques (12). Figure 22.6.11b shows that the cross-flow direction variations of the flow-direction modulus in the 76 × 279-mm plaques are so large that, at each location, the ranges between the maximum and minimum for the 15 values (5 segments × 3 plaques) for the three fill times overlap. In addition, the flow-direction tensile modulus data from all the plaques molded at three injection speeds differ by as much as 1.5 GPa (between 7.5 and 9 GPa). Thus, for the same material and plaque geometry, injection molding processing conditions – injection speed in particular – can cause significant variations in the flow-direction tensile modulus, both within a plaque and from plaque to plaque. The plaque average, maximum, minimum, and standard deviation of the flow-direction tensile modulus of three plaques at three injection times are summarized in Table 22.6.7. Clearly, while the average flow-direction tensile modulus increases with increasing fill time, the modulus variation within each plaque (standard deviation) decreases. In contrast to this, as evident from Table 22.6.8, which lists correlation matrices for pairs of plaques molded at different fill times, the correlation between plaques (plaque-to-plaque repeatability) decreases with fill time or, equivalently, increases with injection speed. 22.6.2

Tensile and Flexural Strength

The previous sections have shown that the local elastic moduli of glass-filled plastics not only vary through the thickness but also along the length and width. Thus, these materials are nonhomogeneous

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Table 22.6.7 Flow-direction tensile moduli of 76 × 279 × 3.05-mm (3 × 11 × 0.12-in) 30-GF-PBT injection-molded plaques. Flow-direction tensile modulus (GPa) Fill time (s)

1.0

1.9

5.6

Plaque

Average

Standard deviation (%)

Maximum

Minimum

X

8.02

0.25 (3.09)

8.62

7.62

XI

8.08

0.28 (3.42)

8.68

7.59

XII

7.97

0.26 (3.21)

8.43

7.51

XIII

8.26

0.24 (2.95)

8.70

7.75

XIV

8.20

0.22 (2.64)

8.64

7.81

XV

8.30

0.23 (2.82)

8.77

7.90

XVI

8.43

0.21 (2.43)

8.85

7.84

XVII

8.51

0.20 (2.41)

8.96

8.03

XVIII

8.33

0.23 (2.72)

8.76

7.77

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

Table 22.6.8 The correlation matrices for flow-direction tensile moduli among 76 × 279 × 3.05-mm (3 × 11 × 0.12-in), 30-GF-PBT plaques injection-molded in fill times 1.0, 1.9, and 5.6 seconds. Fill time (s)

1.0

1.9

5.6

Correlation coefficient

Plaque number

X

XI

XI

0.94

1.00

XII

0.92

0.94

Plaque number

XIII

XIV

XIV

0.86

1.00

XV

0.88

0.89

Plaque number

XVI

XVII

XVII

0.71

1.00

XVIII

0.83

0.64

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

and anisotropic, both across and through the plaque thickness. This complicates the interpretation of tensile test strength data; without knowledge of the local tensile modulus where failure occurs strength data cannot be correlated with the elastic modulus. While understanding failure across the thickness of a nonhomogeneous, anisotropic material is difficult, this is even truer of flexural stiffness and strength in which the nominal stress varies through the thickness. This subsection addresses the variation of the flowand cross-flow-direction tensile moduli in molded plaques of three different thicknesses. Data from tests

Fiber-Filled Material Materials – Materials with Microstructure

on several plaques assess the repeatability of data. Details of how the tests were carried out are important for understanding the complexity of material property variations across plaques. 22.6.2.1 Test Procedures

Only one 6.1-mm thick, 152 × 203-mm plaque was used for failure tests. For the tensile tests, five flow-direction (F1 – F5) and six cross-flow direction (X1 – X6), 19.1-mm (0.75-in) wide rectangular specimens were cut from the plaques (one direction per plaque), as schematically shown in Figures 22.6.12 and 22.6.13, respectively. These rectangular specimens were routed down to ASTM D638 shape having a width of 12.7-mm in the gauge region. The dog-bone specimens were then pulled to failure, under displacement control, at a nominal strain rate of 0.01 s−1 . The strain over a 12.7-mm gauge length was monitored at the center of the specimen by an extensometer, and the tensile modulus for this central portion was calculated from the initial part of the stress-strain curve. 203 mm

F1

F3

152 mm

F2

F4 F5

Figure 22.6.12 Layout of five 19.1-mm wide rectangular strips cut in the flow direction from 152 × 203-mm (6 × 8-in) fan/edge-gated plaque for tensile strength measurements. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

For the flexural tests, 10 flow-direction (F1-F10) and 12 cross-flow direction (X1 – X12), 12.7-mm wide specimens were cut from each plaque, as schematically shown in Figures 22.6.14 and 22.6.15. The flexural tests to failure were done in a three-point bend configuration over a 76.2-mm (3-in) beam span at a constant displacement rate of 2.54 mm s−1 (0.1 in s−1 ). The flexural modulus was calculated from the initial part of the load-displacement curve using P (l∕d)3 4 b𝛿 in which P is the load corresponding to displacement 𝛿 , l is the beam length (span), b its width, and d its thickness. The flexural strength, 𝜎 b was determined by using a linear elastic stress distribution, resulting in 3 Pl 𝜎b = 2 bd2 Eb =

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Introduction to Plastics Engineering

X1

X2

X3

X4

X5

X6

152 mm

203 mm

Figure 22.6.13 Layout of six 19.1-mm wide rectangular strips cut in the cross-flow direction from 152 × 203-mm (6 × 8-in) fan/edge-gated plaque for tensile strength measurements. (Adapted with permission from “Mechanical Properties of Injection-Molded Short-Fiber Thermoplastic Composites. Part 1: The Elastic Moduli and Strengths of Glass-Filled Poly(Butylene Terephthalate) Plaques,” E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

203 mm GATE END

F1

VENT END

F2 F3 F4 F5 F6 F7

152 mm

804

F8 F9 F10

Figure 22.6.14 Layout of 10 12.7-mm wide rectangular strips cut in the flow direction from 152 × 203-mm fan/edge-gated plaque. Three-point bend tests were done on the 76.2-mm gate end (small-dashed lines) and vent-end (large-dashed lines) beam spans. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

Strength variations were characterized by three-point bend tests on two 76.2-mm (3-in) long spans at the gated and vented ends of the 10 flow-direction strips, as indicated by the dotted and broken lines, respectively, in Figure 22.6.14. Along the cross-flow direction, the two 76.2-mm long spans are at the top and bottom ends of the 12 strips as indicated, respectively, by the dotted and broken lines in Figure 22.6.15. For the thinner (3.0-mm, 0.12-in) plaques, strength variations were not investigated across the entire plaque. Instead, only two specimens cut along the flow direction from one plaque were tested.

Fiber-Filled Material Materials – Materials with Microstructure

152 mm

X12

X11

X9

X10

X8

X7

X6

X5

X4

X3

X2

BOTTOM

X1

TOP

203 mm

Figure 22.6.15 Layout of 12 12.7-mm wide rectangular strips cut in the cross-flow direction from 152 × 203-mm fan/edge-gated plaque. Three-point bend tests were done on 76.2-mm top (small-dashed lines) and bottom (large-dashed lines) beam spans. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

22.6.2.2 Directional Tensile and Flexural Strengths

Typical stress-strain curves for tensile tests on standard ASTM D638 dog-bone specimens cut along the flow and cross-flow directions (Figure 22.6.16), exhibit significant nonlinearity; the tensile moduli were calculated from the initial, linear parts of the curves. For injection-molded 30-GF-PBT plaques, the flow direction tensile strength is higher and the failure strain is lower than the cross-flow direction tensile strength and failure strain, respectively.

STRESS (MPa)

120 100 Flow

Cross-Flow

50

0 0

1

2

3

4

5

STRAIN (%) Figure 22.6.16 Typical stress-strain curves for tensile tests on flow- and cross-flow-direction strips cut from injection-molded 152 × 203 × 6.1-mm, 30-GF-PBT plaques. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

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Introduction to Plastics Engineering

The distributions of tensile modulus, strength and failure strain for the five flow-direction and six cross-flow-direction strips are shown in Figure 22.6.17. For these 152 × 203 × 6.1-mm (6 × 8 × 0.25-in) plaques, both the flow direction tensile modulus and the strength are about 50% higher than in the cross-flow direction. (As shown later in Section 22.7.3, this is different from the data for 50% long-fiber filled PA 6,6 plaques, in which the flow and cross-flow tensile moduli are of comparable magnitude and the tensile strength is higher in the cross-flow flow direction.) The flow-direction tensile strength is almost

TENSILE MODULUS (GPa)

9

Flow

Cross-Flow

5

3 1

2

3

4

5

6

STRIP NUMBER (a) 120

TENSILE STRENGTH (MPa)

806

100 Flow

Cross-Flow

60 1

2

3

4

5

6

STRIP NUMBER (b)

Figure 22.6.17 Flow and cross-flow direction mechanical properties of injection-molded 152 × 203 × 6.1-mm, 30-GF-PBTplaques. (a) Tensile moduli, (b) strengths, and (c) failure strains. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

Fiber-Filled Material Materials – Materials with Microstructure

FAILURE STRAIN (%)

7

5

Flow

Cross-Flow

1 1

2

3

4

5

6

STRIP NUMBER (c) Figure 22.6.17 (Continued)

constant (Figure 22.6.17b). These figures show that the cross-flow direction tensile modulus and strength increase with the distance from the gate (with the strip number). This trend is consistent with the general trend exhibited by the cross-flow tensile modulus in the same type of plaques discussed in Section 22.6.1.2. The tensile failure strains shown in Figure 22.6.17c do not exhibit any trends – they vary from 3 to 4% in the flow direction and from 4 to 6% in the cross-flow direction. The statistics for the variations of the tensile moduli, strengths, and failure strains for the flow and cross-flow direction strips cut from two plaques are summarized in Table 22.6.9. The average tensile moduli from these flow and cross-flow strips are slightly lower in comparison to the data in Tables 22.6.4 and 22.6.5, respectively. While both the flow and cross-flow direction tensile failure strains exhibit significant variations, the cross-flow tensile failure strains have larger variations and a larger average value. In general, all the three cross-flow direction properties have larger standard deviations than the flow-direction ones. For the tensile modulus, this trend is not in agreement with the results discussed in Section 22.6.1.4 Table 22.6.9 Tensile properties of 152 × 203 × 6.1-mm (6 × 8 × 0.25-in), 30-GF-PBT injection-molded plaques.

Tensile properties

Direction

Plaque number

Average

Standard deviation

Maximum

Modulus (GPa)

Flow

76 – 19

7.49

0.24 (3.14 %)

7.70 (0.9𝜎 )

7.19 (1.3𝜎 )

Cross-flow

76 – 20

4.94

0.34 (6.87 %)

5.42 (1.4𝜎 )

4.55 (1.1𝜎 )

Flow

76 – 19

113.59

0.57 (0.51%)

114.23 (1.1𝜎 )

112.78 (1.4𝜎 )

Cross-flow

76 – 20

75.73

2.15 (2.84%)

78.10 (1.1𝜎 )

72.62 (1.5𝜎 )

Flow

76 – 19

3.34

0.24 (7.09 %)

3.66 (1.4𝜎 )

3.01 (1.4𝜎 )

Cross-flow

76 – 20

5.06

0.67 (13.3 %)

6.04 (1.5𝜎 )

4.14 (1.4𝜎 )

Strength (MPa) Failure train (%)

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

Minimum

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Introduction to Plastics Engineering

(Tables 22.6.4 and 22.6.5). Note that the number of measurements in this set of tests is about 20 times smaller than in the Tests in Section 22.6.1.4. Because of the way the specimens were cut, and the locations where measurements were made, the data from this set of tests represent the tensile properties only within the gauge length along the flow and cross-flow direction centerlines of the plaques. 22.6.2.3

Variations in Tensile and Flexural Strengths

As shown in Figure 22.6.17a,b, the tensile modulus and strength of specimen X6 cut from a 152 × 203 × 6.1-mm (6 × 8 × 0.25-in) plaque are 19% and 8% higher than the corresponding values of specimen X1. Clearly, the properties vary along the flow direction. To determine the modulus and strength variations across the plaque, two ends of each of the 10 flow-direction strips (Figure 22.6.14) and 12 cross-flow direction strips (Figure 22.6.15) cut from 6.1-mm thick plaques were tested to flexural failure. The distributions of the flexural modulus and strength for two plaques are plotted in Figures 22.6.18 and 22.6.19, respectively. The averages of the flexural modulus and strength and their variations in these flow and cross-flow direction specimens are summarized in Table 22.6.10.

9

FLEXURAL MODULUS (GPa)

808

Flow/Gate

Flow/Vent

5 X-Flow/Top

X-Flow/Bottom

3 1

5 STRIP NUMBER

10

12

Figure 22.6.18 Distributions of the flow and cross-flow direction flexural moduli of injection-molded 152 × 203 × 6.1-mm, 30-GF-PBT plaques. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

A comparison of the flow-direction flexural modulus (Figure 22.6.18) and flexural strength curves (Figure 22.6.19) shows that both the flow-direction flexural modulus and strength are higher at the plaque vented end than at the gated end. Along the cross-flow (width) direction, the flow-direction flexural moduli and strengths of both sets of (gate- and vent-end) specimens are highest in the middle and decrease toward the outer edges; this trend is different from the variation of the tensile modulus (Figures 22.6.8 and 22.6.17a). In contrast to this, the cross-flow flexural modulus and strength at the top and bottom ends (Figures 22.6.18 and 22.6.19) are almost the same for all the 12 specimens, indicating that the material properties are symmetrically distributed with respect to the flow-direction centerline. However, while the cross-flow flexural modulus decreases with increasing distance from the gate (increased strip number), the cross-flow flexural strength increases. This difference is most likely a consequence of the flexural strength being calculated from a linear elastic stress distribution assumption. Depending on the fiber orientation in the skin-core microstructure of the cross section, in fiber-filled thermoplastic parts the flexural stress distribution across the thickness could be highly nonlinear, as exemplified by the flow and cross-flow “yielding” behavior shown in Figure 22.6.16.

Fiber-Filled Material Materials – Materials with Microstructure

FLEXURAL STRENGTH (GPa)

220

Flow/Gate

Flow/Vent

160

X-Flow/Top

X-Flow/Bottom

100 1

5 STRIP NUMBER

10

12

Figure 22.6.19 Distributions of the flow and cross-flow direction flexural strength of injection-molded 152 × 203 × 6.1-mm, 30-GF-PBT plaques. (Adapted with permission from Mechanical Properties of Injection-Molded Short-Fiber Thermoplastic) (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

Table 22.6.10 Flexural properties of 152 × 203 × 6.1-mm (6 × 8 × 0.25-in), 30-GF-PBT injection-molded plaques.

Flexural properties

Direction/location

Plaque number

Average

Standard deviation

Maximum

Minimum

Modulus (GPa)

Flow/gate

76 – 17

7.27

0.42 (5.81%)

7.67 (0.9𝜎 )

6.19 (2.6𝜎 )

7.86

0.28 (3.51%)

8.27 (1.5𝜎 )

7.34 (1.9𝜎 )

4.51

0.12 (2.62%)

4.67 (1.4𝜎 )

4.31 (1.7𝜎 )

4.54

0.10 (2.29%)

4.68 (1.4𝜎 )

4.34 (2.0𝜎 )

196.06

3.30 (1.68%)

200.18 (1.3𝜎 )

191.38 (1.4𝜎 )

206.67

1.41 (0.68%)

208.82 (1.5𝜎 )

204.75 (1.4𝜎 )

137.08

5.12 (3.74%)

144.32 (1.4𝜎 )

130.75 (1.2𝜎 )

137.34

4.31 (3.14%)

144.63 (1.7𝜎 )

132.37 (1.2𝜎 )

Flow/end X-Flow/top

76 – 18

X-Flow/bottom Strength (MPa)

Flow/gate

76 – 17

Flow/end X-Flow/top X-Flow/bottom

76 – 18

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

The data in Table 22.6.10 show that although the average flow and cross-flow flexural moduli are of the same magnitude as the corresponding tensile modulus (Tables 22.6.4 and 22.6.5), the average flexural strengths are significantly higher than the tensile strengths, both in the flow and cross-flow directions. For comparison, the ratios of the flexural to the tensile modulus and strength are listed in Table 22.6.11. While the tensile and flexural moduli differ from each other by less than 10%, the flexural strengths are about 80% higher than the tensile strengths, both in the flow and cross-flow directions. Again, this difference is caused by the linear elastic assumption used in calculating the strength; the flexural stress distribution across the thickness in fiber-filled thermoplastic parts is not only highly nonlinear but also depends on the layer-by-layer fiber orientation.

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Introduction to Plastics Engineering

Table 22.6.11 Average tensile and flexural properties for 152 × 203 × 6.1-mm (6 × 8 × 0.25-in), 30-GF-PBT injection-molded plaques. The terms in the fourth and seventh columns are the ratios of the flexural to the tensile properties. Flow direction Properties

Tensile

Modulus (GPa)

7.49

Strength (MPa)

113.59

Cross-flow direction

Flexural

Ratio

7.57

1.01

201.36

1.77

Tensile

Flexural

Ratio

4.94

4.52

0.91

75.73

137.21

1.81

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

As discussed earlier, the shear-induced fiber orientation is different in injection-molded plaques of different thickness, resulting in the flow- and cross-flow-direction tensile moduli being thickness dependent. To characterize the effects of plaque thickness on the flexural modulus and strength, three-point bend test were done on specimens cut from three 3-mm thick 152 × 203-mm 30-GF-PBT plaques along the flow, cross-flow, and 45° directions (one plaque for each direction). For the 45° direction flexural tests, six 12.7-mm wide rectangular specimens were cut from the plaques, as schematically shown in Figure 22.6.20. Typical load-displacement curves for flexural tests on these specimens (Figure 22.6.21)

203 mm (8 in)

1

152 mm (6 in)

810

2 3 4 5 6

Figure 22.6.20 Layout of six 12.7-mm wide rectangular strips cut at 45° to the flow direction from a 152 × 203-mm

fan/edge-gated plaque, on each of which three-point bend tests were done on 76.2-mm beam spans. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

Fiber-Filled Material Materials – Materials with Microstructure

200 Linear Fit

LOAD (N)

150

100

50

0 0

5

10

15

20

DISPLACEMENT (mm) Figure 22.6.21 Typical load-displacement curves for flexural tests on 12.7-mm wide rectangular strips cut at 45° to the

flow direction from injection-molded 152 × 203 × 3-mm, 30-GF-PBT plaques. (Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

show significant nonlinearity, similar to the tensile test stress-strain curves (Figure 22.6.16); the flexural moduli were calculated from the initial, linear parts of the curves. The flexural results for the flow, cross-flow, and 45° directions are summarized in Table 22.6.12. Table 22.6.12 Flexural properties of 152 × 203 × 3.0-mm (6 × 8 × 0.12-in), 30-GF-PBT injection-molded plaques.

Flexural properties

Modulus (GPa)

Strength (MPa)

Direction

Plaque number

Average

Standard deviation

Flow

71 – 16

8.79

0.28 (3.19%)

X-Flow

71 – 14

4.89

0.06 (1.13%)

45°

71 – 15

5.79

0.38 (6.57%)

Flow

71 – 16

190.64

8.76 (4.60%)

X-Flow

71 – 14

112.83

0.72 (0.64%)

45°

71 – 15

133.00

9.25 (6.96%)

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

The average flow-direction flexural modulus (8.79 MPa) is comparable to the average tensile modulus (8.36 MPa) listed in Table 22.6.4, even though the numbers of tensile and flexural specimens tested were different. As expected, for the same plaque thickness, the cross-flow flexural modulus and strength are lower than those for the flow-direction, and those for the 45° flexure specimens fall in between, being about 15% smaller than the flow and cross-flow averages.

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Introduction to Plastics Engineering

A comparison of the flow and cross-flow properties in Table 22.6.12 with those in Table 22.6.10 shows that while the flow and cross-flow direction flexural moduli increase as the plaque thickness decreases from 6.1 to 3.0 mm, the flow and cross-flow direction flexural strengths decrease. These trends differ from the effects of thickness on tensile moduli, for which the flow-direction modulus increases but the cross-flow direction modulus decrease with decreasing plaque thickness (Tables 22.6.4 and 22.6.5). However, the difference between the flow and the cross-flow direction flexural strength decreases with increasing thickness, as does the difference between the flow and cross-flow direction flexural modulus. Table 22.6.13 lists the average flow- and cross-flow-direction tensile and flexural moduli and strengths, and the ratios of the cross-flow- to flow-direction properties for different sets of tests. The ratios of the cross-flow- to the flow-direction tensile (or flexural) strength or modulus increase with increasing thickness, that is, the differences between the flow and cross-flow properties decrease. Table 22.6.13 Average properties of 152 × 203-mm (6 × 8-in), 30-GF-PBT injection-molded plaques.

Properties

Test

Modulus (GPa)

Tensile

Flexural Strength (MPa)

Average plaque thickness (mm)

Flow

Cross-flow

Ratio

1.9

8.57

4.66

0.54

6.1

7.99

5.19

0.65

6.1

7.49

4.94

0.66

3.0

8.79

4.89

0.56 0.60

6.1

7.57

4.52

Tensile

6.1

113.59

75.73

0.67

Flexural

3.0

190.64

112.83

0.59

6.1

201.36

137.21

0.68

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

22.6.3

Effects of Fiber Aspect Ratio

For short-fiber composites, increasing fiber content and/or fiber aspect ratio can be used to increase mechanical properties, that is, to increase the reinforcing efficiency of the fibers. In general, the longer and thinner the fibers are (the higher the fiber aspect ratios), the more effective the increase in stiffness and strength will be. Because flow-induced shear stresses developed in the screws, runners and mold cavities during compounding and injection-molding processes can bend and break the fibers, the fiber lengths in pellets can be very different from those in molded products. As such, variations in mechanical properties within a part, and from part to part, should not be surprising. The effects of fiber length on tensile properties was characterized by tests on specimens made from two lots of glass-fiber-filled PBT pellets with the same fiber content (30 wt%), but different fiber lengths. The average fiber lengths of the pellets were 159 μm in the first (short) lot and 293 μm in the second (long) lot. The average diameter of these glass fibers was 13 μm with a standard deviation of 1.9 μm. Tensile tests were done on 22 ASTM Type I dog-bone molded specimens of the shorter fiber lot and 25 specimens of the longer fiber lot. The average fiber lengths in these two sets of tensile bars were 155 and 224 μm, respectively, so that the corresponding fiber aspect ratios for these two lots were 12 and

Fiber-Filled Material Materials – Materials with Microstructure

17 for an average glass-fiber diameter of 13 μm. Longer fibers seem more prone to damage than shorter fibers; on average, the fiber length attrition in the shorter and longer lots was 3 and 24%, respectively. Table 22.6.14 summarizes the tensile modulus, maximum stress, and failure strain of the specimens tested. These tensile-bar data indicate that a 42% increase in the average fiber length (or aspect ratio) results in a 14 and 19% increase in the tensile modulus and the tensile strength, respectively. In comparison, the average tensile failure strain is lower for the higher average fiber-length case. For tensile bars, while glass fibers with an average aspect ratio of 12 increase the tensile modulus and tensile strength by 323 and 189%, respectively, the corresponding increases for glass fibers with an average aspect ratio of 17 are 36 and 225%, respectively. The average failure strains for fiber-filled tensile bars are dramatically smaller than those for unfilled tensile bars, but the failure strains for unfilled tensile bars exhibit far more scatter than those for filled bars. Table 22.6.14 Tensile modulus and strength of injection-molded ASTM tensile bars of unfilled PBT and 30-GF-PBT.

Tensile properties

Modulus (GPa)

Strength (MPa)

Failure Strain (%)

Average fiber length

Average

Standard deviation

unfilled

2.60

0.09 (3.29%)

2.73 (1.4 𝜎 )

2.48 (1.4 𝜎 )

155 μm

8.40

0.22 (2.60%)

8.70 (1.4 𝜎 )

8.14 (1.2 𝜎 )

Maximum

Minimum

224 μm

9.60

0.15 (1.54%)

9.95 (2.4 𝜎 )

9.25 (2.4 𝜎 )

unfilled

60.44

0.73 (1.21%)

61.34 (1.2 𝜎 )

59.17 (1.7𝜎 )

155 μm

114.12

0.86 (0.76%)

115.33 (1.4 𝜎 )

112.79 (1.7𝜎 )

224 μm

136.00

0.71 (0.52%)

137.39 (2.0 𝜎 )

134.51 (2.1 𝜎 )

unfilled

18.13

12.71 (70.1%)

3.98 (2.0 𝜎 )

3.29 (1.1 𝜎 )

155 μm

2.76

0.06 (2.34%)

2.91 (2.4 𝜎 )

2.69 (1.1 𝜎 )

224 μm

2.71

0.07 (2.75%)

2.88 (2.3 𝜎 )

2.54 (2.3 𝜎 )

(Adapted with permission from E.W. Liang and V.K. Stokes, Polymer Composites, Vol. 26, pp. 428 – 447, 2005.)

A comparison of the data for molded tensile bars with the data for tensile bars cut from plaques (Table 22.6.14) shows that the tensile moduli and strengths of molded bars lie within the range of the flow-direction moduli and strengths of bars cut from molded plaques. The data-sheet properties in Table 22.6.1 appear to correlate well with the flow-direction tensile properties of 152 × 203 × 6.1-mm plaques. However, the tensile modulus and strength of molded tensile bars are much higher than the cross-flow modulus and strength of specimens cut from molded plaques. Similarly, the data-sheet properties in Table 22.6.1 provide a very inadequate and misleading description of the mechanical properties of the material: differences in the flow and cross-flow direction properties, and the effect of specimen thickness, are not accounted for. 22.6.4

Effects of Matrix Resin

During molding the tumbling action of the fibers is controlled both by the local shear rate, which in turn is affected by the melt viscosity. As such, the local fiber distribution and orientation can be expected to be different for the same part molded in the same way. Some of these differences are described in this section by comparing the results of 30-GF-PBT with those for 33-GF-PA6,6 (ZYTEL 70G33).

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Introduction to Plastics Engineering

These comparisons are based on the same geometries and test methods used for obtaining data for 30-GF-PBT. Following the procedure used in Section 22.6.1.2, the upper half of Figure 22.6.22 shows the variation of the flow-direction tensile modulus in 152 × 203 × 6.1-mm 33-GF-PA6,6 plaques; the lower half shows the corresponding data for 30-GF-PBT from Figure 22.6.3a. The corresponding data for cross-flow modulus variations are compared in Figure 22.6.23, the data for 30-GF-PBT are from Figure 22.6.3b. Both the flow- and cross-flow-direction tensile are systematically higher for the 33-GF-PA6,6 than for the 30-GF-PBT material. Some of this reflects the higher 3 wt% glass content in the glass-filled nylon. And some of the large difference in the moduli for these tow materials could stem from longer glass fibers in nylon. But there is also a qualitative difference in the variations, reflecting the effects of differences in the rheological characteristics of PA6,6 and PBT.

10

FLOW-DIRECTION MODULUS (GPa)

814

33-GF-PA6,6

9 6.1 mm

8

30-GF-PBT

7 0

24

48

72

96

120

FLOW-DIRECTION STRIPS Figure 22.6.22 Comparison of the summaries of the flow-direction tensile moduli in 152 × 203 × 6.1-mm plaques of 33-GF-PA6,6 (upper curve) and 30-GF-PBT (lower curve). Maximum, minimum, and average of each flow-direction strip from the gate end to the end opposite to the gate. (Unpublished data from E.W. Liang and V.K. Stokes.)

To characterize the effects of plaque thickness on the moduli of 30-GF-PBT, the values of the flow-direction moduli were averaged in the cross-flow direction and plotted along the flow direction, as shown for three plaque thicknesses in Figure 22.6.8a. The upper curve in Figure 22.6.24 shows this variation for a 6.1-mm thick 33-GF-PA6,6. For comparison, the lower curve in this figure shows the variation for 30-GF-PBT (from Figure 22.6.8a). While the moduli for 33-GF-PA6,6 are systematically much higher, the pattern of variations are similar for the two materials. However, as shown in Figure 22.6.25, the patterns of variations of the moduli are quite different in 3-mm thick plaques. So, the pattern of modulus variations across an injection-molded plaque does depend on the plastic.

Fiber-Filled Material Materials – Materials with Microstructure

CROSS-FLOW DIRECTION MODULUS (GPa)

6.1 mm

7

33-GF-PA6,6

6

30-GF-PBT

5 0

16

32

48

64

80

96

CROSS-FLOW DIRECTION STRIPS Figure 22.6.23 Comparison of the summaries of the cross-flow-direction tensile moduli in 152 × 203 × 6.1-mm plaques of 33-GF-PA6,6 (upper curve) and 30-GF-PBT (lower curve). Maximum, minimum, and average of each flow-direction strip from the gate end to the end opposite to the gate. (Unpublished data from E.W. Liang and V.K. Stokes.)

22.6.5

Summary of Mechanical Characteristics of Short-Fiber Systems

The results in Section 22.6 show that injection-molded short-glass-fiber-filled thermoplastics are complex nonhomogeneous, anisotropic material systems. Like all fiber-filled materials, they exhibit through-thickness nonhomogeneity (as indicated by differences between tensile and flexural properties) because of oriented layers. The in-plane fiber orientation distribution in these layers causes the material to have in-plane anisotropic mechanical properties. While glass fibers increase both the tensile modulus and strength of unfilled thermoplastics, the increase in the modulus is greater than the increase in strength. However, the average failure strains of fiber-filled tensile bars are dramatically smaller than those of unfilled tensile bars. While fibers with higher aspect

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FLOW-DIRECTION MODULUS (GPa)

10 33-GF-PA6,6

6.1 mm

9

8 30-GF-PBT

7 0

2

4

6

8

10

12

FLOW-DIRECTION DIVISIONS Figure 22.6.24 Comparison of the cross-flow-direction tensile moduli in 152 × 203 × 6.1-mm 33-GF-PA6,6 and 30-GF-PBT plaques. (Unpublished data from E.W. Liang and V.K. Stokes.)

10

FLOW-DIRECTION MODULUS (GPa)

816

33-GF-PA6,6

9

8 30-GF-PBT

3 mm

7 0

2

4

6

8

10

12

FLOW-DIRECTION DIVISIONS Figure 22.6.25 Comparison of the cross-flow-direction tensile moduli in 152 × 203 × 6.1-mm 33-GF-PA6,6 and 30-GF-PBT plaques. (Unpublished data from E.W. Liang and V.K. Stokes.)

Fiber-Filled Material Materials – Materials with Microstructure

ratio increase both the modulus and strength of injection-molded short-fiber composites, they further reduce the failure strain. Data from plaques molded at fill times of 1.0, 1.9, and 5.6 seconds show that the injection speed affects mechanical properties: While the flow-direction tensile modulus increases with fill time (decreases with injection speed), the difference between the corresponding averages is not significant. The mechanical properties of short-fiber filled composites are strongly thickness-dependent. The thinnest plaques exhibit the largest differences between the flow and cross-flow tensile modulus and strength. These differences decrease with increasing thickness. Thus, the thinnest plaques exhibit very high in-plane anisotropy; the thick plaques are less anisotropic. For all plaque geometries and processing conditions, the flow-direction tensile modulus is higher at the edges than in the interior. For all thicknesses, the flow-direction flexural properties – both modulus and strength – are significantly higher than the cross-flow properties. The flexural modulus is comparable to the tensile modulus, but the flexural strength is significantly higher than the tensile strength. Both the flow and the cross-flow direction flexural moduli decrease with increasing plaque thickness. In contrast to this, the flow and cross-flow direction flexural strengths increase with plaque thickness. As with tensile properties, the ratios of the cross-flow- to flow-direction flexural modulus and strength increase with plaque thickness. The flexural modulus and strength of the 45° specimens lie between the flow and cross-flow modulus and strength, respectively, but are lower than the flow and cross-flow averages. The flow-direction tensile strength is almost constant across the plaque width. The cross-flow direction tensile modulus and strength increase with the distance from the gate. The modulus variation pattern within plaques, or the qualitative modulus distribution, is similar for all the plaques: high and low moduli are distributed over similar regions. The modulus variation within each plaque (standard deviation) appears to decrease with plaque thickness. Injection molding processing conditions – injection speed in particular – can cause significant variations in mechanical properties. While the average flow-direction tensile modulus increases with increasing fill time, the modulus variation within each plaque decreases. Data from multiple plaques have shown that the plaque-to-plaque repeatability of local properties is independent of plaque thickness. However, the flow-direction modulus has slightly better repeatability than the cross-flow direction modulus. The plaque-to-plaque repeatability of mechanical properties is also sensitive to injection molding processing conditions. The correlation between properties of plaques decreases with fill time or, equivalently, increases with injection speed. Fiber (length) degradation caused by the complex stress fields developed in the screws, runners, and mold cavities during compounding and injection-molding processes, causes further property variations in injection-molded, short-fiber-reinforced products. As demonstrated by the tensile bars molded from two lots of glass-fiber-filled PBT pellets with the same fiber content (30 wt%), but different fiber lengths, a 42% increase in the average fiber length (or aspect ratio) results in a 14 and 19% increase in the tensile modulus and tensile strength, respectively. Clearly, the single-point mechanical properties given in the manufacturers’ data sheets provide a very inadequate description of the mechanical properties of the material. The data-sheet values are reasonable for flow direction properties, but do not even provide a conservative estimate for the much lower cross-flow properties.

22.7 Long-Fiber Filled Systems In comparison to short-fiber-filled materials normally used for injection molding applications – in which glass fibers typically have diameters and lengths in the ranges of 7 – 1 μm and 0.2 – 3 mm,

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Introduction to Plastics Engineering

respectively – higher stiffness and strength can be obtained by using longer fibers with lengths on the order of 10 mm, having fiber aspect ratio on the order of 1,000. The bridging provided by long fibers also results in better impact properties. However, molding parts with long-fiber filled resins requires special care to reduce fiber breakage from the intense mixing processes during molding. Better fiber-length retention is achieved through proper nozzle, screw, gate, and process designs. The data in Table 22.7.1 compare the tensile strength and modulus of 50% glass-filled, short- and long-fiber reinforced PA6,6 (nylon 6,6), as determined by tests on end-gated, injection-molded ASTM D638 Type 1 tensile specimens, molded in thicknesses of 2.0, 3.2, 4.4, and 6.4 mm. In addition, the last two columns in this table provide skin-to-core thickness ratios. Table 22.7.1 Average tensile moduli and strengths of 2.0-, 3.2-, 4.4-, and 6.4-mm thick injection-molded bars of 50-SGF-PA and 50-LGF-PA. Tensile modulus (Pa)

Tensile strength (MPa)

Percent skin/core ratio

Plaque thickness (mm)

50-SGF-PA

50-LGF-PA

50-SGF-PA

50-LGF-PA

50-SGF-PA

50-LGF-PA

2.0

17

18

198

242

100/0

80/20

3.2

17

18

202

251

100/0

40/60

4.4

17

17

211

243

80/20

50/50

6.4

17

16

215

224

70/30

30/70

(Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

As the thicknesses of the molded specimens increase from 2.0 to 6.4 mm, the average tensile moduli for the short-fiber-filled PA 6,6 (50-SGF-PA66) and the long-fiber-filled PA 6,6 (50-LGF-PA) decrease by 4 and 12%, respectively. For a thickness increase from 2.0 to 3.2 mm, the tensile strengths of SGF-PA and 50-LGF-PA increase by 2 and 4%, respectively. However, for a thickness increase from 3.2 to 6.4 mm, while the strength of 50SGF-PA increases by 7%, that of 50-LGF-PA decreases by 11%. These differences in the behavior of the SGF- and LGFR-materials are partially explained by the observed differences in the skin-core thicknesses of these two materials: The 2.0-, 3.2-, 4.4-, and 6.4-mm thick tensile bars of 50-SGF-PA had skin-to-core thickness percentages of 100/0, 100/0, 80/20, and 70/30, respectively. The corresponding ratios for the 50LGF-PA bars were 80/20, 40/60, 50/50, and 30/70, respectively. Also, the differences in the orientations at the molded edges (side walls) of the 12.7-mm specimens are likely to influence the tensile properties. While the mechanical properties of fiber-filled materials measured by tests on injection-molded ASTM D638 tensile bars (Table 22.7.1) provide a first level quantitative feel for the effects of fiber length on mechanical properties, because the fibers in such bars are preferentially aligned in the length direction, these mechanical properties are likely to be unrealistically high relative to the actual local mechanical properties in molded parts. Also, the fiber orientations at the edges (side walls) of the injection-molded bars are not representative of the fiber orientations in plate-and shell-like structures of which parts are made. And clearly, such tests do not give an idea of the mechanical properties in the lateral, “cross-flow,” directions. Following the methodology used for mapping the properties short-fiber filled materials across molded plaques, the following sections present a phenomenological characterization of the stiffness and strength distributions in injection-molded, long-glass-filled PA6,6 plaques. Measurements of the tensile and flexural elastic moduli and strengths were made on specimens cut from injection-molded plaques of five thicknesses. A methodology for part design with this class of materials is discussed.

Fiber-Filled Material Materials – Materials with Microstructure

22.7.1

Tensile Modulus

This section discusses the distribution of mechanical properties of in long-fiber filled injectionmolded parts. The levels of anisotropy and spatial variation of the mechanical properties in such parts are characterized by data from molded plaques 50 wt% of long-glass-fiber filled PA6,6 (VERTON® RF-700-10-HS), which will be referred to as 50-LGF-PA. The mechanical properties of the VERTON material, both in the dry as-molded condition and for specimens conditioned to 50% relative humidity, as given in the manufacturer’s data sheet, are listed in Table 22.7.2. Note that these properties were measured by tests on injection-molded bars, in which the fibers are preferentially aligned in the length direction. Table 22.7.2 Data-sheet properties of VERTON RF-700 – 10HS from tests on injection-molded bars.

Specimen condition

Flexural modulus (GPa) (106 psi)

Tensile strength (MPa) (103 psi)

Flexural strength (MPa) (103 psi)

Dry

15.9 (2.3)

255 (37)

400 (58)

50% Relative humidity

11.4 (1.65)

186 (27)

262 (38)

(Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

22.7.1.1 Test Procedures

The stiffness and strength of the material were determined through tensile and three-point flexural tests on specimens cut from 152 × 203-mm (6 × 8-in) injection-molded, edge-gated plaques molded in five thicknesses of 2.0, 2.5, 3.0, 3.8, and 6.1-mm (0.08, 0.10, 0.12, 0.15, and 0.24-in). The plaques were molded in a 300-ton injection-molding machine, with a free-flow check ring to minimize fiber breakup during molding, and a fan gate was used to obtain a relatively straight flow front. The molding conditions are listed in Table 22.7.3. Prior to molding, the pellets were predried at an approximate temperature of Table 22.7.3 Processing conditions for molding plaques with five thicknesses on a 300-Ton injection-molding machine with a free-flow check ring. Plaque thickness (mm)

2.0

2.5

3.0

3.8

6.1

Mold temperature (°C)

93

93

99

93

93

Nozzle temperature (°C) Front barrel (°C)

Middle barrel (°C) Rear barrel (°C)

Injection time (s)

271

271

266

271

271

277

277

277

277

277

277

277

277

277

277

277

277

282

277

277

5

8

7

8

14

Cure time (s)

19

22

33

25

40

Screw speed (rpm)

35

35

35

35

35

(Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

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99°C (210°F) as per manufacturer’s specifications. To control the moisture content, the plaques were kept under ambient conditions of 22°C (72°F) and 50% relative humidity. The stiffness and strength of the molded plaque material were characterized in three stages. First, tensile tests on standard ASTM D638 dog-bone specimens and three-point flexural tests on rectangular specimens were used to determine, respectively, the tensile and flexural modulus and strength of the plaque material, both in the flow and cross-flow directions. Second, to characterize variations in tensile strength across the plaque, tensile tests were done on 18 short specimens cut from each plaque. And third, to understand the large in-plane variations in the mechanical properties, the tensile modulus of the material was mapped at 12.7-mm (0.5-in) intervals over 16.13 × 103 mm2 (25 in2 ) regions of plaques. 22.7.1.2

Tensile and Flexural Tests

For the tensile tests, six flow-direction (F1 – F6) and eight cross-flow direction (X1 – X8), 19.1-mm (0.75-in) wide rectangular specimens were cut from the plaques (one direction per plaque), as schematically shown in Figures 22.7.1 and 22.7.2, respectively. (The significance of the dotted and hatched regions is described in the sequel.) These rectangular specimens were routed down to ASTM D638 shape having a 12.7-mm (0.5-in) width in the gauge region. The dog-bone specimens were then pulled to failure, under displacement control, at a nominal strain rate of 10−2 s−1 . The strain over a 12.7-mm wide gauge length at the center (central dotted areas in Figures 22.7.1 and 22.7.2) of the specimen was monitored by an extensometer, and the tensile modulus for this central portion was calculated from the initial part of the stress-strain curve.

F1 F2

152.4 mm

820

F3 F4 F5 F6

203.2 mm Figure 22.7.1 Layout of six 19.1-mm (0.75-in-) wide rectangular bars cut in the flow direction from 152 × 203-mm (6 × 8-in) fan/edge-gated plaque. These bars were routed down to standard ASTM D638 dog-bone specimens having a 12.7-mm gauge width. The dotted area in the center is the region over which an extensometer was used to determine the local tensile modulus. The hatched area is the region in which the specimens failed in the tensile tests. (Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

For the flexural tests, eight flow-direction (F1-F8) and 12 cross-flow direction (X1-X12), 12.7-mm wide specimens were cut from each plaque, as schematically shown in Figure 22.7.3. The flexural tests to failure were done on over a 76.2-mm (3-in) beam span at a constant displacement rate of 2.54 mm s−1 (0.1 in s−1 ). The flexural modulus was calculated from the initial part of the load-displacement curve.

Fiber-Filled Material Materials – Materials with Microstructure

152.4 mm

X1 X2 X3 X4 X5 X6 X7 X8

203.2 mm

152.4 mm

X12

X11

X10

X8

X6

X7

F5

X5

F1

X3

F7 F6

X4

F2

X1

F8

F3

X2

F4

X9

Figure 22.7.2 Layout of eight 19.1-mm (0.75-in) wide rectangular bars cut in the cross-flow direction from 152 × 203-mm (6 × 8-in) fan/edge-gated plaque. These bars were routed down to standard ASTM D638 dog-bone specimens having a 12.7-mm gauge width. The dotted area in the center is the region over which an extensometer was used to determine the local tensile modulus. The hatched area is the region in which the specimens failed in the tensile tests. (Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

203.2 mm Figure 22.7.3 Layout of 8 flow and 12 cross-flow 12.7-mm (0.5-in) wide rectangular bars cut from 152 × 203-mm (6 × 8-in) fan/edge-gated plaque. These bars were used for determining flexural modulus and strength. (Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

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Introduction to Plastics Engineering

22.7.1.3

Strength Variation Study

For the strength variation study, two sets of six, 101.6-mm (4-in) long tensile test specimens were cut from the gated (FN1-FN6) and far ends (FF1-FF6) of plaques, as per the layouts shown in Figure 22.7.4; a third set was cut from the middle (FM1-FM6) of plaques (Figure 22.7.5). The shoulders and the width (12.7 mm, 0.5 in) of these short dog-bone specimens were the same as for standard ASTM D638 dog-bones, but the straight gauge length was only 25.4 mm (1-in) long instead of 76.2 mm (3 in) for the standard specimen. Only two thicknesses were investigated in the strength variation study. The 2-mm (0.08-in) thick specimens were cut from two plaques, that is, one series of three sets of specimens. But for the thicker (3.8-mm, 0.15-in) plaque, two sets of specimens (four plaques) were cut and tested. 2.0-mm Plaque

FN1

FF1

FN2

FF2

FN3

FF3

FN4

FF4

FN5

FF5

FN6

FF6

152.4 mm

822

3.8-mm Plaque

203.2 mm Figure 22.7.4 Layout of 12 19.1-mm (0.75-in) wide by 101.6-mm (4-in) long bars cut in the flow direction from 152 × 203-mm (6 × 8-in) fan/edge-gated plaque; six each were cut from the gated end (FN1-FN6) and the far end (FF1-FF6) of the plaque. These bars were routed down to dog-bone specimens with a 12.7-mm gauge width. The gauge length having this width was 25.4 mm (1 in). The hatched areas show the regions in which the short bars failed in tensile tests. The right-upward and left-upward hatched lines correspond, respectively to failure regions in 2.0- and 3.8-mm thick plaques. The crosshatched areas correspond to failure regions common to 2.0- and 3.8-mm thick plaques. (Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

22.7.1.4

In-Plane Tensile Modulus Variations

In addition to large differences between the local tensile and flexural properties (mainly caused by “layers” of material having different preferred fiber orientations), and difference between the local properties of specimens cut along the flow and cross-flow directions (again attributable to preferred fiber orientation), variations in the properties of specimens cut from different locations on rectangular plaques were unexpectedly large. Also, the plaque thickness affected the differences between the local tensile and flexural properties. These variations and thickness effects likely result from the nonuniform fiber motion and shear-induced fiber degradation during molding.

Fiber-Filled Material Materials – Materials with Microstructure

2.0-mm Plaque

FM1

152.4 mm

FM2 FM3 FM4 FM5 FM6 3.8-mm Plaque

203.2 mm Figure 22.7.5 Layout of six 19.1-mm (0.75-in) wide by 101.6-mm (4-in) long bars, FM1-FM6, cut in the flow direction from 152 × 203-mm (6 × 8-in) fan/edge-gated plaque. These bars were routed down to dog-bone specimens with a 12.7-mm gauge width. The gauge length having this width was 25.4 mm (1 in). The hatched areas show the regions in which the short bars failed in tensile tests. The right-upward and left-upward hatched lines correspond, respectively to failure regions in 2.0- and 3.8-mm thick plaques. The crosshatched areas correspond to failure regions common to 2.0- and 3.8-mm thick plaques. (Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

To resolve these variations, the tensile modulus of the plaque material was determined at 12.7-mm (0.5-in) intervals – both along the flow and cross-flow directions – over approximately 16.13 × −103 mm2 (25 in2 ) portions of plaques of different thicknesses. For this, the plaques were cut into 12.7-mm wide specimens: Lines were first drawn on each plaque at 12.7-mm intervals, to mark 12.7-mm segments for subsequent tensile tests. The 12.7-mm wide specimens (strips) were then cut, at right angles to these lines, starting from the centerline of the plaque and working outward. A thin, high-speed circular saw was used to minimize waste and heat buildup, and to obtain smooth edges for attaching the extensometer. In this way, the plaques were cut into numbered, rectangular 12.7-mm wide specimens with marks at 12.7-mm long intervals. As schematically shown in Figures 22.7.6 and 22.7.7, such specimens were cut in both the flow and cross-flow directions – one direction per plaque. As shown in these two figures, the flow-direction modulus was determined at 10 segments on each of 8 specimens; the cross-flow modulus was determined at six segments on each of 12 specimens. The thickness and the width of each 12.7-mm wide segment on each strip were measured and recorded. Although the test regions represented in these two figures are 127 × 101.6 mm (5 × 4 in) and 152.4 × 76.2 mm (6 × 3 in), respectively, the actual areas from which the specimens were cut is somewhat larger because of the width of the saw used for cutting the specimens (Figure 22.7.7). A servo-hydraulic test machine was used to determine the elastic modulus along each 12.7-mm segment on each specimen. A special purpose MTS extensometer, that can simultaneously measure the strain on two edges of a specimen over a 12.7-mm gauge length, and can be calibrated over a full-scale measurement range of 1% strain, was used for simultaneously measuring the local strains on both edges of 12.7-mm wide specimens. The extensometer was attached to the first segment on a specimen, which was

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Introduction to Plastics Engineering

F1

1

2

3

4

5

6

7

8

9 10

F2

152.4 mm

F3 F4 F5 F6 F7 F8

203.2 mm Figure 22.7.6 Layout of eight 12.7-mm wide bars cut in the flow direction from 152 × 203-mm (6 × 8-in) fan/edge-gated plaque. The dashed lines mark the 10 locations on each specimen at which the tensile modulus was measured at 12.7-mm intervals. (Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 1 2

152.4 mm

824

3 4 5 6

203.2 mm Figure 22.7.7 Layout of 12 12.7-mm wide bars cut in the cross-flow direction from 152 × 203-mm (6 × 8-in) fan/edge-gated plaque. The dashed lines mark the six locations on each specimen at which the tensile modulus was measured at 12.7-mm intervals. (Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

pulled in tension under load control to the prescribed peak load and then unloaded at the same rate. The same procedure was repeated after attaching the extensometer to the second segment, and so on. The maximum load and the left- and right-edge strains were recorded. These data were used to calculate the stress, average strain, and modulus, and to plot contours of the modulus across the plaque. All the tests

Fiber-Filled Material Materials – Materials with Microstructure

were done under load control, at an equivalent strain rate of 5 × 10−3 s−1 , to an approximate strain of 0.2% – the reasons for this choice are discussed next. The range of test parameters over which repeatable measurements can be made on long-glass-filled material without damage – sufficiently large strains for obtaining repeatable measurements, yet not so large as to damage the specimen for subsequent measurements – was determined through a special series of tests: First the tensile stress-strain data from flow and cross-flow tests (Section 22.7.1.2) were examined. Points from the linear portions of the stress-strain curves were used to calculate the expected loads at different strains. Then, two types of exploratory tests were done on excess specimens from the edges of 2- and 6.1-mm thick plaques. For both tests the specimens were pulled under load control at an approximate equivalent strain rate of 5 × 10−3 s−1 . The specimens were also unloaded at the same rate. In the first series of tests, a specimen was loaded/unloaded a minimum of four times at each 12.7-mm segment along the specimen to an approximate strain of 0.15%. After the specimen had been fully scanned, additional tests were repeated at some of the segments. The objective of these tests was to check the repeatability of the tests and to assess any damage that might affect the validity of the scanning procedure. In the second series of tests, a single segment was subjected to increasingly higher load/unload cycles – twice at each load level. For both series of tests, the load-strain curves were examined for linearity. The maximum loads and left and right strains were recorded, from which the stress, average strain, and tensile modulus were calculated. These exploratory tests demonstrated that repeated testing does not damage the material and produces repeatable results so long as the strain is not too high. Based on these tests, subsequent modulus scans were conducted at a constant maximum load that corresponds to a nominal maximum strain of 0.2% for all test conditions. Using the stress-strain data (Section 22.7.1.2), the peak allowable loads were calculated by using the procedure described previously. The peak nominal stresses used for different plaques – based on the constant loads discussed before – and the actual maximum and minimum strains in the specimens are listed in Table 22.7.4. Table 22.7.4 Constant stress used for determining the modulus distribution in 2- and 6.6-mm thick plaques at 12.7-mm intervals. Strain distribution over specimens (%)

Measurement direction

Constant stress on specimens (MPa) (103 psi)

Maximum

Minimum

Standard deviation

2.0 (0.08)

Flow

32.1 (4.65)

0.226

0.188

0.008

2.0 (0.08)

Cross-flow

14.1 (2.05)

0.241

0.174

0.014

6.1 (0.24)

Flow

27.6 (4.00)

0.290

0.203

0.021

6.1 (0.24)

Cross-flow

24.1 (3.50)

0.306

0.175

0.022

Plaque thickness (mm) (in)

(Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

The results of these tests showed an unexpectedly high degree of spatial nonhomogeneity over the plaques. Since an understanding of this nonhomogeneity is helpful for interpreting tensile and flexural moduli and strength data, the spatial and directional variations of the tensile modulus are discussed first.

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Introduction to Plastics Engineering

22.7.2

Spatial and Directional Variations of the Tensile Modulus

To map the variation of the tensile modulus across plaques, the flow and cross-flow direction tensile moduli over 12.7 × 12.7-mm (0.5 × 0.5-in) regions were determined, respectively, across 127 × 101.6-mm (5 × 4-in) and 152.4 × 76.2-mm (6 × 3-in) central portions of plaques. Contours of the flow- and cross-flow direction tensile moduli across these portions of 2.0-mm and 6.1-mm thick plaques are shown in Figure 22.7.8. The two figures on the left, Figure 22.7.8 parts a and b, are for the 2.0-mm thick plaques; those on the right, Figure 22.7.8 parts c and d, are for the 6.1-mm thick plaques. The figures on the top (Figure 22.7.8 parts a and c) and bottom (22.7.8b and d) correspond, respectively, to the flowand cross-flow direction moduli, respectively. The two arrows on the sides of the contour level bars indicate the nominal maximum and minimum moduli in the plaque.

FLOW-DIRECTION MODULUS (GPa) Tensile Test Direction

15

15

15.5–16.5 14.5–15.5

10

12.5–13.5

5 GPa

10

13.5–14.5

(a)

5

11.5–12.5

(c)

GPa

10.5–11.5

2.0-mm-THICK PLAQUES

9.5–10.5

6.1-mm-THICK PLAQUES

8.5–9.5

15

15

7.5– 8.5

10 5 GPa

10

6.5–7.5

(d)

(b) Tensile Test Direction

5 GPa

CROSS-FLOW DIRECTION MODULUS (GPa) Figure 22.7.8 Contours of the flow- (Figures 22.7.8a,c) and cross-flow (Figure 22.7.8b,d) direction tensile moduli for 2.0-mm (Figures 22.7.8a,b) and 6.1-mm (Figure 22.7.8c,d) thick long-glass-filled nylon plaques. (Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

The actual maximum and minimum flow direction tensile moduli for the 2.0-mm thick plaques (Figure 22.7.8a) were 16.8 and 14.1 GPa (2,435 and 2, 045 × psi), respectively, with a standard deviation of 0.6 GPa. These values represent the highest stiffness among all plaques of all the thicknesses tested.

Fiber-Filled Material Materials – Materials with Microstructure

The central region of the plaque was less stiff than the outer edges, which had the highest stiffness. The lowest stiffness was in the region farthest from the gated end. The overall variation in the tensile modulus in these two regions was relatively small (∼2 GPa difference across the plaque) with large areas having small variations. The contours at the end away from the gated end are somewhat different from those for the other flow-direction plaques. This difference appears to be caused by local voids or nonuniformities in glass distribution/orientation that show up as “blemishes” on the plaque surface. The maximum and minimum tensile cross-flow direction moduli (Figure 22.7.8b) were 8.1 and 6.1 GPa (1,176 and 1,176 891 × psi), respectively, with a standard deviation of 0.5 GPa. These values represent the lowest stiffness among all the plaques tested. The central region across the plaque had a slightly higher stiffness than the outer edges. However, the overall variations across the plaque were relatively small (∼2 GPa across the whole plaque) with large areas having small variations. The actual maximum and minimum tensile flow-direction moduli for the 6.1-mm thick plaques (Figure 22.7.8c) were 13.9 and 10.0 GPa (1,447 and 2, 020 × 103 psi), respectively, with a standard deviation of 0.8 GPa. For the thicker plaque, the material stiffness increased with the distance from the gated end; also, the stiffness was higher in the outer specimens. However, the variation across the plaque was relatively small (∼2.8 GPa across the whole plaque) with large areas having small variations. For the thinner plaques, the stiffness appears to decrease with the distance from the gate. The maximum and minimum cross-flow direction tensile moduli (Figure 22.7.8d) were 14.2 and 7.6 GPa (2,055 and 1, 096 × psi), respectively, with a standard deviation of 1.2 GPa. Typically, the material was stiffer starting from the gated end through the center of the specimens. This trend continued more than three-quarters of the way across the plaque. Also, the specimens from the outer lateral edges of the plaques were less stiff, and the end farthest from the gated end had a very low tensile modulus in the central region. The pattern of flow-direction modulus variation across plaques, or the qualitative modulus distribution, is similar for all the plaques in the sense that, in these plaques, high and low moduli are distributed over similar regions. The flow and cross-flow moduli do appear to be correlated – high values of one seem to be accompanied by low values of the other. However, as the thickness increases, the difference between the flow and cross-flow moduli becomes smaller. The large differences in the elastic moduli could result from nonhomogeneous distributions of both fiber content and orientation. The arrows along the contour level bars in Figure 22.7.8, which indicate the nominal maximum and minimum moduli in plaques, provide additional insight into how thickness and direction affect variations in moduli. The arrows in Figures 22.7.8a,b show that although the flow direction moduli for a 2-mm thick plaque are significantly higher than the cross-flow moduli, the variation within each plaque is within a relatively narrow band. In contrast, for the thickest plaque (6.1 mm), the ranges for both directions overlap (Figure 22.7.8c,d). Also, the flow-direction moduli vary within a narrower band than for the cross-flow moduli. One explanation for these large differences between the moduli variations in the 2- and 6.1-mm thick plaques is provided by possible differences in the thicknesses of the skin and core regions. The data for similar materials in Table 22.7.1, in which the skin/core thickness ratios for 2- and 6.4-mm thick molded bars were measured to be 80/20 and 30/70, respectively, would seem to indicate a much higher level of fiber alignment in the thinner specimens. This preferential alignment would explain the large difference between the flow and cross-flow tensile moduli in the thinner specimens and a much smaller variation in the thicker specimens.

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Introduction to Plastics Engineering

22.7.3

Flow and Cross-Flow Mechanical Properties of Injection-Molded Plaques

Tensile tests on standard ASTM D638 dog-bone specimens cut along the flow and cross-flow directions (Figures 22.7.1 and 22.7.2) showed that, in general, the flow direction tensile modulus is larger than the cross-flow direction modulus. In particular, the flow direction tensile modulus in the 2.0-mm thick plaque (the thinnest in the set) is significantly higher than the cross-flow direction tensile modulus. The difference between these moduli decreases with increasing thickness. For the thickest specimens in this set (6.1-mm thick plaques), the flow and cross-flow direction moduli are of comparable magnitude. In contrast, flexural tests on specimens cut from the flow and cross-flow directions showed that the flow-direction flexural moduli for all plaque thicknesses are significantly higher than the cross-flow flexural moduli. While the thickness has a large effect on the flow-direction flexural moduli, it has a very small effect on the cross-flow direction flexural moduli. These trends can be more easily inferred from the average tensile and flexural moduli listed in Table 22.7.5. As shown in Figure 22.7.1a, the difference between the flow and the cross-flow direction average tensile moduli is largest for the thinnest plaques. This difference decreases with increasing thickness; the flow and cross-flow direction average tensile moduli are comparable for the 6.1-mm thick plaques. Figure 22.7.9b shows that, with increasing thickness, the flow-direction average flexural modulus decreases, while the cross-flow direction average flexural modulus almost remains the same. Table 22.7.5 Average flow and cross-flow tensile and flexural moduli for 2.0-, 2.5-, 3.0-, 3.8-, and 6.1-mm thick plaques of long-glass-filled nylon. Flow direction modulus (GPa) (106 psi)

Cross-flow direction modulus (GPa) (106 psi)

Plaque thickness (mm) (in)

Tensile

Flexural

Tensile

Flexural

2.0 (0.08)

15.2 (2.2)

15.9 (2.3)

6.9 (1.0)

5.5 (0.8)

2.5 (0.10)

13.8 (2.0)

14.5 (2.1)

7.6 (1.1)

5.5 (0.8)

3.0 (0.12)

13.8 (2.0)

13.8 (2.0)

9.0 (1.3)

5.5 (0.8)

3.8 (0.15)

13.1 (1.9)

13.1 (1.9)

10.3 (1.5)

5.5 (0.8)

6.1 (0.24)

11.0 (1.6)

11.7 (1.7)

12.4 (1.8)

6.2 (0.9)

(Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

These trends are different from those exhibited by the 30-GF-PBT (Section 22.6.2.2, Figure 22.7.17), in which both the flow direction tensile modulus and the strength are about 50% higher than in the cross-flow direction. And for which the flow-direction tensile strength is almost constant (Figure 22.7.17b). Note that for the 2.0-, 3.0-, and 6.1-mm thick plaques, the average moduli and strengths listed in Table 22.7.5 and shown in Figure. 22.7.9 are averages over all specimens from one plaque (six and eight specimens in the flow and cross-flow directions, respectively). However, for the 2.5and 3.8-mm thick plaques, the reported averages are over all specimens from two plaques for each thickness (12 and 16 specimens in the flow and cross-flow directions, respectively). The open circles in Figure 22.7.9 indicate the flow and cross-flow direction averages for different thicknesses. The bars above and below each open circle represent the maximum and minimum values (the range) of the measurements associated with the averages. Because of overlap in the flow and cross-flow data for the

Fiber-Filled Material Materials – Materials with Microstructure

FLOW DIRECTION

15 10 5

CROSS-FLOW DIRECTION

0 1

AVERAGE TENSILE STRENGTH (GPa)

AVERAGE TENSILE MODULUS (GPa)

20

2

3

4

5

6

20 FLOW DIRECTION

15 10 5

CROSS-FLOW DIRECTION

0

7

1

2

3

4

5

6

7

PLAQUE THICKNESS (mm)

PLAQUE THICKNESS (mm)

(a)

(b)

AVERAGE TENSILE STRENGTH (GPa)

AVERAGE TENSILE MODULUS (GPa)

6.1-mm thick plaque, left and right brackets have been used to indicate the ranges for the flow and cross-flow direction data, respectively. Clearly, the cross-flow direction properties have larger scatter than those for the flow direction.

400 300

FLOW DIRECTION

200 100 CROSS-FLOW DIRECTION

0 1

2

3

4

5

6

7

400

FLOW DIRECTION

300 200 100 CROSS-FLOW DIRECTION

0 1

2

3

4

5

6

7

PLAQUE THICKNESS (mm)

PLAQUE THICKNESS (mm)

(c)

(d)

Figure 22.7.9 Average tensile (Figure 22.7.9a) and flexural (Figure 22.7.9b) moduli, and average tensile (Figure 22.7.9c) and flexural (Figure 22.7.9d) strengths, for flow (circles) and cross-flow (squares) specimens cut from 2.0-, 2.5-, 3.0-, and 6.1-mm thick long-glass-filled nylon plaques. The vertical bars indicate the range of variation. (Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

The plaque thickness has little effect on the flow-direction tensile strength. The corresponding cross-flow strengths are lower and increase with the thickness, with the strength of the 6.1-mm thick plaque being the highest. Note that the average cross-flow tensile strength is actually higher than the average 6.1-mm thick flow-direction strength. Both the flow and cross-flow flexural strengths are affected by thickness. Thicker plaques have higher flow and cross-flow direction strengths. The cross-flow flexural strength, which is fairly uniform across the plaque, is lower than the flow-direction strength for the same plaque thickness. The average tensile and flexural strengths are summarized in Table 22.7.6. As shown in Figure 22.7.9c, the difference between the flow and the cross-flow direction average tensile strengths is largest for the thinnest plaques. This difference decreases with increasing thickness; just as the tensile moduli, the flow and cross-flow direction average tensile strengths have

829

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Introduction to Plastics Engineering

comparable magnitudes for the 6.1-mm thick plaques. Figure 22.7.9d shows that both the flow and cross-flow average flexural strengths increase with increasing thickness. For the flow direction, the ratio of the flexural to the tensile strength increases with the thickness from 1.54 for the 2-mm thick plaque to 2.77 for the 6.1-mm thick plaque. For the cross-flow direction, this ratio almost remains constant at an average value of about 1.36. Table 22.7.6 Average flow and cross-flow tensile and flexural strengths for 2.0-, 2.5-, 3.0-, 3.8-, and 6.1-mm thick plaques of long-glass-filled nylon. The term ratio in the fourth and seventh columns is the ratio of the flexural modulus to the tensile modulus. Flow direction strength (MPa) (103 psi)

Cross-flow direction strength (MPa) (103 psi)

Plaque thickness (mm) (in)

Tensile

Flexural

Ratio

Tensile

Flexural

2.0 (0.08)

179.3 (26)

275.8 (40)

1.54

75.8 (11)

96.5 (14)

1.27

2.5 (0.10)

179.3 (26)

282.7 (41)

1.58

82.7 (12)

117.2 (17)

1.42

3.0 (0.12)

186.2 (27)

303.4 (44)

1.63

96.5 (14)

131.0 (19)

1.36

3.8 (0.15)

172.4 (25)

289.6 (42)

1.68

110.3 (16)

151.7 (22)

1.38

6.1 (0.24)

117.2 (17)

324.1 (47)

2.77

124.1 (18)

172.4 (25)

1.39

Ratio

(Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

Note that while modulus measurements are local to the region where the strain is measured, the strength corresponds to the point where the failure occurs, which can be different for different specimens because each specimen will fail at its weakest point. Thus, in nonhomogeneous materials, the tensile test will only determine the strength of the weakest region. In tensile tests on 13 plaques (78 specimens), 90% of the flow-direction specimens failed between 38 and 70 mm (1.5 and 2.75 in) from the gated end of the specimen. In contrast, in tensile tests on 13 plaques (104 specimens), 94% of the cross-flow specimens failed within 19 and 38 mm (0.75 – 1.5 in) from the center of the specimens. Unlike the flow direction specimens – most of which failed near the gated edge in the hatched region shown in Figure 22.7.1 – the cross-flow specimens failed, randomly, in two narrow bands on both sides of the plaque centerline, in the two hatched regions shown in Figure 22.7.2. The observed large differences in the elastic moduli, described in Section 22.7.2, provide an explanation for the tensile failure locations discussed earlier. The flow and cross-flow modulus scans for the plaques (Figure 22.7.8) show that the material has the lowest tensile modulus in these failure regions. Care must be taken in interpreting and correlating the average elastic moduli and strengths in Figure 22.7.9. First, the tensile moduli were measured over 12.7-mm (0.5-in) intervals on flow and cross-flow strips from the central regions of the plaques. Second, the flexural moduli, which represent an average over 76-mm (3-in) beam lengths, may also be interpreted as the “flexural moduli” at the mid-span of each specimen that, clearly, are located in regions of the plaque that are different from those for the tensile specimens. Third, in contrast, the flow and cross-flow tensile strengths correspond to material from still other regions. Clearly, these failure regions are different from the regions over which the elastic moduli were averaged. And, fourth, the failure locations of the flexural specimens, which tended to fail at mid-span under the load, were totally different from those for the tensile specimen.

Fiber-Filled Material Materials – Materials with Microstructure

A comparison of the data in Table 22.7.2 with the data in Tables 22.7.5 and 22.7.6 shows that the data-sheet properties provide a very inadequate description of the mechanical properties of the material: differences in properties in the flow and cross-flow directions, and the effect of specimen thickness, are not accounted for. The data for 50-LGFR-PA 6,6 in Table 22.7.1 show that the tensile moduli and the tensile strengths of molded bars do vary with the specimen thickness. However, a comparison of this data with the data in Tables 22.7.5 and 22.7.6 shows that molded bars appear to exhibit much higher moduli and strengths than bars cut from molded plaques. Thus, molded bars are not appropriate for generating mechanical properties for design; specimens cut from molded plaques will result in properties that are closer to those in molded parts. 22.7.4

Variations in Strength

As discussed previously, in the tensile tests the standard ASTM D638 dog-bone specimens tend to fail in regions (hatched areas in Figures 22.7.1 and 22.7.2) away from the center of the specimens (dotted areas in Figures 22.7.1 and 22.7.2). Clearly, such tests only determine the strength of the weakest regions of the specimens. To determine the variation of the tensile strength along the plaque, 18 short specimens – six each from each end (Figure 22.7.4) and the middle (Figure 22.7.5) – cut from 2.0- and 3.8-mm thick plaques were tested to failure. For these 2.0- and 3.8-mm thick plaques, the variations of the tensile strength and the tensile modulus can be gauged from the mean values listed in Tables 22.7.7 and 22.7.8, respectively. The results show that in 3.8-mm thick plaques the flow-direction strength increases with the distance from the gate. The strengths of all three sets of specimens from 3.8-mm thick plaques are lowest at the center and increase toward the outer edges. The strengths of the specimens from the middle region fall between the two outer sets of specimens. These trends were not obvious for the 2-mm thick plaques. The regions in which these specimens failed are shown by the hatched areas in Figures 22.7.4 and 22.7.5: The right-upward and left-upward hatched lines correspond, respectively to failure regions in 2.0- and 3.8-mm thick plaques; the crosshatched areas correspond to failure regions common to 2.0- and 3.8-mm thick plaques. The failure regions in the near-gate specimens (FN) are about the same for the 2.0- and 3.8-mm thick plaques; these regions are consistent with the failure regions in the ASTM D638 dog-bone specimens (Figure 22.7.1). Table 22.7.7 Variation of the flow-direction tensile strength in 2.0- and 3.8-mm thick plaques. Tensile strength (MPa) (103 psi) Specimen location: FN

Specimen location: FM

Specimen location: FF

Specimen number

2-mm plaque

3.8-mm plaque

2-mm plaque

3.8-mm plaque

2-mm plaque

3.8-mm plaque

1

186.2 (27)

165.5 (24)

206.8 (30)

193.1 (28)

206.8 (30)

193.1 (28)

2

172.4 (25)

151.9 (22)

213.7 (31)

179.3 (26)

227.5 (33)

199.9 (29)

3

172.4 (25)

137.9 (20)

193.1 (28)

172.4 (25)

193.1 (28)

186.2 (27)

4

172.4 (25)

144.8 (21)

206.8 (30)

172.4 (25)

206.8 (30)

186.2 (27)

5

186.2 (27)

158.6 (23)

213.7 (31)

193.1 (28)

206.8 (30)

186.2 (27)

6

179.3 (26)

165.5 (24)

213.7 (31)

206.8 (30)

213.7 (31)

199.9 (29)

(Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

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Introduction to Plastics Engineering

Table 22.7.8 Variation of the flow-direction tensile modulus in 2.0- and 3.8-mm thick plaques. Tensile modulus (GPa) (106 psi) Specimen location: FN

Specimen location: FM

Specimen location: FF

Specimen number

2-mm plaque

3.8-mm plaque

2-mm plaque

3.8-mm plaque

2-mm plaque

3.8-mm plaque

1

11.7 (1.7)

9.0 (1.3)

12.4 (1.8)

11.7 (1.7)

13.1 (1.9)

11.7 (1.7)

2

12.4 (1.8)

10.3 (1.5)

12.4 (1.8)

11.0 (1.6)

13.1 (1.9)

11.0 (1.6)

3

10.3 (1.5)

10.3 (1.5)

12.4 (1.8)

11.0 (1.6)

11.7 (1.7)

11.0 (1.6)

4

11.0 (1.6)

10.3 (1.5)

11.7 (1.7)

11.7 (1.7)

13.1 (1.9)

11.7 (1.7)

5

11.7 (1.7)

11.7 (1.7)

12.4 (1.8)

11.0 (1.6)

13.1 (1.9)

13.1 (1.9)

6

12.4 (1.8)

11.0 (1.6)

14.5 (2.1)

11.7 (1.7)

13.1 (1.9)

14.5 (2.1)

(Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

22.7.5

Mechanical Properties for Design

Clearly, the single-point mechanical properties of the 50-LGF-PA in the manufacturer’s data sheet provide an inadequate description of the mechanical properties of the material. The data-sheet values are not even conservative. While the 11.4 GPa (data sheet) flexural modulus is lower than the measured flow-direction flexural moduli (Table 22.7.5) of 11.7 – 15.9 GPa, this data-sheet value is not a conservative estimate for the cross-flow direction flexural modulus of 5.5 GPa. Similarly, while the data-sheet flexural strength of 262 MPa is lower than the measured flow-direction flexural strength of 276 – 324 MPa (Table 22.7.6), it is significantly higher than the measured cross-flow direction flexural strength of 97 – 172 MPa. Also, while the published tensile strength of 186 MPa is a reasonable estimate for the measured flow direction strength of all but the thickest plaque, it is significantly higher than the measured cross-flow strength that varies from 76 to 124 MPa. Also, the mechanical properties measured on molded bars of different thicknesses tend to overestimate the properties and underestimate the effect of thickness. And, of course, measurements on molded bars do not provide any information on cross-flow properties. Given the current understanding of the material, the mechanical properties for a conservative analysis of part performance based on a homogeneous, isotropic material model should be tensile and flexural moduli of 6.9 and 5.5 GPa, respectively, and tensile and flexural strengths of 76 and 97 MPa, respectively. These properties clearly will result in an overly conservative design. However, better use of the material requires better use of measured properties. The next step would be to use an equally conservative approach that, while still assuming the material to be homogeneous and isotropic, uses the thickness-based conservative properties listed in Table 22.7.9. Of course, still better material use would require a thickness-based anisotropic material model – the use of which also requires a specification of appropriate local flow and cross-flow directions.

Fiber-Filled Material Materials – Materials with Microstructure

Table 22.7.9 Part-thickness based mechanical properties for use in homogeneous-material based part performance analysis. Elastic modulus (GPa) (106 psi)

Strength (MPa) (103 psi)

Plaque thickness mm (in)

Tensile

Flexural

Tensile

Flexural

2.0 (0.08)

6.9 (1.0)

5.5 (0.8)

75.8 (11)

96.5 (14)

2.5 (0.10)

7.6 (1.1)

5.5 (0.8)

82.7 (12)

117.2 (17)

3.0 (0.12)

9.0 (1.3)

5.5 (0.8)

96.5 (14)

131.0 (19)

3.8 (0.15)

10.3 (1.5)

5.5 (0.8)

110.3 (16)

151.7 (22)

6.1 (0.24)

11.0 (1.6)

6.2 (0.9)

117.2 (17)

172.4 (25)

(Adapted with permission from V.K. Stokes, L.P. Inzinna, E.W. Liang, G.G. Trantina, and J.T. Woods, Polymer Composites, Vol. 21, pp. 696 – 710, 2000.)

22.8 *Fiber Orientation The bulk of this chapter has addressed an empirical, phenomenological description of the complex morphology and mechanical property variations in glass-fiber-filled injection-molded parts. While an interpretation, and even prediction, of such variations is desirable, this cannot be done without using models that require a much higher level of mathematics and mechanics. And even then, the state-of-the-art is not robust enough to predict part performance from the part geometry and molding conditions. This section outlines the complexity of modeling fiber orientation development in a part in several simple steps: First, measures for characterizing three-dimensional fiber orientation are considered; two-dimensional versions of which are then described. Next, results for the motion of a fiber in a two-dimensional flow field are considered. Then, a model for the evolution of fiber orientation in a moving suspension is presented. While the flow field during mold filling results in both in-plane and through-thickness variations in fiber orientation, the bulk of the fibers orient in planes. For this reason, and for mathematical simplicity, a discussion of two-dimensional, planar orientations is useful. 22.8.1

*Orientation of a Single Fiber

A simple approach to fiber orientation is to first consider the motion of rigid fibers in a homogeneous flow in which the strain rate is uniform throughout the flow field. The fiber orientation is then independent of spatial location and the local microstructure. In general, the three-dimensional orientation of a rigid fiber at a point x = (x, y, z), at time t, can be described by a unit vector p(x, t) = ( px , py , pz ), in which its components are functions of (x, y, z, t). Only two of these components are independent; p being a unit vector requires px 2 + py 2 + pz 2 = 1

(22.8.1)

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This orientation may also be characterized by the angles (𝜃 , 𝜙) shown in Figure 22.8.1. These two representations are related through px = sin 𝜃 cos 𝜙,

py = sin 𝜃 sin 𝜙,

pz = cos 𝜃

(22.8.2)

z

p

θ

y

O

ϕ 0 ≤ θ ≤ π x

0 ≤ ϕ ≤ 2π

Figure 22.8.1 Fiber orientation described by an orientation vector or by orientation angles.

In the two-dimensional case, the orientation of a rigid fiber at a point x = (x, y), at time t, can be described by a unit vector p(x, t) = ( px , py ), in which both its components are functions of (x, y, t). This orientation may also be characterized by the angle 𝜙 that the fiber makes with the x - axis, as shown in Figure 22.8.2, in which px = cos 𝜙 and py = sin 𝜙.

y

p px py

ϕ x

O 0 ≤ ϕ ≤ 2π

Figure 22.8.2 Two-dimensional fiber orientation described by an orientation angle 𝜙.

Fiber-Filled Material Materials – Materials with Microstructure

22.8.2

*Fiber Orientation Distribution Function

Instead of tracing the orientation vector for each individual fiber, it is more practical to statistically model fiber orientation at one fixed location. An orientation distribution function 𝜓 ( p, t) = 𝜓 (𝜃 , 𝜑, t) is defined such that the probability of finding fibers with angles between 𝜃 * and 𝜃 * + Δ𝜃 * , and 𝜙* and 𝜙* + Δ𝜙* is given by P(𝜃 ∗ ≤ 𝜃 ≤ 𝜃 ∗ + Δ𝜃 , 𝜙∗ ≤ 𝜙 ≤ 𝜙∗ + Δ𝜙) = 𝜓 (𝜃 ∗ , 𝜙∗ , t ) sin 𝜃 Δ𝜃 Δ𝜙

(22.8.3)

The fiber orientation distribution function must satisfy several conditions. Because fibers are treated as rigid, symmetric rods, a fiber oriented in the direction of p is indistinguishable from that oriented along −p, so that 𝜓 (p) must be an even function of p, that is

𝜓 ( p) = 𝜓 (−p) or 𝜓 (𝜃 , 𝜙) = 𝜓 (𝜋 − 𝜃 , 𝜋 + 𝜙)

(22.8.4)

The distribution function represents the orientation state of a collection of fibers, the numbers of which differ from point to point, so that 𝜓 (𝜃 , 𝜙) must be normalized as ∮

𝜓 (𝜃 , 𝜙) sin 𝜃 d𝜃 d𝜙 =

2𝜋

𝜋

∫0 ∫ 0

𝜓 ( 𝜃 , 𝜙) sin 𝜃 d𝜃 d𝜙 = 1

(22.8.5)

Based on this normalization the distribution function for random, or isotropic, orientation is given by

𝜓 (𝜃 , 𝜙) =

1 4𝜋

(22.8.6)

The distribution function 𝜓 (𝜃 , 𝜙, t) must also satisfy a conservation law for the assembly of rigid fibers that, for a homogeneous flow field, can be shown to reduce to

𝜕 𝜓 (𝜙) 𝜕 𝜕 𝜕 = ( ṗ 𝜓 ) + ( ṗ 𝜓 ) + ( ṗ 𝜓 ) 𝜕t 𝜕 px x 𝜕 py y 𝜕 pz z 1 𝜕 𝜕 =− (𝜓 𝜃̇ sin 𝜃 ) − (𝜓 𝜙̇ ) sin 𝜃 𝜕𝜃 𝜕𝜙

(22.8.7)

This equation basically states that a fiber leaving one orientation state must move to another state. The distribution function at any instant at any location can be experimentally determined by measuring the orientation in a small representative sample of fibers. The number of fibers in the sample with orientation angles (𝜃 , 𝜙) can be counted, and a histogram of the orientation distribution n( 𝜃 * , 𝜙* ) constructed, such that the number of fibers oriented at angles between 𝜃 * and 𝜃 * + Δ𝜃 * , and 𝜙* and 𝜙* + Δ𝜙* is N(𝜃 ∗ ≤ 𝜃 ≤ 𝜃 ∗ + Δ𝜃 , 𝜙∗ ≤ 𝜙 ≤ 𝜙∗ + Δ𝜙) = n(𝜃 ∗ , 𝜙∗ , t ) sin 𝜃 Δ𝜃 Δ𝜙

(22.8.8)

The three-dimensional orientation distribution function can then be obtained as

𝜓 (𝜙) =

n(𝜃 , 𝜙) 2𝜋 𝜋 ∫0 ∫0

n( 𝜃 , 𝜙) sin 𝜃 d𝜃 d𝜙

=

n(𝜃 , 𝜙) Ntotal

(22.8.9)

where Ntotal is the total number of fibers measured. For the two-dimensional case (Figure 22.8.2), the orientation distribution function 𝜓 ( p, t) = 𝜓 (𝜙, t) is defined such that the probability of finding fibers with angles between 𝜙* and 𝜙* + Δ𝜙* is given by P(𝜙∗ ≤ 𝜙 ≤ 𝜙∗ + Δ𝜙) = 𝜓 ( p∗ ) Δ𝜙 = 𝜓 (𝜙∗ ) Δ𝜙

(22.8.10)

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which must satisfy

𝜓 ( p) = 𝜓 (−p) or 𝜓 (𝜙) = 𝜓 ( 𝜋 + 𝜙)

(22.8.11)

Also, with 𝜃 = 𝜋 ∕2 in Eq. 22.8.5, for the 2D case the distribution function 𝜓 (𝜙) is normalized as ∮

𝜓 (𝜙) d𝜙 =

2𝜋

∫0

𝜓 (𝜙) d𝜙 = 1

(22.8.12)

Based on this normalization, the distribution function for random, or isotropic, two-dimensional orientation is given by 1 (22.8.13) 2𝜋 The distribution function 𝜓 (𝜙, t) must also satisfy the conservation law in Eq. 22.8.7, which for the two-dimensional case reduces to 𝜕 𝜓 (𝜙) 𝜕 =− (𝜓 𝜙̇ ) (22.8.14) 𝜕t 𝜕𝜙

𝜓 (𝜙) =

The two-dimensional orientation distribution function can then be obtained as n(𝜙) n(𝜙) 𝜓 (𝜙) = 𝜋 = Ntotal ∫ n(𝜙) d 𝜙

(22.8.15)

0

where Ntotal is the total number of fibers measured. 22.8.3

**Orientation Tensors

While the probabilistic definition of fiber orientation eliminates having to keep track of the orientations of large numbers of fibers, this description still is quite cumbersome. One way to further reduce this complexity is to use orientation tensors, which are weighted averages of the distribution function. However, the use of tensors requires a level of mathematics well beyond that used in undergraduate engineering curricula and, certainly, well beyond the scope of this book. Nevertheless, a short introduction to orientation tensors is being given to show that rational, analytical treatments of fiber-filled systems requires advanced engineering sciences and higher-level mathematics. Most quantities, such as mass, volume, and density are specified by single numbers called scalars. But there are physical quantities such as force and velocity that, besides having magnitudes measured by numbers (scalars), also have associated directions – such as the direction in which a force acts and the direction in which an object is moving. Such objects, which have both a magnitude and an associated direction, are called vectors. Using three-dimensional coordinates, (x, y, z), a vector is specified by its three components vx , vy , and vz in the three respective coordinate directions. A bold symbol is used to differentiate a vector (magnitude and direction) from a scalar; in this representation this vector is written as v = v (vx , vy , vz ) to mean that v is the vector with components vx , vy , and vz in the respective coordinate directions. In a more concise description, the coordinate directions are referred to as the 1-, 2-, and 3-directions and (vx , vy , vz ) are interpreted as (v1 , v2 , v3 ). In this notation, the vector v has the components has the components vi , i = 1, 2, 3. Such vectors have been used in Chapter 6 to describe the velocity field in two-dimensional flow. There are other physical objects called tensors with 9, 27, … components, specifically called second-, third-, … order tensors, respectively. In mechanics, the stress at a point is measured by nine components:

Fiber-Filled Material Materials – Materials with Microstructure

The force acting on each coordinate plane is measured by three components – the normal stress and two shear stresses – so that by accounting for the forces on the other two coordinate planes gives a total of nine components. A second order tensor with nine components aij , i = 1, 2, 3 and j = 1, 2, 3 can then be written as a2 = (aij ) , to mean “a2 is a second order tensor with the nine components aij with the subscripts varying over 1, 2, 3.” Similarly, a fourth order tensor would be written as a4 = (aijkl ) , with each of the four subscripts varying over 1, 2, 3. In the language of tensors, a vector is then a first order tensor. One additional convention is used to concisely write summations of products of tensor components ∑3 by using a summation convention. As an example, a sum such as ti = r =1 air nr is written as ti = air nr , it being understood according to the summation convention that a sum over 1, 2, 3 is implied over any repeated index, such as r in this case; such repeated indices are called dummy indices. This powerful convention makes it is easy to represent very complex summations involving tensors. For instance, in the example that follows, summations over 1, 2, 3 are implied for the two dummy indices r and s, tij =

3 3 ∑ ∑

air ajs nr ns

⇔

tij = air ajs nr ns

(22.8.16)

r =1 s =1

Although the orientation distribution function gives a general description of the fiber orientation state, it is too cumbersome to implement in numerical simulations of molding problems. Instead tensors obtained by taking moments of the distribution function are used to describe and predict fiber orientation in short-fiber composites. While retaining sufficient information, such orientation tensors provide a more concise description of the orientation. The orientation tensors are obtained by integrating tensor products of p weighted by the distribution function 𝜓 over all possible directions. Because 𝜓 is an even function, only tensors of even order exist. Of these, the second order, a2 = (aij ) , and fourth order, a4 = (aijkl ) , tensors are aij =

∮

aijkl =

pi pj 𝜓 (𝜙) d𝜙

∮

pi pj pk pl 𝜓 (𝜙) d𝜙

(22.8.17) (22.8.18)

where, the subscripts i, j, k, l on pi , (aij ), and (aijkl ) have values 1, 2, 3 corresponding to the coordinates x1 = x, x2 = y, and x3 = z. In terms of the second order tensor, the normalization condition for the distribution functions in Eq. 22.8.5 reduces to a11 + a22 + a33 = 1, for the 3D case a11 + a22 = 1, for the 2D case

(22.8.19)

The components of the second order tensor for a random orientation state (see Eqs. 22.8.4 and 22.8.12) for the three- and two-dimensional cases reduce to 1 aij = 𝛿ij , for the 3D case 3 1 (22.8.20) aij = 𝛿ij , for the 2D case 2 where 𝛿 ij is the Kronecker delta (𝛿 11 = 𝛿 22 = 𝛿 33 = 1, and 𝛿 ij = 0 for i ≠ j). Figure 22.8.3 shows the variation of the two-dimensional fiber distribution function 𝜓 = 𝜓 (𝜙) for simple model fiber distributions. The top row in this figure shows the model fiber orientation distribution being considered, the second row shows the corresponding shape of the fiber orientation distribution

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function. And the last row shows the components of the two-dimensional orientation tensor. Note from Eq. 22.8.12 that the area under the curve 𝜓 = 𝜓 (𝜙) must equal 1. In Figure 22.8.3a, the fibers are randomly distributed equally in all directions, so that 𝜓 = 1∕2𝜋 is a constant, and the orientation tensor is aij = 0.5 𝛿 ij . The models in Figure 22.8.3b and c show the special cases in which the fibers are aligned, respectively, along x- (𝜙 = 0) and y- (𝜙 = 𝜋 ∕2) axes; in each of these two cases the orientation tensor only has one nonzero component. Both these are special cases of unidirectional fiber alignment at an angle 𝛼 to 𝜙 = 0, shown in Figure 22.8.3d. In this case all the three components of the orientation tensor are nonzero.

𝜓 1/(2 𝜋)

·0

2𝜋

𝜙

𝜓

𝜓

𝛿(0)

·0

2𝜋

𝜙

·0

𝜓

𝛿(𝜙 – 𝜋 /2)

2𝜋

0.5 0 aij = 0 0.5

1 0 aij = 0 0

0 0 aij = 0 1

(a)

(b)

(c)

𝛿(𝜙 – 𝛼)

·0 𝛼 2𝜋

𝜙

aij =

cos 2 𝛼 sin 2𝛼 2

𝜙

sin 2𝛼 2 sin 2 𝛼

(d)

Figure 22.8.3 Variations of the two-dimensional fiber orientation distribution function 𝜓 = 𝜓 (𝜙) for simple model fiber distributions, together with the corresponding orientation tensor components. (a) Isotropic case in which the fibers are aligned randomly but equally in all directions. (b) All fibers aligned at to the x-axis direction. (c) All fibers aligned at to the y-axis direction. (d) All fibers aligned at an angle 𝛼 to the x-axis.

In Figure 22.8.3, use has been made of Dirac’s delta function 𝛿 (x), defined by { +∞ 0 for x ≠ 0 𝛿 (x) = 𝛿 (x) dx = 1 with ∫−∞ ∞ for x = 0

(22.8.21)

It has the shape of a very narrow infinite spike, such that the area under the curve is unity. With this terminology, 𝛿 (x − 𝛼 ) shifts the “spike” to x = 𝛼 , so that 𝛿 (x − 𝛼 ) = 0 for x ≠ 𝛼 . It can be shown that for any function f(x) +∞

∫−∞

f (x) 𝛿 (x − 𝛼 ) dx = f (𝛼 )

(22.8.22)

Fiber-Filled Material Materials – Materials with Microstructure

22.8.4

*Fiber Orientation Measurement

Fiber orientation measurement plays a critical role in the design and manufacture of fiber-reinforced plastic parts by providing quantitative verification of fiber orientation predictions and relating process optimization to product performance. The techniques for characterizing fiber orientation in composite materials are classified as being direct or indirect. In direct methods, the orientation of each individual fiber is directly measured from an image; the orientation distribution function can then be obtained for a sample containing a sufficient number of fibers. The laborious and time-consuming process of manually making such direct measurements has been automated through the use of computer image analysis. Indirect measurements can be used to compute fiber orientation information from the anisotropy of macroscopic properties caused by the presence of fibers. Although indirect methods are less tedious, quantifying the relationship between orientation data and macroscopic measurements is not easy. 22.8.4.1 Direct Measurement

Optical and X-ray microscopy are two of the direct measurement techniques briefly discussed in this section. Optical Microscopy of Metallographically Polished Surfaces The surfaces of samples cut from fiber-

reinforced parts are first polished using metallographic polishing techniques. Inclined cylindrical fibers intersect the polished cross section and appear as ellipses on the polished surface, as shown in Figure 22.8.4. The out-of-plane orientation angle is defined by the major and minor axes of the ellipse as

𝜃 = cos−1 (b∕a)

(22.8.23)

z

y

ϕ

x

b a

Figure 22.8.4 Fiber cross section on a polished surface of a sample showing the major and minor axes of the approximating ellipse and the in-plane orientation angle 𝜙.

The in-plane angle 𝜙 is obtained by measuring the orientation of the major axis with respect to a preselected reference axis. These angles can be determined by digitizing the four end points of the major

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and minor axes manually or by image analysis. When the three-dimensional data (𝜃 , 𝜙) are obtained from an area sample, an orientation-dependent weighting function has to be used to correct the resulting bias (fibers with 𝜃 close to 0° are more likely to appear on the measured surface than fibers with 𝜃 close to 90°). Although the optical microscopy method is not restricted to translucent matrices or to certain types of fibers, the results are sometimes disappointing because of poor contrast between the fiber and the matrix. In addition, the tedious polishing process can damage the fibers, thereby limiting the accuracy of fiber orientation measurements. Another disadvantage of this technique is that the elliptical cross section does not provide complete information about three-dimensional orientation. The same ellipse shows up on the surface when the fiber orients at angles 𝜙 and 𝜙 ± 180°. As a result, only the components a11 , a12 , a22 , and a33 of the orientation tensor can be completely determined. To determine a13 and a23 , measurements must be made on three perpendicular surfaces. Depending on whether the image is to be used for a two- or three-dimensional characterization of fiber orientation distribution, cross sections or lengths of the cylindrical fibers appear in the images obtained from direct measurement. Several methods for analyzing these images to produce quantitative orientation data are available, such as manual digitization, light diffraction, and computerized image analysis. Manual digitization from a photographic image is tedious. In the light diffraction method, the image of each fiber in a photographic or radiographic negative appears as a transparent “slit” that can be used as a diffraction aperture under coherent illumination. The orientation of the far-field diffraction pattern produced by a single fiber is determined by its orientation. The diffraction pattern resulting from a collection of fibers is related to the orientation distribution of these fibers. Image analyzers have made it possible to rapidly make three-dimensional fiber orientation measurements on polished samples. Using reflected light microscopy, fiber cross sections are obtained by direct imaging of a metallographically polished surface. The image is then digitized into picture elements (pixels), and each pixel is assigned a gray level corresponding to the light intensity averaged over the area represented by the pixel. Typically, a digital image consists of an array of 512 × 512 pixels and the gray level at each pixel is scaled from 0 (black) to 255 (white). High-end image analysis systems allow 1, 024 × 1, 024 pixel resolution. Increasing resolution gives more accurate results, but also requires larger storage space and longer processing times. After a thresholding operation on the gray levels, a binary image is generated to distinguish the fibers from the matrix. X-ray Contact Microradiography (CMR) (CMR) can eliminate the issues of fiber damage and poor contrast

between fiber and matrix, which are commonly encountered in optical microscopy. CMR is not sensitive to the sample surface condition; only the fiber orientation distortion caused by the cutting procedure is of concern. The contrast of the X-ray image between fiber and the background matrix depends on the fiber dimension in the beam direction and the X-ray absorption difference between fiber and matrix. For a given fiber concentration, the thickness of the sample must be so chosen that each fiber can be distinguished in the picture. A sample section cut from the fiber-reinforced part is placed on a fine grain film and exposed to an uncollimated and unfiltered X-ray beam. The fibers lying in the thickness of the section are projected onto the plane of the film, and appear as line segments on the micrograph. The in-plane orientation angle 𝜙 can be measured directly from the micrograph with respect to a reference axis. Assuming the fiber intersects both surfaces of the section, the out-of-plane angle 𝜃 is defined by the projected length of the fiber Lp and section thickness t (Figure 22.8.5)

𝜃 = tan−1 (Lp ∕t)

(22.8.24)

Fiber-Filled Material Materials – Materials with Microstructure

z

θ

t y

ϕ

Lp

tan θ =

Lp t

0 ≤ θ ≤ π x

0 ≤ ϕ ≤ 2π

Figure 22.8.5 Fiber section spanning a sample thickness t. The projected length in the x-y plane is Lp .

Contact microradiography has been shown to be very useful for determining the fiber orientation distribution through the thickness of molded parts. It also has been successfully applied to opaque fiber-reinforced thermoplastics. As in optical microscopy, the X-ray technique cannot fully characterize the three-dimensional orientation, even when the sample thickness is precisely measured, because the same projected length would be obtained from fibers lying at angles 𝜙 and 𝜙 ± 180°. The examination of several sections cut at different angles or use of X-ray Computed Tomography (CT) offer the possibility of obtaining quantitative three-dimensional fiber orientation distributions. 22.8.4.2 Through-Thickness Variations of Orientation Tensor Components

The optical technique describes before can be used to map the through-thickness fiber orientation variations in molded parts, as measured by orientation tensor components. The data in the figures that follow show micrographs of polished, glass-filled specimens cut from the central planes of 80 × 80 × 1.5-mm, injection-molded 30-GF-PP plaques. The dark patches in these micrographs are voids from the molding process. The 30-GF-PP material used has 16-μm diameter glass fibers with a number-average length of 0.80 mm and a weight-average length of 2.11 mm. With reference to Figure 22.4.2, the sprue diameter at x = 0, y = h∕2, and z = 0 is 3.4 mm. All the rectangular specimens examined were cut from the z = 0 plane. During molding the melt moved in the two-dimensional channel bounded by the outer surfaces y = ± h∕2. All the through-thickness specimens examined were cut from the z = 0 surface at four locations: Next to the sprue (x = 0), at 10% of the plaque length, or (x = 8 mm); at 40% of the length, or (x = 32 mm); and at 90% of the length, or (x = 72 mm). The micrograph Figure 22.8.6a shows the through-thickness fiber orientations on the rectangular specimen cross cut from next to the sprue (x = 0); the dark patches are voids from the molding process. The flow is from the left to the right. Figure 22.8.6b shows the variations of the three orientation tensor components a11 , a22 , and a33 across the part thickness. Note that the plaque thickness has been normalized as 𝜂 = 2y∕h, so that the lower and upper plaque surfaces are, respectively, at 𝜂 = −1 and 𝜂 = +1. In the

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case of fiber alignments purely in z-x, or 3-1, planes, a22 ≡ 0, and a11 + a33 = 1. This figure shows that a33 is very small, so for practical purposes the fibers are aligned in the x-z plane, and a11 ≈ 1 − a33 , as appears to be the case in this figure.

1.0

y = + h /2

a33

ORIENTATION TENSOR COMPONENT

842

0.5

a11

100 µm

a22

y = − h /2 x=0

0 – 1.0

– 0.5

0

0.5

1.0

NONDIMENSIONAL PLAQUE THICKNESS (a)

(b)

Figure 22.8.6 (a) Micrograph of specimen cut next to the sprue. (b) Through- thickness variations of the orientation tensor components. (Micrograph and figure courtesy of Rebecca Minnick and Donald Baird, Virginia Polytechnic Institute and State University.)

For the coordinate system being used, Figure 22.8.3 shows that for isotropic fiber orientations a11 = a33 = 0.5, and that when fibers are aligned in the x-direction, a11 = 1, and a33 = 0. Regions close to the upper and lower plaque surfaces have close to isotropic orientations. In the interior regions, low values of a11 indicate that the fibers are aligned more toward the cross-flow direction, that is, along the z-direction. The orientation at each location was obtained by averaging results from several plaques; this is reflected in the error bars on the curves. The error bars on the a22 curve have been removed for clarity. The corresponding micrograph and orientation tensor components for specimens from 10% of the length (x = 8 mm) are shown in Figure 22.8.7. Starting from the plaque bottom (left part of Figure 22.8.7b), the orientation starts from being approximately isotropic (a11 ≈ a33 ≈ 0.5), to more fibers being aligned in the flow direction, to isotropy, to be followed by declining values of a11 indicating greater alignment in the cross-flow direction, with the “same” pattern being repeated in reverse in the upper half of the plaque. Note that a22 has smaller values than in Figure 22.8.6, reflecting that at x = 0 the flow has to turn from a vertical to a horizontal direction.

Fiber-Filled Material Materials – Materials with Microstructure

1.0

ORIENTATION TENSOR COMPONENT

y = + h /2

a33 a11

0.5

100 µm x = 8 mm

a22

y = − h /2 0 – 1.0

– 0.5

0

0.5

1.0

NONDIMENSIONAL PLAQUE THICKNESS (a)

(b)

Figure 22.8.7 (a) Micrograph of specimen cut from x = 8 mm. (b) Through- thickness variations of the orientation tensor components. (Micrograph and figure courtesy of Rebecca Minnick and Donald Baird, Virginia Polytechnic Institute and State University.)

The corresponding micrograph and orientation tensor components for specimens from 40% of the length (x = 32 mm) are shown in Figure 22.8.8. Starting from the plaque bottom, the orientation starts from approximately being isotropic (a11 ≈ a33 ≈ 0.5), to more fibers being aligned in the flow direction, to isotropy, to be followed by declining values of a11 indicating greater alignment in the cross-flow direction, with the “same” pattern being repeated in reverse in the upper half of the plaque. Here again, a22 has a lower value than in Figure 22.8.6. Figure 22.8.9 shows the results for the 90% of the length (x = 72 mm). One difference from the previous cases is that the fiber distribution is not isotopic near the plaque surfaces. Also, a22 has slightly higher values that may result from the complex flow at the plaque end causing motion in the y-direction. The error bars on the orientation tensor curves stem from two sources. The first are measurement errors introduced during the image analysis process being applied to digitizing the data on micrographs. But there is also the inherent variability – not error – that stems from the statistical nature of how fibers will end up at a given location. The data in Section 22.6.1.3 on mechanical property variations in sequentially molded fiber-filled plaques showed that although the pattern of property variations is similar, the actual values vary around a mean. Figure 22.8.10 shows micrographs from the same location in three different plaques. Clearly, the fiber orientations appear different to the naked eye.

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Introduction to Plastics Engineering

1.0

y = + h /2

ORIENTATION TENSOR COMPONENT

844

a11

a33

0.5

100 µm

x = 32 mm

a22

y = − h /2 0 – 1.0

– 0.5

0

0.5

1.0

NONDIMENSIONAL PLAQUE THICKNESS (a)

(b)

Figure 22.8.8 (a) Micrograph of specimen cut from x = 32 mm. (b) Through- thickness variations of the orientation tensor components. (Micrograph and figure courtesy of Rebecca Minnick and Donald Baird, Virginia Polytechnic Institute and State University.)

22.8.4.3

Indirect Measurement

Indirect measurements give fiber orientation information by detecting the anisotropy of mechanical, thermal, optical, or dielectric properties in composites caused by the fiber orientation distribution, fiber volume fraction, and fiber length. For instance, a preferred fiber orientation can be revealed by measuring elastic moduli in several directions through tensile tests. Many techniques are available for the orientation measurements, but some of them are not as versatile as others. For example, X-ray diffraction is restricted to crystalline fibers; whereas light diffusion and diffraction are applicable only if the matrix is translucent. Two nondestructive methods, ultrasonic and thermographic imaging, are described next. Ultrasonic Measurement The relationship of the elastic moduli to 𝜌V 2 (V is the ultrasonic wave speed and

𝜌 is the density of the fiber-reinforced composite), can be used to determine the elastic anisotropy caused by fiber orientation by measuring the wave speed in a number of directions. This is a nondestructive method that can be used for large parts, and measurements are needed on only one part of the surface. It has widely been used for detecting elastic anisotropy, material nonhomogeneity, and defects in metals, plastics, and composites. Thermographic Imaging Anisotropic thermal conductivity resulting from the higher conductivity in the

fiber direction can be determined by a thermographic technique that renders visible the isotherms produced around a point heat source. Such isotherms can be obtained by applying a heat source, such as by a fine tip of a soldering iron normal to the sample surface and then recording a sequence of resulting temperature distributions by an infrared imaging camera. The isotherms in the images appear as ellipses,

Fiber-Filled Material Materials – Materials with Microstructure

1.0

y = + h /2

ORIENTATION TENSOR COMPONENT

a11 a33

0.5

100 μm

a22

y = − h /2 x = 72 mm

0 – 1.0

– 0.5

0

0.5

1.0

NONDIMENSIONAL PLAQUE THICKNESS (a)

(b)

Figure 22.8.9 (a) Micrograph of specimen cut from x = 72 mm. (b) Through- thickness variations of the orientation tensor components. (Micrograph and figure courtesy of Rebecca Minnick and Donald Baird, Virginia Polytechnic Institute and State University.)

100 µm

100 µm (a)

(b)

100 µm (c)

Figure 22.8.10 Micrographs of specimens cut from the same location, x = 72 mm, in three different plaques. (Micrographs courtesy of Rebecca Minnick and Donald Baird, Virginia Polytechnic Institute and State University.)

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the ratio of the lengths of the major to the minor axes of which is proportional to the square root of the ratio of the principal thermal conductivities. This method is more effective for composite materials with long, highly aligned fibers. For short-fiber composites, caution must be exercised in interpreting thermographic imaging data.

22.8.5

**Fiber Orientation Models

A fiber orientation model provides information that, together with a rheological constitutive model, determines the flow field and the resulting fiber orientation distribution. The orientation models reviewed in this section are derived from a balance of forces and moments on a single fiber in a homogeneous flow field. Differences between the various models arise from the particular representation of the angular velocity ṗ chosen in the continuity equation (Eq. 22.8.7). Three analytical models are summarized in the sequel. Existing models for the flow and orientation of suspensions are only valid for certain concentration regimes. Using the fiber volume fraction c or the average distance between a fiber and its nearest neighbor ac , three regimes may be identified: dilute, semi-dilute, and concentrated. A classification of these regimes in terms of the fiber length l, the fiber diameter d, and the fiber number density n, is given in Table 22.8.1, in which ac = 1∕nl 2 . To understand the concentration regimes for glass-fiber-reinforced plastics, the dividing fiber volume fractions for various lengths of glass-fiber are listed in Table 22.8.2. Apparently most commercial composites fall in the semi-dilute and concentrated regimes. However, the processing of long-fiber reinforced materials, such as by injection molding or extrusion, results in a drastic reduction in fiber length as well as in a widening of the range of fiber-length distributions in the material system. Table 22.8.1 Concentration regimes for fiber suspensions. Regime

c

ac

Dynamics

Dilute

c < (d∕l)2

ac > l

Free to rotate; no fiber-fiber interaction

Semi-dilute

(d∕l)2 < c < d∕l

d < ac < l

Free motion; frequent fiber-fiber interaction

Concentrated

c > d∕l

ac < d

Restricted motion; local fiber alignment

Table 22.8.2 Dividing fiber volume fractions for glass-fiber-reinforced composites (d=15 mm is typical of commercial glass fibers). l (mm)

l∕d

(d∕l )2 (%)

d∕l (%)

0.075

3.0

4.0

20.0

0.3

20.0

0.25

5.0

1.5

100.0

0.01

1.0

15.0

1000.0

0.001

0.1

Fiber-Filled Material Materials – Materials with Microstructure

22.8.5.1 Jeffery’s Model

The problem of a rigid ellipsoidal particle moving through a viscous Newtonian fluid was first solved by G.B. Jeffery, and his solution is referred to as the Jeffery model. The equation for fiber motion was obtained with the following assumptions: • The neutrally buoyant single particle considered is rigid, axisymmetric, and large enough for the Brownian motion to be negligible. • The fluid is Newtonian, with the inertia and body forces being negligible. • A no-slip boundary condition is specified at the particle surface. • Away from the local disturbance produced in the immediate neighborhood of the particle, the flow field is steady and varies in space on a scale that is large compared with the dimensions of the particle (the velocity gradient can be assumed constant over the particle size). • The center of the particle moves with the fluid it displaces. In the absence of external moments about the particle centroid, Jeffery described the motion of a particle by the expression ṗ i = − wir pr + 𝜆 (dir pr − drs pr ps pi ),

𝜆=

re 2 − 1 re 2 + 1

(22.8.25)

where dij = (vj, i + vi, j )∕2 and wij = (vj, i − vi, j )∕2 are, respectively, the rate of deformation and vorticity tensors, and re is the effective aspect ratio of the particle. For an ellipsoid, re equals the ratio of the length of the major to the minor axes, that is, re = a∕b (a = length of major axis, b = c = length of the minor axes). For cylindrical fibers (common in molding problems), re can be experimentally determined, or its value assumed to be the length-to-diameter ratio, l∕d. The values of l∕d vary from −1 for a disk (a > b, so that re → ∞). Because it was derived for a single particle in a viscous medium, Jeffery’s solution can be applied to dilute suspension. Although Jeffery’s equation may give good qualitative results for a nondilute suspension flow, a more complete model is required for describing orientation behavior at higher fiber volume fractions, when particle-particle interactions become important. For two-dimensional orientation, Jeffery’s equation can be written as ( ) d11 − d22 ̇ 𝜙 = w12 + 𝜆 d12 cos 2𝜙 − (22.8.26) sin 2𝜙 2 Analytical solutions of this equation are available for a few simple flow fields, for example, simple shear flow and elongational flow. In simple shear flow, the rate of deformation and vorticity tensors are ] ] [ [ 𝛾̇ ∕2 0 0 −𝛾̇ ∕2 , wij = (22.8.27) dij = 𝛾̇ ∕2 0 𝛾̇ ∕2 0 where 𝛾̇ is the shear rate. Substitution of Eq. 22.8.27 in Eq. 22.8.26 followed by an integration results in ( ) cot 𝜙0 re 1− tan 𝛾̇ t re 1 + re 2 (22.8.28) tan 𝜙 = ( ) re cot 𝜙0 + re tan 𝛾̇ t 1 + re 2 where 𝜙 = 𝜙0 at t = 0. For an initial orientation of 𝜙0 = 90°, fiber orientations characterized by angle 𝜙 for three different values of re are shown in Figure 22.8.11. In a simple shear flow, fibers rotate rapidly

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270 re = 5

re = 10

SHEAR ANGLE ϕ

848

180 re = ∞ 90 re = ∞ 0 re = 5

re = 10

90 –30

–20

–10

0

10

20

30

TOTAL SHEAR STRAIN γ· t Figure 22.8.11 Variation of the fiber angle versus the total strain in simple shear flow with the fiber aspect ratio as parameter.

when they are perpendicular to the flow direction (𝜙0 = 90° or 270°) and slowly when they are aligned with the flow (𝜙0 = 0° or 180°). Fiber rotation is periodic in a simple shear flow, with the period of rotation increasing with the fiber length. The period can be easily obtained from Eq. 22.8.28 as ( ) 1 2𝜋 re + (22.8.29) T= 𝛾̇ re and its variation with re is shown in Figure 22.8.12. The shortest period occurs for re = 1, corresponding to a spherical particle. As re → 0, T → ∞, and a stable steady-state solution exists for 𝜙0 = 0° or 180°, as shown in Figure 22.8.11. That is, in a simple shear flow, fibers align along the streamlines. For a planar elongational flow ] ] [ [ 𝜀̇ 0 0 0 (22.8.30) , wij = dij = 0 −𝜀̇ 0 0 and the solution to Jeffery’s equation is

𝜙 = tan−1 [tan 𝜙0 exp (−2𝜆 𝜀̇ )]

(22.8.31)

Figure 22.8.13 shows the rotational motion of fibers for re = 5 and re = 10 for this type of flow. Apparently, the orientation dynamics is not sensitive to the fiber aspect ratio. A fiber at 𝜙0 = 90° is at an unstable equilibrium, and a small perturbation tends to rotate the fiber along 𝜙0 = 0° or 180°. 22.8.5.2

Dinh–Armstrong Model

Dinh and Armstrong developed a single particle model for semi-concentrated suspensions of stiff long fibers in a Newtonian solvent. They assumed that the particle is immersed in an effective homogeneous medium which is a continuum approximation to the fiber suspension. The surface force on the particle is

Fiber-Filled Material Materials – Materials with Microstructure

100

γ˙ T

50

0 5

0

10

15

FIBER ASPECT RATIO re Figure 22.8.12 Variations of the fiber rotation period multiplied by the shear rate in simple shear flow versus the fiber aspect ratio.

FIBER ANGLE ϕ

180

ε˙ < 0

135

90

45

0 –3

re = 5 re = 10 –2

–1

ε˙ > 0 0

1

2

3

TOTAL ELONGATIONAL STRAIN 2 ε˙t Figure 22.8.13 Variation of the fiber angle versus the total strain in elongational flow with the fiber aspect ratio as parameter.

determined by an anisotropic drag exerted by the surrounding medium. Since there are no external body forces acting on the particle and particle inertia was neglected, they concluded that the center of mass translates affinely with the bulk flow. A moment balance about the center of mass results in ṗ i = 𝜅ir pr − 𝜅rs pr ps pi

(22.8.32)

as the equation for fiber motion in a homogeneous flow field, where 𝜅 ij = vi, j is the transpose of velocity gradient tensor. (Note the use of the summation convention in Eq. 22.8.32.) This equation shows that the orientation vector pi changes as though it were a fluid element, except that the fiber cannot stretch. As

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shown in Eq. 22.8.32, the stretching term 𝜅 rs pr ps pi is subtracted from the bulk deformation, 𝜅 ir pr , of the fluid. The bulk stress generated in a suspension of fibers includes two components, one resulting from the viscous dissipation in the fluid and the other resulting from the presence of fibers. The stresses in the suspension are balanced by the drag force, which primarily arises from the relative motion between the fiber and the bulk flow, along the fiber axis. The balance is done inside a representative fluid volume, which is large enough to contain many fibers but small enough to neglect the variation of the velocity gradient. The contribution from the fibers to the stresses in this volume consists of an integral of the drag force over the surface area of each fiber and a sum over all fibers in the volume. Using an estimate for the drag coefficient parallel to the fiber axis, obtained from the behavior of a steady extensional flow, a rheological equation of state for a homogeneous fiber suspension is given by [ ] nl3 𝜋 𝜏ij = 2𝜇 dij + (22.8.33) d a 12 ln(2h∕d) rs rsij where 𝜏 ij is the extra stress, 𝜇 is the viscosity of the fluid, n is the number density for fibers in the volume, h is the average spacing between fibers, and arsij is the fourth order orientation tensor defined in Eq. 22.8.18. In effect, the Dinh–Armstrong model couples the rheological equation of state with Jeffery’s model for an infinitely long fiber to simultaneously determine orientation and flow. 22.8.5.3

Folgar–Tucker Model

Folgar and Tucker developed a model for orientation of rod-like particles in concentrated suspensions. From experiments on concentrated suspension, they found that individual fibers first follow the motion described by Jeffery’s equation for a short time, then quickly reorient to another angle, and again resume the behavior predicted by Jeffery’s model. They argued that the sudden reorientation is caused by fiber-fiber interactions that tend to randomize the fiber orientation distributions. The additional assumptions for their model, over Jeffery’s approach, are • Fiber concentration is homogeneous in space and the centers of mass of all fibers are convected with the fluid. • An interaction occurs when two particle centers pass within one fiber length of each other. • The orientation changes produced by the interactions are identically distributed random variables with zero mean, and are independent of the orientation angles of the interacting particles. • The particle motion is a superposition of the motion of a particle in dilute suspensions and the motion caused by the fiber interactions. The equation for ṗ i , the angular velocity of the fiber axis, is ṗ i = − wir pr + (dir pr − drs pr ps pi ) −

CI 𝛾̇ 𝜕𝜓 𝜓 𝜕 pi

(22.8.34)

where CI is an experimentally determined interaction coefficient and 𝛾̇ = ( 2drs drs )1∕2 is the magnitude of the strain rate tensor, thus consists of two independent components. (Note the use of the summation convention in Eq. 22.8.32.) The first part of their model is Jeffery’s equation with 𝜆 = 1 (re → ∞). The last term in the right-hand side of this equation is an additional phenomenological term for modeling the randomizing effects. This term indicates that fiber interactions cause fibers to turn from regions of higher orientation distribution to regions of lower distribution. In addition, the orientation flux 𝜓 ṗ i is proportional to the distribution gradients 𝜕𝜓 ∕𝜕 pi , through the quantity CI 𝛾̇ , which is equivalent to the

Fiber-Filled Material Materials – Materials with Microstructure

diffusion coefficient in the rotary Brownian motion of extremely small particles, and which characterizes the frequency and strength of the interactions. Thus, CI is assumed to be an intrinsic property of the suspension. Folgar and Tucker showed that the value of this coefficient depends on the concentration of the suspension and the fiber aspect ratio. For each material system, the coefficient has to be determined by fitting the prediction to a simple flow experiment. Together with the tensor description of the orientation and an interaction coefficient CI , the Folgar–Tucker model can be used for predicting fiber orientation in concentrated suspensions. However, the values of CI affect the orientation tensor components and the thickness distribution of the skin/core layers. As such, all the tensor components cannot be fitted to the experiments in one step. 22.8.6

**Fiber Orientation Prediction

Fiber orientation prediction requires the solution of the types of partial differential equations discussed previously. Even the accurate prediction of flow during molding and subsequent part solidification pushes computational techniques and the capabilities of computers. Fiber orientation development during molding adds this complexity. While research and commercial codes are available predicting fiber orientation in molded parts their capabilities are currently limited both by the level of approximation used for simulating flow and the ability to accurately capture the time-dependent motion of fibers in a flow. After fiber distribution and orientation in a part has been determined, this information can, in principle, be used to determine the stiffness and strength of the part. Here too, the capabilities of current computer simulation codes are limited.

22.9 Concluding Remarks The mechanical property data presented in this chapter clearly show that injection-molded, short- and long-glass-fiber-filled thermoplastics are complex nonhomogeneous, anisotropic material systems. Like all fiber-filled materials, they exhibit through-thickness nonhomogeneity (as indicated by differences between tensile and flexural properties). The in-plane fiber orientation distribution in these layers causes the material to have in-plane anisotropic mechanical properties. While glass fibers increase both the tensile modulus and strength of unfilled thermoplastics, the increase in the modulus is greater than the increase in strength. However, the average failure strains of fiber-filled tensile bars are dramatically smaller than those of unfilled tensile bars. While fibers with higher aspect ratio increase both the modulus and strength of injection-molded short-fiber composites, they further reduce the failure strain. The average fiber length in pellets is different from that in molded tensile bars. During compounding and injection molding processes, the flow-induced shear stress developed in the screws, runners and mold cavities orients the fibers, and bends and breaks them, resulting in fiber degradation. As a result, the mechanical properties of injection-molded short-fiber composites not only depend on processing but also on part geometry. Nonuniform flow fields during molding result in nonhomogeneous mechanical properties within the part, and process variation produces part-to-part property variations. Injection speeds also affect mechanical properties. The mechanical properties of all fiber-filled composites are strongly thickness dependent. In short-fiber composites the thinnest plaques exhibit the largest differences between the flow and cross-flow tensile

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moduli and strengths. These differences decrease (the ratios of cross-flow to flow-direction property increase) with increasing thickness. Thus, the thinnest plaques exhibit very high in-plane anisotropy; the thicker plaques are less anisotropic. For all plaque geometries and processing conditions, the flow-direction tensile modulus is higher at the edges than in the interior. For all thicknesses, the flow-direction flexural properties – both modulus and strength – are significantly higher than the cross-flow properties. The flexural modulus is comparable to the tensile modulus, but the flexural strength is significantly higher than the tensile strength. Both the flow and the cross-flow direction flexural moduli decrease with increasing plaque thickness. In contrast to this, the flow and cross-flow direction flexural strengths increase with plaque thickness. In long-glass-filled materials the degree of in-plane nonhomogeneity is unexpectedly high. Clearly, the single-point mechanical properties for fiber-filled materials given in the manufacturers’ data sheets provide a very inadequate description of the mechanical properties of the material. The data-sheet values are reasonable for flow direction properties, but do not even provide a conservative estimate for the much lower cross-flow properties. Also, mechanical properties measured on molded bars of different thickness overestimate properties, underestimate thickness effects, and do not provide any information on cross-flow properties. The brief introduction in Section 22.8 on how fiber orientation can be characterized, measured, and used for predicting fiber orientation in molded parts highlights the complexity of the process, requiring the simulation of the rheology of the flow of very viscous, fiber-filled fluids in complex geometries. This section is largely based on the research of Professor Charles A. Tucker III and his students, most notably F.P. Folgar, W.C. Jackson, and S.G. Advani. The level of engineering science and mathematics required to understand their contributions is well beyond the scope of this book. Because the distribution and orientation of long fibers within parts made of these materials cannot currently be predicted, standard finite element analysis cannot be used to predict part performance. A feature-based methodology for mechanical design may be more useful: The mechanical properties of features such as part thickness, ribs, corners, bosses, and so on determined through experiments – and analyses to assess the sensitivity of mechanical properties to changes in processing conditions and feature geometry – could be used for assessing the performance of complex parts.

853

23 Structural Foams – Materials with Millistructure 23.1 Introduction Parts made from thermoplastic rigid foams, also called structural foams, have complex morphologies consisting of thin “solid” outer skins surrounding porous, or cellular, inner cores. This morphology results in light structural components having relatively high strength-to-weight ratios. The cellular structure of the core provides additional thermal and acoustic insulation. Typical, diverse applications of structural foams include enclosures for audio speakers, automobile load floors and glove box doors, battery trays, material handling pallets, playground slides, refuse bins, satellite dishes, and water skis. Structural foams are not simple materials: they are rigid, cellular material systems in which the local morphology and material properties of parts depend on part geometry and processing conditions. Typically, the walls of foam parts have a cellular core surrounded by thin solid skins. The fracture surfaces of a modified poly(phenylene oxide) (M-PPO) structural foam printer housing, shown in Figure 23.1.1, illustrate the complex morphology in the interior of a foam part. After cutting portion AB at C and then applying bending loads, the arms AC and CB were broken off at A and C, respectively. The resulting mating fracture surfaces at A are shown in Figure 23.1.2a,b, and those at B are shown in Figure 23.1.2c,d. Clearly, the cellular pores tend to be largest at the center and continually become smaller away from the center. While most pores tend to be smaller than a millimeter, some can be much larger – of the order of several millimeters as in some areas of Figure 23.1.2c,d. Because the pores can be seen with the naked eye and can be on the order of a millimeter, such material systems may be referred to as materials with millistructure. Structural foam molding processes (Section 17.8.2) were originally developed to avoid sink marks in injection-molded parts. Essentially, a gas charged melt with a volume smaller than the mold cavity volume is injected into a mold cavity. The hot melt surface contacting the cold mold walls solidifies, resulting in the formation of thin solid skins. The fall in pressure after injection causes the dissolved gas to come out of solution, forming bubbles that continue to grow in size, forcing the thin skin to be in contact with the mold walls, thereby avoiding sink marks, even in relatively thick parts. Structural foams are typically described in terms of a general density reduction resulting from the foaming process. The level of “normal” density reduction is controlled by the process parameters and directly relates to the amount of material injected into the mold cavity. However, as indicated by the fracture surfaces shown in Figure 23.1.2, the actual density reduction can vary over the part. Let the volume of the material injected into a mold cavity having volume v0 be v, and the density of the (unfoamed) resin be 𝜌0 . Then, since the same mass having initial volume v ends up filling the mold cavity volume v0 , by Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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C

A

B

10 cm

Figure 23.1.1 Printer housing made of a modified PPO structural foam. (Photo courtesy of SABIC.)

1 cm

(a)

(b)

(c)

(d)

Figure 23.1.2 Fracture surface showing internal morphology of a modified PPO part. (a) (Adapted with permission from V.K. Stokes, R.P. Nimmer and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1491 – 1500, 1988.) (b – d) (Photos courtesy of SABIC.)

Structural Foams – Materials with Millistructure

neglecting the mass of the dissolved gas or foaming agent, the average density 𝜌 of the foamed material can be defined by 𝜌0 v = 𝜌 v0 . The percentage density reduction is then defined as 100 (𝜌0 − 𝜌)∕𝜌0 = 100 (v0 − v)∕v0 . Structural foam parts are normally designed for nominal density reduction levels in the range of 5 – 25%, although higher density reductions are possible, especially in thicker parts.

23.2 Material Complexity Structural foams are very complex material systems: First, the presence of bubbles, or voids, makes them heterogeneous, noncontinuous materials. Second, the nonuniform, through-thickness bubble distribution makes them nonhomogeneous in the thickness direction. Third, foam parts also exhibit in-plane nonhomogeneity. Finally, the levels of through-thickness and in-plane heterogeneity and nonhomogeneity depend both on processing conditions and on the part geometry. Clearly, to understand how the local morphology – the pore size, shape, and number distributions – affects the properties on a larger scale requires detailed microstructural models that use the detailed local bubble morphology and the properties of the plastic to predict the “macro” properties of the component. Several models with different levels of detail are available. The most detailed models apply the principles of mechanics to idealized bubble geometries to create structural models that predict the macro properties of larger structures. Higher level, less detailed models are based on micromechanical analyses of statistics of the size and number distributions of bubbles. While such detailed models provide insights into how foams work and how their properties can be modified, they are far too detailed and cumbersome for structural analysis of parts. Much of the literature on the mechanical properties of foams is concerned with relating the “flexural modulus” of a rectangular beam to the foam density. A relationship between the local, pointwise (through-thickness varying) modulus and the local, pointwise (through-thickness varying) density has been used to predict the flexural modulus. However, because of the bubbles, there is some question as to the meaning of a pointwise local density for a cellular material, and the dependence of the pointwise local modulus on the pointwise local density is difficult to measure. At a coarser level, it has been suggested that this “flexural modulus” can be simply related to the macroscopic, average foam density – the local density averaged over the thickness. This approach has a major shortcoming, however: The “flexural modulus” is not a material property of structural foam because it depends on the geometry of the test specimen. For example, the “flexural modulus” for rectangular and channel-section beams are different. Furthermore, the “flexural modulus” cannot even be used to predict the stiffness of a foam bar in tension. (The “flexural moduli” of foams are known to be significantly larger than the “tensile modulus.”) Thus, it is unclear as to what stiffness property is appropriate for predicting the structural response of complex structures to general loads. The nonhomogeneous cellular morphology of structural foams therefore raises several questions about their mechanical properties and their use to predict the stiffness and strength of parts: If material properties vary from point to point, then what do tests such as the tensile test measure? How are such “tensile properties” related to those measured in a flexural test? Even more importantly, how can such properties be used to predict the mechanical performance of parts? Should test specimens be molded to a final shape, or be cut from larger pieces? What tests should be used for determining the mechanical properties of foams? One approach is to idealize foams as nonhomogeneous continua. The effect of the bubbles on local behavior is assumed to manifest itself through the local properties of the continuum. In this approach,

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local properties, such as the Young’s modulus, E (x), are assumed to depend on the location x in the part. Mechanics principles are then used to infer the effects of different levels of nonhomogeneity on structural performance. Because the morphologies of individually molded dog-bone foam specimens are not representative of the morphologies of foam parts, specimens cut from molded plaques are recommended. A possible problem with this approach could be inconsistencies in the morphologies of specimens cut from different parts of a plaque, and even differences in specimens cut from the same location in different plaques. An alternative approach, the one explored in the next section, is to treat the foam as continuum with the variation in pore size being accounted for by spatially varying mechanical properties.

23.3 Foams as Nonhomogeneous Continua To explore the consequences of treating a foam part as being made up of a nonhomogeneous continuum, consider the rectangular bar of width b and depth d, shown in Figure 23.3.1, in which the local properties only vary in the y-direction, with no property gradients in the x- and z-directions. Let the Young’s modulus of this nonhomogeneous continuum be E = E ( y).

y

y x

x

d

E = E(y)

b

z (a)

(b)

Figure 23.3.1 Foam bar of breadth b and depth d. (a) Coordinate system. (b) Modulus variation E = E ( y) of equivalent nonhomogeneous continuum. (Adapted with permission from V.K. Stokes, R.P. Nimmer and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1491 – 1500, 1988.)

23.3.1

Nonhomogeneous Bar in Tension

If the bar in Figure 23.3.1 is pulled in tension by a force Fz , the resulting longitudinal strain 𝜀z will be constant, 𝜀z = 𝜀0 , both through the bar thickness and along it (Why?). However, because of variations in the Young’s modulus, the stress distribution 𝜎 z = E ( x) 𝜀0 will be a function of x. Since this through-thickness stress distribution must equal the applied load Fz , it follows that d∕2

Fz =

∫−d∕2

𝜎z dA =

d∕2

∫−d∕2

𝜎z b dy = 𝜀0

d∕2

∫−d∕2

E ( y) b dy

The average stress 𝜎 z = Fz ∕A on the cross-sectional area A = bd is then given by

𝜎z =

Fz 𝜀 d∕2 E ( y) b dy = 0 A A ∫−d∕2

Structural Foams – Materials with Millistructure

and the average “tensile modulus” for the rectangular bar, ETR = 𝜎 z ∕𝜀0 , by d∕2

E TR =

d∕2

1 1 E ( y) b dy = E ( y) dA A ∫−d∕2 A ∫−d∕2

(23.3.1)

Note that this “average tensile modulus” is not a material property. It clearly depends on the geometry of the bar; for example, for a circular foam bar the expression would be different. But it also depends on processing because that affects the distribution E = E ( y). This average modulus is the “modulus” that would be measured by dividing the “stress” 𝜎 z = Fz ∕A by the strain. It is really the modulus of a homogeneous bar with the same stiffness as the nonhomogeneous bar.

23.3.2

Bending of a Nonhomogeneous Bar in the Stiff Mode

Next consider the bending of this rectangular bar of the same idealized material – configured with the “skins,” that is, the stiffest material, being at a uniform distance from the x-axis, as shown in Figure 23.3.2 – by a bending moment Mx . The same arguments as those for the bending of a homogeneous bar (Section 5.2) apply:

y

y z

Mx

d Mx

(a)

x b (b)

Figure 23.3.2 Bending of a foam bar of breadth b and depth d due to a bending moment Mx . (a) Longitudinal arcuate shape. (b) Lateral cross section. (Adapted with permission from V.K. Stokes, R.P. Nimmer and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1491 – 1500, 1988.)

The bar will bend into an arcuate shape, with plane transverse sections remaining plane after bending. The strain will, again, be 𝜀z = y∕R, but the stress 𝜎 z = E 𝜀z = E (y) y∕R no longer varies linearly with y because of the dependence of the Young’s modulus on y. Since this stress distribution must equal the imposed pure bending moment Mx , there cannot be a longitudinal force, so that (why?) Fz =

∫

𝜎z dA = 0

⇒

∫

E ( y) y dA = 0

(23.3.2)

Thus, in general, the neutral axis is no longer the centroidal axis, that is, y = 0 is not the centroid of the cross section. Instead, y = 0 passes through the E-weighted centroid. Even for the symmetric, rectangular section, this E-weighted centroid can be different from the geometric centroid when the distribution E = E ( y) is not symmetric about y = 0. Of course, for this simple rectangular geometry, the E-weighted centroid coincides with the centroid.

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Finally, since the stress distribution must equal the moment Mx , Mx =

∫

⇒

y (𝜎z dA) =

∫

y (𝜎z b dy) =

1 1 E ( y) y2 b dy = E ( y) y2 dA R∫ R∫

Mx 1 = Jyy R

where Jyy = ∫ E ( y) y2 b dy = ∫ E ( y) y2 dA. Thus, for the nonhomogeneous continuum model with E = E ( y) 𝜎z Mx 1 = (23.3.3) = , Jyy = E ( y) y2 dA ∫ Jyy E ( y) y R In this case the bending stiffness – the ratio of the bending moment Mx to the curvature 1∕R – is given by Jyy = ∫ E ( y) y2 dA. For a homogeneous material, for which the Young’s modulus is constant, the expression for the stiffness becomes Jyy = ∫ E ( y) y2 dA = E ∫ y2 dA = E Iyy . Now in homogeneous materials the bending stiffness is the product of the Young’s modulus, which is a material property, and the second moment of area Iyy , which is a geometrical property. In nonhomogeneous materials the bending stiffness cannot be so partitioned. Instead, the bending stiffness results from a synergistic combination of material properties and part geometry. Just as for a rectangular bar in tension, an effective “bending modulus” for the rectangular bar, EBR , can be defined as the “modulus” of a homogeneous bar with the same stiffness as that of the nonhomogeneous bar, that is, M Mx 1 = x ≡ R EBR Iyy Jyy

⇒

EBR =

Jyy Iyy

=

∫ E ( y) y2 dA

(23.3.4)

∫ y2 dA

Clearly, EB differs from ET . From the expressions for these two quantities, it follows that EBR > ETR (EBR < ETR ) when E = E ( y) increases (decreases) with increasing y. Also, EBR not equaling ETR clearly indicates that neither of them are true material properties. As such EBR and ETR will be called the bending modulus and the tensile modulus, respectively, for historical reasons only. The ratio EBR ∕ETR is a measure of how nonhomogeneous the through-thickness variation E = E ( y) is. 23.3.3

Bending of a Nonhomogeneous Bar in a Reduced Stiffness Mode

Next consider the bending of the bar due to a bending moment My , in the configuration shown in Figure 23.3.3 (why is the moment labeled − My ?). Then, following the same arguments as for the bending case discussed previously, the bar will bend into an arcuate shape such that 𝜀z = x∕R and 𝜎 z = E 𝜀z = E ( y) x∕R. As before, the absence of a longitudinal force implies ∫

𝜎z dA = 0 ⇒

∫

E ( y) x dA = 0

so that x = 0 passes through the E-weighted centroid. And, since the stress distribution must equal the moment My , − My =

∫

x (𝜎z dA) =

1 E ( y) x2 dA R∫

⇒

− My Jxx

=

1 , R

or

My Jxx

=

1 −R

(23.3.5)

Structural Foams – Materials with Millistructure

x

x

y

z – My

b

– My d (a)

(b)

Figure 23.3.3 Bending of a foam bar of breadth b and depth d due to a bending moment − My . (a) Longitudinal arcuate shape. (b) Lateral cross section. (Adapted with permission from V.K. Stokes, R.P. Nimmer and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1491 – 1500, 1988.)

where Jxx = ∫ E (y) x2 dA. (The negative sign on R implies that a positive bending moment My will cause the beam to bend concave upwards instead of concave downwards as shown in Figure 25.3.3.) Then, just as for the bending of a bar due to Mx , an effective “bending modulus,” EVBR , can be defined as the “modulus” of a homogeneous bar with the same stiffness, that is, My My 1 = ≡ R EVBR Ixx Ixx

⇒

EVBR =

∫ E ( y) x2 dA Jxx = Ixx ∫ x2 dA

(23.3.6)

where the subscript V on EVBR is used to denote bending in a “vertical” mode, in which the beam bends in the x-z plane. Now ( b∕2 ) d∕2 d∕2 b∕2 Jxx = E ( y) x2 dA = E ( y) x2 dx dy = x2 dx E ( y) dy ∫ ∫−d∕2 ∫−b∕2 ∫−b∕2 ∫−d∕2 ( b∕2 ) ( b∕2 ) d∕2 d∕2 1 1 = x2 (d dz) E ( y) (bdy) = x2 dA E ( y) dA ∫−b∕2 ∫−b∕2 bd ∫−d∕2 A ∫−d∕2 = Ixx ET It then follows that EVBR = ETR ! In retrospect, this result should not be so surprising. For, any thin slice of the beam between y and y + dy has a uniform modulus E (y), so that the beam is made up of thin uniform slices each having a different but constant modulus. The fact that the stiffness of a beam depends linearly on the width allows for an addition of the stiffnesses of the individual slices, resulting in the “vertical bending modulus” being the same as the “tensile modulus.” Thus, the effective “bending modulus” depends on the axis about which the bending occurs; again, a clear indication of these “bending moduli” not being material properties. Rather, such averages determine the stiffnesses of specific geometrical structures. Such moduli would be more useful if they could be used to determine the stiffnesses of other structures – such as, for example, the T-section beam shown in Figure 23.3.4, which is made up of a flat section, with an effective bending modulus EB , and a vertically oriented section having an effective bending modulus of EVBR = ETR .

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y b y0

x

h

t Figure 23.3.4 Cross section of a thin-walled T-beam.

While at first glance it may appear advantageous to obtain a larger effective modulus EB by choosing a material in which E = E ( y) increases with increasing y, this advantage only holds for the “flat” configuration. For the “vertical” bending configuration the effective bending modulus has already been shown to be EVBR = ETR , which is less than EBR . It turns out that the effective bending modulus for non-flat beams is much closer to ETR than to EBR . For example, the effective bending modulus EEffec for a flat beam made by gluing two nonhomogeneous beams, each made of a nonhomogeneous material for which EB = 1.5ET , is given by EEffec = 1.125ET , as shown in Section 23.4. The effective bending moduli of thin-walled prismatic beams are discussed in the next section.

23.4 Effective Bending Modulus for Thin-Walled Prismatic Beams Consider the thin-walled nonhomogeneous beam having the arbitrary cross section shown in Figure 23.4.1, in which the coordinates n and s are measured, respectively, normal to and along the center of thin-walled tube wall. For simplicity, assume that the through-thickness variation E = E (n) of the elastic modulus is the same at all points along the perimeter of the cross section. Then, the local effective tensile modulus ETL and the local effective bending modulus EBL , each of which is a constant, are given by t∕2

ETL =

1 E (n) dn, t ∫−t∕2

t∕2

EBL =

12 E (n) n2 dn t3 ∫−t∕2

(23.4.1)

where t is beam thickness. Let m = EBL ∕ETL be the ratio of these two effective moduli. A detailed analysis of the bending of thin-walled prismatic beams, which is beyond the scope of this book, then shows that the effective bending modulus EEffec of symmetric beams is given by EEffec = ETL [1 + 𝛼 (m − 1)]

(23.4.2)

Structural Foams – Materials with Millistructure

861

s n

Figure 23.4.1 Thin-walled beam of arbitrary cross section. The coordinates n and s are measured, respectively, normal to and along the center of thin-walled tube wall thickness.

where

𝛼=

1 t3 (dx∕ds)2 ds 12Iyy ∫A

is a geometrical property of the cross section. These formulas will now be applied for evaluating the effective bending moduli of several thin-walled sections. Consider the thin-walled rectangular section shown in Figure 23.4.2.

t

y

x

h t1

b

Figure 23.4.2 Cross section of a thin-walled rectangular beam.

Let the thicknesses of the upper and lower flanges be t, and that of the vertical webs be t1 . By symmetry, the E-weighted centroid is the same as the centroid, which has been chosen as the origin of the coordinates. It can be shown that for this section 1 𝛼=[ ] 1 + (t1∕b) (h∕t) 3 + 3 (1 + h∕t) 2 Case (i). h = 2t. There are no webs and the two flanges are “glued” to each other as shown in Figure 23.4.3.

y d x d

b

Figure 23.4.3 Two rectangular nonhomogeneous beams, for each of which EB = 1.5 ET , glued together.

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For this case 𝛼 = 0.25. Even though the adjacent plates are in a bending mode, the effective modulus is considerably reduced. For example, for m = EBL ∕ETL = 1.5, EEffec = 1.125ET ; and for m = EBL ∕ETL = 2, EEffec = 1.25ET . Case (ii). h = 4t (internal gap is 2t), t1 = t. The purpose of this example is to show how rapidly EEffec approaches ETL once the flanges are set apart from each other (by 2t in this case). For this case 𝛼 = 1∕(13 + t∕b) < 0.077 Then, for m = EBL ∕ETL = 1.5, EEffec = 1.038ET . Thus, EEffec is only about 4% larger than ETL even though EBL ∕ETL = 1.5. 23.4.1

I-Section Beam

The box section shown in Figure 23.4.4 is equivalent to an I-section with the same flange widths and thicknesses, the same web height, but a web thickness of 2t1 .

t

y

x

h b

Figure 23.4.4 Cross section of a thin-walled I-beam.

So t1 = t∕2 in Figure 25.4.2 corresponds to the constant thickness I-section shown in Figure 25.4.3. Then, for this case 1 𝛼=[ ] < 0.25 1 + (t1∕b) (h∕t)3 + 3(1 + h∕t)2 23.4.2

T-Section Beam

The T-section shown in Figure 23.4.4 is equivalent is not symmetric about the x-axis. However, since the local tensile modulus ETL has been assumed to be constant, the E-weighted centroid coincides with the centroid of the section. Let the E-weighted centroid, at a distance y0 below the flange, be the origin of the coordinate system. Then ( ) h h+t y0 = 2 h+b and 1 𝛼= 1 + (h∕b) (h∕t)2 + 3(h∕t)2 (1 + t∕h)2 ∕(1 + b∕h)

Structural Foams – Materials with Millistructure

For h ≫ t, and for h and b being of the same magnitude, [ ] 1 2 𝛼 ≅ (t∕h) (b∕h) < (t∕h)2 (b∕h) 1 + 3∕(1 + h∕b) Thus, in any normal T-section design, 𝛼 is on the order of (t∕h)2 (b∕h), so that the effective modulus essentially is ETL .

23.5 Skin-Core Models for Structural Foams While the analysis in Section 23.4 for the bending of thin-walled prismatic beams shows that the effective bending modulus for such beams is essentially the tensile modulus, ETL , that analysis does not address the overall question of the stiffness of geometrically more complex structures. The stiffness of such structures requires the use of numerical methods, such as the finite element method, for which the variation of the through-thickness stiffness, E = E ( y), has to be specified. One approach to specifying this material property is to use experimental bending and tensile moduli to model this through-thickness modulus variation. By defining a new coordinate 𝜂 = 2y∕d for the rectangular shown in Figure 23.3.1, the modulus variation over half the thickness has the shape shown in Figure 23.5.1a. In this coordinate system the expressions for the effective tensile and bending moduli in Eqs. 23.3.1 and 23.3.4, respectively, reduce to 1

ETR =

∫0

E (𝜂 ) d𝜂 1

EBR = 3

∫0

𝜂 2 E (𝜂 ) d𝜂

E(η)

(23.5.1)

Es

E(η)

Es

E(η)

η0 Ec

Ec

0

1 (a)

0

η1 (b)

η2 1

η2 1

0 (c)

Figure 23.5.1 Model for the variation of the Young’s modulus across the plate thickness. (Adapted with permission from V.K. Stokes, Journal of Vibration, Stress, and Reliability in Design, Vol. 109, pp. 82 – 86, 1987.)

23.5.1

Four-Parameter Model

In the piecewise linear modulus approximation shown in Figure 23.5.1b, the modulus variation in terms of the four parameters Ec , Es , 𝜂 1 , and 𝜂 2 is defined by

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Introduction to Plastics Engineering

⎧ Ec , ⎪ ( ) Es − Ec ⎪ E (𝜂 ) = ⎨ Ec + (𝜂 − 𝜂1 ) , 𝜂2 − 𝜂1 ⎪ ⎪ , ⎩ Es

0 ≤ 𝜂 ≤ 𝜂1

𝜂1 ≤ 𝜂 ≤ 𝜂2

(23.5.2)

𝜂2 ≤ 𝜂 ≤ 1

in which Ec , the modulus of the “core” having thickness 𝜂 1 ; and Es , the modulus of the “skin” having thickness 𝜂 0 = 1 − 𝜂 2 . The modulus varies linearly from Ec to Es over 𝜂 1 ≤ 𝜂 ≤ 𝜂 2 . For this model, the expressions in Eq. 23.5.1 integrate to 1 ETR = Es − (Es − Ec ) (𝜂1 + 𝜂2 ) 2 1 EBR = Es − (Es − Ec ) (𝜂1 + 𝜂2 ) (𝜂1 2 + 𝜂2 2 ) (23.5.3) 4 Once ETR and EBR have been determined experimentally, these expressions represent two equations for the four model parameters Ec , Es , 𝜂 1 , and 𝜂 2 . In principle, two additional pieces of information can be provided by inputting the value of 𝜂 2 = 1 − 𝜂 0 by determining the “skin thickness,” 𝜂 0 , from tests, and by assuming that Es = EResin in which EResin is the tensile modulus of the base (unfoamed) resin. Then, the modulus variation shown in Figure 23.5.1b can then be used in for a finite element analysis (FEA) of a structure. 23.5.2

Three-Parameter Model

A particularly simple three-parameter skin-core model is shown in Figure 23.5.1c. With 𝜂 1 = 𝜂 2 = 1 − 𝜂 0 , this simple model is defined by { Ec , 0 ≤ 𝜂 ≤ 𝜂2 E (𝜂 ) = (23.5.4) Es , 𝜂2 ≤ 𝜂 ≤ 1 And Eqs. 23.5.2 and 23.5. reduce to ETR = Es − (Es − Ec ) (1 − 𝜂0 ) EBR = Es − (Es − Ec ) (1 − 𝜂0 )3

(23.5.5)

These expressions can be inverted to give Es m−1 =1+ ETR 𝜂0 (2 − 𝜂0 ) Ec m−1 =1− (23.5.6) ETR (1 − 𝜂0 ) (2 − 𝜂0 ) where m = EBR ∕ETR . For any experimentally determined value of m = EBR ∕ETR , Eq. 23.5.5 determines the parameters Ec and Es . Alternatively, Ec and can be determined from these equations if it is assumed that Es = EResin . The variations of Es ∕ETR and Ec ∕ETR versus the skin thickness 𝜂 0 , with the modulus ratio m = EBR ∕ETR as parameter, are shown in Figure 23.5.2 by solid and dashed lines, respectively. The homogeneous case corresponds to m = 1, for which Es ∕ETR = Ec ∕ETR ≡ 1. For m > 1, Ec ∕ETR starts with a value (3 − m)∕2 at 𝜂 0 = 0 (no skin) and decreases continuously with increases in 𝜂 0 , until finally Ec ∕ETR = 0 at 𝜂 0 = [3 − (4m − 3)1/2 ]∕2. The three-parameter model is not valid for higher values of 𝜂 0 than this initial value because that would result in negative values of Ec . For m = 3, the Ec ∕ETR curve starts at 0, and is negative for 𝜂 0 > 0. Thus, the three-parameter skin-core model will only work for

Structural Foams – Materials with Millistructure

5 m=2 1.5 1.3 1.2 1.1 m=1

4 Es ETR

3 2

Ec E TR

m=

1 2.5

0

2

0

1.3 1.2

1.5

E BR E TR

m = 1.1

1.0

0.5

η0 Figure 23.5.2 Variations of Es ∕ETR and Ec ∕ETR versus 𝜂 0 , with m = EBR ∕ETR as a parameter. (Adapted with permission from V.K. Stokes, Journal of Vibration, Stress, and Reliability in Design, Vol. 109, pp. 82 – 86, 1987.)

1 ≤ m < 3. The curves for Es ∕ETR have steeper variations for low values of 𝜂 0 . The value of Es ∕ETR = m for 𝜂 0 = 1. For low values of m = EBR ∕ETR , there is a wide range of values 𝜂 0 , away from 𝜂 0 = 0 and 𝜂 0 = 1, for which Es and Ec are insensitive to the skin thickness 𝜂 0 . However, Es is very sensitive to small values of 𝜂 0 , while Ec is sensitive to large values of 𝜂 0 . The variation of 𝛽 = Ec ∕Es versus 𝜂 0 , with m = EBR ∕ETR as a parameter, is shown in Figure 23.5.3. Clearly, 𝛽 decreases dramatically with small increases in m.

1.0

m=1

m= m = 1.1

Ec Es

E BR E TR

1.2

0.5

1.3 2.5

1.5

2.0

0 0

0.5

1.0

η0 Figure 23.5.3 Variation of 𝛽 = Ec ∕Es versus 𝜂 0 , with m = EBR ∕ETR as a parameter. (Adapted with permission from V.K. Stokes, Journal of Vibration, Stress, and Reliability in Design, Vol. 109, pp. 82 – 86, 1987.)

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From Eq. 23.5. it follows that 1 − (1 − 𝛽 ) (1 − 𝜂0 )3 EBR , = ETR 1 − (1 − 𝛽 ) (1 − 𝜂0 )

𝛽=

Ec Es

(23.5.7)

The variation of EBR ∕ETR versus 𝜂 0 , with 𝛽 = Ec ∕Es as a parameter, is shown in Figure 23.5.4. Depending on the values of the parameters, EBR could easily be significantly larger than ETR .

3.0

β=

Ec Es

β=0 EBR ETR

2.0 0.1 0.2

1.0 0

0.3

0.4

0.5

0.75 0.5

1.0

η0 Figure 23.5.4 Variation of EBR ∕ETR versus 𝜂 0 , with 𝛽 = Ec ∕Es as a parameter. (Adapted with permission from V.K. Stokes, Journal of Vibration, Stress, and Reliability in Design, Vol. 109, pp. 82 – 86, 1987.)

23.6 Stiffness and Strength of Structural Foams This section explores the stiffness and strength of structural foams. Mechanical properties of polymeric materials are generally determined through tensile tests on flat rectangular bars with cross sections that are 12.7-mm wide and have a thickness that is representative of the application. To facilitate a uniform stress distribution in the test area, test specimens normally have the ASTM D638 dog-bone shape. While such dog-bone specimens – either molded or cut from sheet material – are appropriate for homogeneous resins, molded structural foam dog-bone specimens result in data that overestimates both the stiffness and the strength of foams: Figures 23.6.1a,b show the cross-sectional morphologies of 6-mm thick injection-molded dog-bone specimens of M-PPO structural foams with nominal density reductions of 5 and 15%, respectively. For these two density reductions, the corresponding morphologies of dog-bone specimens cut from 4-mm thick injection-molded plaques are shown, respectively, in Figures 23.6.1c,d. Clearly, the cross sections of the molded specimens are encased in more skin material than those of specimens cut from molded plaques, which are more representative of the morphologies in actual molded parts. All the data discussed in the sequel was obtained through tests on specimens cut from molded plaques.

Structural Foams – Materials with Millistructure

(a)

(b)

(c)

(d)

Figure 23.6.1 Cross-sectional morphologies of 12.7-mm wide M-PPO-SF bars. (a) 6.35-mm thick injection-molded bar with 5% density reduction. (b) 6.35-mm thick injection-molded bar with 15% density reduction. (c) 4-mm thick bar cut from 5% density reduction injection-molded plaque. (d) 4-mm thick bar cut from 15% density reduction injection-molded plaque. (Adapted with permission from V.K. Stokes, R.P. Nimmer and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1491 – 1500, 1988.)

The purpose of the test data described in the sequel is to address the following questions: (i) In view of the randomness of the cross-sectional morphology induced by processing, how repeatable or consistent are the local mechanical properties? (ii) What is the distribution of the mechanical properties across each plaque? And (iii), how does plaque thickness affect the local properties? 23.6.1

Test Procedure for Acquiring Stiffness and Strength Data

The data discussed in the sequel were obtained from specimens cut from 152.5-mm (6-in) wide by 457-mm (18-in) long, molded M-PPO structural foam plaques having thicknesses of 6.35 and 4 mm (0.25 and 0.157-in). The plaques, schematically shown in Figure 23.6.2, were edge-gated at one (left) end with the flow occurring along the length. Eighteen 25.4 × 152.5 mm (1 × 6 in) test specimens – with specimen 1 next to the gated end and specimen 18 at the far end of the plaque – were cut from each molded plaque. As shown, a fan gate was used to create as flat as possible a flow front to limit variations in properties in the cross-flow direction. In this way variations of properties along the 152.5-mm lengths of the test specimens were minimized. The lateral faces of each rectangular bar were machined down to a width of 19 mm (0.75 in). Tests were conducted in the following sequence: First, the average density of each 19 × 152.5 mm (0.75 × 6 in) test specimen was determined by measuring its mass and bulk (external) volume (through

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457 mm

152.5 mm

SPECIMEN 1

SPECIMEN 18

Figure 23.6.2 Layout of eighteen 19 × 152.5-mm (0.75 × 6-in) test specimens cut from 152.5 × 457-mm (6 × 18-in), M-PPO-SF edge-gated, molded plaque. (Adapted with permission from V.K. Stokes, R.P. Nimmer and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1491 – 1500, 1988.)

micrometer measurements of the external dimensions). The average through-thickness density most likely varies along the length of each specimen. And, because of extra skins at the ends of the specimens, this average density may be an overestimate for the actual through-thickness local density. Next, a three-point bend test was used to determine the effective bending modulus, EBR , of each bar: With reference to the three-point bend geometry shown in Figure 5.3.3, the cross-sectional geometry defined in Figure 23.3.2, and Eq. 5.3.4, the bending modulus is determined by EBR =

(L∕d )3 P 4b 𝛿

(23.6.1)

in which 𝛿 is the central beam deflection caused by a central load P acting on a beam of length L, which for all the tests was 76.2 mm (3 in). After determining the bending modulus, each specimen was routed to the shape of an ASTM D638 standard dog-bone specimen having a 12.7-mm (0.5-in) gauge length. The specimens were then tested to failure in a tensile test in which the strains were monitored by an extensometer. These tests made it possible to determine the effective tensile modulus, ETR , the ultimate stress at failure, 𝜎 0 , and the strain at failure, 𝜀0 . 23.6.2

Plaque-to-Plaque and In-Plaque Variations of Material Properties

Figure 23.6.3 shows the variations of the bending modulus, EBR , the tensile modulus, ETR , and the average density, 𝜌, with the specimen position in 4-mm thick, 5% density reduction modified poly(phenylene oxide) structural foams (M-PPO-SF) plaques. The data from three plaques are remarkably consistent. In each plaque, the specimens farthest from the gated end have lower average densities and elastic moduli. The bending modulus is about 30% larger than the tensile modulus. Figure 23.6.4 shows these variations for 4-mm thick, 15% density reduction M-PPO-SF plaques. Although there is somewhat more scatter in the data from three plaques, the results are still remarkably consistent. Except in specimen 18 in one of the plaques, the plaque-to-plaque variations are less than 10%. Here again, specimens farthest from the gated end have lower properties. However, in contrast to 5% density reduction plaques (Figure 23.6.3), the specimen-to-specimen variation within each plaque is much larger for the 15% density reduction plaques. The tensile modulus varies from about 1.93 GPa to about 1.24 GPa across the plaques; the bending modulus varies from 2.76 to 2.07 GPa. These variations in

Structural Foams – Materials with Millistructure

5 M-PPO FOAM NOMINAL DENSITY REDUCTION 5% SPECIMEN THICKNESS 4 mm BENDING MODULUS

EBR

TENSILE MODULUS

ETR

3

2 1.2 1

1.0 AVERAGE LOCAL DENSITY

ρ

0.8

0 0

10 5 15 SPECIMEN NUMBER

DENSITY (g cm−3)

MODULUS (GPa)

4

20

Figure 23.6.3 Variations of the average local density, and the bending and tensile moduli of rectangular specimens cut from 4-mm thick, 5% density reduction, M-PPO-SF edge-gated, molded plaque. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

5 M-PPO FOAM NOMINAL DENSITY REDUCTION 15% SPECIMEN THICKNESS 4 mm

3 BENDING MODULUS

EBR

2 1.2 TENSILE MODULUS

1 AVERAGE LOCAL DENSITY

ETR

1.0 0.8

ρ

0.6

0 0

10 5 15 SPECIMEN NUMBER

DENSITY (g cm−3)

MODULUS (GPa)

4

20

Figure 23.6.4 Variations of the average local density, and the bending and tensile moduli of rectangular specimens cut from 4-mm thick, 15% density reduction, M-PPO-SF edge-gated, molded plaque. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

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the moduli parallel those of the local density. The bending moduli are 30 – 50% higher than the respective tensile moduli for the same specimen. Figure 23.6.5 shows the variations of the ultimate stress at failure, 𝜎 0 , with the specimen number for a 5% density reduction plaque. Except for specimens farthest from the gated end, the ultimate stress does not vary much; it could therefore be used for designing for strength. However, as shown in Figure 23.6.6, the ultimate failure strain, 𝜀0 , varies from 2% to 9%. The general trend is toward higher ultimate strains near the gate. The three load-strain curves in Figure 23.6.7 exhibit load maxima that appear to be fairly consistent across the specimens in Figure 23.6.5 but there is a larger variation of strains at failure beyond the load maxima. This helps to explain the qualitative difference between results in Figures 23.6.5 and 23.6.6. 50 M-PPO FOAM NOMINAL DENSITY REDUCTION 5% SPECIMEN THICKNESS 4 mm

40

30 ULTIMATE STRESS

σ0

20

1.1

10

1.0 AVERAGE LOCAL DENSITY

ρ

0.9

DENSITY (g cm−3)

ULTIMATE STRESS (MPa)

870

0 0

10 5 15 SPECIMEN NUMBER

20

Figure 23.6.5 Variations of the average local density and the ultimate stress of rectangular specimens cut from a 4-mm thick, 5% density reduction, M-PPO-SF edge-gated, molded plaque. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

Figures 23.6.8 and 23.6.9 show the variations of the ultimate stress and ultimate strain, respectively, for a 15% nominal density reduction plaque. Paralleling the trend for the variation of the stiffness, the ultimate stress for the 15% density reduction material exhibits a larger variation in comparison to the 5% density reduction material. Ignoring the data for the first and last specimens, the ultimate stress across the plaque varies between 13.8 and 27.6 MPa. The variation of the ultimate stress across the plaque follows the variation of the average local density. The data for the 15% density reduction material in Figure 23.6.9 shows that the ultimate strain varies more systematically than for the 5% density reduction material. The pattern of variation of the ultimate strain appears to follow the variations of the average density. The strains are lowest away from the gated end. However, the range of variation of the ultimate strain is much larger than for the ultimate stress.

Structural Foams – Materials with Millistructure

NOMINAL DENSITY REDUCTION 5% SPECIMEN THICKNESS 4 mm M-PPO FOAM

ULTIMATE STRAIN

ε0

0.05

1.1 1.0 AVERAGE LOCAL DENSITY ρ

0.9

DENSITY (g cm−3)

ULTIMATE STRAIN

0.10

0 0

10 5 15 SPECIMEN NUMBER

20

Figure 23.6.6 Variations of the average local density and the ultimate strain of rectangular specimens cut from a 4-mm thick, 5% density reduction, M-PPO-SF edge-gated, molded plaque. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

M-PPO FOAM NOMINAL DENSITY REDUCTION 5% SPECIMEN THICKNESS 4 mm

LOAD (kN)

2

SPECIMEN 10 SPECIMEN 8

1

SPECIMEN 14

0 0

0.05 STRAIN

0.10

Figure 23.6.7 Load-strain curves for specimen numbers 8, 10, and 14 cut from a 4-mm thick, 5% density reduction, M-PPO-SF edge-gated, molded plaque. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

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50

40

M-PPO FOAM

ULTIMATE STRESS

30

1.1

σ0

1.0 20 0.9 10

DENSITY (g cm−3)

ULTIMATE STRESS (MPa)

NOMINAL DENSITY REDUCTION 15% SPECIMEN THICKNESS 4 mm

0.8

AVERAGE LOCAL DENSITY ρ

0 10 5 15 SPECIMEN NUMBER

0

20

Figure 23.6.8 Variations of the average local density and the ultimate stress of rectangular specimens cut from a 4-mm thick, 15% density reduction, M-PPO-SF edge-gated, molded plaque. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

NOMINAL DENSITY REDUCTION 15% SPECIMEN THICKNESS 4 mm

1.1 1.0

AVERAGE LOCAL DENSITY ρ

0.9 0.8

0.05

DENSITY (g cm−3)

M-PPO FOAM

0.10

ULTIMATE STRAIN

872

0.7

ULTIMATE STRAIN

ε0

0 0

10 5 15 SPECIMEN NUMBER

20

Figure 23.6.9 Variations of the average local density and the ultimate strain of rectangular specimens cut from a 4-mm thick, 15% density reduction, M-PPO-SF edge-gated, molded plaque. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

Structural Foams – Materials with Millistructure

The ultimate strains of the specimens vary from 4% at the gated end to 1% at the farthest end. The three load-strain curves for specimens 4, 14, and 17 in Figure 23.6.10 provide an explanation for some of the differences between the 5 and 15% density reduction materials. In general, specimens with lower densities in Figure 23.6.9 exhibit a much lower extensional capability; most specimens fail before the appearance of a load maximum. This results in the variation in the ultimate strain in the 15% material more closely paralleling the variation of the ultimate stress than for the 5% density reduction material in which the material undergoes large strains after load maxima has been attained. As such, for purposes of mechanical design, foams with higher density reductions should be treated as materials that cannot undergo further strains after attaining load maxima. M-PPO FOAM NOMINAL DENSITY REDUCTION 15% SPECIMEN THICKNESS 4 mm

LOAD (kN)

2

SPECIMEN 4

1

SPECIMEN 14

SPECIMEN 17

0 0

0.05 STRAIN

0.10

Figure 23.6.10 Load-strain curves for specimen numbers 4, 14, and 17 cut from a 4-mm thick, 15% density reduction, M-PPO-SF edge-gated, molded plaque. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

23.6.3

Effect of Density on Mechanical Properties

Figure 23.6.11 shows the variations of the nondimensional bending and tensile moduli, EBR ∕E0 and ETR ∕E0 , respectively, as functions of the nondimensional density, 𝜌∕𝜌0 . The values of the modulus, E0 = 2.79 GPa, and the density, 𝜌0 = 1.1 g cm−3 , used to normalize the data in Figure 23.6.11 were obtained by tests on solid (unfoamed) M-PPO resin specimens. The tensile and bending moduli appear to be strongly correlated to the foam density. The data for the highest local average density specimens from the 15% density reduction plaques consistently overlap the data from the 5% density reduction plaques. Also, the moduli and densities of specimens from the 15% reduction material have larger variations than the corresponding specimens from the 5% density reduction plaques. Figure 23.6.12 shows the variations of the nondimensional ultimate stress and ultimate strain, 𝜎 0 ∕𝜎 u and 𝜀0 ∕𝜀u , respectively, as functions of the nondimensional density, 𝜌∕𝜌0 , for both 5 and 15% density

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1.5

NONDIMENSIONAL TENSILE AND BENDING MODULI

NOMINAL DENSITY REDUCTION: 5% & 15% SPECIMEN THICKNESS 4 mm M-PPO FOAM

NOMINAL DENSITY REDUCTION

5%

15%

BENDING MODULUS E BR / E 0 TENSILE MODULUS E TR / E 0

1.0

E BR / E 0

E TR / E 0 0.5 0.6

0.8 0.7 0.9 NONDIMENSIONAL DENSITY

1.0

Figure 23.6.11 Nondimensional bending and tensile moduli of 5 and 15% density reduction, 4-mm thick specimens versus the nondimensional average local density of M-PPO-SF. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

1.5 NOMINAL DENSITY REDUCTION: 5% & 15% SPECIMEN THICKNESS 4 mm

NONDIMENSIONAL ULTIMATE STRESS AND ULTIMATE STRAIN

874

M-PPO FOAM

NOMINAL DENSITY REDUCTION

5%

15%

ULTIMATE STRESS

1.0

σ0

ULTIMATE STRAIN

ε0

σ0

0.5

ε0 0 0.4

0.5

0.6

0.7

0.8

0.9

1.0

NONDIMENSIONAL DENSITY Figure 23.6.12 Nondimensional ultimate stress and strain to failure of 5 and 15% density reduction, 4-mm thick specimens versus the nondimensional average local density of M-PPO-SF. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

Structural Foams – Materials with Millistructure

reduction materials. The average values of the ultimate stress, 𝜎 u = 36.9 MPa, and the ultimate strain, 𝜀0 = 0.127, used to normalize the data in Figure 23.6.11 were obtained by tests on solid (unfoamed) M-PPO resin specimens. For nondimensional densities less than about 0.89, the nondimensional ultimate stress and the nondimensional ultimate strain correlate strongly with the nondimensional density. As previously noted, the strain-to-failure data has more scatter. Above nondimensional densities of about 0.89, the nondimensional ultimate strain varies from as little as 0.15 to as much as 0.75, while the nondimensional ultimate stress is relatively constant at about 0.8. Most of this data for 𝜌∕𝜌0 > 0.89 is from 5% density reduction specimens and reflect the characters of the load-strain curves shown in Figure 23.6.7. This class of materials has a maximum in the load-strain curve and exhibits some ductility. This behavior is quite different from that of the solid (unfoamed) base M-PPO resin, whose ultimate strain varied from 0.124 to 0.127 in the six specimens tested. 23.6.4

Dependence of Mechanical Properties on Plaque Thickness

This section compares the mechanical properties of specimens cut from 6.35-mm thick, 5 and 15% density reduction M-PPO plaques with those for the 4-mm thick material discussed in previous sections. Figure 23.6.13 compares the previously described tensile and bending moduli and density data for 4-mm thick, 5% density reduction plaques with data for specimens cut from 6.35-mm thick, 5% density reduction plaques. The corresponding comparisons for 15% density reduction material are shown in Figure 23.6.14. Both these figures show that the moduli for the 4-mm thick specimens are virtually indistinguishable from those for the 6.35-mm thick specimens. Figures 23.6.15 and 23.6.16 compare ultimate stress data from 4- and 6.35-mm thick tensile specimens cut, respectively, from 5 to 15% density reduction plaques. Although the ultimate stresses in the 4-mm 5 SPECIMEN THICKNESS

M-PPO FOAM

4 mm

6.35 mm

4

BENDING MODULUS

3

2 TENSILE MODULUS

ETR

AVERAGE LOCAL DENSITY

1

EBR

ρ

1.1 1.0 0.9

0 0

5

10

15

20

DENSITY (g cm−3)

MODULUS (GPa)

NOMINAL DENSITY REDUCTION 5%

SPECIMEN NUMBER Figure 23.6.13 Comparison of the average local density and the elastic moduli of specimens cut from 4-mm thick, 5% density reduction M-PPO-SF plaques. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

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5 SPECIMEN THICKNESS

M-PPO FOAM

4 mm

6.35 mm

4

3

BENDING MODULUS

EBR

2

E TR

1.1 1.0

1

0.9 AVERAGE LOCAL DENSITY

ρ

0.8 0.7

0 0

10

5

15

DENSITY (g cm−3)

MODULUS (GPa)

NOMINAL DENSITY REDUCTION 15%

20

SPECIMEN NUMBER Figure 23.6.14 Comparison of the average local density and the elastic moduli of specimens cut from 4-mm thick, 15% density reduction M-PPO-SF plaques. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

50 SPECIMEN THICKNESS

M-PPO FOAM

4 mm

6.35 mm

40 NOMINAL DENSITY REDUCTION 5%

30

ULTIMATE STRESS

20

σ0 1.1 1.0

10 AVERAGE LOCAL DENSITY

0.9

ρ

0.8

DENSITY (g cm−3)

ULTIMATE STRESS (MPa)

876

0 0

5

10

15

20

SPECIMEN NUMBER Figure 23.6.15 Comparison of the average local density and the ultimate stress of specimens cut from 4-mm thick, 5% density reduction M-PPO-SF plaques. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

Structural Foams – Materials with Millistructure

thick, 5% nominal density reduction specimens appear to be consistently higher than their counterparts from 6.35-mm thick plaques, the difference is always less than 12%. For the 15% density reduction material, the ultimate stress data do not show any significant and consistent variation that can be attributed to specimen thickness. 50 SPECIMEN THICKNESS

4 mm

6.35 mm

40 NOMINAL DENSITY REDUCTION 15%

30 ULTIMATE STRESS

σ0

20 1.1 1.0 10

0.9 AVERAGE LOCAL DENSITY

0.8

ρ

DENSITY (g cm−3)

ULTIMATE STRESS (MPa)

M-PPO FOAM

0 0

5

10

15

20

SPECIMEN NUMBER Figure 23.6.16 Comparison of the average local density and the ultimate stress of specimens cut from 4-mm thick, 15% density reduction, M-PPO-SF plaques. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

Figures 23.6.17 and 23.6.18 show similar comparisons of ultimate strain data for 4- and 6.35-mm thick specimens cut, respectively, from 5 to 15% nominal density reduction plaques. For the 5% nominal density reduction material (Figure 23.6.17), the average specimen densities are very consistent and, although there is considerable scatter in the ultimate strain data, there is no recognizable dependence on specimen thickness. In the 15% density reduction material (Figure 23.6.18), the densities of the 6.35-mm thick specimens are consistently higher than their 4-mm thick counterparts. The ultimate strains for the 6.35-mm thick specimens are also higher those for the 4-mm thick specimens. The largest differences in the ultimate strain are associated with specimens having the largest differences in specimen density. 23.6.5

Summary Comments

The experimental data discussed in the previous subsections have established results of importance to the mechanical design process: First, the data of specimens cut from any location are consistent with the data obtained from specimens cut from the same location but from several different plaques. This is important because even though the pore structure can be expected to vary from plaque to plaque, local the mechanical properties are consistent within the scatter expected in the properties of such materials. Second, although properties vary along the plaque length, the mechanical properties – such as the elastic

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0.20 SPECIMEN THICKNESS

M-PPO FOAM

4 mm

6.35 mm

0.15

0.10

0.05 ULTIMATE STRAIN

0

ε0

AVERAGE LOCAL DENSITY

0

5

10

1.1 1.0 0.9 0.8

ρ 20

15

DENSITY (g cm−3)

ULTIMATE STRAIN

NOMINAL DENSITY REDUCTION 5%

SPECIMEN NUMBER Figure 23.6.17 Comparison of the average local density and the ultimate strain of specimens cut from 4-mm thick, 5% density reduction, M-PPO-SF plaques. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

0.20 SPECIMEN THICKNESS

M-PPO FOAM

4 mm

6.35 mm

0.15 NOMINAL DENSITY REDUCTION 15%

0.10

0.05

ULTIMATE STRAIN

0

ε0

1.1 1.0 0.9 0.8

AVERAGE DENSITY ρ

0

5

10

15

20

DENSITY (g cm−3)

ULTIMATE STRAIN

878

SPECIMEN NUMBER Figure 23.6.18 Comparison of the average local density and the ultimate strain of specimens cut from 4-mm thick, 15% density reduction, M-PPO-SF plaques. (Adapted with permission from R.P. Nimmer, V.K. Stokes, and D.A. Ysseldyke, Polymer Engineering and Science, Vol. 28, pp. 1502 – 1508, 1988.)

Structural Foams – Materials with Millistructure

moduli and the tensile strength – correlate linearly with the local density. And third, the specimen thickness does not affect local properties: for specimens cut from the same regions there is very little difference in the stiffness or strength data from 4- and 6.35-mm thick specimens. In general, the variations of properties within a plaque increase with the density reduction, with very little variation in the 5% density reduction plaque. While the tensile strengths of bars also correlate linearly with the local density the ultimate strains exhibit significant scatter; at lower nominal density reductions there is little or no additional strain between the point of maximum stress and failure. As such, the ultimate stress appears to be the safest criteria for failure. These results form the basis for the mechanical design of structural foam components: Independent of the local part thickness, all the local mechanical properties (stiffness and strength) correlate with the local average density. Stiffness and strength prediction are then dependent on the local density distribution, methods for predicting, which are currently not available.

23.7 The Average Density and the Effective Tensile and Flexural Moduli of Foams In Section 23.5 the tensile and bending moduli of rectangular bars were correlated to the average density of the bar. As can be seen from Figures 23.6.3 – 23.6.6, 23.6.8, 23.6.9, and 23.6.11 – 23.6.17, the average density of the bars vary across the length of the bar. This density variation is more pronounced in the 15% density reduction material. These variations raise the possibility of the density varying along the length of each bar. This section explores the effects of processing on the local mean density on a 12.7-mm scale and on the effective elastic moduli of foams. Mechanical properties of polymeric materials are generally determined through tensile tests on flat rectangular bars with cross sections that are 12.7-mm wide and have a thickness that is representative of the application. Because of property gradients, the local strain during a tensile test on a nonhomogeneous material (such as structural foam) could be quite different along the two edges of the specimen. Thus, if an extensometer is used to measure strains along an edge, the resulting tensile moduli would depend on which edge it is attached to. These effects can be quite large for nonhomogeneous materials, and raise the issue of what a representative mean modulus is, and how it should be determined. It has been suggested that for such materials the strains be simultaneously measured along both edges of the specimen, resulting in two values, EL and ER , for the local modulus. A representative mean modulus could then be EA = (EL + ER )∕2. In the sequel, the average tensile modulus, EA , will be the average over 12.7 × 12.7-mm regions. 23.7.1

Test Procedure

The data discussed in the sequel were obtained from specimens cut from 152.4-mm (6-in) wide by 457.2-mm (18-in) long, molded polycarbonate structural foam (PC-SF) plaques, for two plaque thicknesses 6.35 and 4 mm (0.25 and 0.157-in), and three nominal density reductions of 5, 15, and 25%. The material used was a 5% glass-filled polycarbonate (PC). In this grade of PC, the glass is used as a nucleating agent to facilitate the foaming process. The plaques, schematically shown in Figure 23.7.1, were edge-gated from the top with the flow direction along the length. Tensile moduli were obtained by tests on long (12.7 × 406.4 mm, 0.5 × 16 in) rectangular specimens: First, parallel lines were marked with ink on a plaque at 12.7-mm (0.5-in) intervals, as shown by dashed

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Introduction to Plastics Engineering

lines in Figure 23.6.1. Then 12.7-mm wide rectangular specimens (shown by solid lines) were cut, resulting in eight 12.7 × 406.4 mm rectangular specimens on which 24 12.7-mm-long segments are delineated by parallel ink lines. For each specimen the 24 segments were numbered consecutively, with segment 1 being closest to the gated edge (numbering system shown in Figure 23.7.1). The mean cross-sectional area of each segment was calculated by determining the mean width and thickness of each individual segment. 152.5 mm

457 mm

406.4 mm

304.8 mm

24

12.7 mm

1

12.7 mm

SEGMENT NUMBER

880

1

8

SPECIMEN NUMBER Figure 23.7.1 Layout of eight 12.7 × 406.4-mm specimens cut from 152.4 × 457.2-mm, edge-gated molded foam plaque. (Adapted with permission from V.K. Stokes, Journal of Materials Science, Vol. 35, pp. 159 – 178, 2000.)

The mean effective tensile modulus, EA , was determined at each 12.7-mm segment by successive tensile tests in which the extensometer was placed at different locations along the specimen. The tensile tests were done at a strain rate of 0.01 s−1 ; the specimens were pulled to a maximum strain of 0.25%. After determining the tensile moduli, three-point bend tests were done on the same set of specimens to determine the mean flexural moduli over 76.2-mm (3-in) spans, at three points along each specimen. The flexural tests were done such that the nominal strain rate in the outermost layers was 0.01 s−1 , and the outermost layers were deformed to a nominal strain of 0.25%. Finally, after determining the elastic moduli, the specimen was cut along the inked lines resulting in 12.7 × 12.7-mm coupons, the densities of which were then determined by mass and volume measurements. This procedure was repeated for each of the 24 specimens. In this way, the local mean density 𝜌 was determined at each 12.7 × 12.7-mm segment over a 101.6 × 304.8-mm (4 × 12-in) area of the plaque.

Structural Foams – Materials with Millistructure

23.7.2

In-Plane Density Variations

The variation of the average foam density across plaques is best visualized through density contours. For this purpose, the local percentage density reduction, 100 (𝜌0 − 𝜌)∕𝜌0 , was calculated by using an unfoamed (solid) density of 𝜌0 = 1.21 g cm−3 for the 5% glass-filled PC. Data from the tests described previously show that the local density reduction can be quite different from the nominal density reduction. For example, in the 6.35-mm thick plaques, for nominal density reductions of 5, 15, and 25%, the actual density reduction varies in the range 3.3 – 8.3, 9.1 – 22.3, and 14.0 – 27.3%, respectively. Thus, plaques with nominal density reductions of 15 and 25% have regions with common local density reductions. Specimen thickness also has an effect. For example, for a nominal density reduction of 5%, while the actual local density reduction varies in the narrow range of 3.3 – 8.3% for the 6.35-mm thick specimens, the 4-mm thick specimens have a much larger variation in the range of 2.5 – 19.8%. Also, while 6.35-mm thick plaques with 5 and 15% nominal density reduction do not have regions with the same density reduction, the 4-mm thick plaques do. The variations of the local density over 101.6 × 304.8-mm (4 × 12-in) regions of the 6.35-mm thick plaques (for nominal density reductions of 5, 15, and 25%) and 4-mm thick plaques (for nominal density reductions of 5 and 15%) are shown in Figure 23.7.2. Clearly, in all cases, the density is the highest at the top (closest to the gate) and drops off with increasing distance from the gate. The variations in the width direction are much smaller. Figure 23.7.2a shows that the 5% density reduction 6.35-mm thick plaque has a relatively small density variation. A comparison of Figure 23.7.2b with Figure 23.7.2c shows that these two plaques with nominal density reductions of 15 and 25%, respectively, have regions with the same density. Also, comparisons of Figure 23.7.2a with Figure 23.7.2d and Figure 23.7.2b with Figure 23.7.2e show that, for the same nominal density reduction, the 6.35- and 4-mm thick plaques have different local density distributions. Besides differences in the materials used – M-PPO foams in Section 3.6 and PC-SF in this section – there are two noteworthy differences on the two data sets: First, while the data on the variations of the density, tensile and bending moduli, and the ultimate stress and strain of M-PPO foams in Section 23.5.5 was obtained by tests on 19 × 152.5 mm rectangular bars cut at right angles to the flow direction, the tensile modulus data for the PC-SF was obtained on specimens cut along the plaque length. Second, while for the M-PPO data the mean density used is the average over each such bar – tacitly assuming that the local density does not vary much across the specimen length – the mean density in the PC-SF is defined over 12.7 × 12.7-mm regions. Although, the M-PPO data does pick up density variations along plaque lengths, and the overlapping of density variations in plaques of different nominal density reductions, the density contours in Figure 23.7.2 show that the local density can vary along the length of specimens, especially in the thinner material. Clearly, as shown in Figures 23.7.2d – j, the longitudinal tensile modulus does vary across the plaque width. The density variations in a plaque correspond to variations in morphology. Variations in the morphology of a 6.35-mm thick plaque with a nominal density reduction of 15% are shown in Figures 23.7.3a – f. The figures on the left (a, c, and e) show the morphologies along the plaque length direction; the morphologies along the sides of the long specimens shown by solid lines in Figure 23.7.1. The figures on the right (b, d, and f) show the morphologies along the width – the morphologies as seen on sections cut along the dashed lines in Figure 23.7.1. The sections shown on the left (a, b, and c) are along the middle of the plaque, that is, along the common interface between specimen numbers 4 and 5. Figures 23.7.3a,b show the morphologies of a specimen cut from near the gated end, Figures 23.7.3c,d correspond to a specimen from near the middle of the plaque, and Figures 23.7.3e,f correspond to a specimen from near the far end of the plaque. To obtain surfaces that would clearly show the morphologies, lines were scribed on the surface of the plaque and the plaque was then bent to crack it open along the scribed lines. Note that in

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Introduction to Plastics Engineering

DENSITY (g cm− 3) 1.12–1.20 1.04–1.12 0.96–1.04 0.88–0.96

(a)

(b)

(c)

(g)

(h)

(d)

(e)

(f)

(i)

( j)

TENSILE MODULUS (GPa) 2.77–3.02 2.52–2.77 2.27–2.52 2.02–2.27

Figure 23.7.2 Density and tensile modulus contours for 101.6 × 304.8-mm regions of 6.35-mm thick GF-PC structural foam plaques for nominal density reductions of 5, 15, and 25%, and for 4-mm thick plaques for nominal density reductions of 5 and 15%. (a – c) Density contours 6.35-mm thick plaques for nominal density reductions, respectively, of 5, 15, and 25%. (d – f) Tensile modulus contours for 6.35-mm thick plaques for nominal density reductions, respectively, of 5, 15, and 25%. (g, h) Density contours for 4-mm thick plaques for nominal density reductions, respectively, of 5 and 15%. (i, j) Tensile modulus contours for 4-mm thick plaques for nominal density reductions, respectively, of 5 and 15%. (Adapted with permission from V.K. Stokes, Journal of Materials Science, Vol. 35, pp. 159 – 178, 2000.)

Figures 23.7.3a,c,e the gated end is on the left, so that during filling the flow occurs from the left to the right. The effect of the flow direction is evident in Figures 23.7.3a,c,e – the shear stresses in the “parabolic” flow front distort the shape of the bubbles. In keeping with the density gradient along the plaque length, these three figures show totally different morphologies near the gate, in the middle of the plaque, and near its far end. Such bubble distortions are not evident in the transverse sections shown in Figures 23.7.3b,d,f. At a given point along the length, the morphology along a transverse section is relatively homogeneous,

Structural Foams – Materials with Millistructure

5 mm

5 mm

(a)

(b)

5 mm

(c)

5 mm

(d)

5 mm

5 mm

(e)

(f)

Figure 23.7.3 Variations in the morphology of a 6.35-mm thick GF-PC structural foam plaque with a nominal density reduction of 15%. (a), (c), and (e) show the morphologies along the plaque length direction. (b), (d), and (f) show the morphologies along the width – as seen on sections cut along the dashed lines in Figure 23.6.1. The sections in (a), (c), and (e) are along the middle of the plaque. (a) and (b) show the morphologies of a specimen cut from near the gated end; (c) and (d) correspond to a specimen from near the middle of the plaque; and (e) and (f) correspond to a specimen from near the far end of the plaque. Note that in (a), (c), and (e) the gated end is on the left, so that flow occurs from the left to the right. (Adapted with permission from V.K. Stokes, Journal of Materials Science, Vol. 35, pp. 159 – 178, 2000.)

corresponding to smaller density variations along in the transverse direction. The morphologies shown in these figures are consistent with the density being the highest near the gate and lowest at the far end. The marked differences in the morphologies in the flow and cross-flow directions should result in anisotropic mechanical properties. Clearly, in a structural foam part, the local density can be very different from the specified nominal density reduction, and may have significant variations across the part. Also, even in parts of the same thickness, the local density at some location in a part with one nominal density reduction may be the same as at some other location in a second part with a different nominal density reduction. Furthermore, the local density can depend on part thickness. Since the local density is known to affect the local mechanical properties of the material, these density variations raise several important questions: For parts of the

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Introduction to Plastics Engineering

same thickness, but of different nominal density reductions, are the mechanical properties of regions having the same local densities the same? That is, for parts of the same thickness, do the local mechanical properties correlate with the local density? Are the local properties of regions of equal local density in parts with different thickness correlated? Such correlations are discussed in the next section.

23.8 Density and Modulus Variation Correlations This section discusses correlations between the tensile modulus and density data as measured on a 12.7-mm scale over a 101.6 × 304.8-mm area of 6.35-mm thick plaques – for three nominal density reductions of 5, 15, and 25% – and 4-mm thick plaques, for nominal density reductions of 5 and 15%. 23.8.1

Density-Modulus Correlation for 6.35-mm Thick Foam

This section addresses tensile modulus and density data for 6.35-mm thick, 5, 15, and 25% density reduction foams. For a 6.35-mm thick, 5%-density-reduction plaque, the average density 𝜌 varied from a minimum of 1.11 g cm−3 to a maximum of 1.17 g cm−3 , the ratio of the maximum to the minimum being 1.05. For this plaque, the minimum and maximum values of EA and the ratio of the maximum to the minimum for this plaque are 2.35, 2.93, and 1.27 GPa, respectively. Because the local density is a measure of the local cellularity, higher densities should correspond to higher elastic moduli. Figure 23.8.1a is a plot of the local average tensile modulus EA versus the local density 𝜌 for this plaque. Except for isolated points, the densities of most segments lie in a narrow band of about 1.1 – 1.15 g cm−3 . Most of the values of EA and 𝜌 are clustered around narrow bands. This figure shows that EA and 𝜌 are quite strongly correlated. For any given 𝜌, the values of EA can be expected to vary depending on the location and orientation of the bubbles. Thus, the apparent scatter in the data can mainly be ascribed to local nonhomogeneities. The solid lines through the data in Figures 23.8.1a – d is the line E = 2 𝜌 + 0.35, in the same units as in the figure. The significance of this line is discussed in Section 23.9. For a 15%-density-reduction plaque, the local minimum and maximum densities were 0.94 and 1.10 g cm−3 . This is a much larger variation than that in the 5%-density-reduction plaque (1.11 – 1.17 g cm−3 ). Also, the local densities in these two plaques do not overlap. For this plaque, the average modulus EA varies from a minimum of 2.08 GPa to a maximum of 2.70 GPa, resulting in a ratio of maximum to minimum of 1.30. Just as in the case of the 5%-density-reduction plaque (Figure 23.8.1a), Figure 23.8.1b shows that the average modulus correlates fairly well with the local density. Here again, the solid line corresponds to E = 2 𝜌 + 0.35. For a 25%-density-reduction foam plaque, the local density varied from a minimum of 0.88 g cm−3 to a maximum of 1.04 g cm−3 , the ratio of the maximum to the minimum density being 1.18. This range of variation, 0.88 – 1.04 g cm−3 , overlaps the density variation range, 0.94 – 1.10 g cm−3 , in the 15%-density reduction plaque. For this plaque, the minimum and maximum values of EA , and the ratio of the maximum to the minimum values are 2.04, 2.71, and 1.33 GPa, respectively. Figure 23.8.1c is a plot of the average tensile modulus EA versus the local average density 𝜌. The densities are distributed over a wider band of 0.88 – 1.04 g cm−3 . The spread of 0.16 g cm−3 is much larger than the spread of 0.06 g cm−3 in the 5%-density-reduction foam (Figure 23.8.1). Clearly, EA correlates strongly with 𝜌, the dependence being approximately linear.

Structural Foams – Materials with Millistructure

3.0 6.35 mm

MODULUS (GPa)

MODULUS (GPa)

3.0

2.5

5% 2.0

6.35 mm

2.5

15% 2.0

0.8

0.9

1.0

1.1

1.2

0.8

−3

1.0

1.1

1.2 −3

FOAM DENSITY (g·cm )

FOAM DENSITY (g·cm )

(a)

(b) 3.0

3.0 6.35 mm

MODULUS (GPa)

MODULUS (GPa)

0.9

2.5

25% 2.0

6.35 mm

2.5

2.0 0.8

0.9

1.0

1.1

1.2 −3

0.8

0.9

1.0

1.1

1.2 −3

FOAM DENSITY (g·cm )

FOAM DENSITY (g·cm )

(c)

(d)

Figure 23.8.1 Average local tensile modulus versus the local foam density for a specimen cut from 6.35-mm thick 5% glass-filled PC-SF plaques. (a) Density reduction of 5%, (b) density reduction of 15%, (c) density reduction of 25%, and (d) consolidated data from Figures 23.8.1a – c. (Adapted with permission from V.K. Stokes, Journal of Materials Science, Vol. 35, pp. 159 – 178, 2000.)

The consolidated data on variations of the average tensile modulus EA versus the local density 𝜌 for all three nominal density reductions of 5, 15, and 25% – Figures 23.8.1a – c, respectively – are shown in Figure 23.8.1d. This figure shows that in 6.35-mm thick plaques with nominal density reductions of 15% and 25%, regions with the same local density have the same tensile modulus within the inherent scatter in this class of materials. Also, the data for the 5% nominal density material follows the same linear dependence of the modulus on the density, even though the local density variations in this material do not overlap those in the 15 and 25% density reduction cases. Thus, the local average modulus correlates with the local density, independent of whether the local properties are for a 5, 15, or 25%-density-reduction foam. This empirical result is important from the perspective of mechanical design, as it suggests that the local material stiffness can be determined once the local density is known. Of course, this study has not considered any anisotropy introduced by the cell structure being affected by the flow field – different cell shapes in the flow and cross-flow directions. Note that the straight line drawn through the data in this figure, EA = 2 𝜌 + 0.35, was not obtained through a least squares fit; rather, it is a visual fit.

885

Introduction to Plastics Engineering

23.8.2

Density-Modulus Correlation for 4-mm Thick Foam

This section addresses tensile modulus and density data for 4-mm thick, 5 and 15% density reduction foams. Figures 23.8.2a,b show the modulus-density data for the 5 and 15% density reduction materials, respectively. The consolidated data for both these density reductions is shown in Figure 23.8.2c. Finally, Figure 23.8.2d shows the data for the 4-mm thick material (Figure 23.8.2c) combined with the data for the 6.35-mm thick material from Figure 23.8.1d. 3.0 4 mm

MODULUS (GPa)

MODULUS (GPa)

3.0

2.5

5% 2.0

4 mm

2.5

15% 2.0

0.8

0.9

1.0

1.1

1.2

0.8

3

0.9

1.0

1.1

1.2 3

FOAM DENSITY (g·cm )

FOAM DENSITY (g·cm )

(a)

(b) 3.0

3.0 4 mm

MODULUS (GPa)

MODULUS (GPa)

886

2.5

4 & 6.35 mm

2.5

2.0

2.0 0.8

0.9

1.0

1.1

1.2 3

0.8

0.9

1.0

1.1

FOAM DENSITY (g·cm )

FOAM DENSITY (g·cm 3)

(c)

(d)

1.2

Figure 23.8.2 Average local tensile modulus versus the local foam density for a specimen cut from 4-mm thick 5% glass-filled PC-SF plaques. (a) Density reduction of 5%, (b) density reduction of 15%, (c) consolidated data for 5 and 15% density reduction foam from Figure 23.7.2a,b, and (d) consolidated data for 4- and 6.35-mm thick structural foam from Figures 23.8.1d and 23.8.2c. (Adapted with permission from V.K. Stokes, Journal of Materials Science, Vol. 35, pp. 159 – 178, 2000.)

The local density for one 4-mm thick, 5%-density-reduction plaque varied from a minimum of 0.97 g cm−3 to a maximum of 1.18 g cm−3 ; the ratio of the maximum to the minimum was 1.22. This 4-mm thick material has a larger variation than that (1.11 – 1.17 g cm−3 ) in the 6.35-mm thick material. The local average tensile modulus EA for this 2-mm thick plaque varies from a minimum of 2.23 GPa to a maximum of 3.10 GPa, resulting in a ratio of maximum to minimum of 1.39. Although (Figure 23.8.2a) EA again appears to correlate linearly with 𝜌, the data are mostly above the solid line EA = 2 𝜌 + 0.35 used to correlate the data for the 6.35-mm.thick material. The local minimum and maximum densities for one 4-mm thick, 15% density reduction plaque are 0.88 and 1.15 g cm−3 , which overlaps the density variation that in the 5% density reduction

Structural Foams – Materials with Millistructure

material (Figure 23.8.2b). Also, this is a much larger local density variation than in the 6.35-mm thick, 15%-density-reduction plaque (0.94 – 1.10 g cm−3 ). The local average tensile modulus EA for this plaque varies from a minimum of 2.02 GPa to a maximum of 2.82 GPa, resulting in a ratio of maximum to minimum of 1.40. Clearly, the average tensile moduli for the 15%-density-reduction material are much closer to EA = 2 𝜌 + 0.35 than is the data for the 5% density reduction material. The consolidated data on variations of the average tensile modulus EA versus the local density 𝜌 for nominal density reductions of 5% and 15% – from Figure 23.8.2a,b. respectively – are shown in Figure 23.8.2c. Just as in 6.35-mm thick plaques, the average modulus appears to correlate linearly with the local density. However, the data for the 4-mm thick material appears to lie above the solid line EA = 2 𝜌 + 0.35, which is a good fit for the 6.35-mm thick material. Figure 23.8.2d shows the combined data from the 4-mm thick (5% and 15% density reduction) and 6.35-mm thick (5, 15, and 25% density reduction) plaques.

23.9 Flexural Modulus After determining the tensile moduli described in the previous sections, the average flexural modulus, EBR , of the same set of specimens was determined by three-point bend tests over 76.2-mm (3-in) spans at three locations along each of the eight specimens, for nominal density reductions of 5, 15, and 25%. In these tests, 76.2-mm long spans of the 12.7-mm wide specimens were centered at the 12.7-mm mark separating two predetermined adjacent segments – the 4-5, 12-13, and 20-21 segment interfaces on each of the eight specimens from each plaque. The arithmetic average of the tensile modulus, ETR , is defined as the average over the 76.2-mm span used in the flexural tests. For 6.35-mm thick plaques, measured values of the average flexural modulus, EBR , over 76.2-mm spans, at three locations – the 4-5, 12-13, and 20-21 segment interfaces – along each of the eight specimens are listed in Table 23.9.1 for nominal density reductions of 5, 15, and 25%. Also listed in this table are values of the arithmetic average of the tensile modulus, ETR , over the 76.2-mm span used in the flexural tests. For each of the three density reductions, the maximum and minimum values of EBR and ETR have been highlighted in bold. The plaque with a 5% nominal density reduction exhibits relatively small variations in EBR along any one specimen, with a maximum and a minimum of 2.84 and 2.62 GPa, respectively. The actual variability is much less than that corresponding to the extreme values. At these extremes, the ratio EBR ∕ETR has the values 1.07 and 0.98, respectively. In this plaque, the highest moduli are not necessarily in the area close to the gate. Surprisingly, in many cases, the highest moduli are in regions farthest from the gate. In the 15% density reduction plaque, the maximum and minimum values of EBR are 3.21 and 2.29 GPa. At these extremes, the ratio EBR ∕ETR has the values 1.30 and 0.99, respectively. In the 25% density reduction plaque, the extreme values of EBR are 2.68 and 2.32 GPa. At these extremes, the ratio EBR ∕ETR has the values 1.14 and 1.09, respectively. At both these nominal density reductions, the moduli are highest near the gated end and generally lowest at the far end of the plaque. Also, in general, for equivalent locations, the moduli are higher in the 15% density reduction plaque. Now the ratio EBR ∕ETR is a measure of how much material is in the outer layers of the material; this ratio is higher for thinner skins. The extreme values of 1.07 and 0.98 for this ratio, both of which are close to unity, indicate very thick skins with a small cellular core. Higher values of this ratio for the 15% and 25% density reduction plaques are consistent with them having thinner skins with thicker cellular cores. Table 23.9.2 lists values of EBR and ETR for 4-mm thick plaques with nominal density reductions of 5 and 15%. The maximum and minimum values of EBR for the 5% density reduction plaque are 2.91 and

887

5

15

25

Mean segment location number

Nominal density reduction (%)

Table 23.9.1 Variations of the average flow-direction flexural and tensile moduli, E BR and E TR , in 6.35-mm thick polycarbonate structural foam plaques, for two nominal density reductions. (Adapted with permission from V.K. Stokes, Journal of Materials Science, Vol. 35, pp. 159 – 178, 2000.)

E BR

E TR

EBR

E TR

E BR

ETR

E BR

E TR

EBR

E TR

E BR

ETR

E BR

E TR

EBR

4-5

2.75

2.64

2.75

2.62

2.75

2.65

2.66

2.62

2.65

2.56

2.65

2.65

2.68

2.63

2.74

2.57

12-13

2.70

2.65

2.75

2.61

2.75

2.70

2.75

2.76

2.62

2.68

2.65

2.69

2.68

2.59

2.75

2.67

20-21

2.75

2.66

2.84

2.65

2.80

2.77

2.75

2.63

2.75

2.62

2.71

2.75

2.75

2.65

2.75

2.74

4-5

2.66

2.53

2.55

2.40

2.62

2.40

2.55

2.36

3.21

2.46

2.66

2.46

2.68

2.44

2.65

2.47

12-13

2.55

2.40

2.59

2.35

2.51

2.42

2.65

2.36

3.14

2.40

2.52

2.39

2.60

2.45

2.52

2.35

20-21

2.38

2.32

2.38

2.24

2.38

2.36

2.29

2.31

3.03

2.27

2.38

2.24

2.38

2.31

2.38

2.21

4-5

2.62

2.32

2.51

2.38

2.52

2.35

2.38

2.35

2.58

2.34

2.56

2.38

2.68

2.35

2.61

2.39

12-13

2.58

2.29

2.58

2.36

2.59

2.25

2.55

2.34

2.58

2.27

2.55

2.30

2.58

2.23

2.58

2.35

20-21

2.38

2.20

2.38

2.27

2.32

2.13

2.38

2.22

2.38

2.21

2.38

2.17

2.38

2.15

2.38

2.16

Average flexural modulus EBR and average tensile modulus ETR (GPa) Specimen number 1

2

3

4

5

6

7

8 E TR

5

15

Mean segment location number

Nominal density reduction (%)

Table 23.9.2 Variations of the average flow-direction flexural and tensile moduli, E BR and E TR , in 4 mm thick polycarbonate structural foam plaques, for three nominal density reductions. (Adapted with permission from V.K. Stokes, Journal of Materials Science, Vol. 35, pp. 159 – 178, 2000.)

E BR

E TR

EBR

E TR

E BR

E TR

E BR

E TR

E BR

ETR

E BR

E TR

E BR

4-5

2.76

2.74

2.87

2.70

2.87

2.81

2.89

2.73

2.73

2.67

2.75

2.65

2.81

2.86

2.91

2.86

12-13

2.79

2.76

2.68

2.76

2.63

2.83

2.60

2.78

2.62

2.72

2.68

2.79

2.49

2.84

2.49

2.79

20-21

2.51

2.66

2.48

2.69

2.34

2.62

2.26

2.57

2.29

2.62

2.34

2.71

2.65

2.64

2.51

2.62

4-5

2.69

2.59

2.54

2.68

2.51

2.59

2.46

2.64

2.55

2.75

2.41

2.69

2.51

2.78

2.75

2.68

12-13

2.62

2.54

2.41

2.58

2.46

2.59

2.62

2.64

2.52

2.66

2.60

2.66

2.58

2.68

2.67

2.65

20-21

2.41

2.32

2.46

2.39

2.25

2.37

2.32

2.36

2.53

2.39

2.57

2.38

2.41

2.38

2.32

2.39

Average flexural modulus EBR and average tensile modulus ETR (GPa) Specimen number 1

2

3

4

5

6

7

8 E TR

E BR

E TR

Introduction to Plastics Engineering

6.35

5

6.35

6.35

4

4

15

25

5

15

Mean segment location number

Nominal density reduction (%)

Table 23.9.3 Ratios E BR ∕E TR of average flexural modulus E BR to average tensile modulus E TR for 6.35- and 4-mm thick polycarbonate structural foam plaques, for three nominal density reductions. (Adapted with permission from V.K. Stokes, Journal of Materials Science, Vol. 35, pp. 159 – 178, 2000.) Specimen thickness (mm)

890

1

2

3

4

5

6

7

8

4-5

1.04

1.05

1.04

1.02

1.04

1.00

1.02

1.07

12-13

1.02

1.05

1.02

1.00

0.98

0.99

1.03

1.03

20-21

1.03

1.07

1.01

1.05

1.05

0.99

1.04

1.00

4-5

1.05

1.06

1.09

1.08

1.30

1.08

1.10

1.07

12-13

1.06

1.10

1.04

1.12

1.31

1.05

1.06

1.07

20-21

1.03

1.06

1.01

0.99

1.33

1.06

1.03

1.08

4-5

1.13

1.05

1.07

1.01

1.10

1.08

1.14

1.09

12-13

1.13

1.09

1.15

1.09

1.14

1.11

1.16

1.10

20-21

1.08

1.05

1.09

1.07

1.08

1.10

1.11

1.10

4-5

1.01

1.06

1.02

1.06

1.02

1.04

0.98

1.02

12-13

1.01

0.97

0.93

0.94

0.96

0.96

0.88

0.89

20-21

0.94

0.92

0.89

0.88

0.87

0.86

1.00

0.96

4-5

1.04

0.95

0.97

0.93

0.93

0.90

0.90

1.03

12-13

1.03

0.93

0.95

0.99

0.95

0.98

0.96

1.01

20-21

1.04

1.03

0.95

0.98

1.06

1.08

1.01

0.97

E BR ∕E TR Specimen number

2.26 GPa, respectively, and the corresponding values of the ratio EBR ∕ETR at these extremes are 1.02 and 0.88. For the higher (15%) density reduction plaque the extreme values of EBR are 2.75 and 2.25 GPa, and the corresponding value of EBR ∕ETR are 1.03 and 0.95. While the extreme values of EBR and the values of EBR ∕ETR at these extremes are on the same order as for the 6.35-mm thick plaque for a density reduction of 5%, this is not true for a density reduction of 15%. Table 23.9.3 lists values of EBR ∕ETR for both 6.35- and 4-mm thick plaques for all the data in Tables 23.9.1 and 23.9.2. While this ratio is larger than unity at most locations, implying that EBR > ETR , there are regions where this ratio is less than unity. For foams with solid skins the expectation is that, locally, EBR > ETR . This ratio being lower than unity could result from two approximations that have been made. First, the flexural moduli are averages measured over a span of 76.2 mm, and therefore could be lower than that corresponding to the local morphology. Second, the tensile modulus used in this ratio is an arithmetic mean over this span. A more representative mean, which would result in lower average tensile moduli and hence in higher values of this ratio, is the harmonic mean.

23.10 **Torsion of Nonhomogeneous Bars Previous sections have considered nonhomogeneous continuum models for structural foams, and have discussed experimental data on the variations of local density, tensile moduli and tensile strength, and

Structural Foams – Materials with Millistructure

bending moduli for such foams. Although test data on torsion of foam bars are not available, nonhomogeneous continuum models for torsion can provide insights into how nonhomogeneity affects torsion. The basic objective is to predict the induced stresses and twist per unit length for a prismatic bar subjected to a twisting moment T. This will be accomplished by first modifying Saint Venant’s theory of torsion to account for spatially varying material properties, and by then applying it to the torsion of thin-walled rectangular bars. 23.10.1

**Basic Equations for Modified Saint Venant’s Theory

Consider the torsion of a long prismatic bar, shown in Figure 23.10.1a, which is subjected to a twisting moment Mz = T about the z-axis. Away from the ends all transverse cross sections are equivalent, so that cross sections may be assumed to rotate about the z-axis at a uniform rate along z, as a result of which a point P on the cross section rotates to the position Q shown in Figure 23.10.1b. Let the constant twist per unit length be 𝜃 0 . Assume that the section at z rotates about the z-axis by an angle 𝜃 0 z. Then, with reference to Figure 23.10.1c, for small rotations 𝜃 0 z the displacements of a typical point P ( x, y) in the x-y plane are given by u = −r z 𝜃0 sin 𝛼 = −yz 𝜃0 v = −r z 𝜃0 cos 𝛼 = x z 𝜃0

(23.10.1)

Because deformations in the axial direction must be independent of z, the z-displacement component has the form w = 𝜃0 𝜓 ( x, y)

(23.10.2)

where 𝜓 ( x, y) is called the warping function.

T

T = Mz

•

•

z

(a) y

Q Q

O

α

P

P: (r, α) P: (x, y)

A r

z θ0

x

α O

(b) Figure 23.10.1 Geometry for the torsion of a prismatic bar.

P y

x (c)

AP = – u AQ = v PQ = rz θ0

891

892

Introduction to Plastics Engineering

For the displacement field described by Eqs. 23.10.1 and 23.10.2, it follows from an extension of the results in Section 3.3 that the only nonzero strain components are ) ( 𝜕𝜓 𝜕u 𝜕w + = 𝜃0 −y 𝛾xz = 𝜕z 𝜕x 𝜕x ) ( 𝜕𝜓 𝜕v 𝜕w + = 𝜃0 +x (23.10.3) 𝛾yz = 𝜕z 𝜕y 𝜕y Let G = G (x, y) be the rigidity modulus relating shear stresses to shear strains for the nonhomogeneous material of the bar. Then the shear stresses are ) ( 𝜕𝜓 −y 𝜏xz = G 𝛾xz = 𝜃0 G( x, y) 𝜕x ) ( 𝜕𝜓 +x (23.10.4) 𝜏yz = G𝛾yz = 𝜃0 G( x, y) 𝜕y which are independent of z. For this stress distribution the equilibrium equations reduce to

𝜕 𝜏xz 𝜕 𝜏yz + =0 𝜕x 𝜕y

(23.10.5)

A substitution from Eq. 23.10.4 in Eq. 23.10.5 results in the equilibrium equation ) ( ) ( 𝜕𝜓 𝜕𝜓 𝜕G 𝜕G G ( x, y) ∇2 𝜓 + −y + +x =0 𝜕x 𝜕x 𝜕y 𝜕y

(23.10.6)

The equilibrium equation in Eq. 23.10.5 has the general solution

𝜏xz =

𝜕Φ , 𝜕y

𝜏yz = −

𝜕Φ 𝜕x

(23.10.7)

where Φ is called Saint Venant’s function for torsion. It has the following properties: (i) The derivative of Φ in any direction in the x-y plane gives the shear stress at right angles to that direction in a clockwise sense. Since the shear stress at the boundary must be parallel to the boundary, it follows that ΦB is constant along the boundary. (ii) If Φ is chosen to be zero at the boundary, then the torque is twice the volume under the Φ = Φ (x, y) surface; that is T = Mz = 2

∫A

Φ dA,

ΦB = 0

From Eqs. 23.10.4 and 23.10.5 ) ( 𝜕𝜓 𝜕Φ = G𝜃0 −y 𝜕y 𝜕x ) ( 𝜕𝜓 𝜕Φ = − G𝜃0 +x 𝜕x 𝜕y Differentiation and addition of these two equations results in the compatibility equation ) ) ( ( 𝜕𝜓 𝜕𝜓 𝜕G 𝜕G 2 +x + 𝜃0 −y ∇ Φ = −2 G𝜃0 + 𝜃0 𝜕y 𝜕x 𝜕x 𝜕y

(23.10.8)

(23.10.9)

(23.10.10)

Structural Foams – Materials with Millistructure

In summary, for nonhomogeneous materials, the displacement field is given by Eqs. 23.20.1 and 23.20.2; the nonzero strains and stress components by Eqs. 23.10.3 and 23.10.4, respectively; and the equilibrium equations by Eq. 23.10.5 or, equivalently by Eq. 23.10.6. The stress function defined in Eq. 23.20.7 satisfies the compatibility equation (Eq. 23.10.10). Finally, the torque T is determined in terms of Φ by Eq. 23.10.8. 23.10.2

**Torsion of Thin-Walled Rectangular Bars

Consider a thin-walled rectangular bar of thickness t and width b, as shown in Figure 23.10.2. The strain and stress fields for this geometry are given, respectively, in Eqs. 23.10.3 and 23.10.4. If G = G ( y), and it is assumed that b ≫ t, then, since all sections parallel to the y-axis – except those near the ends at x = ± b∕2 – are equivalent, the contours of the shear stress 𝜏 will be parallel to the x-axis, except near the ends. Thus, except near the ends, 𝜏 yz ≡ 0, so that from Eq. 23.10.3b

𝜕𝜓 + x = 0 or 𝜓 = − x y + f ( x) 𝜕y

(23.10.11)

where f(x) is an arbitrary function of x. A substitution of this expression in Eq. 23.20.4a gives

𝜏xz = 𝜃0 G ( y) [− 2y + f ′(x)]

(23.10.12)

y

x

t b Figure 23.10.2 Cross section of a thin rectangular bar.

Since 𝜏 xz should not be a function of x except at the ends, f ′ (x) = 0, so that f(x) = constant, which must be zero because, by symmetry, 𝜏 xz = 0 on y = 0. Thus, except near the ends at x = ± b∕2,

𝜓 = −x y

(23.10.13)

𝜏 = 𝜏xz = −2G ( y) y

(23.10.14)

and

From Eq. 23.10.7

𝜏xz =

𝜕Φ = −2G ( y) y 𝜕y

which results in Φ = −2𝜃0

y

∫0

G (𝜉 ) 𝜉 d𝜉 + C1

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where C1 is a constant of integration. The boundary condition Φ = 0 at y = t∕2 implies C1 = 2𝜃0

t∕2

G (𝜉 ) 𝜉 d𝜉

∫0

Thus, except near the ends, Φ = 2𝜃0

t∕2

G (𝜉 ) 𝜉 d𝜉

∫y

(23.10.15)

Let A be the area under the curve Φ = Φ (y). Then t∕2

A=2

Φ dy = 4𝜃0

∫0

t∕2

t∕2

∫0 ∫y

G (𝜉 ) 𝜉 d𝜉 dy

which simplifies to A = 4𝜃0

t∕2

G ( y) y2 dy

∫0

(23.10.16)

The volume V under the Φ surface is then V = Ab, so that the torque T, which is twice this volume, is given by T = 8b 𝜃0

t∕2

G ( y) y2 dy

∫0

(23.10.17)

or, with 𝜂 = 2y∕t, T = bt3 𝜃0

1

∫0

G (𝜂 ) 𝜂 2 d𝜂

(23.10.18)

Now for a homogeneous rectangular bar

𝜏 T = = G 𝜃0 J y

(23.10.19)

where J = bt3∕3. If this formula is used to interpret the results of a torsion test on a nonhomogeneous rectangular bar, then 1

T T = 3𝜃0 G (𝜂 ) 𝜂 2 d𝜂 = GTR 𝜃0 = 3 ∫0 J (bt ∕3)

(23.10.20)

where t∕2

GTR =

∫0 G ( y) y2 dy t∕2

∫0

y2 dy

1

=3

∫0

G (𝜂 ) 𝜂 2 d𝜂

is the average shear modulus that would be determined in such a test.

(23.10.21)

Structural Foams – Materials with Millistructure

23.10.3

**Torsion of Thin-Walled Open Prismatic Sections

As long as the curvature of a thin-walled open section is very large in comparison to its thickness, the equations of Section 23.10.2 may be applied locally. The large-curvature assumption implies that, except at points where the section undergoes a sudden change in direction such as at a bend, it may be considered as a thin rectangular section in which the width b is the perimeter of the section. Thus, with the y-axis normal to the thickness, the shear stress is given by Eq. 23.10.20, in which b is the perimeter of the section. As examples, the effective or equivalent width b for an angle section and a thin-walled circular tube with a slit are shown in Figure 5.5.2. As discussed in Section 5.5, the torsional rigidity of an open section is much smaller than that of an equivalent closed tube. 23.10.4

**Torsion of Thin-Walled Tubes

Consider the thin-walled prismatic tube of arbitrary cross section shown in Figure 23.10.3a, in which the coordinates s and 𝜂 are measured, respectively, parallel and normal to the periphery. The shear stress 𝜏 (s, 𝜂 ) must be parallel to the boundaries at 𝜂 = 0 and 𝜂 = t, where t = t(s) is the tube thickness. Because the thickness t is assumed to be small, the shear stress will generate a tangential force at the periphery.

s

s=0

T (a)

qls + ts +

∂t ds ∂s

τ

τ ds

t

∂(ql) ds ∂s

qds

ql

qds

η (b)

Figure 23.10.3 Forces on an element of a thin-walled tube subjected to torsion.

(c)

l

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Let the force per unit length along s be q(s), so that t(s)

q(s) =

∫q = 0

𝜏 (s, 𝜂 ) d𝜂

(23.10.22)

Then, by considering the equilibrium of the forces acting on the element of the tube shown in Figure 23.10.3c, it follows that ] [ 𝜕 (ql) ds = 0 ql − ql + 𝜕s that is

𝜕q = 0 or q = q0 = constant 𝜕s

(23.10.23)

t(s) q0 1 𝜏 (s, 𝜂 ) d𝜂 = t (s) t (s) ∫𝜂 = 0

(23.10.24)

Equation 23.10.22 may then be used for defining an average shear stress 𝜏 av through

𝜏av =

With reference to Figure 23.10.4, the torque dT generated by the shear stress acting on an element, ds, of the tube is dT = n dF = nq0 ds, or, since n ds = 2 dA, dA is the area of the shaded triangle, dT = 2 q0 dA. It follows that T = Mz = 2Aq0

(23.10.25)

where A is the cross-sectional area enclosed by the tube.

dF = q ds dA

n T = Mz

Figure 23.10.4 Torque due to shear stresses in a thin-walled tube.

Let G = G (s, 𝜂 ) be the modulus of rigidity of the nonhomogeneous material comprising the tube. Then

𝜏 (s, 𝜂 ) = G (s, 𝜂 ) 𝛾 (s, 𝜂 ) t(s)

q0 =

∫0

G (s, 𝜂 ) 𝛾 (s, 𝜂 ) d𝜂

(23.10.26) (23.10.27)

Structural Foams – Materials with Millistructure

Since the tube thickness is small, the shear strain 𝛾 (s, 𝜂 ) may be assumed to be constant across the thickness, that is, 𝛾 (s, 𝜂 ) = 𝛾 (s). Then q0 = 𝛾 (s)

t (s)

∫0

G (s, 𝜂 ) d𝜂

(23.10.28)

so that from Eq. 23.10.24

𝜏av = 𝛾 (s)

t (s)

1 𝜏 (s, 𝜂 ) d𝜂 t(s) ∫𝜂 = 0

(23.10.29)

or

𝜏av = Gav (s) 𝛾 (s)

(23.10.30)

where t (s)

Gav (s) =

1 G (s, 𝜂 ) d𝜂 t (s) ∫0

(23.10.31)

is a thickness averaged shear modulus that varies along the tube periphery. The next step is to relate the toque, T, to the twist, 𝜃 0 , per unit length of the tube. The strain energy density per unit volume of the tube is 𝜏 2 (s, 𝜂 )∕2G (s, 𝜂 ) = G (s, 𝜂 ) 𝛾 2 ( s, 𝜂 )∕2. Consider a unit length of the tube. An element of the tube at (s, 𝜂 ) of length ds and thickness d𝜂 has the volume dV = ds d𝜂 , so that the strain energy in an element ds × t × 1 is then t(s)

ds

∫0

1 G (s, 𝜂 ) 𝛾 2 (s) d𝜂 2

It follows that the total strain energy per unit length of the tube is given by ] [ t (s) 1 2 G (s, 𝜂 ) 𝛾 (s) d𝜂 ds U = ∮ ∫0 2 [ t (s) ] 1 2 𝛾 (s) G (s, 𝜂 ) d𝜂 ds = ∫0 2∮ Use of Eqs. 23.10.28, 23.10.31, and 23.10.24 then gives q0 2 1 T2 1 ds = ds 2 ∮ ∮ 2 t Gav t Gav 8A

U=

(23.10.32)

Since this strain energy must equal the work T𝜃 0 ∕2 done by the torque T, it follows that

𝜃0 =

1 T2 ds 4 A2 ∮ t Ga

(23.10.33)

Now for a homogeneous tube, the 𝜃 0 -T relation is written as 𝜃 0 = T∕GJ, where J=

4A2 ∮ (1∕t) ds

(23.10.34)

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Thus, if the nonhomogeneous tube results are interpreted by a homogeneous-tube type equation 𝜃 0 = T∕JGTW , then the effective thin-wall shear modulus GTW is given by GTW =

∮ (1∕t) ds

(23.10.35)

∮ [1∕(t Gav )] ds

When Gav (s) in Eq. 23.10.31 is independent of s, which could conceivably be true for a tube of constant thickness, then GTW ≡ Gav . For a thin-walled circular tube of constant thickness t, Gav is constant, and the effective shear modulus, GTC , is given by GTC = GTW = Gav . In summary, the torsion of thin-walled tubes is governed by the following equations: t(s)

q0 =

∫0

𝜏 (s, 𝜂 ) d𝜂 = constant

T = 2Aq0

𝜃0 =

T JGTW

where GTW =

∮ (1∕t) ds ∮ [1∕(t Gav )] ds

t (s)

,

Gav (s) =

1 G (s, 𝜂 ) d𝜂 t (s) ∫0

For a constant thickness tube, for which Gav may be assumed to be independent of s, GTW = Gav = GTC , where GTC is the shear modulus for a thin-walled circular tube.

23.11 Implications for Mechanical Design Clearly, in comparison to design with homogeneous materials, mechanical design with structural foams is conceptually more complex because, even in the simplest models in which the foams are treated as continua, nonhomogeneity of the material must be considered. For such materials, in contrast to constant elastic moduli of homogeneous materials, the elastic moduli actually vary through the part thickness; constant elastic moduli are replaced by through-thickness varying functions. For such nonhomogeneous materials tensile and bending tests do not determine true material properties; instead, they determine weighted averages of the through-thickness modulus variation. For foams, in which the moduli of the outer skin layers are larger than for the inner core regions, the effective tensile modulus will be smaller than the bending, or flexural, modulus. Analyses have shown that – except for near flat parts in which the stiffness is measured by the bending modulus – the effective stiffness of a part is better defined by using the effective tensile modulus for determining the part stiffness. Experiments have shown that data acquired from different molded plaques are consistent, and that while properties vary along a part, the elastic moduli and tensile strength correlate linearly with the local density. Moreover, these correlations are not very sensitive to the part thickness. The stiffness of a geometrically complex foam part can be determined at different levels of accuracy. In the simplest analysis treat the part as being made of a homogeneous material that has an elastic modulus equal to the measured effective tensile modulus ETR . If the variation of the average density across the

Structural Foams – Materials with Millistructure

part is known, the density-modulus correlation can be used to allow ETR to vary in a prescribed manner. The tensile-strength-density correlation can then be used to design for strength. In the next level of analysis, measured values of ETR , EBR , the skin thickness, and the modulus of the resin are used to define an appropriate skin-core model, which can then be used for a more accurate determination of the stresses in the part and its stiffness. This procedure again requires knowledge of the variation of the local density across the part. These procedures form the basis for the mechanical design of structural foam components. Unfortunately, such stiffness and strength prediction depend on the local density distribution being known, methods for predicting which in a part are currently not available.

23.12 Concluding Remarks Clearly, structural foams are complex heterogeneous (mixture of resin and small, air-filled cavities), nonhomogeneous (properties vary over the part – both along and across the part thickness) materials. Because of the way foam components are made, as a first approximation the material may be assumed to be nonhomogeneous across the part thickness, and the nonhomogeneity in the length direction may be neglected. In the simplest model for capturing the nonhomogeneity, the material is treated as a nonhomogeneous continuum in which each material property, such as the tensile modulus, varies across the thickness. By applying mechanics principle to bars and beams, the effective tensile and bending moduli – those corresponding to moduli that would be calculated from tensile and bending tests by assuming that the material is homogeneous – can be determined in terms of assumed through-thickness property variations. Such mechanics-based analyses have shown that the effective moduli of nonhomogeneous bars and beams depend on strongly coupled combinations of geometry and material properties. For example, the effective tensile modulus, ETR , and the effective bending modulus, EBR , given by 1

ETR =

∫0

E (𝜂 ) d𝜂 ,

1

EBR = 3

∫0

𝜂 2 E (𝜂 ) d𝜂

are, respectively, the average and the second moments of the through-thickness variation of the modulus E (𝜂 ). When the modulus E (𝜂 ) is larger near the skin (near 𝜂 = 1) than in the core (near 𝜂 = 0), the effective bending modulus EBR is larger than the effective tensile modulus ETR , a clear indication that these moduli are not true local material properties. For a homogeneous material, for which E (𝜂 ) ≡ E0 , it follows that ETR = EBR = E0 . For the shear modulus varying as G (𝜂 ) across the thickness, the effective shear moduli for a thin-walled circular tube and a thin-walled rectangular bar, GTC and GTR , are given, respectively, by 1

GTC =

∫0

G (𝜂 ) d𝜂 ,

1

GTR = 3

∫0

𝜂 2 G (𝜂 ) d𝜂

It is these types of averages – which are not true material properties – that are determined by normal tensile, bending, and torsion tests. Such measured properties can be used to develop models for the variations of E = E (𝜂 ) and G = G (𝜂 ) that can then be used to determine the stiffness of geometrically complex parts by numerical means, such as by using finite element codes. While structural foam parts are normally specified in terms of a nominal density reduction, the data reported in this chapter show that the actual local density in a part can be very different, and may vary significantly across the part. Also, even in parts of the same thickness, the local density at some location

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in a part with one nominal density reduction may be the same as at some other location in a second part with a different nominal density reduction. Furthermore, the local density can depend on the part thickness. Data on two structural foams discussed in this chapter were obtained from two different sets of tests: In the first set, density, tensile and flexural moduli, and tensile strength are determined by tests on 25.4 × 152.5 mm (1 × 6 in) cross-flow bars cut from plaques of 6.35 and 4 mm (0.25 and 0.157-in) thickness of molded M-PPO structural foam. These data show that while the tensile moduli and the flexural strength correlate linearly with the average density of the bar – higher densities correspond to higher moduli and strengths – the data for foams exhibit far more scatter than in equivalent tests on the corresponding homogeneous resin. In the second set of tests, the local tensile modulus of PC-SF was correlated to the average density measured on a 12.7 × 12.7-mm scale. The moduli from 6.35-mm-plaques with nominal density reduction of 15 and 25%, which have regions with overlapping local density reductions, are indistinguishable in regions with common densities, so that for parts of the same thickness the local elastic modulus is determined when the local density is known. The data from 4-mm thick plaques with nominal density reductions of 5 and 15% also exhibit the similar trends. However, while the modulus-density correlation is still linear, the correlation for the thicker plaques appears to underestimate the local modulus at higher densities (lower density reductions). Structural foams normally are used at nominal density reductions higher than 10%, for which the linear dependence of the average modulus on the local density is essentially the same for the 6.35- and 4-mm thick materials. The data discussed in this chapter have established a strong linear correlation between the local modulus and the local density as measured in the plaque molding flow direction. Because the shear field can affect the bubble shape, the cellular morphology can be different in the flow and cross-flow directions. These differences can result in material anisotropy that needs to be characterized. However, the dependence of the cross-flow modulus on the local density is expected to be linear. A complete characterization of the elastic properties also requires a determination of the local Poisson’s ratio and the local shear modulus. Also, the effect of the inherent variability in the modulus of these materials at all densities on the stiffness of parts needs to be quantified. The empirical elastic-modulus density correlation provides the basis for a rational, FEA based mechanical design of structural foam parts. Once the local density distribution in the part geometry has been determined (predicted), the correlation determines the local elastic properties for the FEA. However, although some attempts have been made at predicting the local density of foams, procedures for predicting the density distribution in parts of complex geometry are not presently available. As such, robust procedures for predicting the stiffness and strength of structural foam parts are not available. On the basis of an analysis of the stiffness of thin-walled nonhomogeneous structures, a safe estimate for the stiffness of a foam part can be obtained by a FEA of the structure by assuming it to be made of a homogeneous material having a constant modulus ETR . Even this estimate requires the definition of the foam density corresponding to which an appropriate ETR is chosen.

901

24 Random Glass Mat Composites – Materials with Macrostructure 24.1 Introduction An important class of random glass mat reinforced thermoplastics (GMTs), also called random continuous glass fiber thermoplastic composites, comes in the form of thermoplastic sheets containing nonwoven, random continuous glass fibers. Since the randomly laid glass fibers can uncoil with increasing deformation, heated sheets of this material can undergo large stretches without glass breakage. Blanks of this material can therefore be thermostamped (compression molded or flow formed in dies), at relatively low pressures, to form complex parts with deep-drawn regions, bosses, and ribs. These advantages make this class of materials – typified by Azdel® P100 (Azdel PM 10400-101) composite sheet of 40 wt% random glass mat in a polypropylene matrix – ideally suited for large, stiff, low-cost parts that have high impact strength. GMTs are made by continuously feeding nonwoven, random continuous glass fiber mat, together with molten thermoplastics, such as polypropylene, between two moving metallic belts that are heated and pressurized to maintain a uniform gap. Under the pressure imposed by the belts, the heated resin flows into the random glass mat. The belt then moves into a non-pressurized, unheated region in which the resin-impregnated sheet cools, and then exits the machine as a continuous plate. The width of this plate, which could be as much as 1.2 m (4 ft), is then cut into desired lengths. These rectangular sheets are the materials used in subsequent part forming operations. The process for making GMTs induces a preferential alignment of the glass mat fibers along moving direction – called the machine direction – resulting in different material properties along the machine and cross-machine directions. In an another class of GMTs long chopped glass fibers – on the order of 25 mm (1 in) – are used in place of continuous fibers. Figure 24.1.1 shows a good example of the use of these materials, a GMT load-floor (trunk liner) for a small station wagon.

24.2 GMT Processing GMT parts are formed by placing hearted blanks of the GMT material inside matched molds in a predetermined pattern and thermostamped. In drape molding, the heated blank covers the entire mold area, so that there is little or no lateral flow of the material during the stamping operation and the morphology of the glass mat in the part is essentially the same as that of the original blank. But in many applications Introduction to Plastics Engineering, First Edition. Vijay K. Stokes. © 2020 John Wiley & Sons Ltd. This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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Figure 24.1.1 Thermostamped GMT load-floor (trunk liner) for a small station wagon. The white object in the foreground is 152-mm (6 in) long. (Photo courtesy of SABIC).

the heated blank does not cover the entire mold. Depending on the shape of the part and on the desired material morphology, blanks of different shapes can be placed in different parts of the mold. The stamping charge can also consist of stacks of heated blanks that are nonuniformly distributed over the mold. In such cases the material undergoes a substantial amount of lateral flow during the stamping operation; the moving molten resin “drags” the glass along causing it to be redistributed. This process, in which the glass is said to have flowed, can change the local morphology of the glass, which in turn can result in a significant change in the mechanical properties of the material. Figure 24.2.1 shows radiographs of two 230 × 405-mm plaques that were heated and placed in a mold of the same dimension and then compression molded in the same way as an actual part would be “drape-molded.” In this figure the machine direction is aligned along the length of the plaque. Thus, specimens cut from such plaques would have properties representative of actual drape-molded parts in which the material does not flow laterally during forming; the glass mat in the material just gets compressed. In these radiographs the glass mat shows up as a white “random” network of fibers; the background material in gray is the resin, polypropylene in this case. Figure 24.2.1a clearly shows that the glass is distributed on a macroscale – “cells” on a scale of about of about 10 mm can be seen. Besides the macro structures in the glass network, this figure also shows the random nature of the network. Figure 24.2.1b shows the radiograph of another drape-molded plaque of the same material. Clearly, the glass network pattern looks somewhat different. Thus, in addition to the macrostructure, the glass network patterns appear to be randomly distributed. In both these radiographs, no preferential orientation in the flow and cross-flow directions can be discerned. In each of these two figures the radiograph appears in two halves because the entire plaque could not be accommodated in a single exposure. Figure 24.2.2 shows radiographs of two 230 × 405-mm plaques in each of which two 115 × 405-mm heated blanks were placed in the center of a 230 × 405-mm mold and then compression molded to form a plaque of the original thickness of each blank. Notice the glass orientation induced by the flow of glass fibers. Here again, the machine direction is aligned along the length of the plaque. Clearly, the lateral flow preferentially aligns glass fibers in the lateral direction, which causes material anisotropy. Although lateral glass alignment can be observed in the central core, it remains glass rich, so that in addition to

Random Glass Mat Composites – Materials with Macrostructure

25 mm

(a)

25 mm

25 mm

(b)

Figure 24.2.1 Radiographs showing the random distribution of continuous glass fibers in two 230 × 405-mm drape-molded Azdel plaques. (Photos courtesy of SABIC.)

25 mm

(a)

25 mm

25 mm

(b)

Figure 24.2.2 Radiographs showing the random distribution of continuous glass fibers in two 230 × 405-mm Azdel plaques in each of which two 115 × 405-mm heated blanks were placed in the center and then compression molded. Notice the glass orientation induced by the flow of glass fibers. (Photos courtesy of SABIC.)

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anisotropy, the material becomes nonhomogeneous. The 10-mm, macroscale glass distribution “cells” seen in Figure 24.2.1 appear to have lost their prominence. Figure 24.2.3 shows radiographs of two 230 × 405-mm plaques in each of which three 77 × 405-mm heated blanks were placed in the center of a 230 × 405-mm mold and then compression molded to form a plaque of the original thickness of each blank. Notice the glass orientation induced by the flow of glass fibers. Here again, the machine direction is aligned along the length of the plaque. In this case the large lateral flow of the material preferentially aligns glass fibers in the lateral direction, resulting in large amounts of anisotropy and nonhomogeneity. Although lateral glass alignment can be observed in the central core, it remains glass rich. Here again, the 10-mm, macroscale glass distribution “cells” seen in Figure 24.2.1 have lost their prominence. 25 mm

(a)

25 mm

25 mm

(b)

Figure 24.2.3 Radiographs showing the random distribution of continuous glass fibers in two 230 × 405-mm Azdel plaques in each of which three 77 × 405-mm heated blanks were placed in the center and then compression molded. Notice the pronounced glass orientation induced by the flow of glass fibers. (Photos courtesy of SABIC.)

24.3 Problem Complexity The nonuniform distribution of glass in GMT, which makes this material thermostampable, causes the material to be both anisotropic and nonhomogeneous. As a result, elastic moduli measured by standard ASTM tests show a significant amount of scatter, and the results appear to depend on the specimen size. Also, the flow caused by thermostamping adds to the nonhomogeneity and anisotropy of the material. This behavior of these materials raises interesting questions regarding their use in load-bearing applications: How nonhomogeneous and anisotropic are the mechanical properties? How are structural parts

Random Glass Mat Composites – Materials with Macrostructure

to be designed for stiffness and strength? How does processing, during which the glass is redistributed by the moving molten resin, affect properties? The scatter in the tensile modulus of drape-molded Azdel sheet material was characterized as follows: First, 24 12.7-mm (0.5-in) strips were marked with a pen on the 3.68-mm (0.145-in) thick drape-molded plaque along the cross-machine direction, as shown by dashed lines in Figure 24.3.1. Twelve 12.7-mm strips were then cut from the plaque along the machine direction, as indicated by the solid lines, resulting in 12 12.7 × 230-mm (0.5 × 9-in) specimens with cross-length markers at 12.7-mm intervals, delineating 24 12.7-mm segments. The numbering system for the specimens and the 12.7-mm segments is indicated in Figure 24.3.1. The specimen numbers increase from 1 at the top to 12 at the bottom. The 12.7-mm segments always start with the number 1 at the left and end with the number 24 at the right. 405 mm (16 in)

230 mm (9 in)

1

12.7 mm (0.5 in)

SPECIMEN NUMBER

12.7 mm (0.5 in)

12

1

24 SEGMENT NUMBER

Figure 24.3.1 Layout of 12.7-mm wide specimens cut from a 230 × 405-mm Azdel plaque. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

Tensile tests were then used to determine the tensile modulus of the 12.7 × 405 × 3.68-mm (0.5 × 16 × 0.145-in) strips at 12.7-mm (0.5-in) intervals along the 405-mm length by attaching an extensometer along the left and right (thickness) edge, as indicated by the letters L and R, respectively, in Figure 24.3.2. With the extensometer attached to the right edge of segment 1, the specimen was pulled in tension under strain control, at a strain rate of 10−2 s−1 , to a strain of 0.5%, and the load-strain behavior was recorded, which, by using the cross-sectional area of the specimen, was used to obtain the stress-strain curve, from which the right elastic modulus was obtained from ER = 𝜎 ∕ 𝜀R . After unloading, the extensometer was moved down to the nest 12.7-mm segment, and the test procedure was repeated. In this way the right elastic modulus, ER , was determined at every 12.7-mm interval along a strip. Similarly, by attaching the extensometer along the left edge, the left elastic modulus, EL , was also determined along the strip. These measurements at every 12.7-mm interval along 12, 12.7 × 405-mm strips cut from a 203 × 405-mm (9 × 16-in), 3.68-mm (0.145-in) thick plaque, exhibited rather unexpected patterns: The Young’s

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EXTENSOMETER

L

R

1

2

3

12.7×12.7-mm TEST AREA

TEST SPECIMEN

Figure 24.3.2 Configurations of extensometer for measuring longitudinal strains in a tensile test. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

modulus along one edge (left or right) of a 12.7-mm wide strip could vary by a factor of two over a 12.7-mm distance. The maximum and minimum values over a 150 × 305-mm (6 × 12-in) area were found to be 9.73 GPa (1, 441 × 103 psi) and 3.14 GPa (455 × 103 psi), respectively, a variation by a factor of 3.1. The variation in the modulus appeared to have a random character with no obvious discernible pattern. Furthermore, the strains on the two edges of each 12.7-mm segment were very different and appeared to be uncorrelated. This phenomenon of the macroscale variation of the Young’s modulus raises a number of interesting questions: First, if the modulus varies by a factor of two over a 12.7-mm length scale, what property does the extensometer measure? What average property is measured in a flexural test? Can the flexural properties be correlated with the tensile properties? Second, does the modulus distribution correlate over distance? How repeatable are these data? Third, how can some “standard” properties be defined for this class of materials? And fourth, how are these properties to be used for mechanical design?

24.4 Effective Tensile and Flexural Moduli of Nonhomogeneous Materials To understand the effects of property variations with distance on quantities measured over a finite gauge length, in this section the elastic modulus along a test specimen will be assumed to vary continuously along the length. 24.4.1

Tensile Test

Consider the rectangular bar shown in Figure 24.4.1, in which the Young’s modulus varies as E = E ( x) along the length x-direction. Then, for this one-dimensional geometry, the x-direction strain 𝜀x

Random Glass Mat Composites – Materials with Macrostructure

E = E (x)

x l x0

x0 + l

Figure 24.4.1 Geometry for the extension of a nonhomogeneous rectangular bar.

is related to the stress 𝜎 x through

𝜀x =

𝜎x du = dx E ( x)

(24.4.1)

where u = u( x) is the x-direction displacement. It follows that the extension of the bar between x = x0 and x = x0 + l is given by Δu = u( x0 + l ) − u(x0 ) xo +l

=

∫x o

xo +l

du =

∫x o

𝜎x E ( x)

dx

For this uniform-cross-sectioned bar, which is being pulled by a tensile force, the constant tensile stress will be 𝜎 x = 𝜎 0 . The average strain, 𝜀x , over a gauge length l is then

𝜀0 =

Δu 𝜎0 o = l l ∫x o

x +l

dx E ( x)

It follows that the effective or average elastic modulus, ET , over a length, l, defined by ET = 𝜎 0 ∕𝜀0 , is given by x +l

1 1 o = ET l ∫x o

dx E ( x)

(24.4.2)

By choosing x0 as the origin, and normalizing the distance as 𝜉 = x∕l, the expression for ET reduces to 1 d𝜉 x 1 , 𝜉= = ∫ ET l 0 E (𝜉 )

(24.4.3)

Thus, the effective elastic modulus is the harmonic mean of E = E ( x) over the gauge length, and not the arithmetic mean. Since the harmonic mean of any set of numbers is smaller than their arithmetic mean, the effective tensile modulus will be lower than the average (arithmetic mean) modulus. The harmonic mean tends to weight the mean toward the smaller values in a sample, so that the difference between the harmonic and arithmetic means gives an indication of the fluctuation in the values – larger differences in these means correspond to larger fluctuations in the sample values.

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Introduction to Plastics Engineering

24.4.2

Three-Point Flexural Test

In a flexural or bending test, a beam on two supports is bent by a central load and the resulting deflection is used to calculate the flexural or bending modulus. Consider the centrally loaded beam ABC of length l shown in Figure 24.4.2. Let the y-direction deflection be v ( x). It then follows from the results in Section 5.3 that small deflections of the beam are governed by the Euler–Bernoulli relation E ( x) I ( x)

(24.4.4)

P

y

v

•

d2 v = Mz ( x) dx2

A

B l /2

l /2

• • η=0

x

C

l

η = 2x /l η=1

η=2

Pl /4

Mz

Figure 24.4.2 Geometry for the bending of a nonhomogeneous beam in a three-point flexural test.

where E ( x) is the tensile modulus, both in tension and compression, at x, I ( x) is the second moment of area, and Mz ( x) the bending moment. The deflection v = v( x) of this beam is obtained by integrating this differential equation subject to the boundary conditions v(0) = 0 and v (l) = 0. Now the bending moment is given by Mz = Px∕2 in the region 0 ≤ x ≤ l∕2 and by Mz = P (l − x)∕2 in the region l∕2 ≤ x ≤ l. As a result, Eq. 24.4.4 has to be integrated separately in the two regions. The final solution is obtained by matching the values of v (l∕2) and v ′′ (l∕2) obtained from the individual solutions for the two regions. It can be shown that the deflection, 𝛿 = v(l∕2), under the load is given by [ ] l∕2 P 1 1 2 𝛿= x + dx (24.4.5) 4I ∫0 E ( x) E(l − x) where the second moment of area I is constant for bend tests on rectangular beams. If the effective flexural or bending modulus, EB , is defined by the homogeneous beam bending formula 𝛿 = Pl3 ∕48EB I, then [ ] l∕2 12 1 1 1 2 = 3 x + dx EB E ( x) E(l − x) l ∫0 [ ] 1 2x 1 1 3 2 𝜂 + (24.4.6) d𝜂 , 𝜂 = = 2 ∫0 E(𝜂 ) E(2 − 𝜂 ) l

Random Glass Mat Composites – Materials with Macrostructure

For a variable modulus E = E ( x), a comparison of Eqs. 24.4.3 and 24.4.6 shows that the effective moduli ET and EB are different.

24.5 Insights from Model Materials Clearly, in a material in which the mechanical properties, such as the tensile modulus, are not constant along the material, measured properties will depend on the gauge length – the distance over which the property is measured. For example, in Section 24.4.1 it has been shown that a tensile test for such a material actually measures the harmonic mean of the tensile modulus variations over the gauge length. An assessment of how this harmonic mean is affected by the underlying modulus variation requires a well-defined modulus variation. Two such simple models are considered in the following sections. 24.5.1

Model Material with Sinusoidally Varying Modulus

The simplest model for a varying tensile modulus along a bar is the sinusoidal variation shown in Figure 24.5.1 in which the modulus varies as [ ] 2𝜋 E ( x) = E0 1 + e0 sin (x − x0 ) (24.5.1)

𝜆

TENSILE MODULUS E = E(x) / E0

where 𝜆 is the wavelength of the sinusoid, e0 is its amplitude, and x0 is a measure of the phase such that 0 ≤ x0 ≤ 𝜆.

λ e0 1.0 e0

0 0

x0

x

Figure 24.5.1 Model for sinusoidal variation of the tensile modulus. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

In the following, this modulus variation will be used to assess how the tensile and bending moduli are affected by the parameters that define this variation. 24.5.1.1 Effective Tensile Modulus

By using the normalized variables 𝜉 = x∕l, 𝜂 = 2x∕l, n = l∕𝜆, and 𝜓 0 = x0 ∕𝜆, it can be shown by integrating Eq. 24.4.3 that the effective tensile modulus for a gauge length l is given by

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Introduction to Plastics Engineering

E0 1 = ET n (1 − e20 )1∕2

1 = n (1 − e20 )1∕2

arctan

e0 + tan (n − (1 −

0)

e20 )1∕2

− arctan

(1 − e20 )1∕2 tan n 1 − e0

tan 0 − tan (n − 0 ) 1 − tan 0 tan (n − 0 )

e0 − tan

0

(1 − e20 )1∕2

(24.5.2)

For integer values of n it can be shown from this equation that ET = (1 − e20 )1∕2 , for integer n E0

(24.5.3)

Next consider the case with 𝜓 0 = 0 when n is not an integer. Let n = N + n0 , where N is an integer such that ≤ n0 ≤ 1. Then it can be shown that 1 + n0 ∕N ET (1 − e20 )1∕2 tan n0 = (1 − e20 )1∕2 1 E0 arctan 1− N 1 + e0 tan n0

(24.5.4)

This equation shows that ET ∕E0 approaches (1 − e20 )1∕2 for large values of N. Thus, for large values of n = l∕𝜆, the values of ET will converge to E0 (1 − e20 )1∕2 , which is the value for integer n. Notice that the effective modulus ET , which is the harmonic mean of E = E ( x), is smaller than the arithmetic mean, which for large n converges to unity. The variations of ET ∕E0 versus n, for a phase fraction 𝜓 0 = 0, with e0 as parameter are shown in Figure 24.5.2. Note that n is plotted on a logarithmic scale. For an infinitesimally small gauge length, that is, for l ≪ 𝜆 or n ≪ 1, the mean modulus would equal E ( x)∕E0 at x = 0, which is 1 for 𝜓 0 = 0. ET ∕E0 then begins to increase with n. However, at n = 0.1, ET ∕E0 is not too much larger than unity. After increasing to a maximum, which increases with the value of e0 , ET ∕E0 falls to (1 − e20 )1∕2 , which is shown by solid straight lines, at n = 1. For n > 1, ET ∕E0 oscillates above (1 − e20 )1∕2 for all integer values of n. Of course, ET ∕E0 ≡ (1 − e20 )1∕2 for all integer values of n. The variations are more pronounced for larger values of integer values of e0 . The effects of varying the phase shift on the tensile modulus are shown in Figure 24.5.3 for three phase shifts of 𝜓 0 = x0 ∕𝜆 = 0.25, 0.5, and 0.75. While the shapes of the variations are different for small values of n, for larger values of n they all converge to ET ∕E0 = (1 − e20 )1∕2 . Notice that, while for 𝜓 0 = 0.25 and 0.75 ET ∕E0 oscillate about ET ∕E0 = (1 − e20 )1∕2 , for 𝜓 0 = 0.5ET ∕E0 oscillates below ET ∕E0 = (1 − e20 )1∕2 . Finally, Figure 24.5.4 shows variations of the tensile modulus ET ∕E0 versus the phase shift 𝜓 0 = x0 ∕𝜆 for a normalized amplitude of e0 = 0.5, with the normalized gauge length n = l∕𝜆 as a parameter.

Random Glass Mat Composites – Materials with Macrostructure

2.0

TENSILE MODULUS E T / E 0

e0 = 0.75 e0 = 0.5

ψ0 = 0

e0 = 0.25

1.0

0 0.1

1

10

n = l /λ Figure 24.5.2 Variation of the normalized tensile modulus ET ∕E0 with the normalized gauge length n = l∕𝜆 for a normalized phase shift of 𝜓 0 = x0 ∕𝜆 = 0, with the normalized amplitude e0 as a parameter. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

24.5.1.2 Effective Flexural Modulus

For the model material with a sinusoidally varying tensile modulus (Eq. 24.5.1), the expression in Eq. 24.4.6 for the effective flexural, or bending, modulus, EB , does not result in a closed-form solution; its values must be obtained by numerical integration. Furthermore, in contrast to ET , EB is not constant for integer values of n. The variations of EB ∕E0 versus n, for 𝜓 0 = 0, with e0 as parameter are shown in Figure 24.5.5. Again, EB ∕E0 approaches unity as n → 0. Just as in Figure 24.5.2, EB ∕E0 first increases with n but then decreases to oscillate around (1 − e20 )1∕2 with decreasing amplitude. Here again, larger values of e0 have a larger effect on EB ∕E0 . The effects of varying the phase shift on the bending modulus are shown in Figure 24.5.6 for three phase shifts of 𝜓 0 = x0 ∕𝜆 = 0.25, 0.5, and 0.75. While the shapes of the variations are different for small values of n, for larger values of n they all converge to ET ∕E0 = (1 − e20 )1∕2 . Notice that, while for 𝜓 0 = 0.25 and 0.75 ET ∕E0 oscillate about ET ∕E0 = (1 − e20 )1∕2 , for 𝜓 0 = 0.5ET ∕E0 oscillates below ET ∕E0 = (1 − e20 )1∕2 . Finally, in this series, Figure 24.5.7 shows variations of the bending modulus EB ∕E0 versus the phase shift 𝜓 0 = x0 ∕𝜆, for a normalized amplitude of e0 = 0.5, with the normalized gauge length n = l∕𝜆 as a parameter. For purposes of comparison, the variations of ET ∕E0 and EB ∕E0 versus n are shown in Figure 24.5.8 for e0 = 0.5 and 𝜓 0 = 0.5. For an infinitesimally small gauge length, the effective tensile and flexural

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Introduction to Plastics Engineering

2.0

TENSILE MODULUS E T / E 0

TENSILE MODULUS E T / E 0

ψ0 = 0.25

1.0 e0 = 0.25 e0 = 0.5 e0 = 0.75

0 0.1

2.0 e0 = 0.25 e0 = 0.5 e0 = 0.75

ψ0 = 0.5

1.0

0 0.1

10

1

1

10

n = l/λ

n = l /λ

(b)

(a) TENSILE MODULUS E T / E 0

912

2.0

e0 = 0.75 e0 = 0.5 e0 = 0.25

ψ0 = 0.75

1.0

0 0.1

1

n = l /λ

10

(c) Figure 24.5.3 Variation of the normalized tensile modulus ET ∕E0 with the normalized gauge length n = l∕𝜆 with the normalized amplitude e0 as a parameter, (a) for a phase shift of 𝜓 0 = x0 ∕𝜆 = 0.25, (b) 𝜓 0 = 0.50, and (c) 𝜓 0 = 0.75. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

elastic moduli must equal the local material modulus, which is unity for n = 0. For larger values of n less than unity, ET and EB differ significantly. For 1 < n < 2, the values of EB ∕E0 are out of phase with those of ET ∕E0 by about 𝜋 . As mentioned earlier, both effective moduli converge to (1 − e20 )1∕2 . However, EB ∕E0 converges faster than does ET ∕E0 . The fact that ET and EB converge to the same value E0 (1 − e20 )1∕2 , for sufficiently large values of n is important. It implies that when l = n𝜆 is sufficiently large, the material behaves like a homogeneous material with modulus ET = EB . Although this result has been shown to be true only for the sinusoidal variation shown in Figure 24.5.1, it can be expected to hold for other modulus variations. This is what happens in metals such as steel, in which there are large property variations at the microstructural grain level, but in which the material behaves homogeneously at the macroscopic level.

Random Glass Mat Composites – Materials with Macrostructure

2.0

TENSILE MODULUS E T / E 0

e0 = 0.5 n = 0.1 n = 0.1 n = 0.15 n = 4.5

1.0

0 0

0.5

10

ψ0 = x 0 / λ Figure 24.5.4 Variation of the normalized tensile modulus ET ∕E0 with the phase shift of 𝜓 0 = x0 ∕𝜆 for a fixed normalized amplitude of e0 = 0.5, with the normalized gauge length n = l∕𝜆 as a parameter, (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

24.5.1.3 Effect of Gauge Length on Modulus Distribution Measurement

With reference to Figure 24.4.1, let an extensometer of gauge length l be used to determine the mean tensile modulus at equal intervals h along the length of a test specimen. Then, from Eq. 24.4.2, the mean moduli, over successive segments of length l will be l

1 1 dx = ET1 l ∫0 E ( x) h+l

1 1 = ET2 l ∫h

dx E ( x)

⋮ (r−1) h+l

1 1 = ETr l ∫(r−1) h

dx , E ( x)

r = 1, 2, · · ·

which, on use of X = x − (r − 1)h, becomes l

1 1 dX = , ETr l ∫0 E [ X + (r − 1) h]

r = 1, 2, · · ·

(24.5.5)

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Introduction to Plastics Engineering

2.0 e0 = 0.75 BENDING MODULUS EB / E0

914

e0 = 0.5

ψ0 = 0

e0 = 0.25

1.0

0 0.1

10

1 n = l /λ

Figure 24.5.5 Variation of the normalized bending modulus EB ∕E0 with the normalized gauge length n = l∕𝜆 for a normalized phase shift of 𝜓 0 = x0 ∕𝜆 = 0, with the normalized amplitude e0 as a parameter. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

For the sinusoidal modulus variation in Eq. 24.5.1, this reduces to 1 E0 d𝜉 1 , = ETr l ∫0 1 + e0 sin 2𝜋 [n 𝜉 + (r − 1) 𝛾 − 𝜓0 ]

r = 1, 2, · · ·

(24.5.6)

where 𝜉 = X∕l, n = l∕𝜆, 𝜓 0 = x0 ∕𝜆, and 𝛾 = h∕𝜆. In this equation, 𝛾 = h∕𝜆 is the ratio of the sample interval to the wavelength of the modulus variation. For integer values of 𝛾 , it follows that ET1 = ET2 = · · · ETr , r = 1, 2, · · · . Equation 24.5.6 then integrates to the closed-form expression

E0 1 = arctan ET n (1 − e20 )1∕2 1 + e0

(1 − e20 )1∕2 tan n tan [(r − 1) −

0]

1 + tan [(r − 1) −

+ tan 0 ] tan

(n + (r − 1) − (n + (r − 1) −

0 0

(24.5.7) Figure 24.5.9a shows the variations of ET ∕E0 with position for seven values of n, along a specimen with a sinusoidally varying modulus for 𝛾 = n, e0 = 0.5, and the phase fraction 𝜓 0 = 0. Of course, 𝛾 = n implies that the “extensometer” that “measures” ET is advanced by the gauge length after each “measurement.” The corresponding variations for 𝛾 = n∕2 (the “extensometer” only advanced by half the gauge length after each “measurement”) are shown in Figure 24.5.9b. At a physical level, a segment in Figure 24.5.9a corresponds to a twice-as-large segment number in Figure 24.5.9b. As

BENDING MODULUS E B / E0

BENDING MODULUS E B / E0

Random Glass Mat Composites – Materials with Macrostructure

2.0

ψ0 = 0.25

1.0 e0 = 0.25 e0 = 0.5 e0 = 0.75

0

e0 = 0.25 e0 = 0.5 e0 = 0.75

0 0.1

1

n = l/λ

n = l/λ

(a)

(b) 2.0

ψ0 = 0.5

1.0

10

1

BENDING MODULUS E B / E0

0.1

2.0

e0 = 0.75 e0 = 0.5 e0 = 0.25

10

ψ0 = 0.75

1.0

0 0.1

1

n = l/λ

10

(c) Figure 24.5.6 Variation of the normalized bending modulus EB ∕E0 with the normalized gauge length n = l∕𝜆 with the normalized amplitude e0 as a parameter, (a) for a phase shift of 𝜓 0 = x0 ∕𝜆 = 0.25, (b) 𝜓 0 = 0.50, and (c) 𝜓 0 = 0.75. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

expected, for n = 0.1, the “measured” effective moduli pick up the sinusoidal variation of E = E ( x). The variation of E ( x) is picked up more clearly when 𝛾 = n∕2 than when 𝛾 = n. The sinusoidal variation is not apparent for n = 0.3. The pattern of variation is periodic over a distance equal to the wavelength of the sinusoidal variation. If the values shown in Figure 24.5.9b for n = 0.3 were actually measured in an experiment, it might be tempting to conclude that the modulus variation is approximately sinusoidal, with a wavelength about one-third that of the original sinusoidal variation. For 𝛾 = n, the curves for n = 0.9 and 1.1 (Figure 24.5.9a) show sinusoidal-like variations with the same wavelength as the original variation. However, this pattern is not easily discerned from the curves for the same value of n when 𝛾 = n∕2 (Figure 24.5.9b). In all these curves, the effect of a nonzero 𝜓 0 would be a constant phase shift in the curves, without affecting the pattern of variation. The distance over which the “measured” moduli is periodic depends on the value of n. Figure 24.5.10 shows the variations of ET ∕E0 versus the segment number for 𝛾 = n, e0 = 0.5, and 𝜓 0 = 0 for four values of n. Notice that although the variation of ET ∕E0 has been plotted for twice as many segments as in

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Introduction to Plastics Engineering

2.0

BENDING MODULUS E B / E0

e0 = 0.5 n = 0.1 n = 0.1 n = 0.15

n = 4.5

1.0

0 0

10

0.5

ψ0 = x 0 / λ Figure 24.5.7 Variation of the normalized bending modulus EB ∕E0 with the phase shift of 𝜓 0 = x0 ∕𝜆, for a normalized amplitude of e0 = 0.5, with the normalized gauge length n = l∕𝜆 as a parameter. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

2.0

TENSILE & BENDING MODULI

916

E T / E0

e0 = 0.5 ψ0 = 0

E B / E0

1.0

0 0.1

1

10

n = l /λ Figure 24.5.8 Variations of ET ∕E0 and EB ∕E0 versus n, for e0 = 0.5 and 𝜓 0 = 0. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

Random Glass Mat Composites – Materials with Macrostructure

0

10

0

20 n = 1.3

1.0

20

30

40

50

n = 0.7

1.0

n = 0.5

1.5 1.0 0.5

n = 0.3

1.5

1.5

1.0 0.5 n = 0.3

1.5

0.5

0.5

1.5

n = 0.5

1.5

1.0

n = 0.1

n = 0.7

1.0

1.0

γ=n

n = 0.9

1.0

TENSILE MODULUS E T / E0

1.5

n = 1.1

1.0

n = 0.9

1.0

n = 1.3

1.0

n = 1.1

1.0

TENSILE MODULUS E T / E0

10

1.5

1.5

γ=

1 2

n

n = 0.1

1.5

1.0

1.0

0.5

0.5 0

0 0

10

20

0

10

20

30

SEGMENT

SEGMENT

(a)

(b)

40

50

Figure 24.5.9 Variations of ET ∕E0 versus segment number, for e0 = 0.5 and 𝜓 0 = 0, and (a) 𝛾 = n and (b) 𝛾 = n∕2. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

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Introduction to Plastics Engineering

0

10

20

1.5

30

40

50

n = 1.634

1.0 1.5 TENSILE MODULUS E T / E0

918

n = 1.432

1.0 0.5 1.5

n = 0.735

1.0 0.5

γ= n 1.5

n = 0.438

1.0 0.5 0 0

10

20

30

40

50

SEGMENT Figure 24.5.10 Variations of ET ∕E0 versus segment number, for e0 = 0.5 and 𝜓 0 = 0, and 𝜇 = n. Note how the values chosen for n differ from those in Figure 24.9.9. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

Figure 24.5.9a ET ∕E0 does not exhibit any periodicity for n = 0.438 over a length corresponding to 50 segments. For n = 0.735, the pattern appears to repeat after some 35 segments of “measurements.” An important conclusion can be drawn from the curves for “non-round” values of n in Figure 24.5.9: Even if the underlying variation of the modulus along the length of a specimen is sinusoidal, the values “sampled” by an “extensometer” could exhibit an apparently random variation. As an example, if the modulus has a sinusoidal variation over a unit distance, then an extensometer with a gauge length of 0.438 units would not indicate a repeat pattern for at least 50 units of distance. Thus, care must be taken in interpreting modulus data that appear to have a random variation with distance along a specimen. 24.5.2

Model Material with Rectangular Wave Modulus Variation

To illustrate how the wave form of a model material affects ET and EB , consider the model shown in Figure 24.5.11 in which the modulus varies as a constant plus a rectangular wave. Here again by using

TENSILE MODULUS E = E(x) / E0

Random Glass Mat Composites – Materials with Macrostructure

λ e0 1.0 e0

0 0

x0

x

Figure 24.5.11 Parameters defining the material model with a rectangular wave form tensile modulus. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

the normalized variables 𝜉 = x∕l, 𝜂 = 2x∕l, n = l∕𝜆, and 𝜓 0 = x0 ∕𝜆, ET and EB can be determined by integrating Eq. 24.4.3. The variations of ET ∕E0 and EB ∕E0 versus 𝜓 0 for e0 = 0.5 and n = 0.5 are shown, respectively, in Figures 24.5.12 and 24.5.13, in which solid and dashed lines correspond, respectively, to the square-wave (Figure 24.5.11) and sinusoidal (Figure 24.5.1) modulus variations. Given the differences in these two wave forms, the similarities in the variations of ET ∕E0 and EB ∕E0 with 𝜓 0 are remarkable. Figure 24.5.14 shows the variation of EB ∕E0 versus 𝜓 0 , for e0 = 0.5 and n = 1. Here again, the closeness of the results for the two wave forms is remarkable. 24.5.3

Summary of Lessons Learned from Model Materials

The results of using a model material with a sinusoidally varying tensile modulus to assess the effects of macroscopic property variations on the elastic moduli ET ∕E0 and EB ∕E0 have provided useful insights into how macroscopic variations can affect measured properties: The effective tensile and bending moduli can exhibit large differences when the measurement gauge length is smaller than the wavelength of the sinusoid. However, for gauge lengths greater than about five wave lengths, the predicted tensile and flexural moduli rapidly converge to a value that corresponds to the tensile harmonic mean for one wave length. This has important implications for design. It shows that the material is essentially homogeneous on a scale of several wave lengths, so that for the stiffness design of “large” features, the material can be treated as a homogeneous material with a tensile elastic modulus equal to the harmonic mean. Of course, this finding is strictly true only for the assumed sinusoidal variation. The analysis of model materials has produced an unexpected result: Even when the underlying modulus variation along a specimen length is a sinusoid, the values “sampled” by an “extensometer” can exhibit an apparent random variation. Therefore, care must be taken in interpreting modulus data that appears to have a random variation along the specimen length.

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TENSILE MODULUS E T / E0

2.0 e0 = 0.5 n = 0.5 SINE WAVE SQUARE WAVE

1.0

0 0

10

0.5

ψ0 = x 0 / λ Figure 24.5.12 Variations of the normalized tensile modulus ET ∕E0 versus the phase fraction 𝜓 0 for e0 = 0.5 and n = 0.5. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

2.0

BENDING MODULUS EB / E0

920

e0 = 0.5 n = 0.5

SINE WAVE SQUARE WAVE

1.0

0 0

0.5

10

ψ0 = x 0 /λ Figure 24.5.13 Variations of the normalized bending modulus EB ∕E0 versus the phase fraction 𝜓 0 for e0 = 0.5 and n = 0.5. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

Random Glass Mat Composites – Materials with Macrostructure

BENDING MODULUS EB / E0

2.0 e0 = 0.5 n = 1.0

SINE WAVE

1.0

SQUARE WAVE

0 0

0.5

10

ψ0 = x0 / λ Figure 24.5.14 Variations of the normalized tensile modulus EB ∕E0 versus the phase fraction 𝜓 0 for e0 = 0.5 and n = 1. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 45 – 55, 1990.)

24.6 Characterization of the Tensile Modulus The tensile modulus and strength data used in this section were obtained by tests on 203 × 405-mm (9 × 16-in), 3.68-mm (0.145-in) thick drape-molded Azdel P100 plaques. First, a 203 × 405-mm blank was cut from an Azdel sheet with the 405-mm length aligned along the machine direction. After heating to a nominal temperature of 215°C (419°F), this blank was thermostamped at a nominal pressure of 13.8 MPa (2 × 103 psi) in a mold of the same size as the blank. During this process, called drape molding, while the glass did not “flow,” the plaque was subjected to the thermal and pressure histories that the material undergoes during part forming. Also, it is likely that the number of voids was reduced. The variability in the random distribution of the continuous glass fibers in such drape-molded plaques can be gauged by comparing the radiographs in Figures 24.2.1a,b. 24.6.1

Cross-Machine-Direction Tensile Moduli

The cross-machine-direction tensile modulus of the material was characterized as follows: First, 10 12.7-mm (0.5-in) strips were marked with a pen, along the machine direction, on the 3.68-mm (0.145) thick drape-molded plaque, as shown by dashed lines in Figure 24.6.1. Twenty-two 12.7-mm strips were then cut from the plaque along the cross-machine direction, as indicated by the solid lines, resulting in 22 12.7 × 230-mm (0.5 × 9-in) specimens with cross-length markers at 12.7-mm intervals, delineating 10 12.7-mm segments. The numbering system for the specimens and the 12.7-mm segments is indicated

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Introduction to Plastics Engineering

405 mm (16 in)

12.7 mm (0.5 in)

1

230 mm (9 in)

12.7 mm (0.5 in)

SEGMENT NUMBER

922

10

1

22 SPECIMEN NUMBER

Figure 24.6.1 Layout of 22 12.7-mm wide specimens cut from a 230 × 405-mm drape-molded GMT plaque. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 342 – 353, 1990.)

in Figure 24.6.1. The specimen numbers increase from 1 at the left to 22 at the right. The 12.7-mm segments always start with the number 1 at the top and end with the number 10 at the bottom. Second, the mean cross-sectional area of each 12.7-mm segment on each specimen was determined by measuring the mean thickness and the mean width over that 12.7 × 12.7-mm (0.5 × 0.5-in) region. Third, tensile tests were used to determine the tensile modulus at each of 12.7-mm segment along each strip by attaching an extensometer along the left and right (thickness) edge, as indicated by the letters L and R, respectively, in Figure 24.3.2. With the extensometer attached to the right edge of segment 1, the specimen was pulled in tension under strain control to a strain of 0.5% at a strain rate of 10−2 s−1 ; the load-strain behavior was recorded. After unloading, the extensometer was moved down to the next 12.7-mm segment, and the test procedure was repeated. In this way the right elastic modulus, ER , was determined at every 12.7-mm interval along a strip. Similarly, by attaching the extensometer along the left edge, the left elastic modulus, EL , was also determined along the strip. Finally, after determining the tensile moduli, the 12.7-mm wide specimens were cut along the gauge marks into 12.7 × 12.7-mm coupons. The density of each coupon was then determined by measuring its mass and calculating the volume via measurement of its dimensions. These tests showed significant variations in the two tensile moduli EL and ER over distances of 12.7 mm (0.5 in). Over a 127 × 280-mm (5 × 11-in) area, the maximum and minimum moduli were, respectively: 8.78 GPa (1.273 × 103 psi) and 2.83 GPa (411 × 103 psi) for EL , the ratio of the maximum to the minimum being 3.1; and 9.5 GPa (1.379 × 103 psi) and 2.89 GPa (420 × 103 psi) for ER , the ratio of the maximum to the minimum being 3.3. While EL and ER exhibit similar fluctuations along a specimen, their variations tend to be out of phase; in a specimen segment a high value of one can be accompanied by a low value of the other. These differences in EL and ER in the same segment are likely caused by differences in the glass orientation

Random Glass Mat Composites – Materials with Macrostructure

10

TENSILE MODULUS (GPa)

TENSILE MODULUS (GPa)

across the segment. These differences in EL and ER raise an important question: What constitutes a representative mean modulus for each 12.7 × 12.7-mm segment? One possible measure is the arithmetic mean modulus EA = (EL + ER )∕2. While differences in EL and ER in the same segment can arise from differences in the glass orientation across the segment, differences in EA = (EL + ER )∕2 along specimens are most likely caused by differences in the amount of glass fibers in each segment; differences that should show up as variations in the mean densities of 127 × 12.7-mm segments. The mean local densities of 12.7 × 12.7-mm coupons, across the 127 × 280-mm plaque, varied from a maximum of 1.30 g cm−3 to a minimum of 1.11 g cm−3 . Figure 24.6.2 parts a and b show, respectively, plots of the left and right moduli, EL and ER versus the local mean density 𝜌 for the data from all the 22 specimens. While the overall trend is for the

EL

5

2 1.1

10

ER

5

2

1.3

1.2

1.1

1.3

1.2

DENSITY (g·cm )

DENSITY (g·cm−3 )

(a)

(b) TENSILE MODULUS (GPa)

−3

10

EA

5

2 1.1

1.3

1.2

DENSITY (g·cm−3 )

(c) Figure 24.6.2 Variations of the tensile modulus with density. (a) Left tensile modulus versus density, (b) right tensile modulus versus density, and (c) harmonic mean modulus versus density. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 342 – 353, 1990.)

923

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Introduction to Plastics Engineering

moduli to increase with the density, they do not appear to be strongly correlated. However, as shown in Figure 24.6.2c, the mean modulus EA appears to correlate quite well with 𝜌. Even for the same glass content, and hence the same local mean density, the local modulus will depend on the local glass orientation, thereby providing one explanation for the scatter in the data in Figure 24.6.2a,b. While Figure 24.6.2 shows that EA and 𝜌 are strongly correlated, information on the spatial distribution of 𝜌, and hence EA , is missing. This information is provided by the contour maps of EA and 𝜌 shown in Figure 24.6.3.

DENSITY (g·cm – 3) 1.25–1.30 1.20–1.25 1.15–1.20 1.10–1.15

(a) TENSILE MODULUS (GPa) 6.30–7.30 5.30–6.30 4.30–5.30 3.30–4.30

(b) Figure 24.6.3 Contours of (a) the density 𝜌, and (b) the mean modulus EA over 127 × 280-mm region. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 342 – 353, 1990.)

24.7 Characterization of the Tensile Strength This section describes tensile strength data obtained by tests on 203 × 405-mm, 3.68-mm thick drape-molded Azdel P100 (Azdel PM10400-191) plaques, which underwent the same, drape-molding thermostamping history as applied to plaques used for characterizing tensile moduli. 24.7.1

Test Procedure

For characterizing machine-direction strength, 12.7-mm strips were marked with a pen along the cross-machine direction, as indicated by the dashed lines in Figure 24.7.1. Seven 19-mm (0.75-in)

Random Glass Mat Composites – Materials with Macrostructure

405 mm (16 in)

230 mm (9 in)

19 mm (0.75 in)

1 7

SPECIMEN NUMBER

12.7 mm (0.5 in)

1

23

25.4 mm (1 in)

SEGMENT NUMBER Figure 24.7.1 Layout of seven 292-mm gauge-length, machine-direction dog-bone specimens cut from a 230 × 405-mm drape-molded GMT plaque. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.)

wide strips were cut from the plaque along the machine direction, each of which was then routed to a dog-bone shape with a 12.7 × 292-mm (0.5 × 11.5-in) gauge length, as shown in Figure 24.7.1. This resulted in seven long dog-bone specimens, having a width of 12.7 mm over a 292-mm gauge length, with cross-length markers at 12.7-mm intervals. Then by using the procedure used for characterizing the tensile modulus, the average tensile modulus EA was characterized at each 12.7-mm segment on each of the seven specimens. At each segment, the specimens were pulled at a strain rate of 10−2 s−1 to a strain of 0.5%. Similarly, the EA was characterized along each 12.7-mm segment of 16, long dog-bone cross-machine direction specimens – as shown in Figure 24.7.2 – over a 114-mm (4.5-in) gauge length. The strain rate was the same as for the machine-direction specimens but, because of their inherently higher stiffness causing them to fail at lower strains, the specimens were only pulled to a strain of 0.25%. After characterizing the local average tensile modulus EA in all the specimens, in each specimen one 12.7-mm gauge-length extensometer was attached to the edge having the highest local modulus EA . A second extensometer was attached to the edge of the segment having the lowest average modulus. The dog-bone specimen was then pulled in tension to failure, under a constant displacement rate, at a nominal strain rate of 10−2 s−1 . In this way the stress-strain characteristics of segments having extreme moduli could be monitored during a failure test. This procedure was applied to both the machine- and cross-machine-direction dog-bone specimens. 24.7.2

Machine-Direction Tensile Modulus and Strength Data

Figure 24.7.3 shows the variations of EL , ER , and EA along each of the seven machine-direction specimens at 23 locations. The values for EA are connected by a solid line. At each location, dashed lines connecting

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Introduction to Plastics Engineering

405 mm (16 in) 19 mm (0.75 in)

9

1

16

230 mm (9 in)

1 12.7 mm (0.5 in)

SEGMENT NUMBER

926

25.4 mm (1 in)

SPECIMEN NUMBER Figure 24.7.2 Layout of 16 114-mm gauge-length, cross-machine direction dog-bone specimens cut from a 230 × 405-mm drape-molded GMT plaque. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.)

the values of EL and ER show the spread in the values in relation to EA . The vertical arrows indicate the locations of the segments in which tensile failure occurred. Clearly, while failure in a specimen does not necessarily occur in segments having the lowest value of EA , EA appears to be the single most important indicator for failure. This is also borne out by the location of the failure sites marked by thick bars in the contours of EA shown in Figure 24.7.4. Failure data for the specimens from this plaque, listed in Table 24.7.1, show a remarkable consistency in the stress at failure in the seven specimens, with a maximum and minimum of 82.4 and 68.5 MPa, respectively, and an arithmetic mean of 76 MPa over the seven specimens. The last column in the table is an estimate for the failure strain, obtained by dividing the stress at failure (column 11) in the segment by the tensile modulus (not listed) of that segment. Here again, the ultimate strains exhibit remarkable consistency with a mean of 1.8%. However, this estimate assumes that the stress-strain relationship is linear, which is not always true. This should be apparent from Figure 24.7.5, which shows the stress-stain curves for the highest- and lowest-modulus segments in specimens 4 and 5. In specimen 5, while the stress-strain curve OA for the highest-modulus segment is approximately linear, the curve OB for the lowest-modulus segment is highly nonlinear. Both such curves can be approximately linear as shown by the lines OC and OD for specimen 4. Note that the strains in this figure are not the mean strains across the segment width. Rather, for each specimen, they are, respectively, the strains measured on the left and right edges of two different segments. As such, a comparison of these one-sided strains with the strain estimates based on the mean modulus – which would be better compared with the arithmetic means of the measures left- and right-edge strains in the same segment, if both were available – should be interpreted with care. While the actual

Random Glass Mat Composites – Materials with Macrostructure

9 SPECIMEN-4

8

10

5

EL ER EA

TENSILE MODULUS (GPa)

3 SPECIMEN-3

6

7

SPECIMEN-7 10

5

5

3

3 12 SPECIMEN-2

5

6

7

SPECIMEN-6

5

3

3 EL ER EA

8

SPECIMEN-1

2 SPECIMEN-5

7

14

5

5

2

2 0

10 SEGMENT

20

0

20 10 SEGMENT

Figure 24.7.3 Variations of the machine-direction Young’s moduli EL, ER, and EA at 12.7-mm intervals along specimens 1 through 7. Vertical arrows indicate failure locations in tensile tests. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.)

strains in the highest- and lowest-modulus segments in specimen 5, when the specimen failed elsewhere, were (left edge) 0.83% (column 4) and (right edge) 2.4% (column 8), respectively, the corresponding linear estimates are 1.2 and 1.83%. As before, these linear estimates were obtained by dividing the stress in the segment by EA for the segment. In specimen 4, curves OC and OD for the highest- and lowest-modulus segments, respectively, show that both the curves can be approximately linear. A comparison of the numbers in columns 4 and 5 in Table 24.7.1 shows that the measures strains in the highest- EA segments are much closer to the linear estimates than those for the lowest- EA segments (columns 8 and 9). This is consistent with the high- EA segments having more linear stress-strain curves.

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Introduction to Plastics Engineering

TENSILE MODULUS (GPa) 5.50–6.25 4.75–5.50 4.00–4.75 3.25–4.00

Figure 24.7.4 Contours of the machine-direction Young’s modulus EA over a 133 × 292-mm region of a drape-molded GMT plaque. Tensile failure locations are indicated by solid bars; the tensile tests were conducted in directions normal to the bars (left to right). (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.)

150 MAXIMUM EA SEGMENT MINIMUM EA SEGMENT

STRESS (MPa)

928

100 A

C

D

B

50 SPECIMEN-4 SPECIMEN-5

0O 0

0.01

0.02

0.03

0.04

0.05

STRAIN Figure 24.7.5 Stress-strain curves for regions having the highest (solid lines) and lowest (dashed lines) values of the machine-direction tensile modulus EA in specimens 4 and 5. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.)

Random Glass Mat Composites – Materials with Macrostructure

Table 24.7.1 Machine-direction tensile failure data for drape-molded plaque.

Estimated failure strain (%)

Tensile strength (MPa)

Failure data Segment number

Estimated

Strain at failure (%) Measured

EA (𝐆𝐏𝐚)

Lowest average modulus Segment number

Estimated

Strain at failure (%) Measured

EA (𝐆𝐏𝐚)

Segment number

Specimen number

Highest average modulus

1

23

5.76

1.19

1.36

4

3.91

2.3

1.99

14

77.0

1.8

2

4

5.86

1.44

1.40

7

4.21

2.5

1.95

12

82.4

1.9

3

23

5.67

1.34

1.37

18

3.80

2.3

2.05

6

77.7

2.0

4

20

6.28

1.03

1.21

11

3.86

1.6

1.95

8

74.7

1.7

5

6

5.71

0.83

1.20

8

3.73

2.4

1.83

2

68.5

1.6

6

23

6.17

1.02

1.18

13

4.03

2.0

1.82

6

73.6

1.8

7

5

6.04

1.05

1.31

11

3.83

2.3

2.04

10

77.9

1.8

Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.

Given the nonhomogeneity of the material, the remarkable consistency in failure stress and strain can be explained as follows. Each of the specimens is sufficiently long to include regions with the lowest possible strength for the material. Since a specimen will fail at the weakest link, this test procedure has essentially determined the strength and failure strain of the weakest region of the material. Thus, this material can be assumed to have a lower-bound strength of 76 MPa and a failure strain of 2%. 24.7.3

Cross-Machine-Direction Tensile Modulus and Strength Data

Figure 24.7.6 shows the variations of EL , ER , and EA along specimens 1 through 11 at each of the 9 locations on the 11 cross-flow direction specimens. As before, the vertical arrows indicate the locations of the segments in which tensile failure occurred. Here again, failure in a specimen appears to occur in regions having the lowest value of EA , as should also be apparent from the location of the failure sites shown in Figure 24.7.7 by thick bars in the contours of EA . Failure data for the specimens from this plaque are listed in Table 24.7.2. Column 11 shows that the tensile strengths of the 16 specimens varied from a minimum of 81 MPa, in a segment having EA = 6.42 GPa, to a maximum of 126 MPa, in a segment having EA = 8.44 GPa. The mean strength over the 16 specimens was 103 MPa. These variations in strength are much larger than for the machine-direction specimens (Section 24.7.2) each of which had 23 segments. This can be explained by the cross-machine direction specimens being much shorter, each having only nine segments, because if which each specimen will not have materials with all possible strengths for the material. Stress-strain curves for segments with the highest and lowest EA for specimens 8 (curves OA and OB) and 8 (curves OC and OD) are shown in Figure 24.7.8. The higher modulus curves tend to be more linear. The difference between the maximum and minimum tensile strengths of the machine-direction specimens is smaller than that of the cross-machine direction specimens. The large differences are a consequence of the large differences in the local mechanical properties of these materials. The reasons for these differences are explored in the sections that follow.

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Introduction to Plastics Engineering

15

EL ER EA

4

10

15 EL ER EA

SPECIMEN-6

10

SPECIMEN-11

8

5 8

SPECIMEN-5

10

TENSILE MODULUS (GPa)

930

5 10

0

SPECIMEN-10

5 10

SPECIMEN-4

0

5

6

1

8 5

10

SPECIMEN-3

SPECIMEN-9

5 SPECIMEN-8 6

6 3

10

SPECIMEN-2

9

5

5

SPECIMEN-7 3

10

5

5

2

2 0

5 SEGMENT

3

10

SPECIMEN-1

10

0

5

10

SEGMENT

Figure 24.7.6 Variations of the cross-machine-direction Young’s moduli EL, ER, and EA at 12.7-mm intervals along specimens 1 through 11. Vertical arrows indicate failure locations in tensile tests. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.)

Random Glass Mat Composites – Materials with Macrostructure

TENSILE MODULUS (GPa) 8.0–9.0 7.0–8.0 6.0–7.0 5.0–6.0

Figure 24.7.7 Contours of the cross-machine-direction Young’s modulus EA over a 305 × 114-mm region of a drape-molded GMT plaque. Tensile failure locations are indicated by solid bars; the tensile tests were conducted in directions normal to the bars (top to bottom). (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.)

Table 24.7.2 Cross-machine-direction tensile failure data for drape-molded plaque.

EA (𝐆𝐏𝐚)

Measured

Estimated

Segment number

Tensile strength (MPa)

6

7.42

2.22

1.45

1

6.15

2.01

1.79

3

109

1.73

2

7

7.21

2.20

1.26

3

5.26

1.85

1.73

3

91

1.73

3

2

7.87

2.45

1.34

7

5.94

2.08

1.78

5

104

1.50

4

8

8.81

2.43

1.33

2

7.20

1.86

1.64

0

119

—

5

3

8.88

2.13

1.37

8

6.92

1.60

1.74

8

123

1.77

6

1

10.40

1.60

1.06

9

6.33

2.76

1.71

4

109

1.66

7

3

8.44

2.35

1.49

9

7.06

3.09

1.79

8

126

1.71

8

5

8.00

2.57

1.43

8

5.63

3.91

2.04

6

115

1.61

9

9

7.23

1.61

1.18

1

4.98

2.50

1.75

1

87

1.75

10

6

8.31

1.69

1.16

2

5.85

2.25

1.67

0

98

—

11

5

6.42

1.84

1.25

8

4.80

2.08

1.68

8

81

1.68

12

9

6.59

2.13

1.49

3

5.66

1.77

1.75

1

100

1.59

13

9

7.94

1.58

1.13

2

5.75

1.71

1.58

4

90

1.46

14

7

8.38

1.65

1.03

1

6.11

1.45

1.42

0

87

—

15

1

9.19

1.99

1.31

7

7.17

2.67

1.67

10

118

—

16

5

7.24

1.72

1.25

8

5.64

2.82

1.61

7

91

1.54

Estimated failure strain (%)

Segment number

1

Specimen number

Estimated

Strain at failure (%)

Measured

Strain at failure (%)

Failure data

EA (𝐆𝐏𝐚)

Lowest average modulus

Segment number

Highest average modulus

Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.

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Introduction to Plastics Engineering

150

STRESS (MPa)

932

D C

A

B

100

SPECIMEN-1 SPECIMEN-8

50

MAXIMUM EA SEGMENT MINIMUM EA SEGMENT

0O 0

0.01

0.02

0.03

0.04

0.05

STRAIN Figure 24.7.8 Stress-strain curves for regions having the highest (solid lines) and lowest (dashed lines) values of the cross-machine-direction tensile modulus EA in specimens 1 and 8. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.)

24.7.4

Comparison of Machine- and Cross-Machine Direction Strength Data

An understanding of the variations in EA is important for understanding the large variations in the measured tensile strength. The effective tensile modulus for a specimen over a specified length is the harmonic mean modulus. The variations in the harmonic means EL , ER , and EA for each of the seven machine-direction specimens – over a 292-mm (11.5-in) gauge length – and the 16 cross-machine direction specimens – over a 114-mm (4.5-in) gauge length – are shown in Figure 24.7.9. Clearly, the cross-machine-direction specimens are significantly stiffer than the machine-direction specimens. Given the fluctuations in the 12.7-mm tensile moduli, the harmonic means EL , ER , and EA for each of the seven machine-direction specimens, over a gauge length of 292 mm, is remarkable. The harmonic mean EA has maximum and minimum values of 4.97 and 4.53 GPa, respectively – a variation of less than 10% – and an overall arithmetic mean of 4.75 GPa over the seven specimens. The harmonic means EL , ER , and EA for each of the 16 cross-machine direction specimens, over a gauge length of 114 mm are fairly close, but not as close as in the machine-direction specimens. Also, the variation of EA from specimen to specimen is larger: EA maximum and minimum values of 7.95 and 5.62 GPa, a variation of less than 40%. The arithmetic mean of EA over the 16 specimens is 6.83 GPa. With an average mean of EA = 6.83 GPa, the cross-machine direction stiffness is significantly larger than the machine-direction stiffness of EA = 4.75 GPa. Note that these sets of measurements were made on specimens cut from two different plaques.

Random Glass Mat Composites – Materials with Macrostructure

HARMONIC MEAN MODULUS (GPa)

10 – EL – ER – EA

9

8

7

6 CROSS-MACHINE DIRECTION

5 MACHINE DIRECTION

4 1

5

10

15

SPECIMEN NUMBER Figure 24.7.9 Variations in the harmonic means EL , ER, and EA , of EL , ER, and EA , for 7 machine-direction and 16 cross-machine-direction specimens. (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.)

A likely explanation for the smaller variation in the harmonic means of the machine-direction specimens compared to the variations in the cross-machine direction harmonic means is the larger gauge lengths of the former. Larger gauge lengths increase the possibility of the specimens having the same statistical distribution of properties. Shorter gauge lengths, on the other hand, could result in the specimens being statistically different. The machine-direction tensile-strength data differ from the cross-machine direction data in two respects. First, the tensile moduli and the tensile strength are generally higher for the latter. Second, the strengths of the cross-machine direction specimens show a greater variability than the strengths of the machine-direction specimens, which exhibit remarkable consistency. The higher moduli in the cross-machine direction imply that the material is inherently stiffer in the cross-machine direction than in the machine direction. As discussed earlier, the greater variability in the cross-machine direction strength most likely results from their shorter lengths: the tensile test determines the strength of the weakest portion of the specimen. The longer the specimen, the greater the chance for the specimen to contain material with the lowest possible strength, so that really long specimens should exhibit a more uniform (minimum) strength – as appears to be the case in the machine-direction specimens. In contrast to this, the shorter the specimen, the smaller the chance for the specimen to contain material having the lowest possible strength. In this case, tensile tests will determine the strengths of the weakest part of the specimen, but these strengths will be different.

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Introduction to Plastics Engineering

TENSILE STRENGTH (MPa)

The tensile strength variations of the machine- and cross-machine-digestions specimens versus the left-edge modulus EL , the right-edge modulus ER , and the local mean modulus EA of the segment in which the failure occurred are shown, respectively, in Figures 24.7.10a – c. Clearly, while the local tensile strength does not correlate with EL and ER , it correlates well with the local mean modulus EA . Note that all the machine-direction points are clustered together, whereas the cross-machine direction strength varies widely. Since the local mean modulus EA has been shown to correlate well with the local density, it follows that the local tensile strength also correlates with the local density. 150 M-DIRECTION X--DIRECTION

100

50 3

150

100 M-DIRECTION XM-DIRECTION

50 3

9

5

9

5

TENSILE MODULUS E L (GPa)

TENSILE MODULUS E R (GPa)

(a)

(b)

TENSILE STRENGTH (MPa)

TENSILE STRENGTH (MPa)

934

150 M-DIRECTION XM-DIRECTION

100

50 3

9

5

TENSILE MODULUS EA (GPa)

(c) Figure 24.7.10 Variations of the tensile strength versus the tensile modulus of the failed segment for, (a) the left-edge tensile modulus EL , (b) the right-edge tensile modulus ER, and (c) the mean tensile modulus EA . (Adapted with permission from V.K. Stokes, Polymer Composites, Vol. 11, pp. 354 – 367, 1990.)

24.8 Statistical Characterization of the Tensile Modulus Experimental Data While a small amount of scatter can be expected even in data for homogeneous materials, the large variations in measured GMT property data are a characteristic material property, and should not be looked upon as scatter. This large variation raises many questions: How can two different moduli distributions be compared? Are the distributions of the left and right moduli the same? Is the distribution determined from

Random Glass Mat Composites – Materials with Macrostructure

measurements over 150 × 305-mm area the same as that measured over a smaller area and, if so, over how small an area? Can this distribution be generated by simple tests, such as standard ASTM tests, on a large number of specimens? If so, how many tests are required and how should the specimens be chosen? And finally, how can a statistical characterization of the material properties be used for predicting part performance? An analysis of all the modulus data along many specimens showed that while the large-scale fluctuations of the tensile left and the right elastic moduli are of comparable magnitude, their variations appeared to be uncorrelated. For a statistical characterization of the elastic properties of such materials the elastic modulus is assumed to be a random variable E whose variations at a point r in the material can be modeled by a probability density function f (E, r). For simplicity, consider a material for which the probability density function is independent of position. A good example of such a material would be drape-molded parts for which the elastic modulus can be characterized by a single distribution f (E). 24.8.1

Histograms for Tensile Modulus Data

The first step in determining an appropriate representation for f = f (E) is to arrange the data in the form of normalized histograms, in which the area under each histogram is unity. This makes it easier to compare data sets containing different numbers of measurements. The discussion in the sequel is based on two major sets of tensile modulus data. The first set comprises machine-direction measurements of EL and ER over a 292 × 133-mm region of a plaque at 12.7 × 12.7-mm intervals. The second set comprises cross-machine-direction measurements of EL and ER over a 114 × 305-mm region. In each of these two sets, the EL and ER data can be looked upon as independent data sets. New data sets can be generated by creating smaller subsets, by combining smaller sets into larger ones, and by performing arithmetic operations (such as calculating an arithmetic mean) on the two data sets. Normalized histograms for the left and right machine-direction tensile modulus data are shown, respectively, in Figures 24.8.1 and 24.8.2, and those for the machine-direction data are shown, respectively, in Figures 24.8.3 and 24.8.4. In each case, the range of variation has been divided into 15 equal intervals. Clearly, each of these four histograms is unimodal, skewed, and has lower and upper limits on the modulus. The measured moduli can be no lower than the modulus of the matrix resin (polypropylene) and no higher than that of the fiber (glass). The curves approximating the probability density functions for the four original data sets are shown by the smooth curves in Figures 24.8.1 – 24.8.4. The statistical background for how these smooth curves were obtained, and information on the random nature of the modulus distributions, is discussed in the following sections. 24.8.2

*Moments of the Tensile Modulus Distributions

The question of how different distributions can be quantitatively compared can be answered only when each distribution is uniquely characterized by a set of parameters. Such a set of parameters based on n experimental values E1 , E1 , … En , are the moments, called moments estimates, defined by n 1∑ ′ m1 = E n j =1 j mi =

n 1∑ (E − m1′ ) j , i = 2, 3, · · · n j =1 i

(24.8.1)

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Introduction to Plastics Engineering

MACHINE DIRECTION (16 DATA POINTS)

0.5

0 5

0

10

TENSILE MODULUS EL (GPa) Figure 24.8.1 Histogram showing the normalized distribution of the left machine-direction tensile modulus EL in a drape-molded plaque. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

PROBABILITY DENSITY (1/GPa)

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MACHINE DIRECTION (161 DATA POINTS)

0.5

0 0

5

10

TENSILE MODULUS ER (GPa) Figure 24.8.2 Histogram showing the normalized distribution of the right machine-direction tensile modulus ER in a drape-molded plaque. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

PROBABILITY DENSITY (1/GPa)

Random Glass Mat Composites – Materials with Macrostructure

MACHINE DIRECTION (144 DATA POINTS)

0.5

0 0

5

10

15

TENSILE MODULUS EL (GPa) Figure 24.8.3 Histogram showing the normalized distribution of the left cross-machine-direction tensile modulus EL

PROBABILITY DENSITY (1/GPa)

in a drape-molded plaque. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

CROSS-MACHINE DIRECTION (144 DATA POINTS)

0.5

0 0

5

10

15

TENSILE MODULUS ER (GPa) Figure 24.8.4 Histogram showing the normalized distribution of the right cross-machine-direction tensile modulus ER in a drape-molded plaque. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

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where m1′ and mi are, respectively, the estimates for the moments ⟨E ⟩ and ⟨(E − ⟨E ⟩)i ⟩ defined by ∞

⟨E ⟩ =

∫−∞

E f (E ) dE

+∞

⟨(E − ⟨E ⟩)i ⟩ =

∫−∞

(E − ⟨E ⟩)i f (E ) dE,

i = 2, 3, 4, · · ·

(24.8.2)

in which f (E ) is the probability density function. While two different distributions can have the same arithmetic mean m1′ , the higher the order of the moments estimates (Eq. 24.8.1) to which all the lower moments of two distributions agree, the closer the properties of the two materials are. This property is used to compare sets of experimental data. The higher moments estimates are especially sensitive to data points away from the average m1′ – even a few such data points can have large effects on the higher moments. These differences provide means for filtering suspect extreme values in the data set. Table 24.8.1 lists the arithmetic mean estimates m1′ and the central moments m2 to m9 for the left and right moduli, EL and ER , for both the machine- and cross-machine-direction modulus data. All the data, including questionable data points, were used. Given the nature of the material, the agreement between the values of m1′ and m2 to m9 for the left and right machine-direction modulus data sets is remarkable. This suggests that the distributions for EL and ER are essentially the same, which is what should have been expected: the distinction between left and right depends on which face of the specimen is used as a reference. In the data for the cross-machine-direction, however, the moments m2 to m9 are systematically higher for the left-modulus data. This difference increases with the order of the moment, and is particularly large for m4 to m9 . One reason for this difference could be questionable data points, possibly caused by Table 24.8.1 Comparison of the moment estimates of the left- and right-modulus distributions of the machine- and cross-machine-direction data. Machine direction

Cross-machine direction

Moment

Left

Right

Left

Right

m1′

4.818

4.804

6.790

6.992

m2

0.712

0.733

2.592

1.708

m3

0.337

0.315

5.238

2.202

m4

1.638

1.753

41.34

12.08

m5

2.310

2.034

257

34.8

m6

7.2

7.1

1,971

155

m7

15.2

12.6

15,167

586

m8

41.9

36.7

119,862

2,539

m9

102

80

952,034

10,589

Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.

Random Glass Mat Composites – Materials with Macrostructure

extensometer slip in a few of the large number of tests; a small amount of slip can result in an overestimation of the local elastic modulus. As mentioned before, moment estimates can be used to detect such errors because the higher moments are very sensitive to even a few high values in the data. Table 24.8.2 shows the moment estimates calculated from the same four data sets used in Table 24.8.1, except that the two highest values have been deleted from each of these data sets: The rejection of only two values from each of the two sets of 161 data points for the left and right machine-direction moduli, and the two sets of 144 data points for the left and right cross-machine-direction moduli has resulted in a remarkable reduction in the differences between the left and right moments. Table 24.8.2 Comparison of the moment estimates of the left- and right-modulus distributions of the machine- and cross-machine-direction data, with the two extreme values deleted from each data set. Machine direction

Cross-machine direction

Moment

Left

Right

Left

Right

m1′

4.812

4.799

6.752

6.977

m2

0.654

0.676

2.128

1.539

m3

0.256

0.242

2.058

1.720

m4

1.249

1.376

12.906

8.904

m5

1.49

1.29

31.5

22.0

m6

4.45

4.48

143

90.0

m7

8.39

6.54

516

298

m8

21.8

18.4

2,271

1,169

m9

48.4

33.3

9,620

4,323

Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.

There are two possible explanations for the large differences between the moments for the machineand cross-machine-direction data: First, since the data for the two directions are obtained from tests on specimens from two different plaques, these differences could result from spatial material nonhomogeneities resulting from the lamination process used for manufacturing the material, independent of orientation with respect to lamination direction. Second, these differences could reflect global machine- and cross-machine-direction material anisotropy induced by the lamination process. Here, global anisotropy refers to the orientation dependence of the modulus distribution means and moments. It says nothing about the local anisotropy, which refers to a local (pointwise) dependence of the elastic modulus on direction, which can only be determined by the measurement of the elastic modulus of each (12.7 × 12.7 mm) segment along both directions – something which cannot be done by simple tensile tests. Alternatively, tests on specimens cut from a large number of plaques could be used to rule out spatial nonhomogeneity effects. The next step in the analysis is to use these moment estimates to determine the probability density function for the tensile modulus distribution. This is considered in the next section, where only the first four moment estimates have been used.

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24.8.3

Probability Density Function for the Tensile Modulus

A smooth probability density function that models the main features of the modulus data shown in the histograms in Figures 24.8.1 – 24.8.4 – unimodal, skewed, with upper and lower limits – is given by f̃ (E) = k (E − Emin ) p ( E − Emax ) q

(24.8.3)

defined on the interval Emin < E < Emax . The four parameters Emin , Emax , p, and q can be expressed in terms of the moment estimates m1′ , m2 , m3 , and m4 calculated from the experimental data sets, by Emin = a − 𝛼1 ,

Emax = a + 𝛼2 ,

p=

𝛼1 b2 Δ1∕2

,

p=

𝛼2 b2 Δ1∕2

(24.8.4)

in which the parameters a, b2 , Δ, 𝛼 1 , and 𝛼 2 are defined by m3 (m4 + 3m2 2 ) 2m m − 3m3 2 − 6m2 3 , b2 = − 2 4 , A A (b + 2b2 a)2 − 4b2 (b0 + b1 a + b2 a2 ) , Δ= 1 b2 2 ] ] [ [ b1 + 2b2 a b1 + 2b2 a 1 1 1∕2 1∕2 𝛼1 = , 𝛼2 = Δ + Δ − 2 b2 2 b2 a = m1′ −

(24.8.5)

in which m1 2 (6m2 3 + 3m3 2 − 2m2 m4 ) m m (m + 3m2 2 ) m (4m2 m4 − 3m3 2 ) + 1 3 4 − 2 A A A 2m1 (2m2 m4 − 6m2 3 − 3m3 2 ) m3 (m4 + 3m2 2 ) b1 = − A A A = 10m2 m4 − 12m2 3 − 18m2 3 b0 =

(24.8.6)

The parameter a defined in Eq. 24.8.5 is the mode of the function f̃ (E). The width b of the function f̃ (E), defined as the distance of the two inflection points of the distribution, is given by √ b0 + b1 a + b2 a2 b= b2 − 1

(24.8.7)

Equations 24.8.4 – 24.8.7 were used to calculate the limits Emin and Emax , mode a, inflection points E1 and E2 , and width b. The results for the four data sets used for constructing Table 24.8.2 are listed given in Table 2.8.3. Note that the first inflection point E1 for the left cross-machine-direction is less than Emin , the minimum value of the modulus, and is therefore outside the allowable range of the modulus, so that the width b loses its meaning (indicated by an asterisk in Table 24.8.3). For this case standard deviation should be used to describe the width of the probability density function. In general, this happens when either one of the exponents p or q is less than unity. The smooth curves approximating the probability density functions for the four data sets are shown by the smooth curves in Figures 24.8.1 – 24.8.4.

Random Glass Mat Composites – Materials with Macrostructure

Table 24.8.3 Comparison of the parameters Emin , Emax , a, E1 , E2 , and b.

Limits Data set

Emin

Machine direction Cross-machine direction

Emax

Inflection point

Mode a

E1

E2

Width b

Left

2.97

9.34

4.54

3.61

5.46

1.85

Right

2.52

11.02

4.58

3.70

5.46

1.76

Left

4.46

12.54

5.70

3.74

7.60

3.86*

Right

4.92

17.28

6.21

5.00

7.42

2.42

Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.

24.8.4

Higher Order Moments

The probability density function estimate f̃ (E ) defined in Eq. 24.8.3 can be used to compute the expected value ⟨w⟩ of a function w (E). In general, the expected value ⟨w⟩ =

Emax

∫E

w (E ) f̃ (E ) dE

(24.8.8)

min

exists for w(E ) not being singular in the interval Emin < E < Emax . The arithmetic mean estimate m1 and the central moment estimates m2 to m4 , calculated by using the function f̃ (E ) must equal the respective moment estimates calculated directly from the data sets (see Table 24.8.2). It can be shown that mi (Emin ) = k (Emax − Emin ) p+q+i+1 B( p + q + i + 1, q + 1)

(24.8.9)

where mi (Emin ) is the ith moment calculated about Emin , and B( p, q) is the Beta function. Moments (and their estimates) calculated with respect to one value of a random variable, say x1 , can be related to moments (and their estimates) calculated with respect to one value, say x2 . In general mi (x2 ) =

j=i ( ) ∑ i j

mj (xi ) (x1 − x2 ) i−j

(24.8.10)

j=0

In particular, for x1 = Emin , i = 1, and x2 = 0 m1 (0) = m1′ = Emin + m1 (Emin )

(24.8.11)

For x1 = Emin and x2 = m1′ estimates for the central moments can be related to those calculated with respect to Emin by using mi = mi (m1′ ) =

j=i ( ) ∑ i j

(−1) i−j mj (Emin ) m1 (Emin ) i−j

j=0

where i may take any integer value greater than zero.

(24.8.12)

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Equations 24.8.9 – 24.8.12 were used to compute the moment estimates listed in Table 24.8.4. The moments m5 to m9 in this table, calculated from the probability density function f̃ (E), are in good agreement with those calculated from the raw data (Table 24.8.2). Table 24.8.4 Comparison of the moment estimates m5 to m9 calculated from the probability density function estimate f̃ (E). Machine direction Right

Cross-machine direction

Moment

Left

Left

Right

m5

1.40

1.48

31.4

26.2

m6

4.23

5.07

137

114

m7

7.78

9.44

479

478

m8

21.0

28.3

1,999

2,252

m9

49.1

70.1

8,061

11,058

Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.

Besides the arithmetic mean estimate m1 and a few other central moment estimates, one mean of interest is the harmonic average EH defined by E

max 1 1 = f (E ) dE EH ∫Emin E

(24.8.13)

The harmonic mean exists when Emin > 0 and can be very conveniently expressed as a quickly converging series of central moment estimates mi , i = 0, 1, . . . . The derivation of an expression for the harmonic ̃H involves an expansion of 1∕E in Eq. 24.8.13 as a Taylor series about the arithmetic mean estimate E mean estimate m1 ≠ 0, resulting in ∞ i 1 ∑ (−1) mi 1 = ′ ̃H m1 i = 0 (m1′ ) i E

(24.8.14)

Note that this equation is valid for all probability density functions and not just for that given in Eq. 24.8.3. For the distribution in Eq. 24.8.3 the series expansion may be terminated after n terms, say. For this distribution the central moments can be calculated directly from Eqs. 24.8.9 and 24.8.12. But a direct numerical evaluation of Eqs. 24.8.9 and 24.8.12) may be difficult because of the very high values of the factorials required for calculating central moments of high order. A more desirable approach is to use Eq. 24.8.8 to derive a recurrence relation for the estimates for the absolute moments (about zero), mi′, and the central moments (about the arithmetic mean), mi . This recurrence relation is given by ′ = mi+1

′ [a − (n + 1) b1 ] − nb0 mi−1

(n + 2) b2 + 1

(24.8.15)

where the parameters a, b0 , b1 , and b2 are given by Eqs. 24.8.5 through 24.8.7, m0′ = 1, m1′ is an arithmetic mean estimate, and mi′ denotes the ith absolute moment estimate.

Random Glass Mat Composites – Materials with Macrostructure

Equation 24.8.15 can also be used to calculate central moment estimates mi by replacing parameter a by a − m1′ , parameter b0 by b0 + b1 m1′ + b2 m1′2 , and parameter b1 by b1 + 2b2 m1′ . Table 24.8.5 lists arithmetic and harmonic mean estimates calculated from the original data sets. Equation 24.8.15 gives moment estimates m2 to m4 , which were used to obtain parameters a, b0 , b1 , and b2 . Table 24.8.5 Comparison of the arithmetic and harmonic means obtained from the raw data sets with those obtained from Eq. 24.8.14.

Arithmetic mean

From data

From f̃(E)

Left

4.812

4.681

4.6809

Right

4.799

4.661

4.166144

Left

6.752

6.461

6.46151

Right

6.977

6.777

6.77831

Data set

Machine direction Cross-machine direction

Harmonic mean

Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.

24.9 Statistical Properties of Tensile Modulus Data Sets Data analysis has shown that the distributions of EL and ER in the same plaque are the same. This raises interesting questions: Are the values of E spatially correlated? Does a low (high) value of EL correlate with the value of ER at the same location? Even more importantly, is it possible to obtain the distributions of EA from that of EL (or ER )? 24.9.1

Correlation Between the Left and Right Moduli

Whether or not the modulus measured in a segment is related to the modulus of an adjacent segment is of interest. So, also, would be any correlation between modulus values measured along the two edges of the same segment (left and right-modulus). A possible correlation between EL and ER can be visualized by defining a plane spanned by the leftand right-modulus axes. A point Pi on this plane having coordinates (EL (i), ER (i)) represents the left and right moduli in the ith segment. Such a plot for the two data sets is shown in Figure 24.9.1; clearly, EL and ER appear to be uncorrelated. A quantitative measure of the correlation between the two random variables EL and ER is provided by the correlation coefficient Cov [EL , ER ] (24.9.1) 𝜌EL , ER = √ Var[EL ] Var[EL ] For the two data sets this equation gave a correlation coefficient of

𝜌EL , ER = − 0.1167 This low value of the correlation coefficient suggests that EL and ER are uncorrelated.

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10

RIGHT TENSILE MODULUS E R (GPa)

944

5

0 5

0

10

LEFT TENSILE MODULUS E L (GPa) Figure 24.9.1 Plot of (EL , ER ) pairs for all segments. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

24.9.2

Linear Combination of Two Independent Random Variables

Consider two independent random variables EL and ER specified by their moments ⟨EL j ⟩ and ⟨ER j ⟩, j = 1, 2, . …

(24.9.2)

Now define a new random variable Z by Z = 𝛼L EL + 𝛼R ER

(24.9.3)

in which 𝛼 L and 𝛼 R are arbitrary coefficients. Then, in terms of the moments of the random variables EL and ER , the set of moments ⟨Z i ⟩ = gi (𝛼L , 𝛼R , ⟨EL j ⟩, ⟨ER j ⟩) ,

j = 1, 2, . …

of the random variable Z are given by j=i ( ) ∑ j i−j i i ⟨Z ⟩ = ⟨EL j ⟩⟨ER i−j ⟩ j 𝛼L 𝛼R

(24.9.4)

(24.9.5)

j=0

Note that a determination of the ith moment of random variable Z requires the first i moments of the random variables EL and ER . For 𝛼 L = 𝛼 R = 1∕2 the random variable Z is the arithmetic mean of the two random variables EL and ER . Equation 24.9.5 was used to calculate the first four moments of the random variable Z = (EL + ER )∕2. These moments, together with those directly calculated the experimental data set EA = (EL + ER )∕2, are listed in Table 24.9.1.

Random Glass Mat Composites – Materials with Macrostructure

Table 24.9.1 Comparison of the experimentally and theoretically obtained arithmetic mean estimate m1′ and the central moment estimates m2 to m4 for the average modulus EA data set. Values Moment

Experimental

Theoretical

m1′

4.809

4.806

m2

0.302

0.332

m3

0.041

0.062

m4

0.212

0.328

Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.

The arithmetic mean estimate m1′ and the central moment estimates m2 to m4 were then used to fit a continuous probability density function using the procedure outlined in Section 24.8.3. Figure 24.9.2 shows the histogram for the experimental data together with the computed continuous probability density function. This curve (dashed line) is then compared with the theoretically obtained probability density function in Figure 24.9.3.

PROBABILITY DENSITY (1/GPa)

1.0

0.5

0 0

5

10

AVERAGE MODULUS (GPa) Figure 24.9.2 Histogram and probability density function of experimentally determined distribution of the average modulus EA . (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

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Introduction to Plastics Engineering

1.0

PROBABILITY DENSITY (1/GPa)

946

THEORETICAL EXPERIMENTAL

0.5

0 0

5

10

AVERAGE MODULUS (GPa) Figure 24.9.3 Comparison of experimentally (dashed line) and theoretically (solid line) obtained average modulus distributions. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

The excellent agreement between the theoretically and experimentally obtained probability density functions strongly supports the assumption of the independence of the left and right moduli. It is therefore sufficient to measure the modulus only along one edge of the sample; this information can then be used to obtain the probability function for the average modulus.

24.10 Gauge-Length Effects and Large-Scale Material Stiffness The issue of statistical homogeneity must be considered while obtaining experimental modulus data from specimens cut from plaques. The material would be considered statistically homogeneous when the probability density function characterizing the modulus population is the same at every point in the plaque. The experimental procedure consists of conducting tensile tests on long specimens – lengths much larger than widths – cut from plaques. To obtain a sufficiently large data set for each specimen, the gauge length (segment length on specimen) must be small compared with the overall specimen length. The gauge length, the specimen length, and the plaque size (from which the specimens are cut), must be balanced to obtain data that can be used for determining the probability density function. Assurance of statistical homogeneity requires a plaque that is small in comparison to local features of a part, which in turn requires that the stretching of specimens cut from such plaques be measured over even smaller gauge lengths (smaller gauge-length extensometers). But very small gauge lengths could result in the

Random Glass Mat Composites – Materials with Macrostructure

measurements on adjacent segments not being statistically independent. This introduces a new level of complexity for which many of the simple procedures discussed in this chapter cannot be used: the dependence of adjacent samples must be characterized by a conditional probability function. Also, while the local mechanical properties parts in which the glass has flowed – such as in Figures 24.2.2 and 24.2.3 – can be expected to be affected by the glass flow during processing, such effects have not been considered in this chapter. 24.10.1

Sample Size: Theoretical Considerations

All the data discussed in this chapter were obtained by extension measurements on 12.7-mm segments in long specimens. This 12.7-mm gauge-length data can be used to generate for larger gauge lengths. For example, if two 12.7-mm gauge-length extensometers attached to two adjacent segments in the same specimen measure strains 𝜀1 and 𝜀2 , respectively, so that in terms of the homogeneous tensile stress 𝜎 the effective elastic moduli in these two segments are E1 = 𝜎 ∕𝜀1 and E2 = 𝜎 ∕𝜀2 , respectively, then the effective modulus E(2) for the 25.4-mm (1-in) segment comprising the two adjacent 12.7-mm segments is given by the harmonic mean ) ( 1 1 1 1 = + (24.10.1) E(2) 2 E1 E2 in which the subscript (2) indicates that the effective modulus of the 25.4-mm segment is obtained from moduli from two 12.7-mm segments. It follows that the tensile modulus, E(n) , for a gauge length corresponding to n segment in series would be the harmonic mean n 1∑ 1 1 = E(n) n i =1 Ei

(24.10.2)

in which Ei is the modulus of the ith segment. Using this relationship, 12.7-mm modulus data from one long specimen can be used for to obtain effective moduli corresponding to gauge lengths that are multiples of 12.7 mm. Here again, the elastic modulus is a random variable and each measurement is only one outcome from a continuous rang of possible values. In particular, the randomness of the elastic modulus E(n) is fully characterized only when the probability distribution function f(n) is specified. This distribution can be obtained from f(1) , just as E(n) was obtained from E(1) . Consider a segment made up of two shorter segments connected in series. The moduli E(1) and E(2) of these shorter segments are characterized by the same, known probability density function f(1) where, as before, the subscript (1) indicates that the modulus distribution is based a one-unit segment (12.7-mm in this case). To determine the probability function f(2) (E ) for the effective modulus distribution for two-unit long segments, define the joint probability function f (E1 , E2 ) such that dP = f (E1 , E2 ) dE1 dE2 is the probability that the modulus of the short-segment 1 lies between E1 and E1 + d E1 and that of the short-segment 2 lies between E2 and E2 + dE2 . The joint probability density function f (E1 , E2 ) can be expressed in terms of the probability function f(1) (E1 ) and the conditional probability density function f (E2 |E1 ) as f (E1 , E2 ) = f(1) (E1 ) f (E2 |E1 )

(24.10.3)

where f (E2 |E1 ) is the probability of occurrence of the short-segment 2 modulus, E2 , given the of short-segment modulus, E1 . When the measured modulus on one segment does not depend on measured modulus of an adjacent segment, the conditional probability function f (E2 |E1 ) is just f(1) (E2 ).

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Let P(E(2) ) denote the probability that the effective modulus of two segments connected in series is less than E(2) . Then, by invoking the assumption of spatial independence of adjacent segments P (E(2) ) =

∫∫S(E

f(1) (E1 ) f(2) (E2 ) dE1 dE2 (2)

(24.10.4)

)

where S(E(2) ) denotes the portion of the two-dimensional space (E1 , E2 ) in which ) ( 1 1 1 1 ≥ + E(2) 2 E1 E2

(24.10.5)

A differentiation of P with respect to E(2) results in the probability density function for the effective modulus of the two-unit long segment dP(E(2) ) dE2

f(2) (E2 )

(24.10.6)

This procedure for obtaining the probability density function f(2) for the effective modulus E(2) can easily be extended to more than two segments connected in series. For the n-unit long segment the probability P that the effective modulus is less than E(n) is P(E(n) ) =

∫

…

∫S (E

f(1) (E1 ) f(2) (E2 ) … f(2) (En ) dE1 dE2 … dEn (n)

(24.10.7)

)

where S(E(n) ) is a subregion of the n-dimensional space (E1 , E2 , … En ) in which n 1 1 ∑ 1 ≥ E(n) n i =1 Ei

(24.10.8)

A differentiation of Eq. 22.10.7 gives the desired probability function f(n) . The procedures outlined here can be effectively used to determine the resultant probability density functions. Clearly, f(n) describes the stiffness of a structure consisting of n-unit segments connected in series subjected to uniform stress. 24.10.2

Sample Size: Numerical Experiments

New data sets were generated by combining measured values corresponding to adjacent segments into n-element subsets, and the effective modulus E(n) was calculated for every such data subset. The effective modulus was then calculated for every subset by using Eq. 24.10.1. The probability density function fitting procedure described previously was then applied to every new data set. The results for the leftand right-data set are shown, respectively, in Figures 24.10.1 and 24.10.2. Both these figures show the probability density functions for segment lengths of 12.7 mm (0.5 in) (one unit), 38. mm (three units), and 76.2 mm (six units). Note that the curves corresponding to longer segments are necessarily obtained from a fewer number of points, resulting in coarser estimates. On the other hand, the application of the harmonic mean formula in Eq. 24.10.1 to adjacent segments brings all values closer to the harmonic mean of the population. The widths of the resulting probability density functions decrease as the sample length increases. This implies that the scatter in the measurements based on longer samples would be smaller, and fewer data points would be required to assure a specified level of accuracy. As the number of segments n in an n-unit

Random Glass Mat Composites – Materials with Macrostructure

PROBABILITY DENSITY (1/GPa)

1.5

0.5

n = 6 (76.2 mm)

n = 3 (38.1 mm) 0.5 n = 1 (12.7 mm)

0 5

0

10

AVERAGE MODULUS (GPa) Figure 24.10.1 Comparison Calculated probability distributions of the left effective modulus for three gauge lengths of 12.7, 38.1, and 76.2 mm. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

long segment becomes very large, the measured moduli can be expected to be close to the population harmonic mean. That is, lim f(n) (E) = 𝛿 (E − EH )

n→∞

(24.10.9)

where 𝛿 is the Dirac delta function, and EH is the harmonic mean of the population probability density function f(1) (E). Dirac’s delta function on the right-hand side of this equation is the probability density function of an ideal homogeneous material of modulus EH . The rate of convergence f(n) (E) to 𝛿 (E − EH ) can be evaluated by considering a very long specimen of n segments, with n → ∞. The resulting stiffness E(∞) is then given by 1 E(∞)

n 1 1∑ 1 = lim n→∞ E n→∞ n Ei (n) i =1

= lim

(24.10.10)

For sufficiently large n the moduli Ei , i = 1, 2, … , n will appear in Eq. 24.10.1 with frequencies characterized by the probability density function. Let the range of modulus values be divided into m intervals. The numbers E1 , E2 , … , En can further be arranged into m groups corresponding to the m intervals such that E ( j ) < E1 ( j ) , E2 ( j ) , … , Ekj ( j ) < E ( j + 1)

(24.10.11)

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1.5

PROBABILITY DENSITY (1/GPa)

950

0.5 n = 6 (76.2 mm)

n = 3 (38.1 mm)

0.5

n = 1 (12.7 mm) 0 5

0

10

AVERAGE MODULUS (GPa) Figure 24.10.2 Comparison Calculated probability distributions of the right effective modulus for three gauge lengths of 12.7, 38.1, and 76.2 mm. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

where j = 1, 2, … , m indicates the jth intervals with limits E ( j ) and E ( j+1) , having kj moduli in the interval ( j ). Equation 22.10.10 can then be written as 1 E(∞)

m 1 1 ∑ ki = lim n→∞ E n→∞ n E (i ) (n) i =1

= lim

m ∑ ki 1 = lim n→∞ n E (i ) i =1

(24.10.12)

In the limit ki ∕n can be replaced by the corresponding probability and the summation by an integral resulting in 1 E(∞)

+∞

=

∫−∞

f (E ) 1 dE = E EH

(24.10.13)

This value of EH is the same for every possible mapping of the modulus distribution onto an infinitely long specimen. Since the arithmetic mean of a set of the same numbers is the same number, EH is also the expected value of the modulus for all possible infinitely long specimens.

Random Glass Mat Composites – Materials with Macrostructure

PROBABILITY DENSITY FUNCTION WIDTH

The convergence process expressed by Eq. 22.10.9 is illustrated in Figures 24.10.3 and 24.10.4, which show, respectively, the effect of the specimen size on the width of the probability density function – characterized by the standard deviation 𝜎 – and on the expected value E(n) of the average modulus.

1.0

0.5

0 0

5

10

NUMBER OF GAUGE-LENGTH SEGMENTS Figure 24.10.3 Variation of the probability density function width with the gauge length. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

24.11 Methodology for Predicting the Stiffness of Parts Because no two parts made of GMT will have the same distribution of elastic moduli, different parts having the same geometry will exhibit different stiffnesses. Therefore, for this class of materials, it is not just the performance of one particular part that is of interest, but rather the behavior of all possible parts of a given geometry. The spatial distribution of the elastic modulus results from a random mapping of the elastic modulus population, characterized by a probability density function f (E ). This section discusses a framework for using the statistical properties of GMTs for predicting the stiffness of parts. The goal is to use the probability density function to develop a procedure for obtaining an effective modulus that best characterizes the response of the part.

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Introduction to Plastics Engineering

5.0

EXPECTED VALUE OF E(n)

952

EH

4.5 5

0

10

NUMBER OF GAUGE-LENGTH SEGMENTS Figure 24.10.4 Evolution of the expected value of E(n) with increasing gauge length. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 295 – 308, 1992.)

As an example, consider the rectangular part consisting of 3 × 8 = 24 segments, each 12.7 × 12.7 mm as shown in Figure 24.11.1, that is symmetrically loaded with a force F. To simulate the randomness in material properties, the elastic modulus for each of the segments can be specified by a random sampling of the elastic modulus population characterized by the known probability density function f (E). Consider the x-component of the normal stress. For one distribution of the elastic modulus across the part (Figure 24.11.2a), contours of the stress normalized by the mean stress to which the plaque is subjected show large, random variations of the stress consistent with the random character of the material. However, this particular stress distribution by itself is of little importance from a design standpoint. Contours of the average stress distributions obtained from stress analyses of 10, 100, and 200 similar parts are shown, respectively, in Figures 24.11.2b – d. As the number of parts increases, the average stress patterns become more regular, and increasingly reflect the symmetry of the part and boundary conditions. The range of stress variation in these cases is small. To better understand the part averaging process, it is helpful to introduce a few terms. Let the geometry of a part and the boundary conditions on it be specified. The material properties – here the elastic modulus – can be mapped onto the geometry of the part in an infinite number of ways. Let each of the mappings consist of a simple sampling of the modulus population characterized by the known probability density function f (E). A particular mapping is indicated by the superscript 𝛼 , so that the spatial distribution of the modulus in the 𝛼 th part is E(𝛼 ) (r),

𝛼 = 1, 2, 3, … , ∞

(24.11.1)

Random Glass Mat Composites – Materials with Macrostructure

y

A

B F

D

x

C

Figure 24.11.1 A specimen comprising 3 × 8 = 24 segments subjected to a tensile force F. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 309 – 316, 1992.)

For each such mapping (each different part), a structural analysis of the part can be performed. Among the many quantities of interest (deformation, stress and strain distributions, reaction forces, etc.) consider any particular quantity Q that has the value Q (𝛼 ) for the 𝛼 th mapping. In a majority of the cases, the value of Q will depend on the elastic modulus mapping. This dependence is indicated by the superscript 𝛼 in Q (𝛼 ) = Q(E (𝛼 ) (r))

(24.11.2)

If the probability P (𝛼 ) of the 𝛼 th modulus mapping is known, then the expected value ⟨Q⟩ of Q over all possible mappings 𝛼 is ∑ Q (𝛼 ) P (𝛼 ) (24.11.3) ⟨Q⟩ = 𝛼

The expected value ⟨Q ⟩ is the single number that best characterizes the response of the part. Once the expected value ⟨Q ⟩ is determined, the expected modulus E, if it exists, is the number that, when used as the modulus of a homogeneous material, results in a value of Q that equals the expected value ⟨Q⟩. This can be written as ∑ ∑ Q(𝛼 ) P(𝛼 ) = Q (E (𝛼 ) (r)) P (𝛼 ) = Q (E ) (24.11.4) ⟨Q⟩ = 𝛼

𝛼

Clearly, such an effective modulus E is not a material property because it applies only to a particular geometry and to a particular set of boundary conditions. That the effective modulus E is tied to the geometry and boundary conditions can be illustrated by the following example: Consider a strip consisting of n segments (Figure 24.11.3), similar to the specimen used to obtain the probability density function f (E). For the 𝛼 th modulus mapping, let ui(𝛼 ) denote the displacement of the ith segment in the strip as measured from the left end of the strip. Then un(𝛼 )

=

n ∑

Δ ui(𝛼 )

(24.11.5)

i =1

where Δ ui(𝛼 ) denoted the extension of the ith segment. Let the displacement boundary condition at the right-hand end off the strip be specified as un(𝛼 ) = u0 . Using a one-dimensional elastic relationship between

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Introduction to Plastics Engineering

(d)

200 Parts

> 1.15

100 Parts

1.10-1.15

(c)

1.05-1.10

10 Parts

1.00-1.05

(b)

0.95-1.00

1 Part

0.90-0.95

(a)

< 0.90

954

Figure 24.11.2 Contours of the x-component of the normal stress for the specimen shown in Figure 24.11.1: (a) for 1 part, (b) for 10 parts, (c) for 100 parts, and (d) for 200 parts. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 309 – 316, 1992.)

Random Glass Mat Composites – Materials with Macrostructure

E1(α)

E2(α)

E j (α)

En (α)

(1)

(2)

( j)

(n)

nl0 Figure 24.11.3 A specimen made up of n segments of length l0 connected in series. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 309 – 316, 1992.)

the local (at a segment level) strains 𝜀i(𝛼 ) and the stress 𝜎i(𝛼 ) , u0 =

n ∑ i =1

l0 𝜀i(𝛼 ) =

n ∑

l0

i =1

𝜎 (𝛼 )

Ei(𝛼 )

= l 0 𝜎 (𝛼 )

n ∑ 1 i =1

Ei(𝛼 )

(24.11.6)

where l0 is the length of each segment. The index i has been dropped from the stress 𝜎 (𝛼 ) because all the segments are under the same stress. However, as indicated by the superscript 𝛼 , the stress still depends on the modulus mapping. Let the stress in the strip be chosen as the quantity of interest Q. Then the expected value ⟨𝜎 ⟩ of the stress 𝜎 (𝛼 ) best describes the response of the population of strips. Equation 24.11.16 can be solved with respect to 𝜎 (𝛼 ) yielding u u0 𝜀0 𝜎 (n) = n 0 = = (24.11.7) n n ∑ 1∑ 1∑ (𝛼 ) (𝛼 ) l0 1∕Ei(𝛼 ) nl0 1∕Ei 1∕Ei i =1 n i =1 n i =1 where 𝜀0 is the total strain corresponding to the imposed displacement boundary condition. The expected value of the stress is then given by ∑ ∑ P (𝛼 ) 𝜎 (𝛼) P (𝛼) = 𝜀0 ⟨𝜎 ⟩ = n 𝛼 𝛼 1∑ 1∕Ei(𝛼 ) n i =1

(24.11.8)

This equation can be used to calculate the expected value of stress in the n-segment-long strip, provided that the spatial distribution of the modulus is given. Consider two limiting cases, the first with n = 1 (a short strip) and the second with n = ∞ (a very long strip). In the first, short-strip case, ∑ P (𝛼 ) ∑ (𝛼 ) = 𝜀0 Ei P (𝛼 ) = 𝜀0 EA (24.11.9) ⟨𝜎 (1)⟩ = 𝜀0 (𝛼 ) 𝛼 1∕Ei 𝛼 where EA denotes the arithmetic mean of the modulus over the entire modulus population. From the definition of the effective modulus it then follows that the effective modulus E equals the modulus EA because ⟨𝜎 (1)⟩ = 𝜀0 E = 𝜀0 EA

(24.11.10)

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Introduction to Plastics Engineering

Equation 24.11.8 can also be used to evaluate the effective modulus when the strip under consideration is very long. In this case ⟨𝜎 (∞)⟩ =

∑ 𝛼

𝜎 (𝛼) P (𝛼) = 𝜀0

∑ 𝛼

P (𝛼 ) n 1∑ lim 1∕Ei(𝛼 ) n→∞ n i =1

(24.11.11)

The limit in the denominator of this equation was shown to equal the inverse of the population harmonic mean modulus EH . Therefore ⟨𝜎 (∞)⟩ = 𝜀0

∑ P (𝛼 ) ∑ = 𝜀0 EH P (𝛼 ) = 𝜀0 EH 1∕E H 𝛼 𝛼

(24.11.12)

Thus, for very long strips the effective modulus is the harmonic mean EH . These examples show that the effective modulus depends on the length of the strip and is, therefore, a system rather than a material property. The two effective moduli EA and EH obtained for short and long strips, respectively, can be used to predict the expected stress levels. Because the harmonic mean EH is always smaller than the arithmetic mean EA , for the same strain 𝜀0 the expected value of the tensile stress in the long specimen will always be smaller than in the short one. That is, ⟨𝜎 (1)⟩ = 𝜀0 EA > ⟨𝜎 (∞)⟩ = 𝜀0 EH

(24.11.13)

where ⟨𝜎 (1)⟩ and ⟨𝜎 (∞)⟩ denote the expected values of stress in the short and long strips, respectively. These are only expected values of the stress. Some fluctuations in this random variable can be expected with changes in the stress mapping 𝛼 . Normally, the second central moment is used as a measure of fluctuations. If the random variable 𝜎 (𝛼 ) is characterized by a probability P (𝛼 ) , then ∑ 𝜎 (𝛼 ) P (𝛼 ) = ⟨𝜎 2 ⟩ − ⟨𝜎 ⟩2 (24.11.14) ⟨(𝜎 − ⟨𝜎 ⟩)2 ⟩ = 𝛼

where ⟨(𝜎 − ⟨𝜎 ⟩)2 ⟩ and ⟨𝜎 ⟩2 denote the second central moment and the second moment of the stress, respectively. Equation 24.11.14 can be used to evaluate the fluctuation of stresses in short and long strips. The second moment of the random variable 𝜎 (𝛼 ) for the short strip (n = 1) is ( )2 ∑ (𝛼 ) ∑ ∑ 𝜀0 2 (𝛼 ) 2 (𝛼 ) (𝜎 ) P = P (𝛼 ) = 𝜀0 2 (Ei )2 P (𝛼 ) (24.11.15) ⟨𝜎 ⟩ = (𝛼 ) 1∕E 𝛼 𝛼 𝛼 i The last sum in this equation equals the second moment of the elastic modulus, and is therefore denoted by ⟨E2 ⟩. Substitution of this expression in Eq. 24.11.14 yields ⟨(𝜎 (1) − ⟨𝜎 (1)⟩)2 ⟩ = 𝜀0 2 (⟨E2 ⟩ − ⟨E ⟩2 ) = 𝜀0 2 (E − ⟨E ⟩2 )

(24.11.16)

Thus, the stress fluctuations in the short strip (n = 1) are proportional to the modulus fluctuations, the scaling factor being 𝜀0 . That is, the randomness of the stress in the short strip is the same as the randomness of the modulus. This property of short strips can be better understood by a comparison with the fluctuations in a long strip, for which the expected value of the squared fluctuations

Random Glass Mat Composites – Materials with Macrostructure

(second central moment of the stress in the strip) is zero: Let the second moment 2

(∞)2 =

( )

P ( ) = lim

n→∞

1 n

n i =1

0

1∕Ei(

)

P(

)

(24.11.17)

be first calculated. As before, the expression in the denominator is the harmonic mean EH , so that ∑ ⟨𝜎 (∞)2 ⟩ = 𝜀0 2 EH 2 P (𝛼 ) = 𝜀0 2 EH 2 (24.11.18) 𝛼

A substitution from this equation in Eq. 24.11.14 then determines the expected value of the squared fluctuation of the stress to be ⟨(𝜎 (∞) − ⟨𝜎 (∞)⟩)2 ⟩ = 𝜀0 2 EH 2 − ⟨𝜎 (∞)2 ⟩ = 𝜀0 2 EH 2 − (𝜀0 EH )2 = 0

(24.11.19)

Thus, fluctuations should not be expected in very long strips. A long strip would therefore appear to be made up of a homogeneous material having an elastic modulus equal to the harmonic mean, EH , over the modulus population. 24.11.1

*Effective Structural Stiffness

Analytical solutions of the type obtained for long and short strips cannot normally be obtained for more general cases, for which an effective modulus may not even exist in the sense required by the definition. The discussion on effective moduli will therefore be continued using the terminology and notation commonly used in the finite element method that is suited for complex geometries and boundary conditions. The elastic response of a three-dimensional part can be characterized by [K (𝛼 ) ]{d} = {R}e + {F (𝛼 ) }𝜀0

(24.11.20)

where [K(𝛼 ) ] is a global stiffness matrix, the vector {R}e represents the nodal forces resulting from external concentrated and distributed forces, the vector {F (𝛼 ) }𝜀0 denotes equivalent nodal forces that result from the initial strain field 𝜀0 , and {d} denotes the unknown vector of nodal displacements. The superscript 𝛼 indicates the dependence of the quantity on the elastic property distribution (Young’s modulus and Poisson’s ratio). The dependence on the material mapping comes in through the corresponding equations for the element stiffness, and the element force for an in initial strain 𝜀0 , in the form [K (𝛼 ) ](e) = {F (𝛼 ) }𝜀 = 0

∫V e ∫V e

[B]T [E (𝛼 ) ](e) [B] dV [B]T [E (𝛼 ) ](e) {𝜀0 }dV

(24.11.21)

where [B] is the strain-displacement matrix, [E (𝛼 ) ] is the material property matrix, and the superscript (e) indicates that the quantity is an elemental property. Consider the case in which the initial strain 𝜀0 is identically zero. Furthermore, if the displacement boundary conditions are zero, then Eq. 24.11.20 can be simplified to [K (𝛼 ) ]{d} = {R}e

(24.11.22)

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Introduction to Plastics Engineering

in which the load vector on the right-hand side is independent of the mapping 𝛼 . Equation 24.11.22 can be solved for the nodal displacement vector {d (𝛼 ) } = [K (𝛼 ) ]−1 {R}e

(24.11.23)

The definition of an elastic modulus requires that the quantity of interest be specified. Since a finite element problem is normally formulated in terms of unknown nodal displacements, let them be the quantities of interest. The expected values of the nodal displacement are then ∑ {⟨d⟩} = {d (𝛼 ) }P (𝛼 ) (24.11.24) 𝛼

Also, by definition, the effective global stiffness matrix [K] satisfies [K]{⟨d⟩} = {R}e Then, substitutions from Eqs. 22.11.24 and 22.11.23 in Eq. 22.11.25 results in [ ] ∑ (𝛼 ) −1 (𝛼 ) [K] [K ] P − 1 {R}e = 0

(24.11.25)

(24.11.26)

𝛼

which holds for every external load vector {R}e . This equation can be solved for the global stiffness matrix ]−1 [ ∑ (𝛼 ) −1 (𝛼 ) [K ] P (24.11.27) [K] = 𝛼

This equation is a generalization of the harmonic mean introduced in the previous section. Here, the summation occurs over all stiffness matrices, indexed by 𝛼 , for a specified fixed geometry. The effective structural stiffness therefore depends on the geometry but is independent of the external forces and boundary conditions, provided that the global load vector does not depend on the material property distribution. Consider the case in which the load vector depends on the distribution of the elastic properties of the material, which occurs when the initial strains 𝜀0 are specified (thermal strains, for example) or when nonzero displacement boundary conditions are imposed. In this case the effective structural stiffness [K] does not exist: Let the effective structural stiffness exist, and be independent of the mapping 𝛼 , as required by the definition. Furthermore, assume that the expected that the expected value of the displacement vector exists and that it is also independent of the material mapping. Then, the definition of the effective stiffness requires that [K (𝛼 ) ]{⟨d⟩} = {R (𝛼 ) }

(24.11.28)

The load vector on the right-hand side of this equation depends on the material mapping 𝛼 , as indicated by the superscript. The left-hand side of this equation involves quantities that by definition are independent of the mapping. This contradicts the initial assumption. Thus, the effective structural stiffness exists only when the global load vector is independent of the material mapping. However, the expected values, {⟨d⟩}, of the nodal displacement can always be calculated. It is therefore of interest to determine whether {⟨d ⟩} can be used to obtain expected values of other

Random Glass Mat Composites – Materials with Macrostructure

quantities, such as the displacements, {⟨u(r)⟩}, at a point, or the strain components, {⟨𝜀(r)⟩}, at a point. It is readily shown that {⟨u(r)⟩} = [N(r)]{⟨d ⟩} {⟨𝜀(r)⟩} = [B(r)]{⟨d ⟩}

(24.11.29)

where N(r) is the element shape function matrix. However, the expected value of stress cannot be expressed simply in terms of the expected values of the nodal displacements. The stress at a point must be calculated directly from the definition, requiring a determination of the stresses for all the mappings. A numerical procedure for calculating expected values of various quantities is discussed in the next section. 24.11.2

Numerical Procedure

The mechanical properties considered so far have been random variables for which the probability distributions have been given and are assumed to be equal to the probability that a particular material mapping occurs. In fact, the probability of a specified mapping occurring is zero, since this probability equals 1∕n, with n approaching infinity. Because the probability density function f (E) is based on the extension of predetermined segments under constant stress, the part being analyzed must be partitioned into elements having the same size as the segments used in the experimental characterization of the material. Let each material mapping be done so as to simulate the property distribution in a GMT composite part. The elastic constants for each element can then be chosen randomly from the population of the tensile elastic modulus and the Poisson’s ratio, which are characterized by predetermined probability density functions, denoted by f (E) and g(E), respectively. The desired quantities of interest can then be calculated by using, for example, the finite element method. To obtain the expected value of the quantity Q, this procedure must be repeated a sufficiently large number of times (say, n) to ensure that (24.11.30) |⟨Q⟩ − ⟨Qn ⟩| < 𝜀Q ( ∑n ) (𝛼 ) (𝛼 ) P ∕n is the arithmetic mean of Q after n mappings and 𝜀Q is the specified where ⟨Qn ⟩ = i=1 Q absolute error for the expected value of Q. The existence of such an n is based on the assumption that ∑ Q (𝛼 ) P (𝛼 ) = ⟨Q⟩ (24.11.31) lim ⟨Qn ⟩ = n→∞

𝛼

The procedure outlined previously can be used to calculate more than just the expected value of the quantity of interest. Equations 24.11.30 and 24.11.31 can be extended for higher order moments ⟨Qj ⟩ of Q. A knowledge of the higher order moments can be used to determine the probability density function for the random variable Q. The amount of data stored can be minimized by updating the moments Q after each iteration, in the form (n − 1) ⟨Q j n−1 ⟩ + ⟨Q j n ⟩ (24.11.32) ⟨Q j n ⟩ = n where ) ( n ∑ j j ⟨Q n ⟩ = Q i ∕n i =1

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Introduction to Plastics Engineering

is the jth moment of Q as calculated after n iterations. This numerical procedure includes the random sampling of the elastic property database, which is a set of numbers generated from the elastic modulus probability density function f(E ). This set should be sufficiently large to ensure that the randomly chosen numbers from the database appear with probability density characterized by f(E ). 24.11.3

Some Numerical Results

The procedure outlined in the preceding section was used to obtain the effective stiffness of plaques consisting of m × n 12.7-mm segments (Figure 24.11.4). First a random number generator was used to assign material elastic properties from the probability density function, shown by the smooth curve in Figure 24.8.1, to each segment of the plaque. Each segment was then partitioned into finite elements (so that every finite element in the same segment was assigned the same material property). A constant Poisson’s ratio of 0.29 was used. In these numerical experiments, the left edges of the plaques were fixed while the right edges were displaced by an amount ux (L0 ), where L0 = 12.7n mm is the length of the plaque. Because linear, quadrilateral plane-stress elements were used for the analysis, the boundary conditions on the model were applied through displacement constraints on the edges of the nodes of the finite element mesh. [Nodes on the left edge (x = 0) were free to move only in the y-direction – except the node at (0, 0), which was fixed. The nodes on the right edge (x = L0 ) were subjected to the same x-direction displacement, but were free to move in the y-direction.] The number n was varied from n = 1 (L0 = 12.7 mm) to n = 8 (L0 = 101.6 mm). The plaque width W0 = 12.7m mm was varied from m = 1 (W0 = 12.7 mm) to m = 8 (W0 = 101.6 mm) for each value of n. The average, or the expected, values of displacements and stresses at all the nodes were calculated for each m × n-segment plaque by using the procedure described in the preceding section.

y

W0 = m l0

960

x L0 = n l0 Figure 24.11.4 Geometry of a W0 = mL0 wide by L0 = nl0 plaque made up of m × n segments of length l0 . (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 309 – 316, 1992.)

For each m × n-segment plaque, the expected value ⟨𝜎 x ⟩ of the x-component of the normal stress was used to calculate the effective modulus

𝜎

Em × n = 𝜀 x 0

Random Glass Mat Composites – Materials with Macrostructure

where 𝜀0 = ux (L0 )∕L0 , and the mean 𝜎 x of the expected value of the stress over the width W0 is given by

𝜎x =

W

0 1 ⟨𝜎x ( x, y)⟩ dy W0 ∫0

Note that the expected value of the stress averaged across the width, 𝜎 x , does not depend on x because of equilibrium requirements. The variations of the effective modulus versus the number of length segments n, for three different widths corresponding to m = 1, 4, 8, are shown in Figure 24.11.5. The numbers in parentheses next to the symbols indicate the number of plaques used to obtain the effective moduli. The effective moduli of short, wide plaques (n = 1) are close to the population arithmetic average EA ; these moduli decrease with increasing width (m decreasing from 8 to 1). For each width, the effective modulus rapidly decreases with increasing length (n increasing from 1 to 8), and appears to converge to a constant value for large lengths. As expected, the effective stiffness of a narrow plaque (m = 1) is bounded by the elastic modulus population harmonic and arithmetic means, which in this case were EH = 4.654 GPa (675 × 103 psi) and EA = 4.799 GPa (696 × 103 psi), respectively.

EFFECTIVE STIFFNESS (GPa)

4.85

EA

4.80

m=8

(400) (400) (400)

m=4 (300)

4.70

m =1

(300)

(300)

(300)

(200)

EH (300)

(400) (400) (400)

(400)

4.60 0

5

10

PLAQUE LENGTH (Number of Segments n) Figure 24.11.5 Variations of the effective modulus of an m × n segment plaque versus the length n for three widths m = 1, 4, and 8. (Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 309 – 316, 1992.)

Since the effective moduli are obtained by a random sampling from the population of all plaques, the results for a fixed sample size (fixed number of plaques) will be subject to fluctuations. That is, the averages over the same number of different sets of plaques will result in different values of the effective

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Introduction to Plastics Engineering

modulus. However, with increasing sample size (large number of plaques) these fluctuations are expected to decrease. For the preliminary results presented here, the sample sizes (number of plaques) were varied to check for convergence. The results are listed in Table 24.11.1, in which the first entry represents the plaque size, m × n, and the sample size in parentheses, with the adjacent number being the effective modulus. Fluctuations are evident even in a sample size of 400 plaques. Table 24.11.1 Effective moduli for m × n-size plaques for different sample sizes. m × n=8 × 8 (100) 4.7754 (200) 4.6809 (400) 4.6757

m × n=8 × 1 (1) 5.4830 (100) 4.8379 (400) 4.7756 m × n=4 × 1 (100) 4.7800 (200) 4.7912 (400) 4.7596

m × n=4 × 2 (100) 4.7156 (300) 4.7119

m × n=4 × 3 (100) 4.7275 (300) 4.7021

m × n=4 × 4 (100) 4.6781 (200) 4.6746

m × n=1 × 1 (100) 4.6890 (200) 4.7595 (300) 4.7510 (400) 4.7555

m × n=1 × 2 (100) 4.6329 (300) 4.6353

m × n=1 × 3 (100) 4.6741 (300) 4.6757

m × n=1 × 4 (100) 4.6737 (200) 4.6460 (400) 4.6383

m × n=4 × 8 (100) 4.6754 (200) 4.6724 (400) 4.6739 m × n=1 × 6 (100) 4.6835 (200) 4.6780 (300) 4.6745

m × n=1 × 8 (100) 4.6353 (200) 4.6500 (400) 4.6600

Adapted with permission from W.C. Bushko and V.K. Stokes, Polymer Composites, Vol. 13, pp. 309 – 316, 1992.

24.12 *Statistical Approach to Strength Strength characterization involves three elements. First, a characterization of the material state by parameters that are relevant to strength; the stress state of the material is the most common choice, although other mechanical parameters, such as strains, can be used. Second, a definition of what constitutes material failure, which can depend on the material and on the way it is used in the structure. Theories of failure for solids are discussed in Section 15.7. One mode – brittle failure – is best exemplified by ceramic materials that fracture in a catastrophic manner. For ductile materials, which tend to accumulate significant damage through large permanent deformations prior to fracture, the definition of failure is not obvious and reduces to establishing the amount of permanent deformation that can be tolerated. And third, establishing a general relationship between the first two elements; this can be expressed either by a set of critical parameters characterizing the material state or by a probability distribution function that relates the state of the material under load to the probability of failure. 24.12.1

*State of Material Loading

In contrast to most materials the stresses and strains in a loaded GMT structure are not deterministic; they are random variables that must be characterized by distribution functions. This randomness results from the random nature of the elastic modulus whose distribution function fully characterizes the elastic properties of GMT. However, the effective elastic modulus of a part is a specific average over all possible values of the elastic modulus and may not exist as defined in Section 24.11. As expected, the

Random Glass Mat Composites – Materials with Macrostructure

effective elastic modulus is independent of the load level but it can depend on the geometry and boundary conditions of the problem, and is therefore a system rather than a material property. In fact, all parameters in structural analysis, such as stresses and strains, can be represented by random variables characterized by probability density functions. In particular, for the general three-dimensional case – in which the state of stress is described by the six components 𝜎 i , i = 1, 2, … , 6, where 𝜎 1 , 𝜎 2 , and 𝜎 3 are the normal stresses in the three coordinate directions, and 𝜎 4 , 𝜎 5 , and 𝜎 6 are the corresponding three shear stresses – the stress components constitute a set of random variables. The material loading at point r can be characterized by the probability density function f (𝜎 i , r), such that the probability that the stress at point r is less than 𝜎 i is P (𝜎i , r) =

∫𝜎 ′