Handbook of structural life assessment 9781119135463, 111913546X, 9781119135470, 1119135478

Table of contents :
Content: Acknowledgements xi Introduction xiii Part I Fracture Mechanics Dynamics and Peridynamics 1 1 Fundamentals of Fracture Mechanics 3 1.1 Introduction and Historical Background 3 1.2 Classical Theory of Solid Mechanics 6 1.3 Stress Intensity Factor 12 1.3.1 Analytical Development 12 1.3.2 Evaluation of Stress Intensity Factors 15 1.4 Linear Elastic Fracture Mechanics (LEFM) 17 1.4.1 Griffith s Criterion 18 1.5 Nonlinear Fracture Mechanics 19 1.5.1 Irwin s Modification 20 1.5.2 Crack Tip Element Method 22 1.5.3 Paris Erdogan Law 23 1.5.4 AFGROW Program 25 1.5.5 Energy Release Integrals 26 1.5.6 Mechanisms of Crack Propagation 35 1.6 Boundary?]Layer Effect of Composites 38 1.6.1 Introduction 38 1.6.2 Analytical Treatment 40 1.6.3 Thermal Loading Stress Field 48 1.7 Closing Remarks 55 2 Applications of Fracture Mechanics 59 2.1 Introduction 59 2.2 Fracture Mechanics of Metallic Structures 59 2.2.1 Steel Structures 59 2.2.2 Aluminum Alloys 62 2.3 Damage of Composite Structures 70 2.3.1 Preliminaries 70 2.3.2 Assessment of Composites Mechanics 71 2.3.3 Damage of Sandwich Structures 76 2.3.4 Sandwich Plates under Normal Loads 78 2.3.5 Thermo?]Mechanical Coupling of Sandwich Plates 90 2.3.6 Mechanics of Solid Foams 108 2.4 Closing Remarks 127 3 Dynamic Fracture and Peridynamics 129 3.1 Introduction 129 3.2 Fracture Dynamics 131 3.2.1 Features of Dynamic Fracture 131 3.2.2 Instability of Cracks and Microbranching 133 3.2.3 Experimental Techniques 139 3.2.4 Dynamic Crack Propagation Using Optical Caustics 142 3.3 Fracture Dynamics of Metals 148 3.3.1 Spalling of Metals 148 3.3.2 Dynamic Crack Propagation in Metals 149 3.3.3 Melting Metals 154 3.4 Dynamic Fracture of Composites 155 3.4.1 Functionally Graded Materials and Bi?]Materials 157 3.4.2 Polymer and PMMA Materials 163 3.4.3 Fiber?]Reinforced Composites 169 3.5 Peridynamics 171 3.5.1 Ingredients of Peridynamic Theory 171 3.5.2 Remarks and Restrictions 181 3.5.3 Numerical Simulation 185 3.5.4 Horizon Convergence 190 3.5.5 Application 192 3.6 Closing Remarks 208 Part II Introduction to Structural Health Monitoring 211 4 Structural Health Monitoring Basic Ingredients and Sensors 213 4.1 Introduction 213 4.2 Between Structural Life Assessment and Health Monitoring 213 4.3 Basic Ingredients of SHM 216 4.3.1 Non?]Destructive Evaluation 217 4.3.2 Lamb Waves 220 4.3.3 Acoustic Emission 227 4.3.4 Damage Location using Smart Sensors 241 4.3.5 Electric Resistance and Capacitance Techniques 251 4.3.6 Impact Resonance Method 253 4.3.7 Optimal Sensor Location 256 4.4 Closing Remarks 261 5 Statistical Pattern Recognition and Vibration?]Based Techniques 263 5.1 Introduction 263 5.2 The Statistical Pattern Recognition Paradigm 264 5.2.1 Basic Concept 264 5.2.2 Damage Index and Outlier Analysis 267 5.2.3 Case Study: Impact Tests of Composite Plates 269 5.3 Vibration?]Based Techniques 276 5.3.1 Overview 276 5.3.2 Damage Detection Using Strain Energy Method 278 5.3.3 Damage Detection and Location Using Modal Properties 280 5.3.4 Damage Detection Using Frequency Response Function 285 5.3.5 Damage Index and Modal Assurance Criterion 286 5.3.6 Applications 292 5.3.7 Operational Deflection Shapes/Vibration Deflection Shapes 324 5.4 Closing Remarks and Conclusions 356 Part III Reliability and Fatigue under Extreme Loading 359 6 Fatigue Life and Reliability Assessment 361 6.1 Introduction 361 6.2 Fatigue Life Assessment 362 6.2.1 Fatigue Crack Propagation 362 6.2.2 Fatigue Cumulative Damage 363 6.2.3 Half?]Cycle Fatigue Life Approach 369 6.2.4 Thermal Fatigue 372 6.2.5 Acoustical Fatigue 377 6.2.6 Fatigue of Structural Joints 379 6.2.7 Design Considerations 386 6.3 Design Based on Ultimate Strength of Ship Structures 388 6.3.1 Modes of Ship Failure 394 6.3.2 Modes of Hull Failure 395 6.4 Probabilistic Models of Load Effects 398 6.4.1 Reliability Index 401 6.4.2 Limit Sate Function 403 6.4.3 Risk Analysis 408 6.4.4 Ultimate Limit State (ULS) 410 6.4.5 Reliability and Uncertainty 415 6.4.6 Reliability?]Based Fatigue Assessment 418 6.4.7 Probabilistic Fracture Mechanics Assessment 421 6.5 Climate and Environmental Effects 425 6.6 Closing Remarks 428 7 Structural Reliability and Risk Assessment Under Extreme Loading 431 7.1 Introduction 431 7.2 Historic Extreme Loading Events 432 7.2.1 World Trade Center Towers (Terrorist Attack) 432 7.2.2 Ship Collisions and Grounding 436 7.2.3 Bridges under Extreme Loading 438 7.2.4 Collision of Road Tankers 443 7.3 Structural Life Assessment of Ocean Systems 444 7.3.1 Ship Structural Damage due to Slamming Loads 445 7.3.2 Damage Due to Grounding Accidents 472 7.3.3 Risk Assessment 472 7.3.4 Damage Due to Collisions 478 7.3.5 Reliability under Extreme Loading 487 7.4 Road Tanker Rollover 496 7.4.1 Rollover Scenarios and Metrics 497 7.4.2 Quasi?]Dynamic Approach 506 7.4.3 Rollover of Road Tankers 508 7.4.4 International Standards of Roll Threshold 512 7.4.5 Directional Stability and Dynamics 513 7.4.6 Collision of Vehicles and Structural Fatigue 516 7.4.7 Coupled Dynamics of Liquid Tanker Systems 520 7.4.8 Liquid Vehicle Coupling During Braking 523 7.4.9 Passive Control of Liquid Sloshing 527 7.5 Pipes Conveying Fluids 529 7.5.1 Mechanics of the Linear Problem 530 7.5.2 Mechanics of the Nonlinear Problem 534 7.5.3 Constrained Pipes Conveying Liquid 544 7.6 Closing Remarks 557 Part IV Environment Conditions, Joints and Crack Propagation Control 561 8 Corrosion and Hydrogen Embrittlement 563 8.1 Introduction 563 8.2 Corrosion of Ocean and Aerospace Structures 564 8.2.1 Corrosion of Ocean Structures 564 8.2.2 Problems of Aluminum Ship Structures 578 8.2.3 Corrosion of Aircraft Structures 582 8.2.4 Corrosion Monitoring 588 8.2.5 Corrosion Control 589 8.2.6 Corrosion Fatigue Cracking 593 8.3 Fretting/Wear in Heat Exchangers 598 8.3.1 Analytical and Computational Models 600 8.3.2 Experimental Investigations 603 8.4 Hydrogen Embrittlement 605 8.4.1 Hydrogen Embrittlement Problems 606 8.4.2 Fatigue Crack Enhancement 608 8.4.3 Crack Growth Modeling 610 8.4.4 Hydrogen Cracking Due to Welding 611 8.5 Closing Remarks and Conclusions 613 9 Joints and Weldments 615 9.1 Introduction 615 9.2 Energy Dissipation and Nonlinearity of Joints 616 9.2.1 Friction Characteristics 616 9.2.2 Energy Dissipation 617 9.2.3 Sources of Nonlinearities 621 9.2.4 Nonlinear Identification 623 9.2.5 Force?]State Mapping Technique 626 9.3 Design Considerations 629 9.3.1 Fully and Partially Restrained Joints 634 9.3.2 Sensitivity Analysis to Joint Parameter Variations 636 9.3.3 Stochastic Sensitivity 638 9.3.4 Joint Uncertainties and Relaxation 640 9.3.5 Uncertainty of Boundary Conditions and Material Properties 643 9.3.6 Mechanism of Relaxation and Loosening 645 9.3.7 Case Study A: Elastic Structures with Parameter Uncertainties and Relaxation of Joints 649 9.3.8 Case Study B: Beloiu et al. (2005) Influence of Boundary Conditions Relaxation 662 9.4 Welded Joints 690 9.4.1 Types of Welding Processes 690 9.4.2 Aluminum Welded Panels 694 9.4.3 Fatigue Assessment of Welded Joints 696 9.4.4 Fracture Mechanics Assessment 699 9.4.5 Fatigue Improvement of Welded Joints 704 9.5 Closing Remarks and Conclusions 706 Appendix 708 10 Crack Control 711 10.1 Introduction 711 10.2 Basic Concept and Development of Crack Arresters 711 10.2.1 Basic Concept 711 10.2.2 Crack Arrest Toughness 714 10.3 Crack Arresters of Ship Structures 719 10.3.1 Crack Arresters of Metal Ship Structures 720 10.3.2 Crack Arresters of Composite Ship Structures 730 10.4 Crack Control and Repair of Aerospace Structures 737 10.4.1 Crack Control of Metallic Structures 737 10.4.2 Composite Patches 743 10.4.3 Crack Control of Composite Structures 745 10.5 Pipeline Crack Arresters 752 10.5.1 Transmission Pipelines 752 10.5.2 Buckle Arresters of Pipelines 756 10.6 Closing Remarks 763 References 767 Index 983

Citation preview

Handbook of Structural Life Assessment

Handbook of Structural Life Assessment Raouf A. Ibrahim

Professor Wayne State University, USA

This edition first published 2017 © 2017 John Wiley & Sons Ltd 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 permision to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Raouf A Ibrahim to be identified as the author 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 111 River Street, Hoboken, NJ 07030, USA 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 The publisher and the authors 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 fitness for a particular purpose. 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 every situation. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. The fact that an organization or website is referred to in this work as a citation and/or potential source of further information does not mean that the author or the publisher endorses the information the organization or website may provide or recommendations it may make. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this works was written and when it is read. No warranty may be created or extended by any promotional statements for this work. Neither the publisher nor the author shall be liable for any damages arising herefrom. Library of Congress Cataloging‐in‐Publication Data Names: Ibrahim, R. A., 1940– author. Title: Handbook of structural life assessment / Raouf A. Ibrahim. Description: Hoboken, NJ : John Wiley & Sons, 2017. | Includes bibliographical references and index. | Description based on print version record and CIP data provided by publisher; resource not viewed. Identifiers: LCCN 2016046892 (print) | LCCN 2016055281 (ebook) | ISBN 9781119135494 (pdf ) | ISBN 9781119135487 (epub) | ISBN 9781119135463 (cloth) Subjects: LCSH: Structural analysis (Engineering) | Fracture mechanics. | Materials–Fatigue. | Service life (Engineering) Classification: LCC TA645 (print) | LCC TA645 .I27 2017 (ebook) | DDC 624.1/7–dc23 LC record available at https://lccn.loc.gov/2016055281 Cover image: wx-bradwang/gettyimages Cover design: Wiley Set in 10/12pt Warnock by SPi Global, Pondicherry, India 10 9 8 7 6 5 4 3 2 1

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Contents Acknowledgements  xi Introduction  xiii Part I 

Fracture Mechanics Dynamics and Peridynamics  1

1 Fundamentals of Fracture Mechanics  3 1.1 Introduction and Historical Background  3 1.2 Classical Theory of Solid Mechanics  6 1.3 Stress Intensity Factor  12 1.3.1 Analytical Development  12 1.3.2 Evaluation of Stress Intensity Factors  15 1.4 Linear Elastic Fracture Mechanics (LEFM)  17 1.4.1 Griffith’s Criterion  18 1.5 Nonlinear Fracture Mechanics  19 1.5.1 Irwin’s Modification  20 1.5.2 Crack Tip Element Method  22 1.5.3 Paris–Erdogan Law  23 1.5.4 AFGROW Program  25 1.5.5 Energy Release Integrals  26 1.5.6 Mechanisms of Crack Propagation  35 1.6 Boundary‐Layer Effect of Composites  38 1.6.1 Introduction 38 1.6.2 Analytical Treatment  40 1.6.3 Thermal Loading Stress Field  48 1.7 Closing Remarks  55 Applications of Fracture Mechanics  59 2.1 Introduction  59 2.2 Fracture Mechanics of Metallic Structures  59 2.2.1 Steel Structures  59 2.2.2 Aluminum Alloys  62 2.3 Damage of Composite Structures  70 2.3.1 Preliminaries 70 2.3.2 Assessment of Composites Mechanics  71

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2.3.3 2.3.4 2.3.5 2.3.6 2.4

Damage of Sandwich Structures  76 Sandwich Plates under Normal Loads  78 Thermo‐Mechanical Coupling of Sandwich Plates  90 Mechanics of Solid Foams  108 Closing Remarks  127

Dynamic Fracture and Peridynamics  129 3.1 Introduction  129 3.2 Fracture Dynamics  131 3.2.1 Features of Dynamic Fracture  131 3.2.2 Instability of Cracks and Microbranching  133 3.2.3 Experimental Techniques  139 3.2.4 Dynamic Crack Propagation Using Optical Caustics  142 3.3 Fracture Dynamics of Metals  148 3.3.1 Spalling of Metals  148 3.3.2 Dynamic Crack Propagation in Metals  149 3.3.3 Melting Metals  154 3.4 Dynamic Fracture of Composites  155 3.4.1 Functionally Graded Materials and Bi‐Materials  157 3.4.2 Polymer and PMMA Materials  163 3.4.3 Fiber‐Reinforced Composites  169 3.5 Peridynamics  171 3.5.1 Ingredients of Peridynamic Theory  171 3.5.2 Remarks and Restrictions  181 3.5.3 Numerical Simulation  185 3.5.4 Horizon Convergence  190 3.5.5 Application 192 3.6 Closing Remarks  208

3

Part II  4

Introduction to Structural Health Monitoring  211

Structural Health Monitoring Basic Ingredients and Sensors  213

4.1 Introduction  213 4.2 Between Structural Life Assessment and Health Monitoring  213 4.3 Basic Ingredients of SHM  216 4.3.1 Non‐Destructive Evaluation  217 4.3.2 Lamb Waves  220 4.3.3 Acoustic Emission  227 4.3.4 Damage Location using Smart Sensors  241 4.3.5 Electric Resistance and Capacitance Techniques  251 4.3.6 Impact Resonance Method  253 4.3.7 Optimal Sensor Location  256 4.4 Closing Remarks  261

Contents

5

Statistical Pattern Recognition and Vibration‐Based Techniques  263

5.1 Introduction  263 5.2 The Statistical Pattern Recognition Paradigm  264 5.2.1 Basic Concept  264 5.2.2 Damage Index and Outlier Analysis  267 5.2.3 Case Study: Impact Tests of Composite Plates  269 5.3 Vibration‐Based Techniques  276 5.3.1 Overview 276 5.3.2 Damage Detection Using Strain Energy Method  278 5.3.3 Damage Detection and Location Using Modal Properties  280 5.3.4 Damage Detection Using Frequency Response Function  285 5.3.5 Damage Index and Modal Assurance Criterion  286 5.3.6 Applications 292 5.3.7 Operational Deflection Shapes/Vibration Deflection Shapes  324 5.4 Closing Remarks and Conclusions  356

Part III 

Reliability and Fatigue under Extreme Loading  359

6 Fatigue Life and Reliability Assessment  361 6.1 Introduction  361 6.2 Fatigue Life Assessment  362 6.2.1 Fatigue Crack Propagation  362 6.2.2 Fatigue Cumulative Damage  363 6.2.3 Half‐Cycle Fatigue Life Approach  369 6.2.4 Thermal Fatigue  372 6.2.5 Acoustical Fatigue  377 6.2.6 Fatigue of Structural Joints  379 6.2.7 Design Considerations  386 6.3 Design Based on Ultimate Strength of Ship Structures  388 6.3.1 Modes of Ship Failure  394 6.3.2 Modes of Hull Failure  395 6.4 Probabilistic Models of Load Effects  398 6.4.1 Reliability Index  401 6.4.2 Limit Sate Function  403 6.4.3 Risk Analysis  408 6.4.4 Ultimate Limit State (ULS)  410 6.4.5 Reliability and Uncertainty  415 6.4.6 Reliability‐Based Fatigue Assessment  418 6.4.7 Probabilistic Fracture Mechanics Assessment  421 6.5 Climate and Environmental Effects  425 6.6 Closing Remarks  428

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7

Structural Reliability and Risk Assessment Under Extreme Loading  431

7.1 Introduction  431 7.2 Historic Extreme Loading Events  432 7.2.1 World Trade Center Towers (Terrorist Attack)  432 7.2.2 Ship Collisions and Grounding  436 7.2.3 Bridges under Extreme Loading  438 7.2.4 Collision of Road Tankers  443 7.3 Structural Life Assessment of Ocean Systems  444 7.3.1 Ship Structural Damage due to Slamming Loads  445 7.3.2 Damage Due to Grounding Accidents  472 7.3.3 Risk Assessment  472 7.3.4 Damage Due to Collisions  478 7.3.5 Reliability under Extreme Loading  487 7.4 Road Tanker Rollover  496 7.4.1 Rollover Scenarios and Metrics  497 7.4.2 Quasi‐Dynamic Approach  506 7.4.3 Rollover of Road Tankers  508 7.4.4 International Standards of Roll Threshold  512 7.4.5 Directional Stability and Dynamics  513 7.4.6 Collision of Vehicles and Structural Fatigue  516 7.4.7 Coupled Dynamics of Liquid–Tanker Systems  520 7.4.8 Liquid–Vehicle Coupling During Braking  523 7.4.9 Passive Control of Liquid Sloshing  527 7.5 Pipes Conveying Fluids  529 7.5.1 Mechanics of the Linear Problem  530 7.5.2 Mechanics of the Nonlinear Problem  534 7.5.3 Constrained Pipes Conveying Liquid  544 7.6 Closing Remarks  557 Part IV 

Environment Conditions, Joints and Crack Propagation Control  561

Corrosion and Hydrogen Embrittlement  563 8.1 Introduction  563 8.2 Corrosion of Ocean and Aerospace Structures  564 8.2.1 Corrosion of Ocean Structures  564 8.2.2 Problems of Aluminum Ship Structures  578 8.2.3 Corrosion of Aircraft Structures  582 8.2.4 Corrosion Monitoring  588 8.2.5 Corrosion Control  589 8.2.6 Corrosion Fatigue Cracking  593 8.3 Fretting/Wear in Heat Exchangers  598 8.3.1 Analytical and Computational Models  600 8.3.2 Experimental Investigations  603

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8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.5

Hydrogen Embrittlement  605 Hydrogen Embrittlement Problems  606 Fatigue Crack Enhancement  608 Crack Growth Modeling  610 Hydrogen Cracking Due to Welding  611 Closing Remarks and Conclusions  613

Joints and Weldments  615 9.1 Introduction  615 9.2 Energy Dissipation and Nonlinearity of Joints  616 9.2.1 Friction Characteristics  616 9.2.2 Energy Dissipation  617 9.2.3 Sources of Nonlinearities  621 9.2.4 Nonlinear Identification  623 9.2.5 Force‐State Mapping Technique  626 9.3 Design Considerations  629 9.3.1 Fully and Partially Restrained Joints  634 9.3.2 Sensitivity Analysis to Joint Parameter Variations  636 9.3.3 Stochastic Sensitivity  638 9.3.4 Joint Uncertainties and Relaxation  640 9.3.5 Uncertainty of Boundary Conditions and Material Properties  643 9.3.6 Mechanism of Relaxation and Loosening  645 9.3.7 Case Study A: Elastic Structures with Parameter Uncertainties and Relaxation of Joints  649 9.3.8 Case Study B: Beloiu et al. (2005) – Influence of Boundary Conditions Relaxation  662 9.4 Welded Joints  690 9.4.1 Types of Welding Processes  690 9.4.2 Aluminum Welded Panels  694 9.4.3 Fatigue Assessment of Welded Joints  696 9.4.4 Fracture Mechanics Assessment  699 9.4.5 Fatigue Improvement of Welded Joints  704 9.5 Closing Remarks and Conclusions  706 Appendix  708

9

10 Crack Control  711 10.1 Introduction  711 10.2 Basic Concept and Development of Crack Arresters  711 10.2.1 Basic Concept  711 10.2.2 Crack Arrest Toughness  714 10.3 Crack Arresters of Ship Structures  719 10.3.1 Crack Arresters of Metal Ship Structures  720 10.3.2 Crack Arresters of Composite Ship Structures  730 10.4 Crack Control and Repair of Aerospace Structures  737 10.4.1 Crack Control of Metallic Structures  737 10.4.2 Composite Patches  743 10.4.3 Crack Control of Composite Structures  745

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10.5 10.5.1 10.5.2 10.6

Pipeline Crack Arresters  752 Transmission Pipelines  752 Buckle Arresters of Pipelines  756 Closing Remarks  763 References  767 Index  983

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Acknowledgements The author would like to acknowledge with thanks the following publishers and authors for granting permission to use the following figures: Figures 3.24 and 3.25 are courtesy of Mathematical Sciences publisher Figures  3.26 and 3.27 are courtesy of Professor Walter Gerstle of the University of New Mexico Figures 5.27 and 5.28 are courtesy of Professor Hui Li of Harbin Institute of Technology Figures  7.42 and 7.43 are courtesy of American Society of Naval Engineers and Professor Bilal Ayyub of the University of Maryland Figures 7.54–56 are courtesy of Professor Subhash Rakheja of Concordia University Figures 9.11–13 are courtesy of Professor Chris Pettit of United States Naval Academy

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Introduction The health state of a human‐being is either annually examined or monitored. Likewise, structural life assessment (SLA) periodically evaluates the state and performance of a structural system and provides recommendations for either possible maintenance/ repair or the end of structural service life. On the other hand, structural health monitoring (SHM) detects the state and condition of the structure during its operation. SLA basically relies on the theory of fracture mechanics, fatigue life assessment, and reliability theory. Fracture mechanics deals with the study of the propagation of cracks in a structural element. It seeks to establish the local stress and strain fields around a crack tip in terms of local parameters such as the loading and the geometry of the structure. Structural fatigue life assessment is classified into fatigue crack propagation and cumulative fatigue damage, based on the S–N curve and other probabilistic descriptions. On the other hand, reliability theory describes the probability of a structure to complete its expected function during an interval of time. The opposite of reliability is failure probability per unit time or over time, such as a lifecycle. Note that sustainability and lifecycle of ship systems represent major and fast‐growing challenges for the US Navy (Salvino and Brady, 2008). In the early 1920s the catastrophic failure of a ship hull was averted by the arrest of a rapidly propagating crack. For example, the Majestic and Leviathan liners were perilously close to breaking in two in the North Atlantic. Cracks were propagated across the strength deck and down the ships’ sides and stopped at circular airport openings (Heller et al., 1967; Hinners, 1967). Out of some 2000 Liberty ships built during World War II, more than 100 ships broke in two, while many others had serious cracks (Broek, 1994). Since then, ship design requirements have included mandatory fracture arrest strakes, but intensive action was not initiated until after the epidemic of ship failures originating with the Schenectady and the Esso Manhattan (Boyd, 1969; Heller et al., 1967). As Liu (1998) indicated, it was not until World War II, when many Liberty ships experienced brittle fracture (in the form of sudden splitting into two halves) in the ocean, that engineers began studying and treating fracture seriously. This work has evolved into the present discipline of fracture mechanics. Reliability‐based methods include the analysis of fatigue life of structural details based on the cyclic stress against the logarithmic scale of cycles to failure (S–N curve) approach and on the assumption that fatigue damage accumulation is a linear phenomenon. The S–N curve is based on experimental measurements of fatigue life in terms of cycles to failure for different loading levels and specimen geometries. For some materials there is a theoretical value for stress amplitude below which it will not fail for any

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Introduction

number of cycles, called the fatigue limit, endurance limit, or fatigue strength. Fatigue life is thus specified by the number of stress cycles of a specified character that a specimen sustains before the failure of a specified nature occurs. It is amazing to note that the interest in studying structural fatigue damage is exponentially increasing, as reflected by the review articles dealing with different aspects of fatigue life prediction methods (e.g. Orowan, 1948; Fatemi and Yang, 1998; Yang and Fatemi, 1998; Newman, 1998; Cui, 2002). The strength of structures containing defects is assessed by evaluating the stress concentration caused by the discontinuities (hot spots). At the crack tip, there is a dominant singularity which is common to all solutions of plane problems. The study of crack propagation in structural materials employs methods of analytical and experimental solid mechanics to characterize material’s resistance to fracture. It applies the theories of elasticity and plasticity to the microscopic crystallographic defects found in real materials in order to predict the macroscopic mechanical failure. Fracture mechanics uses fractography to determine the cause of failure in engineering structures by studying the characteristics of a fracture surface. Different types of crack growth produce characteristic features on the surface, which can be used to help identifying the ­failure mode. The theory of fracture mechanics opens the way to analyzing engineering structures that experience predetermined amounts of stable and unstable crack growth (Newby, 1998). Heller et al. (1967) outlined three different ways that can make the structure to be damage tolerant: 1) Select relatively high‐toughness materials that minimize the growth of cracks to a critical size. 2) Use moderate or low‐toughness materials in which cracks are allowed to grow to a critical size and cause fast fracture. However, the structure is designed such that a fast crack is arrested without causing complete loss of the structure. This can be achieved by building a structure consisting of parallel members, one of which may completely fail, or by the use of crack arresters. 3) Install crack arresters in critical locations. These are designed to stop the crack before excessive damage is sustained and contain the structure until repairs can be made. This book deals mainly with different aspects of SLA and its link with SHM. It is divided into four parts. Part I comprises three chapters and addresses the theory of fracture mechanics, dynamics, and peridynamics. Fracture mechanics basically deals with the mechanics of solids containing planes of displacement discontinuities in the form cracks and their growth using energy criteria. It considers failure to be propagating throughout the structure. Chapter  1 presents the theory of fracture mechanics, which may be classified into linear elastic fracture mechanics (LEFM) and elasto‐plastic fracture mechanics (EPFM). The amount of energy available for fracture is usually governed by the stress field around the crack, measured by the stress intensity factor. The value of the stress intensity factor, which depends on the loading mode, is evaluated by different methods, developed by many researchers. For composite structures, the problem of singularity at the free‐corner edge is outlined under room temperature and thermal loading. An assessment of main results documented in the literature is presented. Chapter  2 deals with the main applications of the theory of fracture

Introduction

mechanics to metallic and composite structural components with the purpose of assessing their life expectancy. Note that pure metals are rarely used in engineering applications because they do not exhibit the most desirable properties. Combinations of two or more metals have been developed for providing high strength and durability. For example, steel is carbon dissolved in iron, and using aluminum alloyed with magnesium, copper, or zinc produces strong materials. When grains have purely random orientation and uniform size, the material mechanical properties become independent of the direction of the loading and exhibit isotropic behavior. Most metal alloys are isotropic, and an anisotropic behavior can be induced by deforming the alloy to change the grain structure and orientation. However, composites are heterogeneous combinations of two or more materials, such as carbon fibers in a resin matrix. This chapter begins with fracture mechanics of metals including aluminum alloys and steel structures. This is followed by applications of composite sandwich and foam structural damage. Dynamic fracture mechanics deals with the prediction of a crack’s growth together with its speed and direction of growth. The dynamic stress intensity factor will be introduced in terms of the instantaneous crack tip speed in Chapter 3, where we will also show that dynamic fracture is associated with a number of phenomena such as crack branching, transition from stable to unstable crack growth, and the mirror, mist, hackle sequence of textures on the fracture surface. The theory of dynamic fracture has been employed with metallic and composite structures to examine crack propagation under different types of dynamic loadings. Cracks in structural material form discontinuities, and their modeling requires special formulation. Since partial derivatives do not exist on crack surfaces, the classical equations of continuum mechanics cannot be applied directly when such features are present in the structure. This is the main motive behind the development of a new theory known as peridynamics, which treats internal forces within a continuous solid as a network of pair interactions, similar to springs, which can be nonlinear. Pairs of material points can interact through a spring up to a maximum distance, called the horizon. The basic ingredients of peridynamics are introduced together with its applications. Part II provides a brief account of structural health monitoring and consists of two chapters. Chapter 4 highlights the differences between SHM and SLA. One of the main features of SHM is the comparison between the state of perfect structure characteristics and the state of defected structure characteristics. Early detection of cracks and damage is critical for the safety of the structure, operating personnel, and other resources. Structural health monitoring consists basically of the process of detecting, locating, identifying and diagnosing the damage during system operation. This is usually achieved using smart sensors and non‐destructive evaluation techniques such as Lamb and guided waves, ultrasonic, laser vibrometers, acoustic emission, optical fibers, and electric resistance and capacitance techniques. The problem of optimizing sensor location will be discussed with the aim of maximizing the ability to detect and discriminate relevant data of the structure health state. Chapter 5 then presents two major approaches used in non‐destructive evaluation. These are statistical pattern recognition as a paradigm together with the concept of outliers and the vibration‐based approaches in identifying and locating the damage. These approaches have been supported by several applications reported in the literature. The concept of damage index is demonstrated through a case study of impact tests of composite structures. Vibration‐based methods are demonstrated through two case studies of damage development in T‐joints.

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Part III addresses the problem of structural fatigue life and reliability assessment under extreme loading. Chapter 6 deals with the general problem of structural fatigue and reliability assessment as applied to aircraft and ship structures. Fatigue assessment will be discussed in terms of fatigue crack propagation and cumulative fatigue damage. Other issues include thermal fatigue, acoustic fatigue, and fatigue of metallic, riveted and composite joints, plus related design considerations. Ship structure design is based on ultimate strength, ultimate limit state design, nonlinear analysis considering material and geometric nonlinearities, and initial imperfections. In view of structural parameter uncertainties, probabilistic analysis requires the use of reliability methods for assessing fatigue life by considering the crack propagation process, and the first passage problem, which measures the probability of the exit time from a safe operating regime. Chapter 6 also addresses the climatic and environmental effects on the design of ships and offshore structures. Chapter 7 then deals with the most challenging and difficult problem encountered by designers and engineers who are involved in the design and maintenance of infrastructures, aerospace and ocean vehicles subjected to extreme loading. Selected historical events of extreme loading and its subsequent effects on structures and infrastructures are introduced. These include the terrorist attacks on the twin towers of the World Trade Center, the collision of the Royal Mail Ship Titanic with an iceberg, together with other selected major ship collisions, bridges under extreme loading, and collisions of road tankers carrying hazardous materials. There is then an in‐depth discussion of the main results reported in the literature pertaining to ship structural damage assessments due to slamming loads, liquid sloshing impact loads of liquefied natural gas (LNG) in ship tankers, ship grounding accidents, collision with solid bodies. Under such extreme loadings, structural reliability will be the major issue during the design stage of ocean structures. The main results of road tanker stability and collisions during turning and sudden lane change maneuvers are presented in detail. Liquid sloshing in a highway maneuver can lead to lateral and rollover instabilities, decreased controllability/maneuverability, and increased stress on tank structures. Passive means of controlling liquid sloshing impact are presented, and the chapter ends with the problem of vibroimpact dynamics encountered in pipes conveying fluid in heat exchangers and nuclear plants. The final part of this book comprises three chapters dealing with corrosion, fretting and hydrogen embrittlement; joints and weldments; and crack passive controls. Chapter 8 discusses corrosion, fretting and hydrogen embrittlement in structural systems. These structural defects are a major factor in structural life assessment. Corrosion is the degradation of metals due to chemical reactions with the environment, involving oxygen and/or water. Fretting refers to corrosion damage at the asperities of contact surfaces. It is adhesive in nature, caused essentially by vibration and usually accompanied by corrosion. Then, hydrogen embrittlement is the process by which metals such as steel become brittle and fracture due to the introduction and subsequent diffusion of hydrogen into the metal. This chapter deals with the causes and effects of corrosion on steel and aluminum structures used in ships and aerospace systems. Since structural components made from aluminum and its alloys are vital to the ship and aerospace industries, the influence of environment on aluminum structures and the means of corrosion control and monitoring in both aluminum and other metals is presented. The main analytical, computational and experimental results of fretting/wear in pipes of heat exchangers are considered, and the problem of hydrogen‐induced cracking is presented in terms of fatigue crack enhancement and modeling.

Introduction

Joints and fasteners are used to transfer loads from one structural element to another. They include bolts, rivets, pins, and weldments. In addition they are a source of hot spots of stress concentration. The joint represents a discontinuity in the structure and results in high stresses that often initiate joint failure. Chapter 9 therefore considers the problems associated with joints, such as energy dissipation in bolted joints due to friction models, joint nonlinearity, design considerations, and joint parameter variations. Joint uncertainties and relaxation are the major problems associated with bolted joints and will be addressed in terms of uncertainty of boundary conditions and mechanisms of relaxation and loosening. It is known that joints are sources of high stress concentrations, and under dynamic loading they can result in relaxation and separation of jointed parts. With reference to composite structures, there are two types of joints commonly used in structural components. These are mechanically fastened joints and adhesive bonded joints. Bolted joints are still the dominant fastening mechanism used in joining of primary structural parts for advanced composites. This chapter presents different types of welding processes for steel and aluminum structures, together with associated problems such as fatigue and fracture mechanics assessment of welded joints. Serious structural damage due to brittle fracture has motivated engineers and designers to develop passive controls of crack propagation in the structure, known as crack arresters. The basic principle behind the use of a crack arrester is to reduce the crack‐driving force below the resisting force that must be overcome to extend a crack. The crack arrester can be as simple as a thickened region of metal, or may be constructed of a laminated or woven material that can withstand deformation without failure. Chapter 10 provides different approaches for passive crack control in the form of crack arresters to stop crack propagation before it spreads over a structure component. Crack arresters used in ship structures, aircraft structures, and pipelines are described for both metal and composite materials. The topics addressed in this book have received extensive research activities published in the open literature, which may exceed 30,000 references. This book cites over 4000 of these references, including references of permanent value and important results. The book is directed at graduate students and researchers involved in the design and maintenance of structural systems such as ships, offshore structures, aerospace structures, road tankers, bridges, and multi‐story buildings. Acknowledgements Some results reported in this book were obtained by the author and his coworkers during their work on research projects supported by the National Science Foundation (NSF), the Office of Naval Research (ONR), Air Force Office of Scientific Research (AFOSR), and Institute of Manufacturing Research of Wayne State University. This book is an outgrowth of an invited lecture presented at the International Workshop on Structural Life Assessment of Ocean Structures funded by ONR, Melbourne, Australia, 13–15 August 2012. The lecture was later expanded into six articles published in the SNAME Journal of Ship Production and Design (Ibrahim, 2015a, 2015b, 2015c, 2015d, 2016a, 2016b). I would like to thank Ms Kelly Cooper, who managed the ONR research projects during the period 2004–2012, and also to thank my colleagues involved in the research projects. Particularly, I am indebted to Professors Victor Berdichevsky, Ronald

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Gibson, Nabil Chalhoub, Valery Pilipchuk, Emmanual Ayorinde, and Chris Pettit. Special thanks to my former students Dr Marian D. Beloiu and Dr Ihab F. Grace. I would also like to thank Dr Chuck Farrar of Los Alamos National Laboratory who directed my attention to the problem of hydrogen embrittlement. The author would like to express his gratitude to the editorial staff of John Wiley. In particular, Mr. Paul Petralia, Senior Acquisition Editor, Mechanical and Biomedical Engineering, Hoboken, New Jersey, who originally contacted me and encouraged me to prepare this handbook. Following submitting the manuscript, Mrs. Samanaa Srinivas, Team Leader and Project Editor worked with me to secure copyright of some figures and went over the quality of number of figures. It was Mr. Paul Beverly, Proofreading editor, who did an excellent editorial job. The final stage of the book production was managed by Mrs. Karthika Sridharan and her team. To all these people I like to say: Thank you. It was my wife, Sohair, who has always been supporting me and who encouraged me to accept the challenge and to complete the task at my own pace. I thank her so deeply for her patience and longsuffering during writing this book. Indeed, without her support I could not have completed this book and all my other achievements.

1

Part I Fracture Mechanics Dynamics and Peridynamics

3

1 Fundamentals of Fracture Mechanics 1.1  Introduction and Historical Background Classical fracture mechanics is a direct application of classical continuum mechanics. It  introduces a length parameter into structural assessment known as the size of an existing defect or the crack length. Thus, fracture mechanics is based on the existence of an initial crack and subsequent crack propagation under cyclic loading. An important aspect in the development of classical fracture mechanics is the principle of locality of the state distribution of stresses. Fatigue crack propagation analyzed with fracture mechanics enables us to quantify the residual lifetime of a cracked component. It  requires a combined background of analytical solid mechanics, material science, probability theory, and catastrophe theory1 to calculate the load and stress in structural components. The solution for the stress distribution around a circular hole in a much larger plate under remotely uniform tensile stress was derived by Kirsch (1898). The solution for stresses around an elliptical hole was independently derived by Kolosov (1909) and Inglis (1913). Their solution showed that the concentration of stress could become far greater as the radius of curvature at one end of the hole becomes small compared to the overall length of the hole. The elliptical hole of Kolosov and Inglis defines a crack in the limit when one semi‐axis goes to zero. The Inglis solution was adopted by Griffith (1921) to describe a crack in a brittle solid. Griffith made his famous criterion that spontaneous crack growth would occur when the energy released from the elastic field is in balance with the work required to separate surfaces in the solid. According to Cotterell (2002), the first known works devoted to fracture mechanics were the two seminal papers of Griffith (1921, 1924). Griffith was motivated by the need to understand the effect of scratches on fatigue. It was originally thought that it should be possible to estimate the fatigue limit of a scratched component by using either the maximum principal stress criterion, favored by Lamé and Rankine, or the maximum principal strain criterion, favored by Ponclet and Saint‐Venant. Griffith showed that scratches could increase the stress and strain level by a factor of between two and six. However, Griffith noted that the maximum stress or strain would be the same on a shaft

1  Catastrophe theory is a branch of bifurcation theory in the study of dynamical systems (Thompson, 1982). A catastrophic failure is a sudden and total failure of a structure or system from which recovery is impossible. Catastrophic failures often lead to cascading systems failure. Handbook of Structural Life Assessment, First Edition. Raouf A. Ibrahim. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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Handbook of Structural Life Assessment

of diameter 1 inch whether the scratches were one ten thousandth or one hundredth of an inch deep, provided they were geometrically similar. These conclusions were found to be in conflict with the fatigue results and led Griffith to reject the commonly held criteria of rupture. Wieghardt (1907) had earlier rejected these strength criteria for a different reason, but dealt with the paradox that the stresses at the tip of a sharp crack in an elastic body are infinite no matter how small the applied stress. This fact led him to argue that rupture does not occur when the stress at a point exceeds some critical value, but only when the stress over a small portion of the body exceeds a critical value. Taylor (1965) stated that Griffith was also aware of the paradox. Griffith (1921) adopted energy concepts and realized that a certain minimum work was necessary to produce a fracture, which for an ideal elastic material was the surface free energy. Griffith realized that the fracture problem is an extension of the elastic theory of minimum potential energy. Griffith’s global treatment of the energy balance for a cracked body was described by Taylor (1965) as “the first real advance in understanding the strength of materials.” The practical importance of Griffith’s work lies in his realization that the critical stress depends on a length scale, which is taken as the crack length. Griffith (1921) performed his experiments on a model made of glass material. From his experiments, he estimated the theoretical strength of glass to be about 2 GPa. The observed tensile strength of glass was 170 MPa. Hence Griffith predicted that there were flaws of the order of 5 μm. Griffith believed that the weakness of glass was due to internal flaws, and indeed believed that the surface layers might be of superior strength because flaws would be oriented parallel to the surface (Griffith, 1921). Later, Griffith (1924) stated that the “weakness (in pure silica) is due almost entirely to minute cracks in the surface, caused by various abrasive actions to which the material has been accidentally subjected after manufacture.” Griffith’s evidence was that if a strong silica rod was rubbed lightly with any other solid, it immediately loses its strength. However, he did not state that the weakness in glass was due to surface flaws. Joffé (1928) assumed that surface flaws are responsible for the weakness in glass. Joffé (1924) indicated that the strength of rock salt was due to surface flaws because when the ­surface layer was dissolved in warm water the strength increased. The separation across a surface flaw in glass was found of the order 50 nm. The experiments on mica2 conducted by Orowan (1933) proved that the reduction in strength was due to flaws. The usual tensile strength of mica has a value within the range 200–300 MPa but Orowan (1933) found that the strength is more than 3 GPa by stressing only the central strip of a mica sheet using grips that were much narrower than the sheet. The small value of the usual tensile strength of mica is due to the ­presence of cracks at the edge of the sheet. The first direct evidence for the existence of surface flaws in glass came by chance during experiments conducted on the properties of thin films of metal by Andrade and Martindale (1935). Later, Andrade and Martindale (1937) conducted another series of experiments on various glasses using sodium from a vapor to “decorate” the surface cracks. The reversibility of fracture was experimentally demonstrated by Obreimoff (1930) who studied the fracture of mica using a stable geometry. A glass wedge was used to cleave thin lamellar of mica of

2  The mica group of sheet silicate (phyllosilicate) minerals includes several closely related materials. It has a very pronounced cleavage plane, and almost atomically perfect surfaces can be produced by cleavage.

Fundamentals of Fracture Mechanics

thickness 0.1–0.2 mm from a block of mica. It was found that a crack could grow under the combined effect of mechanical energy and moisture in the air, and demonstrated the reversibility of fracture. The transition temperature from ductile to brittle behavior in structural steel was found to be about 20 °C. In riveted structures, brittle fractures rarely caused catastrophes because a fracture was usually arrested at the edge of the plate. Brittle fracture in steel was studied in a 75 m high by 5 m diameter water standpipe at Gravesend, Long Island, NY in 1898 (SSC‐65, 1953). Without proper fracture control, it is known that welding may cause brittle fracture in steel. In addition, it introduces high residual stresses equal to the yield strength, a heat affected zone adjacent to a weld with a much higher transition temperature than the parent plate, and crack‐like defects. Unstable brittle fracture can easily run through a major part of its section. The first brittle ­fracture in a large welded structure occurred just before World War II in the Vierendeel Truss Bridge in Hasselt, Belgium, followed by failures in similar Belgium bridges during the war (SSC‐65, 1953). Extensive studies of brittle fracture were initiated following the widespread fractures in the welded Liberty ships. There were 145 structural failures in Liberty ships where either the vessel was lost or the hull was so weakened as to be dangerous; a further 694 ships suffered major fractures requiring immediate repair (Biggs, 1960). The size effect on the brittle fracture of steel laboratory specimens was considered by Biggs (1960). Size effect was recognized by Docherty (1932), Irwin (1948), and Shearin et al. (1948). Plates of thickness ¾ in and up to 72 in wide were tested using a 3,000,000 lb hydraulic testing machine (SSC‐3, 1946). Several attempts were made to determine the transition temperature at which the fracture behavior changes from ductile to cleavage. For example, Ludwik (1909) considered the phenomenological transition from ductile to cleavage behavior, and proposed that the cohesive strength was little affected by temperature, but there was a marked increase in the yield strength of low carbon steel as the temperature decreased, so that at a particular temperature, cleavage fracture became easier than yielding. The effect of a notch on the transition temperature was considered to be primarily due to a ­constraint on yielding. Orowan (1945) used Ludwik’s concept and showed that a notch would increase the transition temperature. The Charpy test (Charpy, 1912) is considered as one of the original and the most lasting of the small scale notch bend tests to assess the transition temperature in steel. The first fracture test that fully simulated a welded plate structure was reported by Wells (1956a, 1956b, 1962) who designed a 600 ton testing machine capable of testing 1  in thick 36 × 36 in butt‐welded plates. The welded plates could be tested either as welded or after heat treatment and were welded into the test rig. The plates were cooled with dry‐ice to the desired temperature before testing. Those plates welded with rutile electrodes tended to have pre‐cracks at the saw cut. Fractures were often initiated in pre‐cracked specimens at low stress, or occurred spontaneously on cooling, and these arrested at the edge of the tensile residual stress zone along the weld (Wells, 1961). The transition from high stress fractures at the yield strength to low stress fractures did not always occur at the Charpy transition temperature. This brief historical account constitutes the early state of the theory of fracture mechanics. Over time, this theory has been advanced by the contributions of many researchers. Generally, the theory of fracture mechanics is divided into linear elastic fracture mechanics (LEFM) and elasto‐plastic fracture mechanics (EPFM). LEFM is

5

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Handbook of Structural Life Assessment

convenient for brittle‐elastic materials such as low‐carbon steel, stainless steel, certain aluminum alloys and polymers. Note that plasticity always precedes fracture. The linear theory (LEFM) is governed by a parameter called the stress intensity factor, which determines the entire crack tip stress field and measures the material toughness (Rice, 1972). As indicated by Broek (1994), materials with relatively low fracture resistance fail below their collapse strength and can be analyzed on the basis of LEFM. On the other hand, if fracture is accompanied by considerable plastic deformation, the EPFM is used. The fracture parameters used in EPFM are referred to as the J‐integral, which measures the strain energy release rate,3 and the crack tip opening displacement (CTOD). It should be noted that the present state of classical fracture mechanics has been promoted by the  contribution of numerical methods, which have been developed to meet the demands of understanding the experimentally observed phenomena of fracture, and to transfer fracture mechanics material parameters to structural behavior. An extensive historical account of fracture mechanics and its development is documented by Brocks  and Schwalbe (2016). For a detailed account, the reader may consult those ­references addressing different issues of fracture mechanics (e.g. Broek, 1974, 1994; Hutchinson, 1979; Lawn, 1993; Anderson, 1995; Wang, C.H.,1996; Farahmand et  al., 1997; Liu, A.F., 1998; Tada et  al., 2000; Buckley, 2005; Zehnder, 2007; Gross and Seeling, 2011; Schreurs, 2011). This chapter begins with an introduction to the classical theory of continuum mechanics, which establishes the states of stress that will cause a particular material to fail. This is followed by introducing the stress intensity factor, which is usually used to determine the stress state near the tip of a crack for three linearly independent cracking modes. Methods of evaluating the stress intensity factor using analytical, experimental, or numerical approaches will be discussed. The linear elastic fracture mechanics (LEFM) will be introduced in terms of Griffith’s criterion. This is followed by the nonlinear fracture mechanics (EPFM), which covers Irwin’s criteria, crack tip element method, Paris–Erdogan law nonlinear fracture mechanics, and the J‐, M‐, L, and L‐integrals. Because of its importance, the AFGROW (Air Force Grow), which is a damage tolerance analysis framework developed by the Air Force Research Laboratory, will be briefly outlined. Since layered structural elements may suffer from severe stress concentrations of an interlaminar character, the boundary‐layer effect of composite structures together with the thermal loading stress field will be introduced. The chapter will close with conclusions and closing remarks.

1.2  Classical Theory of Solid Mechanics The principle of locality states that a material point is only directly influenced by, and interacts with, its immediate surroundings. Accordingly, the stress state at a point depends on the deformation at that point only. The resulting crack growth criterion is referred to as local, because attention is focused on a small material volume at the crack tip. The classical theory of continuum mechanics is based on partial differential

3  Note that the term “rate” does not refer to derivative with respect to time. In this context, it refers to derivative with the size of the crack.

Fundamentals of Fracture Mechanics x3(e3) T(2) σ33

σ31 σ13

T(2)

σ32

σ23 σ22

σ12

x2(e2)

σ21 σ11 T(1)

x1(e1)

Figure 1.1  Cartesian components of the stress vector T j , j = 1,2,3 acting on the three faces of the cubic elemental volume dV.

equations whose partial derivatives are continuous. Figure  1.1 shows nine stress ­components at each point of the medium. These stress components are dependent on position and time, i.e. ij ij ( x , t ). The unit vectors e1,  e2, and  e3 are established along  the three Cartesian coordinates x1,  x2,  and  x3, respectively. The stress vector T j , j 1, 2, 3, can be written in the form Tj j 1 e1 j 2 e2 j 3 e13 (1.1) where e1, e2, and e3 denote unit vectors along the coordinate axes 1, 2, and 3, respectively. The stress vector T on a surface element with an outward normal n can be expressed as a linear function of the stress σij. For example the j‐component of the stress ­vector T is K II

xx

2 r

sin

3 2 sin sin 2 2 2

K II

3 cos 1 sin cos 2 2 2 2 r

xy

..., yy

K II

3 sin cos cos 2 2 2 2 r

... (1.2)

...

We can write



T

3

3

Tj e j

j 1

3

ni

j 1 i 1

ij

e j (1.3)

The Cauchy stress is given by the nine components



11

12

13

xx

xy

xz

21

22

23

yx

yy

yz

31

32

33

zx

zy

zz

(1.4)

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Handbook of Structural Life Assessment

The linear momentum principle requires that we set to zero the limit over the closed surface S, i.e. lim 1 / S TdS 0 . S

0

S

In applying the linear momentum principle to an arbitrary finite body, the divergence theorem of multivariable calculus shows that integrals over the area of a closed surface S, may be rewritten as integrals over the volume V enclosed by the surface S, T j dS



n1

S

1j

n2

n3

2j

3j

1j

dS

S

V

2j

x1

3j

x2

x3

dV (1.5)

In order to determine the equations of motion of an elemental volume of the body material consider a small cube of material of side length L centered on some arbitrary chosen point x and the forces along the x1 axis as shown in Figure 1.2. The total body force is f1L3. In the limit as L 0 , upon expanding the stresses in a Taylor series and writing down Newton’s second law, gives 11

21

x1



31

x2

L3

x3

HOT

f1 L3

L3 a1 (1.6)

where a1 is the acceleration along the x1 axis and HOT denotes higher order terms. Canceling L3 on both sides gives 11

21

x1



31

x2

f1

x3

a1 (1.6a)

The other two equations along x2 and x3 axes can be obtained in a similar way in the form 12

22

x1



32

x2

∂σ21 + σ21 + L2 ∂x2 ⃛ ∂σ11 + ⃛ 2 ∂x 1

σ11– L

L

L

2

f2

x3

f1L3

13

a2 ,

x1

23

x2

33

x3

f3

a 3 (1.6b,c)

2

L

∂σ11 + σ11+ L2 ∂x1 ⃛

L

2

L

∂σ21 + L2 σ21– L2 ∂x2 ⃛

Figure 1.2  Forces acting along the x1‐axis on a cube of side length L. Stresses around the cube faces are developed in a Taylor series expansion about their values at x.

Fundamentals of Fracture Mechanics

For plane strain in which all displacements take place in one plane, and in the absence of body forces, the static equilibrium equations (zero inertia forces) are 11

21

x1



12

0

x2

0 (1.7a,b)

22

x1

x2

together with the strain compatibility equation 2

2

22 x12



11 x22

2

2

12

x1 x2

(1.8)

By using the stress–strain relations, the equilibrium equation takes the form 2



11

0 (1.9)

22

Introducing the Airy stress function the absence of body forces, gives 2



11

x22

2

,

12

x1 x2

, and

( x1 , x2 ), which satisfies the plane stress in 2 22

x12

(1.10)

Satisfying the strain compatibility and expressing the strains in terms of stresses by the linear isotropic relations gives 2



11 2

2

where

2

22

2

0 (1.11)

2

. Equation (1.11) is the known bi‐harmonic equation. x12 x22 Before establishing Griffith’s criterion, consider first a circular hole of radius a in a plate whose dimensions are much larger than a and can, for present purposes, be taken as infinite as shown in Figure 1.3. The plate is subjected to a uniform stress in the x2 direction and the boundary of the hole is free of loading. The solution of equation (1.11) was developed to satisfy the conditions 22 and 11 0 as r 0, and that 0 at r a. To establish a unique solution, we have to specify the value of the rr r integral of u / s, where u is the deformation displacement vector, and s is the arc length around the hole. This integral is zero in the present case for which a single‐valued displacement field is required. It is non‐zero if the hole was to represent the core of a dislocation. Accordingly, the solution may be written in the form g (r ) h(r )cos2 , with the functions g(r) and h(r) written the form h(r ) C 4 r 4 C2 r 2 C 0 C 2 r 2. The constants Ci are obtained from the conditions on the stress at r a and r to give single‐valued displacements. The solution for Φ is given by the following expressions for the stresses rr

2

r



1 2

2

a2 r2

1 2 1

1 4 a2 r2

a2 r2

3

a2 r2

3

a4 cos 2 r4

a4 sin 2 r4

1 3

a4 cos2 r4

, (1.12)

9

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Handbook of Structural Life Assessment σ∞

x2 σ22 = 3σ

θ

r x1

σ∞

Figure 1.3  Large plate with a circular hole circular tunnel under remote tensile stress.

x2

(1+ 2b / a)σ x2 x1

2b

σ →∞ x1

2a 2a

Figure 1.4  (a) Elliptic hole and (b) the limit to a flat crack of width 2a.

Equations (1.12) reveal that at r a , 1 2 cos 2 . Of particular interest | 0, 3 , and are the values of the stresses at the special points: 22 | /2 , /2 0 or in . Thus it is expected that failure would begin at 11 a brittle material under remote tensile loading. If the hole is an elliptic cavity, as shown in Figure 1.4(a), with major and semi‐axes a and b, respectively, then the concentration of stress at points of the hole boundary intersected by the x1 axis is

22

1

2b a

1 2 a/

tip

(1.13)

where tip b2 /a is the radius of curvature of the hole boundary at x1 a . Note that the stress induced along the hole boundary where it is intersected by the x2 axis is . With reference to a flat crack lying along the x1 axis with a x1 a as 11 shown in Figure  1.4(b), the stress concentration at the hole results in singularity. To demonstrate this singularity with the linear theory, the crack opening gap u2 u2 u2 ,

Fundamentals of Fracture Mechanics

where + and – denotes the upper and lower crack surfaces, may be given by the expression (Rice, 2010)

u2

4 E

a2

x12 ,

a x1

a (1.14)

The tensile stress transmitted across the x1 axis outside the crack is



11

x1 x12

a

2

over the ranges

x1

a and

a x1

(1.15)

Relations (14) and (15) reveal that at x1 a the displacement of the crack walls is proportional to a2 x12 ; on the other hand, the stress at a small distance ahead of the crack is proportional to 1 / x12 a2 . In linear elasticity, it is well known that stress singularities are prevalent at the corners of geometric boundaries joining dissimilar materials (e.g. Bogy, 1968; Hein and Erdogan, 1971; Kuo and Bogy, 1974). The problem of predicting states of stress that will cause a particular material to fail plays an important role in the design of structural components. Many fractures are appropriately described as being partially brittle and partially ductile, meaning that certain portions of the fractured surface are approximately aligned with planes of maximum shear stress, while others appear granular, as in the case of brittle fracture and are oriented more toward planes of maximum tensile stress. The design and development of structural systems generally involves biaxial (occasionally triaxial) stresses covering a wide range of ratios of principal stresses. In the simple classical theories of failure, it is assumed that the same amount of whatever caused the selected tensile specimen to fail will also cause any part made of the materials to fail regardless of the state of stress involved (Wolf et al., 2003). In the area of structural mechanics, the following stress/strain theories are very useful: Maximum normal stress theory (Rankine) is the simplest of the various theories; it states merely that a material subjected to any combination of loads will experience: 1) yield, whenever the greatest positive principal stress exceeds the tensile yield strength in a simple uniaxial tensile test of the same material, or whenever the greatest negative principal stress exceeds the compressive yield strength, 2) fracture, whenever the greatest positive (or negative) principal stress exceeds the tensile (or compressive) ultimate strength in a simple uniaxial tensile (or compressive) test of the same material. Maximum shear stress theory (also known as the Coulomb or Tresca theory) states that a material subjected to any combination of loads will fail (by yielding or fracturing) whenever the maximum shear stress exceeds the shear strength (yield or ultimate) in a simple uniaxial stress test of the same material (Wolf et al., 2003). Maximum normal‐strain theory (Saint‐Vanent’s theory) states that failure will occur whenever a principal normal strain reaches the maximum normal strain in a simple uniaxial stress test of the same material.

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Handbook of Structural Life Assessment

1.3  Stress Intensity Factor 1.3.1  Analytical Development

The stress intensity factor is usually used to determine the stress state near the tip of a crack. It is applied to homogeneous linear elastic material. It is useful to provide a failure criterion for brittle materials. Figure 1.5 shows that the crack tip polar coordinates such that denotes the 0 is taken as the direction of the crack growth and crack walls. The crack tip field was described by Williams (1952, 1957) and (Rice (2010) who set / t Vcr / x1, where Vcr is the instantaneous speed of crack propagation, to determine the Airy stress function Φ such that r 0 at the crack walls. The solution for Φ was assumed in the form r



2

A cos 

B cos  2

(1.16)

where A, B, and  are constants. The symmetry part of the Airy stress function requires 0 at the crack walls. In other words, the form of the Airy function that r should result in stresses proportional to r  (with values  being multiple of ½). The stresses are: 2



 2  1 r  A cos 

r2

r

r

r

1

B cos  2

1 r  A sin 

/

(1.17a)

B  2 sin  2

(1.17b)

0, which gives ( A B)cos  [ A ( 2) B]sin  0. At , we have r , 3 / 2, 1 / 2, 1 / 2, 3 / 2, This condition gives  with B A( 2). Also  , 3, 2, 1, 0, 1, 2, 3, A. We have to reject all negative values with B F

σyy σxx

y

σxy

r θ

x

F

Figure 1.5  Crack tip coordinates for establishing the stress intensity factor.

Fundamentals of Fracture Mechanics

of , as they would lead to unbounded total strain energy of some finite region and so only the values 1  0 are admissible and allow a singular field meeting the crack 1 / 2, B must be related to A. Thus the singular surface boundary conditions. For  field at the crack tip has one free parameter A which is redefined as 1/ 2 times a parameter designated by KI. The singular distribution of stress at a tensile crack tip is then found in the form: KI cos3 , 2 2 r

r

KI cos2 sin , 2 2 2 r

rr

KI cos 2 2 r

2 cos2

2

(1.18)

where KI is referred to as the mode‐I stress intensity factor, which is proportional to the loading and depends on crack geometry, and gives the stress acting across the plane 0 very near to the crack tip. Irwin (1957, 1958) determined the amount of energy available for fracture in terms of the asymptotic stress and displacement fields around a crack front in linear elastic solids. This asymptotic expression for the stress field near a crack tip is given in terms of polar coordinates, r,  θ as



ij

K 2 r

fij

higher order terms (1.19)

where σij are the Cauchy stresses, r is the distance from the crack tip, θ is the angle with respect to the plane of the crack, and fij are (non‐dimensional) functions that are dependent on the geometry and loading conditions (see Figure 1.5). Irwin called the quantity K as the stress intensity factor. It is seen that equation (1.19) involves singularity close to the tip as r 0. Since the quantity fij is dimensionless, the stress intensity factor can be expressed in units of stress length. Three linearly independent cracking modes are used in fracture mechanics, usually referred as mode‐I, ‐II, or ‐III, as shown in Figure 1.6. Mode‐I is an opening (tensile) mode where the crack surfaces move directly apart. Mode‐II is a sliding (in‐plane shear) mode where the crack surfaces slide over one another in a direction perpendicular to the leading edge of the crack. Mode‐III is a tearing (anti‐plane shear) where the crack surfaces move relative to one another and parallel to the leading edge of the crack. With reference to Figure 1.5, the detailed expressions of σij in the absence of constraint are as follows (Broek, 1994; Fett, 1998).

Mode-I: Opening

Figure 1.6  Modes of crack loading.

Mode-II: In-plane shear

Mode-III: Out-of-plane shear

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Mode‐I (opening mode): xx

KI 2 r

cos

0,

zz

2

3 1 sin sin 2 2

,

KI

yy

2 r

KI

3 cos sin cos 2 2 2 2 r

xy

cos

2

3 1 sin siin 2 2

(1.20)

Mode‐II (sliding mode): For this mode anti‐symmetry of the tangential displacement component and boundary conditions for the crack face stresses allow the determination of the integration constants: K II

xx

2 r

sin

xy



3 2 sin sin 2 2 2

K II 2 r

cos

2

,

yy

3 1 sin cos 2 2

K II

3 sin cos co os 2 2 2 2 r (1.21)

Mode‐III (out‐of‐plane shear): Only the out‐of‐plane displacement is relevant (Schreurs, 2011):

K III

zx

2 r

sin

2

,

K III

zy

cos

2 r

2

, uzz

2 K III

r 2

sin

2

(1.22)

E , E and ν are Young’s modulus and Poisson’s ratio, respectively. 2(1 ) Note that the first term in the series of equations (1.20) is the dominant term close to the crack tip. This term represents the singularity of the stress field, because it becomes infinite when approaching the crack tip. The stress intensity factors for the three loading modes shown in Figure  1.6 are denoted by KI, KII, and KIII. They are formally defined by the expressions (Rooke and Cartwright, 1976): where



K I lim 2 r r

0

yy

r , 0 , K II lim 2 r r

0

yx

r , 0 , K III lim 2 r r

0

yz

r , 0 (1.23)

Mixed mode stress intensity factor may be expressed by the following expression in polar coordinates (Bower, 2012) 3 cos 2 2 r 4

1 3 cos 4 2

KI 1 sin 2 2 r4

sin

KI

r

rr

KI

5 cos 4 2 2 r

3 2

1 3 cos 4 2

K II 3 sin 2 2 r4 K II

1 cos 2 2 r 4 K II

5 sin 4 2 2 r

sin

3 (1.24a) 2

3 3 (1.24b) cos 4 2 3 3 (1.24c) sin 4 2

Fundamentals of Fracture Mechanics

The corresponding expressions in Cartesian coordinates are

xx

yy

xy

zx

KI 2 r KI 2 r

cos cos

2

3 1 sin sin 2 2

K II

2

3 1 sin sin 2 2

K II

KI

3 sin cos cos 2 2 2 2 r

K III

sin

2 r

2

2 r

2

3 (1.25a) 2 cos cos 2 2

3 (1.25b) sin cos cos 2 2 2 2 r

K II 2 r K III

zy

sin

cos

cos

2 r

2

2

3 (1.25c) 1 sin sin 2 2

(1.25d,e)

The displacement components are obtained by integrating the strains with the result



ux uy uz

KI

r 2

1 2

sin 2

KI

r 2

2 2

cos2

2r

K III

2 2

cos sin

2 2

K II

r 2

K II

r 2

2 2 1 2

cos2

2

sin 2

2

sin n (1.26a) 2 co os (1.26b) 2

sin (1.26c) 2

In addition to the stress intensity factor, there is experimental evidence that the stress contributions acting over a longer distance from the crack tip may affect fracture mechanics properties. The constant stress contribution (first “higher‐order” term of the Williams stress expansion, denoted as the T‐stress term) is the next important parameter. The T‐stress term is independent of the distance from the crack tip (Fett, 2008). For remote stresses, the constant stress components xy , 0 0, yy, 0 0, and the stress yx , 0 tensor reads



ij ,0

xx ,0

0

0 0

def

T 0

0 (1.27) 0

where T is known as the T‐stress. The T‐stress has been shown to play a significant role in crack growth under mixed‐mode loading conditions, and also in crack path stability under pure mode‐I loading conditions. Sufficient information about the stress state is available, if the stress intensity factor and the T‐stress are known. In special cases, it may be advantageous to also know higher coefficients of the stress series expansion. This is desirable for the computation of stresses over a somewhat wider distance from a crack tip. 1.3.2  Evaluation of Stress Intensity Factors

Stress intensity factors depend on the geometry of the structural element and loading conditions (tension, bending, thermal stresses, etc.). The stress intensity factor is

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usually determined using analytical, experimental, or numerical approaches. Different methods of stress intensity factor determination are documented in Fett (1998), Fett and Munz (1997), and books by Sih et al. (1965), Sih (1973), Rooke (1986b), Tada et al. (2000), and Sanford (2003). Analytical methods include conformal transformation, where a body is mathematically transformed into another geometry, for which the solution to the stress field is easier to obtain (Muskhelishvili, 1977). The body force method was proposed by Nisitani (1967) for problems with a simple geometry and loading. The problem was formulated as a system of singular integral equations, where the unknown function is the densities of body forces distributed in an infinite body. The experimental methods include photoelastic determination of the mode‐I stress intensity factor (Etheridge and Dally, 1977). This method cannot be used for two‐ dimensional surface cracks as the measurement of stress intensity factor is performed at the surface. The method of caustics,4 known as shadow‐spot method (Kalthoff, 1987 and Rosakis and Ravi‐Chandar, 1986), is an optical way of visualizing the stress distribution close to the crack tip. This method relies on deflection of light rays due to stress‐field gradients. Since the in‐plane stresses near the crack tip are both tensile, the Poisson effect causes a local contraction (or thinning) of the material in the out‐of‐plane direction, which itself acts as a divergent lens and deflects the light. This technique can only be used for edge or through one‐dimensional cracks, with small‐scale yielding at the crack tip. It was demonstrated that strain gauges can be used to give local values of strain, from which the stress field is then inferred (Dally and Sanford, 1987). Stress intensity factors obtained from thermo‐elastic experiments were determined from the cyclic stress field ahead of a fatigue crack by Diaz et al. (2004). The idea of thermo‐elastic experiments is based on the fact that a small change in temperature due to the deformation of the material at the crack tip region can be measured and used to evaluate the stress levels. The change in the compliance of the specimen was used to obtain the stress intensity factor, based on the work‐energy approach (Gallaghar, 1971). Numerical algorithms include the compounding methods in which the stress intensity factor for a complex geometry is obtained as the sum of a series of auxiliary problems (Cartwright and Rooke, 1974). The K value due to each of the geometrical features was determined independently, and the results were compounded, with the addition of an extra stress intensity value, which denotes the interaction of the different boundaries in terms of the stress intensity factor (Cartwright and Rooke, 1974; Rooke, 1986a). The effects of different independent loadings were added to obtain the combined effect. Other numerical approaches include the transform method (Sneddon and Lowengrub, 1969), the Laurent series expansion (Isida, 1971, 1973), the boundary collocation method, and the finite element method (Fett and Munz, 1997; Chahardehi, 2008). The boundary collocation method was used to determine the stress intensity factor from boundary stresses. If the stress‐field disturbance due to a crack is confined to a small region away from the boundaries, its effect on the stresses at the boundaries would be negligible, and therefore undetectable by the boundary collocation method. On the other hand, the finite element method was used for evaluating the stress intensity factor 4  A caustic is the envelope of light rays reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. The caustic is a curve or surface to which each of the light rays is tangential, defining the boundary of an envelope of rays as a curve of concentrated light.

Fundamentals of Fracture Mechanics

for practical purposes. It has two main advantages over the boundary collocation technique. The first is that a priori knowledge of the stress series solution is not needed for the interior of the body. The second is the availability of numerous finite element software packages that have been developed, with many capabilities. Weight functions enable separation of the loading and geometry by considering the effect of each one of these two factors on the stress intensity factor separately. For example, weight functions for one‐dimensional crack problems were considered by Chahardehi (2008). The idea of the weight function approach for calculating the stress intensity factor was explained by Büeckner (1970), who indicated that it generally computes the stress field in the unnotched specimen. Büeckner (1970) showed that for the one‐dimensional crack problem, in which the crack is loaded symmetrically, the stress intensity factor can be expressed using a ‘weight function’, as



K

a

x h a, x dx (1.28)

0

where σ(x) is the stress on the crack surface plane in the uncracked body under the action of the same boundary and body forces, and h(a, x) is the weight function. It should be noted that the weight function is unique for a specific crack‐specimen geometrical configuration and is not a function of the load (Büeckner, 1970; Rice, 1972). Rice (1972) showed that for any symmetrical load system leading to stress intensity factor K and displacement field u, the weight function can be expressed as

H u (1.29) 2K a

h a, x

where H is an appropriate elastic modulus. For an isotropic material H E / (1 2 ) for plane strain, and for plane stress H E. Sih et al. (1965) presented estimated H for anisotropic materials. Equation (1.29) implies that if the stress intensity factor value and the corresponding crack displacement are known under any arbitrary stress distribution, then by use of the weight function, the stress intensity factor for any other stress system acting on the same specimen can be calculated as:



K ii

a 0

ii

x

H ui dx (1.30) 2K i a

The stress intensity factor values for a large variety of crack‐loadings are well documented in several references (e.g. Rooke, 1986b, Murakami, 1987, and Tada et al., 2000).

1.4  Linear Elastic Fracture Mechanics (LEFM) The linear elastic material behavior is an essential assumption in the theory of linear elastic fracture mechanics. Linear elastic fracture mechanics (LEFM) deals with sharp cracks in elastic bodies and is applicable to any material as long as the material is elastic except in a vanishingly small region at the crack tip, brittle or quasi‐brittle fracture, and stable or unstable crack growth. The prediction of crack growth in LEFM theory is based on the concept of energy balance as defined by Griffith’s criterion.

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1.4.1  Griffith’s Criterion

Griffith (1921) developed a linear elastic fracture criterion for brittle materials. He recognized that the difference between the energy released (if a crack was extended) and the energy needed to create new surfaces would result in a force for crack extension. In other words, crack growth will occur when there is enough energy available to generate new crack surface. The energy release rate is an essential quantity in energy balance criteria. Figure  1.4(b) shows a cracked structure with a crack length 2a and subjected to uniaxial loading of stress σ. Griffith estimated the strain energy stored per unit thickness to be Ue



a2 E

2

(1.31)

where E is Young’s modulus of the material, and the minus sign indicates that this energy would be released from the material. The energy associated with the surface area of the crack per unit thickness is Us



2 2a

(1.32)

where γ is the material specific surface energy density (J/m2). Figure 1.7 shows the total energy associated with a crack, which is the sum of the positive energy absorbed to cause the new surface plus the negative strain energy released by allowing the regions near the crack flanks to become unloaded. Note that when the total crack energy reaches its maximum value the structures equilibrium condition becomes unstable and fracture of the structure will occur at the equilibrium condition. Griffith assumed that the crack will propagate under constant applied stress, σ, if an incremental increase in crack length produces no change in the total energy of the surface. In other words, the derivative of the total energy with respect to a vanishes, that is, d Ue U s da



0 (1.33)

Us

Us

Ue

(Ue+Us) ac

(Ue+Us) Ue

a

Figure 1.7  Dependence of strain energy (Ue), surface energy (Us), and their sum of the crack length showing the critical crack length.

Fundamentals of Fracture Mechanics

This condition results in the critical stress, σcr ,



cr

2 E (1.34) ac

The corresponding critical crack length is ac, which is shown in Figure 1.7. Condition (1.34) is known as the Griffith criterion, which states that the change of surface energy must be greater than the change of strain energy in order to maintain the integrity of a structural member. Beyond the critical crack length ac the structure can lower its energy by letting the crack grow still longer. The theory of linear elasticity predicts that the stress at the tip of a sharp flaw in a linear elastic material is infinite. To avoid this singularity, Griffith developed a thermodynamic approach to explain the observed relation. The growth of a crack requires the creation of two new surfaces and hence an increase in the surface energy. The capacity for Griffith’s theory to predict crack growth was questioned by Francfort and Marigo (1998) due to its shortcomings in predicting the crack initiation, crack path, and crack jumps along the crack path. For this reason, they proposed a variational model of quasi‐ static crack evolution in line Griffith’s theory of brittle fracture. However, their proposed model is not itself restricted by the usual constraints of that theory, which require a pre‐existing crack and a well‐defined crack path. In contrast, crack initiation and crack path can be quantified. The numerical implementation of the model of brittle fracture developed in Francfort and Marigo (1998) was employed by Bourdin et  al. (2000). Various computational methods based on variational approximations of the original functional were proposed. They were tested on several anti‐planar and planar examples that are beyond the reach of the classical computational tools of fracture mechanics. Later, Bourdin et al. (2008) revisited Griffith’s theory within the framework of the calculus of variations and developed its rigorous mathematical foundation supported by numerical examples.

1.5  Nonlinear Fracture Mechanics Nonlinear fracture mechanics is also referred to as elasto‐plastic fracture mechanics (EPFM). Elastic‐plastic fracture mechanics deals with ductile fracture and is characterized by stable crack growth (ductile metals). The fracture process is accompanied by formation of a large plastic zone at the crack tip. The elastic crack tip stress reaches an infinite value, which occurs when the distance to the crack tip decreases to zero. Under this condition the material will reach the yield point before the crack tip is reached and the elastic solution is no longer valid. There are some inelastic deformations around the crack front that would make the assumption of linear elastic medium with infinite stresses at the crack tip highly unrealistic (Erdogan, 2000). For ductile materials such as steel, although equation (1.34) still holds, the surface energy, γ, predicted by Griffith’s theory is usually unrealistically high. Irwin (1957) realized that plasticity must play a significant role in the fracture of ductile materials, as described in the next subsection. When σxx reaches its maximum values, i.e. xx max, the other two principal stresses can be written as 2 n 1 and 3 m 1. According to the Von Mises yield criterion, which assumes that yielding occurs when the specific distortional elastic energy Wd

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Handbook of Structural Life Assessment

reaches a critical value Wcd . The critical value is determined from a tensile test and expressed in terms of the yield stress σy and a function of the ratios n and m. The plastic constraint factor is equivalent to the ratio of the maximum principal stress at yield and the yield stress, i.e. 1

max



1 n m n2

y

m2 mn

(1.35)

The requirement of σyy to be not larger than the yield stress, σy, results in loss of equilibrium in the crack plane ( 0). Irwin (1958) developed a solution to this inconsistency, based on the enlargement of the plastic zone, such that the total y‐force equals the force associated with the elastic solution. 1.5.1  Irwin’s Modification

In ductile materials, a plastic zone may develop at the tip of the crack, as shown in Figure  1.8. As the applied load increases, the plastic zone increases in size until the crack grows and the material behind the crack tip unloads. The plastic loading and unloading cycle near the crack tip leads to the dissipation of energy in the form of heat. Hence, a dissipative term has to be added to the energy balance relation devised by Griffith for brittle materials. In physical terms, additional energy is needed for crack growth in ductile materials when compared to brittle materials. Irwin (1957, 1958) divided the energy into the stored elastic strain energy, which is released as a crack grows, and another portion due to the dissipated energy, which includes plastic dissipation and the surface energy. The dissipated energy provides the thermodynamic U / a) is resistance to fracture, and its value per unit area of the crack (G

G p (1.36)

G 2

where Gp is the plastic dissipation (and dissipation from other sources) per unit area of crack growth. The modified version of Griffith’s energy criterion can then be written as f



EG (1.37) a

For brittle materials such as glass, the surface energy term dominates and G 2 2 J / m 2 . For ductile materials such as steel, the plastic dissipation term Plastic zone

Crack

Plane strain

Plastic zone

Crack

Plane stress

Figure 1.8  The plastic zone around a crack tip in a ductile material.

Fundamentals of Fracture Mechanics

dominates and G G p 1000 J/ m 2 . For polymers close to the glass transition temperature the energy dissipation is in the range G 2 J/m 2 to 1000 J/m 2. One basic assumption in Irwin’s linear elastic fracture mechanics is that the size of the plastic zone is small compared to the crack length. However, this assumption is quite restrictive for certain types of failure in steel materials, which can be prone to brittle fracture and catastrophic failures. Note that the crack extension occurs when G 2 R, where R is called the material resistance to crack extension. Depending on how G and R vary with the crack size, the crack growth may be stable or unstable. A plot of R versus crack extension is called the crack growth resistance curve or R‐curve. The corresponding plot of G versus crack extension is the driving force. This curve describes how the resistance to fracture increases with growing crack size in elastic‐plastic materials. The R‐curve is a plot of the total energy dissipation rate as a function of the crack size and can be used to examine the processes of slow stable crack growth and unstable fracture. The condition for stable crack growth is dG dR , while the condition for unstable dR da crack growth is dG dR . A material with a rising R‐curve, however, cannot be uniquely dR da characterized by a single toughness value. According to the condition of unstable crack growth, a flaw structure fails when the driving force curve is tangential to the R‐curve, but this point of tangency depends on the shape of the driving force, which depends on the configuration of the structure. The R‐curve for an ideally brittle material is an invariant property. However, when nonlinear material behavior accompanies fracture, the R‐curve can take a variety of shapes. Materials with rising R‐curves can be characterized by the value of G at initiation of crack growth. This value, however, characterizes only the onset of crack growth and provides no information on the shape of the R‐curve. Ideally, the R‐curve, should only be a property of the material and should not depend on the size or shape of the crack body. If the size of the plastic zone around a crack is small compared to the size of the crack, the energy required to grow the crack will not be critically dependent on the state of stress at the crack tip (Erdogan, 2000). In other words, a purely elastic solution may be used to calculate the amount of energy available for fracture. The energy release rate for crack growth or strain energy release rate may then be calculated as the change in elastic strain energy per unit area of crack growth, i.e.

G

U a

U a

P

(1.38) u

where U is the elastic energy of the system. Subscripts P and u stand for fixed load and fixed displacement, respectively, while evaluating the above expressions. For isotropic, homogeneous, and linear elastic material, Irwin showed that the strain energy release rate, G, for a mode‐I crack (opening mode) is related to the stress intensity factor KI:

G GI

1

K I2 E 2

E

K I2

plane stress (1.39a) plain strain

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Handbook of Structural Life Assessment

where ν is Poisson’s ratio, and KI is the stress intensity factor in mode‐I. Irwin also showed that the strain energy release rate of a planar crack in a linear elastic body can be expressed in terms of the three modes’ stress intensity factors for the most general loading conditions. For pure mode‐II loading, relations (1.39a) are valid by replacing KI by KII. For mode‐III loading, the strain energy release rate is given by the expression

G

2 K III

1 (1.39b) 2G

where G is the shear modulus. Under general loading in plane strain, the strain energy release rate takes the following expression



G

K I2

K II2

2

1 E

2 K III

1 (1.39c) 2G

Irwin made an additional assumption that the size and shape of the energy dissipation zone remain approximately constant during brittle fracture. This assumption suggests that the energy needed to create a unit fracture surface is a constant and depends only on the material. This new material property was given the name fracture toughness and designated by GIc. It is referred to as the critical stress intensity factor, Kc. For mode‐I, fracture occurs when K I K Ic . For the special case of plane strain deformation, Kc becomes KIc and is considered a material property. For mode‐I, the critical value of the stress intensity, KIc, is often used as an engineering design parameter in fracture mechanics. A statistical approach was introduced by Mull et al. (1987) to evaluate the cumulative probability of the critical energy release rate in composite structures. The analysis included the influence of fracture stress, critical crack length, and the angles of crack trajectory inclination, which were taken as random variables. The assessment of damage tolerance of aircraft attachment lugs was developed by Kathiresan et al. (1984). Solutions of stress distributions and stress intensity factors were obtained for various parameters. The parametric variations include lug outer‐to‐inner radius ratio corner and through‐the‐thickness cracks, crack lengths and aspect ratios, with and without interference‐fit bushing and loads above and below yield of the lugs. 1.5.2  Crack Tip Element Method

The material fracture toughness and energy release rate are usually measured by a crack tip opening displacement test. The crack opening displacement (COD) method employs the crack tip opening displacement (CTOD), see Figure 1.9. Crack tip opening displacement (CTOD or δ) is defined as the displacement transverse to the crack tip, which is given by the hypotenuse of a 45°‐45°‐90° right triangle, as shown in Figure 1.9, The apparent advance of the crack tip is known as the crack opening stretch (COS). The CTOD test measures the resistance of a material to the propagation of a crack. CTOD is used for materials that can show some plastic deformation before failure occurs causing the tip to stretch open. Fracture toughness is not a single‐valued property, but it is a set of values that are functions of both the material and the fracture toughness. Experimental measurements of CTOD fracture toughness are supposed to be conducted on specimen thickness

Fundamentals of Fracture Mechanics

CTOD

CGD

Crack tip

COS

Figure 1.9  Definition of crack tip opening displacement (CTOD), crack opening stretch (COS), and clip gage displacement (CGD) or crack opening displacement (COD).

identical to the thickness of the structural element, as recommended by ASTM (1989a, 1989b, 2002) and BSI (1991). The CTOD is estimated from the measurement of the displacement of a clip gage across the crack tips. It is assumed that the CTOD, δ, is the sum of elastic δe and plastic, δp, components i.e. e p. Approximate expressions for CTOD are given in Broek (1994) for LEFM and EPFM as



e

G y

K2 E y

J

LEFM (1.40a)

EPFM (1.40b)

y

where G is the energy release rate (dU/da) and σy is the yield stress. Fracture occurs at a critical value of G (or K) or a critical value of the J‐integral, as will be demonstrated later. 1.5.3  Paris–Erdogan Law

A power law relationship between the crack growth rate during cyclic loading was introduced by Paris and Erdogan (1963) together with the range of the stress intensity factor K K max K min , where Kmax and Kmin are the maximum and minimum stress intensity factors, respectively, in the form

da dN

C

K

m

(1.41)

where N is the number of load cycles, m is the slope between da/dN and ΔK (in log‐log scale) as shown in Figure 1.10. C is the material constant and represents the coefficient at the interception of the log‐log plot. The term on the left side, known as the crack growth rate, denotes the infinitesimal crack length growth per increasing number of load cycles. Figure  1.10 demonstrates three regions: region‐I exhibits a slow crack growing, region‐II represents the power‐law region, and region‐III is the terminal stage whose end defines the ultimate fracture. The Paris law can be used to quantify the residual life (in terms of load cycles) of a specimen for a given crack size, defining the crack intensity factor as

K

Y

a (1.42)

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Handbook of Structural Life Assessment

10–4 da/dN m/cycle (Log scale)

24

Ultimate fracture

Region I: slow crack growth region

10–6

Region II: power-law region

da

10–8

dN

=c(ΔK)m m 1

10–10 10–12

Region III: rapid, unstable crack growth

c

10–14 1 2 4 8 16 32 64 128 Stress intensity factor range ΔK: MPa/√m (Log scale)

Figure 1.10  Typical relationship between the crack growth rate and the range of the stress intensity factor showing three regions of crack development for a given stress ratio.

where σ is a uniform tensile stress perpendicular to the crack plane and Y is a dimensionless parameter that depends on the geometry. The range of the stress intensity factor is

K

a (1.43)

Y

where Δσ is the range of cyclic stress amplitude. Y 1 is taken for the case of a center crack in an infinite sheet. The remaining cycles can be found by substituting this equation in the Paris law

da dN

C

m

K

C

Y

a

m

(1.44)

For relatively short cracks, Y can be assumed to be independent of a, and the differential equation can be solved using separation of variables to give Nf

2

2 m 2

ac

C 2 m

2 m 2

ai

Y

m

(1.45)

where Nf is the remaining number of cycles to fracture, ac is the critical crack length above which instantaneous fracture will occur, and ai is the initial crack length above which fatigue crack growth starts for the given stress range Δσ. If Y strongly depends on a, numerical methods might be required to find reasonable solutions. Fatigue is the most significant failure mode in aircraft and ship structures, offshore platforms, bridges and road tankers. For example, Feng and Ren (2005) presented a design wave approach for fatigue assessment of ship structures. The fatigue assessment of a roll‐on‐roll‐off (Ro/Ro) vessel was computed and the corresponding results were compared with the results from the simplified method and direct calculation method. Yoo et al. (2009) introduced a fatigue analysis model for typical structural parts.

Fundamentals of Fracture Mechanics

Tian and Ji (2011) developed a model to analyze the low‐cycle fatigue strength of ship structures. For aerospace structures, the most significant contribution was developed by the Air Force Grow Program known as AFGROW. A brief outline of AFGROW is given in the next section. 1.5.4  AFGROW Program

A damage tolerance analysis framework, that was developed by the Air Force Research Laboratory and known as AFGROW (Air Force Grow), was documented by Harter (1999). It allows users to analyze crack initiation, fatigue crack growth, and fracture in order to predict the life of metallic structures. The stress intensity factor library provides models for different crack geometries (including tension, bending, and bearing loading for many cases). An advanced multiple crack capability allows AFGROW to analyze two independent cracks in a plate (including hole effects), non‐symmetric corner cracked. Finite element based solutions are available for two non‐symmetric through cracks at holes as well as cracks growing toward holes. This capability allows AFGROW to handle cases with more than one crack growing from a row of fastener holes. AFGROW implements five different material models to determine crack growth per applied cyclic loading. 1) The Walker equation (Walker, 1970) is essentially an enhancement of the Paris equation that includes a means of accounting for the effect of stress ratio (minimum stress/maximum stress, i.e. min / max ) on crack growth rate. This equation takes the form

da dN

C



da dN

C K max 1

K 1

n

m 1

1 m

; for n

; for

0 (1.46a) 0 (1.46b)

Harter (1999) highlighted the reasons for using a different form of the Walker equation when 0. The first is that it is more convenient to use Kmax in place of ΔK for negative values of stress ratio, . If ΔK were used for negative values, the crack growth rate curves would continue to shift to the right as decreases and eventually converge to a factor (1 ) of ΔK at 0. The second reason is that the shift in crack )(m 1) when ) would growth rate is controlled by the term (1 0 for which (1 be less than 1 and thus as m increases, the shift decreases. Generally, m is limited within the range [0,1). 2) The Forman equation (Forman et  al., 1967) is an improvement of the Walker equation and includes a means to account for the upper portion of the da/dN – ΔK curve where the data becomes asymptotic to the value of ΔK at fracture. The Forman equation used in AFGROW is



da dN

1

C Kn Kc

K

(1.47)

The Forman equation is not flexible in modeling data shifting as a function of stress ratio, . There is no parameter to adjust the shift directly. The amount of shifting is controlled by the plane stress fracture toughness of a given material.

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3) Harter T‐method (Harter, 1994) replaced the original name “point‐by‐point Walker shift method.” Harter first developed the method as a means of interpolating and/ or extrapolating crack growth rate data using a limited amount of tabular crack growth rate test data. The data shifting is handled by using Kmax in place of ΔK when 0, and the shifting data is handled by using Walker equation (1.43a). K 0 (1 )(1 m ), for 0.0, At a given da/dN, the relationship reduces to K and K max K 0 (1 )(m 1), for 0.0. This method is simply a way to interpolate/ extrapolate data in log‐log scale by using the exponential form. 4) The NASGRO equation (Forman and Mettu, 1992) was used in NASA’s crack growth life prediction program, NASGRO, Version 3.0. Few additional parameters in the NASGRO equation were introduced by AFGROW (explained later in this section). Forman and Mettu (1992), Newman (1992), and others at NLR and ESA developed the elements of the NASGRO crack growth rate equation: da dN

C

1 1

f

K

p

K th K

1

n

q

K max 1 K crit



(1.48)

where C, n, p, and q are empirical constants, α is the plane stress/strain constant factor, and σmax/σ0 is the ratio of the maximum applied stress to the flow stress, A0 (0.825 34 0.5 2)[cos( Smax / 2 0 )]1/ , A1 (0.415 0.071 ) max , 0 A2 1 A0 A1 A3 A3 2 A0 A1 1, and

f

K op K max

max R, A0

A1 A0

A2 A1

A0 2 A1

2

A3

3

0 2

0. 2

5) A tabular lookup crack growth rate capability is provided in AFGROW to allow users to input their own crack growth rate curves. The tabular data utilizes the Walker equation on a point‐by‐point basis (Harter T‐method) to extrapolate/interpolate data for any value of the stress ratio . The difference in the tabular lookup method is that the user doesn’t have to calculate all of the m values (AFGROW does it internally between each possible pair of input curves). 1.5.5  Energy Release Integrals

Most engineering materials show some nonlinear elastic and inelastic behavior under operating conditions involving large loads. In such materials the plastic zone at a crack tip may have a size of the same order of magnitude as the crack size. Furthermore, the size and shape of the plastic zone may change as the applied load increases and also as the crack length increases. Thus a nonlinear fracture theory is needed for elastic‐plastic materials that can account for the local conditions for initial crack growth such as the nucleation, growth, and coalescence of voids, or decohesion at a crack tip.

Fundamentals of Fracture Mechanics

Numerical algorithms such as the finite element and boundary element methods have been used for estimating the stress intensity factors. These methods have been classified in local or global methods (Ortiz et al., 2006). The local methods employ the local variables defined near the crack tip, whereas the global ones are based on the far field variables or variables associated to the entire domain. The global methods are based on conservation integrals such as J‐, M‐, L‐, and H‐integrals. These integrals constitute a more robust approach since they eliminate the need to solve local singular stress fields very accurately. The next subsections outline the main features of J‐, M‐, L‐ and H‐integrals. The stress intensity factors in mixed‐mode cracks can be obtained using J‐integral. In this case, it is necessary to decouple the stress and strain fields into the symmetric mode‐I and both anti‐symmetric mode‐II and mode‐III (Huber et al., 1993; Rigby and Aliabadi, 1993). The interaction integral (M‐integral) is based on the superposition of the two equilibrium states and was originally formulated by Knowles and Sternberg (1978). The two‐state L‐integral was employed by Choi and Earmme (1992) to compute stress intensity factors for circular arc‐shape cracks. The H‐integral is the conjugate integral due to Büeckner (1973). The H‐integral is derived from the well‐known second Betti reciprocal theorem. It was applied for estimating the stress intensity factor at two‐dimensional corners in isotropic, anisotropic, and dissimilar materials by Sinclair et al. (1984) and Carpenter (1984, 1995). For three-dimensional crack problems a path‐ independent expression was derived by Meda et al. (1998), starting from the H‐integral. All of these methodologies based on path‐independent integrals necessarily require the use of an auxiliary field. The fact that the H‐integral computation requires only evaluation of natural variables like displacements and stresses represents an inherent advantage of this approach, which can imply its better accuracy in comparison with other conservation integrals that require gradients of displacements and/or stresses. 1.5.5.1 J‐Integral

Elastic‐plastic fracture mechanics applies to materials that exhibit time‐independent, nonlinear behavior (plastic deformation). There are two‐parameters characterizing the nonlinear behavior at the crack tip, namely the crack tip opening displacement (CTOD) and the J‐integral. The J‐integral describes the strain energy release rate, dU / da, or energy per unit fracture surface area in a body subjected to monotonic loading. This is true, under quasi‐static conditions, both for linear elastic materials and for materials that experience small‐scale yielding at the crack tip. It may be regarded as a change of potential energy of the structure with an increment of crack extension. Rice (1968) also showed that the value of the J‐integral represents the energy release rate for planar crack growth. With reference to Figure 1.11, the J‐integral is a line integral given by the expression



J

 W x ,y dy T

where W ( x ,y )

ij

ij

d(

ij )

 u ds (1.49) x  is the strain energy density, T

  is the surface traction n

0   vector acting on a segment ds, is the Cauchy stress tensor, n is the normal to the  curve Γ, and u is a displacement vector along arc s.

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n⃗

Γ

y

x

Figure 1.11  J‐integral around a crack tip in two dimensions.

Integral (1.49) is taken as a toughness measure to describe the case where there is sufficient crack tip deformation such that the part no longer obeys the linear‐elastic approximation. It was independently developed by Cherepanov (1967) and Rice (1968). It was shown that an energetic contour path integral is independent of the path around a crack. It assumes nonlinear elastic deformation ahead of the crack tip. This analysis is limited to situations where plastic deformation at the crack tip does not extend to the furthest edge of the loaded part. For plane strain, under mode‐I loading, this relation takes the form:



JIc

GI c

K I2c

1

2

E

(1.50)

where GI c is the critical strain energy release rate, KI c is the fracture toughness in mode‐ I loading. For mode‐II and mode‐III loadings, the relation between the J‐integral and the mode fracture toughness takes the same form after replacing the subscript I by II or III, respectively. An early survey of the analysis of crack tip stress and strain fields for stationary and growing cracks in inelastic solids under monotonic loading was presented by Hutchinson (1982). The J‐integral was defined as a measure of the singularity field from the small strain deformation theory of plasticity. This description of an elastic‐ plastic material breaks down sufficiently near the tip due to effects not modeled by  deformation theory such as strongly non‐proportional plastic deformations, finite strain effects, or microvoiding and cracking. Any appreciable increases in the J‐integral above the value at which crack growth starts are possible with only small amounts of accompanying crack advance. Under these circumstances, the importance of the initiation of crack growth becomes secondary to the point at which a small amount of crack advance becomes unstable. It was indicated by Hutchinson (1982) that the J‐resistance curve approach may be an important range of applications, fairly severe restrictions on its use may be invoked, including limitations to small amounts of crack growth. Consider a stain hardening material for which the nonlinear (plastic) part of the strain, ε, is given in terms of the uniaxial stress, σ, by the power law (Ramberg and Osgood, 1943):

Fundamentals of Fracture Mechanics n



0

y

n

y

y

,

for

0

(1.51)

where σy is the yield stress, 0 y / E , and E is Young’s modulus. In this case, the singularity field defined by Hutchinson (1968a, 1968b) and Rice and Rosengren (1968) is given by the expressions

ij



y

ui ui0

1 n 1

J

y 0 Inr 0r

J

 ij

y 0 Inr

,n , n n 1

ui

ij

0

J

y 0 Inr

n n 1

ij

,n

(1.52a,b,c)

,n

where r and θ are planar‐polar coordinates centered at the tip with θ measured from the line ahead of the crack, as shown in Figure 1.5. The dimensionless θ ‐variations,  ij, ij , and ũi depend on the symmetry of the field and whether plane strain or plane stress holds as the tip is approached, as does the normalizing constant In. For a hardening material with n finite, equations (1.52a) and (1.52b) imply a unique relation between the stress and strain fields sufficiently near the tip. If r0 represents the radius of the region at the tip within the singular field, as given by equations (1.52a) and (1.52b), then it measures the zone of dominance of the singular field. Near the crack tip, it is essential that the region of incipient material separation and the region over which finite strain effects become important be contained within the zone of dominance measured by the radius r0. In most ductile metals the fracture process zone is roughly the same size as the finite strain region near the blunted tip of the crack. Finite element solutions for plane strain mode‐I, in small scale, developed by McMeeking (1977), showed that finite strain effects are only important within a radius of about 2–3 times the crack tip opening displacement, δt. Outside this radius there is little difference between the predictions of small strain theory and finite strain theory. For mode‐I in‐plane strain the zone of dominance of the J‐fields must satisfy the condition

r0

3 t (1.53)

provided that the J‐integral is to be unique measure of crack tip behavior under monotonic loading. Other configurations include the center‐cracked strip in plane strain tension, which lose the high triaxial state of stress ahead of the crack tip under fully plastic yielding. Accordingly, the near‐tip fracture environment cannot be correlated with cases of triaxial. The zone of the singularity field (Hutchinson, 1968a, 1968b; Rice and Rosengren, 1968) of dominance radius r0 approaches zero with zero strain hardening ( n ) for the fully plastic center‐cracked strip under tension. The zone of dominance radius r0 according to Hutchinson (1968a, 1968b) and Rice and Rosengren (1968) the field goes to zero with zero strain hardening ( n ) for the fully plastic center‐cracked strip under tension. Even in the presence of moderate strain hardening, the radius r0 will tend to be very small. Numerical studies of the fully yielded center‐cracked strip by

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McMeeking and Parks (1979) and Shih and German (1981) indicated that R 0.01b where b is the uncracked ligament length, and

b 200

J 0

for n 10 (1.54)

McMeeking and Rice (1975) presented an Eulerian finite element formulation for problems of large elastic‐plastic flow. The method is based on Hill’s variational principle for incremental deformations, and is convenient to isotropically hardening Prandtl–Reuss materials. The formulation was presented in a form convenient for finite element programs, for “small strain” elastic‐plastic analysis, to be simply and rigorously adapted to problems involving arbitrary amounts of deformation and arbitrary levels of stress. Finite‐strain/finite‐element analyses of deeply cracked plane‐strain center‐notch panel and single‐edge crack bend specimens were developed by McMeeking and Parks (1979) who employed non‐hardening and power‐ law‐hardening constitutive laws. The deformation was obtained from small‐scale yielding into the fully plastic range. The J‐integral can be obtained experimentally by analyzing an edge cracked plate in bending, assuming the plastic region spreads over the total ligament length. 1.5.5.2  M‐Integral (Interaction Integral)

The M‐integral is another measure of energy release rate, which can be interpreted as the energy release rate associated with self‐similar expansion within a crack tip bounding contour (Short, 1987). The M‐integral concept was earlier developed by Eshelby (1956, 1970, 1975), Knowles and Sternberg (1972), Budiansky and Rice (1973), Freund (1978), Herrmann and Herrmann (1981), and King and Herrmann (1981). The two‐dimensional mixed‐mode crack problem was reduced to the determination of mixed‐mode stress‐intensity factor solutions in terms of conservation integrals involving known auxiliary solutions. Yau et al. (1980) developed the M‐integral from the J‐integral as a way to extract the stress intensity factors for the three fracture modes from the global energy release rate. The actual field constitutes ij(1) , ij(1) , ui(1) while the auxiliary field is denoted by ij(2 ), ij(2 ) , ui(2 ) . To obtain the M‐integral for linear elastic materials, two solutions were assumed and superposed: 1 ij

ij



2 ij

,

ij

1 ij

2 ij

, ui

ui

1

2

ui , K i

Ki

1

2 K i (1.55)

The formulation of M‐integral due to Song and Paulino (2006) is based on assuming that the crack faces are traction‐free. They introduced a weight function q, which assumes the value 1 at the stress intensity evaluation point and zero on the outer boundary of the domain of integration Γ as shown in Figure 1.12. The q function can be regarded of as a virtual crack extension. The generalized J‐integral takes the form

J

lim s

0

W s

1j

ij ui ,1

nj d

lim s

0

W

1j

ij ui ,1

m j qd (1.56)

where W is the strain energy density, the contour of integration 0 s is  as shown in Figure  1.13, and δ is the Kronecker delta, i.e. ij 1 if i j, otherwise

Fundamentals of Fracture Mechanics

q

1.0 X

Γ

Figure 1.12  Concept of the weighting q‐function. (Adapted from Song and Paulino, 2006) Γt

mj, nj

y

x r Γs

Y

Γ+–

θ nj

mj A

Γ–

Γ0

Crack

Γ=Γ0 +Γ+ – Γs+Γ– X Γu

nj Figure 1.13  Contours near the crack tip showing the normal vectors m j n j for Γ0, and ; and m j on Γs used in the transformation from line integral to equivalent domain integral. (Song and Paulino, 2006)

it is zero. Applying the divergence theorem, equation (1.56) can be transformed into the equivalent domain integral

J

ij ui ,1 A

W

1j

q, j dA

ij ui ,1 A

W

1j ,j

qdA (1.57)

1 1 , and ij ,1 ij ij ij ,1 2 2 ij ui ,1 j. Introducing these expressions into equation (1.57) gives

where “, j” denotes the gradient with respect to xj, W,1 (

ij ui ,1 ) j

ij , j ui ,1

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Handbook of Structural Life Assessment

J



ij ui ,1

W

q, j dA

1j

A

ij , j ui ,1

A

1 2

ij ui ,1 j

ij ,1 ij

1 2

qdA (1.58)

ij ij ,1

The interaction integral utilizes two admissible fields, namely auxiliary and actual fields. The auxiliary field is based on known fields such as Williams’ solution (1957), while actual field utilizes displacements, strains, and stresses obtained by means of numerical methods. The J‐integral of the two states is 1 2 1 2 1 2 1 2 1 J ui ,1 ui ,1 1 j q, j dA ij ij ik ik ik ik 2 A 2 ij , j

A

1 2A

1 ij , j

2 ij ,1

1 ij ,1

where J

1 A

1 ij

2

1 2

1

ui ,1

J

2 A

2 ij

2 ij

A

2 ik

1

1 2A

1 ij

1 ik

2 ik

2 1 ij ,1 ij

1 ij

2 ij

1 2

1 2 ij , j ui ,1 1 2 ij ,1 ij

1

qdA

J

1 2

1 1 ij ,1 ij

1

ui ,1 j

2

J

1 2

M

1 ij

1 ij ,1

qdA

2 ij

1 2

2

ui ,1 j

2 2 ij ,1 ij

1 2

2 ij

2 ij ,1

qdA

(1.60b)

2 ik 2 ij

1 ij

2 2 ij , j ui ,1

A

2

1 ij ,1

2 ij ,1

(1.59)

qdA

(1.60a)

q, j dA

ui ,1

1

ui ,1 j

1 1 ij , j ui ,1

A

1j

2

ui ,1 j 1 ij

q, j dA

1j

1 ij

ui ,1

2 1 ij , j ui ,1

A



1 ik

2 ij

ui ,1

2 ij

1 2

2

ui ,1

M

1

ui ,1

1 ik

1 ik

1

1 ij

ui ,1 j 2 ij

1 ij ,1

2 ik 2

ui ,1 j

1 ij

2 ij ,1

1j

q, j dA

qdA qdA



(1.60c)

The above formulation was obtained for the equilibrium state under stationary condition. The relationship between stress intensity factor and J‐integral for general mixed‐mode problems in two‐dimensions near the crack tip (region surrounded by Γs) may be written in the form:

J local

K I2 * Etip

K II2 (1.61) * Etip

where E* is defined as E E

* Etip



1

for plane stress

2

(1.62) for plane strain

Fundamentals of Fracture Mechanics

For the superimposed fields of actual and auxiliary fields, the relationship between J‐integral and stress intensity factors of actual and auxiliary field is obtained in the form



s J local

1 * Etip

KI

1

KI

2

2

1

K II

2

K II

2

1

J local

2

J local

Mlocal (1.63)

where



Mlocal

2 1 2 KI KI * Etip

1 2 K II K II (1.64)

For a particular choice of auxiliary mode‐I and mode‐II stress intensity factors of actual fields, the stress intensity factors of the actual field are decoupled and were obtained by Yau et al. (1980), Song and Paulino (2006), and Liang et al. (2010) in the form

KI

1

2 1



* Etip

K II

* Etip

2

2

1, and K II

2

0, and K II

Mlocal , with K I

Mlocal , with K I

2

0 (1.65a)

2

1 (1.65b)

A similar technique can be used to formulate a three-dimensional equivalent domain version of the M‐integral, where the integration takes place over a volume (e.g. Wawrzynek et al., 2005). For calculating the stress intensity factors associated with piezoelectric material for an impermeable crack, Banks‐Sills et  al. (2008) derived a conservative integral as an extension of the M‐integral or interaction energy integral for mode separation. The method of displacement extrapolation was extended to validate the results obtained with the conservative integral. An M‐integral describing the mixed‐mode during a creep crack growth process in viscoelastic orthotropic media was developed by Mouton Pitti et al. (2008), based on an energetic approach using conservative laws. The fracture algorithm is implemented in finite element software and coupled with an incremental viscoelastic formulation and an automatic crack growth simulation. This integral provides the computation of stress intensity factors and energy release rate for each fracture mode. The coupling between the M‐integral and an incremental viscoelastic formulation was implemented in finite element software and introduced in an automatic crack growth algorithm, considering the crack lip un‐cohesion and the crack tip vicinity in a process zone. A numerical validation, in terms of energy release rate and stress intensity factors, was performed on a compact‐tension‐shear specimen under mixed‐mode loading for different crack growth speeds. The M‐integral concept was extended by Hu and Chen (2009) to study the degradation of a brittle plane strip caused by irreversible evolution in which the cracks coalesce under monotonically increasing loading. The change of the M‐integral before and after coalescence of two neighboring cracks inclined to each other was monitored. Finite element analyses revealed that different orientations of the two cracks led to different critical values of the M‐integral at which the coalescence occurs. It was concluded that the M‐integral does play an important role in the description of the damage

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extent and damage evolution. It was found that there is an inherent relation between the M‐integral and the reduction of the effective elastic moduli as the orientation of one crack varies, i.e. the larger the M‐integral, the larger the reduction. Furthermore, the M‐integral was found to be inherently related to the change of the total potential energy for a damaged brittle material regardless of the detailed damage features or damage evolution. Liang et al. (2010) applied the M‐integral to calculating stress intensity factor by calculating the stress field and displacement field near to the crack tip, according to the extended finite element method. 1.5.5.3 L‐Integral

The L‐integral is related to the energy release rate due to a virtual rotation. The L‐integral plays an important role in describing the damage content, the damage evolution, and the final failure, especially in the cases where the microvoids are not distributed symmetrically (Hermann and Hermann, 1983). Only when the damage induced from microvoids is symmetrically distributed does the M‐integral play a dominant role without any treatment of the L‐integral. To obtain the L‐integral the invariant curl operator was used, while for the M‐integral the divergence operator was employed. In the two‐dimensional infinitesimal deformation, the L‐integral was defined by Knowles and Sternberg (1972) in the form

L



3ij

wX j ni Ti u j Tk

uk X j ds (1.66a) Xi

where w is the strain energy density, Ti is the traction vector, uj is the displacement, ni is the unit outward normal vector, Xi are the plane coordinate axes, and ε3ij is the two‐ dimensional alternator. Park and Earmme (1986) showed that the L‐integral satisfies the conservation law, i.e. L 0 in a composite body composed of two homogeneous isotropic elastic bodies with a circular interface. They also showed that the integral is a path‐independent. It is related to the J‐integral for a circular arc‐shaped interfacial crack, and thus can be written in terms of mixed‐mode stress intensity factors, i.e. Choi and Earmme (1992)

L

3ij

wX j ni Ti u j Tk

uk X j ds Xi

RJ t

Rh K I2

K II2 (1.66b)

where Γ is the contour enclosing the crack tip with end points on the crack faces, R is the radius of the circular arc crack, Jt the J‐integral defined in the local coordinate 1 1 1 1 2 system (x 1, x2) whose origin is located at the crack tip, and h , 15 1 2 3 j κ j = 3 – 4νj for plane strain, or j for plane stress, and μj and νj (j = 1, 2) are the 1 j shear modulus and Poisson’s ratio, respectively. The property of the M‐integral being rather different in two‐ and three‐dimensions now becomes the most transparent. The relationship between M‐ and L‐integrals was 1 M developed by Chen, Y.H. (2002) in the form, L , where ϑ is the rotation angle for 2 a center crack in an infinite elastic body under remote uniform loading.

Fundamentals of Fracture Mechanics

1.5.5.4 H‐Integral

The path‐independent integral for a two‐dimensional crack referred to as the H‐integral was originally developed by Büeckner (1973). In terms of the actual and auxiliary and elastic fields, the H‐integral may be expressed in the form (Ortiz et al., 2006)

H

1

Ti ui

2

2

Ti ui

1

d (1.67)

where Ti is the traction vector and Γ is an arbitrary path around the corner tip. Note that H 0 when actual and auxiliary elastic solutions have a finite energy in the corner tip neighborhood. The asymptotic series expansion of the displacement field (Aksentian, 1967; Hartranft and Sih, 1969; Andersson et  al., 1995) at the neighborhood was expressed in terms of the angular eigen‐functions and associated stress intensity factors. Based on the property of bi‐orthogonality between the families of the eigen‐ functions associated to positive and negative eigenvalues, Ortiz et al. (2006) showed that the H‐integral given by equation (1.67) is proportional to the stress intensity functions of the actual field, i.e.

K mp

1

Ti ui

2

2

Ti ui

1

d (1.68)

where an adequate normalization factor for the auxiliary field was chosen. A domain‐ independent integral formulation for numerical computation of stress intensity factors along crack fronts and edges in three‐dimensional problems was proposed by Ortiz et  al. (2006). The formulation was implemented as a post‐processing module in a boundary element method code. Note that both LEFM and EPFM outlined in the previous sections deal with metallic structures. However, for the case of composite and sandwich structures the treatment will involve singularities due to the edge effects, which are referred to as the boundary‐ layer effects. These effects will be discussed in Section 1.6. 1.5.6  Mechanisms of Crack Propagation

There are different mechanisms in the way a crack propagates through the material, depending on the material type and its geometry, plus loading and environmental conditions. Schreurs (2011) identified these mechanisms as shear fracture, cleavage fracture, fatigue fracture, crazing, and de‐adhesion. This subsection will address four main mechanisms in fracture mechanics. 1.5.6.1  Fatigue Fracture

Under cyclic loading, the crack tip will travel a very short distance in each loading cycle, provided that the stress is high enough, but not too high to cause sudden global fracture. This mechanism is referred to as fatigue. Because crack propagation is very small in each individual load cycle, a large number of cycles is needed before total failure occurs. The number of cycles to failure Nf is strongly related to the stress amplitude ( max ( max min ) / 2 and the average stress min ) / 2. The mechanisms of fatigue‐crack propagation were examined with particular emphasis on the similarities and differences between cyclic crack growth in ductile materials, such as metals, and corresponding behavior in brittle materials, such as intermetallics

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and ceramics (Ritchie, 1999). The process of fatigue crack growth was considered as a mutual competition between intrinsic mechanisms of crack advance ahead of the crack tip (e.g. alternating crack tip blunting and re‐sharpening), which promote crack growth, and extrinsic mechanisms of crack tip shielding behind the tip (e.g. crack closure and bridging), which impede it (Gilbert et  al., 1999). The mechanisms associated with fatigue‐crack propagation in brittle materials, such as monolithic and composite ceramics and intermetallics, were found to be quite distinct from those commonly encountered in metal fatigue; moreover, their crack‐growth rate (da/dN) behavior displays a markedly higher sensitivity to the applied stress intensity (K) than is observed in most metals (Ritchie and Dauskardt, 1991). As indicated by Ritchie (1988), crack growth is promoted ahead of the crack tip by intrinsic microstructural damage mechanisms, and impeded by extrinsic mechanisms acting primarily behind the crack tip, which serve to ‘shield’ the crack tip from the far‐field driving forces. 1.5.6.2  Shear Fracture

When a crystalline material is loaded, dislocations will start to move through the lattice due to local shear stresses. Also, the number of dislocations will increase. Because the internal structure is changed irreversibly, the macroscopic deformation is permanent (plastic). The dislocations will coalesce at grain boundaries and accumulate to make a void, which may grow until it is transformed into a macroscopic crack. One or more cracks may then grow and lead to failure. Because the origin and growth of cracks is provoked by shear stresses, this mechanism is referred to as shearing. Plastic deformation is essential, so this mechanism will generally be observed in face centered cubic crystals, which have many close‐packed planes. The fracture mechanism of direct shear specimen with guiding grooves in a rock was found experimentally to creating mode‐II fracture (Rao et al., 2001). Macroscale ductile fracture is revealed by obvious changes in cross‐section of the fracture part by shear lips on the fracture surface. Macroscale brittle fractures have fracture surfaces that are perpendicular to the applied load without evidence of prior deformation. Macroscale fracture surfaces can have a mixed‐mode appearance (brittle–ductile or ductile–brittle). The brittle–ductile sequence is more common on the macroscale, while the appearance of the ductile portion is typically microscale in a ductile–brittle sequence. Zr‐based metallic glasses always fail in a pure shear mode, whereas Co‐ based metallic glasses often break into small particles or powder, exhibiting a fragmentation mode (Zhang, Z.F. et al., 2006). The difference in the failure modes for the two glassy alloys indicates that different mechanisms control the fracture processes, which can be described by a combined effect of surface energy, cleavage strength, fragmentation coefficient, and fracture mode factor (ratio of shear stress to cleavage strength). 1.5.6.3  Cleavage Fracture

When plastic deformation at the crack tip is prohibited, the crack can travel through grains by splitting atomic bonds in lattice planes. This is called intra‐ or trans‐granular cleavage. When the crack propagates along grain boundaries, it is referred to as inter‐ granular cleavage. This cleavage fracture will prevail in materials with little or no close‐ packed planes, having HCP (hexagonal close‐packed (HCP) or body‐centered cubic (BCC) crystal structure. It will also be observed when plastic deformation is prohibited

Fundamentals of Fracture Mechanics

due to low temperature or high strain rate. Inter‐granular cleavage will be found in materials with weak or damaged grain boundaries. If the ratio of the atomic cohesive strength, σB, to the macroscopic yield strength, σy, is greater than 4.0, crack‐bridging models within the framework of continuum plasticity were found to predict that the crack blunts, limiting the near‐tip stress to several times the yield strength (Needleman, 1987, 1990a; Varias et al., 1990; Tvergaard et al., 1992). Suo et al. (1993) proposed a theoretical approach for cleavage cracking surrounded by pre‐existing dislocations, which emit from the crack front. The fracture process comprised atomic decohesion and background dislocation motion. A typical micromechanism of brittle fracture is referred to as cleavage, where the atoms are gradually separated by tearing along the fracture plane very rapidly (Pokluda and Andrea, 2010). Brittle fracture in metallic materials occurs only when a pure cleavage or inter‐granular decohesion takes place. Microscopically smooth cleavage cracks, observed in ferrite at very low temperatures, possess a surface energy of 14 J/m2 which is much higher than that of about 1  J/m2 related to the lower‐bound benchmark for ideal cleavage cracks. A satisfactory explanation gives the cleavage mechanism based on alternative short‐range dislocation slip proposed by Knott (2008). The effect of the welding cycle on the fracture toughness properties of high strength low alloy (HSLA) steels was examined by means of thermal simulation of heat‐ affected zone microstructures (Lambert‐Perlade et  al., 2004). Tensile tests on notched bars and fracture toughness tests at various temperatures are performed together with fracture surface observations and cross‐sectional analyses. The influence of martensite‐austenite (M‐A) constituents and of crystallographic bainite packets on cleavage fracture micromechanisms was manifested as a function of temperature. Ravichandran et al. (1991) suggested that cleavage crack retardation at α/β interfaces in Widmanstatten titanium alloys could result from a conjoint action of thickness, ductility, and cyclic softening effects of β phase in addition to strain partitioning, hydride induced cracking effects. The thickness (volume fraction) and ductility of β phase and hydride induced cracking were found to be primarily responsible for cleavage crack arrest at thick β layers. It was revealed experimentally that the critical event for cleavage fracture is the unstable extension of a ferrite grainsized crack in specimens of C‐Mn base and weld steel (Chen, J.H. and Wang, 1994). The main factor promoting the transition from the fibrous crack to cleavage was found to be the increase of the local tensile stress ahead of the crack, which was caused by the increase of the triaxiality of stress and the apparent normal stress in the remaining ligament. The considerable scattering of toughness values in the transition temperature region was found to be due to the random variation of the widths of the tips of the fibrous cracks during their extension and the random distribution of the weakest constituents in the microstructure. 1.5.6.4  Crazing Fracture

Crazing is manifested by a network of fine cracks on the surface of a material. Crazing frequently precedes fracture in some glassy thermoplastic polymers. It occurs in regions of high hydrostatic tension, or in regions of much localized yielding, which leads to the formation of interpenetrating microvoids and small fibrils. If an applied tensile load is sufficient, these bridges elongate and break, causing the microvoids to grow and then cracks begin to form.

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A polymer fails either through deformation without change of volume, referred as shear, or deformation accompanied by volume increase referred as crazing (Kausch, 1987). Crazing was first described morphologically by Kambour (1973) and is initiated when the external stretch causes a microscopic void to open up at a stress concentration created by a pre‐created notch, a heterogeneity in the molecular network, or a foreign particle. The probability of microscopic voids occurring is dependent on the local stress situation. Kausch et al. (2003) studied the effect of intrinsic variables on local molecular motions and on the competition between chain scission, disentanglement, and segmental slip, which in turn determine the dominant mode of instability and plastic behavior. Above a critical molecular weight, toughness depends strongly on the entanglement density; a positive effect of the intensity of sub‐ Tg relaxations (where Tg is the glass transition temperature) and in‐chain cooperative motions on the toughness of these materials is clearly evident. Kramer (2005) proposed that in the craze nucleation mechanism, plane strain is more likely to open up such voids, than plane stress. Van Melick et al. (2003) found that the critical hydrostatic stress is in the order of the van der Waals surface energy of 40 mJ/m2 and it increases with increasing network density. Crazing was initially a phenomenon that was reported solely for glassy, amorphous polymers but it also occurs in semi‐crystalline polymers as reported by Plummer et al. (1994, 2001) and Thomas et al. (2007). It is observed that the typical fibril sizes tend to be an order of magnitude coarser in the case of semi‐crystalline polymers (200 nm) compared to amorphous polymers (20 nm). The experimentally observed presence of two brittle–ductile transitions, one at low temperature or high strain rates, linked with chain scission which dominates crazing, the other at elevated temperatures or low strain rates which involves disentanglement crazing provided an explanation of the polymer network response. The relation between these two brittle–ductile transitions and the major transition temperatures for molecular mobility such as the glass transition and the crystal α relaxation temperature were discussed by Deblieck et al. (2011), who demonstrated that the continuum model and the physicochemical specifications used to describe craze propagation in amorphous glassy polymers are not limited to amorphous polymers.

1.6  Boundary‐Layer Effect of Composites 1.6.1 Introduction

The problem of the boundary‐layer effects is due to interactions of geometric discontinuities of the composite and materials discontinuities through the laminate thickness. It was found to occur only within a very local region near the geometric boundaries of a composite laminate. It is frequently referred to as boundary‐layer effect or free‐edge effect. This problem is unique to composite laminates and not generally observed in homogeneous solids. Layered structural elements in lightweight constructions may suffer from severe stress concentrations of an interlaminar character which is not properly reflected by classical laminate plate theory (Ashton and Whitney, 1970; Bert, 1975; Tsai and Hahn, 1980; Herakovich, 1998a; Jones, 1998; Reddy, 2004). The classical laminate plate theory is a two‐dimensional theory and assumes a layer‐wise plane state of stress in conjunction with the kinematic assumptions of the classical plate theory of Kirchhoff.

Fundamentals of Fracture Mechanics

In the presence of irregularities, the resultant stress fields in general are of a pronounced three‐dimensional nature and on a theoretical basis may even become singular. An example of this class of stress localization problems is the well‐known boundary‐layer or free‐edge effect, which is an example of stress localization and has been the subject of several investigations for many years. The behavior of a multi‐layered fiber‐reinforced composite laminate near its geometric boundaries has been the subject of extensive experimental and analytical studies (e.g. Pipes and Pagano, 1970, 1974; Puppo and Evensen, 1970; Pagano and Pipes, 1971; Pipes and Daniel, 1971; Isakson and Levy, 1971; Rybicki, 1971, Pipes et  al., 1973; Whitney, 1973; Pagano, 1974; Tang, 1975; Tang and Levy, 1975; Hsu and Herakovich, 1977; Rybicki et al., 1977; Wang, J.T.S. and Dickson, 1978, Wang, S.S. and Choi, 1981, 1982a, 1982b; Wang, S.S. and Yuan, 1983). These studies revealed that complex stress states with rapid change of gradients occur along the edges of composite laminates. For example, Pipes and Pagano (1974) considered symmetric angle‐ply laminates under uniaxial tension by approximating the displacement field with Fourier series. Convergence investigations showed that a large number of series terms was needed to achieve reasonably accurate results. This led Pipes and Pagano (1974) to believe the possible existence of a singularity for the interlaminar shear stress at the free edge of angle‐ply laminates. Tang (1975) and Tang and Levy (1975) used layer‐wise series expansions for the stress field with respect to one half of the thickness of the respective layers. An approximation of zero‐order was obtained in closed‐form expressions for the complete stress field in the considered laminates. These expressions satisfied the conditions of equilibrium and traction‐free edges together with continuity of displacements. However, the interlaminar shear stresses did not satisfy the conditions of traction‐free laminate surfaces and were not continuous through the laminate thickness. Conti and De Paulis (1985) extended the work of Pagano and Pipes (1971, 1973) for the stress approximation in angle‐ply laminates and the calculation of interlaminar stress distributions through the laminate thickness. Pagano (1978a, 1978b) developed an approximate analytical approach to predict the stress field within the composite laminates as well as near the free edges. Soni (1982), Pagano and Soni (1983), and Soni and Pagano (1983) developed a global–local model based on a higher‐order theory approach by dividing the laminate into global and local regions. The methodology for the local region was based on the work of Pagano (1978a, 1978b). In this region the approach allows a refined analysis of the stress fields in the vicinity of free edges, whereas the global region is modeled with an equivalent single layer approach introduced by Whitney and Sun (1973). Wang, A.S.D. and Crossman (1977) solved the problem proposed by Pipes and Pagano (1970) by means of the finite element method. A special‐purpose hybrid–stress finite element was presented by Spikler et al. (1981) and Spikler and Engelmann (1986). This hybrid algorithm ensures exact traction‐free conditions at the free edges of symmetric laminates under uniform axial strain. In contrast with the previously established notion about the stress singularity at the free edges, by increasing the mesh refinement they showed that all stress components at the free edges converged to finite values. It was shown that the boundary‐layer effect is three‐dimensional in nature and is considered as one of the most fundamental and important problems in the mechanics and mechanical behavior of composite laminates. The high interlaminar stresses are known to be the dominant factor causing delamination. Wang, S.S. and Choi (1981, 1982a, 1982b) concluded that the boundary‐layer or free‐edge stress field in a composite

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laminate is inherently singular in nature due to the geometric and material discontinuities. Furthermore, the order of boundary‐layer stress singularity can be determined by solving for the transcendental characteristic equation obtained from the homogeneous solution of the governing partial differential equations. The boundary‐layer stress singularity depends only upon material’s elastic constants and the fiber orientations of adjacent plies in composite laminates. Bar‐Yoseph and Pian (1981) developed an analytical method in which the stress field could be found by minimization of the complementary potential energy in the boundary‐layer region. Altus et al. (1980) developed a three‐dimensional finite difference scheme to examine simultaneously all three interlaminar stresses. Kassapoglou and Lagace (1986, 1987) presented an analysis of free‐edge stresses representing the stresses as products of in‐plane exponential terms and polynomials through the thickness. Later, Kassapoglou (1989, 1990) considered free‐edge stresses in bending, and estimated the failure of laminates due to interlaminar stresses by adequate failure criteria. Herakovich (1989b), Kant and Swaminathan (2000), and Mittelstedt and Becker (2004a, 2007b) presented valuable assessments of the problem of free‐edge effects and stress concentration. 1.6.2  Analytical Treatment

Consider a composite laminate of finite width that is subjected to the applied force, Pz, bending moments Mx, and My, and twisting moment, Mt, all acting at the ends, as shown in Figure 1.14. The end effect is neglected by virtue of the Saint Venant principle. In this case, the stresses in the laminate are independent of the z‐coordinate. Introducing the Airy stress functions (Lekhnitskii’s stress potentials), ( x ,y ) and ( x ,y ), which satisfies the plane stress in the absence of body forces, such that (Lekhnitskii, 1963) 2 x

y2

xz

y



, ,

2 y

x2

yz

x

2

,

xy

x y

(1.69a)

(1.69b)

y

Free edge ∂BF

r

Mt Layer (m) My

O

θ Mx

Pz Z

Interface ∂BI τyz

x

Layer (m+1) τxz

σz

Figure 1.14  Schematic diagram of composite laminate of finite width subjected to the applied force, Pz, bending moments, Mx and My, and twisting moment, Mt, all acting at the ends.

Fundamentals of Fracture Mechanics

Wang, S.S. and Choi (1981) showed that the governing equations of equilibrium are given by the partial differential equations

L3

L2



L4

L3

2 A4

A2 S35 (1.70a)

A1 S34

0 (1.70b)

where L2, L3, and L4 are linear differential operators given by the expressions



L2

2

S44

L3

x2 3

S24

L4

S22

2S45

2

x y

S25

x3 4

2S26

x4

2

S55

S46

y2

(1.71a)

3

S14

x2 y

4

x3 y

2S12

S66

3

S56

x y2

4

x2 y2

2S16

S15

3

y3

4

x y3

(1.71b)

S11

4

y4

(1.71c)

where Sij is the compliant tensor defined by the stress–strain relationship

Sij

i

j,

i, j 1, 2, 3, 4 , 5, 6 (1.72)

where repeated subscript indicates summation. With reference to Figure 1.14, let the edges of a composite laminate, BF , be traction free and the interface of the mth and (m + l)th plies be a straight line meeting the traction‐free edge at a right angle. In this case, we can obtain the following boundary conditions along BF :

x

xy

0 (1.73)

xz

The conditions at the ends of the composite laminate may have the form, from the statically equivalent loads, xz dxdy

0,

B



yz dxdy

0,

B z ydxdy

B

Mx ,

z dxdy

Pz ,

B

B

z xdxdy

My ,

B

yz x

xz y

dxdy

Mt

(1.74)

where B denotes the entire area of the cross‐section. Now consider a portion of the laminate cross‐section composed of the mth and (m + l) th fiber‐reinforced laminae, as shown in Figure  1.14. Assuming that the plies are perfectly bonded along the interface B, we can immediately establish the continuity conditions of the stresses and displacements along the interface as

m x m xy

nxm nxm

m xy

nym

m y

nym

m 1 x m 1 xy

nxm nxm

1

1

m 1 xy

nym 1 (1.75a)

m 1 y

nym 1 (1.75b)

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Handbook of Structural Life Assessment m xz



nxm

m yz

nym

m 1 xz

nxm

m 1 yz

1

nym 1 (1.75c)

m m 1 um um 1, , w m w m 1 , (1.75d) where the superscripts denote the mth and (m + l)th plies in a composite laminate. nx and ny are components of unit outward normal to the interface. The governing equations (1.70a) and (1.70b) are coupled linear partial differential equations with constant coefficients related to the anisotropic elastic constants of each individual lamina. The homogeneous solution of these equations subject to the near‐ field boundary conditions and interface continuity conditions was obtained by Wang, S.S. and Choi (1981). The homogeneous boundary conditions and interface continuity conditions also provide the information for determining the free‐edge stress singularity in a composite laminate. According to Lekhnitskii (1963), the homogeneous solution for the governing partial differential equations may be written in the general form 6

x ,y



k

k 1

x

ky

,

6

x ,y

k 1

k

k

x

ky

, (1.76)

where a prime denotes differentiation with respect to its argument, and the coefficients μ are the roots of the following algebraic characteristic equation 4



0 (1.77a)

23

2

and k



3

k

4

k

2

k

3

k

(1.77b)

where 2 3

4

S55

2

2S45

S15

3

S14

S11

4

2S16

S44 2

S56 3

S25

S46

S66

2

2S12

S44 S26

S22



It was indicated by Wang and Choi that equation (1.77a) cannot have a real root (thus, μk have to appear as complex conjugates), and Φk, are analytic functions of the complex i i )/(1 k ) with k (1 i k )/(1 i k ). The r and θ variables Zk x k y r (e ke are components of polar coordinates. Substituting equations (1.76) into equations (1.69a) and (1.69b), the homogeneous solutions for stresses may be expressed in terms of Φk(Zk), i.e.



6

h x

h yz

k 1 6 k 1

2 k

2 k

’’ k

'' k

Zk ,

6

h y

Zk , h xz

k 1 6

k 1

k k

Zk , (1.78a,b)

'' k

’’ k

Zk ,

h xy

6 k 1

k

’’ k

Zk , (1.78c,d,e)

Fundamentals of Fracture Mechanics

We can express the stress functions Φk(Zk) in the form Zk

k



C k Zk

2

/

2 , (1.79)

1

where Ck and ϖ are arbitrary complex constants to be determined. Substituting equation (1.79) into equations (1.78) gives h x h yz

3 k 1 3 k 1 3

h xy

2 k Zk

Ck

k 1

Ck

3

2 k Zk

,

h y

,

h xz

Ck

k Zk

Ck

3 k Zk

Ck

k Zk

Ck

3

k Zk

3 k 1 3 k 1

C k Zk Ck

k

C k 3 Zk , k Zk

Ck

3 k

k Zk

, (1.80)

,

where the superscript (h) denotes homogeneous solution and the overbar denotes the complex conjugate of the associate variable. Wang, S.S. and Choi (1981) obtained the corresponding expressions for displacements. The homogeneous solutions are required to satisfy the homogeneous boundary conditions and interface continuity conditions. This leads to a standard eigenvalue problem for determining the values of ϖ. It is noted that ϖ generally appears as a set of complex conjugates, which enable us to make equations (1.66) real functions by superposition. Furthermore, the value of ϖ is required to satisfy the condition Re 1 to ensure bounded values of displacement components at the origin, where Re represents the real part of ϖ. Equations (1.80) can be transformed into polar coordinates (r, θ) to take the form 3 k 1 3 r



rz

k 1 3 k 1

3

C k H 1k Zk

C k 3 H 1k Zk ,

z

C k H 3 k Zk

C k 3 H 3 k Zk ,

rr

C k H 5 k Zk

C k 3 H 5 k Zk ,

k 1 3 k 1

C k H 2 k Zk

C k 3 H 2 k Zk ,

C k H 4 k Zk

C k 4 H 2 k Zk , (1.81)

where Zk are defined in the polar coordinates and H1k ( k sin cos )2 , H 2 k k ( k sin cos ), H 3 k ( k sin cos )( k cos sin ), and H 4 k ( k cos sin )2. The boundary conditions (73) along the free‐edges of the mth and (m + l)th plies in terms of polar coordinates take the form m



m z

m 1

r

m

m 1 z

0 on r

m 1

/2

n 0 on

(1.82) /2

The continuity conditions (75) along the ply interface take the form m



,

m z m 1

, ,

r

m

, ur m , u

m 1 z

,

r

m 1

m

, uzm

, ur m

1

,u

m 1

, uzm

1

on

0 (1.83)

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The homogeneous boundary conditions (82) and the continuity conditions (83) can be explicitly written in the form 3

m

k 1



m

C k H jk

3

Ck

k 1

m 1

m k

2 m 1

H jk

Ck

2 m 1 k

2

m m 3 H jk

Ck

2

m k

2 m 1 3

0 (1.84a)

2

m 1

H jk

2

m 1 k

2

0 (1.84b)

3



Ck

k 1

m

m rk

Ck

m 3

m rk

Ck

m 1

m 1 rk

Ck

m 1 3

m 1 rk

0 (1.84c)

where j 1, 2, 3; r 1, 2, 3, 4 , 5, 6 . H (jkm ) ( 2) and H (jkm ) ( ) are the values of Hjk evaluated at ( 2) and ( 2), respectively. The functions (km ) ( ) are defined by the i e )/(1 k ), 1k 1, 2k expression k ( ) (e i pk , 5k qk, k k, 3k k , 4k 2 p S S S S S , q S S / S24 k / , and tk k 11 k 12 14 k 15 k k 16 k k 12 k 22 k 6k S S t S S / S / S S , and . From equation (1.84c) k 25 k 26 k 14 k 24 k 44 k k 45 k 46 we can write the coefficients C k(m ) in terms of C k(m 1) in the form Ck



m

a ksC k

m 1

, k , s 1, 2, , 6 (1.85)

In this case equation (1.84a) takes the form

6

s 1

Cs m

1

3 k 1

{ H jkm

2

a ks

m k

m

2

H jk

2

a

k 3 s

m k

2

}

0 (1.86)

Equations (1.84b) and (1.86) constitute a system of homogeneous linear algebraic equations in C k(m 1). The existence of a non‐trivial solution for C k(m 1) requires the vanishing of the coefficient determinant

0 (1.87)

where Δ|ϖ| is a 6 6 matrix involving ϖ in a transcendental form. Thus, equation (1.83) is a transcendental characteristic equation for the standard eigenvalue problem. It has a very complicated structure, as can be seen from the coefficients of C k(m 1) in equations (1.84) and (1.86), and the detailed expression for Δ|ϖ| may be obtained using Mathematica. The investigation of the characteristic equation requires the employment of standard numerical techniques. The eigenvalues ϖn obtained from the numerical solution of equation (1.87) give important information concerning the behavior of the edge stresses and displacements. Due to the positive definiteness of strain energy of the elastic structure and the condition Re[ ] 1, the eigenvalue of ϖn is bounded by the condition

1 Re

n

1 0 (1.88)

Fundamentals of Fracture Mechanics 0.03

–ϖ1

0.02

0.01

0.0 0°

30°

60°

90°

Θ

Figure 1.15  Dependence of the eigen‐value ϖ1 on the fiber orientation angle Θ of ply graphite‐epoxy composite. (Wang and Choi, 1981)

This condition characterizes the order of the inherent singularity of the boundary‐ layer or free‐edge stresses in a composite laminate. For the commonly used angle‐ply graphite‐epoxy composite the order of the boundary‐layer stress singularity is a function of the fiber orientation Θ. Wang, S.S. and Choi (1981, 1982c) numerically estimated the values of ϖ1 for each of the fiber orientations and their results are plotted in Figure 1.15, which reveals that the composite free edge associated with the laminate of an approximately (±51°) ply orientation possesses the strongest boundary‐layer stress singularity. As the Θ changes to either directions, the order of the stress singularity ϖ1 decreases rapidly. Its value converges to zero for the cases of 0 or 90 , since the two plies become identical with orthotropic elastic properties. Pipes et al. (1973), Herakovich et al. (1979), and Sun and Zhou (1990) found that the high stresses developed in the boundary‐layer region coupled with the low interlaminar strength are responsible for the initiation and growth of local heterogeneous damage in the forms of interlaminar (delamination) and intralaminar (transverse cracking) fracture in composite laminates under static loading. Christensen (1979) and Wilkins, et  al. (1982) found that these stresses have significant effects on the long‐term strength of composite laminates under cyclic fatigue loading. Yin (1994a, 1994b) used a variational approach involving Lekhnitskii’s stress functions for the assessment of free‐edge stresses in laminates under uniaxial extension, bending, and torsion. The fundamental nature of the boundary layer effects in multi‐layer composite laminates was studied by Michel et al. (1996). The structure of the boundary layer field, based on the theory of anisotropic elasticity and on Lekhnitskii’s (1963) complex variable stress potentials were presented. The order of singularity included in the formulation of the stress and displacement fields at composite laminate edges

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Handbook of Structural Life Assessment

was studied through some examples of structures and materials including delamination and free edges. The results revealed the strong influence of several parameters on singularity orders and therefore on stresses. The finite‐difference approach was used in the investigation of the edge effect in a composite laminate subjected to a piecewise constant load by Guz et  al. (1986). The evaluation of the zone of the edge effect was obtained in terms of the ratio of the elastic properties of the isotropic layers and their geometric characteristics. An eigen‐function expansion was developed for the determination of the three‐dimensional stress field in the neighborhood of the intersection of the free edge of a hole and an interface in a laminated composite plate by Folias (1992). For transversely isotropic laminae, the stress field was shown to possess a weak singularity whose strength depends on the material constants, the fiber orientations of the two adjacent laminae, and the polar angle. Results for [0°/90°], [0°/70°], [0°/45°], and [0°/20°] were presented, and the best and worst fiber orientations were identified. The circumferential stress was shown to possess a small jump across the interface. The edge‐zone equation of Mindlin–Reissner plate theory for composite plates laminated of transversely isotropic layers was studied by Nosier et al. (2001). Analytical solutions were obtained for both circular sector and completely circular plates with various boundary conditions. The interlaminar stresses close to the free edges of general cross‐ply composite laminates based on higher‐order equivalent single‐layer theory was determined by Sarvestani and Naghashpour (2014). The laminates with finite dimensions were subjected to a bending moment, an axial force, and/or a torque for investigation. Full three‐dimensional stresses in the interior and the boundary‐layer regions were determined. The computed results were compared with those obtained from Reddy’s layer‐wise theory. It was found that higher‐order equivalent single‐layer theory precisely predicts the interlaminar stresses near the free edges of laminates. The three‐dimensional stress field, developed at the free‐edge of an externally loaded composite laminated plate, was found to exist in a thin layer close to the free‐edge layer. It may cause delamination, well before the expected failure of the matrix or fibers. It is mainly explained by the mismatch of the elastic material properties between two adjacent dissimilar laminate layers. The free‐edge effect was characterized by the concentrated three‐dimensional and singular stress fields at the free edges in the interfaces between two layers of composite laminates. An assessment of modeling techniques and the effect of stress field for symmetric laminates subjected to different load condition was presented by Soni and Pagano (1987), Murthy and Chamis (1989), Bar‐Yoseph and Ben‐David (1991, 1995), and Mittelstedt and Becker (2004f, 2004 g, 2007b). It was found that the edge effect is more dominant in tension than in bending loading for symmetric and unsymmetric laminates, and more pronounced for symmetric angle‐ply than for unsymmetric angle‐ply laminates. The main difficulty of analyzing unsymmetrically laminated shells is due to the coupling of different modes of loading and deflection. The response of the shell to constant internal pressure was studied by Preissner and Vinson (2004), with particular attention given to the bending boundary layer near the ends using the ABAQUS finite element program. It was found that the extent of the bending boundary layer in the non‐circular case is 2.5–4 times longer than that predicted by the classical equation, but the intensity of the bending boundary layer was

Fundamentals of Fracture Mechanics

reduced. An unusual compounding effect in boundary layer response for short non‐­ circular shells was described. Gu and Reddy (1992) developed a finite‐element model based on the quasi‐three‐ dimensional elasticity theory of Pipes and Pagano (1970, 1974) to examine the effect of geometric nonlinearity on free‐edge stress fields in composite laminates subjected to in‐plane loads. It was found that the qualitative nature of the stresses remains the same as those obtained in the linear analysis, but the nonlinear stresses are larger in magnitude by 5–40%, depending on the laminate. However, in most cases the difference was found to be about 10%. Davi (1996) examined the stress fields in general laminates under uniform axial strain using the integral equation theory and boundary element method. An analytical, parametric study of the attenuation of bending boundary layers in balanced and unbalanced, symmetrically and unsymmetrically laminated thin cylindrical shells was presented by Nemeth and Smeltzer (2000) for nine contemporary material systems. It was found that the effect of anisotropy in the form of coupling between pure bending and twisting has a negligible effect on the size of the bending boundary‐layer attenuation length of symmetrically laminated cylinders. Moreover, the results showed that the coupling of the membrane and flexural anisotropy and the anisotropy caused by unsymmetric lamination is generally unimportant with regards to the primary effect of the individual shell anisotropies on the bending boundary‐ layer decay length. Numerical studies of corner singularities and possible fracture phenomena in corner regions were presented by Labossière and Dunn (2001) and Dimitrov et  al. (2001, 2002a, 2002b, 2002c). For example, stress singularities in a laminated composite wedge under real three‐dimensional corner effects were studied by Dimitrov et  al. (2001, 2002a), who developed a numerical approach for the asymptotic analysis of the linear‐ elastic solution in the neighborhood of some three‐dimensional singular points. Their results revealed a strong dependence of the singular exponents on the wedge angle: for wedge angles smaller than π (convex wedges) the singularity is relatively weak, whereas for angles greater than π (concave wedges) the dominant singularity is significantly stronger and reaches quickly its minimum near 0.5. This means, that holes with sharp edges or concave corners are much more dangerous for composite structures than convex corners or edges. Carrera and Demasi (2002a, 2002b) presented an assessment of the accuracy of the finite‐element mixed layer‐wise formulations using the Reissner mixed variational theorem by comparing various results (including the interlaminar stresses) of several finite‐element models and elasticity theory within composite laminates and sandwich plates. Note that asymptotic solutions for a class of elasticity problems including stress singularities using an eigen‐function expansion method were developed by Chaudhuri and Xie (1998, 2000), Xie and Chaudhuri (1998, 2001), and Chiu and Chaudhuri (2002) based on the work of Williams (1952). Performing a separation of variables for the displacements and formulating a power‐law series for the radial components of the displacements, the order of the assumed power‐law singularities was finally calculated from a set of eigen‐equations that resulted from boundary and continuity conditions. Müller et al. (2002a, 2002b, 2003) expressed the in‐plane stress and displacement field in the vicinity of unsymmetrically laminated bi‐material notches by two complex analytical potentials.

47

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Handbook of Structural Life Assessment

The boundary layer effect in rectangular laminated plates was studied by Kumari and Kapuria (2011) and Kapuria and Kumari (2012), who used an efficient third‐order zigzag theory. A Levy‐type solution was obtained for rectangular plates, with two opposite edges hard‐simply supported. While strong edge effects were observed at the free edges and soft‐simply supported edges, no edge effect appears at the hard‐simply supported edge. The effect of length to thickness ratio and aspect ratio on the boundary layer is investigated. The results are compared with those of the smeared third‐order theory to ascertain the effect of inclusion of layer‐wise terms in the displacement approximation of the zigzag theory. The effect of electromechanical coupling and electric boundary conditions on the distributions of stress resultants near the edges was studied. The effect of inclusion of layer‐wise terms in the displacement approximation of the efficient layer‐wise theory was examined by comparing the results with those of the coupled smeared third‐order theory. 1.6.3  Thermal Loading Stress Field

The free‐edge effect is characterized by three‐dimensional and singular stress fields at the interfaces between dissimilar layers at the free edges of laminates under thermal, mechanical, or hygroscopic loadings. Free‐edge stress fields exhibit steep stress gradients and rapidly decay with increasing distance from the laminate edges, until finally they vanish in the inner laminate regions. The amount of moisture held by hygroscopic materials is usually proportional to the relative humidity. Wang, S.S. and Choi (1982c) obtained complete solutions for the hygroscopic stresses in graphite–epoxy laminates with various fiber orientations and ply thicknesses. Both variables were shown to have significant effects on the development of in‐plane and interlaminar hygroscopic stresses. Important parameters characterizing the boundary‐layer behavior such as hygroscopic edge stress singularities and stress intensity factors were considered. Thermal loadings were introduced by Webber and Morton (1993) and Morton and Webber (1993) who presented failure analyses of composites under mechanical and thermal load by applying the force‐balance method in combination with adequate failure criteria. Additional terms for the consideration of the discontinuous change of the elastic material properties in the interfaces were introduced by Rose and Herakovich (1993). The additional terms took into account the local mismatches in Poisson’s ratio and coefficient of mutual influence between adjacent layers. The influence of thermally induced residual stresses on the strength of composite materials together with the influence of adhesive layers between two adjacent laminate plies on free‐edge stress fields with respect to the free‐edge effect was studied by Wu (1990). Later, Wu (1992) examined [ ] ‐symmetric laminates under combined thermal and mechanical loading. His analysis included the physical nonlinear behavior and the influence of fibreless transition layers. For the case of [ ] ‐symmetric layups, the thermal stresses were found to have strengthening effect on the ultimate load of the laminate. The influence of combined thermal and mechanical loads on free‐edge stress fields was studied by Yin (1994c, 1997) and Kim and Atluri (1995). Yin considered the variation of thermal loads through the thickness as well as along the coordinate parallel to the interface. Thermal loads were assumed to vary linearly only through the thickness. Free exponential parameters in the assumed stress shapes were estimated by

Fundamentals of Fracture Mechanics

minimization of the total laminate complementary potential. The applicability of the described method was found to be limited to thin laminates with few layers. This limitation is due to the fact that by increasing number of plies the minimization of the energy functional requires the simultaneous solution of an increasing number of distinctly nonlinear equations for the unknown free parameters. On the other hand, Kim and Atluri (1995) employed an approach based on admissible function representations of stresses, which account for the effects of both global and local mismatches in Poisson’s ratio and coefficient of mutual influence by applying respective mismatch terms in the stress representations. The unknown stress functions were determined by application of the principle of minimum complementary energy of the laminate. Zhang and Yeh (1998) used stress functions and a variational approach to the free‐edge effect in symmetric laminates under combined mechanical and thermal load. Significant experimental evidence of singular stress fields in the vicinity of free laminate corners was reported by Herrmann and Linnenbrock (2002). Free‐edge effects in symmetric and unsymmetric cross‐ply laminates under thermo‐ mechanical loading conditions were considered by Tahani and Nosier (2003) who used a layer‐wise C0 ‐continuous displacement formulation based on the work of Nosier et al. (1993). They used the layer‐wise theory to study the interlaminar stresses near the free edges of the laminates. Based on the reduced elasticity displacement field of a long laminated composite plate, Nosier and Bahrami (2006, 2007) studied interlaminar stresses in anti‐symmetric angle‐ply laminates under extension and torsion. The interlaminar stresses near the free edges of generally laminated composite plates under extension were calculated by Nosier and Maleki (2008). The constant parameter appearing in the reduced displacement field, which describes the global rotational deformation of a laminate, was obtained by employing an improved first‐order shear deformation theory. The effects of end conditions of laminates, fibers orientation angles and the stacking sequences of the layers within laminates, and geometric parameters on the boundary‐ layer stresses were considered. Free‐edge effects in laminated, circular, cylindrical shell panels subjected to hygrothermal loading were studied by Nosier and Miri (2010) who used a displacement‐based technical approach. Appropriate reduced displacement fields were determined for laminated composite shell panels with cross‐ply and anti‐ symmetric angle‐ply layups. An equivalent single‐layer shell theory was employed to determine the constant parameters appearing in the reduced displacement fields. The distributions of transverse shear and normal stresses in various shell panels under a uniform temperature change were numerically estimated. Becker et  al. (1999) and Mittelstedt and Becker (2003a) adopted a force‐balance approach for the stress assessment at rectangular free corners of cross‐ply laminates under thermal load. Mittelstedt and Becker (2004a, 2004b) presented a closed‐form analytical method for approximate calculations of the stress fields in the vicinity of free rectangular corners of thermally loaded laminates with arbitrary layup. Their approach is essentially based on the force‐balance method of Kassapoglou and Lagace (1986, 1987) and of Becker et al. (1999). Closed‐form analytical methods for rectangular corners were developed by Becker et al. (1999, 2000, 2001), Mittelstedt and Becker (2003b, 2004a) employing the force‐balance method formulation for cross‐ply laminates and laminates with arbitrary layups, respectively. Mittelstedt and Becker (2003a) used displacement based equivalent single‐layer theory approach for rectangular cross‐ply laminates, and Mittelstedt and Becker (2004c, 2004e) displacement based layer‐wise

49

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Handbook of Structural Life Assessment x

z

y

d

Figure 1.16  Schematic diagram of a laminate of arbitrary layup with n plies and the total thickness d showing the coordinate frame at the free‐corner edge.

approach for rectangular cross‐ply laminates based on the work of Reddy (1987). Numerical approaches employing the finite element method were carried out by Griffin (1988), Icardi and Bertetto (1995), and Herrmann and Linnenbrock (2002). The asymptotic analysis of the free‐corner situation in layered structures was considered by Koguchi (1997) who used the boundary element approach. Finite element analysis was employed by Labossière and Dunn (2001), Nied and Ayhan (2001), Dimitrov (2002), Dimitrov et  al. (2001, 2002a, 2002b, 2002c), and Dimitrov and Schnack (2002). Mittelstedt et  al. (2004), and Mittelstedt and Becker (2004f, 2004 g) used the scaled boundary finite element method. Mittelstedt and Becker (2004b) considered a laminate of arbitrary layup with n plies and the total thickness d and is exposed to a constant temperature change ΔT, as shown in Figure 1.16. The three‐dimensional stress field was studied in the interface between two arbitrarily chosen plies k and k 1 in a corner region. The functional dependence of all stress components was assumed in the product form



k xx

1

1

k yy

1

1e

k xy

1 e

x e 2x

1x

x

1

1

1e

1

A1 z

k

A2

y

A3 z

k

A4

y e

2y

1 e

2y

k

A5 z

A6

k k

(1.89)

k

where φ, φ1, φ2, λ1, λ2, η, ψ1, and ψ2 are constants to be determined using the variational principal. These constants describe the decaying rate of stress away from the free corner and edges of the laminate. The assumed in‐plane stress distributions must satisfy the homogeneous boundary conditions for the intralaminar stresses, i.e.

k xx

|x

0

k yy

|y

0

k xy

|x

0

k xy

|y

0

0 (1.90)

For the limiting case, x , y , conditions for the stress field must also be satisfied. The exponential terms in equations (1.89) must vanish for x , y . The constants Ai were obtained by the following adjustments:

Fundamentals of Fracture Mechanics



A1

k

A3

k

A5

k

1 d d d

k

1

k

1

k

k xxu

k xxl

,

A2

k

k xxl

k yyu

k yyl

,

k A4

k yyl

k xyu

k xyl

,

A6

k

k xyl

zk

1

k xxu

k xxl

1

k yyu

k yyl

1

k xyu

k xyl

d zk

k

d zk

k

d

k

(1.91)

where the overbar denotes the classical laminate plate theory stresses, the subscripts u and l denote the values at the upper and lower interfaces of the k‐th ply, respectively, z( k 1) is the thickness coordinate of the lower interface of the (k − 1)‐th layer, and d(k) is the thickness of the k‐th ply. The three‐dimensional equilibrium equations are k xy

k xx

x

k xz

y

0,

z

k yz

k xz

x

k zz

y

z

0,

k xy

k yy

k yz

x

y

z

0 (1.92)

Equations (1.92) are used to determine the unknown interlaminar stress components by layer‐wise integration: k xz k yz



k zz

z d /2 z d /2 z d /2

k xx

x

k xy

x k xz

x

dz dz dz

z d /2 z d /2 x3 d /2

k xy

y k yy

y k yz

y

dz dz (1.93) dz

The continuity of the interlaminar stress components requires the following conditions to be satisfied: k xz k yz



k zz

x, y, z x, y, z x, y, z

zk

k 1 xz

x, y, z

zk

k 1 yz

xx , y , z

zk

k 1 zz

x, y, z

zk z k (1.94) zk

The traction‐free surfaces of the laminate must satisfy the following conditions: 1 xz 1 yz



1 zz

x, y, z x, y, z x, y, z

d /2

n xz

x, y, z

d /2

d /2

n yz

x, y, z

d /2 (1.95)

d /2

n zz

x, y, z

d /2

The interlaminar shear and normal stresses may be expressed in terms of linear combinations of stress functions obtained from integration of the in‐plane stresses:

51

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Handbook of Structural Life Assessment k xz

2

x

xe

1

2y

1e

1 k 2 A1 z 2

k

A2 z

B1

k

1 k 2 k k A5 z A6 z B5 2 1 k 2 k k 1 1e 2 x 2 ye y A3 z A4 z B3 2 1 k 2 k k 1x 1 e 2y A5 z A6 z B5 1e 2 1 k 3 1 k 2 k 2 3 A1 z A2 z B1 z e x xe x 1 1e 2 y 6 2 1 k 3 k k k 1x 2y A5 z A6 z 2 2 B5 z B6 1e 2e 3 1 e

k yz

k zz



1x

2e

2y

(1.96)

B2

k

where the constants Bi are given by the following expressions



B1

k

B2

k

B3

k

B4

k

B5

k

B6

k

k i 1 k i 1 k i 1 k i 1 k i 1 k i 1

1 2 z 2 i 1 2 z 3 i 1 2 z 2 i 1 3 z 3 i 1 2 z 2 i 2 3 z 3 i

1

A1

i 1

A1

i

1

A1

i 1

A1

i

1

A3

i 1

A3

1

i A3

1

A5

i 1

1

A5

i

i

i 1 A3

zi

1 2 z 2 i zi

A2

1

A2

i

A2

i 1

A4

A2

1

i A4

1

zi

1

A6

i 1

z 2i

1

A6

A5

i

i 1

i 1 A4

i 1

i

A5

i

i

A4

1

1 2 z 2 i

i 1

(1.97)

i

A6

i 1

A6

The homogeneous boundary conditions for the interlaminar shear stresses along the free edges are

k xz

|x

k yz

0

|y

0

0 (1.98)

As stated earlier, the constants φ, φ1, φ2, λ1, λ2, η, ψ1 and ψ2 are determined using the principle of minimum total complementary energy . For a thermo‐elastic orthotropic material, the strain–stress tensor under thermal loading can be written in the form s11

j

s12

j

s13

j

0

0

s16

j

j

s22

j

yy

s12

s23

0

0

s26

zz

s13

j

s23

j

s33

j

0

0

s36

zz

j s45

0

yz xz

xx

j j

yz

0

0

0

j s44

xz

0

0

0

s45

j

s55

j

0

j

s26

j

s36

j

0

0

s66

xy



j

s16

j

xx

11t

yy

22t

xy

T

33t

0 0 0

(1.99)

Fundamentals of Fracture Mechanics

For the case of an n‐plied laminate with linear‐elastic material properties under thermal loading, the complementary energy is 1 2

n



j 1

j

S

j

j

j

j

d

jT

t

j

j

T

j

d

j

Min (1.100)

where Ω(j) denotes the volume of the j‐th layer of the laminate, S ( j ) is a symmetric off‐axis compliance matrix, ΔT is the constant temperature change and (t j ) are the coefficients of thermal expansion. The components of Cauchy’s stress tensor are given in the vector representation: j



j xx

j yy

j yy

j yz

j xz

T

j xy

(1.101)

Under the assumption of linear elastic material, the elastic parameters of the j‐th layer ( j) are given by the compliance matrix S and the compiled off‐axis coefficients of ( j) thermal expansion t :

j

S



t

s11

j

s12

j

s13

j

0

0

s16

s12

j

s22

j

s23

j

0

0

s26

s13

j

s23

j

s33

j

0

0

s36

0

0

0

s44

j

s45

j

0

0

0

0

s45

j

s55

j

0

j s16

j s26

j s36

0

0

s66

j

j xxt

j yyt

j zzt

0 0

j j j

(1.102)

j

T

j xyt

(1.103)

Note that the elastic compliances sij in an off‐axis system may be transformed in terms of the values sij given in the on‐axis system using the elementary tensor transformations as s11

s11 cos 4

s22 sin 4

2 s12

s66 cos2

sin 2

s22

s11 sin 4

s22 cos 4

2s12

s66 cos2

sin 2 , s33

s12



s11

s22

s66 cos

2

s23 sin

2

sin

2

s13

s13 cos

s16

2 s11 cos2

s22 sin 2

s26

2 s11 sin

2

2

s36

2 s13

s45

s55

s66

4 s11

s22 cos

,

s23 cos sin , s44 cos sin , s22 2 s12 cos2

2

s23

s12 sin s13 sin

4

2

cos sin cos sin s44 s55 sin 2

cos

s23 cos2 2 s12 2 s12

s44 cos2 s44 sin 2

s33

4

s66 cos sin 3 s66 cos

3

sin

cos3

sin

cos sin 3

s55 sin 2 s55 cos2

s66 cos2

sin 2

2



53

Handbook of Structural Life Assessment

The coefficients of thermal expansion of the orthotropic material need to be transformed to the transformation rules as 11t 12 t



11t

cos2

22 t

2 cos sin

sin 2 ,

11t

22 t

22 t

,

11t

23t

sin 2 0,

13t

22 t

cos2

33t

33t



A detailed evaluation of the complementary energy given by equation (1.86) is documented by Mittelstedt and Becker (2004b) based on mathematical optimization procedures outlined by Vanderplaats (1984). The distribution of stress field near the free corner of symmetric laminates consisting of four plies made of transversely isotropic carbon fiber‐reinforced plastic was estimated by Mittelstedt and Becker (2004b). Each ply has a thickness of 0.5 mm, which leads to a total laminate thickness of 2.0 mm, and the plate is exposed to a uniform temperature rise of T 100K (°F = K × 9/5 − 459.67). The interlaminar normal stress σzz is commonly considered to be the predominant cause for the onset of delamination. Figures. 1.17(a)–(c) show three‐dimensional plots of the interlaminar stresses σzz over the region 0 x 2.0 mm, 0 y 2.0 mm, and z 0.4 mm for orientation angles of the laminate plies [0°/90°]‐, [0°/60°]‐ and [0°/30°]‐ symmetric laminates, respectively. These figures reveal distinct dependence of the interlaminar stress fields on the variation of the orientation angle. In the pure cross‐ply (a)

(b) 40

40

20

20

σzz(MPa)

σzz(MPa)

0 –20 –40 2 1 y(mm)

2 0 0

0 –20 –40 2

2

1 y(mm)

1 x(mm)

0 0

1

x(mm)

(c) 40 σzz(MPa)

54

20 0 –20 –40 2 1 y(mm)

2 1 0 0

x(mm)

Figure 1.17  Interlaminar stress σzz over the plane xy for a uniform temperature rise of T 100K for: (a) [0°/90°]‐symmetric laminate, (b) [0°/60°]‐symmetric laminate, and (c) [0°/30°]‐symmetric laminate. (Mittlestedt and Becker, 2004b)

Fundamentals of Fracture Mechanics

layup [0°/90°]‐symmetric laminates, the distribution of the peeling stress σzz is nearly perfectly symmetric with respect to the corner tip, i.e. the occurring stress fields along both in‐plane coordinate directions x and y exhibit close similarities, with a reversed sign. After displaying some maximum value at the laminate edge, σzz changes its sign once before vanishing in the inner laminate regions. It is seen that the resultant distribution of the interlaminar normal stress σzz throughout the whole corner region of Figure  1.17(a) may essentially be constructed by the superposition of the two corresponding free‐edge effects. Note that the highest tensile edge stresses σzz of all considered symmetric layups are encountered. This is to be expected since the mismatch in the elastic properties of the adjacent layers then also reaches its maximum, which is the primary reason for the occurrence of this class of stress concentration phenomenon. However, the situation changes with the [0°/60°]‐ and [0°/30°]‐symmetric laminates shown in Figures.  1.17(b) and (c). With decreasing the second orientation angle, the resultant distributions of σzz lose their symmetry properties and the resultant free‐edge effects display different behavior along their respective edges. This observation is easily explained with the differing material properties along the two laminate edges which, in all cases, is the basic reason for stress concentrations at the free edges of layered structures. The dependence of the free‐corner effect on the orientation angles of the laminate plies was further studied for the cases of [30°/Θ2] ‐symmetric laminates and [60°/Θ2] ‐ symmetric laminates by varying the fiber orientation Θ2. Figures.  1.18(a) and (b) show the distributions of σzz over the interval 0 y 2.0 mm at x 0.0 and z 0.4 mm. It is seen from Figure 1.16(a) that the tensile stress σzz reaches its maximum value for the case of the [30°/0°]‐symmetric laminate. For the case [30°/90°]‐symmetric laminate the corner stresses remain higher than the free‐edge stresses. On the other hand, the case of [60°/Θ2]‐symmetric laminate, shown in Figure  1.17(b), reveals ­significant dependence of the resultant stresses on the angular orientation of the laminate plies.

1.7  Closing Remarks This chapter has introduced the basic ingredients of the classical theory of fracture mechanics. The applications of this theory in aerospace and ocean structures will be considered in Chapter 2. Fracture mechanics criteria have their limitations, as described earlier. For example, one basic assumption in Irwin’s linear elastic fracture mechanics is that the size of the plastic zone is small compared to the crack length. However, this assumption is quite restrictive for certain types of failure in steel materials, which are prone to brittle fracture and catastrophic failures. Furthermore, the J‐integral is also limited for situations where plastic deformation at the crack does not extend to the furthest edge of the loaded part. The limitations of LEFM were discussed in detail by Bouchbinder et al. (2013). LEFM falls short of explaining the fast dynamics of a crack once it deviates from a perfectly straight path. Thus, high‐velocity path instabilities, most notably the side‐branching and the oscillatory instabilities, remain open problems in this framework. Furthermore, when fracture phenomena on a timescale for which inertial resistance of the material to motion is significant (i.e. hyperelasticity), the influence of crack tip plasticity and material strain rate sensitivity cannot be handled by

55

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Handbook of Structural Life Assessment

(a) σzz(MPa) 35

[30°/90°]

30 [30°/75°]

25 20

[30°/60°]

15 [30°/45°]

10 5

[30°/0°]

0 –5

[30°/15°] 0

1 y (mm)

2

(b) σzz(MPa) 35

[60°/0°] [60°/15°]

25

[60°/90°] [60°/75°]

[60°/30°]

15

[60°/45°]

5 –5 –15 –25

0

1 y (mm)

2

Figure 1.18  Interlaminar tensile stress σzz over the interval 0 y 2 mm, at x 0 and z 0.4 mm for a uniform temperature rise of T 100K : (a) [30°/Θ2]‐symmetric laminates and (b) [60°/Θ2]‐symmetric laminates. Mittlestedt and Becker (2004b)

fracture mechanics theory. The importance of the M‐integral will be demonstrated in dynamic fracture energy release due to interaction energy of mixed modes. These effects are only accounted for in the theory of dynamic fracture and peridynamics, which will be addressed in Chapter 3. When a significant region around a crack tip experiences plastic deformation, other approaches can be used to determine the possibility of further crack extension and the direction of crack growth and branching. A simple technique, incorporated into numerical calculations, is the cohesive zone model method which is based on concepts proposed independently by Dugdale (1960) and Barenblatt (1962). The relationship between the Dugdale–Barenblatt models and Griffith’s theory was first discussed by Willis (1967). The equivalence of the two approaches in the context of brittle fracture

Fundamentals of Fracture Mechanics

was shown by Rice (1968). Interest in cohesive zone modeling of fracture was extended following the work of Xu and Needleman (1994) and Camacho and Ortiz (1996) on fracture dynamics. The deterministic and stochastic modeling of fatigue crack damage in metallic structures for on‐line diagnostics and health monitoring of operating machinery was considered by Patankar and Ray (1998) and Ray and Patankar (1998). A dynamical model of fatigue crack propagation was developed in the deterministic state‐space setting based on the crack closure concept under cyclic stress excitation. The model state variables were crack length and crack opening stress. For the case of stochastic state‐space model of fatigue crack propagation, the state‐space model was found capable of capturing the effects of stress overload and underload on crack retardation and sequence effects. The non‐stationary statistics of the crack growth process under (tensile) variable‐amplitude load were obtained in a closed form without solving the governing stochastic differential equations of Itô’s type. Teng et al. (2008) presented a statistical study on plasticity and fracture properties of a ductile aluminum casting. The test data fitted was found to fit well with both normal and Weibull probability functions. Shear fracture strains spread in a narrower range than tensile ones. The theory of fracture mechanics does not account for corrosion and hydrogen embrittlement. However, some recent attempts have considered environment‐assisted cracking, or environmentally induced cracking in the form of corrosion that produces a brittle fracture in alloys. These efforts will be discussed in Chapter 8.

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2 Applications of Fracture Mechanics 2.1 Introduction The classical theory of fracture mechanics has been extensively applied to assess the life of structural components made of different materials such as metals and composites. Research activities of these applications have been motivated by serious and catastrophic events of aircraft, ships, bridges and multi‐story buildings. Fracture toughness and damage tolerance are equivalently used to imply life/durability. These are usually measured by the number of load cycles, which will induce fracture in structures without and/or with assumed defects. Other important issues are dealing with the critical size that an assumed defect will grow to an imminent structural fracture under applied service loading conditions and the defect size that a loaded composite structure can safely withstand (Chamis, 1984). This Chapter presents an overview of the main results of these research activities in two main Sections. Section II deals with fracture mechanics of metallic structures such as steel and aluminum alloys commonly used in aircraft and ship structures. Section II is devoted to the fracture mechanics of composites, sandwich structures, and foams. Damage of sandwich structures under normal loads constitutes of damage formation modeling, experimental investigations. The problem of thermo-mechanical coupling of sandwich plates involves analytical modeling, exact solution of the boundary‐value problem, numerical and asymptotic solutions of thin skin, and characterization of thermal, mechanical and geometrical modeling. Section II also covers the experimental investigation of solid foams under cyclic load‐unload tests and the phenomenological modeling of the foam response under compressive loading.

2.2  Fracture Mechanics of Metallic Structures 2.2.1  Steel Structures

During the design stage of metallic structures, the designer has to consider several ­factors based on strength, weight, and optimum configurations (Farahmand, 2001; McCormac and Csemak, 2011). The selected design must withstand environmental conditions without causing failure, and this requires comprehensive structural ­analyses of static, dynamic, fatigue, and fracture to ensure the integrity of the structure. Handbook of Structural Life Assessment, First Edition. Raouf A. Ibrahim. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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The associated problems include structural failures resulting from cracks that are inherent in the material, or defects that are introduced in the part due to improper handling or rough machining, and this must be assessed through fracture mechanical concepts. The applicability of plane strain fracture toughness values, KIc, as a possible design parameter was examined by Amateau and Steigerwald (1965, 1967), who compiled available data on plane strain fracture toughness, measurement of KIc for a variety of materials and determination of certain processing history and section size effects on KIc. The toughness parameter KIc can serve as a relative rating parameter, which would help in selecting the most appropriate material. In addition, quantitative estimates can also be made of the load‐carrying capacity of components containing possible defects. The selection of metallic materials for many applications is usually decided by the simple characterization of the stress‐corrosion‐cracking sensitivity in terms of the linear‐elastic threshold stress intensity KIscc (Judy and Goode, 1975). The application of fracture mechanics theory for selecting metallic structural materials was documented in a research monograph by Campbell and Underwood (1982). Some progress was made to establish crack growth rate relations to explain different fatigue phenomena (e.g. McEvily et al., 1999; Sadananda et al., 2001). For example, the modified constitutive relation developed by McEvily et al. (1999) was found to account for the effects of initial crack size and load sequence. This model was further modified by Cui and Huang (2003) who introduced an unstable fracture condition and defined a virtual strength to replace the yield stress. Further extension was made by Wang et al. (2008), who considered the slope of the fatigue crack growth rate curve as a variable rather than a fixed value of 2 for different materials. The extended McEvily model was further improved by Yi et al. (2008), Wang and Cui (2009, 2010a, 2010b), Wang et al. (2010), and Chen, F.L. et al. (2010, 2011). The improvement of the crack growth rate model based on the concept of partial crack closure was introduced by Cui et al. (2011), who developed a unified fatigue life prediction method for marine structures and established a correct crack growth rate relation based on the concept of partial crack closure. Stress analysis and fatigue life prediction of ship structures was examined by Petershagen (1995). The crack propagation theory based on fatigue assessment of ship structures was considered by Niu et al. (2009). The stress intensity factor based on finite element analysis was calculated and the Paris model used to predict crack propagation life. Nominal local effects such as hot spots (structural discontinuities) and the cyclic stress/strain concept were introduced. Liu, Y.H. et al. (2010) studied the tensile fatigue test and fatigue crack propagation on 907A steel plate specimens with V‐type gap. The fatigue crack propagation model was adopted for analyzing and predicting crack ­propagation situation and the remaining life of damaged warship in waves. The initiation and propagation of fatigue cracks in compact tension specimens of mild steel in a tensile residual and compressive cyclic stress condition was reported by Greasley et al. (1986). It was shown that fatigue cracks grow several millimeters at a diminishing rate. Fracture toughness data can be used to calculate the size of a crack that would initiate a brittle fracture under certain stress conditions at a particular temperature. It is also possible to analyze the stress that would cause a certain sized crack to give a brittle fracture at a particular temperature. It is important to consider the loading rate effect and the effect of temperature on the initiation of fracture toughness of steel plates in terms of crack tip opening displacement. These effects were described by Barsom and Rolfe (1999) and SSC‐430 (2003). For example, Figures. 2.1(a) and (b)

Applications of Fracture Mechanics

(a) 2 1.8 1.6 CTOD. (mm)

1.4 1.2 1 0.8 0.6 0.4 0.2 0 –120

–100

–80

–60

–40

–20

0

20

40

Temperature (°C)

(b) 1.6 1.4

CTOD. (mm)

1.2 1 0.8 0.6 0.4 0.2 0 –100

–80

–60

–40

–20

0

20

Temperature (°C)

Figure 2.1  Crack tip opening displacement dependence on temperature for (a) ABS grade B steel plate, (b) ABS grade EH 36 steel plate: under intermediate (♢), quasi‐static (QS ♦) and 0.25 mm CTOD rate loadings (SSC‐430, 2003).

show the dependence of the crack tip opening displacement (CTOD) on temperature for ABS grade B and grade EH 36 steel plates. These figures show the effect of strain (loading) rate and temperature on the transition behavior of a ship plate. For the intermediate rate loading (6.5 × 103 MPa m/s) the curves were shifted to the right of the quasi‐static rate transition. At 0.25 mm CTOD lower bound, a design temperature of 0 °C is safe for quasi‐static loading rate. At intermediate loading rate, the EH and B grades give CTOD values lower than 0.25 mm. Full thickness fracture toughness of conventional ship plate grades, including modern high‐strength steels, was determined at loading rates representing quasi‐static, intermediate, and impact conditions (Pussegoda et al., 1996). It was shown from the CTOD

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transition temperature curves that the transition temperature increase from quasi‐static to intermediate rate (a three‐order rate increase) is much greater than the increase from intermediate to impact (a two‐order rate increase). Correlations between nil‐ductility transition (NDT)1 temperature and dynamic fracture toughness transition temperature were reported in SSC‐430 (2003). Early work was presented to show that the termination of the lower shelf of the stress intensity factor, K, base toughness at about 103/s strain rate is close to the NDT temperature (Barsom and Rolfe, 1999). The crack initiation and crack propagation events during the fracture toughness measurements at static and dynamic loading with respect to perfectly brittle, quasi‐brittle, and ductile behavior of material was considered by Krasowsky (2001). The effect of temperature, loading rate, stress state mode and specimen size on the stretched zone formation, as well as on the crack initiation was studied. If plane‐strain conditions are satisfied, the stretched zone parameters were found to be independent of the specimen size variation. A correlation was obtained of the stretched zone height and width, with fracture toughness of materials. A plane strain transition temperature for a given specimen thickness was defined for ferritic steels as the temperature along the KIc transition curve where the ASTM validity limit is reached. This temperature is linearly related to the specimen thickness logarithm under both static and dynamic loading. For a given steel, transition temperatures deduced from other tests were found to correspond to various characteristic values of the non‐dimensional crack tip plastic zone size. The different relationships were developed to deduce transition temperatures of various tests from the knowledge of either the KIc transition curve or from two other transition temperatures. A full‐scale fracture test to identify the effects of material toughness and plate thickness on the plastic deformation capacity governed by brittle fracture was conducted by Kuwamura et  al. (2003). They employed the artificial hot spot method and multiple critical point method for the experiment to simulate brittle fracture in steel members undergoing plastic deformation and to obtain reliable data from a small number of specimens. The artificial hot spot method was found with absolute certainty to originate brittle fracture at an intended location, and the multiple critical point method provided a satisfactory consistency in the statistical estimation of pre‐fracture ductility. The test results demonstrated that material toughness in terms of Charpy impact energy has a substantial influence on the ductility governed by brittle fracture, while the plate thickness is of less importance within a test range greater than an inch. 2.2.2  Aluminum Alloys

The rapidly increasing use of aluminum alloys in aerospace and ship structures requires a fundamental understanding of mechanisms and mechanics of fracture that govern stiffened panels. The development of structural integrity and damage tolerance analyses of aluminum structures is vital in light of the increased use of aluminum alloys in 1  Nil ductility transition (NDT) temperature of a metal represents the point at which the fracture energy passes below a predetermined point (for steels typically 40 J – for a standard Charpy impact test). Ductile– brittle transition temperature (DBTT) is important since, once a material is cooled below the DBTT, it has a much greater tendency to shatter on impact instead of bending or deforming. The Charpy impact test, also known as the Charpy V-notch test, is a standardized high strain rate test, which determines the amount of energy absorbed by a material during fracture. This absorbed energy is a measure of a given material’s notch toughness and acts as a tool to study temperature-dependent ductile–brittle transition.

Applications of Fracture Mechanics

aircraft and marine structures. The fracture of structural aluminum alloys in terms of plane‐strain stress intensity factor was studied by Kaufman et al. (1971). If the material is so tough that the yielded region ahead of the crack extends to the far edge of the ­specimen before fracture, the crack is no longer an effective stress concentrator. Instead, the presence of the crack merely serves to reduce the load‐bearing area. In this regime the failure stress is conventionally assumed to be the average of the yield and ultimate strengths of the material. Lucke and Brown (1971) presented a method for predicting the fatigue life of 2024 T3 and 6061 T6 aluminum alloys subjected to either constant amplitude sinusoidal or wideband random fatigue loadings. The fatigue life was determined by calculating the critical value of the crack length when the structure can no longer support its load. Rahman et  al. (1990) developed a Markov model to evaluate mode‐I fatigue crack propagation in structures subjected to time varying random load processes. Their analysis involved phenomenological models for material behavior and is based on the theories of linear elastic fracture mechanics and compound Poisson processes. For high strength aluminum alloys of fairly thick sections the plane‐strain fracture‐toughness concepts and the values of the critical plane‐strain stress intensity factor were reported. Hermann (1994) studied mode‐I fatigue crack growth in notched specimens of 7017‐T651 aluminum alloy subjected to fully compressive cyclic loads. From residual strain models it was found that the fatigue crack growth is confined to a region of tensile cyclic stress within the residual stress field. Kasaba et al. (1998) described a possible fatigue crack growth under fully compressive stresses. Woelke (2004) reported on a comprehensive study of research needs for aluminum structures identified the key research areas, such as material behavior, fracture evaluation and design of welded aluminum structures subjected to dynamic loading. He then proposed an accurate and efficient analysis and limit state based design methodology for welded aluminum ship structures for lightweight high speed aluminum vessels. The proposed methodology accounted for anisotropic, nonlinear, and rate dependent behavior of aluminum sheets, subjected to dynamic loads causing fracture and failure of structural components. Experimental and analytical investigations of the fatigue crack growth and fracture response of aluminum reinforced compact tension specimens were performed by Farley and Newman (2004). This showed that selective reinforcement significantly improves these responses, primarily through load sharing by the reinforcement. With the appropriate combination of reinforcement architecture and mechanical properties, fatigue cracks can be arrested using selective reinforcement. For both fatigue crack growth and fracture, the three most influential properties were reinforcement width, reinforcement stiffness, and interface stiffness. Crushing characteristics of full‐scale extruded aluminum panels under a four‐point bending load configuration were studied experimentally and numerically by Zheng et al. (2004). Tests were run all the way to first fracture, which occurred inside buckling induced dimples on the compressive side. The location of the first fracture and the corresponding load were predicted using a fracture criterion. Based on a simplified beam model, a closed form analytical solution of the force–deflection response for the pre‐ fracture stage was generated. Later, Zheng et al. (2008, 2009) presented experimental and numerical investigations on deformation and fracture of the large‐scale welded thin‐walled AA 6061 panels. Under mode‐I, the proposed model was capable of ­capturing crack initiation, initial growth, and a more complicated phenomenon where the crack experiences a sudden jump to a new location and steadily propagates.

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The fatigue crack growth rates and the threshold of stress intensity factor range were experimentally measured. Some attempts were made to predict the fatigue crack growth rates from the tensile properties of the material (e.g. Kang et al., 2006; Firrao et al., 2007; Farahmand and Nikbin, 2008; Tiryakioglu and Hudak, 2008). An approximate estimation approach of model parameters introduced by Cui et al. (2011) was validated for a wide range of alloys, such aluminum alloy (Paris et al., 1999; Newman et al., 1999; Zhao et  al., 2008), titanium alloy (Jha and Ravichandran, 2000), and steel (Wang and Cui, 2010b; Dinda and Kujawski, 2004; Taheri et  al., 2003). The model parameters were derived from experimental data under the stress ratio 0.1 and then min / max used to estimate the crack growth rate curves under other stress ratios. Figures 2.2(a)– (c) show comparisons between measured and predicted crack growth rate curves and their dependence on the stress intensity factor range for 2324‐T39 aluminum alloy, 300 M steel, and titanium alloy Ti‐10v‐2Fe‐3Al, respectively, for different values of stress ratio, . Damage of welded aluminum thin‐walled structures was experimentally examined by Zheng (2006). A local fracture criterion was validated on two types of aluminum components without welds. The first was S‐rails under quasi‐static and dynamic axial loading; while the second was large‐scale extruded aluminum panels under four‐point bending. A wide range of aluminum weldments were examined from the point of view of microstructure, hardness distributions, and stress–strain relations. It was concluded that aluminum weldments exhibit very different mechanical characteristics from comparable steel weldments. A unique series of large‐scale mode‐I and mode‐III fracture tests was performed on full‐scale welded panels. The mode‐I simulation was found to have the ability to model a sudden jump of the crack from the weld zone to the heat‐ affected zone. A comprehensive tool consisting of application of advanced fracture models, material calibration, and validation through component testing was provided by Galanis (2007), with the purpose of increasing the survivability envelope and speeding up the development process of new vessels. The effect of stiffening configurations on fracture of aluminum structures was examined by studying the structural response of various stiffened plates, which were compared with unstiffened plates represented by small‐scale compact tension specimens. It was shown that mapping of crack patterns in stiffened plates is feasible and can enable ship designers to evaluate critical areas within a structure with respect to crack initiation, propagation, and optimum material usage. The fatigue crack growth resistance and fracture toughness of three aluminum alloys (5083‐H321, 5086‐ H116, 5383‐H116) used in marine structural applications were studied experimentally (SSC‐448, 2007). AA5083 is one of the aluminum alloys most widely used for the plate components of high‐speed craft. Although AA5083 performs well in its marine applications, it was not developed specifically for this environment but AA5383 was specifically developed to help optimize aluminum behavior in the marine environment. The difference in fatigue crack growth rate in air was found to be negligible. In simulated seawater environment, AA5086 showed a slightly superior performance. In addition, all samples showed the same ranking of toughness, with 5086 showing the highest toughness, followed by 5083, and then 5383. The dependence of the J‐integral on the crack extension, Δa, for the three aluminum alloys and two different sample sizes, 2‐in and 4‐in, revealed that the larger sample size exhibits slightly higher values of toughness at a given increment of crack extension.

(a) 0.10 0.01

da/dN (m/cycle)

1E–3 1E–4 1E–5 1E–6 1E–7 1E–8 0.10

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da/dN (m/cycle)

1E–5

1E–6

1E–7

1E–8

1E–9

10

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ΔK (MPa√m)

Figure 2.2  Comparison between measured and predicted crack growth rate curves for different values of stress ratio : a) 2324‐T39 aluminum alloy (Paris et al., 1999, Cui et al., 2011) 0.1 ■ measured, ___ ‐ ___ predicted 0.3 ▲ measured, ‐ ‐ ‐ ‐ predicted 0.5 ♦ measured, ___ ‐ ___ predicted 0.7 ► measured, predicted 1.0 • measured, predicted b) for 300 M (Dinda and Kujawski, 2004) 0.05 ■ measured, ___ ‐ ___ predicted 0.3 ▲ measured, ‐ ‐ ‐ ‐ predicted 0.5 ♦ measured, ___ ‐ ___ predicted 0.7 ► measured, predicted

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(c) 0.10 0.01 da/dN (m/cycle)

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1E–3 1E–4 1E–5 1E–6 1E–7 1.0

10.0

100.0

ΔK (MPa√m)

Figure 2.2  (Continued) c) Titanium alloy Ti‐10 V‐2Fe‐3Al (Jha and Ravichandran, 2000). 0.1 ■ measured, ___ ‐ ___ predicted 0.5 ▲ measured, ‐ ‐ ‐ ‐ predicted 0.8 ♦ measured, ___ ‐ ___ predicted

Under localized impulsive loading, Lee and Wierzbicki (2005) analyzed the deformation and fracture of thin plates. A calibration procedure for aluminum alloy 2024‐T351 using a damage plasticity theory was presented by Xue and Wierzbicki (2009). The pressure sensitivity and the Lode angle2 dependence were included in a fracture envelope. Bai and Wierzbicki (2010) revisited the Mohr–Coulomb fracture criterion to determine the ductile fracture of isotropic crack‐free solids. This criterion correctly accounts for the effects of hydrostatic pressure as well as the Lode angle parameter. It turns out that these two parameters, which are critical for characterizing fracture of geo‐materials, also control fracture of ductile metals (Wilkins et  al. 1980; Barsoum 2006; Barsoum and Faleskog, 2007; Xue 2007; Bai and Wierzbicki 2008). It was found that the Mohr–Coulomb fracture locus predicts almost exactly the exponential decay of the material ductility with stress triaxiality in agreement with theoretical analysis of Rice and Tracey (1969) and the empirical equation of Hancock and Mackenzie (1976), and Johnson and Cook (1985). The Mohr–Coulomb criterion also predicts a form of Lode angle dependence which is close to parabolic. Test results of 2024‐T351 aluminum alloy and TRIP RA‐K40/70 (TRIP690) high strength steel sheets, were used to calibrate and validate the proposed Mohr–Coulomb fracture model. A phenomenological modified Mohr–Coulomb ductile fracture model, developed by Bai and Wierzbicki (2010), was employed to predict failures in a stretch‐bending operation on AHSS sheets by Luo and Wierzbicki (2010). Beese et al. (2010) incorporated the effect of plastic anisotropy on the fracture modeling of aluminum alloy 6061‐T 6 sheets. 2  The Lode angle provides a measure of the magnitude of the intermediate principal stress in relation to the minor and major principal stresses.

Applications of Fracture Mechanics

Plane strain toughness kIC (Ksi √in)

100

D6ac

80 TI 6–4 60 TI 6–6–2 40

Steel Aluminum

20

0

0

100

Titanium

200

300

Yield strength (Ksi)

Figure 2.3  Dependence of fracture toughness on yield stress for three types of metals. (Wood, 1974)

Significant factors leading to the development of damage tolerance criteria and i­ llustrated the role of fracture mechanics in the analysis and testing aspects necessary to satisfy these requirements were reviewed by Wood (1974, 1975). The need for efficient aircraft structures has resulted in the selection and use of high strength alloys in ­primary members, with little regard for the general decrease in fracture toughness associated with increased yield strength (Figure 2.3). Sophistication in design and analysis techniques and closely monitored weight‐saving programs have afforded some the opportunity to exploit conventional alloys such as 7075 aluminum far beyond the practical limits, with the result being higher allowable design stresses with each aircraft structure. These general practices, of course, have reduced the tolerance of the structure to both initial manufacturing defects and service produced cracks. Critical flaw sizes are often on the order of the part thickness and often much less, making positive detection during normal field service inspections improbable. Higher design stresses, of course, increase the likelihood of early fatigue cracking in‐service and may result in loss of fleet readiness and expensive maintenance and/or retrofit programs. For aircraft structures, the designer must select a material of reasonably high strength, in order to meet static strength requirements and still achieve minimum weight. For evaluating structural efficiency, the ratio of yield stress to the material density was selected as a design parameter. The achievement of maximum yield strength and maximum fracture toughness is often difficult to achieve, as reflected in Figure  2.3. This figure reveals that toughness decreases with increasing yield strength for aluminum, titanium, and several selected steels. In view of the dramatic decrease in KIC, the designer must limit the upper bound of acceptable range of yield strength. The material selection process is therefore a tradeoff procedure, since many concurrent requirements must be satisfied. The designer must establish a criterion for accepting either a reduced toughness or strength level. The choice might be dictated by overall flaw tolerance. For example, Figure 2.4 shows the dependence of a crack size parameter expressed as the ratio (KIC/σy)2 on the yield stress. Since structures are designed to withstand (statically) a percentage of the yield strength, this parameter may be conveniently used

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Crack length parameter (KIC/σy)2

Handbook of Structural Life Assessment 0.3

Titanium

0.2

Steel

0.1 Aluminum 0.0

0

100

200

300

Yield strength (Ksi)

Figure 2.4  Dependence of crack length parameter on yield stress for three different metals. (Wood, 1974) 0.3 Aluminum Crack length parameter (KIC/σy)2

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Desired trend 0.2

Ti ta ni um

0.1

Hypothetical limit Hypothetical manufacturing defect limit

Steel

0.0

0

500

600

700

800

900

1000 1100 1200

Structural efficiency parameter (σy / ρ)

Figure 2.5  Dependence of crack length parameter on structural efficiency parameter for three different metals. (Wood, 1974)

to illustrate flaw tolerance sensitivity. Figure 2.4 indicates that a more dramatic reduction in the crack length parameter, with increased yield strength. The same trend is repeated in Figure 2.5; however, the yield strength is normalized to the material density ρ. The parameter σy/ρ is one form of structural efficiency used to select materials. Note that material ranking has changed, with titanium being superior to steel. One exception illustrated is 18% Ni maraging steel and 9Ni 4C, which fall beyond the bounds illustrated. Maraging steels (“martensitic” + “aging”) are iron alloys that are known for possessing superior strength and toughness without losing malleability (ability to deform), although they cannot hold a good cutting edge. Aging refers to the extended heat‐treatment process. These steels are a special

Applications of Fracture Mechanics

class of low‐carbon, ultra‐high‐strength steels that derive their strength not from carbon, but from ­precipitation of intermetallic compounds. The principal alloying element is 15–25% nickel. There are recognizable limits on both the values of (KIC/σy)2 and (σy/ρ) for ­currently used materials as illustrated in Figure 2.5. If fracture is assumed to occur at the design limit stress, the value of critical crack length, ac, can  be computed. For  many  aircraft structures, design limit stress is of the order of L 0.6 y ( K IC /0.6 y )2 / ( K IC / y )2. Damage tolerance criteria, results of crack resistance testing of wing/body skin ­materials, and predicting fatigue crack propagation rates, based on linear classic fracture mechanics were reported by Nesterenko (1993). Damage tolerance is the property of a structure to retain its safety level for a specified service life after partial or complete fracture of primary components due to fatigue, corrosion, and/or accidental operational damages. Damage tolerance criteria, material crack resistance, stress–strain state, and stress intensity factors for primary airframe components were identified. The static and  cyclic crack resistances were determined for airframe materials. The fatigue crack growth rate as a function of the stress intensity factor range was obtained from experimental data for the crack growth rates of 10 7 mm/cycle to 1 mm/cycle. The  experimental results revealed that the fracture toughness of materials of ageing airplanes is 20–50% less, and the crack growth duration is 4–7 times less in comparison with new materials of the same grades made. The required crack resistances were determined for aluminum alloys used in the skin of the aircraft: fracture toughness > 155 MPa m , the required values of the fatigue crack growth rate of 0.75–1.5 mm/cycle at the stress intensity factor range of 31 MPa m . General and analytical stochastic crack growth analysis methodologies were proposed by Yang and Manning (1990) for predicting the statistical crack growth damage accumulation in metallic structures of airframes. The “general” approach employed a numerically defined deterministic (median) crack growth curve directly. The “analytical” approach approximates the median crack growth curve with one or more compatible crack growth segments. Analytical predictions for both approaches were correlated with experimental test results. Stress analyses of cracked aircraft components were reviewed by Newman (2000) in an attempt to understand the fatigue and fatigue‐crack growth process. The prediction of residual strength of complex aircraft structures with widespread fatigue damage was also considered. Observations of small‐crack behavior at open and rivet‐loaded holes and the development of small‐crack theory were used to the prediction of stress‐life behavior for components with stress concentrations under aircraft spectrum loading. Fatigue‐crack growth under simulated aircraft spectra can be predicted with the crack‐closure concept. Residual strength of cracked panels with severe out‐of‐plane deformations (buckling) in the presence of stiffeners and multiple‐ site damage was predicted with advanced elastic‐plastic finite‐element analyses and the critical crack‐tip‐opening angle fracture criterion. A general overview of the approaches of fatigue life of aerospace structures based on fracture mechanics was presented by Kardomatea and Geubelle (2010). Metallic glasses are amorphous metals in which the structure is not crystalline (as it is in most metals), but rather is disordered, with the atoms occupying more or less random positions in the structure. Considerable progress has been made to understand the deformation response of metallic glasses. Metallic glasses are characterized by low fracture toughness. Detailed experimental studies coupled with complementary

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numerical simulations have provided insights into the micromechanisms of failure, as well as the nature of crack tip fields, and have established the governing fracture criteria for ductile and brittle glasses. Mechanics‐based approaches for understanding the fracture behavior of bulk metallic glasses have been reviewed by Schuh et al. (2007), Xu et al. (2010), and Narasimhan et al. (2015). Bulk metallic glasses that exhibit shear bands in mediated plastic flow are referred to as ductile, as they exhibit high levels of KIc. In spite of the high KIc values, the bulk metallic glasses hardly show any crack growth resistance curve (the R‐curve measures the resistance to fracture increases with growing crack size in elastic‐plastic materials.) This is due to the lack of microstructural features such as grain boundaries, which can offer resistance to crack growth. Furthermore, unlike crystalline metals and alloys where a one‐to‐one correspondence between ductility and toughness can typically be observed, no such correlation exists in bulk metallic glasses. For ductile bulk metallic glasses, the measured KIc values are comparable to crystalline counterparts. According to Pan et al. (2007), Wang et al. (2009) and Narayan et al. (2014), most bulk metallic glasses exhibit low KIC values in the range (0.5 15 MPa m ) and hence are termed as ‘brittle’. Some tough bulk metallic glasses can be made brittle through annealing at ­temperatures below Tg, a process referred to as structural relaxation (Duine et al., 1992; De Hey et  al., 1998). Hassan et  al. (2008) examined the effects of temperature and ­loading rate on fracture toughness and mechanism in Vitreloy‐1. They found that as  temperature changes from 0.8Tg to Tg, there was a sharp increase in the fracture toughness with the behavior changing from plane strain (flat fracture) to plane stress (shear fracture). In general, increasing the strain rate caused decrease in the toughness, especially at high temperatures (above 0.9Tg).

2.3  Damage of Composite Structures 2.3.1 Preliminaries

Composite structures are known to possess high strength and high stiffness‐to‐weight ratio. Structural failure may be induced by mechanical, aerodynamic, hydrodynamic, acoustic, or aero‐thermal loads. Structural integrity, failure prediction and service life assessment are of major interest to engineers and designers of structural components made of non‐traditional materials, such as composites. Prediction of the complex loading environment in high‐speed operation and constitutive/fracture models, which can adequately describe the nonlinear behavior exhibited by advanced alloys and composite materials, are critical in analyzing the nonlinear structural response of modern ­aerospace vehicles and ship structures. A classical plate theory version of Irwin’s virtual crack closure method was developed and used for the energy release rate, first for plane strain and then for three‐dimensional deformations. It was shown that the classical plate theory does not provide quite enough information pertaining to decomposition of energy release rate into its opening and shearing mode components. This theory is an analytical approach to determine total strain energy release rate. It also provides a damage prediction methodology through linear elastic fracture mechanics theory and classical laminate plate theory. Prediction of the energy release rate and its components for mixed‐mode delamination of

Applications of Fracture Mechanics

composite laminates was discussed by Schapery and Davidson (1990). The crack tip element is not restricted by material types and loadings. Furthermore, it requires less computational time for three‐dimensional finite element modeling and can be experimentally validated as described by Davidson (1998). The crack tip element approach expresses strain energy release rate in terms of the loadings at the crack tip, that is, the forces and moments. Davidson et al. (1995) extended the earlier crack tip element theory to solve for the oscillatory singularity at the crack tip that frequently occurs in composite laminates. The approach was referred to as non‐singular field, which ignores the condition of the damage zone at the crack tip (Davidson, 1998, 2001). It allows the prediction of fracture toughness for different geometries and loadings for the same material system. Two‐ and three‐dimensional crack tip elements were introduced by Davidson et al. (1996, 2000) and Yu and Davidson (2001), to obtain the energy release rate and mode mix in practical geometries using plate theory based near‐tip forces and moments. Davidson and Yu (2005) determined the total energy release rate and the individual energy release rate components in skin–stringer delamination problems, using the three‐dimensional crack tip element and three‐dimensional finite element approaches. It was found that, excluding areas near free edges where results are highly sensitive to the local mesh, predictions by the two approaches are quite close. The distribution of the total strain energy rate, GT, according to the crack tip element analysis of T‐joints was determined by Dharmawan et al. (2008). 2.3.2  Assessment of Composites Mechanics

The assessment of composite structural mechanics may be represented in terms of the selected combination of micromechanics, macro‐mechanics, combined stress failure, laminate theory, singularity mechanics, life/durability, and structural analysis. In the area of mechanics of composite structures, life/durability is generally used to describe how long a composite structure will survive its monotonic or cyclic load service environment. There are different types of damage mechanisms in composite materials under stress loading. Among these are fiber or second‐phase failure, delamination, matrix cracking, and debonding of phases. Compact tension specimens of rigid polyurethane foam were tested in fatigue by Noble (1983). Crack growth was monitored using acoustic emission (AE). During the load cycle the AE activity was decomposed into four regions: the crack faces “un‐sticking,” fracture events at or close to peak load, a period of zero AE just after peak load, and AE associated with crack closure during the unloading part of the cycle. The fracture was found to increase rapidly with crack length. Microcrazing3 in the struts of flexible polyurethane foams was observed during compressive deformation and observed directly in the scanning electron microscope (Kau et al., 1992). Attributed to this phenomenon was the decrease in stress at maximum compression and the intensity of acoustic emission during compressive cycling. The higher content of styrene‐acrylonitrile copolymer in these foams resulted in higher modulus, more severe microcrazing, increase in AE activity, and a decrease in the stress at maximum compression as cycling progressed. 3  Crazing is a network of fine cracks on the surface of a material. It is a phenomenon that frequently precedes fracture in some glassy thermoplastic polymers.

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Simplified predictive methods and models to evaluate fiber/polymer‐matrix composite material for determining structural durability and damage tolerance were presented by Chamis and Ginty (1987). An integrated computer code referred to as CODSTRAN (COmposite Durability STRuctural ANalysis) developed by Chamis and Smith (1978) was used to evaluate local damage occurrence, cumulative damage, and progressive fracture. Fatigue toughness of composite structure was predicted using composite mechanics in conjunction with fracture mechanics concepts and finite element analysis. The general procedure was described by Murthy and Chamis (1985) who developed three‐dimensional finite‐element modeling for end‐notch and mixed mode flexure. They were able to quantify composite damage tolerance in terms of strain energy release rate and crack length. Interlaminar fracture mechanics is useful for characterizing the onset and growth of delaminations in terms of the total strain energy release rate, GT, due to interlaminar tension (mode‐I), GI, interlaminar sliding shear (mode‐II), GII, and interlaminar scissoring shear (mode‐III), GIII (e.g. O’Brien, 1982, 2001; Sun and Jih, 1987; Martin, 1998; Tay, 2003). A quasi‐static mixed‐mode fracture criterion was obtained by plotting the dependence of the interlaminar fracture toughness, GC, versus the mixed‐mode ratio, GII/GT, by using pure mode‐I double cantilever beam (DCB), GII /GT 0, pure mode‐II end notched flexure (ENF) (GII /GT 1), and mixed mode bending (MMB) tests of ­varying ratios. A curve fit of this data was generated to determine a mathematical relationship between GC and GII/GT. Benzeggagh and Kenane (1996) introduced the relationship

GC

GIC

GIIC

GIC GII / GT

(2.1)

where GIC and GIIC are the experimentally determined fracture toughness data for mode‐I and mode‐II, respectively, η is a factor to be determined using a curve fit. Failure is expected to occur when, for a given mixed mode ratio, GII/GT, the calculated total energy release rate, GI, exceeds the interlaminar fracture toughness, GC. The interlaminar fracture toughness values were determined experimentally over a range of mixed modes from pure mode‐I loading to pure mode‐II loading (Davies et al., 1999; Martin and Davidson, 1999; ASTM, 2000a, 2000b). O’Brien (2001) introduced a methodology based on fracture mechanics for characterizing the onset and growth of delaminations in composites. The method was adopted by Krueger et al. (2000, 2002a) to investigate delamination onset and debonding in simple laboratory coupon4 type specimens. The virtual crack closure technique was found to be very useful for computing energy release rates (Rybicki and Kanninen, 1977) during delaminations in laminated composite structures. Krueger (2004) presented an overview of the virtual crack closure technique. Results based on continuum two‐dimensional and solid three‐dimensional finite element analyses were found to provide the mode separation required when using the mixed‐mode fracture criterion. For three‐dimensional problems, delamination onset or growth can be predicted based on the entire failure surface, GC GC GI , GII , GIII . Several specimens were suggested for the measurement of the mode‐III interlaminar 4  S–N curves are derived from tests on samples of the material to be characterized (often called coupons) where a regular sinusoidal stress is applied by a testing machine, which also counts the number of cycles to failure. This process is sometimes known as coupon testing.

Applications of Fracture Mechanics

fracture toughness property (e.g. Martin, 1991; Robinson and Song, 1992). The state‐of‐ the‐art in fracture toughness characterization and interlaminar fracture mechanics analysis tools was presented by Krueger (2006). The existence of an initial delamination crack near the free edge was assumed by Rybicki et al. (1977), who considered the free‐edge delamination process as that of a stable crack growth when the total strain energy release rate for the crack exceeds certain critical value. Various experimental investigations (Wilkins et al. 1982; Donaldson and Mall 1989), using split cantilever beam specimens with unidirectional layup, revealed that the critical energy release rates for modes I, II, and III differ from each other. The mixed‐mode stress intensity factors for a free‐edge delamination crack in a laminate under tensile loading conditions were determined by Chow and Atluri (1998) using a mutual integral approach based on the application of the path‐independent J‐integral to a linear combination of three different solutions. It was found that the calculated mixed‐mode stress intensity factors of the free‐edge delamination crack remain relatively constant as the crack propagates into the laminate. The stress intensity factor KI due to water‐ice inclusion in both transversely isotropic and orthotropic matrices was analytically obtained by Roy et al. (2002). It was found that the stress field in the vicinity of an elliptic inclusion was an acceptable analytical idealization to obtain the stress intensity for a crack with ice inclusion. The mixed‐mode stress intensity factors of composite materials using the crack opening displacement (COD) were determined by Ju and Liu (2007). Series solutions of the composite material with a crack were used to evaluate COD values, and the least‐squares method was used to calculate mixed‐ mode stress intensity factors. Both finite element simulations and laboratory experiments were applied to validate this least‐squares method with acceptable accuracy if the even terms of the series solution are removed. The prediction of fatigue delamination onset life was studied by Murri et al. (1998, 2006) and Krueger et  al. (2002b). A standard test was documented for the mode‐I ­double cantilever beam test (ASTM, 2000c). Hansen and Martin (1999) generated mixed‐mode onset data. Hoyt et al. (2002) employed the interlaminar fracture mechanics to characterize the extension or growth of delaminations when subjected to fatigue loading. As indicated by Krueger (2006) delamination growth rate can therefore be expressed as a power law function. However, the exponent is typically high for composite materials compared to metals, as indicated by Martin and Murri (1990) and König et al. (1997). Layered composites may experience delamination and debonding phenomena caused by incompatibility of the microscopic modes of deformation associated with individual layers. A finite element method was employed by Goswami and Becker (2000) to study the causes and effects of debonding phenomena between the face‐sheet and the core of a sandwich plate under in‐plane loading. It was found that under an applied in‐plane loading, there is a significant stress concentration at the junction of three cell walls and the face‐sheet, which easily leads to the generation of cracks and their growth. It was reported that there is a significant amount of energy release rate even in the case of a very small or virtually no crack. This phenomenon indicates that the glue used to attach the face‐sheet and the cell must withstand a non‐zero energy release rate, even in the intact situation without any debonding. Two concepts for analysis and assessment of the delamination induced by the deformation mode incompatibility were presented by Hohe and Becker (2001) and Hohe et  al. (2003). The first concept is a microscale

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approach based on the analysis of local stress singularities. The second concept was defined on the mesoscale, using the average energy release rate as a fracture parameter. Both concepts were demonstrated for a structural sandwich panel with hexagonal honeycomb core. Mittelstedt and Becker (2004a) presented an overview of the classical free‐edge effect based on selected results published over the period 1967 through 2004. The results included approximate closed‐form analytical methods for the stress analysis in the free‐ edge effect situation. In their subsequent work, Mittelstedt and Becker (2004b) developed a refined closed‐form analysis method for the calculation of interlaminar stress concentrations in the vicinity of rectangular wedges of thermally loaded composite laminates with arbitrary layup. The interlaminar stresses were derived from the three‐ dimensional equilibrium conditions in combination with the exact fulfillment of the given homogeneous boundary conditions of traction‐free laminate facings and the requirement of continuity of the interlaminar stresses at the ply interfaces. The far field conditions of recovery of the stress results by classical laminate plate theory in the inner laminate regions with increasing distance from the laminate corner were considered. An asymptotic expansion of stress singularities in the vicinity of interfaces at straight free edges and also at free corners of composite laminates where two straight free edges intersect was presented by Mittelstedt and Becker (2005a, 2005b). The computations were performed using the boundary finite element method, which was also was employed by Müller et  al. (2005) for predicting directions of cracks emerging from notches at bi‐material junctions under arbitrary loading conditions. Results for potential crack initiation angles of both homogeneous and bi‐material media were presented with multiple examples of different wedge angles and different loading combinations. It was found that fracture starts inside the softer material, and with small increments of  the arbitrary loading ratio (tensile to shear loading) the propagation angle shifts inside the second material. An asymptotic analysis of the mechanical fields was performed near laminate ­reinforcement patch corners by Wigger and Becker (2007a, 2007b). Various configurations of interface corners were examined and their effect on the singular characteristics of the cross‐sectional force field was studied. It was found that for a characterization of the singular behavior of the in‐plane forces, each singular in‐plane force term has to be considered and that the corresponding displacement modes are useful for understanding this behavior. The mechanical in‐plane fields of anisotropic reinforcement patch corners were examined using the semi‐analytical boundary finite element method. The study highlighted a relation between the number of singular in‐plane force exponents and the character of the associated displacement mode. A closed‐form approach for the calculation of the occurring interlaminar stress concentrations in the surroundings of rectangular reinforcement patch corners of thermally loaded composite cross‐ply laminates was developed by Wigger and Becker (2008). The interlaminar stresses were evaluated using a perturbing stress field which comprises separable stress functions. It is known that the presence of a hole results in a detrimental stress concentration in the vicinity of the hole with a strength degradation and premature failure of the structure. Reinforcements by elliptic doublers, as well as doublers adapted to reinforcement requirements in a layer‐wise manner were considered by Engels et al. (2000, 2002) in an  attempt to avoid overloading. A mathematical structural optimization technique was  employed to determine the optimal design of the reinforcement, whereas the

Applications of Fracture Mechanics

finite‐element method was used for the structural analysis. The boundary finite‐ element method was employed by Lindemann and Becker (2002) to examine the free‐ edge stresses around a circular hole in laminates. As in the case of boundary element method, only the boundary needs to be discretized, whereas the element formulation in essence is finite‐element based. Numerical results for the concentration of interlaminar stresses at holes in composite laminates revealed good agreement with comparative finite‐element calculations. The boundary finite element method was employed by Mittelstedt and Becker (2006, 2013) to examine three‐dimensional stress singularities, which occur at notches and cracks in isotropic half‐spaces as well as at free edges and free corners of layered plates. Special emphasis was given to studying the stress concentration phenomena as they occur at straight free edges and at free corners of arbitrary opening angles in composite laminates. The scaled boundary finite‐element method, which combines the advantages of the finite‐element and boundary‐element methods, was developed by Wolf and Song (1996) and Wolf (2003). In the case of finite‐element method no fundamental solution is required and thus expanding the scope of application. On the other hand, in the boundary‐element method the spatial dimension is reduced by one as only the boundary is discretized with surface finite elements that the boundary conditions at infinity are satisfied exactly and that no approximation other than that of the surface finite elements on the boundary is introduced. Several two‐dimensional examples were analyzed by Goswami and Becker (2012) for crack and notch situations. A number of three‐dimensional cases were considered for different crack configurations that yield a high order of singularity. Parametric studies were conducted for structures with bi‐ material interfaces. It has been shown that the scatter in damaged composite test data is significantly lower than in unnotched composite (Tomblin and Seneviratne, 2011). Scatter analysis of fatigue data may be described using such approaches as Weibull, joint Weibull, and Sendeckyj wear‐out models. Recent advances in the technologies dealing with the assessment of useful life for composite aircraft fatigue‐critical, flight‐critical components and structure was presented by Makeev and Nikishkov (2011). These include non‐destructive sub‐surface measurement shift from just detection of defects to three‐ dimensional measurement of defect location and size and fatigue structural analysis techniques with the ability to capture multiple damage modes and their interaction. Mittelstedt and Becker (2003a, 2003b, 2004c, 2004d) presented closed‐form analysis of displacements, strains, and stresses in the vicinity of free rectangular corners of ­symmetric cross‐ply laminates under uniform thermal load by means of a layer‐wise C‐continuous displacement approach. The laminate was discretized into an arbitrary number of mathematical layers through the thickness. The method is based on the concept of a single‐layer formulation for the displacement field. The in‐plane components of the displacement representations were determined by formulating equilibrium in an integrated form and fulfilling the given boundary conditions, which can also be done in an integral sense. Later, Mittelstedt and Becker (2007a) addressed the development of a layer‐wise variational approach to free‐edge effects in layered plates with arbitrary layups under thermo‐mechanical load. The physical layers of the laminate were discretized into an arbitrary number of sub‐layers through the plate thickness. In addition to terms corresponding to classical laminate plate theory, the layer‐wise displacement fields consist of unknown in‐plane functions in the layer interfaces and a linear interpolation

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through the layer thickness. The boundary conditions of traction‐free laminate edges were satisfied in an average sense using an integral formulation. A hybrid approach utilizing finite fracture mechanics was presented by Hebel and Becker (2008) to model crack initiation at stress concentrations. The model was demonstrated to cover relevant effects for simple configurations as well as for glass‐ceramic sealing joints in solid oxide fuel cells subject to thermo‐mechanical loading. 2.3.3  Damage of Sandwich Structures

The failure modes of various foam core composites at the skin, skin‐to‐core interface, and the core under three‐point loading were the subject of several studies. Foam core composites with different types of skin architectures were considered by Vaidya et al. (1994). In situ AE and post‐test microscopic analysis of quasi‐static and flexural fatigue testing of sandwich composites indicated core damage to be the predominant failure activity while fiber rupture served as a precursor to catastrophic failure (Shafiq and Quispitupa, 2006). The study revealed that multiple crack initiation sites were observed under fatigue loading conditions in the vicinity of the notch tip. Both modes‐I and mode‐II cracking were observed in the core and along the interface between the core and the face‐sheets. Damage assessment of foam core sandwich composite panels due to slamming water waves was studied by Charca and Shafiq (2010) as a function of slamming energy (161–779 J) and deadrise angle (0°–45°). They found that higher slamming energy and lower deadrise angle resulted in greater damage to the material. Catastrophic failure was observed to occur beyond a threshold strain of 0.0035 mm/mm. Core shear along the interface with the face‐sheets and local buckling of the face‐ sheet and resin fragmentation were observed to be the dominant modes of failure under slamming. Quasi‐static and flexural fatigue tests on polyurethane foam core sandwich composites monitored in situ with AE sensors were conducted by Charca et al. (2010). The test results revealed that core shear and tearing were the predominant failure mechanisms that led to eventual catastrophic failure. The foam core sandwich composite lifetime underwent substantial reduction as a function of increasing stress level according to the S–N curve. Flexural fatigue endurance was beyond a 500,000 cycle arbitrarily established cut‐off below 60% of the ultimate static load. Tensile failure experiments on polyurethane foams by imposing a constant strain rate were reported in Deschanel et al. (2006). For heterogeneous materials, the failure was found not to occur suddenly but can develop as a multi‐step process through a succession of microcracks that end at pores. The acoustic energy and the waiting times between acoustic events were found to follow power‐law distributions. This was found to remain true while the foam density is varied. Tensile tests at different temperatures (from room temperature down to −65 °C) were performed on polyurethane foams of relative density 0.58. It was found that the foam becomes increasingly brittle as the temperature was decreasing. The Young’s modulus was found to increase as the temperature decreases, likewise the maximum stress, which almost doubled between room temperature and −65 °C. It was reported that the plastic plateau disappears with decreasing temperature and that the failure strain is less important as the material becomes more brittle. The behavior in AE at room temperature and at −10 °C were almost identical, where acoustic activity begins late when the material is already in the plastic plateau

Applications of Fracture Mechanics

stage and a divergence of the number of AE events occurs at the end. The microcracks nucleate, concentrate, and coalesce at the end of the test, producing the final failure. By contrast, for the tensile tests at −30 °C and −56 °C, the AE activity was found to start at the very beginning, indicating the early occurrence of damage. The number of events was found to rise gradually as the load increases. There has been a longstanding controversy as to what is the weakest element of sandwich plates, the skin‐core bonding or the foam core. The problem is that the cracks, developed in the plate, go through both the core and the skin–core bonding. The cracks propagate so fast that it is hard to detect the place where the crack was nucleated, in the core or in the skin‐bonding area. The damage mechanisms in sandwich plates with polymer foam cores were investigated by Ayorinde et al. (2012) using four‐point bending tests of a sandwich plate monitored by a high speed camera. It was found that in all cases cracks are nucleated inside the polymer foam core. First, several separated small cracks develop in the core, then the small cracks are connected and form the main crack that propagates through the core. The main crack reaches the skin–core bonding area and moves through this area causing the skin detachment. The influence of sub‐zero temperature of the ultimate load of each material was examined at discrete values of sub‐zero temperature. The results revealed that both materials become increasingly brittle with decreasing temperature, in agreement with experimental observations on PU foams under tensile loading reported in the literature (Deschanel et al., 2006; Soni et  al., 2009). For example, Figure  2.6 shows the dependence of the ultimate load on temperature over the range (−60 °C to +20 °C). It is seen that there is a trend of the effect of temperature on the ultimate load, which reveals increased hardness, and the tendency to be more brittle as the temperature decreases for both types of material tested, namely carbon‐fiber‐reinforced polymers (CFRP) and glass‐fiber‐reinforced epoxy (GFRP). It is noteworthy to refer to the work of Kalarikkal (2004), who tested graphite/epoxy 300 250

Ultimate load (lb)

200 150 100 50 CFRC GFRC

–80

–60

0 –40

–20

0

20

40

Temperature (°C)

Figure 2.6  Dependence of ultimate load on temperature for carbon and glass sandwich beams. (Ayorinde et al. 2012)

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laminates of different architectures. It was pointed out that the fabric (woven) architecture specimens resisted the deterioration in fracture toughness better than the unidirectional ones. Since the skins are respectively of woven glass fiber/epoxy and unidirectional graphite/epoxy, it is possible that apart from other causes, this difference in architecture is contributing to the difference in sub‐zero temperature behavior. 2.3.4  Sandwich Plates under Normal Loads

Sandwich plates, with so‐called construction foam cores, have become a common ­element of engineering structures. Possible effects of hidden damage formation in hard‐ skin and foam‐core sandwich plates under dynamic transverse loading were studied analytically and experimentally by Pilipchuk et al. (2013b). The load can be applied to a smooth surface of the plate from some external source or can be the result of interaction between structural subcomponents, as shown in Figure 2.7. The mechanics and geometry of such damages essentially differ from those observed under bending loads. Damages illustrated by Figure 2.7 were developed during the unloading phase, when the hard but flexible skin tends to take its undeformed/flat position and thus creates a ­tensile load on the foam cells crumpled during the compression phase. Since the skin remains undamaged, core damages are localized and hidden under the hard skin of the sandwich plate. For damage identification purposes, it is important therefore to develop reliable models describing very small geometrical changes of skin surfaces due to ­damages in core materials. Note that skin–core debond failure of sandwich plates was considered in many studies from different viewpoints (e.g. Vizzini and Lagace, 1987; Carlsson et al., 1991; Kim and Dharan, 1992; Davies and Cantwell, 1996; Li and Carlsson, 1999; Viana and Carlsson, 2003; Vadakke and Carlsson, 2004). It is usually assumed that skin–core debond is the result of skin sheet buckling under in‐plane compression. Specifics of debond failure may differ in sandwich structures with low‐ and high‐density foams (Li and Carlsson, 1999 and Viana and Carlsson, 2003). In specimens with a low‐density foam, failure tends to occur cohesively in the foam. Examination of the failure surfaces showed that the amount of core material adhered to the face‐sheet decreased with increasing foam

Figure 2.7  Sandwich plate damaged due to transverse localized loading created by the T‐joint in a slow compression cycle. (Pilipchuk et al., 2013a)

Applications of Fracture Mechanics

density, indicating increasing tendency for core–resin interfacial failure, as reported by Vadakke and Carlsson (2004). Closed‐cell polymer foams have the solid material in edges and faces, so each cell is sealed off from its neighbors. That is why closed‐cell foams have higher compressive strength, due to their structures. Arezoo et al. (2011) obtained experimental data pertaining to the quasi‐static mechanical response of polymethacrylimide (PMI) foams of density 50–200 kg/m3. It was shown that foams of low density collapse by cell wall buckling, while foams of high density undergo plastic cell wall bending. As a result, both the elastic and plastic macroscopic responses of foam display a tension/compression asymmetry. The behavior of sandwich panels consisting of brittle composite skins supported by a ductile core under low velocity impact was examined by Besant et al. (2001). The modeling considered nonlinearities caused by large deflections, plastic deformation of the core, and in‐plane degradation of the composite skins. The analyses revealed that honeycomb is a good absorber of impact energy. The honeycomb was shown to absorb energy by a combination of local core crush under the impactor and through‐thickness shear yielding. Minakuchi et al. (2008b) formulated a segment‐wise model for theoretical simulation of static indentation loading and unloading responses in honeycomb sandwich beams with composite face‐sheets. The honeycomb sandwich beam was divided into many segments, based on the periodic shape of the honeycomb, and the complicated through‐thickness behavior of the core was integrated into each segment. Further experimental verification was presented by Minakuchi et  al. (2008a), using specimens with different types of honeycomb cores. The damage growth mechanism under indentation load was clarified from the viewpoint of the reaction force from the core to the face‐sheet. The case of foam‐core sandwich beams subjected to static indentation and subsequent unloading was considered by Zenkert et  al. (2004). The analytical model was based on elastic‐perfectly‐plastic compressive behavior of the foam core. Upon removal of the load, an elastic unloading response of the foam core was assumed. In the finite element analysis of static indentation and unloading of sandwich beams the foam core was modeled using the crushable foam material model. The response of the foam core was experimentally characterized in uniaxial compression, up to densification, with subsequent unloading and tension until tensile fracture. Both models were able to predict load–displacement response of sandwich beams under static indentation and a residual dent magnitude in the face‐sheet after unloading, along with residual strain levels in the foam core at the unloaded equilibrium state. The sandwich upper skin under normal loading will be described in the next section within the theory of thin linearly elastic plates, whereas the core is represented by the Winkler foundation, whose characteristic was obtained by fitting the test data. Therefore, shear deformations of the core material are assumed to be insignificant as compared with the vertical tension. In addition, the lower skin of the sandwich is assumed to lie on a stiff smooth base in compliance with the actual test setup, in which the damage shown in Figure 2.7 was observed. Within such a model, occurrence and further evolution of damages are fully determined by the nonlinear properties of stress– strain response of the foam material during the load‐unload cycle under compression (e.g. Ashby et al., 2000; Shen et al., 2001; Zhao and Tao, 2009). Note that, despite some common qualitative features, a universal stress–strain characterization of foam structures seems to be impossible. The next section is based on both physical properties of

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raw materials and technologies of manufacturing of foams. In addition, standard ­compression tests show that results depend also on the geometry of specimens. 2.3.4.1  Damage Formation Model

In general, the model developed here relates to typical approaches dealing with indentations (debond) damages in sandwich plates (e.g. Abrate, 1997; Zenkert et  al., 2004; Shuaeib and Soden, 1997). Usually, due to significant complexity of three‐dimensional foam models, the core of sandwich is represented by some foundation with piecewise linear stress versus strain characteristics followed by an assumed scenario of damage formation (Pilipchuk et al., 2013b). The skin is usually represented by a thin elastic plate or beam. Figure  2.8 shows a schematic diagram of a sandwich plate whose core is molded between two thin elastic plates of thickness hs each. Under normal compression loading to the upper skin surface the core foam acts as a Winkler foundation. The loading is assumed to be slow enough so that the inertia and viscosity terms can be ignored. Under such conditions, the partial differential equation of the upper skin takes the form ,T



Ds

4

w

q t , x, y

(2.2)

in the square domain shown in Figure 2.8, G



0 x

Lx , 0

y

Ly

(2.3)

where w w(t , x , y ) is the vertical upward directed coordinate of the middle surface of the plate, q q t , x , y is (downward) pressure on the upper skin, 4 is the bi‐harmonic operator of differentiation with respect to the Cartesian horizontal coordinates, x and y, Ds Ehs3 /[12(1 s2 )] is the cylindrical bending stiffness of the skin plate in terms of the standard notations of the theory of thin elastic plates, E is the skin Young’s modulus, νs is Poisson’s ratio, and t , x , y is the vertical compressive strain of the foam, which is calculated as w (2.4) hc



Ly z

x

Lx

y

hs

0

hc

Figure 2.8  Schematic diagram of a sandwich plate resting on a stiff base showing the model geometry and coordinate frame.

Applications of Fracture Mechanics

where hc is the thickness of the core (foam); as the external pressure q is directed down0. Also, in equation (2.2), ward and therefore w 0, the compressive strain is positive σ(ε, T) is the normal stress, which depends on the strain ε and temperature T. This dependence will be discussed in the next section. At this point, we can assume that under compressive loading the foam in the presence of the skin may respond in a different way from its behavior in the absence the skin, even though the load is distributed quite homogeneously around the entire loading area of a specimen. In addition, this effect may also depend on the specimen size and other factors. This contributes ­inevitable uncertainties and complexity of quantitative studies. Boundary conditions along the boundary G of the domain G given by equation (2.3) are specified below. The plate surface is perfectly horizontal, i.e. w 0, at the initial time, t 0. Note that both the strain and the plate coordinate must obey the same boundary and initial conditions due to relationship (2.4). Taking into account equation (2.4) and eliminating the coordinate w from equation (2.2), gives the following differential equation governing the strain ε: 4



1 hc Ds

,T

1 q t , x, y hc Ds

(2.5)

Now consider the specific case when the plate is infinite in the y direction, Ly , while the other two edges, x 0, x Lx, are free. The load is perfectly localized with respect to y coordinate, as shown in Figure  2.9, but remains constant in x. Under ( y ). such  assumptions, the plate model is actually reduced to the beam case, Nevertheless, keeping in mind the two‐dimensional case given by equation (2.2) is still convenient for further discussion of experimental results and possible generalizations. Furthermore, the foundation (core) pressure σ(ε, T) is represented by Hooke’s law, E0 , where E0 is the effective modulus of elasticity within the elastic segment of foam response. This assumption is justified by the present experimental results, ­showing that the debond damage at low temperature is due to brittle effect, which is triggered at the very end of elastic segment. Above the brittle temperature range, the foam compression is described by more complicated relationships associated with ­specific changes in the solid foam structure. In such cases, however, it is found that the foam under the present study is usually smashed in a “plastic way” with no debond or Figure 2.9  Beam model geometry and coordinate system.

w Q0 –a

Damage area

a

y

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separation effects inside the sandwich structure. Therefore, it is assumed that Hooke’s law is applicable within the range 0 max , where εmax is the strain at yielding point, whereas the stress jumps to zero slightly above the yielding strain; in other words, E0 0



if 0 max if max

max

cell

(2.6)

where εcell is a very small number, whose order of magnitude is estimated by that of the relative foam cell size in terms of the sandwich thickness. Note that the discontinuity of stress–strain response imposed by the relationship (2.6) is valid within a small enough neighborhood of the yielding point. This approximation would be quite questionable in the case of continuous material or high temperature loading, when significant plastic deformations of cell walls may take place. However, foam cells have finite size. As a result, brittle collapse of cells resembles snap‐through effects with a sudden pressure drop. This interpretation seems to comply with numerous experimental observations as will be demonstrated shortly. As the load increases, the upper skin will eventually contact undamaged layers of the foam, so the stress will come back to the yielding level. As a result, the horizontal plateau area, which is formed under further advance of the strain, may appear to have a non‐smooth shape. On average, however, the plateau can be approximated by a straight horizontal line such that relationship (2.6) is replaced by a typical elastic‐plastic characteristic before the densification segment is reached. This type of the foam characterization was adopted by others (e.g. Zenkert et  al., 2004) for modeling the advanced damages under normal environmental conditions. A discontinuous tooth‐wise characterization similar to (2.6) was assumed by Minakuchi et al. (2008b), where the downward pressure jump is limited by a horizontal straight line representing some constant pressure σcrush corresponding to a perfectly plastic behavior. Note that replacing zero by a constant number in equation (2.6) would lead to no significant formal complications from the standpoint of derivations of the present analysis. It must be noted that the stress–strain law with the related core crush criteria constitutes the major part of the diversity of debond modeling. It may happen that sandwich plates represent very individual mechanical structures whose properties may be affected by such factors as technological details of manufacture, the corresponding chemical ingredients of the materials, and operating/testing conditions. As a result, generalization or comparison of quantitative and even some qualitative results obtained by different authors may be not feasible. So the present analytical model is developed specifically to describe test results obtained in the present study for certain types of sandwich plates under sub‐zero temperatures and quasi‐static normal compressive loading. It will be seen that equation (2.6) leads to an adequate match with the test data in terms of the structural response on localized loads before and after the low temperature brittle cell crush. Within the segment 0 max , the elastic state of sandwich is well described by the differential equation



d4 dy 4

where k 4

k4

q0

y (2.7)

E0 / hc Ds , q0

Q0 / Lx hc Ds , and δ indicates the Dirac delta function.

Applications of Fracture Mechanics

Since the load is localized at zero, the following conditions at infinity are assumed:



y

:

0,

d dy

0 (2.8)

The solution of the boundary value problem (2.7) and (2.8) is obtained in the form Q0 exp 2 Lx hc Ds k 3

y ,Q0



k y

k y

sin

2

4

2



(2.9)

Solution (2.9) and its first two derivatives with respect to y are continuous at y 0, but its third derivative has a step‐wise discontinuity whose jump is exactly q0. This provides singularity, which is necessary to compensate the delta‐function on the right‐ hand side of equation (2.7). Therefore, the presence of modulus y , while incorporating the fact of symmetry with respect to the coordinate y, provides necessary non‐smoothness as required by the localized loading. Note that solution of problem (2.7) and (2.8) in the form (2.9) can be obtained by introducing non‐smooth substitution of argument, y in equation (2.6), as described by Pilipchuk (2010). The corresponding skin deflection is obtained from relation (2.4); see Figure 2.10 for illustrations. It is seen that solution (2.9), represented by Figure 2.10, has a symmetric bell‐shaped graph with maximum at y 0. The damage size a can be found from the condition a,Q0 max . At fixed εmax, this transcendental equation has a solution, a a Q0 , if Q0 Qmax . An approximate explicit solution can be obtained under the assumption ka 2, which enables us to use truncated Maclaurin’s expansion for solution (2.9) by including only quadratic terms. This gives the following estimate for the damage area



y

a Q0

2 k2

4 2 Lx hc Ds k

max

Q0



(2.10)

–0.0000 –0.0001

w [m]

–0.0002 –0.0003 –0.0004 –0.0005 –0.0006 –0.04

–0.02

0.00

0.02

0.04

y [m]

Figure 2.10  Skin deflection profiles under the loads: Q₀ = 200 N (dashed line), and Q₀ = 1000 N (solid line). (Pilipchuk et al., 2013a)

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Note that equation (2.10) is obtained from the model of yet undamaged structure based on the critical strain criterion. Nevertheless, it provides a lower‐boundary ­estimate for the damage size as soon as the damage remains relatively small, causing negligible effects on the middle surface of the plate. In contrast to the external loading, which is perfectly localized, the foundation response (under the skin) is disturbed. Therefore, its load variation due to the damage depends on the size of damage, and can be neglected as soon as the damage is sufficiently small. This asymptotic consideration, which is in line with the power series expansion of solution (2.9), helps to significantly ease the damage size estimate. Taking into account the core relaxation effect caused by the damage requires two different pieces of solution to match at the damage’s yet unknown boundaries. This would lead to somewhat high damage size estimate, but at the cost of essential technical complications. During the loading phase, the load Q0 gradually increases, and this is associated with growing the damage size, as reflected by equation (2.10). After the load has reached its reverse point Q0 Q0,r , the unloading phase begins, during which the sandwich upper skin tends to restore its shape. At the first stage of this reverse process, the upper skin, although separated, is still in elastic contact with the smashed layers of the foam. This is why the dependence of load on displacement will initially follow the linearly elastic law, as will be seen below from the test diagrams. Then some area of the skin near the center y 0, usually including few layers of the foam with adhesive material, is disconnected from the rest of the foam. Starting from this point, the debond damage becomes increasingly visible. After the separation area has reached the size of damage, a a Q0,r ar , stress concentrations near the crack edges will likely destroy more foam cells, leading to the increase of the damage size. Such scenario of the unloading is obviously nonlinear. However, the above described details of unloading phase are ignored in the present analytical formulation, which assumes that the disconnected area momentarily takes the size a ar and then remains constant until the end of unloading. This results in the linear unloading law from the reverse point {εr, Q0,r} to zero. This simplification of the reverse path is justified by the fact that the major portion of damage has a low‐temperature brittle nature and it happens during the loading phase near the yielding point. However, a detailed description of the unloading phase may appear to be of ­interest under other conditions, such as high speed dynamic or impact loads at higher temperatures. Once the damage takes place, the model is described inside and outside the damaged area by the following two equations, respectively:

d4 dy 4

q0

y ,

y

a

d4 dy 4

k4

0,

y

a (2.12)

(2.11)

and

Arbitrary, constants of integration must be chosen to satisfy the continuity conditions for the strain ε and its slope dε/dy as follows: a 0

a 0

and

d dy

y

a 0

d dy

(2.13) y

a 0

Applications of Fracture Mechanics

Performing necessary algebraic manipulations, gives the following two solutions inside and outside the damage area, respectively: y ,Q0

Q0 a3 y 2 24 Lx Ds hc a 6 a2 k 2

if y

y a

3

y a

2

1

2 tan ka / 2 ak ( 2 ka tan ka / 2

6 4 2 ak a2 k 2 tan ka / 2

ak ak



2

(2.14)

2 ak tan ka / 2

a and a2Q0 exp k a y ,Q0



y / 2 sin

2 2 Lx hc Ds k ak cos ka / 2

k y 4

2

2 ak sin ka / 2

(2.15)

if y a|. Recall that, in equations (2.14) and (2.15), a a Q0 as given by equation (2.10) ­during the loading phase, but a ar during unloading. In order to visualize the damage formation, it is assumed that, during the loading, equations (2.14) and (2.15) relate to both the upper skin and the core surface, which is attached to the upper skin. During the unloading, the skin separates from the core within the damaged area y ar , therefore two functions are needed inside this area to describe the clearance between the skin and core:



skin

y ,Q0 ;

core

y ,Q0 r ;

y y

ar , a ar ar , a ar



(2.16)

where the function ε is defined by equation (2.14). According to equations (2.4), (2.14), and (2.16), the skin is moving upward as the load reduces from its maximum value Q0,r to zero, whereas the surface of core remains at the same level. For demonstration sake, consider the following numerical values of parameters were  selected: E0 62.0 106 N/m 2 , hc 0.0125m, max 0.027, Es 26.0 109 N/m 2 , hs 10 3 m, s 0.065, Lx 0.0254 m, where the effective yielding strain, εmax, at which frozen foam collapses, is found from the test. With reference to Figure 2.11, in terms of slopes, the load versus strain diagrams appear to be a very good match in both regions before and after the foam collapse critical point. However, near the critical (yielding) point associated with the slope braking, the experimental curve has a discontinuity with  the load drop, which cannot be captured by the present model. This happens because the model considers that the damage is growing smoothly in size starting from zero at y = 0. Practically, however, the crack develops very quickly to some observable size by affecting a long array of cells. Figures 2.12(a) and 2.12(b) illustrate the geometry of the sandwich beam at the final moment of the loading phase and that at the end of unloading cycle, respectively. The maximum load was Q0 ,r 1000 N. The vertical edges of the crack occurred due to the fact that the foundation (core) model ignores shear interactions between the vertical

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Handbook of Structural Life Assessment 1000 800

Model

600

Q (N)

86

Test

400 200 0 0.00

0.02

0.06

0.04 ε

0.08

Figure 2.11  Load versus strain response of the PMI beam under localized compressive loading; analytical solution and test result. (Pilipchuk et al., 2013a)

(a) –0.015

w –0.010

–0.005 –0.002 –0.004 –0.006 –0.008 –0.010 –0.012

(b) –0.015

y 0.005

0.010

0.015

0.005

0.010

0.015

w –0.010

–0.005 –0.002 –0.004 –0.006 –0.008 –0.010 –0.012

y

Figure 2.12  Damage formation under the load–unload cycle according to the analytical solution: (a) load, and (b) unload. (Pilipchuk et al., 2013b)

layers of the foam. In order to account for these effects, the corresponding generalizations for foundation models must include spatial derivatives of the strain, ε. 2.3.4.2  Experimental Results

Two different cases of the test setup were used, as shown in Figure 2.13(a) and (b). In Figure  2.13(a), a piece of sandwich plate of the same size is tested during the cyclic compression–tension loading under different temperatures. Both upper and lower skins of the specimen are fixed to the moving plate and base, respectively, by glue, so that the tensile load can be applied. Finally, the setup of Figure 2.13(b) is designed to test pieces of sandwich plates, whose one dimension is much larger than other two (beams), under “localized” loading, by using a cylindrical body of a relatively small diameter. The major difference between cases of Figure 2.13(a) and (b) is that, in the latter case, the

Applications of Fracture Mechanics

(a)

Compressive – tension cycle

(b)

Compressive load – unload cycle

Glue

1” × 1” piece of sandwich plate

1” × 5” piece of sandwich plate

Figure 2.13  Test setup diagrams: (a) sandwich plate specimen including skin prepared for compression ‐ tension loading, and (b) a narrow piece of sandwich plate (beam) prepared for localized load‐unload cycle.

upper skin of the specimen is under bending, so bending rigidity of skin contributes to the load versus effective strain response. The case of Figure  2.13(b) is similar to the setup as described by Zenkert et al. (2004). In Figure 2.13(b), there is a specific point on the upper surface indicating maximum in the load profile, whereas in Figure 2.13(a) the load is practically uniformly distributed over both upper and lower surfaces of the specimen. The sandwich plate specimens are made of polymethacrylimide (PMI‐71 IG) foam material. The skins are made of two glass fibers of type 3‐ply 7781 style E‐glass fiber/ epoxy fabric laminate with weave pattern of 8 HS (harness satin) of which the warp and fill yarns run at 0° and 90°, to the length, respectively. There is a wide range of density of this type of foam, but the one used in the present study has an average density of 75 kg/m3. All the specimens were cut directly from a PMI 71 foam sheet. The tests are conducted using an Endura‐Tec servo‐pneumatic testing machine. The temperature inside the chamber was controlled by Sun PC100 controller system whose typical operating temperature can vary from −200 °C to +365 °C. The load was applied in a quasi‐static way, such that the loading speed is much lower than those under which any inertia effects can develop in the specimen. Most of the results of the present study, as well as other results related to mechanical properties of foam materials, are represented in the form of stress–strain responses. It must be noted that interpretation of strain, while calculated in the same way, may be different for continuous (homogeneous) materials and foams. In the latter case, the standard expression h/hc , where Δh is the displacement of the loading plate, gives some effective (average) strain over the specimen thickness. For the core material, local strain may significantly vary from one layer to another due to the mechanical specifics of the foam compression, which represents a propagating front of collapsing cells. In particular, the local strain after the front is much higher than before the front. Within the elastic segment, however, different layers of the foam specimen are under same conditions, and therefore some strain magnitude, ε, is applicable to different layers of the specimen. Note that factors such as the presence of skin, temperature, and loading speed appear to play significant roles on the test results. In particular, the type of damage was found

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to be the result of brittle low temperature effects. It is important to note, however, that the brittle effects developed in specimens with and without skin are quite different. The stress–strain diagram at low temperature under compression of the foam without skin still develops three typical segments widely known in the literature (elastic, plateau, and densification) as the effective strain gradually increases from zero to about 0.7, as reported by Grace et al. (2012). However, it was noticed that lowering the temperature would result in some convexity of the plateau segment. A simple visual analysis of the plateau and specimen after the test shows possible accumulation of a large number of brittle microdamages inside the foam material under loading at low temperature. In  particular, the foam material, while still preserving a unit piece, acquires quite ­obvious damage. The presence of skin results in a different stress–strain response from the case of solid foam material, as will be demonstrated later in this chapter. In particular, when reaching the yielding point, the foam collapses by developing a large‐scale brittle crack, in such a way that no plateau segment occurs. The only possible reason for such a different behavior is that the skin with adhesive material, penetrating through the foam, imposes certain geometrical constraints of deformation of foam cells by preventing their lateral displacements at least within some layers near the skin. As a result, the entire structure actually represents a combination of substructures with different spatial scale mechanical properties. During compression, boundaries between such substructures represent the source of shear stress accumulation and large‐scale brittle cracks. At room temperature, T = 22 °C, however, the compression phase develops in a regular way with a plateau segment without brittle crack. The tension phase reveals that no significant damage occurs during compression, and the foam resists tension through the cell alignment segment until it breaks apart. A more detailed experimental study was conducted for localized load‐unload cycles, in accordance with Figure 2.13(b). In order to obtain a visual representation of structural changes in the sandwich material during low temperature load‐unload cycle, a Phantom v12.1 camera was focused on the loading area through the window of an environmental chamber. Typical snapshots during the loading and unloading phases are shown in Figure 2.14, where snapshots (a) and (b) correspond to the beginning and the end of loading phase, respectively. Then, snapshots (c) and (d) were taken shortly after the beginning and before the end of unloading phase, respectively. Usually a brittle crack damage occurs at high enough compressive load during the loading phase. Such a crack is barely seen about 1.0–2.0 mm below the cylinder’s surface in Figure 2.14(b), which was taken during the loading phase. The initiation of such cracking was accompanied by significant audio effects, and cannot be missed. However, the damage becomes obvious only during the unloading process, when the upper skin tends to restore its straight horizontal shape by releasing its potential energy of elastic deformations accumulated during the compression phase. During the unloading phase, the crack may continue to gradually grow, and as a result the final size of the crack may exceed that which initially occurred during the compression phase. The dependence of load on the effective strain and the corresponding residual shapes of sandwich beams under different temperature and loading conditions are shown in Figs 2.15 and 2.16, for two different values of loading speed. Generally, the diagrams confirm what has already been mentioned, namely all the three factors – environmental temperature, loading speed, and maximal effective strain – play an important role in the

Applications of Fracture Mechanics

(a)

(b)

(c)

(d)

Figure 2.14  Snapshots of the crack formation during ‘localized’ load–unload test of the sandwich beam under low temperature conditions T = −60 °C, loading speed v = 0.005 in/s, and maximal effective strain max 0.3 , (a‐b) loading phase, and (c‐d) unloading phase. (Pilipchuk et al., 2013b)

damage formation. The low temperature represents a decisive factor triggering the damage formation under low‐speed compressive loading. With reference to Figure 2.15, the damage usually occurs at a temperature between −20 °C and −30 °C, which is well below the PMI glass transition point, at about +125 °C. However, comparison with the diagrams of Figure  2.16 reveals that increasing the loading speed shifts the upper boundary of critical temperatures upward. Also, the intensity of compression, that is maximal strain achieved during compression, has some effect, but mostly on the damage size. Note that the load–strain diagrams in Figs 2.15 and 2.16 have specific distinctive features as compared to typical stress–strain diagrams obtained for foam material alone under homogeneously distributed pressure. Namely, the “plateau” is not horizontal any more, but has a clear positive slope. This seems to be a combined result of two major factors. In addition to the contribution of bending rigidity of the skin, geometrical specifics of localized loading cause some widening of the front of collapsing cells, as the strain grows, by making it harder for the loading to proceed. Brittle cracking effects are accompanied by sudden load dropping, as most clearly seen in the case of lower speed and lower temperature in Figs  2.15(a) and (b). The increase in loading speed results in some smoothing effect on the response, as follows from the comparison of Figure 2.15 and 2.16.

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Handbook of Structural Life Assessment

(a)

(b)

Q (N)

800

T = –60°C εmax = 0.3

1500 εT = –60°C = 0.6 max

V = 0.005 in/s

V = 0.005 in/s

Q (N)

1000 600 400

1000 500

200 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 ε

(c)

0 0.0

0.1

0.2

0.3 ε

0.4

0.5

0.3 ε

0.4

0.5

(d)

T = –60°C

1200 εmax = 0.3 1000 V = 0.005 in/s 800 600 400 200 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 ε

1500 Q (N)

Q (N)

90

T = –60°C εmax = 0.6 V = 0.005 in/s

1000 500 0 0.0

0.1

0.2

0.6

Figure 2.15  Results of low speed (v 0.005in/ sec ) ‘localized’ load–unload tests of sandwich beams under different sub‐zero temperatures and maximal effective strains: (a) T 60 C, max 0.3, (b) T 60 C, max 0.6, (c) T 30 C, max 0.3, (d) T 20 C, max 0.6. (Pilipchuk et al., 2013b)

2.3.5  Thermo‐Mechanical Coupling of Sandwich Plates 2.3.5.1  Historical Background

The design of structural components exposed to variable climatic conditions requires special attention for the dynamic behavior of structures under combined effects of external loading, temperature variations, inertia forces, and elastic forces. The mathematical modeling describing the interaction between inertia forces, external forces, temperature variations, and internal stresses is not a simple task. The principal effects of temperature variations, especially from extreme cold regions to warm regions, include a reduction in stiffness due to softening of the material, and development of internal stresses within the sandwich material due to mismatch in thermal expansion coefficients of the core and skin materials. These effects in turn affect the dynamic response of the structure. There has been a tremendous advancement in the science and technology of new materials that are characterized by low density, high strength, and high stiffness‐to‐ weight ratios. For composite materials, the problem of thermo‐elasticity is complicated because bending, twisting, and extensional deformations can be fully coupled in composite structures. Thus, there are significant differences in mechanical behavior between the new materials and conventional metals, and these differences must be

Applications of Fracture Mechanics

(a)

(b) 1500

1200 800 600

T = –30°C εmax = 0.3

400

V = 0.5 in/s

Q (N)

Q (N)

1000

1000

T = –30°C εmax = 0.6

500

V = 0.5 in/s

0

200 0 0.00

0.05

0.10

0.15 ε

0.20

0.25

0.1

0.2

0.3 ε

0.4

0.5

(d)

1500

1000 T = –20°C εmax = 0.3

500

Q (N)

Q (N)

(c)

0.0

V = 0.5 in/s

0

1000

T = –20°C εmax = 0.6

500

V = 0.5 in/s

0 0.00

0.05

0.10

0.15 ε

0.20

0.25

0.0

0.1

0.2

0.3 ε

0.4

0.5

Figure 2.16  Results of higher speed (v 0.5in/ sec) ‘localized’ load–unload tests of sandwich beams under different moderate sub‐zero temperatures and maximal effective strains: (a) T 30 C, 0.3, (b) T 30 C, max 0.6 , (c) T 20 C, max 0.3, (d) T 20 C, max 0.6. (Pilipchuk max et al., 2013b)

considered in developing the analytical modeling. The effects of thermal stresses on mechanical behavior of composite materials have been studied by many authors. The basic theory is well documented by Boley (1972) and Rokne et  al. (1980a, 1980b). Thermal stresses can arise due to sudden changes in the ambient thermal conditions (temperature, heat flux, etc.). Nguyen et  al. (1987), Gundappa et  al. (1987), and Hasselman et al. (1992) analyzed thermal cycles in plates and cylinders subjected to a single and multi‐thermal cycles. Gundappa et al. (1987) analyzed the effect of a finite boundary conductance on the magnitude of thermal stresses in a flat plate subjected to symmetric and asymmetric conductive heat transfer from an infinite, mechanically non‐interacting medium. Nguyen et al. (1987) found that the magnitude of the stresses encountered during the cooling part of a cycle decreases with decreasing the duration of the heating part of the cycle. It was also shown that at any specific value of the Biot number,5 Bi, the magnitude of maximum thermal stress within any cycle decreases with 5  The Biot number is the ratio of the heat transfer resistances inside of and at the surface of a body. This ratio determines whether or not the temperatures inside a body will vary significantly in space, while the body heats or cools over time, due to a thermal gradient applied to its surface. Values of the Biot number smaller than 0.1 imply that the heat conduction inside the body is much faster than the heat conduction away from its surface, and temperature gradients are negligible inside the body.

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increasing frequency. At any given frequency the magnitude of the peak stress exhibits a maximum value during the first cycle and decreases to a constant value from cycle to cycle with increasing number of cycles. The flexural response of thermally induced (self‐excited) bending vibrations of uniformly heated beams in still air was studied by Murozono (1996). The analyses considered the nonlinear dependence on the velocity of an unsteady component of the coefficient of heat transfer at the beam surface. The stability boundary derived from the theoretical analysis was found to be dependent on the phase difference between velocity fluctuations and thermal bending moments, the damping ratio, and a non‐dimensional parameter related to the heating rate. Experimental measurements showed that this type of thermally induced vibration occurs only in air and does not occur in a vacuum. Deflection responses of the beam were measured for various values of system parameters, and experimental stability boundaries were compared with those predicted analytically. For long and narrow beams, only first‐mode bending vibration occurred, but under certain circumstances higher mode bending vibrations were also excited simultaneously on a relatively short beam. The problem of thermal buckling of orthotropic and isotropic plates has been studied extensively (e.g. Biswas,1976; Chen and Chen, 1987a, 1987b, 1989a, 1989b, 1990, and 1991). For example, Biswas (1976) extended the Berger approximate equations to heated plates in order to obtain the large deflection of a heated semicircular plate with simply supported edges under stationary temperature distribution. Kamiya (1978) adapted the Berger method to the problem of large thermal bending of sandwich plates. The governing field equations were derived by the variational calculus for a thermal stress field. The equations were presented in the quasi‐linear form with respect to the deflection and the differences of in‐plane displacements on each face. Yarovaya (2006) used the hypothesis of ‘broken normal’ to derive equations for thermo‐elastic bending of a circular sandwich plate on a deformable Winkler’s foundation. The system of equilibrium equations was solved exactly in terms of special functions. In a series of studies, Chen and Chen (1987a, 1987b, 1989a, 1989b, 1990, and 1991) examined the thermal buckling of laminated composite plates subjected to a temperature change. The displacement equations of equilibrium were used, and Galerkin’s method was employed to determine the critical buckling temperature. They also considered the thermal post‐buckling behavior of composite laminated plates subjected to a non‐uniform temperature field using the finite element method. The results revealed that the thermal post‐buckling behavior of composite laminated plates is influenced by lamination angle, plate aspect ratio, modulus ratio, and the number of layers. It was shown that the influence of ­temperature‐dependent mechanical properties on the thermal buckling behavior is ­significant. Frostig and Thomsen (2007, 2008a, 2008b) studied the buckling response and nonlinear behavior of sandwich panels subjected to either thermally induced deformations or to simultaneous thermal and mechanical loads. Their analysis was based on soft core possessing thermal dependency of mechanical properties. The interaction between elevated temperatures and mechanical loads was found to change the response from a linear to an unstable nonlinear one when the degradation of the mechanical properties due to temperature changes was considered. A finite element approach was used for studying the behavior of sandwich plates with different anisotropic composite facings due to aerodynamic and thermal fields by Weinstein et al. (1983). The inherent coupling phenomenon of stretching and bending

Applications of Fracture Mechanics

in such non‐symmetric composite sandwich plates was considered. The results of three different examples revealed the sensitivity of the stress and displacement fields to the class of heterogeneity and anisotropy of the considered sandwich plates. Kharitonov et al. (1987) obtained an expression for the temperature field and investigated the fluctuations induced by a thermal impact. Birman and Simitses (2000) proposed an analytical approach for sandwich box type composite shells designed to withstand a combination of thermal loading, internal pressure, torsional, and axial loads. The facings of the shell were considered dissimilar to maximize their efficiency according to the loads acting on each facing. Their analysis includes the formulation of the equations of motion based on a first‐order shear deformable version of Sanders’ shell theory. The facings were treated as a thin geometrically nonlinear plate or shell on an elastic foundation using von Karman’s approach. An enhanced micromechanical constitutive formulation based on the incorporation of the effect of the thermo‐mechanical coupling on the material properties and temperature was presented. Later, Birman (2005) used a first‐order shear deformable theory to study the wrinkling of a simply supported sandwich panel based on linear‐dependency of material properties on temperature. The first‐order shear deformable theory was also employed by Birman et al. (2006) and Liu et al. (2006) to study the response of a sandwich panel and a sandwich beam with temperature‐dependent properties subjected to elevated temperatures. The disadvantage of the first‐order shear deformable theory is that it does not take into account the changes in the height of the core during deformation. Sandwich‐structures usually comprise a thermal interface connected to a hollow plate lamination. Each laminate is made of a hollow aluminum plate having a series of mm‐scale channels or compartments that are filled with phase change material. Simple semi‐empirical models describing the thermal response of plate‐like thermal energy storage‐structures were developed by Wirtz et al. (2004). They found that the aluminum structure has an excellent performance‐to‐weight in terms of mechanical properties of the structure. A generalized thermal response model was developed to couple the thermal response of the thermal interface to that of the aluminum/phase change material lamination. Rakow and Waas (2005, 2007) presented experimental and analytical results for the thermo‐mechanical response of actively cooled metal foam sandwich panels. Inconel foam sandwich panels were subjected to through‐the‐thickness thermal ­gradients, defined by fixed temperature conditions on opposite face‐sheets. The experimental measurements were used to develop a sequentially coupled thermo‐mechanical finite element model. The numerical model provided a strain‐temperature history for metal foam sandwich panels under through‐the‐thickness thermal gradients. The nonlinear equations governing the large thermal axis‐symmetric deformations of circular sandwich plate in terms of the middle plane’s displacements based on sandwich plate theory were derived by Yang et al. (2007). It was shown that the boundary conditions and the stiffness greatly effect critical buckling loads. The thermo‐elastic bending analysis of functionally graded ceramic‐metal sandwich plates was developed by Zenkour and Alghamdi (2008) under the effect of thermal loads. The sandwich plate faces were assumed to have isotropic, two‐constituent material distribution through the thickness. The core layer was assumed to be homogeneous and made of an isotropic ceramic material. Displacement functions satisfying the boundary conditions were used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. The axial stress was found to

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take the maximum compressive (minimum tensile) at the top (bottom) surface of the core layer. However, the transverse shear stress was found to take the maximum value at a point on the plate mid‐plane. Wu et al. (2009) developed a global‐local higher‐order analytical approach by considering the transverse normal strain under thermo‐mechanical loading. Numerical results provided the distributions of displacements and stresses for angle‐ply laminated plates under temperature loads. The dependence of the eigenvalues of a layered plate on the thermo‐elastic coupling was studied by Minagawa (1987). The layered plate was exposed to a uniform sudden heating or cooling. The basic equations were transformed into a system of matrix equations, and the problem was formulated as a nonlinear eigenvalue problem of a polynomial matrix. Matsunaga (2007) presented a two‐dimensional global higher‐order deformation theory for the free vibration and stability problems of angle‐ply laminated composite and sandwich plates, subjected to thermal loading. The method of power series expansion of continuous displacement components was used to obtain a set of fundamental governing equations involving the effects of both transverse shear and normal stresses. Critical temperatures were obtained by increasing the temperature until the natural frequency vanishes. The free vibration and damping characteristics of plates consisting of composite stiff‐layers and an isotropic viscoelastic core under thermal loads were studied by Vangipuram and Ganesan (2007, 2008). The results revealed some interesting phenomena such as shifting of modes with temperature. Frostig and Thomsen (2009a, 2009b) developed an analytical formulation of the free vibration analysis of sandwich panels with a flexible core that possesses temperature‐dependent mechanical properties. Their analysis is based on the high‐order sandwich panel theory approach. The resulting reduction of the modulus of elasticity and the shear modulus with the elevated temperature resulted in a corresponding reduction of the eigenvalues as the thermal gradient increases. This decrease was found to be significant for the lowest vibration modes. The response of panels made of temperature‐dependent mechanical properties was studied by Gu and Asaro (2005). They used a power fitted function to model the degradation of the mechanical properties due to rising temperature of sandwich panels made of functionally graded material. The small‐ and large‐amplitude vibrations of compressively and thermally post‐buckled sandwich plates with functionally graded material face‐sheets in thermal environments were studied by Xia and Shen (2008) and Shen and Li (2008). The material properties of functionally graded material face‐sheets were assumed to be graded in the thickness direction according to a simple power‐law distribution. The numerical results showed that, as the volume fraction index increases, the fundamental frequency increases in the pre‐buckling region, but decreases in the post‐ buckling region. In contrast, the nonlinear frequency ratio was found to decrease in both the pre‐ and post‐buckling regions with increasing volume fraction index. The results also revealed that the substrate‐to‐face‐sheet thickness ratio and temperature changes have a significant effect on the fundamental frequency, but only have a small effect on the nonlinear frequency ratio. One should mention the work of Pagano (1969) who developed exact solutions within the framework of the linear theory of elasticity. The solutions applied to unidirectional and bidirectional layered anisotropic plates under cylindrical bending. In another work, Pagano (1970) constructed three‐dimensional elasticity solutions for rectangular laminates with pinned edges. The lamination geometry comprised an arbitrary number of layers.

Applications of Fracture Mechanics

An exact solution of linear elasticity theory for bending of sandwich plate‐like beams due to temperature difference at the plate faces was developed by Pilipchuk et al. (2010). It is based on the assumption that the heat flux is stationary, material is linearly elastic, and displacements are small. The exact solution yields the temperature profile, stress and displacement distribution across the plate thickness. The analytical results are complemented by an example of a simply supported sandwich beam. The next section outlines a case study of the linear analysis of sandwich plate‐like beams due to temperature difference at the plate faces based on the work of Pilipchuk et al. (2010). 2.3.5.2  Analytical Modeling of Thermomechanical Coupling

Consider a two‐dimensional elastic layer occupying a strip in the undeformed state, L/2 x L/2, and h/2 y h/2, where x and y are the Cartesian coordinates in the strip, L is the length the strip, and h is its thickness. The strip contains three isotropic elastic layers, as shown in Figure 2.17. The outer elastic layers are identical and are symmetric with respect to the mid‐line y 0. The thicknesses of the center layer and the outer layers are denoted by hc and hs, respectively. Indices c and s refer to the inner layer (core) and in the outer layers (skin), respectively. The total thickness of the sandwich plate is h hc 2hs. The material properties of the plate are characterized in terms of Lamé constants6 λ (Lamé’s first parameter) and μ (second parameter or shear modulus), thermal expansion coefficient, α, thermal conductivity coefficient, κ, and the normal coordinate, y. The role of the first parameter in mechanics of sandwich plates was ­considered by Berdichevsky (2010a, 2010b). These parameters are classified as follows: y



y

c s c s

y

hc /2

hc /2 y

y

hc /2

hs

hc / 2

hc /2

y

hc /2

hs

, ,

y y

c s c s

y hc /2 y hc /2

hc /2 y

hc /2

hs

hc /2

hs

hc /2 y

(2.17)

The projections of the displacement vector on x and y axes are denoted by u and υ, respectively. The third component of the displacements is assumed to be zero for plate‐ like beams. The plate deformation is caused by the temperatures imposed at the faces of the plate,

y h/2

,

y

h/2



(2.18)

where θ is the difference of the current temperature, T, and the temperature of the stress‐free state, T0, i.e. T T0 . The solution for the thermal displacements, u and υ, can be presented as a superposition of two solutions, describing independently bending and extension components of 6  In linear elasticity, the Lamé parameters (λ,  μ) are two material-dependent quantities that arise in stress–strain relationships, which satisfy Hooke’s law in three-dimensions of homogeneous and isotropic materials, i.e. Tr I 2 , where σ is the stress, Tr(ε) is the trace of strain matrix, I is the identity matrix, and ε is the strain tensor. λ is referred to as the first Lamé parameter, and has no physical meaning, but it serves to simplify the stiffness matrix in Hooke’s law, while μ is known the second Lamé parameter and is equivalent to the shear modulus, G.

95

96

Handbook of Structural Life Assessment y θ=[ θ] / 2 hs Skin

v(x,y) u(x,y)

y x

0 Core

x

hc

μc, λc, αc, Kc hs

Skin

θ = –[ θ ] / 2

μs, λs, αs, Ks

Figure 2.17  Schematic diagram of sandwich plate cross‐section showing the coordinate frame.

the plate. In the bending state, the longitudinal displacement is an odd function of the coordinate y, while the vertical displacement is an even function of y. In the extension state, the longitudinal displacement is an even function of y, while the vertical displacement is an odd function of y. With reference to the bending problem, the boundary conditions given by equations (2.18) are replaced by the conditions

/2,

y h/2

y

/2,

h/2

2



(2.19)

where [θ] is assumed to be constant. The equations governing the deformed state are 1) equilibrium equations xy

xy

yy

(2.20) 0 2) equation of stationary heat conduction for the heat flux in the y direction, q xx

x



q y

0,

y

x

y

0

(2.21)

3) constitutive equations for the components of stress components σxx, σxy and σyy, and heat flux, q, xx yy



xx

q

xx

y

yy

2

xx

yy

2

yy

,

xy

2

xy

(2.22)

(2.23)

where the coefficient β relates to the heat expansion coefficient α through the rela3 2 tionship . 4) geometrical relations for the strain components, εxx, εxy, and εyy,

xx

u , x

xy

1 2

u y

x

,

yy

y

, (2.24)

Applications of Fracture Mechanics

5) compatibility of the displacements, stress components, temperature, and the heat flux at the interface y hc /2, u

1 hc 0 2

y xy

y

1 hc 0 2

1 hc 0 2 xy

1 y hc 0 2

y



u

y

1 hc 0 2

y

y

,

1 hc 0 2

1 hc 0 2

,

, q

yy

y

,

1 y hc 0 2

yy

q

,

1 hc 0 2

y

1 hc 0 2

y

y

1 hc 0 2

1 hc 0 2

,

(2.25)

6) vanishing stress components on the faces of the plate yy



0,

xy

0, at y

h / 2 (2.26)

Under bending, u, σxy, εxy, and q are even functions of y, while υ, θ, σxx, σyy, εxx, and εyy are odd. Therefore, it is sufficient to satisfy only the compatibility conditions (2.25) on the interface y hc /2; then the compatibility conditions on the interface y hc /2 will be satisfied automatically. It is convenient to seek for a solution that is similar to the Saint Venant solution of the theory of elastic beams. In this case, for the time being, there is no need to specify conditions at the ends x L/2. Once the solution is obtained, it is possible to address the issue of satisfying the various boundary conditions by choosing the involved arbitrary constants. For stationary heat conduction, we can write q c constant, where the value of the constant c can be found from equation (2.23) and the boundary conditions (2.19). This is achieved by multiplying (2.23) by 1 and integrating with respect to y to give c

h/2

1

dy

,

h/2

or c

h/2

h/2

1 1

dy

and so y

h/2 h/2



1 1

dy

y

1

dy (2.27)

0

Since κ is a piecewise constant function, the temperature θ will be piecewise linear odd function of y. We need to seek a solution for which

yy

0 (2.28)

From the second equilibrium equation (2.20), σxy is a function of only y, which is denoted by the function B(y), i.e.

xy

B y (2.29)

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Handbook of Structural Life Assessment

From the first equilibrium equation (2.20), A y

xx



xB y , (2.30)

where A(y) is another function of y and a prime denotes differentiation with respect to y. Assuming that A(y) and B(y) known, we can express εxx, εyy and εxy in terms of A(y) and B(y): 2

2

2

2

yy

xx

2

2

yy

y

2 A y

u y

xy

v x

xx

yy

xx

yy

(2.31)

B y xB

A y

2





u x 2

2

xx

(2.32)

xB y

B y (2.33)

From equation (2.31) we can find u(x, y): u x ,y

u0 y

x



2

4

A y

1

2

4

x2 B y (2.34) 2

2

where u0(y) is an arbitrary function. From equations (2.33) and (2.34) we can write v / x in the form

x

1

u y

B y 2

x 2



4

1

2

B y

u0 y

x

2

4

A y

2

1 (2.35)

B y

Note that υ(x, y) is described by the two equations (2.32) and (2.35). The functions u0(y), A(y) and B(y) are determined from the condition that these two equations are consistent, i.e. x

A y

4 1

B y

u0 y

x

xB y 4

2

4 A y

2

1

x2 2 4

2

B y

Applications of Fracture Mechanics

Equating terms of equal powers of x gives the following three equations:







1

B y

4

4

4

2

A y

2

u0 , (2.36)

B y 1

2

0 (2.37)

0 (2.38)

B y

This is a system of three ordinary differential equations with piecewise constant coefficients for functions, A, B, and u0. Generally, this kind of differential equation must be interpreted in terms of generalized functions. However, within an arbitrary layer of the sandwich, the coefficients are constant and therefore equations (2.36)–(2.38) become quite easy to solve for the bending deflection and stress state, as described in the next section. 2.3.5.3  Exact Solution for Sandwich Plate

Consider an arbitrary layer of the sandwich with number i, such that i 1 and i 3 indicate the lower and upper skins, respectively, and i 2 indicates the core layer; in particular, i



i 1, 3 , i 2

s c

i 1, 3 i 2

s

i

c



As a result, equations (2.36)–(2.38) take the form



u0 ,i y

4

3



Bi y

i

2

Ai y 0

4

i

i

i i

2

i

Bi y

i i

y

i



(2.39)

The general solution of equation (2.39) has the polynomial form



3 i 4 24 i i 2 i i

u0 ,i y

c0 i

c1i y

Ai y

a0i

a1i y

Bi y

b0i

b1i y b2i y 2

i

i

i i

y

3b1i y 2

2b2i y 3 (2.40)

where c0i, c1i, a0i, a1i, b0i, b1i, and b2i, i 1, 2, 3 is a set of 21 arbitrary constants of integration.

99

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Handbook of Structural Life Assessment

Substituting (2.40) into (2.34) and (2.35), gives ui x ,y

c0 i

ui x ,y

c1i y

c0 i

c1i y

2 a0i x 2 x a1i

b2i x y 2 b2i y 3

b1i x 2 3 y 2

2 a0i x 2 x a1i

8 i b2i x y 2 b2i y 3

b1i x 2 3 y 2

8 i b2i x y 2b2i y 3 3b1i x 2 3 y 2

6a0i x 6 x a1i

24 i

p0i

vi x , y

2 i i x 2 12b0i b2i x 2 24

i

where

y

y

y

2 i c1i

i

8 i 3 y b1i

3a1i x

i

2a0i

i

2

i

i

a1i y y i

b2i y

i

1 24 i i i 4 6b0i b2i x 2

6 i c1i

(2.41)

i

s ds, and p0i are three additional arbitrary constants due to integra-

0

tion of differential equation (2.35) for three layers of the sandwich. Therefore, the final form of the general solution in terms of the displacements has 24 arbitrary constants – eight for each of the three layers. Substituting equations (2.40) into equations (2.28)–(2.30), gives the following stress components



yy

x ,y

0

xx

x ,y

a0i

a1i y x b1i

xy

x ,y

b0i

b1i y b2i y 2

2

b2i y

i

i

2

i i

y (2.42)

In the case of bending, the following conditions are used for determining the arbitrary constants: 1) Symmetry conditions: u2 x , y

2

x, y

u2 x ,y , u1 x , y 2

x ,y ,

1

x, y

u3 x,y x ,y

3

(2.43)

Since equalities (43) must hold for any x and y within their domain, then substituting solutions (2.41) in equations (2.43) and matching coefficients of the same powers of coordinates, gives 13 linear algebraic equations for the arbitrary constants. 2) Free boundary conditions:

xy ,1

x , h/2

0,

xy ,3

x , h/2

0

(2.44)

This is reduced to a single equation for arbitrary constants due to the symmetry imposed. 3) Continuity of shear stress:

xy ,1

x , hc /2

xy ,2

x , hc /2 ,

xy ,2

x , hc /2

xy ,3

x , hc /2

(2.45)

Applications of Fracture Mechanics

This is also reduced to a single equation for arbitrary constants due to the imposed symmetry. 4) Continuity of displacements: u2 x , hc /2 2



u3 x , hc /2 , u2 x , hc /2

x , hc /2

x , hc /2 ,

3

2

u1 x , hc /2

x , hc /2

1

x , hc /2

(2.46)

Matching the coefficients of the same powers of x in equations (2.46), gives 11 more equations, so that the total number of equations with respect to 24 arbitrary constants becomes 26. However, only 20 equations appear to be linearly independent. As a result, four arbitrary constants will remain undetermined. From the mechanical standpoint, this is due to the fact that no conditions were imposed yet at the ends x L/2. As an example, consider the case of a simply supported plate by introducing the ­bending moment and mean vertical displacement as, respectively, hc /2

M x



xx ,1

x ,y ydy

h/2

hc /2 xx ,2

h/2

x ,y ydy

x ,y ydy

xx ,3

hc /2

(2.47)

hc /2

and



mean

x

hc /2

1 h

v

hc /2

x ,y dy

1

2

h/2

h/2

x ,y dy

3

hc /2

x ,y dy

(2.48)

hc /2

Then the boundary conditions give four more independent equations for determining four unknown constants left in the solution

M

L/2

mean

M L/2

0,

mean

L/2

L/2

(2.49) 0.

(2.50)

The final solutions for displacements u and υ are obtained using Mathematica. The important characteristics of thermo‐elastic state are given below. First, introduce the following characteristics of the model geometry and thermo‐elastic properties hs , r1 hc



c s

, r2

c

s

s

s

c

c

In particular, the parameter



h , hs 1 2

hc

, r3

s

s

c

c

c

c

s

s

(2.51)

determines the core and skin thicknesses:

h (2.52) 1 2

Based on equation (2.27), the temperature profile is described by

y

S1 y

1

y

S2 y

2

y

S3 y

3

y

(2.53)

101

102

Handbook of Structural Life Assessment

where Si(y) is equal to unity within the i‐th layer, but takes zero value outside the i‐th layer:

S1 y S3 y

H y h / 2 H y hc / 2 , S2 y H y hc / 2 H y h / 2

H y hc / 2

H y hc / 2 ,

and



1

y

0

r1

y h

1 1 r1 2 1 2

3

y

0

r1

y h

1 1 r1 2 1 2

,

2

y

0

y , h

(2.54)



where H is the unit‐step Heaviside function, and 0 1 2 / 1 2r1 the integrals in (2.48), gives the mean vertical displacement in the form

mean

x

0

1 4



x2 L2

1 2 r3 3 3

1 2 1 2r1

1 r1

1 2 r2r3 3 6

4 r1 4

2

. Calculating

(2.55)

2

2 where 0 / 16h c is an amplitude of the mean displacement that the cL c plate would have in case 0. Finally, the only non‐zero stress component is described by



xx

x ,y

S1 y

xx ,1

x ,y

S2 y

xx ,2

x ,y

S3 y

x ,y

xx ,3

(2.56)

where

xx ,i

x ,y

2 s0

1 2 r3 3 3 (1 r1 4 r1 y r1 r2 h 1 r2 r3 3 6 4 2

1 2 1 2r1

1

i 1 /2

2



2

1 r1 1 2

(2.57)

for i 1, 3 and

xx ,2



x ,y

4 s0

1 2 1 2r1

3 1 r2

3

1 r1 2r2

1 2 r2 r3 3 6

4 r1 r2 4

2

2

y (2.58) h

where s0 [ ] s s /( s 2 s ). Therefore, the main characteristics of the thermo‐elastic state of sandwich plates are obtained exactly in closed analytical form as given by equations (2.53)–(2.58). Note that vanishing of all stress components except σxx is a peculiarity of the simple support boundary conditions, and for other boundary conditions all stress components can be non‐zero.

Applications of Fracture Mechanics

0.4

θ (deg)

0.2 0.0 ϵ = 0.50 –0.2

ϵ = 0.10 ϵ = 0.02

–0.4 –0.4

–0.2

0.2

0.0

0.4

y/h

Figure 2.18  Temperature profiles for three different values of skin‐to‐core thickness ratio. (Pilipchuk et al., 2010)

2.3.5.4  Numerical Example and Asymptotic of Thin Skin

Consider an example of the thermo‐elastic coupling in sandwich plates made of glass  fiber/epoxy skin and Rohacell 70 closed cell polymethacrylimide (PMI) foam core.  The  following parameters were taken for numerical implementation of the ­solution: L 1.0(m ), h 0.015 (m ), [ ] 1.0 C, Es 26 109 Pa (elasticity modulus for skins), s 0.065 (Poisson ratio for skins), s 3.64 10 8 m/m/m/m/ C (thermal expansion coefficient for skins), k s 0.04 (N m/s)/(m C ) (thermal conductivity for  skins), Ec 0.092 109 Pa (elasticity modulus for core), c 0.3 (Poisson ratio for  core), c 7.5 10 8 m/m/ C (thermal expansion coefficient for core), and kc 0.045 (N m/s)/(m C )k (thermal conductivity for core). The parameters involved in the solution are calculated using the following standard expressions of Lamé’s constants



s

c

1 1

Es s

Ec c

s

1 2

s

c

1 2

c

,

s

Es 2 1

s

,

c

Ec 2 1

c

,

s

,

c

3 3

s

c

2 2

s

c

s

c



Figure 2.18 shows the estimated temperature profiles at three different values of skin‐ to‐core thickness ratio 0.02, 0.1, 0.5. The corresponding deviation of temperature profiles from the homogeneous case is shown in Figure 2.19. The temperature gradient profiles for three different values of skin‐to‐core thickness ratio is demonstrated in Figure 2.20. In particular, Figure 2.18 gives the temperature profiles, which appear to be very close to linear, but, as can be seen from Figure 2.20, non‐smooth. Figure 2.19 shows the deviation of temperature profiles from the homogeneous case, in which θH denotes the temperature of the homogeneous case. This representation clarifies the influence of discontinuities in the structure cross‐section. The temperature gradients have step‐ wise discontinuities when passing from core to skin layers of the sandwich. Generally, the gradients through the whole sandwich are higher for thinner skins.

103

Handbook of Structural Life Assessment 0.015

ϵ = 0.50

0.010

θ–θH [deg]

ϵ = 0.10 0.005

ϵ = 0.02

0.000 –0.005 –0.010 –0.015

–0.4

–0.2

0.0

0.2

0.4

y/h

Figure 2.19  Deviation of temperature profiles from the homogeneous case. (Pilipchuk et al., 2010)

74

ϵ = 0.1

72 de/dy [deg/in]

104

ϵ = 0.5

70 68

ϵ = 0.02

66 64 –0.4

–0.2

0.0

0.2

0.4

y/h

Figure 2.20  Temperature gradient profiles for three different values of skin‐to‐core thickness ratio. (Pilipchuk et al., 2010)

Figure 2.21 illustrates the non‐zero stress component σxx as a function of the vertical coordinate y. Due to the apparent oddness of stress with respect to y, the sandwich is in a pure bending state. Also, the stress appears to be a piecewise linear, discontinuous function of y, whose jump magnitudes quite strongly depend on . Moreover, as follows from the set of plots, the jump magnitudes must have a minimum within the range 0.02 0.5. From that standpoint, an optimal skin‐to‐core thickness ratio should exist. Figure 2.22 shows the distribution of the mean vertical displacement over the longitudinal coordinate x, as given by equation (2.55). It is seen that the amplitudes at the middle of the plate, x 0, converge for higher . This effect becomes more explicit from the diagram in Figure 2.23, which shows the displacement amplitude as a function of ε. In particular, the amplitude becomes as much as twice smaller in a very narrow range of  near zero, but then the decay becomes very slow as the skin thickness grows.

ϵ = 0.02

20 ϵ = 0.5

σxx (Pa)

10

ϵ = 0.1

0 –10 –20 –0.4

–0.2

0.0

0.2

0.4

y/h

Figure 2.21  Stress profiles at the cross‐section x 0.4

ε = 0.1 ε = 0.02

/V0

0.3

0. (Pilipchuk et al., 2010)

ε = 0.5

0.2

0.1

0.0 –0.4

–0.2

0.0 X/L

0.2

0.4

Figure 2.22  Mean vertical displacement profile at three values of skin‐to‐core thickness ratio. (Pilipchuk et al., 2010) 1.0

Vmax/V0

0.8 0.6 Asymptotic 2

0.4

Exact

Asymptotic 1

0.2 0.0

0.0

0.1

0.2

ε

0.3

0.4

0.5

Figure 2.23  Amplitude of the mean vertical displacement as a function of skin‐to‐core thickness ratio. (Pilipchuk et al., 2010)

106

Handbook of Structural Life Assessment

In addition, Figure 2.23 demonstrates the behavior of amplitudes given by the following two asymptotic estimates asymp1



asymp 2

x

x

x2 L2

0

1 4

0

x 2 1 2 1 3r3 1 4 2 L 1 2 r1 3r2 r3



1 2 1 r1 3 1 r2 r3 O O

O

2

(2.59)

2 2

(2.60)

The asymptotic estimate given by equation (2.59) is based on the linear part of the power series expansion of equation (2.55) with respect to . Equation (2.60) is the ­estimate obtained by keeping the linear parts separately in the numerator and denominator of equation (2.55), which rather corresponds to the so‐called Padé approximants (Andrianov, 1991). Padé approximant is known as the “best” approximation of a function by a rational function of given order. This approximant gives a better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. The diagram shows quite significant difference between estimates (2.59) and (2.60) as the parameter increases. In particular, equation (2.60) appears to be a much better match with the exact solution given by equation (2.55) because the parameter leaves the narrow interval near zero. Generally, the curves in Figure 2.23 show that thermo‐elastic behavior of the sandwich is very sensitive to the skin thickness variation, while the skin thickness is such that hs (1/20)hc. However, outside this value, the effect of skin thickness variation is quite insignificant. 2.3.5.5  Combined Thermal, Mechanical and Geometrical Model Characterization

In order to clarify the combined influence of thermal, mechanical, and geometrical property variations, it is useful to introduce the following dimensionless parameters



V

mean

L

c

0

hc hs

,

c

1

s

c s

, a

s c

(2.61)

where V is the dimensionless amplitude of mean vertical displacement, the ratio Λ may serve as a characteristic of the relative (core‐to‐skin) longitudinal stiffness, and a is the relative (skin‐to‐core) characteristic of thermal expansion. Taking into account expressions (55) and (61), gives V



1L 1 2 8 h 1 2r1 2 3 6

1

2a 3 3 1 r1

c

4

2

1

c

1

s

1

4 r1

2

1

c

1

s

1

c

1

s

(2.62)

Figure  2.24 illustrates the dependence of the amplitude V on the stiffness para­ meter  Λ  for three different values of thickness‐to‐length ratio h/L in the “hard skin”

Applications of Fracture Mechanics 80

h/L = 0.001

60

V 40

20

h/L = 0.01

h/L = 0.1 0 0.0

0.2

0.4

0.6

0.8

1.0

Λ

Figure 2.24  Dependence of the amplitude of averages vertical displacement on the relative stiffness at three different values of thickness per length ratio: hard skin region. (Pilipchuk et al., 2010)

150 h/L= 0.001 100 V 50 h/L= 0.01

h/L= 0.1 0 0

20

40

60

80

100

Λ

Figure 2.25  Dependence of the averaged vertical displacement on the relative stiffness at three different values of thickness per length ratio: transition from hard to soft skin. (Pilipchuk et al., 2011)

interval 0 1. It is seen from the right‐hand side of equation (2.62), that the amplitudes parameter V is linearized with respect to Λ near zero, so that no singularity develops as 0; see Figure 2.24. As the parameter Λ increases, and therefore skin softens, the function V(Λ) becomes asymptotically constant, as demonstrated in Figure  2.25, although such cases are close to a mono‐layer plate and therefore do not comply with the very idea of sandwich design. The above treatment provided a closed form analytical solution of the problem of thermo‐mechanical coupling in sandwich plates, obtained for the case of bending caused by different temperatures on the outer surfaces of the plate. The plate’s mean transverse deflection has been estimated for different values of skin‐to‐core thickness

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ratio and the relative (core‐to‐skin) longitudinal stiffness Λ. The influence of g­ eometrical and physical properties of the layers on the bending intensity has been illustrated by a series of diagrams. In particular, it was shown that the bending intensity is quite sensitive to variations of the relative (core‐to‐skin) longitudinal stiffness or skin‐to‐core thickness, when such parameter ratios are small as compared to unity. The mechanics of the core structure alone has received extensive studies from ­material scientists and structural mechanics. The next section outlines the mechanics of solid foams commonly used in sandwich structures. 2.3.6  Mechanics of Solid Foams 2.3.6.1 Background

Solid foams represent an important component of lightweight composite structures. Many types of foam serve as a core subcomponent in sandwich composite materials. Solid foam exists either in the form of metallic‐based or polymer‐based or hybrid‐based materials. Examples of polymer‐based materials include polymethacrylimide (PMI), expanded polystyrene (EPS), and polyetherimide (PEI). Hybrid‐based material foams include bioactive glass‐polyvinyl alcohol (PVA), phenolic foams reinforced with glass and rubber, and nanoclay reinforced hybrid syntactic foams. Material scientists are dealing with the mechanics of foam microstructure. For example, Williams (1968) showed that the average number of faces per cell is close to 14 and the average number of edges per face is 5.1. Mills and Zhu (1999) proposed a tetrakaidecahedral cell having six square and eight hexagonal faces. If cell faces do not rupture, there are contributions to the mechanical properties from the compression of the cell gas and from the stretching of cell faces. Young’s modulus was found to depend linearly on the relative density (foam density to the polymer density) as indicated by Renz and Ehrenstein (1982). Rusch (1970) suggested that the cell gas contribution to the stress would take place in the post‐yield hardening of compressed cell foam. Cell faces were treated as membranes. Maiti et al. (1994) argued that cell edges would touch each other at high strains, as the length of the buckling element is decreased while the stress required for buckling would increase (densification). The nonlinear face analysis was performed by Mills and Zhu (1999) for the simplest loading direction along the z-axis in a regular lattice of cells, in which all cells have the same collapse stress. The elastic‐ plastic model predicts that the polymer contribution to the foam stress reaches a ­maximum then declines significantly when the edges are buckled. Closed‐cell foams have the solid material in edges and faces so each cell is sealed off from its neighbors. That is why closed‐cell foams have higher compressive strength due to their structure. Based on the results reported in the literature (e.g. Gibson and Ashby, 1997; Goga, 2011; Fatima et  al., 1995; Flores‐Johnson et  al., 2008; Wierzbicki and Doyoyo, 2003) a typical stress–strain curve of solid foams exhibits three definite regions: linear elasticity, plateau, and densification. Arezoo et al. (2011) obtained experimental data pertaining to the quasi‐static mechanical response of PMI foams of density ranging from 50 to 200 kg/m3. It was shown that foams of low density collapse by cell wall buckling, while foams of high density undergo plastic cell‐wall bending. As a result, both the elastic and plastic macroscopic responses of foam display a tension/compression asymmetry. An effective stress analysis of cellular foam materials in the finite strain regime was developed by Hohe and Becker (2003). The homogenization of the

Applications of Fracture Mechanics

microstructure is performed by means of a strain energy based representative volume element procedure. The procedure assumes macroscopic equivalence of a representative volume element for the given microstructure and a similar volume element consisting of the effective medium. The equivalence is valid if the average strain energy density in both volume elements is equal, provided that the deformation gradient with respect to both elements is equal in a volume average sense. Disordered microstructures were considered by means of a randomized periodic model in conjunction with a stochastic approach. The model was applied to an analysis of the effective stress–strain behavior of two‐dimensional model foams with periodic and disordered microstructure. Special interest was directed to effects of the geometric nonlinearity. The acoustic emission (AE) responses during compression and indentation were examined by Kádár et al. (2004) for two kinds of metal open cell foams (sponges): an aluminum–silicon foam (AlSi10) and Alporas foam (AlCa5Ti3) were examined by Kádár et al. (2004). They reported two types of AE signals, which can be distinguished according to the rise time of the amplitude of the AE signal. The signal revealed basically two different modes of deformation: fracture and plastic yield. Later, Kádár et al. (2007) found that the yield stress of foam and deformation mechanisms during tension are governed by the cell‐edge material and by the structure of the foam. Different deformation mechanisms were found to take place depending on the pore‐size and the cell‐edge thickness. In the case of large pores with thicker cell‐edges, the deformation mechanisms were reported to be controlled by the structure, while for small pores (with thinner cell‐edges) the microstructure of the cell‐edge material governs the deformation. Ramsteiner et al. (2001) carried out a number of tests to study the mechanical properties of open‐cell and closed‐cell foams. They showed that the foam density is the dominating parameter influencing its behavior. Brothers et al. (2006) used AE methods to study the nature and evolution of microfracture damage during uniaxial compression of ductile amorphous and brittle crystalline metal foams made from a commercial Zr‐based bulk metallic glass. They compared the results with those obtained for aluminum‐based foam of similar structure. For the amorphous foam, acoustic activity revealed evolution of the damage process from diffuse to localized damage through the foam stress plateau region, and reversion back towards diffuse damage in the foam densification region. The effect of temperature as well as strain rate on the quasi‐static compressive response of polyvinyl chloride (PVC) closed‐cell foams was examined by Thomas et al. (2002). Foams with densities 75, 130, and 300 kg/m3 were tested at room and elevated temperatures. A reverse trend in failure modes was observed when moving from room to elevated temperatures at high strain loading, which was not found in quasi‐static testing at elevated temperatures. Post‐impact tests were conducted to evaluate the residual strength of the foam cores subject to elevated temperatures and high strain rate. The energy absorption of all cores was found to be proportional to the strain rate. At elevated temperatures, the change in the foam stiffness was found to be proportional to the relative density. Deschanel et al. (2006) reported tensile failure experiments on polyurethane (PU) foams by imposing a constant strain rate. For heterogeneous materials the failure does not occur suddenly and can develop as a multi‐step process through a succession of microcracks that end at pores. The acoustic energy and the waiting times between acoustic events follow power‐law distributions. This remains true while the foam density is varied. Tensile tests at different temperatures (from room temperature down to −65 °C) were performed by Deschanel et al. (2006) on PU foams of relative

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density 0.58. It was found that the foam becomes increasingly brittle with temperature decreasing. The Young’s modulus was found to increase as the temperature decreases, likewise the maximum stress which almost doubles between room temperature and −65 °C. It was reported that the plastic plateau disappears with decreasing temperature, and that the failure strain is less important as the material becomes more brittle. The behavior in AE at room temperature and at −10 °C was found to be almost identical, where acoustic activity begins late when the material is already in the plateau stage and a divergence of the number of AE events occurs at the end. By contrast, for the tensile tests at −30 °C and −65 °C the AE activity was found to start at the very beginning, indicating the early occurrence of damage. The number of events rises gradually as the load increases. The coupled effects of sea water and low temperature on the mechanical properties of closed cell PVC closed‐cell H100 foam core material were reported by Siriruk et al. (2012). Interfacial delamination fracture response for the sandwich structures due to  the combined effects of sea water and low temperature was evaluated and the ­associated degradation in critical energy release rate for delamination was found to be  substantial. Experimental results for H100 foam cores associated with moisture induced expansion strains were also reported. The following sections describe the work of Grace et  al. (2012) who experimentally examined the compressive behavior of closed‐cell PMI foam used as a core component of sandwich structured composite materials. A series of tests were carried out to study the deformation behavior of PMI foam in compression under relatively slow (quasi‐static), still non‐stationary, loads. Specific effects of minor fluctuations of the loading speed on the foam response were analyzed. The results were used for phenomenological modeling of the foam displacement–load response at different compression speeds and temperatures. 2.3.6.2  Experimental Investigations

Experimental investigations were conducted by Grace et  al. (2012) on test specimens made of PMI, Rohacell 31‐IG (industrial grade), foam. This material is homogeneous and isotropic with closed cells. There is a wide range of density of this type of foam, but the one used in the present study has an average density of 32 kg/m3, modulus of elasticity of 36 MPa, shear modulus of 13 MPa, tensile strength at break of 1.0 MPa, compressive strength of 0.400 MPa, and flexural strength of 0.800 MPa (see Evonic Degussa Rohacell® at Matweb.com). All the specimens were cut directly from a PMI foam sheet with dimensions 50.8 × 50.8 × 12.7 mm. The tests were conducted using an EnduraTec servo‐pneumatic testing machine. The EnduraTec Win‐Test software was used to control the applied load to preserve the prescribed crosshead speed. To study the effect of temperature on the mechanical properties of foam material, tests were carried out inside the environmental chamber at three different values of temperature within the wide range from −60 °C to +60 °C. The temperature inside the chamber was controlled by a Sun PC100 controller system whose typical operating temperature can vary from −200 °C to +325 °C. The effect of loading speed on the compressive behavior was studied by conducting the tests at three different values of loading speed, ν = 0.127, 1.27, and 12.7 mm/s. For verification purposes, at least three specimens were tested for each case. Miniature Physical Acoustics Co. (PAC) pico‐AE piezoelectric sensors were attached to gather acoustic emission (AE) data, which is synchronized with mechanical load–deflection data during compression tests. The AE data acquisition together with its control unit is

Applications of Fracture Mechanics

a DisP‐4 four‐channel Physical Acoustics AE‐Win system. The foam compression process was monitored by Phantom V12.1 high speed camera supported by Phantom 640 software at a speed of 50 × 103 frames per second. Foam specimens were also subjected to 100 successive loading/unloading cycles to different values of specific engineering strain: 0.1, 0.35, and 0.75. The loading‐unloading cycle has the sawtooth (triangular wave) temporal shape, in which the specimen was loaded to the specific strain. Then the specimen was unloaded at the same loading rate. Different phases of typical strain–stress foam responses on quasi‐static compressive loading for three different samples tested under the same condition are shown in Figure 2.26. It is seen that the strain–stress plots of each sample do not coincide with those of other samples, owing to the non‐homogenous nature of the foam material. As mentioned, the strain– stress curves are composed of three major segments, corresponding to physically different processes developed during the compression – elastic, plateau, and densification. At small strains, less than 8%, the behavior is linearly elastic, with a slope equal to the foam’s effective Young’s modulus. As the load increases, the foam cells begin to collapse by elastic buckling, plastic yielding, or brittle crushing, depending on the mechanical properties of the cell walls. Collapse progresses at roughly constant load, giving a stress plateau. This phenomenon was well described by Kádár et al. (2008), who observed the formation of deformation bands with associated strain hardening during compression tests. The dependence of wavy character of the acoustic emission (AE) count rate on strain was found to be correlated to the formation of deformation bands. The compressive deformation of foams did exhibit signs of strain localization in deformation bands. It was assumed that each layer of cells has the same strength, and therefore the deformation strength remains almost constant until the entire process is completed (e.g. Weaire and Fortes, 1994; Bart‐Smith et al., 1998; Bastawros and Evans, 2000). This region is associated with relatively small elastic deformations of cell walls. However, as the load increases, the foam cells begin to collapse. Cell collapse progresses at roughly constant load as displayed by the stress plateau. Note that all the curves are coherent and smooth in the elastic and densification areas, whereas some deviation between the curves with fluctuations in each of them 600 Densification

500

σ(kN/m2)

400 Plateau 300 Elastic deformation

200 100 0

0

0.1

0.2

0.3

0.4 ε

0.5

0.6

0.7

0.8

Figure 2.26  Assembly of stress–strain curves for different samples showing variation across multiple samples at T 22 C under a loading speed 1.27 mm/s. (Grace et al., 2012)

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is observed within the plateau. This is simply explained by sensitivity of the plateau process to the typical non‐homogeneity of the foam material and some deviations from the desired loading speed due to the specifics of the speed controller, friction between the machine parts, and other natural imperfections. Such sensitivity is not surprising because, during the plateau phase, there are always clusters of cells near the critical state whose walls are about to lose stability. In other words, the plateau process can be viewed as propagating instabilities in the foam microstructure. Since critical states are known to be sensitive to even very small perturbations of external loads and inherent parameters, the response will always exhibit some limits of uncertainty. Nevertheless, as can be seen from Figure 2.26, the layer of uncertainty during the plateau is narrow enough for the given material. Therefore, only one sample for each case is selected for illustration purposes. When all cells are completely collapsed, the opposing walls in the cells are compacted, leading to the final region of foam densification. At this stage, the cell walls are fully crushed so that the air is forced out of the compacting cells, causing the stress to increase steeply. The selected snapshots of the compression process taken by the high speed camera are shown in Figure 2.27.

t=0

t = 86620

t = 141720

t = 212600

t = 294500

t = 352750

t = 440950

t = 529120

Figure 2.27  Snapshots of propagating (from the top) front of collapsing foam cells during compression at different time instances shown in µs units; see the next figure for details. (Grace et al., 2012)

Applications of Fracture Mechanics

The time of each snapshot is shown in µs. In this particular case, the foam collapse triggered near the upper (moving) surface of the specimen propagates downward towards the fixed end until all the cells are collapsed. Figure 2.28 shows high resolution images of the foam structure near the front of collapsing cells obtained by the Keyence VK‐9700 3D laser scanning microscope at different magnitudes of the effective7 strain. Figure 2.28(b) is taken with an effective strain of 0.1, and it is very hard to see any trace of collapsed cells. Note that all cells are randomly oriented, and as the effective strain increases, say 0.5 as shown in Figure 2.28(c), some cells are visibly collapsed as those highlighted within rectangular regions, while cells in the remaining regions are oriented differently, and their projection by the microscope will not reveal their collapse. As the strain increases further, say 0.75 as shown in Figure 2.28(d), most of the cells are collapsed near the front surface. (a)

(b)

(a) (c)

Region of collapsed cells

(d)

Almost all cells are collapsed

Figure 2.28  Microstructure of PMI foam near the front of collapsing cells within the area 2 mm ×2mm shown for different strain values: (a) ε = 0, (b) ε = 0.1, (c) ε = 0.5, and (d) ε = 0.75. (Grace et al., 2012) 7  Note that, during the plateau phase, the definition of strain as calculated from the displacement of the upper surface in the specimen thickness can no longer serve as any local characterization of the material state.

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In order to clarify the effect of environmental temperature on the foam response, compressive tests were also conducted at three different values of specimen’ temperatures: 60 °C, 22 °C (room temperature), and −60 °C. Figure 2.29(a), (b), and (c) show the effect of temperature on the load–displacement behavior of PMI for three different values of loading speed: ν = 0.127, 1.27, and 12.7 mm/s, respectively. It is seen that low (a) 1600

–60°C

22°C

Load (N)

1200 800 60°C 400 0 0

2

4

(b)

6 Displ (mm)

8

10

1600

–60°C

Load (N)

1200 22°C

800

60°C

400 0 0

2

4

(c) 1600

6 Displ (mm)

8

10

22°C

1200 Load (N)

114

–60°C

800

60°C

400 0 0

2

4

6

8

10

Displ (mm)

Figure 2.29  Effect of temperature on the PMI load‐displacement response at different values of loading speed: (a) 0.127 mm/s, (b) 1.27 mm/s, and (c) 12.7 mm/s. (Grace et al., 2012)

Applications of Fracture Mechanics

temperature yields somewhat stiffer foam in the elastic region, as reflected by higher slopes of the low temperature curves. This effect, however, becomes less observable at higher loading speeds as follows from Figure 2.29(b) and (c). It can also be seen that lowering the temperature affects both the length and shape of the plateau segment. For example, at −60 °C the plateau segment becomes longer and acquires convexity, such that the foam experiences strain hardening up to almost the middle of the plateau segment. In contrast, at 22 °C and 60 °C, the plateau segment resembles perfectly plastic behavior. An unusual low temperature effect, however, is represented by significant spike‐wise load drops during the plateau phase; see Figure 2.29(a). For better understanding of this phenomenon, a number of compression tests was performed at different sub‐zero temperatures but fixed loading speed 0.127 mm/s. The results are illustrated by the series of diagrams shown in Figure 2.30. It is seen that lowering the temperature to −10 °C has no significant effect on the foam behavior. However, at −20 °C, the degree of load fluctuation is increased, as observed in Figure  2.30(d). Further temperature drop to −25 °C leads to random excessive load spikes, associated with b ­ rittle crushing of arrays (clusters) of foam cells, as shown in Figure 2.30(e). Further temperature decrease intensifies the effect of brittle crushing, as follows from Figure 2.30(f )–(h). Therefore, we can conclude that for PMI foams, there exists a critical temperature at which clusters of foam cells are synchronously crushed in a brittle way under compressive loads above the yielding point during the plateau phase. The importance of determining this temperature is due to the fact that these materials are practically used in many structures operating at low temperatures such as navy ships navigating in Arctic regions. This mechanism of spiking load behaviors was supported by recording the acoustic emission (AE) responses from three sensors installed on the specimen sides. Figure 2.31 shows the time evolutions of the load and acoustic emission count density at −60 °C and loading speed 0.127 mm/s The figure shows an excellent correlation between the load fluctuations and the associated acoustic emission count density over the plateau segment. This implies that the load drop manifested by the observed spikes is accompanied by progressive collapse of the foam cells due to their brittleness acquired at low temperature. The effect of loading speed was also studied for different values of temperature level and represents the results of tests conducted at the three different loading speeds (0.127, 1.27, and 12.7 mm/s). Note that, despite of their wide range, these speeds should be considered as slow (quasi‐static) as compared to the speed of elastic waves inside the foam structure. However, this range of speeds appears to be of the same temporal scale as the speed of waves of collapsing cells, as will be demonstrated shortly. The influence of loading speed was found to develop mostly within the plateau segment and appears to be different at positive and sub‐zero temperatures. In particular, at positive ­temperatures, 22 °C and 60 °C, increasing the speed from 0.127 to 12.7 mm/s elevates the plateau by about 40% of the plateau load. This effect has a simple physical explanation: since more cells collapse per unit time then higher force is required to support the process. This explanation is in agreement with previous studies. For example, Gibson and Ashby (1997) developed one of the pioneering and well‐known micromechanical models of foams represented by an array of cubic cells. The elastic modulus was described by the combination of stretching lamellae and bending of struts. It was assumed that the elastic modulus is proportional to n‐th power of the relative density of foam (the relative density

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(b) 600

600

500

500

400

400

300 T = 22°C

200

σ (kN/m2)

σ (kN/m2)

(a)

100 0

(c)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

(d)

0

500

500 σ(kN/m2)

600

300 200 100 0

T = –10°C 0

(e)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ε (f)

300 200 100 0

T = –22°C 0

600

500

500

400

400

300 200

T = –25°C

100 0

0

(g)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

400

600

σ (kN/m2)

σ (kN/m2)

T = –5°C

200

600 400

σ (kN/m2)

300 100

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ε

200 0

T = –30°C 0

(h)

600

600

500

500

400

400

300 200 T = –40°C

100 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ε

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ε

300 100

σ (kN/m2)

σ(kN/m2)

116

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ε

300 200 T = –60°C

100 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ε

Figure 2.30  Effect of sub‐zero temperature on compression behavior of PMI foam for a loading speed 0.127 mm/s. (Grace et al., 2012)

is the ratio of foam density to the density of the solid skeleton). For closed‐cell foams, the model was modified to include an expression representing the internal gas pressure of the closed cells. The modified model also included similar expressions for shear modulus and elastic collapse stress (plateau stress). Kanahashi et  al. (2000) applied that

Applications of Fracture Mechanics 1335.0

5 × 104 Load 4 × 104

890.0

3 × 104

667.5

Count

2 × 104

445.0

1 × 104

222.5 0.0

Count density

Load (N)

1112.5

0

10

20

30

40

50

0

Time (sec)

Figure 2.31  Measured time evolution of load and acoustic emission count density for 0.127 mm / s at T 60 C. (Grace et al., 2012).

Gibson–Ashby model to describe the effect of strain rate on the plateau stress of open‐ cell aluminum foam. They showed that the response of open‐cell foam is sensitive to strain rate. Hamada et al. (2009) found that the plateau stress by the static compression test increases as the density increases, and the plateau stress increases further as the test piece volume increases, as a result of combining the d ­ uctility of the cell wall and the effects of the inner gas. Accordingly, there is a scale effect of test pieces. At low temperature, −60 °C, increasing the speed has little effect on the plateau level despite the presence of significant downward load spikes. Note that some reduction of the plateau fluctuations due to the speed increase was observed under positive temperatures. At sub‐zero temperature, however, this effect becomes dominating while the mean shape and level of the plateau remains practically unchanged. Although the temperature range of the present study (from −60 °C to +60 °C) is well below the glass transition point of the PMI material, the above qualitative changes in the foam response still indicate a strong dependence of the foam microstructural properties on temperature. Namely, the foam structure, composed of thin‐walled cells, appears to be more sensitive to brittle phase than compact materials. In particular, the plateau convexity points to some qualitative changes in physical mechanisms of the cell collapse process. Also, the elimination of spikes by speed increase indicates sensitivity of such mechanisms with respect to non‐temperature factors. Note that the “catastrophic” spike‐wise clustering of collapsing cells was triggered by  very small fluctuations of the displacement of loading machine, as shown in Figure  2.32(a). Practically, the rate of displacement fluctuates about some average value, rather than being perfectly constant. This conclusion was verified by recording the acoustic emission (AE) responses using two sensors: the first sensor was placed on the upper cap of the testing machine, Figure 2.33(a), and the second sensor was placed on the foam side face, Figure 2.33(b). It can be seen from Figure 2.33 that the AE count density produced from the upper cap of the testing machine is much higher than that from the foam itself. We may conclude that the spikes in the plateau segment reflect the excessive response of the foam structure to the fluctuations in loading speed. In more detail, the foam behavior under fluctuating speed was

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Handbook of Structural Life Assessment 5 Actual displacement

Disp (mm)

4.5 4 3.5 3

Axial command

2.5 20

25

30

35

40

Time (s) 0.06

Displ (in)

118

Average speed = 0.127 mm/s

0.04 δD

δD 0.02

Constant speed = 0.127 mm/s

0 3

5

7

9

11

Time (s)

Figure 2.32  Time evolution of axial displacement: (a) Comparison between axial command and actual displacement, (b) Definition of displacement increment for 0.127 mm/s. (Grace et al., 2012)

examined at room temperature for an average speed 0.127 mm/s. In this case, the values of speed fluctuations were intentionally imposed as

1, 2

D

D / t (2.63)

where D is the displacement, and δD is the displacement increment defined in Figure 2.32(b). Figure 2.34(a) shows that, as the value of speed fluctuation increases, the magnitude of spike‐wise load drop also increases; meanwhile, there is no influence of such fluctuations on the length of plateau segment. Figure 2.34(b) shows that the spikes occur at the same frequency as the speed fluctuations. This confirms the fact that the  fluctuations in the plateau segment are due to sensitivity of foam to the loading conditions. 2.3.6.3  Cyclic Load‐Unload Tests

The main objective of these series of tests is to explore the foam capacity for absorbing the energy during repeatable cyclic loading (hysteresis). The tests are conducted at the room temperature, 22 °C, and at 60 °C, under which the foam can withstand multi‐cyclic loading to different levels of strain within the plateau segment. Each displacement control test is conducted at the loading‐unloading displacement rate 0.127 mm/s. Foam specimens were subjected to 100 of successive loading‐unloading cycles to different

Applications of Fracture Mechanics

(a) 300

3

Load (lb)

200

2

150 Count 100

1

Count density*106

Load

250

50 0

0 0

10

20

30 40 Time (sec)

50

60

70

(b) 24

300

16

Load (lb)

200 Count

150

8

100

Count density *103

Load

250

50 0

0 0

10

20

30

40

50

60

70

Time (sec)

Figure 2.33  Acoustic emission response for 0.127 mm / s at T 22 C as recorded by: (a) sensor placed on the upper cap of testing machine, and (b) sensor placed on the foam side face. (Grace et al., 2012)

values of specific engineering strain 0.1, 0.35 and 0.75. The loading‐unloading cycle has the saw‐tooth (triangular wave) temporal shape, in which the specimen is loaded to the specific strain. Then the specimen was unloaded at the same loading rate. During the unloading phase, the specimen attained most of its possible recovery. Figure  2.35(a) shows the stress–strain curves for the first two high temperature loading and unloading cycles. The amount of permanent deformation is obtained at zero stress. Figures 2.35(b) and (c) show the stress–strain curves for the first two and 100th cycles at 60 °C and 22 °C, respectively. Residual strain at the end of each cycle indicates permanent damage of the foam cells. The stress–strain curves exhibit hysteresis during loading and unloading cycles. The largest hysteresis loop corresponds to the first cycle in which the energy loss is mostly due to the work done in destroying the cells. All the first cycles have largest residual strains. The number of destroyed cells during a cycle is drastically decreased already during the second cycle.

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(a) 1600 δD = 0.254 mm

Load (N)

1200

800 δD = 0.762 mm δD = 1.72 mm

400

0 0

2

4

6

8

10

8

10

Displ (mm)

(b) 1600 Constant speed

1200 Load (N)

120

800 δD = 0.254 mm

400

0 0

2

4

6 Displ (mm)

Figure 2.34  Compression under fluctuating speed (average speed 0.127 mm/s), (a) effect of changing value of the speed, (b) comparison with result at constant speed. (Grace et al., 2012)

Tests at low temperatures were conducted on sandwich plates with the same PMI foam material. The presence of skin results in a different stress–strain response are demonstrated in Figure 2.36, obtained from quasi‐static cyclic loading tests at sub‐zero temperature, −60 °C. In particular, when reaching the yielding point, the foam collapses by developing a large scale brittle crack, in such a way that no plateau segment occurs. The only possible reason for such a different behavior is that the skin with adhesive material, penetrating through the foam, imposes certain geometrical constraints of deformation of foam cells by preventing their lateral displacements at least within some layers near the skin. As a result, the entire structure actually represents a combination of substructures with different spatial scale mechanical properties. During compression, boundaries between such substructures represent the source of shear stress accumulation and large‐scale brittle cracks. However, at room temperature, 22 °C, the compression phase develops in a regular way, with a plateau segment without brittle crack. The tension phase reveals that no significant damage occurs during compression, and the foam resists tension through the cell alignment segment until it breaks apart.

Figure 2.35  Compressive cyclic stress–strain curves: (a) maximum strain 0.35 at T 60 C , (b) maximum strain 0.1, 0.35, and 0.75 at T 60 C , (c) maximum strain 0.1, 0.35, and 0.75 at T 22 C Cycle, Cycle 2, Cycle 100. (Grace et al., 2012)

600

σ(kN/m2)

500 400

Cycle 1

300 200

Cycle 2

100 0

0.2 ε

0

0.1

0

0.2

0.4 ε

0.6

0.8

0.2

0.4

0.6

0.8

0.3

0.4

800

σ(kN/m2)

600 400 200 0

800

σ (K–N/m2)

600 400 200 0

0

ε

T = –60°C

Plateau

σ (kN/m2)

1000

0

T = 22°C

–1000 –2000

Cell wall allignment

–0.02

0.00

0.02

ε

0.04

0.06

0.08

0.10

Figure 2.36  Stress versus effective strain diagrams of sandwich specimens obtained during cyclic compression–tension tests under room and low temperature conditions and loading speed 0.127 mm/s.

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Now most of the energy loss is due to plasticity and viscosity effects. Quantitatively, the dissipated energy is given by the area enclosed in the hysteretic loop, i.e.



E



d

max

[

l

n,

ul

n, ]d (2.64)

0

where εmax is the maximum strain reached, n is a cycle number, subscripts l and ul denote loading and unloading phases, respectively. The dissipated energy for different cycles is calculated and shown in Figure 2.37. It is seen that the amount of energy dissipated for each cycle is reduced at higher temperatures. 2.3.6.4  Modeling the Foam Response to Compressive Loading

Modeling of solid foams may be divided into two groups: phenomenological and micromechanical models (Avalle et al., 2007). One of the pioneering and well‐known micromechanical models is that presented by Gibson and Ashby (1997) who modeled the foam as an array of cubic cells. The elastic modulus was described by the combination of stretching lamellae and the bending of struts. This section presents a phenomenological model of the foam load–displacement response at different values of compression speed and temperature. Note that the suggested model fits the integral behavior of the foam response by ignoring possible localized effects of spiking, as discussed in the previous sections. The noticed spiking effects must be investigated locally based on microstructural modeling. As noticed in the literature and the present test results, a typical load–displacement curve of solid‐foam materials essentially differs from that usually observed for compact materials. As mentioned earlier, the presence of qualitatively different regimes of the corresponding dependencies reflects different structural states of the foam under progressive compression. Conditionally, three major segments in the load versus displacement curves can be specified as follows: 1) Linear elastic region before first foam cells collapse. This is well described by Hooke’s law, q ko x , where x and q are displacement and load, respectively, and ko is an ­elastic stiffness coefficient. 2) The plateau segment, which is attributed to a propagating front of collapsing cells.  This is almost a kinematic process  –  a noticeable specific feature of foam ­materials – that develops under practically constant load, as soon as the rate of cell collapse per unit displacement remains unchanged. However, a series of tests at sub‐ zero temperatures revealed some deviations from the plateau shape by a “convex component.” This phenomenon is well captured by the following parabolic law q do d2 ( x x0 )( x x1 ), where do is the plateau base level, x0 and x1 are the ­beginning and the end of the plateau, respectively, and d2 is a positive parameter controlling deviation from the horizontal level. 3) Finally, by assuming that all cells are collapsed and fully compressed, we might expect that Hooke’s law would work again. However, the full compression to compact (no emptiness) material takes some time and requires significant loads. As a result, the load versus displacement curve resembles a rather strengthening characteristic. Nevertheless, the following linear law will still be used q k1 ( x xr ), where k1 is an effective stiffness, and xr may be qualified as some residual displacement, which is assumed to happen after unloading the foam; see Figure  2.38 for the geometrical

Applications of Fracture Mechanics Number of cycles

(a) 15

ΔE(KJ/M3)

10 7 5 3 2 1.5 1 1

2

5

10

20

50

100

Number of cycles

(b) 100

ΔE(KJ/M3)

50 20 10 5 2

1

2

5

10

20

50

100

Number of cycles

(c)

ΔE(KJ/M3)

200 100 50 20 10 1

2

5

10

20

50

100

Figure 2.37  Energy loss in logarithmic scale during load–unload cycles at displacement speeds 12.7 mm /s for maximum strain: (a) 0.1, (b) 0.35, and (c) 0.75; solid line: T 22 C, and dashed line: T 60 C, (Grace et al. 2012).

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d2 > 0 d0

d2 = 0

Figure 2.38  Geometry of the load versus displacement foam model: 50– solid bold line, and 5 – thinner solid and dashed lines. (Grace et al., 2014)

k1

q

k0 x0

x1

xr x

1.0

α = 50

Figure 2.39  Smoothing function at two different magnitudes of the parameter α.

α=5

0.8 0.6 p 0.4 0.2 0.0 –4

–2

2

0

4

x

meaning of the parameters. This assumption is always valid in a narrow enough interval, which may be sufficient for practical purposes. Alternatively, the power‐ form, or maybe exponential dependence, can be used for a broader area. Note that transition between the above listed states never happens suddenly. In order to provide smooth transitions between different loading regimes, we can use the smooth step‐wise function,

p

,x

1 1 tanh 2

x



(2.65)

where α is a positive parameter. Function p(α, x) is illustrated in Figure  2.39 for two different values of the parameter α. Note that equation (2.65) describes the unit‐step discontinuous function as and can be used to build the unit‐form expression describing the entire compression process, q x, , T

ko x 1 p p



, x xo

, x xo

k1 x xr

p p

do

d2 x xo

, xo

p

, x x1

, x x1

p

, x1

x x1 p

, x1



(2.66)

Applications of Fracture Mechanics

where all the parameters are assumed to be linear functions of the temperature T and loading speed, dx/dt , ko do x1

koo 1 ko1T k02 , k1 k10 1 k11T k12 , ao 1 a1T a2 , doo 1 do1T d02 , d2 d20 1 d21T d22 , xo xoo 1 xo1T x02 x10 1 x11T x12 , and xr xr 0 1 xr1T xr 2

,

(2.67) Generally speaking, linear dependencies in expressions (2.67) represent the leading 0. These can order terms of the corresponding Maclaurin’s expansions near T 0 and always be used under relatively small variations of the temperature and loading speed. For the specimen size used in the experimental study, the following optimized coefficients are obtained to fit the test data under different values of loading speeds and temperatures:



k00 10022, k01 0.0047, k02 0.4508, k10 3580, k11 0, k12 0.4 , a0 38.8, a1 0.0021, a2 0.58, d00 279.5, d01 0.0005, d02 0.15, d20 1142.8, d21 0.02145, d22 0.46413, x00 0.0173, (2.68) x01 0.006, x02 0.1932, x10 0.370056, x11 0.0022, x12 0.03, xr 0 0.211242, xr1 0.003485, and xr 2 0.07.

The fit of load–displacement is illustrated by Figure  2.40, where the diagrams are represented in metric units. Introducing numerical factors of transition to metric units, gives the family of stress σ versus strain ε curves in the form

, , T

4.4482216 q( x , v , T ) (2.69) A

where x 39.3701L , v 39.3701L , L 0.0127 m and A 0.00258064 m 2 are the specimen thickness and area, respectively. A dot denotes time derivative. Thus, the stress σ is given in pascal (N/m2) units. The numerical factors are due to conversion from US customary system, which is inherited by the dependence q(x, v, T) from the test machine, into SI units. The effect of temperature on the shape of the plateau segment as described earlier is represented in equation (2.66) by the term d2 ( x xo )( x x1 ). It was shown that the ­plateau segment follows almost a straight line for positive temperature cases and becomes convex at sub‐zero temperature. The coefficient d2 is temperature dependent and is given by the fifth expression in equation (2.67), where d21 has a negative value as  given in (2.68). Hence, for sub‐zero temperatures, the term d2 ( x xo )( x x1 ) in ­equation (2.66) becomes positive giving the convex shape of the plateau segment. Figure 2.41 shows the foam stress–strain response according to relation (2.69) under different loading speeds and temperatures. The developed model enables us to conduct detailed analyses for variation of stress–strain response shapes due to the temperature and loading speed parameters. Although, in many cases of modeling the mechanical behavior of foams, simplified piecewise‐linear diagrams of perfect elastic‐plastic

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(b) 2500

2500

2000

2000

1500

Load [N]

Load [N]

(a)

Model

1000 Test

500 0

0

2

1500 1000 500

4

6

8

0

10

0

2

2500

2500

2000

2000

1500 1000 500 0

6

8

10

1500 1000 500

0

2

4

6

8

0

10

0

2

Displ. [mm]

4

6

8

10

8

10

Displ. [mm] 2500

2000

2000 Load [N]

2500

1500 1000

1500 1000 500

500 0

4

Displ. [mm]

Load [N]

Load [N]

Displ. [mm]

Load [N]

126

0

2

4

6

Displ. [mm]

8

10

0

0

2

4

6

Displ. [mm]

Figure 2.40  Illustration of fitting the load versus displacement data at different speed loading: (a) 0.127 mm / s , and (b) 12.7 mm / s each for three different values of temperature T 60o C (upper row), T 22o C (middle row), and T 60o C (lower row). (Grace et al., 2012).

materials are sufficient (Zenkert et al., 2004), detailed geometrical characterization of the response curves may appear to be important when dealing with stability problems. This is due to the high sensitivity of structural response to any small mechanical property variations near critical states. In closing this section, the stress–strain curves exhibited three major segments of different physical meaning: linear elasticity, plateau, and densification. These reflect different structural states of the foam under progressive compression. It was found that the elastic strength and the level of plateau both increase when the specimen’s temperature is lowered. However, this effect is reduced as loading speed increases. Tests revealed that even minor fluctuations of the loading speed can trigger significant sharp (spike‐wise) load drops during the plateau phase as the temperature goes

Applications of Fracture Mechanics

(a)

(b) 600

600 T = –40°C

550

500

v=12.7

450

v=0.127 v=127

400

σ (kN/m2)

σ (kN/m2)

550

400 350 300

0.4 ε

0.6

0.8

(c)

0.0

0.2

0.4 ε

0.6

0.8

(d) 600

600

v = 0.127 mm/s

550 T=–40

500

σ (kN/m2)

550 σ (kN/m2)

v=1.27

v=0.127

300 0.2

v=12.7

450

350

0.0

T = 40°C

500

450 T=0 T=40

400 350

T=–40

500 T=0 T=40

450 400 350

300 0.0

v = 17.7 mm/s

300 0.2

0.4 ε

0.6

0.8

0.0

0.2

0.4 ε

0.6

0.8

Figure 2.41  Foam stress–train response under different conditions of loading: (a) different loading speeds at low temperature, (b) different loading speeds at high temperature, (c) different temperatures at low speed, and (d) different temperatures at high speed. (Grace et al., 2012)

below some critical level, which is about −25 °C for the given foam material. Quite significant spike amplitudes indicated significant sensitivity of the loading process in  the plateau segment to variations of loading conditions. This sensitivity can be  attributed to the propagating microstructural instabilities of collapsing foam cells – the leading process within the plateau segment. In other words, as the front of collapsing cells propagates, there are always layers of cells at critical states near the front that are ready to collapse under very small external perturbations. Cyclic compression tests to specified maximal strains showed that the energy dissipated, in permanent cell deformation, is predominantly during the first cycle. The amount of energy dissipated for each cycle was reduced at higher temperatures due to possibly lower influence of viscosity and brittle effects. Based on the test data a phenomenological model describing the foam load–displacement response was developed. The model described the compressive stress–strain curves at different loading speeds and temperatures.

2.4  Closing Remarks The applications of the theory of fracture mechanics to studying the mechanical characteristics of metal alloys and composite structures under quasi‐static loading and thermal heating have been presented in this chapter. The main results published in the

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literature have been reviewed, together with a number of case studies. The selection of metallic materials for many applications is usually decided by the simple characterization of the stress‐corrosion‐cracking sensitivity in terms of the linear‐elastic threshold stress intensity. Some progress was made to establish crack growth rate relations to explain different fatigue phenomena. The development of structural integrity and damage tolerance of aluminum structures was found to play an important role in the design of aluminum alloys used in aircraft and marine structures. The fracture of structural aluminum alloys in terms of plane‐strain stress intensity factor received particular attention. It was found that the presence of the crack merely serves to reduce the load‐ bearing area. Characterizations of composite structures have received extensive research studies in both aerospace and ocean structures. Of particular interest is the influence of environmental factors such as temperature and static loading. An exact solution of linear elasticity theory for bending of sandwich plate‐like beams due to temperature difference at the plate faces has been obtained, based on the assumption that the heat flux is stationary, material is linearly elastic, and displacements are small. The exact solution yields the temperature profile, stress, and displacement distribution across the plate thickness. The influence of geometrical and physical properties of the layers on the bending intensity revealed that the bending intensity is quite sensitive to variations of the relative (core‐to‐skin) longitudinal stiffness or skin‐to‐core thickness, when such parameter ratios are small as compared to unity. Uniaxial compressive tests conducted on PMI foam specimens at different values of loading speed and temperature below the glass transition point exhibit three major segments of different physical meaning: linear elasticity, plateau, and densification. These reflect different structural states of the foam under progressive compression. Quite significant spike amplitudes indicate significant sensitivity of the loading process in the plateau segment to variations of loading conditions. This sensitivity can be attributed to the propagating microstructural instabilities of collapsing foam cells – the leading process within the plateau segment. In other words, as the front of collapsing cells propagates, there are always layers of cells at critical states near the front that are ready to collapse under very small external perturbations. These results reflect that there is a strong need to study fracture mechanics and dynamic loading as well as extreme loading. This will be the main subject of the next chapter.

129

3 Dynamic Fracture and Peridynamics 3.1 Introduction It is believed that Freund (1986) introduced the basic theory of dynamic fracture, which deals with fracture phenomena on a timescale for which inertial resistance of the material to motion is significant. The deformable body typically contains a dominant crack or other stress concentrating defects, and the phenomena of primary interest are those associated with conditions for the onset of extension of a crack, or its arrest. Material inertia can have a significant effect in a variety of ways. Load transfer from the rapidly loaded boundary of a body to the region of a crack edge can occur by means of stress waves. Likewise, a rapidly running crack emits stress waves, which can be geometrically reflected or scattered back to the region of the crack. Dynamic fracture is a significant failure of structures subjected to impulsive loads such as impact, explosion, thermo‐mechanical shock, or earthquake waves in their operating environments. The dynamic fracture of ductile metals is known to occur through the nucleation and growth of microscopic voids. As the voids grow, the surrounding metal is plastically deformed to accommodate the change in void volume. In the presence of high loading rates, the dynamic fracture process exhibits some peculiarities that differ from those of static fracture. In dynamic fracture theory, the material deformation close to crack tips is extremely large, leading to significant changes of local elasticity, referred to as hyperelasticity. The fundamental theory of dynamic fracture is well documented by Freund (1990). Dynamic fracture deals with structural fracture on a timescale for which inertial resistance of the structure to motion is significant. Material inertia can have a signi­ ficant effect in a variety of ways. These were described by Freund (1986) based on the following ground: A rapidly running crack emits stress waves which can be geometrically reflected or scattered back to the region of the crack. It is through such waves that a rapidly running crack senses the nature of the imposed loading on the body through which it runs, as well as the configuration of the body. Material inertia may also lead to effects more subtle than those associated with load transfer. Crack tip fields are usually distorted from their equilibrium forms during rapid crack growth. Inertial resistance to motion on a very small scale near the edge of a

Handbook of Structural Life Assessment, First Edition. Raouf A. Ibrahim. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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crack may make the material appear more resistant to separation than it is due to its strength alone. There is a wide range of physical mechanisms by which materials separate on the scale of material microstructure and, in cases where multiple mechanisms are competing, inertial effects can have an influence on which is operative. The effect of material inertia and strain rate sensitivity on the initiation and propaga­ tion of fracture from a pre‐existing crack tip has been a central issue in the dynamic failure of engineering materials under stress‐wave‐dominated loading conditions (Lee and Prakash, 1999; Rosakis and Ravichandran, 2000). For a material particle at a small distance away from the moving crack tip, the local stress field will depend not only on the instantaneous values of the crack tip speed and stress intensity factors, but also on the past history of these time‐dependent quantities. For cracks propagating along curved paths the stress field is also expected to depend on the nature of the curved crack path (Liu and Rosakis, 1994). Dynamic fracture in solids has attracted the interest of engineers and physicists due to its technological interest and to inherent scientific contribution. In his research monograph, Ravi‐Chandar (2004) indicated that rapidly applied loads are encountered in a number of technical applications such blasting, min­ ing, and accidental conditions. The theory of peridynamics depends crucially upon the non‐locality of the force interactions and does not explicitly involve the notion of deformation gradients. It is based on integral equations and thus does not require spatial derivatives to be evaluated within the structure body. Since partial derivatives do not exist on crack surfaces, the classical equations of continuum mechanics cannot be applied directly when such features are present in the structure. This is the main motive behind the development of a new theory known as peridynamics, which treats internal forces within a continuous solid as a network of pair interactions, similar to springs, which can be nonlinear. The purpose of this chapter is to present the basic ingredients of the theories of dynamic fracture and peridynamics together with their applications. This chapter is organized as follows. Section 3.2 introduces the theory of fracture dynamics. This includes the instability of cracks and the definition of dynamic stress intensity factor together with early controversies pertaining to the prediction of instability speed of cracks. The basic features of dynamic fracture will be discussed as proposed by physicists in terms of birth, childhood, and crisis. This is followed by introducing the formation of crack microbranching, which will ultimately lead to material failure. The role of molecular dynamics will be outlined in an attempt to understand the conditions leading to the rapid creation of displacement discontinuities. Dynamic crack propagation using optical caustics is then introduced as a useful method for high speed photography applications. The applications of the theory of dynamic fracture to metal and composite structures will be treated in Sections 3.3 and 3.4, respectively. Section 3.5 is devoted to the theory peridynamics and its applications for damage prediction. This section will cover numerical simulations and horizon convergence. Applications of peridynamics will include metal structures, concrete structures under extreme loading, composite structures, and electronic packages. This chapter is closed by conclusions and closing remarks.

Dynamic Fracture and Peridynamics

3.2  Fracture Dynamics 3.2.1  Features of Dynamic Fracture

Cracks form at the atomic scale, extend to the macroscopic level, are irreversible, and travel far from equilibrium (Marder and Fineberg, 2004). There is a very special set of forces between atoms as reported originally by Slepyan (1981). These forces make it possible to develop analytical solutions for cracks moving in lattices. The behavior of cracks in these models has the following three features proposed by Marder and Gross (1995), Heizler et al. (2002) and Marder and Fineberg (2004): Birth: There is a range of velocities starting at zero until around 20% of the speed of sound at which steady crack motion is forbidden. Above this range crack motion becomes possible. Childhood: Above the birth range of velocity, a steady stable crack motion is allowed and perfectly stable. At exactly the same externally applied stress, however, a stationary crack could also be stable. Crisis: Above a critical velocity steady crack motion becomes unstable. As the crack speeds up, the relativistic contraction discovered by Yoffe (1951) becomes more and more important, until eventually horizontal bonds above the crack line begin to snap. The results of numerical simulations are shown in the upper right of Figure 3.1. The crack might decide to build tree‐like patterns of sub‐surface cracks once steady motion has become impossible. Figure 3.1 also shows the computer simulations of the evolution of crack velocity in a simple model at the atomic scale, showing a transition between smoothly moving cracks and a violent branching instability that is similar to experiment. The transition is a function of the energy stored per unit length to the right of the crack. 0.5

Velocity

0.4 0.3 0.2 0.1 0.0

0

100

200

Time

Figure 3.1  Computer simulations in a simple model at the atomic scale showing a transition between smoothly moving cracks and a violent branching instability that is similar to experiment. The transition is a function of the energy stored per unit length to the right of the crack. (Marder and Fineberg, 2004)

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The use of continuous fiber reinforced titanium alloy composites within aerospace engines offers the possibility of large reductions in the component section and, as a consequence, large savings in engine weight. The sub‐critical crack growth and cata­ strophic failure in a continuous SiC fiber reinforced Ti‐6A1‐4V composite system was studied by Ibbotson et al. (1991). The crack growth resistance of this type of composite system is known to be dependent on the crack size, and is controlled by the number of fibers bridging the crack faces. Physically, initial crack growth occurs rapidly through the titanium alloy matrix. Increasing crack length leads to a greater number of fibers bridging the crack faces and hence to an increase in crack closure forces. These effects were found to result in a reduction in crack tip stress intensity even though the crack length is increasing. This reduction in crack tip stress intensity produces reduced crack growth rates, da/dN, with increasing crack length. For this particular composite system, it was found that dominant mode‐I cracks are produced, and these cracks extend until catastrophic failure, under the test conditions employed. A series of tests measured the matrix crack lengths using the direct current potential drop techniques. The use of these techniques coupled with low test frequencies allowed the detection of single fiber failure events. The tests highlighted the role of applied stress intensity range, ΔK, in promoting first fiber failure. They also indicated the subtle balance that can exist for stable/unstable crack growth transitions under cyclic loading. The potential of titanium matrix composites (Ti MMCs) for the transition from stable to unstable fatigue crack growth was experimentally characterized and reviewed by Bowen (1996). The transitional K0,max above which unstable fatigue crack growth occurs was predicted and compared with experimental measurements of titanium Tiβ21s/SCS‐6 composite by Liu and Bown (1999). Figure  3.2 shows the dependence of transitional K0,max on the initial‐notch‐to‐specimen‐width ratio a0/W for different values of fiber bundle strengths, σu. The solid circle points represent test results of unstable fatigue 50 1 40 K0,max MPa√m

132

2 3

30

4 5

20

10

0 0.0

0.1

0.2

0.3

a0 /W

Figure 3.2  Dependence of the transitional K0,max on the initial‐notch‐to‐specimen‐width ratio a0/W as predicted (lines) and measured (symbols) in 4‐mm‐wide titanium Tiβ21s/SCS‐6 composite specimens under bending: Predicted line 1: u 4.0 GPa, 2: u  3.5 GPa, 3: u  3.0 GPa, 4: u  2.5 GPa, and 5: 2.0 GPa. ⚬: stable, ⦁: unstable. (Liu and Bown, 1999) u

Dynamic Fracture and Peridynamics

crack growth, while open circle points belong to test results of stable fatigue crack growth. It is seen that the predicted results reveal that the transitional K0,max decreases with a decrease of a0/W, and in agreement with experimental trends. The predicted results shown in Figure 3.2 suggest that the in‐situ fiber bundle strength in the Tiβ21s/ SCS‐6 composite is approximately 3.5 GPa when compared with experimental measurements. It was indicated that an appropriate fiber strength distribution is essential to maintain optimized fatigue crack growth resistance in Ti MMCs. A two‐parameter criterion of body failure in unstable crack propagation was formulated on the power balance basis for the Griffith two‐dimensional problem by Matvienko (2003, 2006) and Matvienko and Priymak (2006, 2007). 3.2.2  Instability of Cracks and Microbranching

The mechanics of crack tip plasticity in dynamic crack growth influences two modes of dynamic fracture: (a) cleavage and (b) microvoid nucleation, growth, and coalescence. The relationship between the crack driving force and the crack tip speed was developed in terms of crack tip plastic fields by Freund (1987). The problem was treated by Mataga et al. (1987) using both the continuum theory of viscoplasticity and dislocation dynamics. In each approach the crack is traveling through the material with a relatively high density of pre‐existing mobile dislocations. The dependence of the critical stress intensity factor on the instantaneous crack tip speed by K Id (a ) was obtained by Freund (1986). For crack face loading, the tensile stress intensity factor was given in the form:

K Id t

C I * 2 cd t (3.1)

2(1 2 ) , ν is Poisson’s ratio, cd is the propagation speed of plane tensile (1 ) waves, t is the time and σ* is the acting normal stress. The dynamic fracture toughness is in general a material‐dependent function of the crack tip speed. A dynamic fracture criterion was proposed by Freund (1990) in the form where C I



K Id (a(t ), t , Load ) K D

(3.2)

where a(t) is the time‐dependent crack length, a (t ) is the crack velocity, and K Id is d the instantaneous stress intensity factor. K I measures the strength of the near‐tip fields, which drive crack propagation. KD is the dynamic fracture toughness representing the resistance of the material to crack propagation. Three numerical techniques are used in structural mechanics to extract fracture parameters for linear elastic, nonlinear, and dynamic fracture mechanics. These tech­ niques were reviewed by Beskos (1987) and include the finite element, finite difference, and boundary element. These algorithms were used for determining two‐ and three‐ dimensional stress intensity factors, crack opening displacements, J‐integrals for two‐dimensional, stable crack growth, ductile fracture, and the generation mode for obtaining dynamic elastic fracture parameters. Finite difference and boundary element methods were used for analyzing the elasto‐dynamic and elastic‐plastic dynamic states in fracturing two‐ and three‐dimensional problems. As the fracture energy approaches zero, a crack propagating at its asymptotic velocity is equivalent to a disturbance moving

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along a free surface. Stroh (1957) predicted the crack’s limiting velocity to be the Rayleigh wave speed, VR, which is the highest speed at which a wave can move along a free surface. Dynamic stress intensity factor was determined for homogeneous materials using different numerical algorithms. These methods include Lagrangian finite difference method (Chen, 1975, Lin and Ballmann, 1993), modified path‐independent J‐integral (Kishimoto et al., 1980), explicit time scheme finite element method (Murti and Valliappan, 1986), and time domain boundary element method with singular quarter‐point bound­ ary elements (Dominguez and Gallego, 1992; Fedelinski et  al., 1994; Fedelinski, 2004; Sladek et al., 1997, 1999). The mixed mode problem of a semi‐infinite plate containing an edge crack under an impact loading was considered by Lee and Freund (1990) who determined dynamic intensity stress factors through linear superposition of several stress wave propagation solutions. In studying fracture behavior in two‐ and three‐ dimensional cracked bodies under impact loading, Enderlein et al. (2003) used the finite element method to evaluate pure mode‐I dynamic stress intensity factors. Early studies of crack speed revealed some discrepancies. For example, Yoffe (1951) predicted that the instability speed of cracks is about 73% of the Rayleigh wave speed, VR, as the circumferential hoop stress exhibits a maximum at an inclined cleavage plane for crack speeds beyond this critical crack speed (see also Broberg, 1990; Freund, 1990). On the other hand, experimental measurements showed that the critical instability speed can be much lower than that value for different types of materials. For example, Fineberg et al. (1991) and Sharon et al. (1995) found experimentally that the instability speed is about one third of the Rayleigh wave speed. Later, Gao (1993) showed that Yoffe’s model is consistent with a criterion of crack kinking into the direction of maxi­ mum energy release rate. Some important features of dynamic fracture were described by Ashurst and Hoover (1976) using atomistic simulations of fracture. The simulation comprised only 64×16 atoms with crack lengths around ten atoms. Abraham et al. (1994) performed molecular‐ dynamics1 simulations of fracture in systems of up to 500,000 atoms. The dynamic energy release rate of a rapidly moving crack was found to enhance the possibility for the crack to split into multiple branches at a critical speed of about 50% of the Raleigh speed, as indicated by Eshelby (1969) and Freund (1972). Marder and Gross (1995) presented an analysis based on the discreteness of atomic lattice (Slepyan, 2002) and found that instability occurs at a speed similar to those indicated by the models of Yoffe, Eshelby, and Freund. Abraham et al. (1997) proposed that the onset of instability can be understood from the point of view of reduced local lattice vibration frequencies due to softening at the crack tip. Later, Abraham (2005) described the onset of the instability in terms of the secant modulus. Close to crack tips, material deformation was found to be extremely large, leading to significant changes of local elasticity, referred to as hyperelasticity. Gao (1996, 1997) provided insight into the reduced instability speed based on the concept of hyperelasticity. It was indicated that the atomic bonding in real materials tends to soften with increasing strain, leading to the onset of instability when the crack 1  Molecular dynamics deals with studying the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamical evolution of the system. In the most common version, the trajectories of atoms and molecules are determined by numerically solving the equations of motion for a system of interacting particles, where forces between the particles and their potential energies are calculated using interatomic potentials or molecular mechanics force fields.

Dynamic Fracture and Peridynamics

speed becomes faster than the local wave speed. Buehler and Gao (2007) emphasized that the hyperelasticity is the key to understanding the existing discrepancies between theory, experiments, and simulations on dynamic crack instability. As indicated in Chapter 1, the crack tip stress field is known to have an inverse square root singularity at the crack tip, with the angular variation of the singular field depend­ ing weakly on the instantaneous crack tip speed and with the instantaneous stress intensity factor being a scalar multiplier of the singular field. The influence of the transient process on the stress field in the immediate vicinity of the crack tip during non‐steady growth was studied by Freund and Rosakis (1992) and Liu and Rosakis (1994). Transient mixed‐mode elasto‐dynamic crack growth along arbitrary smoothly varying paths was studied by Liu and Rosakis (1994). A representation of the crack tip fields in the form of an expansion about the crack tip was described in terms of powers of radial distance from the tip. The higher order coefficients of this expansion were found to depend on the time derivative of crack tip speed, the time derivatives of the two stress intensity factors, and the instantaneous value of the local curvature of the crack path. This representation was used to interpret some experimental observations, with the conclusion that the higher‐order expansion provides an accurate description of crack tip fields under fairly severe transient conditions. Some experimental observations of fracture mechanics were reported by Fineberg and Marder (1999). It was shown that once the flux of energy to a crack tip passes a criti­ cal value, the crack becomes unstable, and it propagates in increasingly complicated ways. As a result, the crack cannot travel as quickly as theory predicted, fracture surfaces become rough, it begins to branch and radiates sound, and the energy for the crack motion increases considerably. Earlier, Ravi‐Chandar and Knauss (1984a, 1984b, 1984c, 1984d) showed that cracks do not reach the terminal velocity predicted by theory, and that they have an unexplained instability at a critical velocity. When energy flux to a crack tip passes a certain critical value, efficient steady motion of the tip becomes unstable to the formation of microcracks that propagate away from the main crack. According to Fineberg and Marder (1999) as the crack undergoes a hierarchy of insta­ bilities, the ability of the crack tip to absorb energy is enormously increased. The energy flux through the crack tip was assumed to be converted into fracture energy with a time delay, τ, due to the development of a viscoelestic process zone in front of the crack tip as demonstrated by Alvis and Lobo (2006). This delay was found to exist, and has a well‐defined timescale, which is a characteristic property of the mate­ rial. Its magnitude was found to be of the order of the viscoelastic relaxation time of the sample material under local fracture conditions. The onset of instability observed in the velocity of dynamic crack growth is due to the time delay. This time delay factor, τ, implies the possibility of deriving an equation for the crack growth velocity in the form of a logistical map equation. Alvis and Lobo (2006) showed that the most familiar mod­ els in fracture mechanics are intrinsically incomplete. A simple case of dynamic crack growth of a semi‐infinite body with plane strain condition was to derive an expression for the crack velocity in the form of a logistic equation and map. It is important to understand the formation of crack microbranching, which will ultimately lead to the failure of the material. The modeling of crack microbranching for materials undergoing dynamic fracture where branching is understood as the phenom­ enon where a single propagating crack is divided into two or multiple crack branches. Microbranching is a form of instability of fracture dynamics (Fineberg et  al. 1996).

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The phenomenon of microbranching instability is an important problem that has not been fully resolved in the fracture of brittle amorphous materials. The dynamics and structure of simple rapid cracks in a brittle material are well understood (Freund, 1990; Marder, 1991; Sharon and Fineberg, 1999; Sharon et  al., 2002; Goldman et  al., 2010; Bouchbinder et al., 2010b; Livne et al., 2010). The onset of microbranching was exam­ ined by Boué et al. (2015) who measured the real‐time dynamics and symmetries of the strain fields produced by rapid tensile cracks in brittle gels. It was found that once a simple tensile crack is subjected to shear perturbations, cracks undergo the micro­ branching instability above a finite velocity‐dependent threshold. A distinct relation between the microbranching and the oscillatory instabilities of rapid cracks was estab­ lished. It was shown that a finite mode‐II perturbation is directly correlated with the transition from a straight crack to microbranching. This transition is hysteretic over a range of crack velocities, where both microbranches and straight cracks coexist. Microbranching is an intrinsic instability of dynamic fracture, for a velocity less than the critical propagation velocity associated with the onset of the oscillatory instability. In real three‐dimensional materials, tensile cracks form a planar surface whose edge is a rapidly moving one‐dimensional singular front. It was found that perturbations to a crack front in a brittle material result in long‐lived and highly localized waves referred to as front waves (Sharon and Fineberg, 1996; Sharon et al., 2001). These waves exhibit a unique characteristic shape and propagate along the crack front at approximately the Rayleigh wave speed (Ramanathan and Fisher, 1977; Morrissey and Rice, 1998, 2000). These front waves are intrinsically three‐dimensional, and cannot exist in conventional two‐dimensional theories of fracture (Freund, 1990). Front waves can transport and dis­ tribute asperity‐induced energy fluctuations throughout the crack front. They may help to explain how cracks remain a single coherent entity, despite repeated interactions with randomly dispersed asperities. It was shown by Sharon and Fineberg (1996) that front waves can be generated by either externally placed asperities or, beyond the critical velocity for the microbranching instability, spontaneously. They were found to be initiated by localized microscopic branching events, which effectively create a local perturbation of the fracture energy. Front waves arriving at sample edges are reflected, with insignifi­ cant loss of amplitude. They also form mirror images on opposing crack faces. Thus the tracks are formed by deflection of the crack front normal to the fracture plane. An overview of the dynamics of fast fracture in brittle amorphous materials was pre­ sented by Fineberg (2006). The review provided some details on the numerous features commonly observed in dynamic fracture resulting from an intrinsic (microbranching) instability of a rapidly moving crack. This instability was found to result in large velocity oscillations, the formation of non‐trivial fracture surface structure, a large increase in the overall fracture surface area, and a corresponding sharp increase of the fracture energy with the mean crack velocity. It was demonstrated that the loss of translational invariance resulting from crack‐front interactions with localized material inhomoge­ neities causes both localized waves that propagate along the crack front and the acquisi­ tion of an effective inertia by the crack. Crack‐front inertia coupled with the microbranching instability provided an explanation of the chain‐like form of the micro­ branch induced patterns observed both on and beneath the fracture surface. At high levels of stress intensity factor (e.g. at a round notch) a crack propagates at higher stress levels, and sequential crack branching can occur (Bowden et al., 1967). An energy‐based fracture model for multiple crack‐branching, or crack tip shattering, was

Dynamic Fracture and Peridynamics

proposed and formularized for the mode‐I crack tip condition in isotropic solids by Xie et al. (2013). An analytical solution of the energy release rate for multiple crack‐branching from a mode‐I crack tip was developed, and cracking angles were analytically derived. It was shown that Griffith’s theorem and conservation law can be applied to both the conventional mode‐I crack extension and multiple crack‐branching. The predicted mul­ tiple crack‐branching phenomena were found to be in good agreement with the experi­ mental observations reported in the literature. Cylindrical vessels with artificial surface defects under internal pressure were tested to study patterns of rapid crack propagation and branching in carbon steel (Alexeev et al., 2013). It was found that macroscopic frac­ ture patterns are common for the model made of polymethylmethacrylate and structural materials made of carbon steel materials. In carbon steel, fracture with rapid crack propagation follows the trans‐granular cleavage mechanism. The lateral dimension of the crack formation zone was found to increase along the crack path and to decrease immediately before branching. Crack branching in both materials occurs when its veloc­ ity V reaches its limiting value V* at which the flow of elastic strain energy entering the crack tip G exceeds the energy G* expended by the material to hold off the growth of the single crack (i.e. at G G* as a necessary condition and V V* as sufficient condition). The concept of mixed‐phase simulation with fracture‐path prediction mode was introduced by Nishioka et al. (2001) using a postulated propagation direction criterion together with experimentally obtained crack propagation history. An automatic moving finite element method that incorporates the Delaunay automatic triangulation was developed. The mixed‐phase simulation with fracture‐path prediction mode was performed for mixed‐mode impact fracture tests. Various dynamic fracture parameters were evaluated by the path‐independent dynamic J‐integral. It is known that higher loading rates would result in an increase of the velocity of crack propagation through the body. This would in turn increase the supply of energy flux to the crack tip and enhance the appearance of microbranching, eventually leading to macro‐cracks and resulting in the ultimate failure of the specimen. The cracks were modeled using the finite element method (Linder and Armero, 2007) and the method was extended to dynamic fracture problems by Armero and Linder (2009) and Linder and Armero (2009). Later, a numerical approach was developed by Raina and Linder (2010) to describe the complex physical phenomena associated with crack branching in brittle materials. This was achieved by numerically modeling the failure zones within the individual finite elements based on the concept of the embedded finite element method. This approach resulted in the redundancy of the branching criterion based on crack tip velocity, and so both microbranching and macrobranching can be modeled. Dynamic fracture deals with conditions leading to the rapid creation of displacement discontinuities (opening and/or shear) resulting in the creation of new surfaces. Dynamic shear localization deals with conditions leading to the rapid creation of velocity (displace­ ment gradient/strain rate) discontinuities across narrow rapidly expanding regions, accompanied by rapid loss of stress carrying capability and intense heating. Shear bands often serve as sites of initiation and growth for subsequent dynamic fracture. Dynamic fracture in brittle materials involving the limiting speed of cracks, crack tip instabilities, and crack dynamics at interfaces were studied using large‐scale molecular dynamics by Buehler and Gao (2007). The local elasticity was found to govern the dynamics of fracture, in which case the assumption of linear elastic material behavior becomes insufficient to describe the physics of fracture, as indicated by Buehler et al.

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(2003) and Buehler and Gao (2004, 2006). The dynamics of fracture that lead to material failure were found to be governed entirely by the material behavior at the smallest scales (Buehler et  al., 2003; Fineberg, 2003). Hyperelastic deformation near a crack tip was found to provide explanations for a number of phenomena including the mirror‐mist‐hackle instability widely observed in experiments, as well as supersonic crack propagation in elastically stiffening materials. The relation of stress and strain in real solids was found to be strongly nonlinear near a moving crack tip. Buehler and Gao (2006, 2007) showed that hyperelasticity plays an important role in dynamic crack tip instabilities. It was found that the dynamic instability of cracks can be regarded as a competition between different instability mechanisms controlled by local energy flow and local stress field near the crack tip. The result of a large‐scale molecular dynamics simulation illustrating the mirror‐mist‐hackle transition is shown in Figure 3.3. Figure 3.3(a) shows the transi­ tion process of the dynamic mirror‐mist‐hackle as the crack speed increases, while Figure 3.3(b) shows the time evolution of the crack velocity normalized by the Rayleigh wave speed. Mirror

Mist

0%

30%

50% Hackle

1 0.8 Tip speed vt /cR

138

0.6

50%

0.4

30%

0.2 0

0

200

400

600

800

Non-dimensional time t

Figure 3.3  Crack propagation showing: (a) the dynamical mirror‐mist‐hackle transition as the crack speed increases, and (b) the crack velocity history (normalized by the Rayleigh‐wave speed). (Buehler and Gao, 2007).

Dynamic Fracture and Peridynamics

Fracture surfaces in brittle materials usually have the feature of mirror‐mist‐hackle. This feature is characterized by the crack face morphology changes as the crack speed increases, and is referred to as dynamic instability of cracks. Up to a critical speed of about one third of the Rayleigh wave speed, the crack surface is atomically flat (mirror regime). For higher crack speeds the crack starts to roughen (mist regime) and eventu­ ally becomes very rough (hackle regime), accompanied by extensive crack branching and perhaps severe plastic deformation near the macroscopic crack tip. The phenom­ enon of mirror‐mist‐hackle was found to be a universal behavior that appears in various brittle materials, including ceramics, glasses, and polymers. This dynamical crack instability was also observed in computer simulations performed by Abraham et al. (1994), Marder and Gross (1995), Gumbsch et al. (1997), Holland and Marder (1998), and Deegan et al. (2003). The problem of representing crack speed behavior in terms of a stress intensity factor was studied by Knauss and Ravi‐Chandar (1986). The study confirmed the fact that instantaneous crack speed depends on the past history of the crack tip stress field. Dynamic fracture mechanisms and the methods of modeling were reviewed by Ravi‐ Chandar and Knauss (1998) and Ravi‐Chandar (2003a, 2003b). Dynamic fracture may involve various mechanisms of nonlinear damage, many of which act at the microscopic scale, as indicated by Cox et al. (2005). The material point method, a particle method based on finite elements, was used in fracture dynamic analysis. The method was used as an alternative to dynamic finite element methods to simulate large material deforma­ tions since there is no re‐meshing required by the material point method. Crack tip parameters, dynamic J‐integral vector, and mode‐I, mode‐II, and mode‐III stress inten­ sity factors were calculated by Guo and Nairn (2006) from the dynamic stress solution. Failure phenomena in the presence of high local strain rates were classified into dynamic fracture and dynamic shear localization or adiabatic shear banding (Rosakis and Ravichandran, 2000). A common characteristic of these failure phenomena is the rapid loss of stress carrying capability in timescales such that inertial and/or material rate sensitivity effects are important. An assessment of research activities in dynamic failure mechanics was presented by Rosakis and Ravichandran (2000). The assessment included dynamic crack initiation and growth in brittle materials, elastic‐plastic solids, and heterogeneous solids, such as layered materials and composites, and adiabatic shear banding in ductile materials. 3.2.3  Experimental Techniques

Experimental investigations using direct optical techniques such as the method of caustics, photoelasticity, and coherent gradient sensing were conducted to measure KD(υ). For example, Brickstad (1983) conducted dynamic fracture experiments on a high strength steel where the crack velocity and boundary displacements were experi­ mentally measured and were used as input to a numerical analysis to deduce the variation of KD(υ). It was found that KD(υ) does not show a significant dependence on the crack acceleration, , consistent with experimental observations by Dally (1979). Kanazawa et al. (1981) obtained an extensive set of data concerning KD(υ) at various test temperatures for a Si‐Mn steel. Angelino (1977) obtained KD(υ) for SAE 4340 steel from boundary load and displacement measurements. Bilek (1980) used double canti­ lever beam specimens made of 4340 steel subjected to wedge loading and measured

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crack speed and boundary displacements. This information was used in the numerical solution of an analytical model in which the double cantilever beam specimen was modeled as a Timoshenko beam on a rigid foundation. It was shown that KD(υ) reaches a minimum before rapidly rising as a function of υ. This phenomenon, of KD(υ) going through a minimum at a non‐zero velocity, is of great significance in crack arrest consid­ erations. It was linked to material rate sensitivity by the work of Freund and Hutchinson (1985) and Freund et al. (1986). A dynamic drop‐weight test was conducted by Pandolfi et al. (2000) for assessing the cohesive models of fracture in dynamic crack growth. Coherent gradient sensing was used to study dynamic fracture in C300 maraging steel. The specimens were subjected to three‐point bend impact loading under a drop weight tower. High‐speed photographs of the coherent gradient sensing interferograms were analyzed to determine the crack tip location, the velocity, and the dynamic fracture toughness as a function of time. Examination of the specimens after testing revealed the fractography of the fracture surfaces, including the development of shear lips. Numerical simulations, using a three‐dimensional finite element model, validated a number of observed features such as the crack growth initiation time, the trajectory of the propagating crack tip, and the formation of shear lips near the lateral surfaces. For dynamic crack propagation in the two‐dimensional tensile mode, the predicted caustic curves and corresponding initial curves were studied within the framework of plane stress and small‐scale yielding conditions. These curves were found to have geometrical features, which are quite different from those for purely elastic crack growth. An attempt was made by Rosakis and Freund (1981) to include crack tip plasticity and inertial effects in the analysis underlying the use of the optical method of caustics to determine the dynamic stress‐intensity factors. Rosakis (1983) extended the applicability of the optical method of caustics to materials that exhibit substantial amounts of plastic deformation prior to or during crack growth. Shadow spot patterns were photographed with a high speed camera to study fracture initiation and dynamic fracture propagation in structural metals for which both plasticity and inertia effects cannot be considered negligible. The method of caustics was used to determine directly the value of the J‐integral at the tip of a stationary crack in an elastic‐plastic strain hardening material. The caustics obtained in this test are formed by reflection of light rays from points well within the crack tip plastic zone, conveying direct infor­ mation about the near tip plastic singularity. The dynamic crack propagation experi­ ments were performed using specimens of an austenitized, quenched, and tempered 4340 steel. Values of the dynamic stress intensity factor were determined by the use of the method of caustics. This method is useful for high speed photography applications. The method was first used in experiments involving very rapid crack propagation and stress wave loading by Kalthoff et al. (1976), Katsamanis et al. (1977), Theocaris (1978), and Goldsmith and Katsamanis (1979). It was assumed that the elastic stress field in the vicinity of a rapidly propagating crack tip has precisely the same spatial variation as the elastic stress field near the tip of a stationary crack, and the influence of inertial effects on the spatial distribution of the crack tip field was not taken into account. Kalthoff et al. (1978) intro­ duced an approximate correction factor to account for the error introduced when the static local field is used in the interpretation of caustic patterns. Rosakis (1980) presented the exact equations of the caustic envelope for elastic specimens containing rapidly

Dynamic Fracture and Peridynamics

growing cracks. He also presented the caustic equations for the case of mixed mode plane stress crack propagation. It was assumed that the deformation field near the propagating crack tip is dominated by the dynamic stress intensity factor, K 1d , and the stress field at a finite region near the crack tip can be approximated accurately by the elasto‐dynamic asymptotic singular solution. Numerical algorithms and experimental investigations play a central role in the study and understanding of the dynamic fracture process, where analytical solutions are difficult. Experimental investigations enable us to observe directly the instantaneous stress intensity factor in terms of crack speed. Rosakis et al. (1983) presented a description of dynamic crack propagation experiments on double cantilever beam specimens of high strength steel. Measurements of the crack tip deformation field and of crack speed during propagation were obtained using the optical method of caustics in reflection. The inherent time dependence of crack propagation experiments requires that many sequential measurements of field quantities be made in an extremely short time and in a way that does not interfere with the process to be observed. Dynamic crack propagation experiments were conducted on wedge loaded double cantilever beam specimens of an austenitized, quenched, and tempered 4340 steel by Rosakis et al. (1984) using the opti­ cal method of caustics. The instantaneous value of the dynamic stress intensity factor KdI was obtained in terms of crack tip velocity. Some fundamental aspects of dynamic crack growth in structural steels were described by Rosakis and Zehnder (1985). In particular, the role of material inertia and plasticity in the dynamic crack growth process was clarified. The extent of the region of three‐dimensionality of the crack tip stress field was examined by Rosakis and Ravi‐Chandar (1986) using reflected and transmitted caustics on plexiglass and high‐strength 4340‐steel compact tension specimens. At each thickness, measurements were taken at different values of distance r from the crack tip, ranging from r / h 0 to r / h 2, where h is the specimen thickness. The results showed that plane‐stress conditions prevail at distances from the crack tip greater than half the specimen thickness, while no significant plane‐strain region was detected. At r / h 0.5, the experimental results were found to be in good agreement with those predicted using the crack tip boundary‐layer solution of Yang and Freund (1986), and the numerical results by Levy et al. (1970). The value of the J‐integral for planar cracked structures of arbitrary geometry and loading using direct optical method of caustics was presented by Zehnder and Rosakis (1986). Zehnder and Rosakis (1988) and Zehnder et al. (1990) examined the dynamic fracture initiation and propagation in 4340 steel using the optical method of reflected caustics combined with high speed photography. During crack propagation, the crack tip velocity and stress intensity factor time records varied smoothly and repeatedly. The experiments showed that for the particular heat treatment of 4340 steel used, the dynamic fracture propagation toughness depends on crack tip velocity through a relation that is a material property. A relation between the caustic diameter, D, and the value of the J‐integral was obtained experimentally and numerically for a particular statically loaded specimen geometry of 4340 steel. During the high speed propagation of cracks, large temperature increases were detected at the crack tip due to the intense dissipation of plastic work. An experimental investigation of the temperature fields at the tip of dynamically propagating cracks in 4340 steel was performed by Zehnder and Rosakis (1990) using a focused array of high speed infrared detectors. Temperature fields

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were measured for cracks growing at speeds of 700–1900 m/s. Maximum temperature increases were found to be as high as 465°C. Some experimental investigations of dynamic crack initiation, propagation, and arrest were carried out based on the assumption that the deformation field near the propagating crack tip is K 1d ‐dominant (Krishnaswamy and Rosakis, 1991b). It was found that the analysis of caustics based on K 1d ‐dominance may not always adequately characterize the behavior of the deformation field in the vicinity of a transiently propagating dynamic crack tip. For this reason, Freund and Rosakis (1992) suggested that under fairly severe transient conditions, a representation of the crack tip field in the form of a higher‐order expansion (involving time derivatives of crack tip velocity and stress intensity factor) should be used to interpret the experimental observations. Later, Liu et al. (1993b) re‐examined the optical method of caustics by considering non‐uniform crack growth histories. The exact mapping equations of caustics and the initial curve equation were derived for a non‐uniformly propagating dynamic crack based on the theoretical results of Freund and Rosakis (1992) and Rosakis et al. (1991). These results allowed both the crack tip speed and the dynamic stress intensity factor to be arbitrary differentiable functions of time. They developed an explicit relation between the dynamic stress intensity factor, K 1d , and two geometrical dimensions of the caustic pattern. It was shown that the classical analysis of the caustics is a special case of this result under the condition of strict K 1d ‐dominance. 3.2.4  Dynamic Crack Propagation Using Optical Caustics

Consider a plate specimen of uniform thickness, h, in the undeformed state, its mid‐plane occupies the x1,  x2 plane of an orthonormal Cartesian coordinate system. As the specimen is subjected to applied loads, non‐uniform gradients in the optical path of light transmitted through it, or reflected from its surface, are generated. For a transparent specimen, the gradients in the optical path are due to non‐uniform changes in the thickness of the plate and also due to stress‐induced gradients in the refractive index of the material in the specimen interior. For an opaque specimen, the gradients in the optical path are due to non‐uniform surface elevations of the plate. A collimated beam of light is travelling in the x3 direction, normally incident on the plate, as shown in Figure 3.4. Under certain stress gradients, the reflected or refracted rays will deviate from parallelism and create an envelope in the form of a three‐dimensional surface in space. This caustic surface is the locus of points of maximum luminosity in the reflected or transmitted light fields. The deflected rays are tangential to the caustic surface. If a screen is positioned parallel to the x3 0 plane, so that it intersects the caustic surface, then the cross‐section of the surface can be observed on the screen as a bright curve (the caustic curve) bordering a dark region (the shadow spot). Suppose that the incident ray, which is reflected from or transmitted through point p(x1, x2) on the specimen, intersects the screen at the image point P(X1, X2). The (X1, X2) coordinate system is iden­ tical to the (x1,  x2) system, except that the origin of the former has been translated by a distance z0. The position of the image point P will depend on the gradient of the optical path change ΔS(x1, x2) introduced by the specimen, as well as on the distance z0, and was given by Rosakis and Zehnder (1985) by the mapping equation:

X

x z0

S x1 , x2 (3.3)

Dynamic Fracture and Peridynamics

(a)

Virtual screen

Specimen

(b)

Z0

x2

Z0

D/2

Crack front

r – r(z0)

Real screen

Specimen

x2 r – r(z0)

x2 x1

D/2 Crack front x2

x1 D/2

D/2

Figure 3.4  Caustic formation in (a) reflection, and (b) transmission. (Liu et al., 1993)

where X X i ei , x xi ei , i 1, 2, ei denotes a unit vector, and is a two‐dimensional gradient operator. Equation (3.3) describes the mapping of the points on the specimen onto the points on the screen. The resulting caustic curve on the screen is the optical mapping of the locus of points for which the determinant of the Jacobian matrix of mapping equation (3.3) is zero, which is a necessary and sufficient condition for the existence of a caustic curve. All points inside and outside this curve map outside the caustic (Rosakis and Zehnder, 1985). Since the light transmitted through or reflected from both the interior and the exterior of the initial curve maps only outside the caustic, the area within the caustic remains dark and is customarily referred to as the shadow spot. The shape of the caustic curve depends on the near‐tip normal displacement, u3, of the plate surface, initially at x3 h / 2 , where h is the undeformed specimen thickness. For transparent specimens the optical path change, ΔS, depends on both local changes in thickness and on local changes in the refractive index. The conditions of generalized plane stress dominate in thin cracked plates at distances from the crack tip larger than half of the specimen thickness (Rosakis and Ravi‐Chandar, 1986; Yang and Freund, 1985), which implies that if the initial curve is kept outside the near‐tip three‐dimensional zone, the resulting caustic could be inter­ preted as the basis of a generalized plane stress analysis. It was also assumed that the initial curve is always kept outside the plastic zone and the fracture process zone. Under these conditions, Rosakis (1992) obtained the following expression for the optical path difference: ˆ22 x1, x2 (3.4) S x1, x2 hc ˆ11 x1, x2 where c is a material constant given by

c

D1

E

n 1 E

c

for transmission for reflection



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where D1 is the stress optical constant, E and ν are Young’s modulus and Poisson’s ratio of the material, respectively, and cσ is the stress‐optical coefficient. ˆ11 and ˆ22 are the average stress components of the solid thickness. Consider a planar, mode‐I crack that grows through a two‐dimensional, homogeneous, isotropic, linearly elastic solid, with a non‐uniform speed υ(t), along the positive x1 direction. Note that (x1,  x2) is a coordinate system that translates with the moving crack tip. The scaled polar coordinate for longi­ tudinal and shear, (rl, θl) and (rs, θs), respectively, are defined by rl2, s t



2 l ,s

where

x12 t

2 l ,s 2

1

t x22 , t

l ,s

t

tan

l ,s

1

t x2

(3.5)

x1

, and cl and cs are the longitudinal and shear stress wave veloci­

cl2, s

ties of the elastic material, respectively. The modified method developed by Liu et al. (1993b) yielded the following expression for the dynamic stress factor D

2 2 3 l hcz0 F

K ld t

5/ 2

gl

l

2 g1

1

D

gl

l l

5/ 2

D

1 2

D

1 4

2  t DG2 4 2 l cl

G1

l l

5/ 2

D 2

D

(3.6) where the angle

X D

fl

l

l

Gl

D

D

l

D



is obtained from the root of the following trigonometric equation: f2

1 2

1 4

2 t X

1 2

1 4

2  t D G2

4 2 l cl

f1

4 2 l cl

G1

D

l

2

D

l

D

l l

(3.7) 2

D

where a dot denotes time derivative and the prime denotes differentiation with respect D to the argument l . The functions appear in equations (3.6) and (3.7) are given by the following expressions: G1

( D) l

g1

( D) l

2

( D) l

2 1 3 l

( D) l

1 3 l

f1 f2

g1

( D) l sin l( D ) l

2 g1

( D) l

tan

( D) l /2

( D) 2 sin 3 l , 3 2 ( D) 2 1 sec l 2 2 l

4 3sec

( D) l

2

cos

( D) l

, G2 g2

( D) l

1 cos ( D) l

2

( D) l

4 5sec

g2

( D) l

2 3sin 3

( D) l

2 g2

sin 5

2 2 sec l 3 ( D) l

2

( D) l

cos 5

( D) l

2

( D) l

2

tan

( D) l /2

( D) l

2

cos 3

( D) l

2

Dynamic Fracture and Peridynamics

The dynamic stress intensity factor corresponding to the classical analysis was deter­ mined by Liu et al. (1993b) and given by the expression

K ld t

C

D 5/2 (3.8) 10.7 z0 hc

where C(υ) is a function of the crack tip velocity given by the expression C

3.17

1 lF

g1

l

5/ 2

2 l

, F

D

4

l

2 s s

1 1

2 s 2 2 s

(3.9a,b)

The asymptotic stress around the tip of a non‐uniformly propagating dynamic crack was presented by Rosakis et al. (1991). An expression for r0/υt with respect to time after crack initiation and z0 was obtained in the form



r0 t

z0 h cs t

4 /5

3 c 4 2

1 1

2 s

1

I

/ cs 2 s

2/5

(3.10)

Figure 3.5 shows the dependence of the normalized dynamic stress factor with refer­ ence to the classical value on the radius non‐dimensional parameter (r0/υt) for different values of the crack velocity parameter υ/cs . Any value deviating from unity meas­ ures the estimated error incurred by the classical interpretation of caustics during crack growth. It is seen that as r0 / t 0 the classical approach becomes accurate.

Figure 3.5  Dependence of the normalized dynamic stress intensity factor with respect to the classical theoretical value on the radius non‐dimensional parameter (r0/υt) for Poisson’s ratio 0.3, different crack tip velocities (υ/cs). Square points (◽) are obtained according the classical analysis, circle points (⚪) according to the modified method. (Liu et al., 1993).

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Figure 3.6  Time evolution after crack initiation of the normalized dynamic stress intensity factor with respect to the classical theoretical value for different crack tip velocities (υ/cs), for 4340 steel Poisson’s ratio 0.3, /E 10 4 , z 0 2.0 m, h 0.01m, c s 3200 m / sec . Square points (◽) are obtained according the classical analysis, circle points (⚪) according to the modified method. (Liu et al., 1993)

However, as r0 / t , there are large deviations of the dynamic stress intensity factor, as predicted by the caustic approach from the theoretical values as indicated by the square points (◽). The normalized dynamic stress factor predicted by the modi­ fied approach (equations (3.6) and (3.7)) indicated by circle points (O) was found to less than 5%, error. This error was found to be caused by the assumption of circularity about the initial curve. Figure 3.6 shows the time evolution of the normalized dynamic stress factor with reference to the classical value on from the moment of crack initia­ tion for different parameters. This figure was generated for different values of the crack tip velocity parameter for material parameters corresponding to 4340 steel, z0 2.0 m and specimen thickness h 0.01 m. It is seen that the classical analysis of caustics becomes accurate only after a certain time from crack initiation. Figure 3.7 shows the same time evolution of the dynamic stress intensity factor ratio for different values of load level / E but for PMMA material parameters. It is seen that for a higher load level, the transient effect is much more significant than for the lower load level, especially at the time near the crack initiation. At a given time, and for fixed /E implies larger initial curve radii, as revealed from value of z0, higher values of equation (3.10). It is seen from these two figures that as t 20 s, big errors have been observed when the classical analysis is used. Note that the plane stress problem of a crack growing symmetrically from zero initial length at constant velocity under uniform remote tensile stress was analyzed by Broberg (1990). In the plane of deformation, the crack lies in the interval. Figure 3.8 shows the same time evolution of the dynamic stress intensity factor ratio for different values of z0. It shows that as z0 is decreased (the initial curve shrinks to the crack tip) the value of K Id caustic obtained by the classi­ cal analysis of caustics slowly approaches K Id theor .

Dynamic Fracture and Peridynamics

Figure 3.7  Time evolution after crack initiation of the normalized dynamic stress intensity factor with respect to the classical theoretical value for three different load levels /E, Poisson’s ratio 0.3, z 0 2.0 m, h 0.01m, c s 1200 m / sec, and crack tip velocity ratio / c s 0.3. Square points (◽) are obtained according the classical analysis, circle points (⚪) according to the modified method. (Liu et al., 1993)

Figure 3.8  Time evolution after crack initiation of the normalized dynamic stress intensity factor with respect to the classical theoretical value for different values of the distance from the screen z0; / E 10 4 , Poisson’s ratio 0.3, h 0.01m, c s 1200 m / sec, and crack tip velocity ratio / c s 0.3. Square points (◽) are obtained according the classical analysis, Circle points (○) according to the modified method. (Liu et al., 1993)

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The next two subsections present the application of fracture dynamic to structural elements made of metals and composites.

3.3  Fracture Dynamics of Metals A relationship between size and fracture probability for metals and ceramics was devel­ oped by Staverding (1971). This relationship is formally identical with Weibull’s law. However, it is not a consequence of the statistics of flow distribution but follows from a least action law for dynamic fracture. Staverding and Werkheiser (1971) showed that the dynamic fracture criterion for metals is equivalent to the failure criterion of a Voigt body containing a non‐Newtonian dashpot. Integration of the differential equation leads to the concept of a critical incubation time for fracture. Some consequences of the dynamic fracture law were applied to dynamic metal behavior under conditions of elevated temperatures and shock loads. The explanation of transition between ductile and brittle modes of fracture based on the incubation time concept was proposed by Gruzdkov et al. (2009). Based on this proposed concept the relation between the influ­ ences of different loading parameters (strain‐rate, temperature) on the mode of fracture was developed. 3.3.1  Spalling of Metals

Spall refers to flakes of a material that are broken off a larger solid body, and this can be produced by a variety of mechanisms such as projectile impact, corrosion, or excessive rolling pressure. Schmidt (1973) reported the results of a hollow cylindrical specimen loaded on the inner diameter by a polyethylene‐coated exploding wire, which resulted in a circumferential spallation pattern. Prior to fracture, the spall surface was found to undergo biaxial deformation with a total hoop strain of the same order of magnitude as the total radial strain. High speed photographs of the coated exploding wires and dynamic‐flash X‐rays of the specimens demonstrated the axi‐symmetry of both the loading and the spall phenomenon. The fundamental mechanical aspects of dynamic fracture in metals were presented by Meyers and Aimone (1983). Special attention was given to the problem of spalling produced by the interactions of shock and reflected tensile waves. The process of spalling was described as a sequence of nucleation–growth–coalescence of voids or cracks. Quantitative models predicting the extent of damage were compared with experimental observations. A number of metallurgical aspects of importance were dis­ cussed including failure initiation sites, crack propagation paths, strain‐rate‐dependent ductile‐to‐brittle transition, grain size effect and inter‐granular versus trans‐granular spalling. Of particular importance in iron and steels is the change in spall morphology when the 13 GPa stress is exceeded. Buchar et  al. (1985) examined the formation of failure under impact loading in its most frequently occurring form of spalling. The main parameters of the material, which characterize its fracture behavior in shock loading and their relationship with the conventional quantities of fracture mechanics were identified. Free surface velocity measurements showed that the spall strength of metals is not sensitive to the initial temperature or peak pressure of a shock‐wave pulse until the

Dynamic Fracture and Peridynamics

thermodynamic parameters of the damaged material are far from the melting point, and decrease sharply near it. Assuming localized melting, the expressions for threshold temperature and for the dependence of spall strength on temperature were derived by Bogash (1998). The threshold temperature and the spall strength were found to depend on the stress gradient in the shock pulse to a power of 1/3 for Al, Mg, Sn, Pb, Zn, and Mo. The nucleation, growth and coalescence of microvoids in pure aluminum, OFHC copper, and Ti‐6Al‐4V alloy were studied experimentally by Sun et al. (1998). It was also analyzed using numerical simulations based on the nucleation and growth model. Loading of very high strain rates in the range of 105–107/s was applied using an electric gun and a high power laser beam. The statistical data of voids in recovered specimens revealed that the void nucleation behaviors as well as the localization of the damaged band are markedly different in the three tested metals. The results of nucleation and growth model showed that the initial nucleation rate is material and strain rate sensi­ tive, and can be spread over 3 to 4 orders of magnitude from aluminum to titanium alloy. The number of voids in aluminum was reduced. Gas gun recovery experiments were used by Belak et al. (2005) to study incipient spallation fracture in light metals such as Al, Cu and V. The void size and spatial distribution were determined directly from X‐ray tomography. The single crystal samples showed a bimodal distribution of small voids with large (50–100 µm) well separated voids. The plastically damaged region sur­ rounding the large voids was quantified using optical and electron backscattering microscopy. Microhardness measurements indicated this region to be harder than the surrounding metal. 3.3.2  Dynamic Crack Propagation in Metals

The steady‐state dynamic crack propagation in elastic‐perfectly plastic solids under mode‐I plane stress and small‐scale yielding condition was studied numerically by Deng and Rosakis (1991) using the finite element method. The ratio of the crack tip plastic zone size to that of the element nearest to the crack tip is of the order of 1.6 × 104. It was found that the crack tip strain and velocity fields possess logarithmic singularities, which is consistent with the assumptions in the asymptotic analysis by Gao (1987). The asymptotic stress and deformation fields of dynamic crack extension in materials with linear plastic hardening under combined mode‐I and mode‐III were examined by Yuan (1994). Regardless of the degree of the mixed modes of the crack tip field and the strain‐ hardening, the in‐plane stresses under the combined mode‐I and mode‐III conditions were found to have stronger singularity in the whole mixed mode steady‐state crack growth than that of the anti‐plane shear stresses. The crack propagation velocity was found to reduce the singularities of both pure mode and perturbed crack tip fields. The domain of dominance of mode‐I asymptotic elasto‐dynamic crack tip fields was experimentally studied by Krishnaswamy and Rosakis (1991a, 1991b) for the cases of dynamically loaded stationary cracks as well as dynamically propagating cracks. Three‐ point bend specimens loaded dynamically using a drop‐weight tower were used. The results of two‐ and three‐dimensional elasto‐dynamic finite‐element simulations of the drop‐weight experiments revealed that the asymptotic elasto‐dynamic field is not an adequate description of the actual fields prevailing over any sizeable region around the crack tip.

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Dynamic shear loading of pre‐cracked plates may induce either dynamic tensile cracking or dynamic shear banding as a direct result of the competition between mate­ rial ductility, rate, and fracture toughness (e.g. Lee and Freund, 1990; Needleman and Tvergaard, 1995; Ravi‐Chandar, 1995; Zhou et al., 1996). The use of a coherent gradient sensing apparatus was explored by Lambros et  al. (1991) and Mason et  al. (1992) in studying dynamic fracture mechanics. The ability of the method to accurately quantify mixed‐mode crack tip deformation fields was tested under dynamic loading conditions. The coherent gradient sensing system’s measurements of KI and KII were found to be in good agreement with the predicted results. Later, Guduru et  al. (1998, 2001) used coherent gradient sensing in their studies of dynamic crack initiation in ductile steels. The dynamic crack initiation in ductile Ni‐Cr steel and 304‐stainless steel under differ­ ent loading rates was studied by Guduru et al. (1998), who developed dynamic failure criteria. Pre‐cracked steel specimens were subjected to impacting dynamic three‐point bending loading. During the dynamic deformation and fracture initiation process the time history of the transient temperature in the vicinity of the crack tip was monitored using a high‐speed infrared detector. The dynamic temperature trace in conjunction with the Hutchinson (1968a) and Rice and Rosengren (1968) solution was used to determine the time history of the dynamic J‐integral, Jd(t), and to establish the dynamic fracture initiation toughness, J cd . The study was extended by Guduru et al. (2001) to examine the initiation and propagation characteristics of dynamic shear bands in C300 maraging steel. The optical technique of coherent gradient sensing was employed to study the evolution of the mixed mode stress intensity factors. The temperature field evolution during the initiation and propagation of the shear bands was detected using a two‐dimensional high speed infrared camera. The infrared images revealed the transi­ tion of crack tip plastic zone into a shear band and also captured the structure of the tip of a propagating shear band. Figure 3.9 shows a sequence of coherent gradient sensing images, illustrating the propagation of the shear‐band following its initiation. Figures 3.10 and 3.11 show the advance of the shear‐band and its velocity, respectively, for five different impact speed experiments with velocity uncertainty of ±80 m/s. It is seen that the shear‐band velocity is highly transient and is a function of the impact speed. Furthermore, it is seen that the maximum shear‐band speed is about 1100 m/s. The development and evolution of the mode‐II plastic zone at the tip of the initial crack was visualized using imaging of the temperature field detected by the infrared camera at the tip of the fatigue crack on the left‐hand side of Figure 3.12. For impact speed 35 m/s, Figure  3.12 shows a sequence of thermal images revealing the development of the temperature field as a function of time. The black lines on these images were artificially superimposed to represent the approximate location of the initial crack tip. The posi­ tion of the crack was inferred from the temperature patterns. It is seen that plasticity gradually builds up, with the characteristic shape of a mode‐II dominated plastic zone. Starting at about 21 μs, the central hot region extends to the right, as indicated by the contour lines, signifying the process of shear localization. The measured highest tem­ perature rise within the plastic zone when this happened was found to be at least 80 K. Failure mode switching and failure mode transitions were replicated numerically by Li, S. et  al. (2001, 2002) in an asymmetrically impact‐loaded pre‐notched plate. For intermediate impact speed (25 m/s < V ≤ 30 m/s), the numerical results showed that a cleavage crack initiates from the tip of the dynamic shear band, indicating a dominance of brittle failure mode, and a failure mode switch. For high impact velocities (V > 30 m/s),

Dynamic Fracture and Peridynamics

Figure 3.9  A sequence of snapshots of coherent grading sensing showing shear band propagation in C300 Maraging steel. (Guduru et al., 2001)

Shear-band advance (mm)

20

36.0 m/s 38.0 m/s 35.3 m/s 32.0 m/s 39.0 m/s

16

12

8

4

0

0

10

20

30

40

50

Time (μs)

Figure 3.10  Time evolution of shear‐band advance for different for different impact speeds in C300 Maraging steel. (Guduru et al. 2001)

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Handbook of Structural Life Assessment

36.0 m/s 38.0 m/s 35.3 m/s 32.0 m/s 39.0 m/s

1200

V (m/s)

152

800

400

0

0

10

20

30

40

50

Time (μs)

Figure 3.11  Time evolution of shear‐band velocity as a function of time for different impact speeds in C300 Maraging steel. (Guduru et al. (2001)

Figure 3.12  A sequence of thermal images showing the transition of crack tip plastic zone to a shear band in C300 Maraging steel. (Guduru et al., 2001)

the numerical results revealed that a dynamic shear band penetrates through the speci­ men without a trace of cleavage‐type fracture, which is a ductile failure mode. The numerical simulation provided the details of the adiabatic shear band to a point where the periodic temperature profile inside the shear band at μm scale can clearly be seen. Experimental investigations to characterize the dynamic fracture characteristics of 2024‐T3 aluminum thin sheets with thicknesses of 1.63–2.54 mm were conducted by Owen et al. (1998), Zhuang (1998), and Chow (2001) to determine the dependence of the dynamic crack initiation toughness of aluminum 2024‐T3 on loading rate. The

Dynamic Fracture and Peridynamics

critical dynamic stress intensity factor K cd was determined over a wide range of loading rates. The dynamic crack propagation toughness, KD, was measured as a function of crack tip speed using high sensitivity strain gages. A dramatic increase in both K cd and KD was observed with increasing loading rate and crack tip speed, respectively. These rela­ tions were found to be independent of specimen thickness over the range 1.5–2.5 mm. d The dynamic initiation fracture toughness K IC of a bulk metallic glass system 2 (Vitreloy‐1) and its β‐phase composite were investigated by Rittel and Rosakis (2005). Both the coherent gradient sensing interferometry and one‐point impact techniques d revealed very similar trends in the K IC –K Id relationship for Vitreloy‐1. A drastic increase in initiation toughness with the stress intensity rate was reported. Meng et al. (2008) presented the fractographic evolution from vein pattern, dimple structure, and then to periodic corrugation structure, followed by microbranching pattern, along the crack propagation direction in the dynamic fracture of a tough Zr41.2Ti13.8Cu12.5Ni10Be22.5 (Vit.1) bulk metallic glass (BMG) under high velocity plate impact. A model based on fracture surface energy dissipation and void growth was proposed to characterize this fracture pattern transition. It was found that once the dynamic crack propagation velocity reaches a critical fraction of Rayleigh wave speed, the crack instability occurs together with crack microbranching. An experimental study of the temperature–time laws of fracture of Cd, Sn, and Zn was presented by Pavlovskii et  al. (1991). The experiments were conducted under thermal shock produced using pulses of relativistic high‐current electron beams of sub‐microsec­ ond range as the temperature of the samples is varied. It was found that dynamic fracture is a thermo‐fluctuation process, and the dependence of the fracture time on the absorbed energy has an exponential form. The ductile–brittle transition temperatures of modified 9Cr‐1Mo steel welds from two different weld positions, namely down hand (1G) and overhead (4G), were evaluated under dynamic loading by Moitra et al. (2008). The refer­ ence temperatures were found to be higher for the 4G position than the 1G position. Xu, Z.J. and Li (2011) studied the variation of dynamic fracture toughness with the loading rate for two high‐strength steels, namely 30CrMnSiA and 40Cr. Experiments were conducted on the modified split Hopkinson pressure bar apparatus, with striker velocities 9.2–24.1 m/s and a constant value of 16.3 m/s for 30CrMnSiA and 40Cr, respectively. It was observed that for 30CrMnSiA, the crack tip loading rate increases with the increase of the striker velocity, while the fracture initiation time and the dynamic fracture toughness simultaneously decrease. A model of dynamic tensile frac­ ture of metals applicable for a wide range of strain rates was developed by Mayer (2013). The model considered a stage of thermal‐fluctuation nucleation of voids and stages of void growth and aggregation. The model was described in terms of specific free energy of the metal surface and a distribution parameter for weakened zones of material. These parameters were obtained for Cu, Al, Fe, Ti, Ni, and Mo by fitting with the experimental data and molecular‐dynamics simulations. It was shown that there are two regions with different slope in the strain rate dependence of strength. The first one is at strain rate < 108/s, in which voids are nucleated in weakened zones. The second region is 2  Liquid-metal and Vitreloy are commercial names of a series of amorphous metal alloys. Liquid-metal alloys combine a number of desirable material features, including high tensile strength, excellent corrosion resistance, very high coefficient of restitution and excellent anti-wearing characteristics, while also being able to be heat-formed in processes similar to thermoplastics.

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where strength grows up relatively fast with strain rate at strain rate > 10−8/s, over which the number of weakened zones becomes insufficient, voids are nucleated predominantly in perfect material, and the strength growth is decelerated. 3.3.3  Melting Metals

It is known that fast energy absorption in metal leads to heating, melting, propagation of shock waves, and deformation. This occurs under the influence of one of the following sources: 1) Intense laser irradiation (e.g. Anisimov et al., 1984; Meshcheryakov and Bulgakova, 2006; Tarasenko et al., 2006; Povarnitsyn et al., 2007; Nedialkov et al., 2007; Amikura et al., 2008; Alonso et al., 2009; Ganeev et al., 2010). 2) Irradiation by the beams of charged particles, including electrons (e.g. Chistjakov et al., 1989; Boiko et al., 1999; Rotshtein et al., 2004; Bardakhanov et al. 2011; Pham et al., 2011). In the case of electron irradiation, a region of enhanced pressure with a characteristic thickness of from hundredths of a millimeter to a few millimeters is formed due to a rapid pulse heating. The unloading of this region leads to the thermal expansion of a melt, first under pressure and then by inertia. As a result, a region of tensile stresses arises in which the melt turns into a metastable state, which is destroyed by cavitation, that is, by the formation and growth of voids filled with metal vapor. After that, the substance turns into a two‐phase state. The aggregation of voids leads to the fracture of the melt and its fragmentation into droplets. 3) Explosive loads including electric explosion of conductors (e.g. Caldirola et al., 1974; Bushman et al., 1988; Kanel et al., 1996; Sedoi and Valevich, 1999; Krasik et al., 2008). A metal under irradiation experiences structural changes, phase transitions, and fracture. Along with surface evaporation and fracture, which occur under laser irradiation and in some electron irradiation regimes, bulk boiling may occur in a metal, as indicated by Sanchez and Mengüç (2008). Bulk boiling may occur not only due to the thermal expan­ sion as in the case of femtosecond laser (Agranat et al., 2010; Ashitkov et al., 2012) or high‐current electron irradiation (Rotshtein et al., 2004), but also under tensile stresses in a melt that occurs during the reflection of shock waves from the free surface of the melt, as demonstrated by Signor et al. (2008) and Chen et al. (2012). Experimental studies of fracture of melts were conducted by Signor et  al. (2008), Chen et  al. (2012), and Ashitkov et  al. (2012), who found difficulties in fixing the parameters of the substance in the extreme state. Molecular dynamics simulations were extensively used to analyze structural changes on the atomic level and to obtain the parameters of a material (the viscosity and surface tension coefficients), which are subsequently used in continuum simulation (e.g. Bazhirov et al., 2008; Kuksin et al., 2011, 2012; Shao et al., 2013). Dynamic fracture experiments were performed on fully amorphous Liquid Metal‐1 (LM‐1), a Zr‐based BMG by Sunny et al. (2013) with the purpose of studying fracture initiation and propagation in notched specimens. The results revealed that the critical dynamic stress intensity factor achieved by the notched LM‐1 specimens was of order 110 MPa.m½. In situ high‐speed camera images of the notched sample during the dynamic loading process showed multiple fracture initiation attempts and subsequent arrests prior to catastrophic fracture initiation.

Dynamic Fracture and Peridynamics

Mayer and Mayer (2015) described three evolution stages of a melt under intense deformation. These are (a) condensed liquid metal; (b) liquid metal with vapor bubbles; and (c) metal vapor with droplets of the liquid phase. The transition from stage (a) to (b) was found to be associated with thermo‐fluctuation formation of bubble nuclei in a metastable liquid phase (cavitation) as reported by Kuksin et al. (2011, 2012). At stage (b), a substance represents a bubble liquid. The transition from stage (b) to (c) is associated with the fracture (fragmentation) of the simply connected liquid phase into isolated regions that turn into spherical droplets under surface tension, while vapor fills the simply connected region. The dynamics of the substance was described by the equa­ tions of mechanics of a two‐phase heterogeneous continuum (Nigmatulin, 1987). The idea is to apply a continuum approximation or the approximation of interpenetrating continua; at every point of the space, both phases are described by continuous fields of variables such as the concentrations and the radii of bubbles or droplets, pressure, temperature, density, velocity, and the volume fractions of the liquid and vapor phases. The liquid is a simply connected carrier phase at stages (a) and (b), and the vapor is a simply connected carrier phase at (c). Vapor bubbles and liquid droplets represent a multiply connected dispersed phase at stages (b) and (c), respectively. A continuum model of deformation and fracture of metal melts under high‐current electron and ultrashort laser irradiation was developed by Mayer and Mayer (2015). The model was based on the equations of mechanics of a two‐phase continuum and the equations of the kinetics of phase transitions. The change (exchange) of the volumes of dispersed and carrier phases and of the number of dispersed particles was described. The energy and mass exchange between the phases due to phase transitions was taken into account. Molecular dynamic simulations were carried out with the use of the LAMMPS program (large‐scale atomic/molecular massively parallel simulator). The continuum model was verified by molecular dynamic calculations and experimental results. It was shown that an increase in the strain rate leads to an increase in the strength of a liquid metal, while an increase in temperature leads to a decrease in its strength. This was confirmed by referring to Figure 3.13(a)–(c), which show the dependence of the tensile strengths of aluminum, copper, and nickel melts, respectively, on tem­ perature, as calculated by the continuum model (shown by curves) and determined on the basis of molecular dynamics simulations (shown by solid circles and diamonds). The dashed curves (and solid circles) belong to strain rate of 109/s while the solid curves (and diamonds) are for strain rate of 1010/s. It is seen that there is a quantitative agree­ ment between the results of molecular dynamics simulations and continuum models, which can be interpreted as a mutual verification of the parameters of the model.

3.4  Dynamic Fracture of Composites Non‐metallic materials are very varied, and include non‐homogeneous (such as functionally graded materials), polymer‐based materials, bi‐materials, laminated composites, and sandwich structures. It is known that laminated composites with uni­ directional reinforced plies are the most popular form of composites that allow high volume fraction of fibers and therefore achieve high utilization of fiber properties. However, laminated composites are prone to delaminate due to edge stresses and under impact loading. Cracks leading to fracture occur at interfaces between two different

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(a)

(b)

–Ps (Gpa) 6

–Ps (Gpa) 8

6 4 4 2 2

0

1000

2000

3000

4000

0 2000

T (K)

3000 4000 T (K)

5000

6000

(c) –Ps (Gpa) 8

6

4

2

0

2000

3000

4000

5000

T (K)

Figure 3.13  Dependence of strength of different metals melt on temperature (Mayer and Mayer, 2015): (a) Dependence of strength of aluminum melt on the temperature at strain rates of 109 s 1 (‐ ‐ ‐ and •) and 1010 s 1 (_____ and ⬧) as calculated by the continuum model (curves) and molecular dynamics simulations (• and ⬧) (b) Dependence of strength of copper melt on the temperature at strain rates of 109 s 1 (‐ ‐ ‐ and •) and 1010 s 1 (_____ and ⬧) as calculated by the continuum model (curves) and molecular dynamics simulations (• and ⬧) (c) Dependence of strength of nickel melt on the temperature at strain rate of 109 s 1 (‐ ‐ ‐ and •) and 1010 s 1 (_____ and ⬧) as calculated by the continuum model (curves) and molecular dynamics simulations (• and ⬧)

Dynamic Fracture and Peridynamics

constituents such as a fiber and the matrix in a composite. The study of the dynamics of non‐metallic materials has been treated analytically, numerically, and experimentally to examine the instability of crack tip; time evolution of crack extension and velocity, and dynamic stress intensity factor. The main results of these studies are summarized in this section. 3.4.1  Functionally Graded Materials and Bi‐Materials

Functionally graded materials (FGMs) are a special class of composites and character­ ized by variation in composition and structure, resulting in corresponding changes in the properties of the material. These materials can be designed for specific functions and applications. Various approaches based on the bulk (particulate processing), pre­ form processing, layer processing, and melt processing have been used to fabricate functionally graded materials. The concept of functionally graded material is achieved by varying the microstructure from one material to another, with a specific gradient. This enables the material to have the best of both materials. If it is for thermal or cor­ rosive resistance or malleability and toughness, both strengths of the material may be used to avoid corrosion, fatigue, fracture, and stress corrosion cracking. Due to lack of symmetry in material properties, their fracture mechanics is inherently mixed‐mode when a crack is not parallel to the direction of material property variation. Bi‐materials, on the other hand, consist of two materials of different properties. For example, a metal shell and a plastic bearing surface. Common combinations include steel‐backed PTFE‐ coated bronze and aluminum‐backed Frelon. Wang and Meguid (1995) proposed a theoretical and numerical treatment of a finite crack propagating in an interfacial layer with spatially varying elastic properties under anti‐plane loading conditions. Parameswaran and Shukla (1999) developed expressions for the first stress invariant, and studied the effect of non‐homogeneity on dynamic crack growth. By using the boundary integral equation method, Zhang et  al. (2003) analyzed the effects of material gradients on dynamic mode‐III stress intensity factors. Chalivendra and Shukla (2005) and Chalivendra (2007) developed the transient crack tip field expansions for straight and curved cracks propagating in a functionally graded material. Later, Jain and Shukla (2007) investigated the transient behavior of mixed‐ mode cracks in FGMs, both analytically and experimentally. Tests on the experimental characterization of crack behavior in FGMs were conducted on different models (e.g. Parameswaran and Shukla, 1998, 2000; Lambros et al., 2000). These models include particulate epoxy composite with a spatially varying reinforce­ ment concentration, discrete property variation made by casting a polyester resin mixed with varying amounts of plasticizer, and UV‐sensitive material whose stiffness and fracture toughness depend strongly on the amount of UV exposure. The experimental studies of dynamic crack propagation behavior in functionally graded materials were limited due to the inability to analytically describe the stress and deformation field at the tip of a dynamically growing crack. The quasi‐static mixed‐mode crack propagation in a functionally graded beam under offset loading was experimentally studied by Abanto‐Buento and Lambros (2006) and Jin et al. (2009). The dynamic fracture behavior of FGMs was experimentally studied by Jain and Shukla (2006). Kirugulige and Tippur (2006) conducted mixed‐mode dynamic fracture experiments on FGM samples made of compositionally graded glass‐filled epoxy plates with initial edge‐cracks along the

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material gradient. Rousseau and Tippur (2001) presented an experimental study of the crack tip deformation and dynamics stress intensity factors in compositionally graded glass‐filled epoxy under low velocity impact loading. These dynamic stress intensity factors are important fracture parameters in understanding and predicting dynamic fracture behavior of a cracked body. Wu et al. (2002) extended the J‐integral to incorporate material gradients and dynamic effects. They evaluated the J‐integral for a single edge cracked FGM panel under step loading in conjunction with the element‐free Galerkin method. These research activities were restricted to mode‐I stress intensity factor in conjunction with either the J‐integral or the displacement correlation technique. The finite element method was used to numerically simulate dynamic mixed‐mode fracture behavior of FGMs (e.g. Kim and Paulino, 2004; Kirugulige and Tippur, 2008; Zhang and Paulino, 2005; Bayesteh and Mohammadi, 2013). Konda and Erdogan (1994) obtained mixed‐mode stress intensity factors for an unbounded non‐homogeneous elastic medium containing an arbitrarily oriented crack by solving integral equations. Young’s modulus was assumed to vary exponentially along the x direction as E ( x ) E (0)e x , and a constant Poisson’s ratio of 0.3 and β is the material non‐homogeneity parameter. The dynamic stress intensity factors were evaluated by Song and Paulino (2006) for both homogeneous and non‐homogeneous materials. Various homogeneous and non‐homogeneous cracked bodies under dynamic loading were employed to investigate dynamic fracture behavior such as the variation of dynamic stress intensity factors for different material property profiles. In the interaction energy integral method, auxiliary fields were introduced and super­ posed on top of the actual fields that come from the solution to the boundary value problem. The interaction energy integral is a crack tip contour integral surrounding a point on the crack front defined in the limit as the contour is shrunk onto the crack tip (see the M‐integral in Chaper 1). Song and Paulino (2006) extended the interaction integral (M‐integral) to incorporate material non‐homogeneity and dynamic effects for evaluation of dynamic stress intensity factors. In the interaction energy integral method, auxiliary fields are introduced and superposed on top of the actual fields that come from the solution to the boundary value problem. Auxiliary fields are based on known fields such as Williams’ solution (Williams, 1957), while actual fields utilize quantities such as displacements, strains, and stresses obtained by means of numerical methods such as the finite element method. The interaction energy integral is a crack tip contour integral surrounding a point on the crack front defined in the limit as the contour is shrunk onto the crack tip. The interaction energy integral approach for computation of mixed‐mode stress intensity factors in two‐dimensional crack problems was introduced by Stern et  al. (1976). The method was commonly used for the extraction of stress intensity factors in two‐dimensional and three‐dimensional bi‐material crack problems by Shih and Asaro (1988), Nakamura and Parks (1989), Nakamura (1991), Nahta and Moran (1993), Gosz et al. (1998), and Gosz and Moran (2002). Wawrzynek et al. (2005) developed and demonstrated a methodology for computing stress intensity factors in generally anisotropic materials. Various mixed‐mode problems under dynamic loadings to compute dynamic stress intensity factors for mode‐I and mode‐II were adopted by Song and Paulino (2006). For exponentially graded materials in the x direction, Song and Paulino (2006) obtained dynamic stress intensity factors for different values of the material gradation parameter

Dynamic Fracture and Peridynamics

β = 0.0, 0.05, 0.1, and 0.15. The ratio of material properties at the left and right bounda­ ries ranged from 1.0 to 90.0. Figure  3.14 demonstrates the time evolution of mixed mode dynamic stress intensity factors at the left and right crack tip locations. Both crack tips have the same initiation time for the different material gradations because Young’s modulus and mass density follow the same exponential function. It is seen that as the gradation parameter increases, the magnitude of KI(t) at the right crack tip increases. At the left crack tip, up to around 15 µs, the magnitude of KI(t) is larger for (a)

Normalized of K1 of left crack tip

1.4 β = 0.15

1.2

β = 0.10 β = 0.05

1.0

β = 0.0 0.8 0.6 0.4 0.2 0.0

0

4

8

12

16

20

Time (μs)

(b) Normalized of K1 of right crack tip

3.0 2.5

β = 0.15 β = 0.10 β = 0.5 β = 0.0

2.0 1.5 1.0 0.5 0.0

0

4

8

12

16

20

Time (μs)

Figure 3.14  Time evolutions of mixed mode dynamic stress intensity factors for different values of material gradation parameter β: (a) normalized KI(t) at the left crack tip, (b) normalized KI(t) at the right crack tip, (c) normalized KII(t) at the left crack tip, (d) normalized KII(t) at the right crack tip. (Song and Paulino, 2006)

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(c)

Normalized of K11 of left crack tip

0.4

0.0

–0.4

–0.8 β= 0.15 β= 0.10 β=0.05 β= 0.0

–1.2

–1.6

0

4

8

12

16

20

16

20

Time (μs)

(d) 0.5 Normalized of K11 of right crack tip

160

0.0

β = 0.15 β = 0.10 β= 0.05 β = 0.0

–0.5

–1.0

–1.5

–2.0

0

4

8

12 Time (μs)

Figure 3.14  (Continued)

smaller values of β and after that time, the magnitude of KI(t) becomes smaller for smaller values of β. On the other hand, as β increases, the absolute magnitude of KII(t) at both crack tips first decreases and then increases. Moreover, the absolute value of maximum KI(t) at the right crack tip is higher than that at the left crack tip as β increases. Time evolutions of normalized mixed‐mode dynamic stress intensity factors at both the right and left crack tip locations are shown in Figure 3.15. The mixed mode stress intensity factors are normalized with respect to K s a , where σ0 is the magnitude 0 of the applied stress and 2a is the crack length. The figure reveals that the dynamic stress intensity factors at the right crack tip are first initiated, and then the dynamic stress

Dynamic Fracture and Peridynamics

(a)

(b) 3

Normalized KI & KII

p(t)

K1 at right crack tip K1 at left crack tip

2 1 0

–1 –2

p(t)

K11 at right crack tip

0

10

K11 at left crack tip

20

30

40

Time (μs)

Figure 3.15  Characteristics of non‐homogeneous materials: (a) graded material geometry along the x direction, (b) time evolutions of normalized dynamic stress intensity factors at the left and right crack tips. (Song and Paulino, 2006)

intensity factors at the left crack tip are initiated. The interpretation of this sequence is that the initiation time of dynamic stress intensity factors depends on the dilatational wave speed. At any given time of the transient response, the magnitude of KI(t) at the right crack tip is higher than that at the left crack tip for the non‐homogeneous case. This is due to the fact that the values of material properties at the right crack tip are higher than those at the left crack tip. Assessments of dynamic fracture studies on FGMs were presented by Shukla et al. (2007) and Tippur (2010). Shukla et al. (2007) focused on experimental investigations dealing with the dynamic crack growth in model FGMs using the optical method of reflection photoelasticity and high‐speed photography. The results established a gener­ alized relationship between the crack velocity and the dynamic mode‐I stress intensity factor. On the other hand, Tippur (2010) reviewed those contributions dealing with the coherent gradient sensing technique to the advancement of dynamic fracture behavior to understand fracture of homogeneous materials, bi‐materials, particulate composites, fiber reinforced composites, and FGMs. Rupture along a bi‐material interface exhibits remarkable dynamic properties as reported by Ranjith and Rice (2001) and Cox et al. (2005). For example, Ranjith and Rice (2001) showed that steady frictional sliding along an interface between dissimilar elastic solids with Coulomb friction acting at the interface is ill‐posed for a wide range of mate­ rial parameters and friction coefficients. The connection between the ill‐posedness and the existence of a certain interfacial wave in frictionless contact was established. It was shown that a friction law with no instantaneous dependence on normal stress but a simple fading memory of prior history of normal stress makes the problem well‐posed. An oscillatory stress singularity was observed at the tip of a crack lying on an interface between two elastic solids (Williams, 1959). Some explicit solutions of the stress field in interfacial fracture problems were obtained by Sih and Rice (1964), England (1965), Erdogan (1965), Rice and Sih (1965), Rice (1988), and Shih (1991). These studies provided asymptotically an oscillatory stress singularity, confirming Williams’ result.

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Most of the research activities on interface fracture are dealing with the quasi‐static failure of bi‐material systems. Further theoretical studies were reported by Goldshtein (1967), Brock and Achenbach (1973), Willis (1971, 1973), Atkinson (1977), Yang, W. et al. (1991), and Deng (1993). Experimental investigations on transient elasto‐dynamic crack growth in a bi‐material interface were conducted by Tippur and Rosakis (1991) and Liu, C. et al. (1993a). The dynamic photoelasticity, optical caustics, and the finite element method were employed by Fang et al. (1999) to study the transient response and debonding failure of a cantilever beam of bi‐material. The beam has an edge crack in the center part of the main beam terminating at the interface with the reinforced beam. The shadow patterns of optical caustics demonstrated the response of the dynamic stress intensity factors of the crack tip in the main beam to illustrate the effect of the reinforcement. The stress intensity factors of interface crack dynamics were extracted from finite element field solutions using the M‐integral by Lo et al. (1994). This integral was used to extract the mode‐I and mode‐II contributions of the mixed‐mode crack tip field. The ratio of the mixed‐mode factors, or the phase angle was used as an essential parameter in defining the fracture resistance of dynamic interface cracks. A computational analy­ sis was performed for an interface crack in an infinite plane under either remote tensile or shear loading. The phase angle and the energy release rate of the growing interface crack with different crack tip velocities were obtained for several bi‐material combina­ tions. For a short time after the initiation of a growth, the energy release rate and the phase angle were found to be very sensitive to the crack tip velocity. They are much less sensitive to the mismatch in bi‐material properties. At late time, the phase angle merges closer to the corresponding quasi‐static values. Highly transient elasto‐dynamic fracture processes in both homogeneous and bi‐ material systems were studied by Liu (1994). Note that the dynamic stress intensity factor Kd‐dominant field adequately describes crack tip deformations. It was found that due to the wave character of the mechanical fields during transient and dynamic crack growth, the customary assumptions of steady state and Kd‐dominance may be violated. By permitting the crack tip speed and the dynamic stress intensity factor to be arbitrary functions of time, the transient asymptotic elasto‐dynamic field near the moving crack tip was established in the form of higher‐order expansion for both homogeneous solids and bi‐material systems. The Kd‐dominance during dynamic crack initiation and tran­ sient crack growth was investigated by solving a particular transient initial/boundary value problem. The crack was initiated under the influence of the wave, and then propa­ gated dynamically. This full field solution and the equivalent Kd‐dominant field revealed that the higher‐order transient representation provides a very good description of the actual stress field. Plate impact experiments exhibited the strong effects of transients on the interpretation of the measured results. Transient effects were found to be magnified by the material property mismatch between the constituent solids. High speed interferometric measurements on dynamically propagating interfacial cracks were reported by Lambros and Rosakis (1995a, 1995b) for polymethylmeth­ acrylate (PMMA)/steel bi‐material specimen. Impact loadings, using either a drop weight tower device or a high speed gas gun, were used. In gas gun experiments, termi­ nal crack tip speeds of up to 1.5C sPMMA, where C sPMMA is the shear wave speed of PMMA, were measured. Very large dynamic effects were observed in all dynamic bi‐material tests. It was concluded that the whole process of interfacial crack initiation and growth

Dynamic Fracture and Peridynamics

in these tests is driven by energy “leaking” from the metal side to the PMMA side of the bond. Furthermore, very severe transient effects occurred during the early stages of crack growth. Dynamic complex stress factor histories were obtained by fitting the experimental data to asymptotic crack tip fields. A dynamic crack growth criterion for crack growth along bi‐material interfaces was proposed. In the subsonic regime of crack growth, it was found that the opening and shearing displacements behind the propagat­ ing crack tip remain constant and equal to their value at initiation. The dynamic frac­ ture toughness of PMMA compact compression specimen under transient loading was studied by Rittel and Maigre (1996). The evolution of both the mode‐I and mode‐II stress intensity factors was assessed from the onset of loading until early crack propaga­ tion detected by a fracture gage. Dynamic fracture toughness was taken as the value of the mode‐I stress intensity factor at fracture time. The fracture toughness was observed to increase markedly with the stress intensity rate. Fractographic examination showed the existence of a characteristic rough zone directly ahead of the notch‐tip of dynami­ cally fractured specimen. The deflection/penetration behavior of dynamic mode‐I cracks propagating at vari­ ous speeds towards inclined weak planes/interfaces of various strengths in otherwise homogeneous isotropic plates was examined by Xu et al. (2003). A dynamic wedge‐ loading mechanism was used to control the incoming crack speeds, and high‐speed photography and dynamic photoelasticity were used to observe the failure mode transition mechanism at the interfaces. Simple dynamic fracture mechanics concepts used in conjunction with a postulated energy criterion were applied to examine the crack deflection/penetration behavior to predict the crack tip speed of the deflected crack. It was found that if the interfacial angle and strength are such as to trap an incident dynamic mode‐I crack within the interface, a failure mode transition occurs. This transition was found to be characterized by a distinct speed jump as well as a dramatic crack speed increase as the crack transitioned from a purely mode‐I crack to an unstable mixed‐mode interfacial crack. The problem was experimentally examined by Chalivendra and Rosakis (2008) to measure the deflection/penetration of dynamic mode‐I cracks propagating at slower and faster crack velocities toward inclined weak interfaces of three dissimilar angles: 30°, 45°, and 60°. A simple wedge‐loading specimen configuration, proposed by Xu et al. (2003), made of brittle Homalite‐100 was used. A modified Hopkinson bar setup was used to achieve well‐controlled impact loading conditions. 3.4.2  Polymer and PMMA Materials

An elastomer (or elastic polymer) material is a polymer with viscoelasticity (combining viscosity and elasticity). This type of material possesses very weak inter‐molecular forces with low Young’s modulus and high failure strain compared with other materials. Elastomer material is often used interchangeably with the term rubber. Elastomers are usually thermosets (requiring vulcanization) but may also be thermoplastic. Polymethylmethacrylate (PMMA) material is also known as acrylic or acrylic glass and has the trade names Plexiglas, Acrylite, Lucite, and Perspex. This material is a transparent thermoplastic often used in sheet form as a lightweight or shatter‐resistant alternative to glass. Thermosetting polyester display filters and are produced under two types by Homalite: H‐100 and H‐101.

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Complex fracture processes of cohesive finite elements have been demonstrated by many investigators (e.g. Needleman, 1990b; Xu and Needleman, 1994; Ortiz and Pandolfi, 1999; Pandolfi et al., 2000; Ruiz et al., 2001; Nguyen et al., 2001). Siegmund and Needleman (1997) studied numerically the dynamic crack growth for a plane strain block with an initial central crack subject to impact tensile loading acting on an isotropically hardening elastic‐viscoplastic solid. A cohesive surface constitutive relation relating the tractions and displacement jumps across the crack plane was developed. Crack growth and crack arrest emerged naturally as outcomes of the imposed loading. The effective stress inten­ sity factor was found to increase dramatically at a certain value of the crack speed. This speed was found to depend on the cohesive surface strength, the material flow strength, the characterization of strain rate hardening, and the impact velocity. Several attempts have been made to understand various aspects of the high velocity crack tip instabilities in the framework of linear elastic fracture mechanics by Adda‐ Bedia (2005), Katzav et  al. (2007), Bouchbinder et  al. (2005), and Bouchbinder and Procaccia (2007). Direct measurements of the deformation surrounding the tip of dynamic mode‐I cracks propagating in brittle elastomers at velocities of 0.2–0.8 times the shear wave speed were performed by Bouchbinder et al. (2008, 2009, 2010a, 2010b). The measurements demonstrated how linear elastic fracture mechanics (LEFM) breaks down near the tip of a crack. This breakdown was quantitatively described by extending LEFM to the weakly nonlinear regime, by considering nonlinear elastic constitutive laws up to second‐order in the displacement‐gradients. It was shown that the scale of the near‐tip region is delineated by a dynamic length‐scale, ℓnl, from the crack tip. At  this scale, the weakly nonlinear theory was formulated to provide an excellent description of the measured deformation fields. The dynamic length‐scale, ℓnl, is an important scale as it denotes the scale where LEFM breaks down, second‐order dis­ placement‐gradients become non‐negligible compared to the first‐order ones, and deformation‐dependent material behavior is initiated, as demonstrated by Gao (1996), Buehler et al. (2003), Buehler and Gao (2006), and Bouchbinder and Lo (2008). Crack dynamics in brittle materials were found to be governed by dynamical instabili­ ties of the crack tip (e.g. Fineberg et al., 1991). Gross et al. (1993) reported experimental measurements of acoustic emission, crack velocity, and surface structure. The results demonstrated quantitatively similar dynamical fracture behavior in PMMA and soda‐ lime glass samples. This unexpected agreement suggests that there exist universal fea­ tures of the fracture energy that result from dissipation of energy in a dynamical instability. Improved measurements with high resolution measurements of the crack’s velocity at 1/20 µs intervals for about 10,000 points throughout the duration of an experiment with velocity resolution of ±5 m/s were reported by Gross (1995) and Marder and Gross (1995); these made it possible to follow the long‐time dynamics of a crack in more detail. In applications to the fracture of PMMA a spatial resolution between measurements of order 0.2 mm was obtained by Fineberg et al. (1992), Sharon et al. (1995), and Fineberg and Marder (1999). Figure 3.16 shows the time evolution of the measured crack velocity propagating in PMMA, as reported by Fineberg and Marder (1999). It is seen that the crack first accelerates abruptly, over a time of less than 1 µs, to a velocity on the order of 100–200 m/s. Above the critical velocity, vc, the crack velocity exhibits rapid oscillations. As the crack’s velocity increases, these oscillations increase in amplitude. The crack begins at rest, and the tip has ample time to become slightly blunted making it difficult for the crack to begin moving.

Dynamic Fracture and Peridynamics

Figure 3.16  Measured time evolution of a crack tip velocity in PMMA material. After an initial jump to about 150 m/s, the crack accelerates smoothly up to a critical velocity vc shown by the dotted horizontal line. Beyond this velocity, strong oscillations in the instantaneous velocity of the crack develop and the mean acceleration of the crack slows. (Fineberg and Marder, 1999)

The critical fracture velocity, VC, was found to be roughly linearly dependent on the lowest crack acceleration rates in PMMA and glass, as reported by Livne et al. (2005) and Bouchbinder et  al. (2010a), see Figure  3.17(a). However, it shows large apparent scatter in the values of the critical fracture velocity, VC, for a given value of the crack acceleration. This uncertainty is believed to be because the microbranching instability in gels undergoes a sub‐critical bifurcation (hysteretic transition) from a single crack to a multiple‐crack state as shown in Figure 3.17(b). The reverse transition from a crack state with microbranches to a single‐crack state occurs at velocities far less than VC. Once the system falls within the bistable region of velocities, either a single or multi‐crack state can exist. Bouchbinder et al. (2010a) indicated that the microbranching instability can be suppressed by reducing the sample thickness h in the Z direction. When the thickness h is sufficiently reduced the total number of “noise” (activation) sources was found to be significantly reduced. In addition, any microbranch chain soon encounters a sample edge and disappears. When this occurs, Livne et  al. (2007) found that a new and unexpected oscillatory instability is observed at a critical velocity of 0.9VR , as shown

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Handbook of Structural Life Assessment

(a)

Vc(m/s)

4.0

3.0

2.0

1.0 0

4000

2000 Acceleration (m/s2)

(b) Single crack

Micro-branching state

3.0 2.5 Vc(m/s)

166

2.0

V/VR 0.6 0.5 0.4

1.5

0.3 Vc(min)

1.0

0.2 0.1

0.5 5

10

15

20

25

Crack length (mm)

Figure 3.17  (a) The transition to the micro‐branching instability is a roughly a linear function of the crack acceleration prior to the transition, (b) large hysteresis observed when the micro‐branching state, shown by the shaded region, undergoes a reverse transition to the single‐crack state. (Bouchbinder et al., 2010a)

in Figure 3.18. The characteristic scales of this instability such as oscillation wavelengths, λ, or amplitudes, A, are dependent on sample geometry or dimensions. Although such a high velocity oscillatory instability was shown to occur in LEFM, Bouchbinder and Procaccia (2007) suggested that the predicted oscillation wavelength must scale with the sample dimensions in the LEFM framework because no other scale exists. It was believed that these observations indicated a new intrinsic/dynamical scale, which is needed to describe these dynamics. The oscillatory instability shown in Figure 3.18(a) contains a sequence of photographs of a propagating crack with an interval of 0.69 ms between each shot. The first top two photo frames indicate that the crack is smooth and then undergoes transition to oscillatory motion in the subsequent frames at approximate speed of 0.9VR when microbranching is suppressed. Figure 3.18(b) shows

Dynamic Fracture and Peridynamics

(b)

(a)

Y

5 mm

X

5 mm

0.2 mm

Z X

(c) 10 mm

0.2 mm

(e)

(d) 0.9

8

0.3

λ (mm)

λ (mm)

0.8

0.6

Amp. (mm)

Amplitude (mm)

8

0.6 0.4

4

0

8

12

Time (ms) 8 4 Time (ms)

0

12 4

0.0 10

6 5

6

0.2 0

7

15 σ (k.Pa)

20

10

15

20

σ (k.Pa)

Figure 3.18  Oscillatory instability of a crack: (a) a sequence of photographs of a propagating crack, (b) photographs of XY profile (top) and (XZ) fracture surface (bottom), of a 0.2 mm thick gel sample, (c) the fracture surface is micro‐branch dominated, (d) steady‐state amplitude of oscillations versus the applied stress, (e) wavelengths of the oscillations as a function of the applied stress. (Livne et al., 2007, Bouchbinder and Procaccia, 2007, and Bouchbinder et al., 2010a)

two frames, the top is the XY profile, while the bottom is the XZ fracture surface of a 0.2 mm thick gel sample where oscillations are developed. Figure 3.18(c) is for the case of a 2.0 mm thick gel where the crack preserves its straight line trajectory. The shown fracture surface in Figure 3.18(c) is dominated by microbranching, while the oscillating crack of Figure 3.18(b) is a mirror surface. Figures 3.18(d) and (e) show the steady state amplitude and wavelength as functions of the applied stress, respectively, for the gel compositions used in Figs 3.18(a)–(c). The transition time evolution of the oscil­ lation amplitude and wavelength are shown in the insets of Figs 3.18(d) and (e), respec­ tively. The square symbols are for the sample of dimensions X × Y ~ 125 mm × 115 mm; the triangle symbols belong to 130 mm × 155 mm, and the circles are for 125 mm × 70 mm.

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The experimental measurements and LEFM predictions of crack‐tip opening displace­ ment (CTOD) were compared by Livne et al. (2008), and the results revealed that the discrepancies become significant as the crack velocity increases. Goldman et  al. (2010) conducted an experimental investigation on polyacrylamide gels – which are transparent, brittle, and incompressible elastomers – with the purpose of studying the crack behavior when it acquires inertia. The inertia was found to increase with crack velocity until it effectively becoming infinite as a crack’s limiting velocity is approached. It was demonstrated that this behavior is in quantitative agreement with an equation of motion derived in the framework of a crack propagating in an infinitely long strip by Marder (1991). Such an equation was derived in closed form by finding the energy flux to the tip of a slowly accelerating crack in a brittle strip. It is known that cracks create large stress amplification at their tips, leading to large material deforma­ tion. Livne et  al. (2010) used a brittle neo‐Hookean3 material and performed direct measurements of the near‐tip structure of rapid cracks. The measurements revealed a hierarchy of linear and nonlinear elastic zones through which energy is transported, before being dissipated at a crack’s tip. This result provided a comprehensive picture of how remotely applied forces drive material failure in the most fundamental of fracture states: straight, rapidly moving cracks. The capability of the cohesive finite element to reproduce complex dynamic fracture phenomena, such as crack nucleation and branching was presented by Arias et  al. (2007). Figure 3.19 shows snapshots of the simulations of a square pre‐notched PMMA plate subjected to a high initial uniform tensile strain rate. It is seen that a dynamic crack nucleates and propagates from the notch. Then crack accelerates until it becomes unstable, leading to recursive branching and eventually fragmentation. Xu et al. (2003) examined other dynamics cracking events such as the crack deflection/penetration behavior of cracks approaching inclined interfaces. A systematic integrated numerical– experimental approach to verify and validate numerical simulations of dynamic fracture along weak planes was presented by Arias et al. (2007). The validation test specifications were developed based on the consideration of a brittle material to decouple fracture from

Figure 3.19  Snapshots (increasing times to the right) of the fracture process of a square pre‐notched PMMA plate subjected to an initial uniform tensile strain rate. Color intensity indicate level of a relevant stress. (Arias et al., 2007) 3  A neo-Hookean solid is a hyperelastic material model, similar to Hooke’s law, that can be used for predicting the nonlinear stress–strain behavior of materials undergoing large deformations.

Dynamic Fracture and Peridynamics

constitutive response. A combined experimental–numerical–analytical inversion tech­ nique for an accurate determination of the cohesive law was developed. Crack branching in Homalite‐100 sheets of ⅛″ and ⅜″ thickness was studied using dynamic photoelasticity by Bradley and Kobayashi (1971a, 1971b) and Kobayashi et al. (1972a, 1972b). Dynamic stress intensity factors, crack velocities, and branching angles were measured. Dynamic stress intensity factors were found to reach a peak value at branching, with a value of three times larger than the fracture toughness of the material, and preceded the actual branching. The dynamic stress intensity factor after branching was found to drop and then increases again to the maximum stress intensity at which point branching occurs again. Roughness of the fracture surface can be related to a dynamic stress intensity factor and crack velocities near the branching stress intensity factors and terminal crack velocities, respectively. The stress history due to a propagating crack at points away from the prospective crack line was determined based on the asymptotic crack tip stress distribution by Ravi‐Chandar (1983). The problem of dynamic crack propagation was examined exper­ imentally by Ravi‐Chandar and Knauss (1984a, 1984b, 1984c, 1984d). Crack initiation and arrest were considered in thin sheets of Homalite‐100. It was found that as the rate of loading increases to as high as 105 MPa/s, the stress intensity factor required to initiate crack growth increases markedly. Crack arrest resulting from a simulated pres­ surized semi‐infinite crack in an unbounded medium was found to occur abruptly. The crack was found to always stop at a constant value of the stress intensity factor, which was lower than the stress intensity factor required for quasi‐static crack growth initia­ tion. The occurrence of microcracks at the front of the running main crack, which control the rate of crack growth, were found to grow and turn away smoothly from the direction of the main crack in the process of branching. The experimental study con­ firmed the rapid crack propagation at constant velocity, even though the stress intensity factor varied considerably during this propagation. This velocity was determined by the initial stress wave loading on the crack tip. The effect of stress waves on the behavior of running cracks was also examined. The stress waves were induced by reflection from the boundary through a judicious choice of specimen geometry and also through use of a second stress wave generator. It was found that these waves affect the direction of propagation, the crack speed, and the process of branching. It was also found that while stress waves modify the process of crack branching they do not by themselves consti­ tute the reason for crack branching. 3.4.3  Fiber‐Reinforced Composites

Experimental fracture mechanics studies of fiber‐reinforced composites showed that global load and/or displacement measurements to study quasi‐static fracture of com­ posites were used by Tirosh (1980), Donaldson (1985), and Liu et al. (2001). The crack tip field measurement approaches include the use of strain gage methods to evaluate stress intensity factors under quasi‐static loading conditions (Shukla et al., 1989). The method was extended to measure the dynamic stress intensity factors by Khanna and Shukla (1994, 1995), who used multiple gages for point‐wise measurements. Full field optical measurement methods were found to provide complete information on deforma­ tion components at various load levels and can be used to locate the crack tip at various time instants. Full‐field methods include the optical methods of caustics Rosakis (1980),

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photoelasticity (Kobayashi and Dally, 1980), Moiré interferometry (Kokaly et al., 2003), and thermo‐elastic fracture analysis (Lin et  al., 1997). They were used to study the dynamic fracture of isotropic and anisotropic materials by Yao et al. (2004). Coherent gradient sensing is a lateral shearing interferometer full‐field measurement technique developed by Tippur and Rosakis (1991) and Tippur (1992). The digital image correlation technique has become relatively popular in dynamic fracture measurements (e.g. McNeill et al., 1987; Bruck et al., 1989; Tan et al. 2005; Hild et al., 2006; Abanto‐ Bueno and Lambros, 2006; Mekky and Nicholson, 2006; Han and Siegmund, 2006). For example, McNeill et al. (1987) demonstrated digital image correlation for extracting quasi‐static mode‐I stress intensity factors using crack opening displacements. The digital image correlation method was extended by Abanto‐Buento and Lambros (2002) to the study quasi‐static mixed‐mode crack growth in non‐homogeneous, isotropic inelastic polymer sheets with graded mechanical properties. Strain rate effects on frac­ ture behavior of single‐edge notched multi‐layered unidirectional graphite composites (T800/3900‐2) under static and dynamic loading conditions using a digital speckle correlation method were studied by Lee et al. (2010). A two‐dimensional digital image correlation method was used to obtain time‐resolved full‐field in‐plane surface dis­ placements when specimens were subjected to quasi‐static and impact loading. Stress intensity factor and crack extension histories for pure mode‐I and mixed mode cases were extracted from the full‐field displacements. When compared to the dynamic stress intensity factors at crack initiation, the static values were found to be consistently lower. The stress intensity factor histories exhibit a monotonic reduction under dynamic loading conditions whereas an increasing trend was observed after crack initiation under quasi‐ static loading cases. Preliminary results of optical experiments performed on dynamically deforming thick polymeric‐composite laminate plates made of unidirectional graphite/epoxy fiber‐ reinforced composite were presented by Lambros and Rosakis (1997). Edge‐notched plates were impact loaded in a one‐point bend configuration using a drop‐weight tower. Crack growth speeds of 900 m/s and significant dynamic effects were observed through emission of stress waves from the propagating crack tip. Further dynamic fracture exper­ iments on unidirectional graphite‐epoxy composite were conducted under in‐plane, symmetric and asymmetric, impact loading by Coker and Rosakis (1998) and Coker (2001). It was found that mode‐I cracks propagated subsonically, with crack speeds increasing to the neighborhood of the Rayleigh wave speed of the composite. The dependence of the dynamic initiation fracture toughness on the loading rate was found to be constant for low loading rates and then to increase rapidly. The dynamic crack propagation toughness was observed to decrease with crack tip speed up to the Rayleigh wave speed of the composite. For asymmetric types of loading the results revealed highly unstable and intersonic shear‐dominated crack growth along the fibers for mode‐II. These cracks propagated with unprecedented speeds reaching 7400 m/s. Karashov (2013) analyzed the dynamic fracture behavior of laminated composites with nanoreinforced interfaces. The analysis provided quantitative dynamic interlaminar fracture treatment under mode‐I, mode‐II, and mixed‐mode loadings. Crack propaga­ tion was evaluated in specimens with and without nanofiber reinforcement. Significant improvements in fracture resistance were observed as a result of nanoreinforcement. In particular, a possibility of suppression of ultrafast intersonic mode‐II crack into the subsonic regime as a result of nanoreinforcement was demonstrated. It was shown that

Dynamic Fracture and Peridynamics

a single dynamic experiment on a newly developed mixed mode test specimen may provide critical dynamic fracture resistance parameters for a broad range of mixed modes, a potentially highly efficient new method of dynamic material fracture characteri­ zation. A numerical model for composite laminate subjected to side impact was developed. The model utilized the data from the new mixed‐mode test. Laminate delamination under impact was studied numerically, and significant reduction in delamination area was demonstrated as a result of nanoreinforcement.

3.5 Peridynamics4 3.5.1  Ingredients of Peridynamic Theory

The classical theory of continuum mechanics is based on partial differential equations whose partial derivatives are continuous. Since partial derivatives do not exist on crack surfaces and other singularities, the classical equations of continuum mechanics cannot be applied directly when such features are present. Cracks in structural materials form discontinuities, and their modeling requires special formulation. Peridynamic theory treats internal forces within a continuous solid as a network of pair interactions, similar to springs, which can be nonlinear. The response of the springs depends on their direction in the reference configuration, and their length. Pairs of material points can interact through a spring, up to a maximum distance called the horizon. This formulation was proposed by Silling (2000) who developed the peridynamic model of continuum mechan­ ics with fracture discontinuities. The peridynamic theory is based on integral equations and thus does not require spatial derivatives to be evaluated within the structure body (Silling, 2000, 2003, 2012; Silling et al., 2003; Silling and Askari, 2004, 2005; Silling and Bobaru, 2005; Weckner and Abeyaratne, 2005). Peridynamics unifies the mechanics of continuous and discontinuous media within a single set of equations. Peridynamics is a recently developed theory in solid mechanics since it replaces the partial differential equations of classical continuum theories with integro‐differential equations. The basic equation of peridynamics is usually written in the form:

x u x ˘t

f u xt

u x t dVx ' b x t (3.11)

H

where x is a point in a body horizon H, and u is the displacement vector field. The vector valued function, f, is the force density that x ′ exerts on x as shown in Figure 3.20. This force density depends on the relative displacement and relative position vectors between x ′ and x. This force describes how the internal forces depend on the deformation. The term b represents the body force density field. The interaction between any x and x ′ is called a bond. The force density f is assumed to vanish if the point x ′ is outside a neigh­ borhood of x in the undeformed configuration, which is called the horizon. Note that bonds can break irreversibly and broken bonds carry no force. The origin of the theory of peridynamics and its applications in different engineering problems is documented in a research monograph by Madenci and Oterkus (2014). 4  The term “peridynamic,” an adjective, comes from the prefix peri, which means all around, near, or surrounding; and the root dyna, which means force or power.

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Figure 3.20  Volume region (horizon) in a solid showing the bond between two points and the force density vector (pairwise) applied at both points according to Silling (2012).

x′ f(x′, x, t) x

H

Non‐local theories in continuum mechanics are believed to trace back to the works of Kroner (1967), Eringen (1972, 1992), Eringen and Edelen (1972), Kunin (1983), and Rogula (1982). These theories described certain effects such as the unreasonable infinite stresses occurring at a crack tip in local linear elasticity that are not captured accurately in the corresponding local formulation. Non‐local approaches have been developed by Altan (1989, 1991), Bažant and Pijaudier‐Cabot (1988), Bažant and Jirasek (2002), Pisano and Fuschi (2003), Polizzotto (2001), Silling (2000), Wang and Dhaliwal (1993a, 1993b), Chen et al. (2004), and Lei et al. (2005). The theory of peridynamics, developed originally by Silling (2000), is considered to be a strongly non‐local method due to the presence of an integral operator, as opposed to higher‐order gradient operator in the equation of motion. A multi‐scale modeling framework combining peridynamics and atomistic models was developed by Rahman et al. (2014). Typically, atomistic models are governed by molecular dynamics schemes. In this framework, peridynamics models at higher length scale act as external environment for the peridynamics models at smaller length scale. The mathematical formulation of peridynamics based framework for hierarchical multi‐scale modeling was carried out. It showed that the reflection between different models at different length scales is absent in peridynamics based framework for hierarchical multi‐scale modeling. It was shown that the displacement field has a strong correlation with the length scale of material. In peridynamics, particles interact non‐locally through a bond across the distance between them, much as in molecular dynamics (Silling and Lehoucq, 2010). The term non‐local implies that points separated by a finite distance may exert force upon each other. Values of some quantity at a point are strongly influenced by values of the field in a neighborhood of that point. This is in contrast to the classical partial differential equa­ tion models in which the particles interact locally through direct contact with each other. The notion of a peridynamic stress tensor derived from non‐local interactions was defined by Lehoucq and Silling (2008). At any point in the body, this stress tensor was obtained from the forces within peridynamic bonds that geometrically go through the point. The peridynamic equation of motion was expressed in terms of this stress tensor, and the result was formally identical to the Cauchy equation of motion in the classical model, even though the classical model is a local theory. This stress tensor field was established to be unique in a certain function space compatible with finite element approximations. Silling (2009b, 2010b) presented an approach for deriving peridynamic material models for a sequence of increasingly coarsened descriptions of a body. Each successively coarsened model excludes some of the material present in the previous

Dynamic Fracture and Peridynamics

model, and the length scale increases accordingly. This excluded material, while not present explicitly in the coarsened model, is nevertheless taken into account implicitly through its effect on the forces in the coarsened material. Numerical examples demonstrated that the method accurately reproduces the effective elastic properties of a composite as well as the effect of a small defect in a homogeneous medium. Two overviews of some of the results concerning the mathematical analysis and numerical solution of the governing equation of motion of peridynamics were presented by Emmrich and Weckner (2006) and Weckner et al. (2007). The question of energy conservation and the comparison of elastic energy in both the peridynamic and the classical theory were discussed. Comparing the elastic energy density associated with homogeneous deformations in peridynamics to the corresponding energy in classical elasticity shows how the connection to experimentally measurable material properties such as the Young’s modulus and the Poisson ratio can be established. The response of a state‐based peridynamic material was considered by Silling (2010a) for a small deformation superposed on a large deformation. The appropriate notion of a small deformation restricts the relative displacement between points. The material properties governing the linearized material response were expressed in terms of the modulus state. This determines the force in each bond resulting from an incremental deforma­ tion of itself or of other bonds. Conditions were derived for a linearized material model to be elastic. If the material is elastic, then the modulus state can be obtained from the second Fréchet derivative of the strain energy density function. Peridynamic material models for a sequence of increasingly coarsened descriptions of a body were developed by Silling (2010b). These models include small scale linearized state‐based description. Each successively coarsened model excludes some of the material present in the previous model, and the length scale increases accordingly. This excluded material, while not present explicitly in the coarsened model, is nevertheless taken into account implicitly through its effect on the forces in the coarsened material. The peridynamic theory depends crucially upon the non‐locality of the force interac­ tions and does not explicitly involve the notion of deformation gradients. The linear bond‐based non‐local peridynamic models with reference to problems associated with non‐standard non‐local displacement loading conditions were studied by Zhou and Du (2010). Both stationary and time‐dependent problems were considered for a one‐dimensional scalar equation defined on a finite bar and for a two‐dimensional system defined on a square. The study was supported by applications to the numerical analysis of the finite‐dimensional approximations to peridynamic models. A review of peridynamic models including the ordinary bond‐based, state‐based models and a non‐ordinary triclinic model was presented by Zhou (2011). Later, a functional analytical framework for a linear peridynamic model of a spring network system in any space dimension was developed by Du and Zhou (2011). Different properties of the peridy­ namic operators were examined for general micromodulus functions. These properties were utilized to establish the well‐posedness of both the stationary peridynamic model and the Cauchy problem of the time‐dependent peridynamic model. The classical (local) theory of solid mechanics is based on the assumption of a con­ tinuous distribution of mass within a body. It further assumes that all internal forces are contact forces (Truesdell, 1977) that act across zero distance. The mathematical description of a solid that follows from these assumptions relies on partial differential equations that additionally assume sufficient smoothness of the deformation for the

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partial differential equations to make sense in either their strong or weak forms. The classical theory was demonstrated to provide a good approximation to the response of real materials down to small length scales, particularly in single crystals, provided these assumptions are met (Maranganti and Sharma, 2007). A spatial approximation for the numerical solution was proposed, based on the quadrature formula method. Long‐range interactions for linearly elastic media resulting in nonlinear dispersion relations were modeled by Emmrich and Weckner (2007a) and Weckner and Silling (2011) using an initial‐value problem for an inte­ gro‐differential equation that incorporates non‐local effects. A numerical approxi­ mation based upon quadrature was proposed and carried out for two examples, one involving jump discontinuities in the initial data corresponding to a Riemann‐like problem. Different spatial discretization methods for solving the peridynamic equa­ tion of motion were proposed by Emmrich and Weckner (2007b). The proposed methods were tested for a linear microelastic material of infinite length in one spatial dimension. Moreover, the conservation of energy was studied for the continuous as well as discretized problem. Later, Emmrich et  al. (2013) presented a survey on important analytical and numerical results and applications of the peridynamic theory. An existence theory was developed by Bellido and Mora‐Corral (2014) based on minimization of the non‐local energies appearing in peridynamics. The direct method of the calculus of variations was employed in an attempt to find minimizers of the energy of a deformation. In order to demonstrate the differences between the local and non‐local theories of continuum mechanics, we can adopt the treatment presented by Weckner et al. (2009b) using Green’s functions. Starting from the general equation of motion of the material point x at time t in one spatial dimension for both theories in the general form u x ,t



b x ,t (3.12)

L u x ,t

where ρ is the material density, a dot denotes differentiation with respect to time, L[.] is the linear operator acting on the displacement field, u(x, t), accounting for internal forces, and b(x, t) represents the external forces. Weckner et  al. (2009a) showed that Fourier transformations can be used to develop a solution of the general inhomogene­ ous initial value problem for the three‐dimensional linear bond‐based peridynamic formulation. Several examples illustrated this approach and showed the importance of the peridynamic horizon. According to Weckner et al. (2009b), the form of the operator L[.] is different for local and non‐local theories. For the local theory, the internal forces are written by the differential operator L



L u x ,t

E

2

u x ,t x2

(3.13)

where the superscript L stands for local, and E is Young’s modulus. For the non‐local theory and in particular the peridynamics theory for an infinite linear microelastic material the operator takes the integral form NL



L u x ,t

c x

x u x ,t

u x ,t

dx (3.14)

Dynamic Fracture and Peridynamics

where c( x x ) is the micromodulus function which is an even function, x and x ′ are two points in a volume region called horizon. It was shown by Weckner and Abeyaratne (2005) that the solution of equation (3.12) may be written in the integral form u x ,t

u0 x xˆ g xˆ ,t dxˆ

v0 x xˆ g xˆ ,t dxˆ

t

b x xˆ , t tˆ

ˆ ˆ g xˆ ,tˆ dxdt

0

(3.15)

Where g x ,t NL

k

1 2

eikx

sin

1 cos k

k t k

dk , L

k

c0 k , c0

E/

speed of sound, and

1/2

c

d

/

.

where k is the wave number. The solutions of one‐dimensional boundary value problems corresponding to classical, fractional, and non‐local diffusion on bounded domains were compared by Burch and Lehoucq (2011). The latter two diffusions were found viable alternatives for anomalous diffusion, when Fick’s first law5 is an inaccurate model. In the case of non‐local diffusion, a generalization of Fick’s first law in terms of a non‐local flux was demonstrated to hold. For the three‐dimensional case, the deformation kinematics can be established with the aid of Figure 3.21, for two material particles X and X ′ in which the material particle X at time t 0 is represented by its position vector x R3 and the particle X ′ is repre­ sented by its position vector x ′. At time t, the particle position assumes the position y ( x ,t ) x u( x ,t ), where u(x, t) is the displacement field, while the other particle assumes the position y ( x ,t ) x u ( x ,t ). The velocity of the particle X may be defined as ( x ,t ) y ( x ,t ) u ( x ,t ). Note the relative position of the two particles X and X ′ is x x and is referred to as the peridynamic bond. The corresponding relative posi­ u( x , t ) u( x ,t ), where n is the unit tion is y ( x ,t ) y ( x ,t ) and = n vector pointing from X to X ′. For smooth deformation, one can introduce the deforma­ tion gradient F( x ,t ) ( y ( x ,t ))T , where T denotes transpose. 5  Fick’s first law relates the diffusive flux to the concentration under the assumption of steady state. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative). In one (spatial) dimension, Fick’s first law is written in the form:

Cd

x

Where ψ is the “diffusion flux” (amount of substance per unit area per unit time, (mol/m.s) and measures the amount of substance that will flow through a small area during a small time interval, Cd is the diffusion coefficient or diffusivity (length time-1), Φ (for ideal mixtures) is the concentration (amount of substance per unit volume, mol/m2), x is the position. Cd is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles.

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Handbook of Structural Life Assessment X′

ξ+η f = fn ξ + η X

u′ ξ

u

y′ y x′

x O

Figure 3.21  Kinematics of deformation in three‐dimension. (Adapted from Weckner et al., 2009b)

An upscaling scheme for the passage from atomistic to continuum mechanical models for crystalline solids was proposed. It is based on a Taylor expansion of the deformation function, and allows us to capture the microscopic properties and the discreteness effects of the underlying atomistic system up to an arbitrary order. The resulting continuum mechanical model involves higher‐order terms and gives a descrip­ tion of the specimen within the quasi‐continuum regime (Arndt and Griebel, 2005). Furthermore, the convexity of the atomistic potential is retained, which leads to well‐ posed evolution equations. The non‐local force that particle X ′ exerts on X is the f x , x , t n . The equations of motion of the material point central force f x , x , t in an infinite linear elastic, isotropic, homogeneous body according to the local contin­ uum mechanics as given by the Navier equations (Elmore and Heald, 1985), with initial conditions u 0 ( x ) u( x ,0) and v 0 ( x ) u ( x ,0), are

 x ,t u



x ,t

2

x ,t

x ,t

1 2

u x ,t



x ,t

b x ,t (3.16a) x ,t I (3.16b)

Tr

u x ,t

T

(3.16c)

where μ and λ are the Lamé constants, which can be expressed in terms of the Young’s modulus E (3 2 ) / ( ) and the Poisson ratio / 2( ). For the case of non‐local peridynamic formulation of continuum mechanics, the equation of motion for an infinite, isotropic, homogeneous, linear microelastic, pair­ wise equilibrated material, takes the form (Silling 2000)  x ,t u



C

H x,

C

u x

,t

u x ,t

dV

b x ,t (3.17a)

(3.17b)

Dynamic Fracture and Peridynamics

where H(x, δ) is known as the horizon and is taken to be the sphere with center x and radius C CT is the symmetric (0, ] as shown in Figure  3.20, C micromodulus tensor, and Λ(ξ) is the micromodulus function, which contains all constitutive information. The general equation of motion in three‐dimension can be written for local and non‐local peridynamics in the form  x ,t u



L

b x ,t (3.18a)

L u x ,t

L u x,t

NL



u x ,t (3.18b)

u x ,t

L u x ,t

H x,

C

u x

,t

u x ,t

dV (3.18c)

The main difference between the local and non‐local formulation is that there is a jump condition for linear momentum formulated across a moving discontinuity surface A(t). The spatial point y on that surface momentarily occupies the material point x(t), such that y A (t ) y( x A (t ), t ), which has a velocity A (t ) F x A . x A A (t ) F The corresponding linear momentum is



nA

v vA v

nA

0

S

local (3.19) non-local

where nA is the surface normal,  is the mass density in the actual configuration, which is related to the reference density by det( F )  . The jump conditions for mass and continuity of displacement are identical in both formulations, i.e.

nA F

A



0 balance of mass (3.20a) 0 continuity (3.20b)

x A

The non‐local condition given by the second equation of (3.19) takes the form after substituting the balance of mass condition of equation (3.20a)

nA

A

0 (3.21)

In a non‐local formulation with both balance of mass and linear momentum, the normal component of the velocity field is always continuous, and any discontinuous surface moves like a material interface in the normal direction. The only possible velocity jumps would lie in the tangent plane that locally coincides with the generally curved discontinuity surface. Thus a jump in the velocity field is always accompanied by a jump in the displacement field. Another important result is that for the elastic energy density for any homogeneous deformation to be identical in both local and non‐local elasticity would lead the restriction

or the Poisson ratio

1 (3.22a) 4

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This condition relates the micromodulus function Λ to the Lamé constant λ as follows



r r r dr 0

15 (3.22b) 2

The above restriction only exists in the bond‐based formulation and no longer valid in the state‐based peridynamics formulation as indicated by Silling (2000) and Silling et al. (2007). In local continuum mechanics, the phase velocity does not depend on the wavelength for either pressure or shear waves. On the other hand, peridynamics always leads to wave dispersion. A non‐ordinary state‐based peridynamic method to solve transient dynamic solid mechanics problems was developed by Warren et  al. (2009). ¼ as with the This method was shown to be not restricted to a Poisson’s ratio of bond‐based method. Non‐local nodal deformation gradients that are used to define nodal strain tensors were obtained and were used with the nodal strain tensors to obtain rate of deformation tensors in the deformed configuration. For the case when all points of the material interact, the micromodulus function may be assumed to be either exponential e

/ 7 8 /

3/2

t

/  7 15 /

2

or trigonometric

/

e

2

(3.23a)

sin x / 

x /  cos x /  e

/

2

(3.23b)

where  is the length scale measuring the degree of non‐locality. The exponential form of the micromodulus function (3.23a) leads to wave dispersion for any finite wavelength 2 / k , while the trigonometric micromodulus function (3.23b) behaves like a low‐pass filter. Note that waves with a wavelength larger than c 2 travel with the same phase velocity as in the local formulation, while waves smaller than the cut‐off wavelength χc are dispersed. The exponential form of the micromodulus function results in the following phase velocities as a function of the normalized wave k : number NL

p ||

4 2e

2

NL

p

4 4e

2

2

/



2

2

/4

1

1 16

5

3

2

/



/4

2

2

16 3 O

2

O

2

(3.24a)

(3.24b)

The trigonometric micromodulus gives NL



p ||

/

3 5

3 5

2

1 (3.25a) 1

Dynamic Fracture and Peridynamics

NL

1

p

5

/



3

1

5

5

2

1

(3.25b) 1

It was shown by Weckner et al. (2009b) that large wavelength expansion confirms the convergence towards local elasticity for materials with . The equation of motion (3.18a) can be written in terms of the acoustic tensor N(k) by applying the Fourier transform with respect to the coordinate x. Before performing this process, it is useful to shed some light on the acoustic tensor N by introducing the concept of shear bands. Shear bands or “localized deformations” usually develop within a broad range of ductile materials (alloys, metals, granular materials, plastics, polymers, and soils) and even in quasi‐brittle materials (concrete, ice, rock, and some ceramics). The relevance of the shear banding phenomena is that it precedes failure, since extreme deformations occurring within shear bands lead to intense damage and fracture (Bigoni, 2012). Shear band formation is an example of a material instability, corresponding to an abrupt loss of homogeneity of deformation occurring in a solid sample subject to a loading path com­ patible with continued uniform deformation. For example, consider an infinite body made up of a nonlinear material, quasi‐statically deformed in a way that stress and strain may remain homogeneous. The incremental response of this nonlinear material is assumed for simplicity to be linear, so that it can be expressed as a relation between a stress incre­   ment and a strain increment , through a fourth‐order constitutive tensor C as   C (3.26) where the fourth‐order constitutive tensor C depends on the current stress, strain and, possibly, other constitutive parameters. Conditions for the emergence of a surface of dis­ continuity (of unit normal vector n) in the incremental stress and strain are identified with the conditions for the occurrence of localization of deformation. In particular, incremen­ tal equilibrium requires that the incremental tractions (not the stresses) remain continuous   n n (3.27) (where + and − denote the two sides of the surface) and geometrical compatibility imposes a strain compatibility restriction on the form of incremental strain: 



1 g 2

n+n

g (3.28)

where the symbol denotes tensor product and g is a vector defining the deformation discontinuity mode (orthogonal to n for incompressible materials). A substitution of the incremental constitutive law given by equation (3.26) and of the strain compatibility (3.28) into the continuity of incremental tractions (27) yields the necessary condition for strain localization:

C g

n n = 0 (3.29)

The second‐order tensor N(n) is defined for every vector g as

N n g C g

n n (3.30)

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N(n) is the so‐called the acoustic tensor, which defines the condition of propagation of acceleration waves. This condition for strain localization coincides with the condition of singularity (propagation at null speed) of an acceleration wave. This condition represents the so‐called loss of ellipticity of the differential equations governing the rate equilibrium. The equation of motion (3.18a) may be written in terms of the acoustic tensor N(k)  k ,t u



N k



N|| k nk nk

L

N|| k

L

N

NL



b k ,t (3.31)

N k u k ,t

k Pnk (3.32)

N

k 2 (3.33a)

2

k = k 2 (3.33b)

N|| k

r r 4 A1 kr dr (3.33c)

4 0

NL



N

k

r r 4 A2 kr dr (3.33d)

4 0

where





A1 x

sin x

1 3

2 cos x x2

x

j

1 x2 j

2 sin x x3

j 0

2

2j 2 ! 2j 5

x2 O x4 10 (3.34)

A2 x

1 3

cos x

sin x

x

x

j

1 x2 j

3

j 0

2

2j 3 ! 2j 5

x2 O x 4 (3.34b) 30

The transformed initial conditions are u 0 ( x) e

0



u (k )

i 2 xk

dx ,

0

(k )

0

( x) e

i 2 xk

dx (3.35a,b)

Introducing equations (3.22b), (3.34a) and (3.34b) in the non‐local acoustic tensor (3.33c), yields the same acoustic tensor of the local case as the wave number k 0 for materials with . Emmrich and Weckner (2007a), Silling and Lehoucq (2008), and Gunzburger and Lehoucq (2010) showed that the convergence of the non‐local peri­ dynamic equation (3.17a) towards the local Navier equations (3.16a) can be achieved for linear bond‐based formulation and nonlinear state‐based formulation. The solution of equation for non‐local static case is

u k

N

1

k b k (3.36)

This solution is subject to a point load at the origin, b( x ) P ( x ), and takes the form

u k

f nx x nx nx

f Pnx x Pnx

P (3.37)

Dynamic Fracture and Peridynamics

1

f nx x



2



2

1

f Pnx x

2

a1 xk 0

N

a2 xk

2

0

k2

N

k2

sin kx

k

N|| k

k2

k2

k

kx

N

k2

sin kx

N|| k

kx

N

dk (3.38a)

k

k2

k

dk (3.38b)

where x x . Note that in the non‐local case, the integrals given in equations (3.38a,b) do not converge since the last term in the integrands is unbounded for large k. Unlike in the local case the acoustic tensor is constant in the limit: NL N|| (k ), NL N (k ) NL N . The reason for this divergence is the presence of a Dirac distribution in the solution. Equations (3.38a,b) can be explicitly revealing the presence of a Dirac distribution in Green’s function in the non‐local formulation NL

f nx x

NL

x

N

2

1

a1 xk

2

0

k2 NL N k

k2 NL N|| k

sin kx kx

k2 NL N k

k2 NL N

dk

(3.39a) NL

f Pnx x

NL

x N

2

1

2

a2 xk 0

NL

k2 N k

NL

k2 N|| k

sin kx kx

NL

k2 N k

k2 N

NL

dk

(3.39b) With reference to the exponential micromodulus function, the results of the numerical integration was obtained by Weckner et al. (2009b), and Figures 3.22(a) and (b) show the dependence of the normalized displacement components along the direction of x, NL fnx(x)/μ, and orthogonal to x, NLfPnx(x)/μ, respectively, for different degrees of non‐ locality . It is seen that in the limit 0, together with the local solution 0 with of vanishing horizon 0 the non‐local solution converges towards the local solution almost everywhere. However, for any finite horizon 0, the displacements under the point load remain bounded when ignoring the Dirac distribution In contrast, the displacements in local elasticity are unbounded due to the presence of the 1/x singular­ ity in the solution. This is perhaps not surprising as the motivation in some of the earlier work on weakly non‐local methods was to remove the presence of the unphysical 1/ x singularities in the stress field surrounding a crack tip. 3.5.2  Remarks and Restrictions

Restrictions imposed by the second law of thermodynamics on peridynamics in both the bond‐based and the state‐based formulations were studied by Ostoja‐Starzewski et al. (2013). The study was carried out in the framework of thermo‐mechanics with internal variables. In bond‐based peridynamics, there are two possible thermo‐mechanical interpretations of the dissipation function. One interpretation only admits a thermody­ namic orthogonality of Ziegler, while the other admits powerless forces within a repre­ sentation theory of Edelen. The latter interpretation was shown to be admissible in

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Local solution δ = 0

6

δ = 1/80

5 NLfnx(x) 4 μ

δ = 1/40

3

δ = 1/20

2 1 0 0.0

5 4 NLfPnx(x)

μ

0.05

0.10 x

0.15

0.20

0.15

0.20

Local solution δ = 0 δ = 1/80

3

δ = 1/40

2

δ = 1/20

1 0 0.0

0.05

0.10 x

Figure 3.22  Dependence of the normalized displacement components: (a) along the direction of x, (b) orthogonal to x. (Adapted from Weckner et al., 2009b)

state‐based peridynamics. An adjoint design sensitivity analysis based on the peridy­ namics of the reformulated continuum theory was developed by Moon et al. (2014) for the solution of dynamic crack propagation problems using the explicit scheme of time integration. The problems of non‐shape design sensitivity analysis were considered for the dynamic crack propagation, including the successive branching of cracks. Since both original and adjoint systems possess time‐reversal symmetry, the trajectories of systems were found to be symmetric about the u axis. The accuracy of analytical design sensitivity was verified by comparing the results with those obtained by the finite difference. It was demonstrated that the peridynamic adjoint sensitivity involving history‐dependent variables can be accurate only if the path of the adjoint response analysis is identical to that of the original response. The original peridynamic formulation used a functional that relates forces on neigh­ boring material points through a pairwise linear potential. This had some drawbacks including a resulting Poisson ratio of ¼. Recent developments have improved the origi­ nal formulation through the use of peridynamic force‐vector states. These force‐vector states handle the constitutive behavior of the material, allowing neighboring material points to interact with each other in any fashion desired, not just pairwise. Force‐vector states also do not have the restriction of linearity or symmetricity imposed upon the

Dynamic Fracture and Peridynamics

stress tensor. Foster et  al. (2009) presented dynamic fracture experiments where modeling using an elastic viscoplastic peridynamic constitutive model. The study was supported by using experimentally collected rate‐dependent fracture toughness meas­ urements as inputs that determine crack initiation. Another method for implementing a rate‐dependent plastic material within a peridynamic numerical model was proposed by Foster et  al. (2010, 2011). The resulting material model implementation was fitted to rate‐dependent test data on 6061‐T6 aluminum alloy. It was shown that with this material model, the peridynamic method accurately reproduces the experimental results for Taylor impact tests over a wide range of impact velocities. The resulting model retained the advantages of the peridynamic formulation regarding discontinuities. A failure criterion was presented by analyzing the energy required to break all bonds across a plane of unit area (energy release rate). This crite­ rion enabled one to determine the critical energy density required to irreversibly fail a single bond. By failing individual bonds, this allows cracks to initiate, coalesce, and propagate without a prescribed external crack law. This was demonstrated using exper­ imentally collected fracture toughness measurements to evaluate the energy release rate. The capability of the micropolar peridynamic theory to analyze elastic behavior of plates with various length and width has been examined by Ferhat and Ozkol (2010). The displacement fields of these structures were computed using the micropolar peri­ dynamic model while Poisson’s ratios are kept constant. The results were compared both to the analytical solution of the classical elasticity theory and to the solution of displacement‐based finite element methods. In the peridynamic theory the constitutive model contains only central forces and can be applied only to the materials having ¼ Poisson’s ratio. The strict Poisson’s ratio limitation of the peridynamic theory was overcome using the proposed micropolar peridynamic theory. The response of a state‐based peridynamic material was studied by Silling (2009a, 2010a) for a small deformation superposed on a large deformation. A state‐based peri­ dynamic material model describes internal forces acting on a point in terms of the col­ lective deformation of all the material within a neighborhood of the point. The appropriate notion of a small deformation restricts the relative displacement between points, but it does not involve the deformation gradient, which would be undefined on a crack. The material properties that govern the linearized material response were expressed in terms of a new quantity called the modulus state to determine the force in each bond resulting from an incremental deformation of itself or of other bonds. Conditions were derived for a linearized material model to be elastic, and to satisfy balance of angular momentum. If the material is elastic, then the modulus state can be obtained from the second Frechet derivative of the strain energy density function. The equation of equilibrium with a linearized material model is a linear Fredholm integral equation of the second kind. The fatigue failure of materials due to cyclic loading was predicted by Oterkus et al. (2010c) using the peridynamic theory. It was found that failure occurs when and where it is energetically favorable. Fatigue life prediction is a natural extension of crack initia­ tion and growth while allowing material degradation. The crack growth phase is viewed as a quasi‐static series of discrete crack growth steps. Crack growth process was found to be controlled by the critical value of material stretch, and that cyclic loading caused degradation in the critical stretch. When the critical amount of stretch is reached, stable crack growth occurs. The energy and momentum conservation laws of the peridynamic

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non‐local continuum theory using the principles of classical statistical mechanics were derived by Sears and Lehoucq (2009) and Lehoucq and Sears (2011). The peridynamic laws allow the consideration of discontinuous motion, or deformation, by relying on integral operators. The derivation employed a general multi‐body interatomic potential, avoiding the standard assumption of a pairwise decomposition. The integral operators were expressed in terms of a stress tensor and heat flux vector under the assumption that these fields are differentiable, demonstrating that the classical continuum energy and momentum conservation laws are consequences of the more general peridynamic laws. An important conclusion was that non‐local interaction is intrinsic to continuum conservation laws when derived using the principles of statistical mechanics. A generalization of the original peridynamic framework for solid mechanics was proposed by Silling et al. (2007). This generalization allowed the response of a material at a point to depend collectively on the deformation of all bonds connected to the point. A mathematical object called a deformation state was defined, a function that maps any bond onto its image under the deformation. A similar object called a force state was defined, which contains the forces within bonds of all lengths and orientation. The relation between the deformation state and the force state was found to be the constitutive model for the material. In addition to providing a more general capability for reproducing material response, the new framework provided a means to incorpo­ rate a constitutive model from the conventional theory of solid mechanics directly into a peridynamic model. It allowed the condition of plastic incompressibility to be enforced in a peridynamic material model for permanent deformation analogous to conventional plasticity theory. The application of peridynamics theory for crystal plasticity simula­ tions was presented by Sun and Sundararaghavan (2014), based on the state‐based theory of peridynamics developed by Silling et al. (2007). The stress tensor at a particle was computed from strains calculated by tracking the motion of surrounding particles. A quasi‐static implementation of the peridynamics theory was developed, which employs an implicit iterative solution procedure similar to a nonlinear finite element implementation. Peridynamics results were compared with crystal plasticity finite element analysis for the problem of plane strain compression of a planar polycrystal. The peridynamic formulation was employed by Dayal and Bhattacharya (2006) to study the motion of phase boundaries in one dimension. A nucleation criterion was derived by examining nucleation as a dynamic instability. The induced kinetic relation was found by analyzing the solutions of impact and release problems. The interaction of a phase boundary with an elastic non‐transforming inclusion in two dimensions was also studied. The phase boundaries were found to remain essentially planar with little bowing. Furthermore, a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion was found while the original phase boundary slows down or stops. Transformation was found to pro­ ceed as the freshly nucleated phase boundaries propagate, leaving behind some untrans­ formed martensite around the inclusion. A condition for the emergence of a discontinuity in an elastic peridynamic body was proposed by Silling et  al. (2010). The condition resulted in a material stability condition for crack nucleation. The condition was derived by determining whether a small discontinuity in displacement, superposed on a possi­ bly large deformation, grows over time. Stability was shown to be determined by the sign of the eigenvalues of a tensor field that depends only on the linearized material properties. This condition for nucleation of a discontinuity in displacement can be

Dynamic Fracture and Peridynamics

interpreted in terms of the dynamic stability of plane waves with very short wavelength. A numerical example illustrated that cracks in a peridynamic body form spontaneously as the body is loaded. A method to couple peridynamic theory and finite element analysis was introduced by Kilic and Madenci (2009a, 2010c). Peridynamics is used in the regions where failure is expected, and the remaining regions are modeled utilizing the finite element method. The study was demonstrated through a simple problem, and predicted results were compared with those the peridynamic theory and finite element method. The damage simulation results for the present method were demonstrated by considering a plate with a circular cutout. A systematic analytical treatment of peristatic and peridynamic problems for a one‐dimensional infinite rod were analytically studied by Mikata (2012). Dispersion curves and group velocities for the materials with three different micromoduli were also studied. It was found that some peridynamic materials can have negative group velocities in certain regions of wavenumber. This indicates that peridynamics can be used for modeling certain types of dispersive media with anomalous dispersion. 3.5.3  Numerical Simulation

The peridynamic theory was numerically simulated for damage prediction of many problems. For example, Silling (2003) and Silling and Askari (2004, 2005) considered the Kalthoff–Winkler experiment in which a plate having two parallel notches was sub­ jected to impact by a cylindrical impactor, and the peridynamic simulations captured the angle of crack growth observed experimentally under impact. The basis of the peri­ dynamic theory and its numerical implementation in a three‐dimensional code called EMU6 (E for Young’s modulus and MU for shear modulus) were outlined by Silling and Askari (2004) for modeling the problem of impact damage. The study was supported by simulations of a Charpy V‐notch test, accumulated damage in concrete due to multiple impacts, and crack fragmentation of a glass plate. A new constitutive model was introduced for tearing and stretching of rubbery materials by Silling and Askari (2005). The oscillatory crack path was predicted when a blunt tool is forced through a membrane. A numerical approach was developed by Weckner and Emmrich (2005) to calculate the one‐dimensional dynamic response of a non‐local, peridynamic bar composed of non‐homogeneous linear material. The principal physical characteristic of the response was manifested under long‐range forces leading to nonlinear dispersion relations. Askari et al. (2006) and Colavito et al. (2007a, 2007b) utilized peridynamic to predict damage in laminated composites subjected to low‐velocity impact and static indentation. Xu, J.F. et al. (2007b) and Kilic (2008) considered notched laminated com­ posites under biaxial loads. An implicit time integration algorithm for a non‐local, state‐based, peridynamics plasticity model was developed by Mitchell (2011). The flow rule was proposed to establish the first ordinary, state‐based peridynamics plasticity model. Integration of the flow rule utilizes elastic force state relations, an additive decomposition of the deformation state, an elastic force state domain, a flow rule, loading/unloading 6  EMU is a peridynamic computer program that assumes the material particles to be rigid bodies of finite, rather than infinitesimal, size. This modeling assumption requires a huge number of such material particles even for modeling relatively simple structures.

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conditions, and a consistency condition. It was shown that the resulting constitutive model does not violate the second law of thermodynamics. A non‐local yield criterion that depends upon the yield stress and horizon for the material was developed. A perfectly matched layer absorbing boundary layer was formulated and numerically applied to peridynamics in two dimensions by Wildman and Gazonas (2012a, 2012b). An absorbing boundary layer is characterized by exponentially decaying waves without introducing reflections at the boundary between the computational region and the absorbing layer. Wildman and Gazonas (2012b) adopted state‐based peridynamics as perfectly matched layers are considered essentially anisotropic absorbing materials. Bessa et  al. (2014) established the link between the mesh‐free state‐based peridynamics method and other mesh‐free methods, in particular with the moving least squares reproducing kernel particle method. It was concluded that the discretization of state‐based peridynamics leads directly to an approximation of the derivatives that can be obtained from the kernel particle method. The development of discretized bond‐based peridynamics for solid mechanics was presented by Liu W.Y. (2012) and Liu W.Y. and Hong (2012a, 2012b) who also addressed the connection between the classical elasticity and the discretized peridynamics in terms of peridynamic stress. Numerical micromoduli for one‐ and three‐dimensional models were derived. The elastic responses of one‐ and three‐dimensional peridynamic models were also studied. A pairwise compensation scheme was introduced for simula­ tions of an elastic body of Poisson ratio not equal to ¼. Numerical studies were con­ ducted to investigate the responses of brittle and ductile material models. A comparison of stresses and strains between finite element analyses and peridynamic solutions was performed for a ductile material. An approach to couple the discretized peridynamics and the finite element method was developed to take advantage of the generality of peridynamics and the computational efficiency of the finite element method. The coupling of peridynamic and finite element sub‐regions was achieved by means of interface elements. Two types of coupling schemes were introduced, namely the VL‐ coupling scheme (the whole volume of the interface element is subjected to coupling forces) and the CT‐coupling scheme (the interface between the peridynamic and finite element sub‐regions is similar to a contact surface). A numerical scheme for the contact‐impact procedure ensuring compatibility between a peridynamic domain and a non‐peridynamic domain was developed. Hong (2012) presented a literature review and development of a peridynamic solver together with the development of an auto­ matic volume calculation routine using the partition of unity principle. The study included the development of peridynamic and finite element coupling module using parallelization of the solvers. A numerical method for solving dynamic problems within the peridynamic theory was described by Silling and Askari (2005). Accuracy and numerical stability were dis­ cussed, and this illustrated the properties of the method for modeling brittle dynamic crack growth. The discontinuous Galerkin method was derived for elasto‐statics based on the peridynamic theory by Aksoy and Senocak (2011). Numerical analyses were performed for different problems, and the numerical results were compared with those of known exact solutions. Yu, K. et al. (2011) and Tisan and Du (2013) presented differ­ ent numerical solutions of non‐local constrained value problems associated with linear non‐local diffusion and non‐local peridynamic models. Standard finite element methods and quadrature‐based finite difference methods were used. The similarities and

Dynamic Fracture and Peridynamics

differences of the resulting non‐local stiffness matrices were illustrated and discussed. The issue of convergence in both the non‐local setting and the local limit was con­ sidered. Integration of the peridynamic force over different intersection volumes was calculated using an adaptive trapezoidal integration scheme with a combined relative‐ absolute error control. Numerical examples were presented to demonstrate the accuracy of the proposed approach. Brothers et al. (2014) established the suitability of a method for numerical computation of tangent‐stiffness operators, referred to as complex‐step, and compared the results with other techniques for numerical derivative calculation: automatic differentiation, forward finite difference, and central finite difference. The complex‐step method was implemented in a massively parallel computational peridy­ namics code for the purpose of this comparison. The forces and a particle equation of motion for the prototype microelastic brittle material model were developed by Parks et al. (2008a). The prototype microelastic brittle model was originally introduced by Silling and Askari (2005) and does not allow “healing” because once a bond between two particles is broken, the bond remains broken. The assumption is that once the underlying material fractures, then the material remains fractured. This is in contrast to molecular dynamics, where force interactions between atoms may be zero or non‐zero over time. A document for peridynamics with LAMMPS (Large‐scale Atomic/Molecular Massively Parallel Simulator) developed by Parks et al. (2008b, 2011) presented the implementation of a discrete peridynamic model within the LAMMPS molecular dynamic code. This document provides a brief over­ view of the peridynamic model of a continuum, discusses how the peridynamic model is discretized, and outlines the LAMMPS implementation. Parks et al. (2008a) described the implementation of peridynamic theory within a molecular dynamics framework so enabling mesoscale7 and macroscale modeling. This adds a computational mechanics capability to a molecular dynamic code, enabling simulations at mesoscopic or even macroscopic lengths and timescales. Peridynamics and molecular dynamics have simi­ lar discrete computational structures, as peridynamics computes the force on a particle by summing the forces from surrounding particles, similarly to molecular dynamics. Parks et al. (2008a) considered the impact of a rigid sphere of diameter 0.01 m and velocity of 100 m/s directed normal to the surface of the target on a homogeneous block of brittle prototype microelastic material model of a cylinder of diameter 0.074 m and thickness 0.0025 m, and containing 103,110 particles. The region defining a peridynamic material was discretized into particles forming a cubic lattice with lattice constant a. The effects of symmetry‐breaking and the peridynamics horizon δ upon the solution is demonstrated in Figure 3.23, which was obtained numerically by Parks et al. (2008a) for four different values of δ 1.5a, 2.0a, 2.25a, and 3.0a , where a is the side length of the cubic lattice, and the initial mesh is either perturbed or unperturbed. It is seen that the results do not qualitatively depend on regularity of the lattice for large enough δ. However, a symmetric initial mesh acted upon by a symmetric projectile produces a symmetric solution. It was demonstrated by Seleson et al. (2009) that the peridynamics 7  Materials of an intermediate length scale, as being between the size of a quantity of molecule and of materials measuring micrometers. At the micrometer level are bulk materials. Both mesoscopic and macroscopic objects contain a large number of atoms. Whereas average properties derived from its constituent materials describe macroscopic objects, as they usually obey the laws of classical mechanics, a mesoscopic object is affected by fluctuations around the average, and is subject to quantum mechanics.

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Figure 3.23  Top monolayer of target after impact for different values of δ, and for initially unperturbed and perturbed meshes. For sufficiently large δ, crack growth is arbitrary. Perturbation of the initial mesh acts only to break symmetry of the solution. (Parks et al., 2008a)

model can be cast as an upscaling of molecular dynamics. The extent to which the solu­ tions of molecular dynamics simulations can be recovered by peridynamics was dis­ cussed. An analytical comparison of the equations of motion and dispersion relations for molecular dynamics and peridynamics was presented along with supporting com­ putational results. In order to take advantage of the computational robustness of the finite element method, Macek and Silling (2007) implemented the peridynamic model in a conven­ tional ABAQUS finite element analysis code by representing the peridynamic interac­ tions with truss elements and using embedded element technique for the overlap region. Kilic and Madenci (2010c) adopted the peridynamic theory since it uses displacements rather than displacement derivatives. They developed an approach to combine the peri­ dynamic theory and finite element analysis in one treatment. The regions where failure is expected were modeled using peridynamics while remaining regions were modeled utilizing the finite element method. The coupling introduced an overlap region in which both the peridynamic and finite element equations are used simultaneously. The peridynamics method was examined by Becker and Lucas (2011), who studied deformation conditions commonly observed before and after dynamic material fail­ ure. They used a combination of full simulations, targeted numerical techniques, and analytic solutions. The results revealed numerical dispersion and boundary condition artifacts expected with non‐local methods, anomalous volume‐shear coupling related to the numerical grid, and limitations of the commonly used material model imple­ mentations for large strain and impact simulations. Differences and agreements between peridynamics and conventional finite element formulations were addressed.

Dynamic Fracture and Peridynamics

Both peridynamics and classical finite element analysis were applied to simulate mate­ rial response under dynamic blast loading conditions by Littlewood (2010). A com­ bined approach was utilized such that the portion of the simulation modeled with peridynamics interacts with the finite element portion of the model via a contact algorithm. The peridynamic interface to the constitutive model was based on the calculation of an approximate deformation gradient, requiring the suppression of possible zero‐energy modes. The classical finite element portion of the model utilizes a Johnson–Cook constitutive model. Later, Littlewood et al. (2012) employed peri­ dynamics in combination with modal analysis for the prediction of characteristic frequency shifts throughout the damage evolution process. The application of modal analysis to peridynamic models enabled the tracking of structural modes and characteristic frequencies over the course of a simulation. Shifts in character­ istic frequencies resulting from evolving structural damage were isolated and utilized in the analysis of frequency responses observed experimentally. A methodology for quasi‐static peridynamic analyses, including the solution of the eigenvalue problem for identification of structural modes, was developed. Repeated solution of the eigen­ value problem over the course of a transient simulation yielded a dataset from which critical shifts in modal frequencies can be isolated. The computed natural frequencies of an undamaged simply supported beam were found to agree with the classical local solution. Analyses in the presence of cracks of various lengths were shown to reveal frequency shifts associated with structural damage. A collocation approach was presented by Evangelatos and Spanos (2012) for spatial discretization of the partial integro‐differential equation arising in a peridynamic for­ mulation in stochastic fracture mechanics. Nodes were distributed inside the domain forming a grid, and the inverse multi‐quadric radial basis functions were used as inter­ polation functions inside the domain. The peridynamic stiffness was generated in a manner similar to the finite element method. Any discontinuity in the domain was included and affected only the peridynamic stiffness of the adjacent nodes. The proba­ bility density function of the energy release rate was determined at a given crack tip point for all possible crack paths. Thus, the crack propagation direction can be proba­ bilistically identified. This was accomplished by numerical evaluation of the requisite Neumann expansion using pertinent Monte Carlo simulations. Liu, D. and Jia (2012) showed that peridynamic simulation of the damage process does not require any knowledge of the damage location and orientation prior to the simulation. On the other hand, finite element analysis requires knowledge of damage location and orientation in advance to impose a special finite element mesh, such as initial damage elements and cohesive zone layers, for damage simulations. Based on these differences, peridynamics should be more suitable for simulating dynamic damage process in composite materials, which have different properties in different locations and different orientations. A two‐dimensional model of composite damage process was proposed by Jia and Liu (2013) using peridynamic theory. Two computational algorithms were introduced and incorporated into peridynamic programming to significantly improve its computational efficiency. The uses of the two‐dimensional peridynamic model for simulating dynamic damage process in composite materials were favorably validated by experiments. A four‐parameter peridynamic model was subsequently presented for investigating orthotropic materials. The model was verified by analyses involving uniaxial tension and vibration. It was also validated with

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single‐edge‐notch testing results. Later, Jia and Liu (2014) presented the simulation of wave propagation in a split Hopkinson’s pressure bar based on peridynamics. 3.5.4  Horizon Convergence

As stated earlier, the peridynamic model of solid mechanics is a non‐local theory containing a length scale. It is based on direct interactions between points in a contin­ uum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. Silling and Lehoucq (2008) addressed the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. It was shown that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridy­ namic stress tensor converges in this limit to a Piola–Kirchhoff stress tensor8 that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting Piola–Kirchhoff stress tensor field is differentiable, and its divergence repre­ sents the force density due to internal forces. The limiting, or collapsed, stress–strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions. Adaptive refinement algorithms, for the non‐local method peridynamics for discontinuities and long‐range forces were intro­ duced by Bobaru et al. (2009). These algorithms used scaling of the micromodulus and horizon. The particular features of adapting peridynamics for which multi‐scale modeling and grid refinement are closely connected were discussed. Three types of numerical convergence for peridynamics were presented in an attempt to obtain uniform conver­ gence to the classical solutions of static and dynamic elasticity problems in one‐dimen­ sional in the limit of the horizon going to zero. Continuous micromoduli lead to optimal rates of convergence independent of the grid used, while discontinuous micromoduli produce optimal rates of convergence only for uniform grids. The relative error for the static and dynamic solutions obtained using adaptive refinement were found to be signifi­ cantly lower than those obtained using uniform refinement, for the same number of nodes. The peridynamic analysis of dynamic crack branching in brittle materials was pre­ sented by Ha and Bobaru (2010). The study includes results of convergence studies under uniform grid refinement (m‐convergence) and under decreasing the peridynamic horizon (δ‐convergence). Comparisons with experimentally obtained values were made for the crack tip propagation speed with three different peridynamic horizons. The influence of the particular shape of the micromodulus function and of different materials (Duran 50 glass and soda‐lime glass) on the crack propagation behavior was presented. It was shown that the peridynamic solution captures the main features observed experi­ mentally pertaining to dynamic crack propagation and branching. The branching patterns also correlated with tests published in the literature that showed several branching levels at higher stress levels reached when the initial notch starts propagating. The strong influence reflecting stress waves from the boundaries on the shape and 8  The first Piola–Kirchhoff stress tensor relates forces in the current configuration to areas in the reference configuration. On the other hand, the second Piola–Kirchhoff stress tensor relates forces in the reference configuration to areas in the reference configuration. The force in the reference configuration is obtained via a mapping that preserves the relative relationship between the force direction and the area normal in the reference configuration.

Dynamic Fracture and Peridynamics

structure of the crack paths in dynamic fracture was reported. Bobaru and Ha (2011) proposed adaptive refinement algorithms for the bond‐based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multi‐scale modeling, and in peridy­ namics changing the horizon is directly related to multi‐scale modeling. The peridy­ namic theory with a variable horizon size was implemented in two dimensions, and the results revealed convergence to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero. Ha and Bobaru (2011a, 2011b) employed a bond‐based peridynamic model to study the characteristics of dynamic brittle fracture observed experimentally; crack branch­ ing, crack‐path instability, asymmetries of crack paths, successive branching, and sec­ ondary cracking at right angles from existing crack surfaces. The source of asymmetry in the crack path in numerical simulations with an isotropic material and symmetric coordinates about the pre‐crack line was analyzed. Asymmetries in the order of terms in computing the nodal forces were found to lead to different round‐off errors for sym­ metric nodes about the pre‐crack line. This resulted in inducing the observed slight asymmetries in the branched crack paths. A dramatically enhanced crack‐path instabil­ ity and asymmetry of the branching pattern were obtained when fracture energy values that change with the local damage were used. Bobaru et al. (2012a, 2012b) developed a peridynamic model for a multi‐layered glass system and obtained results for a high‐ velocity impact test. The main features reported in the experimental results were cap­ tured in the peridynamic results. The model predicted both diffuse damage zones, indicative of closely spaced cracks, and crevice cracks that develop in the system due to bending deformations. The peridynamic model was found to uncover the time‐evolu­ tion of damage and the dynamic interaction between shock waves, propagating cracks, interfaces, and bending deformations in three‐dimensions. Bobaru and Hu (2012) dis­ cussed the peridynamic horizon, which defines the non‐local region around a material point, its role, and its practical use in modeling. An example of crack branching in a nominally brittle material (homalite) was adopted, and it was shown that crack branch­ ing takes place without wave interaction. Well‐posedness and structural properties of the peridynamic equation of motion were established for the linear case corresponding to small relative displacements by Emmrich and Weckner (2007c). Chen and Gunzburger (2010) used piecewise constant and discontinuous piecewise linear functions in regions of discontinuities, and used continuous piecewise linear function in areas where the solutions is smooth. The choice of the horizon radius to implement the peridynamic model more accurately was consid­ ered, and cases when radius is fixed as a constant or as a function of grid distance were tested. Du et  al. (2013a) proposed an adaptive finite element algorithm for the numerical solution of a class of non‐local models which correspond to non‐local diffusion equa­ tions and linear scalar peridynamic models with certain nonintegrable kernel functions. The convergence of the adaptive finite element algorithm was rigorously derived with the help of several basic ingredients, such as the upper bound of the estimator, the estimator reduction, and the orthogonality property. Well‐posedness results for the state‐based peridynamic non‐local continuum model of solid mechanics were estab­ lished with the help of a non‐local vector calculus by Du et  al. (2013b) using basic

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operators of the non‐local calculus developed by Du et al. (2013c). The peridynamic strain energy density for an elastic constitutively linear anisotropic heterogeneous solid was expressed in terms of the field operators of that calculus. The peridynamic Navier equilibrium equation was then derived as the first‐order necessary conditions and this is shown to reduce, for the case of homogeneous materials, to the classical Navier equation as the extent of non‐local interactions vanishes. Using standard results, well‐ posedness was also established for the time‐dependent peridynamic equation of motion. Mengesha and Du (2014) generalized the previous well‐posedness results that were formulated for very special kernels of non‐local interactions. They provided a more rigorous treatment to the convergence of solutions to non‐local peridynamic models to the solution of the conventional Navier equation of linear elasticity as the horizon parameter goes to zero. The results were valid for arbitrary Poisson ratio, which is a characteristic of the state‐based peridynamic model. Several meshing strategies for use with a common peridynamics solution scheme were considered by Henke and Shanbhag (2014). The study included convergence behavior, the effects of model parameters, and sensitivity to perturbations. A qualitative comparison of the fracture patterns that result, and suggestions for best practices for generating meshes that lead to efficient, high‐quality numerical simulations of peridy­ namic models was presented. A two‐dimensional state‐based peridynamics model was presented by Le et al. (2014) and verified by simulating a two‐dimensional rectangular plate with a round hole in the middle under constant tensile stress. Dynamic relaxa­ tion and energy minimization methods were used to find the steady‐state solution. The model showed m‐convergence and δ‐convergence behaviors when m increases and δ decreases. Simulation results exhibited a close quantitative matching of the displacement and stress obtained from the two‐dimensional peridynamics and a finite element model. 3.5.5 Application 3.5.5.1 Metals

As an application of peridynamic formulation of elasticity theory, Silling et al. (2003) and Zimmermann (2005) considered the deformation of an infinite bar subjected to a self‐equilibrated load distribution. The bar problem was formulated as a linear Fredholm integral equation and solved using Fourier transform methods. The solution was found to exhibit features such as decaying oscillations in the displacement field and progres­ sively weakening discontinuities that propagate outside of the loading region. It was argued that these features, when present, are guaranteed to decay, provided that the wave speeds are real. This would lead to a one‐dimensional version of St Venant’s prin­ ciple for peridynamic materials that ensures the increasing smoothness of the displace­ ment field remotely from the loading region. The peridynamic result was found to converge to the classical result in the limit of short‐range forces. Silling et  al. (2003) highlighted some advantages of the peridynamic model. For example, it allows for the spontaneous emergence of discontinuities, in contrast to the classical theory, which predicts deformations with infinite smoothness in typical (elliptic) materials. Furthermore, the model includes long‐range forces between material particles, unlike the classical theory, which generally deals only with contact forces between particles. The peridynamic theory was employed for damage prediction of many problems.

Dynamic Fracture and Peridynamics

For example, Silling (2003) considered the Kalthoff–Winkler experiment in which a plate having two parallel notches was subjected to impact by a cylindrical impactor, and the peridynamic simulations successfully captured the angle of crack growth observed in the experiments. Impact damage was also predicted using peridynamics by Silling and Askari (2004, 2005) who considered a plate with a center crack to show the conver­ gence of their numerical method. Silling and Bobaru (2005) applied peridynamics theory to study stretching and tearing of membranes. They also studied string‐like structures that sustain tensile loads while interacting with each other through inter‐molecular and contact forces. Their work is an early effort to apply peridynamic theory at the nanoscale. The one‐dimensional dynamic response of an infinite bar composed of a linear “microelastic material” was studied by Weckner and Abeyaratne (2005) in which the constitutive model accounted for the effects of long‐range forces. The general theory was developed independently by Kunin (1983), Rogula (1982), and Silling (2000), and is called the peridynamic theory. The general initial‐value problem was solved, and the motion was found to be dispersive as a consequence of the long‐range forces. The result converges, in the limit of short‐range forces, to the classical result for a linearly elastic medium. Explicit solutions in elementary form were given in a broad class of special cases. Initially, the displacement field is continuous, but it involves a jump discontinuity for all later times, the Lagrangian location of which remains stationary. For some mate­ rials the magnitude of the discontinuity‐jump oscillates about an average value, while for others it grows monotonically, presumably fracturing the material when it exceeds some critical level. The dynamic crack initiation toughness of a 4340 steel was studied by Foster (2009) who conducted an experimental technique for measuring KIc in metals using the Kolsky (1949) bar by using pulse shaping techniques to ensure a constant loading rate applied to the sample before crack initiation. Dynamic crack initiation measurements were reported on a 4340 steel at two different loading rates. The steel was shown to exhibit a rate dependence, with the recorded values of KIc, being much higher at the higher loading rate. A failure criterion was introduced to model the dynamic crack initiation toughness experiments. The failure model was based on an energy criterion and employed the KIc values recorded experimentally as an input. The failure model was validated, and revealed good agreement with experimental results. The double torsion technique is a powerful experimental method for fracture param­ eter characterization of brittle materials. In particular, the geometry provides a crack length independent test configuration. The double torsion technique employs traditional approaches such as cohesive elements, extended finite elements, and analytically enriched finite elements. Becker and Turner (2013) considered the double torsion geometry with the use of peridynamics to model the fracture of brittle materials. Peridynamics was found to overcome several of the challenging aspects of modeling the double torsion geometry. For example, discontinuities are naturally part of the solution, the crack path can be arbitrary, and the formulation is robust for a wide variety of mate­ rial properties. The double torsion technique using peridynamics was compared with experimental results in terms of the obtained load, displacement, and cracking behavior in order to evaluate the efficacy of this approach for fracture test geometry analysis. A peridynamic state based model to represent the bending of an Euler‐Bernoulli beam was developed by O’Grady and Foster (2014). The model was non‐ordinary and derived from the concept of a rotational spring between bonds. While multiple

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peridynamic material models capture the behavior of solid materials, the proposed model was claimed to be the first one‐dimensional state based peridynamic model to resist bending. For sufficiently homogeneous and differentiable displacements, the model was shown to be equivalent to Eringen’s non‐local elasticity. As the peridynamic horizon approaches 0, it reduces to the classical Euler–Bernoulli beam equations. The influence function in the peridynamic theory was used to weight the contribu­ tion of all the bonds participating in the computation of volume‐dependent properties. Seleson and Parks (2011) used influence functions to establish relationships between bond‐based and state‐based peridynamic models. They demonstrated how influence functions can be used to modulate non‐local effects within a peridynamic model inde­ pendently of the peridynamic horizon. The effects of influence functions by studying wave propagation in simple one‐dimensional models and brittle fracture in three‐ dimensional models were explored. An approach to incorporate classical continuum damage models in the state‐based theory of peridynamics was proposed by Tupek et al. (2013). The approach provided the description of the damage evolution process in peridynamics according to well‐established models. It is based on modifying the peridynamic influence function according to the state of accumulated damage. The proposed peridynamic damage formulation was implemented for the particular case of a well‐established ductile damage model for metals. The model was applied to the simulation of ballistic impact of extruded corrugated aluminum panels and compared with experiments. An extension of the constitutive correspondence framework of peri­ dynamics was proposed by Tupek and Radovitzky (2014). A solution that is rooted directly within the non‐local theory was proposed by introducing generalized non‐local peridynamic strain tensors. Lipton (2014) provided a link between peridynamic evolu­ tion and brittle fracture evolution for a broad class of peridynamic potentials associated with unstable peridynamic constitutive laws. Distinguished limits of peridynamic evo­ lutions were identified that correspond to vanishing peridynamic horizon. The limit evolution was found to be both bounded linear elastic energy and Griffith surface energy. The limit evolution corresponds to the simultaneous evolution of elastic displace­ ment and fracture. For points in space‐time not on the crack set, the displacement field was found to evolve according to the linear elastic wave equation. The wave equation provides the dynamic coupling between elastic waves and the evolving fracture path inside the media. The elastic moduli, wave speed, and energy release rate for the evolu­ tion were explicitly determined by moments of the peridynamic influence function and the peridynamic potential energy. 3.5.5.2  Concrete Structures under Extreme Loading

The response of civil structures to severe impact, such as the accidental or terrorist impacts of large commercial aircraft, has prompted the interest of studying the impact dynamic characteristics of reinforced concrete structures. Most concrete structures can withstand significant damage, and analytically capturing the failure mechanisms in reinforced concrete, including severe cracking, crushing, shear plug formation, and rebar rupture or bond slip and pullout, is a formidable challenge. The state of the art of finite element based constitutive modeling and computational methodology of rein­ forced concrete with emphasis on severe damage modeling and failure evaluation was presented by James et al. (2003). Verification and validation of the developed method­ ology was illustrated using high‐velocity impact tests conducted in the USA and Japan.

Dynamic Fracture and Peridynamics

Later, James and Rashid (2005) developed constitutive modeling of reinforced concrete and analytical methods with emphasis on damage evaluation from severe impact loads. Hanus et  al. (2010) presented experimental results of the mechanical response of concrete. Dynamic three‐point bend tests on small beam specimens and shots of soft projectiles on a reinforced concrete slab revealed different failure modes (bending or shear fracture, single or multiple fractures) depending to the loading rate. The non‐local peridynamic model assumes “micromaterial” constants of a linear “spring” connecting any two particles for small relative displacements. The spring breaks when the relative tensile displacement between the two particles exceeds u *. The interac­ tion takes place only among the particles closer together than a distance, δ (the material horizon), in the deformed configuration. Gerstle et  al. (2005, 2007a) extended Silling’s model by including moment densities to act between infinitesimal particles. These infini­ tesimal frame elements were named as micropolar peridynamic links. The infinitesimal material particles move in accordance with Newton’s second law, in response to both external forces and internal forces which result from the deformations and rotations of the ends of the peridynamic links. The peridynamic theory was applied to the quasi‐static deformation of concrete by Gerstle and Sau (2004) and Gerstle et al. (2005). Due to the complexity of the interaction between the concrete and the reinforcing steel, a large number of cracks can form at significantly different size scales. Using the finite element method, cracks can be modeled either discretely or in a smeared fashion. Gerstle et al. (2007c) indicated that neither the smeared crack nor the discrete crack approach is fully satisfactory for modeling the behavior of reinforced concrete struc­ tures at all size scales. Gerstle et al. (2007c) presented a peridynamic model for simulating the behavior of reinforced concrete structures. It was shown that as the grid is refined spatially, the stable time step size is also decreased. Thus, the number of computations required increases dramatically with grid refinement. In 1993, Sandia National Laboratories conducted an impact test in which an F‐4 Phantom aircraft impacted a massive, essentially rigid reinforced concrete wall (Sugano et al. 1993). The purpose of the test was to determine the impact force as a function of time. Additional objectives of the test were to determine the crushing behavior of the aircraft, to determine if the engines broke away from the aircraft before their impact, and, if so, to measure their impact velocity, and to record the dispersal of fuel after impact. Water was used instead of jet fuel for this test. The target was a block of rein­ forced concrete 3.7 m thick in the direction of impact and 7.0 m square perpendicular to the impact direction. It was mounted on a platform. The impact was perpendicular to the target at a speed of 215 m/s. The impacting mass of the aircraft and fuel surrogate was slightly more than 18,000 kg. The length of the F‐4 is about 1.77 m. Demmie and Silling (2007) were motivated by the results of the Sandia National Laboratories tests and considered the problem of extreme loading of structures using peridynamics with an application to extreme loadings on reinforced concrete structures by impacts from massive objects, and using the EMU computer code. Peridynamic theory was extended to model composite materials, fluids, and explosives. Figure 3.24 shows a top and side view of materials in the EMU model at time zero and materials at 0.05 s and 0.09 s dur­ ing the simulation of the experiment. Figure 3.25 shows the corresponding sequence of damage to the concrete structure. The results did not reveal perforation of the target because of the target’s strength. Only crush‐up of the aircraft occurs during the simulation, as was observed during the experiment.

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Figure 3.24  Top and side views of time sequence of materials damage during F‐4 impact simulation. (Adapted from Demmie and Silling, 2007)

Figure 3.25  Side and front views of time sequence of concrete damage during F‐4 impact simulation. (Adapted from Demmie and Silling, 2007)

The peridynamic model was generalized by Gerstle et al. (2007b) by adding pairwise peridynamic moments to simulate linear elastic materials with varying Poisson’s ratios. This model, referred to as the micropolar peridynamic model, was formulated within a finite element context to enable efficacious application of boundary conditions and

Dynamic Fracture and Peridynamics

efficient computational solutions using an implicit solution algorithm. The algorithm is suitable for quasi‐static simulation of damage and cracking in concrete structures. Very simple tensile damage mechanisms at the microstructural (peridynamic) level were found sufficient to explain a great deal of the microcracking and fracture mechanics observed in concrete structures. The formulation of the micropolar peridynamic model was documented by Sau (2008) who proposed different constitutive damage models for quasi‐brittle materials. Conventional finite elements were used in regions of the problem where the displacement is expected to be continuous and the peridynamic model was used in regions where discontinuities develop. The second program imple­ ments an explicit model that uses the dynamic relaxation method to solve the governing micropolar peridynamic equations. The micropolar model developed by Gerstle et al. (2007b) assumed that the pairwise force function between particle i and any other particle j depends not only upon the axial stretch, s, of a link between particles i and j, but also upon the maximum stretch, smax, of any other peridynamic link connected to either particle i or particle j. With reference to Figure 3.26, the link remains linearly elastic as long as its stretch, s, exceeds a compressive stretch limit, sc, and is less than a tensile stretch limit, st. However, if s exceeds the tensile stretch limit, st, the tensile peridynamic force suddenly drops to β times its former value, after which it remains constant with increasing stretch until s exceeds another specific limit, αtst, after which the pairwise force drops to zero. It is also assumed that the micropolar moments are instantly reduced to zero if the stretch s exits the linear region sc s st . If s is less than a compressive limit, sc, two possibilities exist. If the maximum transverse stretch is less than st, the link remains linear elastic, while if it is greater than st, the compressive force remains constant until s exceeds another specific limit, αcsc, above which it drops to zero. The sensitivity to the transverse stretch was reported to be caused by the loss of lateral support and consequent com­ pressive instability of the peridynamic link. The plateau in the compressive regime simulates energy dissipation due to friction. Figure  3.27 shows the predicted stress– strain plots for different values of transverse stress as computed from plain strain f cst β(cst) αcsc

sc

s st

αtst

c 1 smax>st smax70

30 40 Time (sec) >35

>20

50 >9

60

S1

>4

Figure 4.8  Distribution of acoustic emission energy in (μJ) over the beam length for carbon fiber sandwich (showing location of acoustic emission sensors S1–S4). (Ayorinde et al., 2012)

significantly higher stresses in this region rather than in that between the loading rollers is reflected in the grayscale coding of the energy levels. The load–deflection plot combined with the acoustic emission energy time evolution for the glass fiber sandwich beam is shown in Figure 4.9(a). Comparing with the load– deflection curve of the carbon sandwich shown in Figure 4.4(a) reveals that the peak load occurs at a higher deflection than that of Figure 4.4(a). The peak load for the glass sandwich (about 175 lb) is lower than that of the carbon sandwich (about 203 lb). Figure 4.9(b) displays the evolution of acoustic emission amplitude. The load and time at which acoustic emission is detected is significantly higher than those of the carbon sandwich beam. In other words, the material appears to be not as acoustically active as the carbon sandwich. In fact, the acoustic emission amplitude plot is easier to use here in comparing data with the load–deflection curve. Also, on account of the differences in material behavior of the two specimens, there is no exact one‐to‐one correspondence in the milestone posts “a” to “g”, used for the carbon sandwich. Figure 4.10 shows the waveform of acoustic emission during the occurrence of the load‐drop, point “g” in Figure 4.9(b). Burst emissions are observed. Such emissions are symptomatic of cracks, and prominent ones such as shown in Figure 4.10 signify large cracks. The peak frequency plot for the glass sandwich material in Figure 4.11 shows more continuity in the medium frequency range than that for the carbon sandwich, once acoustic emission starts. The test truly progresses to catastrophic failure more quickly than that of the carbon sandwich. The low frequency band is also densely populated and is rich in the low amplitude events that show large‐scale cracking. The peak frequency attained by damage events is also lower than that of the carbon sandwich, being only about two‐thirds (approx. 800–1200 kHz). It appears that any large strains or micro damage in the glass fibers, or associated micro damage events, respond at lower frequencies in the glass material case than for the carbon sandwich. Figure  4.12 shows

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(a) 200

3 Load

150

d

2

e 100

a 1

AE energy

Load (lb)

Log [Energy (μJ)]

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Figure 4.9  Time evolution of (a) AE energy and load–deflection plot for glass fiber sandwich beam, (b) AE amplitude–time record of the glass fiber sandwich beam. (Ayorinde et al., 2012)

Figure 4.10  Waveform of emission at location around point “g” on the glass fiber sandwich response of Figure 4.9(b). (Ayorinde et al., 2012)

0.03 0.02 Magnitude (V)

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0.01 0 –0.01 –0.02 –0.03

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Structural Health Monitoring Basic Ingredients and Sensors

Peak frequency (kHz)

800 600 400 200 0

0

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Figure 4.11  Acoustic emission peak frequency versus time for the glass fiber sandwich beam. (Ayorinde et al., 2012) Figure 4.12  Three‐dimensional plot of peak acoustic emission frequency versus time and amplitude for glass sandwich. (Ayorinde et al., 2012) 12000

Peak freq. (kHz)

9000 6000 3000 0

100 80

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three‐dimensional plot of the evolution of acoustic emission frequency and amplitude for the glass sandwich. It shows the major crack signal at low frequency and high‐amplitude. Figure 4.13 shows the distribution of the location and energy levels of the acoustic emission events recorded in the damage region. Comparing with Figure 4.8, it is seen that there are fewer and lower‐intensity emissions in the GFRC sample than in the carbon fiber reinforced composite (CFRC) sample. This shows that the CFRC sample tends to be harder and less ductile than the GFRC counterpart. In‐situ surface observations are often used to examine damage processes for an interpretation of the underlying mechanisms. Real‐time damage detection using high‐speed photography provides a detailed sequence of damage development and its original initiation. High speed imaging is an important tool in a wide variety of scientific research applications. Imaging requirements such as frame rate, exposure time, number of

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y (mm) 150 S4

140

S3

130 120 110 100 90

S2 0

10

20

30

40

50

60

Time (sec) >70

>35

>20

S1 >9

>4

Figure 4.13  Distribution of AE energy in (μJ) over the beam length for glass fiber sandwich (showing location of acoustic emission sensors S1–S4). (Ayorinde et al., 2012)

images in a sequence, image resolution and quality, tri‐color versus monochromatic images, etc., depend on the application. In common usage, the term ‘high speed camera’ is used for cameras capable of capturing image sequences as well as for single shot cameras. A single shot high speed camera is a camera capable of capturing a high speed image (i.e. an image with a very short exposure time) that appears to freeze the motion of a moving object. The speed of such a camera simply refers to the inverse of the exposure time. On the other hand, high speed cameras are capable of capturing a sequence of high speed images with very short inter‐frame separation. The speed, or frame rate, of such a camera refers to the inverse of the inter‐frame time, but it is naturally understood that, for all practical purposes, the exposure time is less than, or at most equal to, the inter‐frame time. A variety of techniques can be used to overcome this limitation on the maximum framing rates imposed by the readout time. One of these techniques relies on reducing the size of the image to be read out (i.e. the pixel resolution) in order to reduce the readout time and therefore increase the frame rate. The damage development and its location are monitored by a Phantom v7.3 camera supported by Phantom 640 software, and 61,000 frames per second are selected. The Phantom v12.1 was also later used, at a speed of 100,000 fps. Figure 4.14(a) includes a set of slides showing the initiation and development of the crack. Figure 4.14(b) shows magnifications of two slides, one in the early stage of crack initiation (in the form of invisible microcracks) and the second is in the nearly developed stage of crack, which appears as a discontinuous line of crack at about 45°. Note that the crack in this particular carbon sandwich is initiated in the foam core, not in the skin. However, the skin showed some indentation without crack. Figure 4.15 shows selected magnified slides generated from the Phantom v12.1 camera at a speed of 100,000 fps. The last crack

Structural Health Monitoring Basic Ingredients and Sensors

(a)

(b) Loading roller

Lower skin

Upper skin

Foam core

Line of discontinuous core crack

Support roller

Figure 4.14  (a) Assembled CFRC sandwich beam failure slides (61,000 fps, Phantom video camera), (b) magnification of two slides of (a). (Ayorinde et al., 2012)

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(a)

Figure 4.15  Sample CFRC sandwich beam failure slides (100,000 fps, Phantom video camera). (Ayorinde et al., 2012) (a) Slides sequence of failure of CFRC beam, case 1 (b) Slides sequence of failure of CFRC beam, case 2

Structural Health Monitoring Basic Ingredients and Sensors

(b)

–800

–705

–797

–704

–779

–703

–743

–702

–708

–701

–707

–600

–706

Figure 4.15  (Continued )

shows that the crack occurs as a discontinuous line at 45° near the right roller. Figure 4.16 displays a set of slides showing the development of crack in the glass fiber Rohacell composite (GFRC) beam using the Phantom v7.3 video camera. 4.3.4  Damage Location using Smart Sensors 4.3.4.1  Smart Sensors

A sensor is a device designed to acquire information from an object and transform it into an electrical signal. A traditional integrated sensor can be divided into three parts

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Figure 4.16  Sequence of slides of the test specimen showing the development of crack in the glass fiber Rohacell composite (GFRC) beam using the Phantom video camera. (Ayorinde et al., 2012)

(Kirianaki et al., 2002; Spencer et al., 2004; Nagayama, and Spencer, 2007): (1) the sensing element (e.g. resistors, capacitor, transistor, piezoelectric materials, photodiode, etc.), (2) signal conditioning and preprocessing (e.g. amplification, linearization, compensation, and filtering), and (3) a sensor interface (e.g. the wires, plugs, and sockets to communicate with other electronic components). The essential difference between a smart sensor and a standard integrated sensor is its intelligence capabilities, i.e. the onboard microprocessor. The microprocessor is typically used for digital processing, analog to digital or frequency to code conversions, calculations, and interfacing functions, which can facilitate self‐diagnostics, self‐identification, or self‐adaptation. Smart sensors have four important features: (1) an onboard microprocessor, (2) wireless communication, (3) a small size, and (4) a low cost. Smart sensing technology may be the only way to fulfill the objective of SHM using a densely distributed sensor network. Some of the developed smart sensors were reviewed by Gao and Spencer (2008). The term “motes” refers to a general class of robust and versatile sensors. Some refer to mote/sensor line of products as “Smart Dust” implying that one day the technology will reach the nanoscale. Commercial‐off‐the‐Shelf Dust (COTS Dust) is a device that incorporates communications, processing, sensors, and batteries into a package about a cubic inch in size (Hollar, 2000). A sensor node, also known as a mote, is a node in a sensor network capable of performing some processing, gathering sensory information, and communicating with other connected nodes in the network. A mote is a node but a node is not always a mote. The MICA mote is the first commercial smart sensor generation and was named as “Rene Mote” (Culler et al., 2002). The basic MICA hardware uses a fraction of a watt of power and consists of commercial components a square inch in size. The hardware consists of a small, low‐power radio and processor board (known as a mote processor/radio  –  MPR  –  board) and one or more sensor boards (known as a mote sensor – MTS – board). The combination of the two types of boards

Structural Health Monitoring Basic Ingredients and Sensors

forms a networkable wireless sensor. The MPR board includes a processor, radio, A/D converter, and battery. A wireless modular monitoring system, consisting of a microprocessor, radio modem, data storage, and batteries was designed by Kiremidjian et  al. (1997), Straser and Kiremidjian (1996, 1998), and Straser et al. (1998). The system was designed to acquire and manage the data and to facilitate damage detection diagnosis. Agre et  al. (1999) introduced a smart sensor that supports bidirectional, peer‐to‐peer communications with a small number of neighbors. Lynch et  al. (2001) developed a proof‐of‐concept smart sensor that uses a standard integrated circuit component. Key features of the unit include wireless communications, high‐resolution 16‐bit digital conversion of interfaced sensors, and a powerful computational core that can perform various data interrogation techniques in near real‐time. The sensor unit was validated through various controlled experiments in the laboratory. Spencer et al. (2004) provided a brief account of smart sensing technology and identified some issues and associated challenges. The multi‐scale SHM framework using wireless smart sensor incorporates hardware for high‐sensitivity multi‐metric sensing, software for long‐term and robust operation of a self‐powered wireless smart sensor network, multi‐scale and hybrid SHM strategies for optimized use of the information obtained from the SHM systems, and full‐scale implementation to validate the potential of its practical applications (Jo and Spencer, 2015). Hybrid SHM strategy combining numerical modeling and multi‐metric physical monitoring has allowed comprehensive prediction of structural responses at arbitrary locations. For example, a high‐sensitivity accelerometer board was developed for low‐ level ambient acceleration measurements, which incorporates 16‐bit A/D converter, low‐noise analog accelerometer, and carefully designed analog signal processing circuit. To increase performance, flexibility, and versatility of the wireless smart sensor networks, Sim and Spencer (2015) considered decentralized modal analysis and efficient decentralized system identification in the wireless smart sensor networks, together with multi‐metric sensing. The performance of the decentralized approaches and their software implementations were validated through full‐scale application of the Jindo Bridge, a 484‐meter‐long cable‐stayed bridge located in South Korea. Various types of sensors such as piezoelectric sensors and fiber optics have been extensively used for structural integration and adaptation within monitored composite structures, as demonstrated by Boller, 1996. For example, Egusa and Iwasawa (1993) used piezoelectric rubbers and paints while Seydel and Chang (2000) adopted smart panels. Passive and active approaches have been used for damage detection in composite materials. In the passive approach, sensors are bonded or embedded in structures in order to monitor impact strain data. The strain data is then used for detection and location of impacts. Damage extent is correlated with impact energy, with no damage occurring below a certain energy threshold. Previous studies in this area include: impact detection (Staszewski, 1996a, 1996b), impact location (Boller, 1996; Jones et al. 1995; Weems et  al., 1991; Gunther et  al., 1992; Schindler et  al., 1995; Staszewski, 1996a); impact detection, location, and estimation of impact energy using the modal analysis approach (Seydel and Chang, 2000); neural networks (Boller, 1996; Gunther et al., 1992; Staszewski, 1996b; Staszewski et  al. 1996), and optimal sensor location techniques (Staszewski et al., 1996; Wong and Staszewski, 1988; Staszewski et al., 2000b; Staszewski and Worden, 2001). The active approach is based on ultrasonic or acousto‐ultrasonic waves introduced to a structure by a probe at one point and sensed by another probe at

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a different position. Damage in a structure can be identified by a change of the response signal and often signal attenuation is sufficient for detection. A method for the non‐destructive detection and location of damage using parameterized partial differential equations and Galerkin approximation techniques was presented by Banks et al. (1996). The use of beam‐like structures with piezoceramic sensors and actuators to perform identification of those physical parameters was used to detect the damage. Experimental results were presented from tests on cantilevered aluminum beams damaged at different locations and with damage of different dimensions. It was demonstrated that the method can sense the presence of damage, and locate and characterize the damage to a satisfactory precision. Later, a two‐dimensional model for in‐ plane vibrations of a cantilever plate with a non‐symmetrical damage was used by Banks and Emeric (1998) in the context of defect identification in materials with piezoelectric ceramic patches bonded to their surface. These patches can act both as actuators and sensors in a self‐analyzing fashion, which is a characteristic of smart materials. The natural frequency shifts due to the damage were estimated numerically and compared to experimental data obtained from tests on cantilever aluminum plate‐like structures damaged at different locations with defects of different depths. The damage location and extent are determined by an enhanced least‐square identification method. Beard and Chang (1997) used built‐in sensors and actuators for active damage detection in filament wound composite tubes. Whelan (1999) discussed the application of optical fiber strain sensors and pulsed digital speckle pattern interferometry to the analysis of a vibrating composite panel. A prototype system for condition monitoring of composite structures was developed which relies on the on‐line measurement of vibration characteristics in order to detect any deterioration in performance due to the accumulation of damage. Vibration modes of the excited panel were determined using an out‐of‐plane pulsed‐digital speckle pattern interferometry system. An interferometric fiber optic sensor using ordinary single‐mode fibers to detect acoustic emission for damage assessment of composite materials was developed by Liu, K. et al. (1990). The fiber sensor was embedded in both graphite/epoxy and Kevlar/ epoxy composite specimens and used to produce fast direct correlation of acoustic emission with their concomitant forms of damage, such as matrix crack or material fiber rupture. A fiber‐optic Fabry‐Pérot acoustic emission sensor system based on the improved double wavelength stabilization technique was developed by Jianghai et al. (2007), with the purpose of detecting structural damage. Kaya and Kaya (2002) combined the acoustic emission technique and the forced resonance as non‐destructive damage evaluation of cyclic‐fatigued alumina fiber‐reinforced mullite ceramic matrix composite. The two approaches provided information about the damage initiation and progression in real time. It was concluded that the composite cyclic fatigue damage at room temperature begins with multiple crack formation within the brittle mullite matrix and is followed by delamination. Final failure of the composite is caused by fiber fracture with extensive cyclic sliding along the fiber/matrix interface. An attempt to identify the damage location in advanced grid structures was made by Amano et al. (2005), who obtained the structural strain distribution measured by fiber Bragg grating sensors embedded in all ribs. The advanced grid structures is a structure that uses carbon fiber reinforced plastic unidirectional composites as ribs. When some damages occur in the advanced grid structures, the structural strain distribution in the advanced grid structures changes accordingly. The damage location was identified by

Structural Health Monitoring Basic Ingredients and Sensors

considering the tendency of change. Fiber Bragg grating sensors were embedded into advanced grid structures and three‐point bending test was conducted. The results showed that these embedded sensors could detect the strains applied to the corresponding ribs. Then, a low velocity impact test was carried out, which revealed that only fiber breakage appeared in the advanced grid structures. The beam element model was used for damage location identification. Strain distributions in a structure including damages were calculated with this model. Damage detection and its location based on fiber Bragg grating sensors was experimentally studied by Betz et al. (2007). Bragg gratings were used for sensing ultrasound by detecting the linear strain component produced by Lamb waves. A tuneable laser is used for the interrogation of the Bragg gratings to achieve high sensitivity detection of ultrasound. The interaction of Lamb waves with damage, such as the reflection of the waves at defects, allowed the detection of damage in structures by monitoring the Lamb wave propagation characteristics. As the reflected waves produced additional components within the original signal, most of the information about the damage can be found in the differential signal of the reference and the damage signal. Making use of the directional properties of the Bragg grating the direction of the reflected acoustic waves can be determined by mounting three of the gratings in a rosette configuration. Two suitably spaced rosettes were used to locate the source of the reflection by taking the intersection of the directions given by each rosette. Chau et al. (2007) discussed the development of both hardware and algorithms to detect, locate and quantify delamination in composite laminated beam structures. They presented an integrated structural health monitoring system with the capability of interrogating over 50  FBG sensors simultaneously with sub‐picometer (1 × 10−12 m) resolution at over 50 kHz, together with an FBG‐sensor/piezo‐actuator matrix smart skin design and methodology. They used damage detection location and quantification algorithms. Coverley and Staszewski (2003) proposed a method of impact location in composite materials based on a classical sensor triangulation methodology. The method combines experimental strain wave velocity analysis with an optimization genetic4 algorithm procedure. The method was validated on a composite panel with embedded piezoceramic sensors. It was shown that strain data from only three piezoceramic sensors provides good impact location results, avoiding learning and modeling difficulties associated with other techniques. In studying impact damage detection, Mahzan et al. (2010) compared two signal processing methods for impact location in composite aircraft structures. The first method is based on a modified triangulation procedure and genetic algorithms, while the second technique applies artificial neural networks. A series of impacts was performed experimentally on a composite aircraft wing‐box structure instrumented with bonded piezoceramic sensors. The strain data was used for learning in the neural network approach. The triangulation procedure utilizes the same data to establish impact velocities for various angles of strain wave propagation. The study demonstrated that both approaches are capable of good impact location estimates in this complex structure. In order to locate the delamination damage, Chang et al. (2009) used the fiber Bragg grating (FBG) strain sensors network embedded in the solid rocket motor composite 4  A genetic algorithm is an algorithm used originally to solve both constrained and unconstrained optimization problems based on a natural selection process that mimics biological evolution. The algorithm repeatedly modifies a population of individual solutions.

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shell. They performed a multi‐step approach of delamination damage location, based on strain energy, in which the strain field was measured in a scatter grid by the FBG strain sensors network in hydraulic testing. The continuous displacement and strain field were reconstructed with relative sensors data, using a moving least square mesh‐ free fitting method. The strain energy of each sub‐region was calculated from the reconstructed data. Finally the damaged sub‐region was identified successfully by singularity value of strain energy. The results of simulation and experiment indicated that the damage identification and location only required the measure of strain field of the composite shell. The assessment of structural damage location in composite honeycomb sandwich panels was considered by Zanarini (2005) using an experimental non‐destructive approach on a pre‐damaged sample. Full field displacement maps were acquired by means of optical non‐contact electronic speckle pattern interferometry technology in order to obtain high spatial definition and locate small defects on the sample, such as debondings, material separations, voids, cracks, and delaminations. Four different loading approaches were adopted to detect the flaws: acoustic, thermal, static, and harmonic excitation. The displacement maps depicted with high accuracy the inhomogeneous local behavior of the structure induced by the defects. A structural neural system using firing of sensor neurons was proposed by Kirikera et al. (2006, 2008) to reduce the number of data acquisition channels needed for damage detection. Fatigue testing of a composite specimen on a four‐point bending fixture was performed, and the structural neural system was used to monitor the specimen for damage in real time. The structural neural system has two channels of signal output that are digitized and processed. The first output channel tracks the propagation of waves due to damage, and the second output channel provides the combined acoustic emission responses of the sensors. The data from these two channels was used to predict the location of damage and to qualitatively indicate the severity of the damage. The structural neural system was used to indicate where and when a more sensitive inspection is needed, which can be done using ground‐based NDE techniques. The onboard NDE approach used a fine coverage of neurons for highly sensitive NDE, which continuously listens for damage and provides real‐time processing. 4.3.4.2  Application of Fiber Bragg Grating

To demonstrate the application of fiber Bragg grating, the bolt pre‐load loss monitoring is considered in this section as reported by Todd et al. (2007). The hybrid Bragg grating 3 × 3 interrogation system was used to help assess bolt pre‐load loss in a hybrid material joint. The bolt fastens a steel member to a composite material member, and over time (particularly under elevated temperatures or mechanical loads), the composite material creeps, leading to loosening, pre‐load decrease, and loss of functionality. Todd et  al. (2001b) and Nichols et al. (2003a, 2003b) utilized a vibration technique for structural damage detection, based on chaotic attractor property analysis. The technique involves applying a chaotic dynamic excitation to the structure and extracting properties of the resulting chaotic structural response. The structural response may be represented in its state‐space, where each state variable traces out some trajectory in time. Under a stationary input, the state approaches an attractor, which may be thought of as a geometric subset of the state‐space. As damage progresses on the structure, the dynamics are affected in such a way as to affect the properties of the attractor. Essentially, the method

Structural Health Monitoring Basic Ingredients and Sensors

uses a model‐building approach to predict and compare attractors between damaged and undamaged structures. This approach was extended by Todd et al. (2004) to predict and compare relationships between attractors on the structure and to observe how this relationship changes as damage progresses, rather than just how a specific attractor changes as damage progresses. As the connectivity is gradually lost between structural members, it is expected that relative dynamics between locations on either side of the damaged area may be optimally sensitive to such damage. Sensor responses to chaotic excitation were measured at various locations, and corresponding state‐space models were constructed. A functional relationship between any two pairs of attractors, Xi and Yi, was generated to reveal its evolution with time. The relationship was established by using one attractor to forecast the evolution of the error and build an error metric (“prediction error”) between them as the forecasting ability breaks down. Prediction errors close to unity indicate a lack of a continuous functional relationship. The existence of this relationship was mainly due to strong coupling between the two structural locations chosen for attractor reconstruction, and that damage resulted in a change of the relationship. This approach was employed on a composite beam of length 1.219 m, width 0.1715 m, and thickness 0.01905 m. The beam was bolted at both ends to two steel plates, and the entire assembly was clamped to a fixed base, as shown in Figure 4.17. The response of the structure was detected on either side of the connection using FBG strain sensors interrogated by the hybrid 3 × 3 system (Todd et al., 2001b). One sensor was located on the steel plate (sensor 1) while the other was located approximately 7.5 cm from the bolts on the composite beam (sensor 2), as shown in Figure 4.17. The composite material utilizes a quasi‐isotropic layup consisting of (0/90) and (±45) knit E‐glass fabric. The specific layup is six sets of [(±45),(0/90)]  –  symmetric plies stacked on top of each other in the first half of the laminate. The structure is bolted to a steel frame 19 × 19 cm with ALD‐Dynagage instrumented bolts of length 8.9 cm, capable of measuring axial force. Excitation was provided by means of a B&K

To data acquisition Steel block

Composite

Optical fiber Steel block

Fixed base

Shaker

Sensor 2

Sensor 1

Composite

Projection of the dashed section

Fixed base

Figure 4.17  Schematic diagram of the bolted joint showing the instrumented bolts (•) and the fiber Bragg gratings (▪). (Adapted from Todd et al., 2007)

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(a)

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Figure 4.18  Excitation plots: (a) time history of excitation signal, (b) corresponding phase space plot. (Adapted from Todd et al., 2007)

electrodynamic shaker, located at the mid‐span of the beam and oriented such that the stinger was pressing down on the beam. Between the stinger and the beam is an Omega LCFD‐25 load cell for recording the input signal. A sample excitation waveform is shown in Figure  4.18, along with its corresponding phase space representation. The plots show the characteristic “lobes” of the Lorenz system used to excite the beam. Essentially the system oscillates between two fixed points at banded, but varying, frequencies. The structure in the phase portrait is a hallmark of chaotic systems. Ten different levels of bolt pre‐load (NdZ10) were assigned to the bolts at one end of the beam, starting at approximately 50 kN for the baseline case. The torque on every bolt could not be exactly controlled to 50 kN, due to variations in the individual connections and uncontrollable creep in the composite material. Figure 4.19 shows the progression of axial load for each of the four instrumented bolts whose locations are displayed in Figure 4.17. The bolts on the undamaged end of the

Structural Health Monitoring Basic Ingredients and Sensors 100

Boltaxial load (kN)

80 Undamaged end bolts 60 40 20

Damaged end bolts

0 Damage progression

Figure 4.19  Progression of measured axial load for each of the instrumented bolts. (Adapted from Todd et al., 2007)

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Figure 4.20  Attractors reconstructions for (a) sensor 1 and (b) sensor 3 for undamaged case. (Adapted from Todd et al., 2007)

beam remain relatively unchanged for the duration of the test. At each pre‐load level, five experiments were conducted. Sample attractor reconstructions for the two systems are shown in Figure 4.20 for the undamaged case (pre‐load at 50 kN). Data collected from the steel plate shows a slightly more “blurred” attractor, since the sensor at this location was further from the driving signal. For each attractor pair, a total of 1000 individual cross prediction errors were computed, resulting in 25,000 total values for each pre‐load level. The resulting sets of prediction error were resampled so that confidences could be generated based on the mean values. Based on the resulting normal distribution, confidence intervals of 95% were constructed and are displayed in Figure  4.21. The notation 2/1 denotes the usage of sensor 2 data to forecast data taken from sensor 1 and vice versa. Figure 4.21a shows the progression of the confidence intervals with damage, while the lower plot shows the corresponding probability density functions. Confidence limits for the undamaged case

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Mean prediction error

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Mean prediction error

Figure 4.21  Performance of the nonlinear cross‐prediction error (a) mean prediction error showing confidence intervals, (b) corresponding probability density functions.Left column of (a) and (b) belongs to sensor 2 data to forecast data taken from sensor 1, while the right column is for sensor 1 data to forecast data from sensor 2. (Adapted from Todd et al., 2007)

are highlighted in gray. Any subsequent interval that lies inside this area is commensurate with the no‐damage and damage cannot be detected at the 95% confidence level. On the other hand, intervals which do not overlap this region manifest the occurrence of dynamical change to the system. Thus pre‐load loss is detectable at approximately 17.8 kN of pre‐load. The mean prediction error begins to climb, indicating the occurrence of damage. At the 22.2 kN level, damage is detected at approximately the 50% confidence level, possibly indicating that the bolt is beginning to come loose. The prediction error is 3.5% of the signal variance indicating the existence of functional relationship exists. As the connection degrades, the functional relationship is diminished, thus reducing the ability of one sensor to predict the dynamics of the other. This is evidence of a reduction in the strength of the coupling between the dynamics on either side of the connection with damage. In the maintenance of aerospace, naval, and civil structures, the ability to evaluate the integrity of the structure is becoming an important technology. Assessing the structural condition without removing the individual structural components is known as non‐ destructive evaluation (NDE) or non‐destructive inspection. Some techniques are based on visual observations of cracks, such as visual inspection and dye‐penetrant inspection. Some are based on the electromagnetic properties of the material, such as magnetic particle inspection and eddy‐current inspection. Still other techniques are based on the interpretation of the structural condition by observing the change in the mechanical properties of the structure. Vibration (or modal) testing is used for determining the global mechanical characteristics of a structure, because techniques for modal data reduction and analysis are well developed for other applications. Modal techniques for NDE are typically implemented using finite element model (FEM) update. FEM update methods are based on modifying the FEM stiffness, mass,

Structural Health Monitoring Basic Ingredients and Sensors

and/or damping matrices to minimize some measure of error, which is typically a function of the FEM matrices and the measured modal parameters. A review and a detailed discussion of the overall field of FEM update are presented by Hemez and Farhat (1993). When using FEM update for model refinement, it may be desirable to constrain the material properties of several different elements to be the same, and then update this common parameter. However, for NDE, it is desirable to let these properties vary independently because the changes are to reflect isolated incidents of damage and not overall errors in modeling assumptions. This makes FEM updating for NDE more difficult than FEM updating for other applications. In an attempt to identify the damage location in advanced grid structures (AGS), Amano et al. (2005) obtained the structural strain distribution measured by fiber Bragg grating (FBG) sensors embedded in all ribs. The AGS is a structure that uses carbon fiber reinforced plastic (CFRP) unidirectional composites as ribs. When some damages appear in the AGS, the structural strain distribution in the AGS changes accordingly. The damage location was identified by considering the tendency of change. FBG sensors were embedded into the AGS, and a three‐point bending test was conducted. The results showed that these embedded sensors could detect the strains applied to the corresponding ribs. Then, a low velocity impact test was carried out, which revealed that only fiber breakage had appeared in the AGS. The beam element model (BEM) was used for damage location identification. Strain distributions in the structure including damages were calculated with this model. 4.3.5  Electric Resistance and Capacitance Techniques

Changes of electric resistance and capacitance as a result of crack and damage of composite structures have been utilized as a means of detecting and locating structural damages. The possibility of in‐situ detection of damage in unidirectional carbon‐fiber‐ reinforced polymers (CFRP) by means of electric resistance measurements was considered by Abry et  al. (1998a, 1998b, 1999, 2001). The conducting paths in unloaded specimens were investigated by changing the electrode location. It was shown that conduction occurred both along the fiber and in the transverse direction, by virtue of fiber‐to‐fiber contacts. Monotonic and cyclic flexural tests were carried out with unidirectional carbon/epoxy laminates of different fiber volume fractions. Monotonic tests under post‐buckling bending conditions were performed on cross‐ply [0/90], and [90/0], carbon/epoxy laminates. The monitoring of electrical resistance and capacitance changes, linked to the modifications of the conduction paths in the composite, and occurrence of voids during loading, allowed the detection of damage growth. The electrical potential was applied to detect and locate impact damage in CFRP plates of Hexcel T300/914 composite by Angelidis et  al. (2002, 2005). The potential field across the surface of the laminates was measured using arrays of electrical contacts. These results were compared with those calculated using three‐dimensional finite element simulation of current flow in CFRP laminates. A comparison of potential distribution on the top surface for damaged and undamaged laminates showed a substantial difference in the potential field around the impact damaged area. Todoroki and Tanaka (2002) and Todoroki et  al. (2002a, 2002b) employed an electric resistance change method for identification of delamination location and size of beam‐type and plate‐type specimens fabricated from cross‐ply laminates. On the specimen surface,

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multiple electrodes were mounted by co‐curing copper foil to measure electric resistance changes. The electric resistance changes were measured using a conventional strain gage amplifier. Response surfaces were adopted as a tool for solving inverse problems to estimate location and size of the delamination crack from the measured electric resistance changes of all segments between electrodes. The method provided estimates of the location and size of the embedded delamination for graphite/epoxy laminated composites. An electric potential based damage detection technique utilizing an artificial neural network was presented by Anderson et al. (2003). The scheme is applicable to electrically conductive composite structures and electro‐active membranes. The scheme consists of feed‐forward back‐propagation multi‐layer neural network architecture to determine the damage region from the electric potential readings taken at the boundary. A strain measurement system utilizing the electric capacitance change of steel wire reinforced tires was proposed by Todoroki et al. (2003a, 2003b). A specimen cut from a commercial tire was connected to an oscillator circuit, and the change of the capacitance caused a change in oscillating frequency of the oscillating circuit. Measurement of the frequency of the oscillating circuit was used to measure the strain of the tire wirelessly. A tension test was performed, and the frequency of the oscillating circuit was measured during the test, and this was used for passive wireless strain monitoring of tires. Shimamura et al. (2005, 2006) experimentally demonstrated the detection of bearing failure in CFRP bolted joints using an electrical resistance change method. It was shown that fiber breaking and delamination‐induced permanent increase in the electric resistance of the bolted composite joints is effective for detecting bearing failure. Detailed analyses were carried out to examine the detectability in terms of the damage size and the distance between damage and electrodes. The results revealed that bearing failure of less than 10 mm square causes the electric resistance change and that the electrodes can be mounted more than 10 mm from a bolt hole. Takahashi et al. (2007) performed statistical diagnosis using electrical resistance changes to detect a delamination crack in a CFRP beam. They proposed a measuring method of multiple electrical resistance changes to perform statistical diagnosis. The proposed method measures electrical resistance changes of multiple segments in a CFRP beam, although electrical interference must be considered when multiple voltages are charged at once. A delamination crack was detected by the change of relative relationship between multiple electrical resistance changes due to damage occurring. An electrical resistance change method with integrated probes on a single side of the surface of a CFRP composite structure was adopted by Todoroki et al. (2006a, 2006b) to detect the matrix cracking of the laminated composites. The method employed finite element analyses to search for the best placement of probes for matrix crack detection using a rectangular plate. A four‐probe method was adopted for measuring the electrical resistance change. The finite element analysis revealed that the electrical resistance increases linearly with increase in the number of matrix cracks inside of the probes. Todoroki et al. (2007) and Todoroki (2008) employed an electric impedance change method for the identification of damage location and dimension of the damaged area of thick beam‐type specimens fabricated from cross‐ply laminates of 36 plies and carbon‐ fiber reinforced polymer laminates. The damage caused a drop in electrical impedance. A residual stress relief model was proposed to explain the decrease. Relationships between the electrical impedance changes and damages were obtained by means of

Structural Health Monitoring Basic Ingredients and Sensors

response surfaces. The response surfaces estimated the damage location and dimension of the damaged area exactly, even for the thick CFRP laminates. The electrical impedance change method can be used as an appropriate sensor for measurement of residual stress relief due to damages of thick CFRP laminates. A wireless sensor for detection of internal delamination in a CFRP laminate was introduced by Matsuzaki et al. (2008). The method utilizes a simple electrical resistance change in CFRP, and so monitors delamination at only one location. For monitoring of large‐scale structures, however, many sensors have to be distributed to cover the structure. A major problem with using many sensors is time synchronization between sensors. To overcome the problem and enable strain/damage to be monitored at multiple locations with time synchronization, Matsuzaki et al. (2009) developed a simple wireless strain/damage sensor that consists of a bridge circuit, voltage‐controlled oscillator, and amplifiers. Since the sensor does not need A/D conversion procedures or memory storing, there is no time delay. Each sensor has an original basic frequency that changes in accordance with the electrical resistance. It was shown that the system was capable of measuring applied strain and detecting fiber breakage at multiple locations in CFRP laminates with time synchronization. 4.3.6  Impact Resonance Method

It is known that Rayleigh waves can be used to detect sub‐surface flaws in thick plates, while Lamb waves are used for inspection of thin plates and can explore the entire thickness of the plate and propagate a long distance without appreciable attenuation. However, Lamb wave testing is generally complicated due to the coexistence of at least two modes at any given frequency and by the strongly dispersive nature of these modes at high frequency. Furthermore, a single and pure Lamb mode may generate a variety of other modes, either by interacting with a surface or sub‐surface flaw or by crossing the interface between two materials of different impedance (Viktorov, 1967). Accordingly, the output signal becomes richer, but often very difficult to interpret, and a detailed physical analysis of the propagation mechanism may be of great interest (Alleyne and Cawley, 1992; Guo and Cawley, 1993). Within the framework of the European project Damage Assessment in Smart Composite Structures (DAMASCOS), Agostini et al. (2000) simulated damage phenomena in composite materials in order to optimize the interrogation frequency and use of propagating modes. They simulated the interaction of Lamb waves with various kinds of defects using the local interaction simulation approach reported by Delsanto et al. (1992, 1994a, 1994b, 1998, 1999) and Schechter et al. (1994). It is believed that Sansalone and Carino (1986, 1989) and Carino et al. (1986) were the first who measured resonated waves produced by an impact on the surface of concrete structures. They developed the impact‐echo method for the detection of cracks in concrete. The concept of detecting defects in concrete specimens using the impact resonance method was demonstrated by Grosse and Reinhardt (1996) – see Figure 4.22. The wave field radiated by the impactor propagates into the object along spherical wavefronts. After a short travel time, the direct wave meets the transducer “a”. Depending on the distance between impactor and transducer, at a later time the background echo also reaches the transducer. A waveform of the wavelength twice the thickness of the specimen is developing, consisting of a constructive interference. This means that a stationary wave is dominating the recording, showing a rather low frequency signal with

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S1

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Impactor

b

a υp

d

fr Specimen

Figure 4.22  Schematic diagram of impact resonance method of detecting defects in a concrete specimen. (Grosse and Reinhardt, 1996)

frequency fr. An analysis of the time signal by a fast Fourier transform shows the f­ requency spectrum of the primary wave plus or minus the resonance modes according to the thickness of the specimen

d

p

/ 2 fr

(4.7)

where υp is the P‐wave5 velocity. In the presence of defects such as spalls or horizontal flaws, the spectrum does indeed show additional maxima that are more or less clear. However, normally the main peak in the resultant spectrum is due to the background echo (discontinuity) as illustrated in Figure 4.23(b). A wave applied on the surface of a concrete slab can be detected by a piezo‐transducer at a distance of 5 cm. Figure 4.23(a) shows the complete signal in the time domain. The signal decays after approximately 20 ms. The maximum amplitude in the spectrum near 6.7 kHz corresponds to a thickness of the concrete slab of 30 cm and a wave velocity of 4020 m/s, according to equation (4.7). Note that the defect location does not need to be known a priori, as the receiver and impact point are near the ends of the structural member. This means that structural damages are globally determined rather than locating individual defects. The method was also demonstrated by Gheorghiu et al. (2005). A decreasing trend of P‐wave velocities with increasing fatigue was reported. The use of the impact resonance method for evaluating the structural health of thermal‐cycled reinforced concrete beams with and without externally strengthened CFRP pultruded plates was outlined by Ward et al. (2008). The specimens were 1.2 m long and subjected to 55 thermal cycles 5  The P-waves are a type of elastic wave that can travel through gases (as sound waves), solids, and liquids. The name P-wave is often said to stand either for primary wave, as it has the highest velocity and is therefore the first to be recorded; or pressure wave (Milsom, 2003) as it is formed from alternating compressions and rarefactions. The velocity of P-waves in a homogeneous isotropic medium is given by K p

4 3

2

where K is the bulk modulus (the modulus of incompressibility), μ is the shear modulus or modulus of rigidity (the second Lamé parameter), ρ is the density of the material through which the wave propagates, and λ is the first Lamé parameter.

Structural Health Monitoring Basic Ingredients and Sensors (a)

Amplitude

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Figure 4.23  Detected transducer signal in (a) the time domain, and (b) frequency domain. (Grosse and Reinhardt, 1996)

ranging from +23 °C to −18 °C. Fatigue loading consisting of up to two million cycles at high and low stress levels was performed. At predetermined load cycle intervals, the loading was stopped and the impact resonance method was performed on the specimens. The FFT spectrum of the specimens’ vibration, modal fundamental frequencies, and dynamic properties were used to assess damage in the specimen in monitoring the health of strengthened and unstrengthened reinforced concrete beams subjected to thermal and fatigue cycles. The passive impact detection system based on FBG sensors which can be either embedded or surface mounted on a composite structure was presented by Staszewski et al. (2000a), Staszewski and Found (2002), and Staszewski and Boller (2004). The intelligent signal processing of the optical fiber sensor data was discussed and demonstrated using a series of simple impact tests. Their studies included the problem of impact damage detection in composite structures using acousto‐ultrasonic sensors. The effectiveness of piezoelectric polyvinylidene fluoride (PVF2) film sensors embedded within the face‐sheet of a sandwich panel in detecting low‐velocity impact and damage was examined by Anderson et al. (2001). Four sensors were constructed and located in the corners of a rectangular test coupon and an instrumented dropped‐ weight impact test apparatus was utilized to record the sensor and impact force signals during the impact event. It was demonstrated that the sensors are capable of producing high‐level signals at low levels of impact energy, irrespective of the location of impact. Furthermore, comparisons between the contact force and sensor data revealed that the sensors are capable of indicating if and when impact damage occurs during the low‐ velocity impact event. The time‐of‐arrival information derived from each sensor was also utilized to determine if the location of impact could be found.

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4.3.7  Optimal Sensor Location

Early detection of faults is critical for the safety of the structure, operating personnel, and other resources. When a fault occurs in a system, it can propagate and affect several process variables. Sensor location is one of the important issues in fault detection in distributed parameter systems. The performance of a diagnostic system strongly depends on the number and location of sensors. It is necessary to develop an algorithm for identifying the optimal number, type, and location of the sensors for fault detection and diagnosis. Several optimization techniques for sensor allocations for SHM have been proposed in the literature. Optimal sensor–actuator placement strategies work to maximize the ability to detect and discriminate relevant data features given these limited resources. With reference to active sensing approach there are some constraints such as the number of actuators and sensors available for the network and the power available for interrogation. A combined neural network/combinatorial optimization approach was addressed by Worden and Burrows (2001) and Staszewski et al. (1996), with the purpose of minimizing the probability of mis‐location of structural faults in a metal plate. Use of optimal sensor allocation strategies during the design phase can ensure better diagnostics at a reduced cost for a system incorporating a high degree of built‐in testing. Staszewski (1998) made an attempt to assess some of the recent advances in wavelet analysis for damage detection. Later, Staszewski (2002) addressed the importance of intelligent signal processing for damage identification in composite materials. A number of examples of damage detection in composite materials were presented, such as data denoising techniques, feature extraction and selection, pattern recognition, and optimal sensor location procedures. Azam et al. (2004) proposed multiple fault diagnosis and optimization techniques for optimal sensor placement for fault detection and isolation in complex systems. The problem of sensor placement for impact detection on composite structures was treated using information theory by Wong and Staszewski (1998). The information content of sensor data was assessed, and estimated arbitrary dependences between features obtained from different sensor locations. The information assessment was made on the sensor ability to detect the amplitude of impacts. The method was validated using a simple impact test experiment on a composite plate. The study showed that the sensor location problem can be solved before the actual damage detection procedure is performed. Staszewski et al. (2000b) studied the problem of optimal sensor placement for impact detection and location in composite materials. The time‐varying strain data was measured using piezoceramic sensors. An effective impact detection procedure was established using a neural network approach. The procedure determines the location and amplitude of impacts. A genetic algorithm was employed to determine the optimum sensor positions for a diagnostic system. It was shown that genetic algorithms combined with neural networks can be effectively used to find near‐optimal sensor distributions for damage detection. The development and optimization of a new piezoelectric sensor using Lamb waves for aerospace structures health monitoring was presented by Rguiti et  al. (2006). To identify all the generated Lamb waves for the purpose of damage location is to measure the Lamb wave signal at different locations along the propagation direction. The development of a distributed sensing technology using metallic multi‐electrodes deposited

Structural Health Monitoring Basic Ingredients and Sensors

on a piezoelectric substrate was a key element, which built the bridge between the ­sensors signals and the structural integrity interpretation. Piezoelectric properties were measured by the electrical resonance technique. The sensitivity of this new sensor to multi‐wave generation and damage detection were demonstrated in the case of an ­aluminum plate. A strategy for minimum number of piezoelectric patch sensors and their locations to detect the presence and extent of damage in a composite wing‐box was presented by Yan et al. (2007b), using a genetic algorithm. A damage index using all differences in voltage signals decomposed by wavelet transform was proposed. An approach for determining optimal sensor and actuator placement for SHM was presented by Frazier et al. (2008). The approach was based on using a linear dynamic finite‐element model of a structure, first‐order sensitivities of measures of the observability and controllability of the model, and genetic algorithms. A robust design was performed using a game theory approach, whereby one population in the genetic algorithm attempted to find out what model error can cause the monitoring system to perform the worst, while the other population tried to find the sensor and actuator locations that cause the monitoring system to perform the best. The selection of sensor locations from a set of possible candidate positions for achieving the best identification of modal frequencies and mode shapes was examined by Huang et al. (2009). The sensor layout problem was taken as a combinatorial optimization problem and a genetic algorithm was adopted to solve the problem. The dual‐structure coding, partially matched crossover, inversion mutation, and adaptive probabilities of crossover and mutation were adopted to solve the sensor layout problem. An optimum number and location of strain sensors to achieve the most appropriate strain pattern that produces minimum error of damage classification in a statistical sense was considered by Singh and Joshi (2009). A stochastic objective function, based on weighted sum of mean error and information entropy of error associated in classification of structural damages, was formulated. A real coded genetic algorithm was developed and devised with simple uniform crossover and mutation operators to optimize the location as well as the number of sensors in a single optimization framework. A sensor placement method fro maximizing the performance of an SHM system with a minimal number of sensors for detection of impact in structures, particularly for structures made of fiber‐reinforced composite materials was presented by Markmiller and Chang (2010). The performance of the SHM system was evaluated based on the probability of detection. This optimization problem was formulated to maximize the probability of detection through selection of optimal sensor locations for a given sensor network. A genetic algorithm was adopted and integrated with the SHM system to ­perform the optimization process. The problem of optimal sensor placement for fault detection and isolation consists of determining the set of sensors that minimizes a predefined cost function satisfying at the same time a pre‐established set of fault detection and isolation specifications for a given set of faults. An algorithm for model‐based fault detection and isolation sensor placement based on formulating a mixed integer optimization problem was proposed by Sarrate et al. (2008). Fault detection and isolation specifications were translated into constraints of the optimization problem, considering that the whole set of analytical redundancy relations have been generated. The optimal location of sensors based on the expected cost specifications was proposed by Duan et al. (2011), who considered the

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minimal expected diagnosis cost as the objective function for the sensor optimization with a fixed number of sensors. The fault tree model was used, making full use of the qualitative and quantitative information provided by reliability analysis. With the framework of fault detection and isolation, the optimization of the data acquisition process would increase the reliability of structure diagnostic. The analytical redundancy relations that were generated together with the hypothetical signature matrix was presented by Du et al. (2011). The optimal sensor placement problem was mapped onto a special case of the 0–1 integer programming problem, which was, in the end, solved by the algorithm of binary particle swarm optimization. The sensor placement problem was considered by Chi et al. (2012), who determined an optimal minimal isolation set associated with the least number of sensors. A dedicated genetic algorithm was developed to solve the formulated sensor placement problem of fault detection of a two‐tank system. The genetic algorithm and particle swarm optimization algorithm were analyzed for multi‐class sensor placement optimization method in research on mechanical power and transmission system fault diagnosis (Pengcheng et  al., 2012). Various optimal sensor placement algorithms were used by Vosoughifar et al. (2013). A genetic algorithm was adopted for the optimization formulation in selection of the best sensor placement according to the blast loading response of the cold‐formed steel structure. A numerical algorithm was proposed for optimal sensor placement procedure, which utilized the exact value of the structural response under blast excitation. A scheme for prioritizing sensor locations for structural damage detection was outlined by Sun et al. (2010) with the purpose of finding the sensitivity for local damages and the independence of the target mode. The two different optimization criteria lead to an inconsistency of the optimal result. Considering the structural response changes during damage development, the relationship between the structural response and damage was deduced from the structural equation of motion by a quasi‐analytical mode. The optimal sensor placement of compliant sheet metal parts for the fixture fault diagnosis was considered by Liu et al. (2011). The Bayesian network approach for fixture fault diagnosis was proposed to construct the diagnostic model. Given the desired number of sensors, the diagnostic ability of the sensor set was evaluated based on the mutual information of the nodes. A new sensor placement method outlined and validated with a real automotive sheet metal part. The proposed method was used to perform the fixture fault diagnosis and sensor placement optimization effectively, especially in a data‐rich environment. Patan (2012) extended the existing methods of sensor location and developed the link between the properties of the measurement process and optimization of fault location in distributed parameter systems. Combination of fault evolution sequences and the magnitude ratio information for effective fault diagnosis were outlined by Mobed et al. (2015). This was achieved by including the idea of artificial sensors that represent pairwise sensors from the original list of possible sensors. A probabilistic approach for identifying the optimal number and locations of sensors for SHM was proposed by Azarbayejani et al. (2008). The method developed the probability distribution function that identifies the optimal sensor locations such that damage detection is enhanced. The approach was based on using the weights of a neural network trained from simulations using a priori knowledge about damage locations and damage severities to generate a normalized probability distribution function for optimal sensor allocation. A pre‐stressed concrete bridge was selected as a case study

Structural Health Monitoring Basic Ingredients and Sensors

to demonstrate the effectiveness of the proposed method. The results show that the proposed approach can provide a robust design for sensor networks that are more efficient than a uniform distribution of sensors on a structure. Teo et al. (2009) studied the scattering of stress waves due defects in representative aircraft structures with multi‐ layered construction and geometry variation. The probability of detection of non‐­ surface‐penetrating defects in the structure, as well as minimizing the contributions of multi‐layered construction and geometry variation to false indications, were presented. The problem of selecting the appropriate frequency and location of the sensor in monitoring sub‐surface defects on the structure required the determination of the optimal combination of frequency and sensor location. The optimal placement of sensors for state estimation based continuous SHM was achieved using three different approaches (Van der Linden et al., 2010). The first is by minimizing the static estimation error of the structure deflections, using the linear stiffness matrix derived from a finite element model. The second approach is by maximizing the observability of the derived linear state space model. The third approach is by minimizing the dynamic estimation error of the deflections using a linear quadratic estimator. A simple search‐based optimization implementation was demonstrated on a model of the long‐span New Carquinez Bridge in California. A Bayesian experimental design approach was implemented by computing the total posterior expected cost of detection over the entire monitoring region (Flynn and Todd, 2010a, 2010b, 2010c). A semi‐analytical modeling approach for wave scattering within the Bayesian probabilistic framework was implemented in order to optimally place active sensors for detecting cracks of unknown location, size, and orientation. The implementation involved a set of a priori probability distributions on the three unknowns and defining spatial distributions of cost associated with type I and type II detection errors. These parameters were dependent on the geometry, material, in‐service structural loading, and performance requirements of the structure. The optimization of actuator‐sensor arrays, making use of ultrasonic wave propagation for detecting damage in thin plate‐like structures, was also considered. A concave‐shaped plate was densely instrumented and they applied artificial reversible damage to a large number of randomly generated locations, acquiring active sensing data for each location. The algorithm was used to predict optimal subsets of the dense array. Sensor distribution and placement was optimized in studying the damage identification for a bridge structure using vibration experimental modal analysis (Li, A.N. et al., 2004; Li, D.S et  al., 2008; Jin and Song, 2008). The Wenhui cable‐stayed bridge was taken as an example, and the modal data was calculated by the finite element model using the effective independence method, modal assurance criterion method, and an eigenvector sensitivity based method to optimize the placement procedure of the sensors. The corresponding modal data at different locations was employed to identify the damages based on the finite element model. The problem of optimal sensor placement on a spatial lattice structure with the aim of maximizing the data information so that structural dynamic behavior can be fully characterized was considered by Liu et  al. (2008). Based on the criterion of optimal sensor placement for modal tests, an improved genetic algorithm was introduced to find the optimal placement of sensors. The modal strain energy and the modal assurance criterion were adopted so that three placement designs were produced. A computational simulation of a 12‐bay plain truss model was considered to demonstrate the feasibility of the optimal algorithms. The obtained

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optimal sensor placements using the improved genetic algorithm were compared with those gained by exiting genetic algorithm using the binary coding method. The damage identification technique for bridge structure by vibration mode analysis based on the precision of modal experiment was presented by Yu and Hai (2008). The study included the number and distribution of sensors to determine the appropriate information about changes of the structural state of the bridge. Taking an actual long span bridge as an example, and calculating the modal data by the finite element model, a method based on the eigenvector sensitivity, the effective independence method and the modal assurance criterion method were used to optimize the placement procedure of the sensors in this paper. The numerical example showed that the eigenvector sensitivity based method is an effective method for optimal sensor placement to identify vibration characteristics of bridges. The problem of locating sensors on a bridge structure with the aim of maximizing the data information so that structural dynamic behavior can be fully characterized was considered by Meo and Zumpano (2004, 2005). The study employed four different optimal sensor placement techniques, two based on the maximization of the Fisher information matrix and two on energetic approaches. Mode shape displacements were taken as the measured dataset and two comparison criteria were employed. The first was based on the mean square error between the finite element model and the cubic spline interpolated mode shapes. The second criterion measured the information content of each sensor location to investigate the strength of the acquired signals and their ability to withstand the noise pollution, while keeping intact the information relative to the structure properties. The results revealed that the effective independence driving‐point residue method provides an effective method for optimal sensor placement to identify vibration characteristics of the studied bridge. An optimal sensor placement algorithm as a combinatorial optimization problem, which is solved using a swarm intelligence technique called particle swarm optimization, was presented by Rama Mohan Rao and Anandkumar (2007). The proposed algorithm achieved the best identification of modal frequencies and mode shapes. The various aspects of experimental investigations of the structural dynamics of a composite material fuselage panel were considered by Luczak et  al. (2008). A large number of vibration based damage detection algorithms were used to detect deviations in structural parameters. An optimal approach involving intensive experimental tests was carried out on an aircraft fuselage panel made of composite material. Several excitation and measurement techniques were applied with the purpose of studying the variability of test data and to identify best practices in dynamic testing on composite material with reduced number of sensors. The influence of sensor placement on the measurement data quality and, subsequently, on the effectiveness of machine SHM was studied by Gao et  al. (2004). The signal propagation process from the defect location to the sensor was analyzed. Numerical simulations using finite element modeling were then conducted to determine the signal strength at several representative sensor locations. The analytical and numerical results showed that placing sensors closer to the component being monitored resulted in higher signal‐to‐noise ratio, thus improving the data quality. The global optimization of sensor locations and a sensitivity analysis based on the minimization of interferences due to wireless communications between sensors were studied in the presence of additive white Gaussian noise by Cotae et al. (2007). A Gram matrix

Structural Health Monitoring Basic Ingredients and Sensors

approach for robust determination of sensor locations by minimizing the interferences among sensors for engine health monitoring systems was adopted, together with an iterative algorithm for maximizing the determinant of the Gram matrix. The analytical results were verified by simulations providing details concerning numerical implementations. An acoustic based sensor network with optimal sensor placement was designed to characterize the existence of damage in composites by Das et al. (2009) using piezoelectric sensors and actuators. Initially, the sensing region and the certainty region of a sensor‐actuator pair were estimated, based on experimental data. The parameters of actuator position, excitation intensity, material properties of the host structure, the sensitivity property of the transducers, and detectable perturbations were considered in the analysis. In multi‐sensor applications, the placement of two neighboring sensors were found to create an overlap between their certainty regions. The overlap of the neighboring sensors was determined next and the minimum overlapping criteria was imposed in the design of sensor sets.

4.4  Closing Remarks SHM is a multidisciplinary topic as in the case of SLA. Early detection of cracks and damage is critical for the safety of the structure, operating personnel, and other resources. Basic ingredients of structural health monitoring constitute the process of detecting, locating, identifying, and damage diagnosis during the system operation. However, in the early stage of placing a structure in service, it is very hard to detect the initiation of cracks or defects, which inherently exist, even when compared with its original state. This of course depends on the type of sensors, their number, and the strategy of placing them. This is an optimization problem, which has been extensively addressed in the literature. The results reported in chapter have been selected from thousands of published papers and technical reports. This chapter has presented the  basic ingredients of non‐destructive evaluations used in SHM. The next chapter will  cover other SHM issues dealing mainly with statistical pattern recognition and vibration‐­based techniques.

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5 Statistical Pattern Recognition and Vibration‐Based Techniques 5.1 Introduction The concept of statistical pattern recognition is a valuable paradigm in structural health monitoring (SHM). Statistical models are usually employed to determine the state of damage, particularly its existence and location. Statistical procedures can be used to better determine the type of damage, the extent of damage, and the remaining useful life of the structure. Damage classification has been established based on the damage index, which represents the difference between the original input waveform and the reconstructed signal. Outlier detection is the primary class of algorithms applied in unsupervised learning applications. An outlier appears to deviate markedly from other members of the sample. Outliers, being the most extreme observations, may include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low. Vibration‐based damage detection methods are used in structural systems to identify their global dynamic response. The effects of damage on a structure can be classified as linear or nonlinear. Vibration‐based damage detection techniques are based on observing changes to the structure’s dynamic characteristics to detect the presence of damage and determine its location. In addition, operational deflection shapes and vibration deflection shapes are very valuable tools to detect damage occurrence. This chapter has two main sections. Section 5.2 deals with the statistical pattern recognition as a paradigm and the role of outliers in detecting damage in terms of damage index. The case study of impact tests on a composite plate reported by Sohn et  al. (2007a) is included to demonstrate the idea of statistical pattern and damage index. Section 5.3 constitutes the major part of this chapter and addresses the vibration‐based techniques in detecting, locating, and identifying structural damage. Vibration‐based methods include strain energy method, modal analysis, frequency response function, modal assurance criterion, and operating or vibration deflection shapes. Selected applications as reported in the literature include composite structures, T‐joints, bridges, and concrete structures.

Handbook of Structural Life Assessment, First Edition. Raouf A. Ibrahim. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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5.2  The Statistical Pattern Recognition Paradigm 5.2.1  Basic Concept

The development of statistical models for differentiating between features from the undamaged and damage structures involves the implementation of algorithms that operate on the extracted features to quantify the damage state of the structure. Pattern recognition is the assignment of a label to a given input value. In other words, it provides information for all possible inputs and to perform most likely matching of the inputs, taking into account their statistical variation. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes (for example, to determine whether a given state of the structure exceeds a prescribed value). The processes of automatic recognition, description, classification, and grouping of patterns constitute the main features of statistical pattern recognition. The process of statistical pattern recognition has four components (Farrar et al., 2000; Sohn et al., 2001): 1) Operational evaluation, which deals with the implementation of a damage detection capability. 2) Data acquisition involves selecting the types of sensors to be used, including required bandwidth and resolution, the locations where the sensors should be placed, the number of sensors to be used, and the data acquisition/storage/transmission hardware. It also includes data cleansing, which involves selectively choosing data to pass on to or reject from the feature selection process. 3) Feature extraction is the process of separating changes in sensor reading caused by damage, from those caused by varying operational and environmental conditions. The feature extraction process is based on fitting some model, either physics‐based or data‐based, to the measured system response data. 4) Statistical model development deals with the implementation of the algorithms that operate on the extracted features to quantify the damage state of the structure. In particular, the feature extraction may involve correlating measured system response parameters, such as vibration amplitude or frequency. Alternatively, it may involve developing features for damage identification by applying engineered flaws, similar to ones expected in actual operating conditions, to systems and developing an initial understanding of the parameters that are sensitive to the expected damage. The flawed system can also be used to validate that the diagnostic measurements are sensitive enough to distinguish between features identified from the undamaged and damaged system. Damage accumulation testing can also be used to identify appropriate features. This process may involve induced‐damage testing, fatigue testing, corrosion growth, or temperature cycling to accumulate certain types of damage. The application of statistical analysis procedures to the vibration‐based damage detection problem was presented by Fugate et al. (2001). The damage detection process was cast in the context of a statistical pattern recognition paradigm. An autoregressive model was fitted to the measured acceleration–time histories from an undamaged structure. Structural damages were detected by judging statistical difference of data of intact state and present state (Iwasaki et al., 2002, 2005). The method requires data of the undamaged state and does not require the complicated modeling and large amounts

Statistical Pattern Recognition and Vibration-Based Techniques

of data for training. The damages were detected from the change of strain data measured on the specimen surface by the statistical tools such as response surface and F‐statistics. Ethernet local area network technologies were adopted for the sensor integration, together with a statistical unsupervised damage detecting method for automatic damage diagnosis. Damage was detected from changes in a set of data measuring loads on the turnbuckles of the jet‐fan. In another study, Iwasaki et al. (2007) proposed a statistical diagnostic method for structural damage detection, which employs system identification using a response surface, and damages were automatically diagnosed by testing the change of the identified system by statistical F‐distribution. The statistical diagnostic method consists of a learning mode and a monitoring mode. The monitoring mode was performed to diagnose the structural condition. A reference response surface was calculated from the reference data using the response surface method. In the monitoring mode, data was measured from a structure to be diagnosed, and a measured response surface was calculated. The statistical similarity of the reference response surface and the measured response surface was tested using the F‐test for the damage diagnosis. The statistical pattern recognition algorithms fall into the general classification referred to as supervised learning based on the available data from both the undamaged and damaged structure. The term “pattern” is used to denote the p‐dimensional data vector x = {x1, …, xp}T of measurements whose components xi are measurements of the features of a structure, where T denotes transpose. Thus the features are the variables used for classification. It is assumed that there exist C groups or classes, denoted by g1,  …, gc and associated with each pattern x. The objective is to establish decision boundaries in the feature space, which separate patterns belonging to different classes. The decision boundaries are determined by the probability distributions of the patterns belonging to each class, which must either be specified or learned (Duda and Hart, 1973; Devroye et al., 1996). Thus, a pattern vector x belonging to group gi is viewed as an observation drawn randomly from the group‐conditional probability function p(x|gi). A number of well‐known decision rules, including the Bayes decision rule, the maximum likelihood rule (which can be viewed as a particular case of the Bayes rule)1 are used to define the decision boundary (Watanabe, 1985; Jain et  al., 2000). Group classification and regression analysis are categories of the supervised learning algorithms. Unsupervised learning refers to algorithms that are applied to data not containing examples from the damaged structure. Supervised algorithm uses class information to design a classifier referred to as discrimination, while unsupervised algorithm is allocating data to groups without class information, referred to as clustering (Webb and Copsey, 2011). The SHM is fundamentally one of statistical pattern recognition and a paradigm. Statistical process control (Montgomery, 1997) is a process based rather than structure based and uses a variety of sensors to monitor changes in a process. A coherent strategy for intelligent fault detection was presented by Worden and Dulieu‐Barton (2004) who  provided a precise definition of what constitutes a fault and specification for 1  Bayes’ rule relates the odds of event A1 to the odds of event A2, before (prior to) and after (posterior to) conditioning on another event B. The odds on A1 to event A2 is simply the ratio of the probabilities of the two events. By definition, this is the ratio of the conditional probabilities of the event B given that A1 is the case or that A2 is the case, respectively. The rule simply states: posterior odds equals prior odds times Bayes factor.

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operational evaluation, which makes use of a hierarchical damage identification scheme. Farrar et al. (2001) defines the SHM process in terms of a four‐step statistical pattern recognition paradigm: (1) operational evaluation; (2) data acquisition, normalization, and cleansing, which involves selecting the excitation, the sensor types, number and locations, and the data acquisition/storage/transmittal hardware; (3) feature extraction and information condensation; and (6.iv) statistical model development for feature discrimination. The damage state of a structure involves damage existence, location, type, degree of severity, and prognosis. Statistical models are usually employed to determine the state of damage, particularly its existence and location. The statistical procedures can be used to better determine the type of damage, the extent of damage, and the remaining useful life of the structure. Farrar and Worden (2007) indicated that statistical models are also used to minimize false indications of damage, which fall into two categories. The first is false‐positive damage indication (indication of damage when none is present). The second is the false‐negative damage indication (no indication of damage when damage is present). Errors of the first type are undesirable, as they will cause unnecessary downtime and consequent loss of revenue as well as loss of confidence in the monitoring system (for more details see, e.g. Hayton et  al., 2007; Sohn, 2007; Worden and Manson, 2007). Statistical damage detection in a structure operating under different temperatures via vibration testing was considered by Hios and Fassois (2009) using a stochastic global model based approach. The approach relies upon global models of the functionally pooled form, which are capable of describing the dynamics at any temperature, and statistical decision‐making. Its effectiveness was validated via a large number of experiments performed on a smart composite beam at different temperatures within the range [−20 °C, +20 °C]. Outlier detection is the primary class of algorithms applied in unsupervised learning applications. An outlier is an observation that is numerically distant from the rest of the data. It is one that appears to deviate markedly from other members of the sample in which it occurs. Outliers can occur by chance in any distribution, but they are often indicative either of measurement error or that the population has a heavy‐tailed distribution. In the former case we need to discard them or use statistics that are robust to outliers, while in the latter case they indicate that the distribution has high kurtosis and that one should be very cautious in using tools or intuitions that assume a normal distribution. Outliers, being the most extreme observations, may include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low. However, the sample maximum and minimum are not always outliers, because they may not be unusually far from other observations. There are some general guidelines to follow when carrying out a classification. These include detecting and removing outliers. Equally important is to perform preprocessing of the data with the aim of reducing the number of variables or standardizing the variables to zero mean and unit variance. The possibilities of multivariate statistics for structural health monitoring were examined by Worden (1997, 1998) and Worden et al. (1999). They demonstrated the use of kernel density estimation  –  a means of approximating the probability density function for multivariate datasets. Once the probability density function is established, it can be used to flag departures of data from the normal condition. The concept of discordancy (not in harmony or in agreement) from the statistical discipline of outlier analysis was used to signal deviance from the norm. Manson et al. (2001) examined the

Statistical Pattern Recognition and Vibration-Based Techniques

influence of environmental changes on a composite panel behavior under changing conditions of humidity and the detection of structural damage. Measurements of Lamb wave profiles and the outlier analysis were used for damage detection. Later, Manson et al. (2002) presented a demonstrator system that first constructs a network of detectors, which are trained to be sensitive to change in the Lamb wave excitations in a composite panel. The output from detectors was then used to train an artificial neural network to recognize which area of the composite panel is being subjected to damage. 5.2.2  Damage Index and Outlier Analysis

Damage classification was established based on the damage index (DI), which represents the difference between the original input waveform and the reconstructed signal: t1

DI

1

t0 t1 t0

2

I t V t dt

I t

2

t1

dt V t t0

(5.1) 2

dt

where I(t) and V(t) refer to the known input and reconstructed signals, and t0 and t1 represent the starting and ending time points of the baseline signal’s first A0 mode. The value of DI becomes zero when the time reversibility of Lamb waves is preserved. The root square term in Equation (5.1) becomes 1.0 if and only if V(t) = βI(t) for all t where t0 ≤ t ≤ t1 and β is a non‐zero constant. If the reconstructed signal deviates from the input signal, the DI value increases and approaches 1.0, indicating the existence of damage along the direct wave path. When the DI value exceeds a threshold value, the corresponding signal indicates the occurrence of damage. To set the threshold value, Sohn et al. (2007a) suggested that it was necessary to collect the DI values from the baseline condition of the system and to characterize the distribution of the DI values. A threshold value can then be established for a user‐specified confidence level. To accomplish this task, consecutive outlier analysis was employed to identify damage without relying on the past baseline data. The issue of minimizing damage misclassification was addressed by Sohn et  al. (2007a) who developed an instantaneous damage detection scheme that does not rely on past baseline data. The proposed instantaneous damage diagnosis based on the concepts of time reversal acoustics and consecutive outlier analysis, and the proposed damage diagnosis were tested for detecting delamination in composite plates. The time reversal method was originally developed in modern acoustics to compensate the dispersion of Lamb waves and improve the signal‐to‐noise ratio of propagating waves (Prada and Fink 1998; Ing and Fink 1996, 1998; Fink 1999). The concept of time reversal acoustics is based on reconstructing an input signal at an excitation point if an output signal recorded at another point is reemitted to the original source point after being reversed in a time domain. Damage detection using the time reversal process is based on the fact that if there is any nonlinear defect along a wave propagation path, the time reversibility breaks down. By examining the deviation of the reconstructed signal from the known input signal, certain types of damage can be identified without requiring any past baseline signals.

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A statistical method for damage detection was presented by Worden et al. (1999). The DI values were instantaneously computed from a questionable state of a structure, and their values from all the paths were then sorted in ascending order. The largest DI value was tested for discordance against the remaining DI values. This is repeated for the second largest outlier, the third, and so on until a predetermined number of DI values are tested for discordance. Note that each DI value is tested for discordance with respect to the other simultaneously obtained DI values rather than with respect to those obtained from the baseline condition of the structure. Sohn et al. (2005) indicated that the measured data often do not conform to normal distribution, and the outlier analysis for other types of distributions is of considerable practical importance. For example, outliers in exponential samples arise when Poisson processes are employed (Barnett and Lewis, 1994). According to the central limit theorem, which states that if {X1, X2, …, Xn} is a set of random variables with arbitrary distributions, the sum variable X X1 X 2 Xn will have a Gaussian distribution as n approaches infinity. If the problem deals with the tails of a distribution, then the most relevant statistic for studying the tails of the parent distribution is the maximum operator,2 max({X1, X2, …, Xn}), which selects the point of maximum value from the sample vector. This statistic is relevant for the right tail of a univariate distribution only. For the left tail, the minimum should be used. Extreme value statistics deals with the extreme deviations from the medium of probability distributions. It seeks to assess, from a given ordered sample of a given random variable, the probability of events that are more extreme than any previously observed events. Extreme value distributions are the limiting distributions for the minimum or the maximum of a very large collection of random observations. The theorem of extreme value statistics states that in the limit as the number of vector samples tends to infinity, the induced distribution on the maxima of the samples can only take one of the three forms: Gumbel, Weibull, or Frechet (Fisher and Tippett, 1928). Furthermore, these three forms of extreme value distributions can be unified into a single generalized extreme value distribution (GEV) for the maxima and minima values, respectively (Jenkinson, 1955):





Maxima:

x; , ,

exp

Minima:

x; , ,

1 exp

1/

x

1

1

x

, 1/

x

1

,

1

0 (5.2)

x

0 (5.3)

where and are the generalized extreme values for maxima and minima, μ, σ, and γ are the mean (location) standard deviation (scale) and shape parameters of the generalized extreme values, respectively.3 Sohn et  al. (2007a) used the generalized extreme values to convert any extreme value distribution to an exponential distribution so that consecutive outlier analysis can be used. Consecutive outlier analysis was formulated 2  The element maximum operator () compares each element of matrix-1 to the corresponding element of matrix-2. The larger of the two values becomes the corresponding element of the new matrix. 3  These symbols should not be confused with those used in the theory of fracture mechanics of Part I.

Statistical Pattern Recognition and Vibration-Based Techniques

for exponential samples. An exponential distribution with a scale parameter b and an origin at a was given by the following probability density distribution: 1 exp b

x

x a b

0



for x a (5.4) for x a

An outlier test for the single smallest sample in an exponential sample is first formulated. A test statistic for the smallest potential outlier is defined as (Lewis and Fieller, 1979):

Y1 a , for non-zero origin (5.5a,b) Yi na

Y1 , for zero origin, or T Yi

T

where samples Y1, Y2, …, Yn are arranged sorted in an ascending order, and n is the size of the samples. It was shown that this test statistic has the probability density function ψn(t):

n

t

n n 1 1 nt

n 1

for t

0



for 0 t 1 n

1 n (5.6)

The test statistic has a recurrence relationship with the cumulative density function n 1 (t ):

n

t

nb t

1

1, n 1

n 1

t

1 t

(5.7)

where b(t )r ,s [ B(r ,s)] 1 t r 1 (1 t )s 1 is a beta density function with parameters r and s, and B(r, s) is a binomial distribution. The significance probability associated with an observed value t of a discordance statistic T is denoted by SP(t)



SP t

nF2 ,2 n

1

n 1 t 1 t

(5.8)

where Fv,u(x) is an F‐cumulative distribution function with v and u degrees of freedom, SP(t) is the probability that T takes values more discordant than t. An outlier probability OP(t) may be defined by the equation



OP t

1 nF2 ,2 n

1

n 1 1 t

t (5.9)

The analysis is consecutively conducted, starting from the smallest value to the ­second smallest value, the third, and so on, until all outliers are identified. 5.2.3  Case Study: Impact Tests of Composite Plates

The concept of damage index in the presence of outliers was demonstrated by Sohn et al. (2007a), who conducted impact tests on a composite plate shown in Figure 5.1.

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4

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Figure 5.1  Layout of the sensing and actuating PZT wafers for detecting delamination on a composite plate of 2 ft × 2 ft and 0.25 in thickness. The composite laminate contains 48 plies stacked according to the sequence [6(0 / 45 / 45 / 90 )]sym, consisting of Toray T300 graphite fibers and a 934 epoxy matrix. (Sohn et al., 2007a)

The surface damage on the composite plate was visually inspected after each impact test. A very small (~5 mm diameter) dent was barely visible on the impact side of the plate. On the back side of the plate, there was a very small crack (less than 1 mm thickness and about 2 cm long), in the form of an internal delamination. Figure 5.2(a) shows the actual impact location, while the identification of the damaged paths shown in Figure  5.2(b) was established such that if there is any defect along the wave propagation path, the time reversibility of Lamb waves breaks down. To identify the location and area of the delamination, a damage localization algorithm developed by Sohn et al. (2003b) was used. The delamination location and size estimated by the active sensing system is shown in Figure 5.2(c). Figure 5.3 shows the time reversibility of Lamb waves when the original input was inserted at PZT wafer #6, and the response was reemitted at PZT wafer #9 after time reversed (Figure 5.1). Figure 5.3 shows the violation of the time reversibility due to delamination. Figure  5.3(a) reveals that the reconstructed signal (the dotted curve) is very close to the original input signal (the solid curve) except near the tails of the A0 mode. On the other hand, Figure 5.3(b) reveals the distortion of the reconstructed signal due to the internal delamination. The process of the Lamb wave propagation was repeated for different combinations of actuator–sensor pairs. A total of 66 different path combinations were investigated. Based on the time reversibility shown in Figure 5.3, the DI was computed for all 66 paths when there was no delamination on the plate. The resulting DI values are shown in Figure 5.4(a). This procedure was repeated after the delamination was introduced to the impact location indicated in Figure 5.2(a). Sohn et al. (2007b) established a threshold value shown in Figure 5.4(a) by fitting one of the extreme value distributions to the baseline DI values. When the

Statistical Pattern Recognition and Vibration-Based Techniques

(a)

(b)

(c)

Figure 5.2  Damage localization and quantification based on a time reversal process: (a) actual impact location, (b) actuator‐sensor paths affected by the internal delamination; and (c) the damage size and location. (Sohn et al., 2007a)

DI values were computed again after the impact test, this predetermined threshold value was used to identify five outliers shown in Figure 5.4(b). These five outliers correspond to the five wave propagation paths crossing the delamination location shown in Figure 5.2(b). The values of DI after the impact arranged in an ascending order are shown in Figure 5.5(a). The outlier probability is shown in Figure 5.5(b) according to equation (5.9) for the first 12 largest DI values. The maximum possible number of outliers was set to k 12, based on the assumption that the damage is localized and only a few wave propagation paths are affected by delamination. For a given sample size, n, an upper limit on the maximum possible number of outliers needs to be estimated. Note that the concept of outlier analysis becomes meaningless if the maximum possible number of outliers becomes close to or larger than half of the total sample number. In this case, the outlier detection problem turns into a classification problem between two classes of data. For a consecutive outlier analysis of the kth largest value, the sample size of the n k remaining samples should be large enough so that a generalized extreme value

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(a) 1.0

0.5

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Figure 5.3  Comparison between the original input signal (solid) and the restored signal (dotted) during the time reversal acoustic process between exciting PZT wafer # 6 and sensing PZT wafer #9: (a) the time reversibility of Lamb waves for the intact composite plate and (b) the breakdown of the time reversibility due to the internal delamination. (Sohn et al., 2007a)

can be fit to the data. The outlier probability for the largest DI values was found to be  about 49.4%, and the maximum value of the outlier probability was found to be 0.99999900891873 for the fifth largest DI value. For a given confidence level of 99.9%, the fifth largest DI value was classified to be an outlier, and this implies that the larger DI values X i , i 63, 64 , 75, 66 are outliers as well. Once the number of actual outliers

Statistical Pattern Recognition and Vibration-Based Techniques

(a) 1.0

Damage index

0.8 0.6

A threshold (=0.243) corresponding to A 99.9% confidence level

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Figure 5.4  Threshold levels and subsequent damage identification using the baseline data: (a) threshold value using damage index values obtained from the pristine composite plate, (b) identification of damaged paths using the previously established threshold value. (Sohn et al., 2007a)

was identified, the automated procedure for the selection of the best‐fit generalized extreme value and parameter estimation described earlier was applied to the remaining DI values X i (i 1, 2, , n 6). Using the parameter estimation technique for the generalized extreme value, the model parameters, μ,  σ, and γ of the best‐fit generalized extreme value were estimated to be 0.0331, 0.04 , and 0.067, respectively. From the estimated generalized extreme value, a threshold to a 99.9% confidence level was computed to be 0.251, as shown in Figure 5.5(d). This value is close to the threshold value of 0.243 previously estimated from the baseline DI values. Note that the correct outliers were identified using the consecutive outlier analysis described in Figure 5.5(b), and this additional

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(a) Outliers

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0.8 0.6 Fit an extreme distribution to the remaining data

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(b) The outlier probability reaches its maximum value at the 5th largest damage index value 1.0 0.8 Outlier probabilities

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Figure 5.5  Threshold and instantaneous damage identification without using prior baseline data: (a) sorted DI values in an ascending order and the damaged paths (outliers) are identified, (b) the computed outlier probability for the first 15 largest damage index values; it reaches its maximum value at the 5th largest damage index value, (c) damage distribution function after excluding the five outliers identified in (b), the GEV is fitted to the remaining 63 DI values to estimate a new threshold value (=0.251); and (d) damage index versus wave propagation paths showing the use of the new threshold value (0.251 vs 0.243) confirms that the 5 largest DI values are outliers and the associated paths are influenced by internal delamination. (Sohn et al., 2007a)

step of establishing the threshold value in Figures 5.5(c) and (d) were simply taken to substantiate the findings of the consecutive outlier analysis. An outlier analysis for damage detection of railroad tracks using a macro‐fiber composite impedance‐based wireless structural health monitoring system was outlined by

Statistical Pattern Recognition and Vibration-Based Techniques

(c)

Damage distribution function

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0.8 Empirical cumulative density function

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Figure 5.5  (Continued )

Park et  al. (2008a). An outlier analysis based on Mahalanobis squared distance4 was proposed by taking root mean square deviation values of impedance signatures as a damage‐sensitive feature vector. Optimal threshold values for both root mean square deviation and Mahalanobis squared distance were determined through the proposed 4  The Mahalanobis (1936) distance of an observation x { x1 , { 1 , , n }T and covariance matrix S is defined as: D( x )

(x

)T S 1 ( x

,xn }T from a set of observations with mean )

This distance is zero if the point is at the mean, and grows as the point moves away from the mean. It measures the number of standard deviations from the point to the mean of D.

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outlier analysis. It was shown that the use of Mahalanobis squared distance improved the damage detection capability with a lower threshold level as compared to that of root mean square deviation.

5.3  Vibration‐Based Techniques 5.3.1 Overview

As stated in Chapter 4, a damage or a fracture in a given structure results in the creation of new surfaces and boundary conditions. These new boundary condition will affect the dynamic characteristics of the structure, particularly in its normal mode properties, frequency response function, and shift in the peaks of the fast Fourier transform. Vibration‐based damage detection methods are used in structural systems to identify their global dynamic response. The effects of damage on a structure can be classified as linear or nonlinear. A linear damage is characterized by the fact that the initially linear‐ elastic structure remains linear‐elastic after damage. The changes in modal properties are due to changes in the geometry and/or the material properties of the structure, but the structural response can still be modeled using linear equations of motion. On the other hand, nonlinear damage is manifested by the fact that the initially linear‐elastic structure behaves as a nonlinear system after the damage has occurred. Vibration‐based damage detection techniques are based on observing changes to the structure dynamic characteristics to detect the presence of damage and determine its location. There are still some doubts regarding the validity of modal techniques, since significant frequency shifts may be caused by changes in environmental conditions. In addition, operational deflection shapes and vibration deflection shapes are very valuable tools to detect damage occurrence. Vibration‐based damage detection techniques usually do not require any prior knowledge of the damage location. These techniques are classified based on the type of the measured data used and the method used to identify the damage. Sub‐surface structural failure was detected by Vandiver (1975) who measured changes in the natural frequencies in the first two bending modes and the first torsional mode of offshore light station tower structure. Statistical energy analysis was introduced as a method of predicting the dynamic response of a wide variety of fixed and floating offshore structures to random wave forces. It was found that failure of most members produces a frequency change greater than 1%, and thus damage in most of the elements is detectable. A vibration technique for non‐destructively evaluating the integrity of structures using measurements of the structural natural frequencies was proposed by Adams et al. (1978) and Cawley and Adams (1979). It was shown that measurements made at a single point in the structure can be used to detect, locate, and quantify the damage. The method was applied on an aluminum plate and a cross‐ply carbon‐fiber‐reinforced plastic plate. Excellent agreement was reported between the predicted and actual damage sites together with the magnitude of the defect. The change of the element stiffness was related to the change of the modal frequency (Stubbs, 1985; Stubbs and Osegueda, 1987; Stubbs and Osegueda, 1990a, 1990b). Experimental investigations were conducted by Spyrakos et  al. (1990), Chen et  al. (1995), and Brincker et al. (1995) to establish the dependence of the modal frequency

Statistical Pattern Recognition and Vibration-Based Techniques

changes due to damage in the structure. It was found that the frequency decreases when the damage extent increases. Salawu (1997) indicated that a 5% frequency shift might be required to detect structural damage with confidence when using frequencies as an indication of the damage occurrence. However, frequency shifts alone might not necessarily indicate that damage has occurred in the structure. For example, Aktan et  al. (1994) showed significant frequency shifts (exceeding 5%) caused by changes in ambient conditions measured for bridges in a single day. Additionally, different damage locations can produce the same degree of the frequency shift; therefore, using only frequency changes might not be sufficient to uniquely determine the damage location. Changes in natural frequencies and the occurrence of anomalous mode shape measurements were used as indicators of structural damage in a series of tests on structures made of glass reinforced plastic space‐frame lattice (Adams, R.D. et al., 1991). A vibration‐based technique was used to identify multiple discrete cracks in a structure by Hu and Liang (1993). Two damage modeling techniques were integrated to provide a crack‐ detection technique that utilizes the global vibration characteristics of a structure but offers local information on each individual crack, including location and extent of the cracks. The first involves the use of massless, infinitesimal springs to represent discrete cracks and the second employs the continuum damage concept. In the spring model, the Castigliano’s theorem and the perturbation technique were used to derive a theoretical relationship between the natural frequency changes and the location and extent of the discrete cracks. In the continuum damage model, the effective stress concept coupled with the Hamilton’s principle was used to derive the similar relationship. In the proposed integrated approach, the continuum damage model can be used first to identify the discretizing elements of a structure that contain cracks. Then, the spring damage model was used to quantify the location and severity of the discrete crack in each damaged element. An example of a simply supported beam containing two discrete cracks was given to illustrate the application and accuracy of the proposed approach. Friswell et al. (1998) used low frequency measured vibration data to detect and locate damage in structures. Since a structure can be damaged in many possible locations, the equations used for an inverse approach are usually ill‐conditioned and must be regularized. One approach is to assume that the damage changes only a small number of the candidate parameters, and the identification is based on this reduced parameter set. Lu and Hsu (2000) adopted the wavelet transform to detect the existence and location of structural damage. Minor localized damage, which generates vibration signals that are almost indistinguishable from those of an undamaged system were found to induce significant changes in the wavelet coefficients of the vibration signals. In addition, the maximum difference between wavelet coefficients of the damaged and original systems was found to occur at the location of the damage. Space geometry changes extracted from a nonlinear time series was presented by Todd et al. (2001a) within the context of vibration‐based damage detection in a structure. Peeters et al. (2001) discussed two practical issues in the application of vibration‐ based health monitoring to civil engineering structures: the excitation source and the effect of temperature. The results of different excitation types were compared: band‐ limited noise generated by shakers, an impact from a drop weight, and ambient sources such as wind and traffic. The effect of temperature on measured natural frequencies was demonstrated, and a methodology was proposed to distinguish these temperature effects from real damage events. The temperature sensitivity of some vibration‐based

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damage identification techniques was considered by Serker et al. (2009). With the help of a simply supported beam with different damage levels, reliability of these techniques for damage identification in a changing environmental temperature condition was studied. The temperature effect was considered to result in changing the modulus of elasticity of the material. It was shown that both damage localization and quantification can be affected in the presence of the temperature effect which can lead to unreliable damage detection. Some review articles addressing vibration‐based techniques from different aspects and applications were reported in the literature (e.g. Doebling et al., 1998; Zou et al., 2000; Alvandi and Cremona, 2006; Montalvão et al., 2006; Zhu, H.P. et al., 2011). For example, Doebling et al. (1998) provided an overview of methods to detect, locate, and characterize damage in structural and mechanical systems by examining changes in measured vibration response. The methods were also described in general terms including difficulties associated with their implementation and their fidelity. A summary of developments in the area of vibration‐based damage identification based on changes in vibration characteristics was presented by Farrar et  al. (1999, 2001). Vanlanduit et al. (2000) studied the feasibility of using nonlinear vibration features to detect and locate damage in structures. A comparison of linear and nonlinear damage detection features was performed on measurements of a cracked plexi‐glass panel and a delaminated composite plate. Zou et al. (2000) presented a review of vibration‐based model‐dependent methods with piezoelectric sensor and actuator incorporated into composite structures. These methods utilize finite element analysis techniques, together with experimental results, to detect damage. They locate and estimate damage events by comparing dynamic responses between damaged and undamaged structures. According to the dynamic response parameters analyzed, these methods can be subdivided into modal analysis, frequency domain, time domain, and impedance domain. Model‐dependent methods are able to provide global and local damage information. Montalvão et al. (2006) reviewed some of the latest advances in structural health monitoring and damage detection, with an emphasis on composite structures on the grounds that this class of materials currently has a wide range of engineering applications. The state‐of‐the‐art of dynamic methods for structural health monitoring and damage detection in structures was presented in a book by Morassi and Vestroni (2008). Zhu, H.P. et al. (2011) reviewed the vibration‐based damage identification methods based on dynamic characteristics and recognition algorithms. 5.3.2  Damage Detection Using Strain Energy Method

The strain energy of a vibration mode is referred to as the modal strain energy of that mode. The total modal strain energy is the sum of the modal strain energy contributions of all modes considered. The modal strain energy is calculated by linking the deformation of a structure to the strain. When a particular vibration mode stores a large amount of strain energy in a particular structural load path, the frequency and displacement shape of that mode are highly sensitive to changes in the impedance of that load path. Thus, strain energy is a logical choice of criterion in model update mode selection. Stubbset al. (1992) introduced a damage identification method based on the observation that local changes in the modal strain energy of the vibration modes of a structure are a sensitive indicator of damage. A distinction must be made between axial, flexural, and

Statistical Pattern Recognition and Vibration-Based Techniques

torsional deformation–strain relations (Duffey et al., 2001; Loendersloot et al., 2009). In the case of structures with dominant global behavior, such as a cantilever beam, the lowest frequency modes may also contain the best overall distributions of structural strain energy. However, for structures that are dominated by local behavior, such as a free truss with distributed mass, the strain energy distribution is not as concentrated in the lowest frequencies. In many cases, the modes that contain vital information about the damage occur at higher frequency, as indicated by Kashangaki et al. (1992). A mode selection strategy based on maximum modal strain energy was adopted by Doebling et al. (1997) for structural damage detection. They considered strategies that use the strain energy stored by modes in both the undamaged and damaged structural configuration. It was demonstrated that more accurate results are obtained when the modes are selected using the maximum strain energy stored in the damaged structural configuration. Since damage alters the dynamic characteristics of a structure, Sampaio et al. (1999) proposed four steps of damage detection, namely existence, localization, extent, and prediction. The frequency response function curvature method encompasses the first three steps based on only the measured data, without the need for any modal identification. The method was described theoretically, and compared with two of the most referenced methods in the literature. Otieno et al. (2000) presented a study dealing with the sensitivity of the modal strain energy method to the stiffness matrix and its accuracy in detecting the location and extent of damage. The modal strain energies for each element of the undamaged structure were computed for each mode using the original analytical matrix and measured modal data. Modal data from the damaged case was used to update the stiffness matrix by a simplified matrix update scheme. Modal data was obtained from experiment using scanning laser vibrometer to extract the mode shape vectors from the experimental results. It was found that the modal strain energy change method does not give an accurate location of the damage. A free vibration method was evaluated for damage detection in thick composite plates by Penn et  al. (1999). Vibration was initiated by a single external mechanical pulse and was sensed by piezoelectric patches bonded to the surface of the composite plate. The investigation was conducted on unidirectional fiber glass/epoxy plates approximately 0.387 in thick, containing controlled delaminations. The free vibration method was found relatively insensitive for identifying small to medium delaminations in thick plates. Several frequency computations using a finite element model were performed to determine the influence of delamination size and location for a single delamination. Cornwell et al. (1999) used measured modal parameters to detect and locate damage in plate‐like structures. A method based on the changes in the strain energy of the structure was applied to beam‐like structures characterized by one‐dimensional curvature. The method was generalized to plate‐like structures characterized by two‐ dimensional curvature. This method requires the mode shapes of the structure before and after damage. Worden and Manson (2001) presented an experimental verification of structural health monitoring approach using the strain energy method to detect and locate damages in structures. A stiffened panel whose upper surface made of aluminum sheet was used to carry out a complete top surface modal analysis with 19 measurement points for each stage of damage. Finite element simulations were used to extract natural frequencies and mode shapes and to compute modal damage indicators for each level of damage. The results from the wing‐box test signified that it was possible to detect damages

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by careful choice of modes. The problem of identification of damage locations for plate‐ like structures using strain mode technique was considered by Li, Y.Y. et al. (2002). The strain modal analysis based on the Rayleigh–Ritz approach of a damaged plate was performed to obtain strain mode shapes. Sazonov and Klinkhachorn (2005) presented an analysis of determining the optimal sampling interval that would minimize the effects of measurement noise and truncation errors on the calculation of the curvature and strain energy mode shapes, thus maximizing sensitivity to damage and accuracy of damage localization. The low frequency vibration excitation for damage detection in thin layered composite material was employed by Ajaykumar (2006). Thin layered laminates were tested before and after the introduction of damage. With the help of a simply supported beam with different damage levels the reliability of vibration‐based damage identification techniques was examined by Alvandi and Cremona (2006). These techniques are based on measured modal parameters, which use only a few mode shapes and/or modal frequencies of the structure that can be easily obtained by dynamic tests. By realizing two  sets of dynamic measurements, corresponding to two moments of the structure lifetime, the dynamic modal parameters was obtained. In order to properly assess the performance of these techniques, different noise levels were randomly introduced to the response signals of a simulated beam excited by a random force. It was concluded that among the evaluated techniques the strain energy method presents the best stability regarding noisy signals. The change in mode shape curvature, change in flexibility, and change in flexibility curvature methods were also found capable of detecting and localizing damaged elements, but in the case of complex and simultaneous damages these techniques show less efficiency. Two identification algorithms for assessing structural damages using the modal test data were proposed by Hu, N. et al. (2001). The algorithms are similar in concept to the subspace rotation algorithm or best achievable eigenvector technique. A quadratic programming model was set up for the two approaches to predict the damage extent. Manson et al. (2002) presented a demonstrator system that first constructs a network of detectors trained to be sensitive to change in the Lamb wave excitations in a composite panel. The output from these detectors was then used to train an artificial neural network to recognize which area of the composite panel is being subjected to damage. Worden et al. (2009) demonstrated a neural network approach on the damage location problem for an aircraft wing. The results were compared with a probabilistic classifier based on a multi‐layer perceptron neural network and shown to give similar results. Shi, H.W. et al. (2009, 2011) proposed multi‐criterion method incorporating modal flexibility and modal strain energy method for multiple damage assessment in beam and plate structures and truss bridges based on the vibration characteristics of the structure. 5.3.3  Damage Detection and Location Using Modal Properties

Modal techniques for non‐destructive evaluation are typically implemented using finite element model update based on modifying the stiffness, mass, and/or damping matrices to minimize some measure of error. A review and a detailed discussion of the overall field of finite element model update were presented by Hemez and Farhat (1993, 1995b). When using finite element model update for model refinement, it may be desirable to constrain the material properties of several different elements to be the same and then

Statistical Pattern Recognition and Vibration-Based Techniques

update this common parameter. However, for non‐destructive evaluation, it is desirable to let these properties vary independently because the changes are to reflect isolated incidents of damage and not overall errors in modeling assumptions. This makes finite element model updating for non‐destructive evaluation more difficult than finite element model updating for other applications. The modal analysis approach was used to investigate the possibility of detecting damages in composite material structures by Crema et al. (1985). The case history of a glass fiber blade of a small wind turbine was considered with the purpose of achieving significant data by the changes in the modal parameters due to damages in the structure induced by loads above the working design loads. The damage induced by loads well above the working load was found to cause a small decrease in the natural frequencies for several modes. Methods for finite element model update used for non‐destructive evaluation include optimal matrix update (e.g. Zimmerman and Kaouk, 1992a), sensitivity‐based matrix update (e.g. Hemez and Farhat, 1993; Farhat and Hemez, 1993; Doebling et al., 1993; Ricles and Kosmatka, 1992), and eigen‐structure assignment (e.g. Zimmerman and Kaouk, 1992b; Kashangaki, 1992; Lim, 1995; Lim and Kashangaki, 1994). The first few modes of the structure were used in finite element correlation, since they are generally the best identified modes. However, in some situations, the higher frequency modes are critical to the location of structural damage. Many modes below these in frequency were found to undergo no significant modification as a result of the damage, so they contribute to the computational burden without contributing significantly to the location of the damage. The number of modes was found to be limited by the inherent ill‐conditioning and statistical bias associated with large‐order update problems (e.g. Cogan et al., 1995; Hemez and Farhat, 1995a). It was concluded that it is necessary to develop systematic criteria for selecting which modes are most indicative of the structural damage. Mode selection was studied by Lim, T.W. (1991), Lim, K.B. (1992), and Kashangaki (1995). For example, Kashangaki (1995) presented the modal sensitivity parameter of the finite element model eigen‐solution to changes in each of the elemental stiffness parameters. This sensitivity parameter was then used as a pre‐test analysis tool for determining which modes should be targeted in the test. Lim, T.W. (1991) presented a method of mode selection based on modal cost, which is a measure of the level of energy required to excite a certain mode. Other studies by Kashangaki et al. (1992) and Chen and Garba (1988) revealed the importance of strain energy in the identification of structural behavior and location of structural damage in terms of the structural load paths. Damage‐induced changes in natural frequencies, damping ratios, and mode shapes can be detected using experimental modal analysis by Nicholson and Alnefaie (2000). A parameter was defined, which was calculated using the measured changes in eigen‐ parameters. This parameter was found to exhibit a significant “jump” at the damage location and whose value is a direct measure of the severity of the damage. This damage‐sensitive parameter was utilized to localize the damage and to assess its severity. A  modal parameter analysis for damage detection of composite structures was presented by Hwang, H.Y. and Kim (2004), Yam et al. (2005), and Just‐Agosto et al. (2007). Rucevskis et al. (2009) presented a vibration‐based damage detection method based on the idea that a damage is a combination of different failure modes in the form of loss of local stiffness in the structure altering its dynamic characteristics. The study focused on the identification of damage location and size in a laminated composite beam by

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extracting mode shape information obtained from vibration experiments. A low‐velocity impact was used in order to introduce damage in a beam. The modal frequencies and the corresponding mode shapes were obtained by using a scanning laser vibrometer with a piezoelectric actuator. Wosu and Daniel (2008) carried out an experimental analysis to detect damage in composite beams. The analysis incorporated finite element analysis of delamination and its effect on the natural frequencies of the composite beam using cohesive zone elements of a commercial finite element software material model. The results revealed a significant increase in the delaminated region as the percentage deviation in the natural system frequencies increases. The amount of deviation was different for different types of delamination. A stress wave propagation technique was used by Ali et al. (2005) to detect the precursor to the buckling failure in the form of early changes in the local curvature of the blade. Two different techniques were used to detect the precursors to buckling in the blade. The first is the change in the vibration shapes and natural frequencies of the column. The second is the change in the characteristics of diagnostic Lamb waves during the buckling deformation. Experimental results revealed that very small changes in curvature during the initial stages of buckling are detectable using the structural health monitoring techniques. Finite element analysis was used to identify the frequencies that are most sensitive to buckling deformation, and to select suitable locations for the placement of sensors that can detect even small changes in the local curvature. A small series of all‐composite test pieces emulating wings from a lightweight all‐composite unmanned aerial vehicle (UAV) were developed to support damage detection and SHM research by Oliver et al. (2006). Detailed finite element models were generated for four structural components and the assembled structure. Each wing component piece was subjected to modal characterization via vibration testing using a shaker and scanning laser Doppler vibrometer before assembly. The correlation process led to the following average percentage improvement between experimental and finite element frequencies of the first 20 modes for each piece: top skin 10.98%, bottom skin 45.62%, main spar 25.56%, aft spar 10.79%. The assembled wing model with no further correlation showed an improvement of 32.60%. To detect surface cracks in various composite laminates Hu et al. (2006, 2010) and Hu and Wu (2008, 2009) performed modal analysis to obtain the mode shapes from both experimental and finite element analysis results of a plate with crack damage on one side of the plate. The mode shapes were then used to calculate strain energy using the differential quadrature method. The strain energies of laminated plates before and after damage were used to define a damage index which identified the surface crack location. Experimental modal analysis was conducted on an aluminum alloy 6061 thin plate to obtain the mode shapes before and after damage under a completely free boundary condition. A damage index was defined based on strain energy ratio of the plate before and after damage to identify the location of surface crack in plate structure. For all measured mode shapes, a damage index is defined by using the ratio of modal strain energies of the plate before and after damage. Experimental results showed that the scanning damage index identifies a surface crack location by using only a few measured mode shapes of the aluminum plate. Weber et  al. (2005) presented a vibration‐based damage detection method, which updates a finite element model from measured eigen‐frequencies and mode shapes. Changes in stiffness of individual elements were interpreted as damage. An appropriate

Statistical Pattern Recognition and Vibration-Based Techniques

finite element model was proposed by Yam et al. (2005) to analyze the dynamic characteristics of different types of structures made of honeycomb sandwich plates and multi‐ layer composite plates, with internal cracks and delamination. Natural frequencies, modal displacements, strains, and energy were evaluated for the determination of damage severity and location. Vibration measurements were carried out using piezoelectric patch actuators and sensors for comparison and verification of the finite element model. The mechanism of mode‐dependent energy dissipation of composite plates due to delamination was manifested. Experimental results showed the dependence of changes of modal parameters on damage size and location. The results showed that the measured modal damping change combined with the computed modal strain energy distribution can be used to determine the location of delamination in composite structures. Both numerical and experimental findings were found essential as a guideline to size and location identification of damage in composite structures. Minor impacts, perforations, and debonding damages in sandwich composites were studied by Just‐Agosto et al. (2007). Vibration mode shape curvature and damping matrix identification were used together two techniques based on transient temperature response (geometrical interpretation of convex hulls and thermographic heat patterns). Based on the robustness and promise in actual service implementation, only the results of the mode shape curvature and thermographic heat patterns were implemented into a properly trained neural network. The influences of delamination on the natural frequencies, frequency, and voltage responses of smart non‐uniform thickness laminated composite beams was examined by Ghaffari et al. (2009). A higher‐order finite element model, taking into account the electromechanical coupling effect, was used and found to accurately and efficiently represent the dynamic responses of the structure for forced/random stimuli. Damage in composites was identified by Zwink et al. (2010) using vibration measurements. The approach is based on the presence of modulation between the low‐frequency harmonic vibration response that is excited by the rotor dynamics and the high‐frequency acoustic response that is excited by a ceramic actuator. The hypothesis is that modulation between these two forced response components occurs if the composite material is damaged because the damage introduces nonlinear stiffness or damping restoring forces. For example, a delamination will produce clapping, buckling, or rubbing of the composite material when it is loaded, depending on the direction and frequency range of the static and dynamic loads. The low frequency vibration is of sufficient energy to “pump” the composite structure to stress the damaged region whereas the high frequency acoustic waves are of small enough wavelength to “probe” the structure to detect the damage. The effect of a single crack in an axially vibrating thin rod was found to cause the nodes of the mode shapes to move toward the crack (Dilena and Morassi, 2002, 2003a). Detecting a small open crack in an axially vibrating beam with viscous boundary conditions using non‐destructive dynamical measurements was studied by Dilena and Morassi (2003b). The damage was simulated by an equivalent linear elastic spring. It was shown that the measurement of the changes in a suitable pair of eigenvalues leads to the solution of the diagnostic problem, namely identification of crack location and its severity. Later, Morassi and Dilena (2006) considered the identification of a single defect in a discrete spring–mass or beam‐like system by measurements of damage‐induced shifts in resonance frequencies and anti‐resonance frequencies. For initially uniform

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discrete systems, it was shown that the measurement of an appropriate set of frequencies and anti‐resonances permits unique identification of the damage. The theoretical results were confirmed by comparison with numerical and experimental tests. Panigrahi et al. (2009) formulated an objective function for the genetic search optimization procedure along with the residual force method for the identification of macroscopic structural damage in a uniform strength beam. They considered variable width of the beam in order to keep the strength of beam uniform throughout, and in another case they considered both width and depth to vary to represent a special case of a uniform strength beam. The experimental results were simulated numerically by using finite element models of structures with inclusion of random noise on the vibration characteristics in an attempt to identify the damage occurrence. Yoon et al. (2009) used only the mode shapes from the damaged structure without a priori knowledge of the undamaged structure and introduced a “globally optimized smooth shape” with an analytic mode shape function for identifying the location of structural damage in a beam. The damage detection ability was enhanced by the summation of normalized damage indexes. An efficient flexibility‐based method for locating damage in plate structures was proposed by Yoon et al. (2004). A procedure for locating variability in structural stiffness was presented by Yoon et  al. (2005). The procedure locates regions in a structure where the stiffness varies. A variability index was generated for each test point on the structure. Increased variability is due either to structural stiffness features or damage. A statistical treatment of the indices enables discrimination of areas with significant stiffness variability. Providing the damaged areas are sufficiently small compared to the total surface area, their indices will be statistical outliers. Radzienski et al. (2011) introduced a method for structural damage detection based on experimentally obtained modal parameters of an aluminum cantilever beam. The damage was practically realized as saw cuts of different sizes and at different locations. The analysis included an attempt at damage identification with the most often used damage indicators based on measured modal parameters. New damage indicators such as frequency change based damage indicator and the hybrid damage detection method were introduced, utilizing the change of natural frequencies and any mode shape because the measurement of frequencies is much less time‐consuming in comparison to total mode shape measurement. It was found that the proposed technique is suitable for damage localization in beam‐like structures. A noise analysis method together with the Monte Carlo technique was proposed by Yu et al. (2006) for assessing the influence of experimental noise of modal data on sensitivity‐based damage detection methods. A one‐story portal frame was adopted to evaluate the efficiency of both the proposed noise analysis technique and the improved modal sensitivity based method. The assessment of results showed that the proposed statistics‐based noise analysis technique is effective and is more suitable for the vibration‐based damage identification. Kim, J.T. et al. (2007) proposed a generic algorithm based damage detection method using a set of combined modal features. The algorithm used was formulated for beam‐type structures. Experimental modal tests were performed on free‐free beams. Modal features such as natural frequency, mode shape, and modal strain energy were experimentally measured before and after damage in the test beams. Experimental results revealed that the damage detection is the most accurate when frequency changes combined with modal strain energy changes were used as the modal features for the proposed method.

Statistical Pattern Recognition and Vibration-Based Techniques

Through computer simulation and experimental investigation of a simply supported beam, Adewuyi et al. (2009) comparatively evaluated the performance of these techniques for practical civil structural health monitoring by using displacement modes from accelerometers and long‐gage distributed strain measurements. Dynamic computer simulation techniques were developed by Shih et al. (2009), using non‐destructive methods for damage assessment in beams and plates, which are important flexural members in building and bridge structures. In addition to changes in natural frequencies, this multi‐criterion procedure incorporates the modal flexibility and the modal strain energy method, which are based on the vibration characteristics of the structure. Using the results from modal analysis, algorithms based on flexibility and strain energy changes before and after damage were obtained and used as the indices for the assessment of the status of the structural health. 5.3.4  Damage Detection Using Frequency Response Function

The frequency response function is the quantitative measure of the response spectrum of a structure to response to sine wave or random excitation. It is used to characterize the dynamics of the structure in the frequency domain. The frequency response function provides a measure of response amplitude and its phase relative to the excitation. For linear time‐independent systems, the frequency response also will not vary with time unless the structure experiences a damage or fracture. In this case, the peaks of the response FFT will experience a shift, and this can be best revealed by wavelet transform. A frequency response function‐modal energy diagnostic method was proposed by Hu and Jiang (1993) for damage detection of composite laminated plates. It was found that, in the case of small damage, the ratio of each order generalized modal energy contribution in an element or area is approximately equal to the ratio of change rate at each order frequency response function (FRF) resonant peak of undamaged plate in the corresponding damaged element or area. Hwang, H.Y. and Kim (2004) presented methods to identify the locations and severity of damage in structures using frequency response function by taking only a subset of vectors from the full set of frequency response functions for a few frequencies and calculating the stiffness matrix and reductions in explicit form. The development of a minimum rank perturbation theory was proposed by Kaouk and Zimmerman (1993a), as an approach for system health monitoring and model correlation. Algorithmic approaches to enhancing system health monitoring capability were developed by Zimmerman et al. (1993) and Kaouk and Zimmerman (1993b), who utilized direct frequency response function data. The advantages of using direct frequency response function measurements include reduction in testing time, in that typically data is required at only a small number of frequency values. The frequency response function of a structure was measured by Mastroddi (2000) with the purpose of either identifying its numerical spatial (and then spectral) model, or directly localizing and quantifying possible damage and failure in term of mass, stiffness, and damping variations. An on‐line vibration‐based damage identification technique for fiber‐reinforced laminated composites and their sandwich construction was developed by Kim, H.Y. (2003). This technique uses the structural dynamic system reconstruction method exploiting the frequency response functions of a damaged structure. The delamination extent for the sandwich beams and the fatigue damage level for the

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laminated composites were identified in terms of the changes in natural frequencies and damping ratios of the reconstructed frequency response functions for damaged composite structures. Oliver and Kosmatka (2005) examined the viability of modal analysis techniques for detecting skin–core disbonds in carbon fiber Nomex honeycomb sandwich panels through laboratory testing. A series of carbon fiber prepreg and Nomex honeycomb sandwich panels were fabricated by means of autoclave co‐cure. A laser vibrometer was used to capture frequency response functions of all panels, and real and imaginary frequency response functions at different locations on each plate and operating shapes for each plate were compared. Preliminary results suggest that vibration‐based techniques hold promise for damage detection of composite sandwich structures. Oruganti et al. (2008a, 2008b) discussed the applicability of various proven methods and techniques in the analysis of vibration signatures obtained from damaged structures under dynamic conditions to identify the location and extent of damage. The study comprised testing a carbon–epoxy composite beam with an embedded delamination with a mechanical actuator and a scanning laser vibrometer as a sensor for recording the frequency response and finding suitable means to interpret the acquired signals. A damage detection technique based on the use of compactly supported sub‐band space/frequency and time/frequency analysis using local vibration characteristics was proposed by Medda and DeBrünner (2009). This technique allows the sensor array to “scan” local portions of the structure, resulting in accurate spatial selectivity on the array and high signal‐to‐ noise ratio for any given scan direction. Sub‐band analysis and adaptive beam‐forming were integrated in a wavelet packet sub‐band framework. Using a cross correlation function of vibration responses under white noise excitation, the concept of inner product vector was introduced by Wang, L. et al. (2009) and Yang et al. (2009a, 2009b, 2009c). Based on the concept of the inner product vector, a damage detection method was proposed, and the measuring noise resistance ability of the proposed method was also analyzed theoretically. Under white noise excitation, the inner product vector of a structure is the superposition of the vibration modes of the structure, and the contribution of each mode shape only depends on the modal parameters of the structure. The damage location is identified by the abrupt change of inner product vector curve of the structure. Numerical simulation examples of delamination damage detection for a composite laminated beam demonstrated the effectiveness and the anti‐noise ability of the proposed method. 5.3.5  Damage Index and Modal Assurance Criterion

The level of damage in a structure can be controlled by controlling its maximum response. Thus we need to establish limits to the maximum or cumulative demands of some response parameters, and to supply the structure with mechanical characteristics that help control its response within the established limits (TeraGilmore, 1997). Damage indices establish analytical relationships between the maximum and/or cumulative response of structural components and the level of damage they exhibit. The response index most often used to estimate damage in reinforced concrete ductile members is that developed by Park et al. (1987). A methodology for automatic damage identification and localization was developed by Banerjee et al. (2009a) using a combination of vibration and wave propagation data.

Statistical Pattern Recognition and Vibration-Based Techniques

The structure was assumed to be instrumented with an array of actuators and sensors to excite and record its dynamic response, including vibration and wave propagation effects. A damage index, calculated from the measured dynamical response of the structure, was introduced as a determinant of structural damage. The index was used to identify low velocity impact damages in increasingly complex composite structural components. Banerjee et al. (2009b) presented a damage index, DI, approach for damage detection and localization based on high frequency wave propagation data and low frequency vibration measurements. The motion produced by the two types of sources is acquired by multiple sensors located on the surface of the structural component. The damage index, DI, was defined by the following expression: f s /2

DI

1

FD 2 f k

fk 0 f s /2

FI

fk 0

2

(5.10) fk

where fk are the frequencies where the spectra are evaluated, FI and FD are the magnitudes of the frequency response functions or spectra for the undamaged and damaged structures, respectively, and fs is the sample rate. The DI assumes non‐zero values only if any change in the measured dynamical response of the structure occurs. It becomes zero if the experimental measurements are identical. Using the initial measurements performed on an undamaged structure as baseline, damage indices were evaluated from the comparison of the frequency response of the monitored structure with an unknown damage. The sensitivities of static and dynamic parameters to damage occurrence in plate‐like structures were examined by Yam et al. (2002). Corresponding damage indices were proposed to analyze their identification capabilities. For static analyses, damage indices were formulated using the out‐of‐plane deflection, and its slope and curvature based on a finite element model. For dynamic analyses, two damage indices related to the curvature mode shape and the strain frequency response function were proposed. Detecting the presence of delamination damage in composite panels based upon their higher‐frequency structural response was studied by Gherlone et al. (2005). Two alternative damage indices were examined that facilitate the identification of the location and extent of delaminations. The damage indices do not require vibration measurements to be performed on the undamaged structure. Use was made of the bending and twisting curvatures corresponding to the higher frequency mode shapes that are post‐ processed via two different smoothing techniques. A vibration‐based approach to detect crack damage in a cantilever composite wing‐ box was studied using the Hilbert–Huang transform by Chen, H.G. et al. (2006, 2007a, 2007b) and Yan et al. (2007a). The dynamic responses of intact wing‐box and damaged wing‐box were determined using improved the Hilbert–Huang transform. A feature index vector of structural damage, representing the variation quantity of instantaneous energy, was constructed. It was shown that the proposed damage feature index vector is more sensitive to a small damage than those in traditional signal processing. The results revealed that the damage feature index vector is more sensitive to a small damage. Another vibration‐based method, which uses an index called the empirical mode

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decomposition energy damage index was presented by Briand et al. (2010). Field testing was conducted on a mechanical pipe joint on a condensation line. Piezoceramic sensors, placed at various locations around the joint were used to monitor the free vibration of the pipe imposed through the use of an impulse hammer. Multiple damage progression scenarios were established, each having a healthy state and multiple damage cases. Subsequently, the recorded signals from the healthy and damaged joint were processed through the empirical mode decomposition energy damage index in an effort to detect the inflicted damage. A comparative test was carried out on a simple unstiffened woven composite plate (530 × 530 × 3 mm) shown in Figure 5.6. A reversible damage was simulated with a 20 g mass added to the panel. Actuator 1 was used to excite the structure, and the healthy and damaged frequency response functions were measured with both the piezo‐patches and laser vibrometer at the other seven locations, shown in Figure 5.6(a), for the same baseline and damaged configuration. The signal provided by the amplifier to the piezo‐ patch acting as a source was employed as input function for both cases. The voltage measured by the piezo‐patch acting as a sensor was considered as the input function, whereas the velocity measured by the vibrometer was considered as the output function. The laser vibrometer acquisitions were made at eight locations as close as possible to the piezo‐patches for comparison purposes. Using the DI definition given by equation (5.10), the results are shown in Figure 5.7. The laser vibrometer was used to construct the DI map by performing only one scan for 92 acquisition points over the panel for both the healthy and damaged configurations. Figure 5.7 shows the accuracy of the predicted “damage” location through the distribution of the DI over the panel. The change in mode shapes has been used as a method of detecting damage occurrence (West, 1984). The analysis of empirical modal models based on the modal assurance criterion (MAC) was demonstrated for the case of the Space Shuttle Orbiter aft bulkhead. Modal models of the bulkhead were constructed before and after environmental acoustic tests equivalent to 30 or 100 missions and compared to detect significant structural changes; subdivision of the modal vector followed by computation of the MAC for each part was used to localize the changes. Numerical and experimental investigations of a beam showed that the MAC value is not so sensitive to damage and suggested that graphical comparison of the mode shapes might be a good way to locate damage (Fox, 1992). The measured responses measured by the laser vibrometer were used to extract the modal natural frequencies and mode shapes in the frequency range 150–1000 Hz and to calculate the DI from equation (5.10) over different frequency ranges. In order to spatially correlate the modal shapes at each corresponding natural mode on the healthy and damaged structure, the MAC (Ewins, 2001; Allemang, 2003) was used to compare and quantify the changes in the modal parameters evaluated from the measurements carried out on the healthy and damaged structure and was defined by the expression: MAC

n,

e

T n T n

e

e

2 t e

n

(5.11)

where {ψn} and {ψe} are the healthy and damaged eigenvectors (mode shapes) at corresponding mode numbers, respectively, and T denotes transpose. The MAC can assume

Statistical Pattern Recognition and Vibration-Based Techniques

(a) Actuator 1

Added mass

Actuator 8

(b)

Figure 5.6  The laser vibrometer scanning setup for composite panel: (a) The piezo‐patches are numbered while the damage is simulated with a 20 g mass added, (b) schematic diagram of the laser beam showing acquisition points. (Banerjee et al., 2009b).

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0.6

0.5

0.4

0.3

0.2

0.1

Figure 5.7  Damage index distribution over the panel due excitation generated by actuator‐1 over frequency range 150–1000 Hz. The 92 small circles represent the acquisition points. The 20 g added mass is located at the black solid circle. (Banerjee et al., 2009b) 1.0

Modal assurance criterion (MAC)

290

0.8

0.6

0.4

0.2

0.0

I

II

III

IV

V

VI VII Mode number

VIII

IX

X

XI

Figure 5.8  The modal assurance criterion (MAC) evaluated with excitation given by actuator‐1 for the damaged and healthy panels over frequency range150–1000 Hz. (Banerjee et al., 2009b)

a value ranging from 0 (no correlation) to 1 (full correlation) and a value of 0.9 ensures a reasonably good experimental correlation, indicating that no change in the modal parameters implies the absence of damage to the structure. Figures 5.8 and 5.9 show that the MAC values predict a good correlation up to about 500 Hz for two different excitation locations, actuators 1 and 8, respectively. Figure 5.10 shows the values of the

Statistical Pattern Recognition and Vibration-Based Techniques

Modal assurance criterion (MAC)

1.0

0.8

0.6

0.4

0.2

0.0

I

II

III

IV

V

VI VII Mode number

VIII

IX

X

XI

Figure 5.9  The modal assurance criterion (MAC) evaluated with excitation given by actuator‐8 for the damaged and healthy panels over frequency range 150–1000 Hz. (Banerjee et al., 2009b)

0.7 0.6 0.5 0.4 0.3 0.2 0.1

Figure 5.10  Damage index distribution over the panel under excitation of actuator number 8 over frequency range 500–1000 Hz. The small circles represent the acquisition points. The 20 g added mass is located at the black solid circle. (Benerjee et al., 2009b)

DI calculated over frequency range 500–1000 Hz. It was shown that the frequency response functions approach can detect damage at frequencies higher than 500 Hz. When the analysis was carried out in the frequency range 150–500 Hz the damage detection and localization was not achievable from the DI formulation. This is in

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agreement with the good correlation given by the MAC shown in Figures 5.8 and 5.9. In other words the damage (added mass) does not produce an appreciable effect below 500 Hz. From 500 Hz to 1 kHz a poor modal correlation is found, as reflected from Figures 5.8 and 5.9 and consequently the DI distribution over the panel provides a clear localization of the damage, as shown in Figure 5.10. Finally, it is worth noting by comparing Figures 5.7 and 5.10 that even performing the DI evaluation over the entire frequency domain, where some modes are well correlated to each other and some others do not, a good estimation of the damage location can be achieved. Schallhorn (2012) considered a vibration‐based damage detection algorithm that localizes the sensor arrangements such that irregularities within the structural system can be detected, located, and quantified. A plate specimen and a connection specimen were analyzed numerically and experimentally. The quantification of the damage was based on a supervised learning method in which original and damaged states are known. 5.3.6 Applications 5.3.6.1  Damage Detection in Composites

Sabin and Shenoi (2003) studied the effectiveness of the combination of changes in natural frequencies and curvature mode shapes as input for artificial neural networks for location and severity prediction of damage in fiber‐reinforced plastic laminates. Different damage scenarios were introduced by reducing the local stiffness of the selected elements at different locations along the finite element model of the beam structure. The use of modal parameter analysis for damage detection of structures made of composites was employed by Yam et al. (2003, 2005) and Xue, S. et al. (2005). An appropriate finite element model was proposed to analyze the dynamic characteristics of different types of structures made of composites, such as honeycomb sandwich plates and multi‐layer composite plates, with internal cracks and delamination. Natural frequencies, modal displacements, strains and energy were analyzed for the determination of damage severity and location. Vibration measurements were carried out using piezoelectric patch actuators and sensors for comparison and verification of the finite element model. Experimental results showed the dependence of changes of modal parameters on damage size and location. The damage detection for honeycomb sandwich composite beams by using a combined damage detection method was examined experimentally by Yang et al. (2009b). The method was incorporated with the correlation function amplitude vector (CorV) analysis and the continuous wavelet transform. It was proved that the CorV of a structure is related to the frequency response function. The change between the CorVs of the undamaged and damaged structure can be used as the damage index. Only the time domain vibration responses of the intact and damaged structures were required in the damage detection procedure. The CorV of the beam was calculated by using the measured vibration responses at all the measured points. These CorVs of intact and damaged beam were analyzed by bi‐orthogonal continuous wavelet transform. The results demonstrated that the debonding damage of the structure can be identified by comparing wavelet coefficients of the intact and damaged structures. It was indicated that the damage region can be located through the two maximum moduli of the difference of the coefficients obtained from intact and damaged structures. Yang et al. (2008, 2010) proposed a hybrid method combining the CorV and the continuous wavelet transform to

Statistical Pattern Recognition and Vibration-Based Techniques

detect damage of composite structures. The feasibility and effectiveness of the proposed hybrid method is verified by numerical simulations of delamination damage detection of a composite beam and a laminate plate. Yang et  al. (2009) and Wang et  al. (2010, 2011) proposed a damage detection method using the inner product vector. The inner product vector was shown to be a weighted summation of the mode shapes and can be directly calculated from the vibration responses. The effectiveness of the CorV and inner product vector method was demonstrated by delamination damage detection for a composite laminate beam. A vibration‐based damage detection method for a static laminated composite shell, partially filled with fluid, was presented by Cheng et al. (2007a). The crack damage was simulated using advanced composite damage mechanics in a dynamic finite element model, in which the interaction between the fluid and the composite shell was considered. The accuracy of finite element model was validated by comparing the computed and measured structural frequency response function. Structural damage indices were constructed and calculated based on energy variation of the structural vibration responses decomposed using wavelet package before and after the occurrence of structural damage. Cheng et al. (2007b) adopted the eigenvalue perturbation theory to obtain the eigenvalues and eigenvectors of the damaged structure for reducing the computation load. Two artificial neural networks were trained based on the response data simulated using the finite element method and perturbation theory enhanced finite element method, respectively. Zhou, W. et al. (2008) described a damage detection method and applied it to assess the composite structure filled with fluid A vibration‐based structural damage identification technique, which represents damage as a change in the internal mechanical forces within a structural component, was validated by Adams and White (2010) who compared analytical, numerical, and experimental results for a simple beam component. Mass and stiffness changes in a beam model were shown to cause shifts in the internal force amplitudes and frequencies. Impact damage on the front side of a more complicated AI‐AI sandwich honeycomb panel using active sensor measurements taken on the back side of the panel was identified. Load cells installed beneath the actuators were used to directly measure the input excitation instead of indirectly using the drive voltages as inputs to the damage identification algorithm. Kesavan et al. (2008, 2010) presented experimental and analytical results dealing with the autonomous damage detection technique for a two‐dimensional polymeric composite T‐joint, used in maritime structures. Two methods of damage detection were considered, namely a statistics based outlier technique and one using artificial neural networks (ANNs). The SHM using ANNs system was found to be capable of not only detecting the presence of multiple delaminations in a composite structure. It was also found to be capable of determining the location and extent of all the delaminations present in the T‐joint structure, regardless of the load (angle and magnitude) acting on the structure. The system developed relies on the examination of the strain distribution of the structure under operational loading. A finite element model of a unidirectional carbon–PEKK (a matrix poly “ether‐ketone‐ketone” polymer with aligned, continuous unidirectional fiber reinforcement) composite T‐beam, including a delamination, was implemented and validated by experimental results (Loendersloot et  al., 2009). The finite element model was used to investigate the effect of the size and the location of a delamination on the detection and localization. It was shown that the bending modes are well capable of detecting a delamination of 10 mm, even though it consists of the

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minimum number of two elements. The torsion modes were hardly affected by the presence of even a large delamination of 100 mm. This is attributed to the location of the delamination with respect to the shear center of the cross‐section. 5.3.6.2  Damage Detection in T‐Joints

The function of the T‐joint is to transmit flexural, shear, and tensile/compressive loads from one panel (web) to the other (flange). In the composite structures of a ship, the hull and bulkhead are the primary structures maintaining the ship’s stiffness under various loadings. Sandwich panels constitute a major structural element in both hull and bulkhead. These elements consist of two stiff skin sheets and a relatively weak, but lightweight core. For example, a typical joint between the hull and bulkhead used in mine countermeasure vessels (MCMVs) is the T‐joint. Failure prediction, damage tolerance, optimum design, and stitched joining technology for composite materials have been documented by Tong and Steven (1999). This book discussed failure mechanisms and predictions of metal and composite bonded joints using both strength of materials and fracture mechanics methods. The design of a sandwich T‐joint with reduced weight but with the same strength as, or higher strength than, the existing design was presented by Hawkins et al. (1993) and Toftegaard and Lystrup (2005). The nonlinear deflection response of highly stressed sandwich T‐joints using large deflection and plasticity analyses was studied by Theotokoglou (1997). The finite element algorithm was based on a plane model of the joint, and accounted for plastic deformation of the core and geometric nonlinear effects. The algorithm was also used to identify internal stress states, in various regions of the joint and for different attachment configurations, leading to a failure. Later, Theotokoglou (1999) numerically estimated the J‐integral in studying the fracture mechanics of sandwich T‐joints formed by two panels connected by lap joints subjected to lateral loads. Seven different T‐joint sandwich designs made of balsa wood cores and carbon fiber reinforced polymer (CFRP) facings were compared under quasi‐static tensile and compressive loading until failure by Brünner and Paradies (2000). The performance of the designs is assessed based on failure loads and deformations, acoustic emission data, and visual inspection. The design improvements yield an increase of the failure load from around 50 kN to at least 140 kN (corresponding to the failure of the horizontal plate). Acoustic emission monitoring revealed an early onset of damage and significant damage accumulation at loads around 70% of the ultimate failure load. For some T‐joints, there are indications of significant damage in zones outside the region of the final failure. The performance of composite sandwich T‐joints subjected to both static and dynamic loading commonly used in large panels for naval applications was studied by Zhou et al. (2008). The failure modes of the T‐joints were found to be sensitive to the core shear failure strength of the base panel. As the main load‐bearing structure of ship, the mechanical and reliability designs of T‐joints are the important aspects of the ship structure design. To improve the properties of composite T‐joints, Wang, Z.G. et al. (2011) studied the properties of the bolted composite T‐joints. The influences of bolt‐ clamping load and friction coefficient on the stress distribution of bolt‐hole edge were examined. A detailed finite element analysis was presented by Shenoi and Hawkins (1992) to predict stress distributions and load transfer mechanisms within joints of various geometries under representative static loads. Numerical and experimental studies have

Statistical Pattern Recognition and Vibration-Based Techniques

been reported in the literature dealing with different aspects of the design and behavior of T‐joints (e.g. Burchardt, 1995; Theotokoglou and Moan, 1996; Shenoi et al., 1998; Read and Shenoi, 1999; Turaga and Sun, 2000). Turaga and Sun (2000) outlined various modes of failure of different T‐joints of composite sandwich panels. The key modes of failure include debonding between the two sandwich components to be joined, debonding between the attachment and the sandwich panel, and cracking in the core of the sandwich. A new type of T‐joint incorporating an aluminum U‐channel in the web sandwich was found to provide much improvement over the conventional circular fillet T‐joint. It was shown that using bolts in a circular fillet joint could cause early failure in the core and would not help much to increase the ultimate joint strength. Junhou and Shenoi (1996) and Mouritz et  al. (2001) reviewed the performance characteristics of joints in fiber‐reinforced plastic (FRP) marine structures. Finite element analysis was performed for T‐joints by Dharmawan et al. (2004) and Li et al. (2006a), based on linear elastic static conditions. For this case, the load for thin T‐joints using the crack tip element method was found to be 20 times higher than that analyzed using the virtual crack closure method. This difference was attributed to the fact that the depth ratio of the T‐joint analyzed using the crack tip element was also 20 times higher than with the virtual crack closure method, while the bulkhead thickness remained the same. The fracture behavior of a glass fiber reinforced plastic (GFRP) T‐joint was studied numerically and experimentally by Li et al. (2006a). It was shown that skewed loading affects the strain energy release rate at the crack tips. The work was extended by Li et  al. (2006b) who used embedded fiber Bragg grating sensors. They developed a technique based on signal processing and statistical outlier detection. Herszberg et al. (2005) considered the damage assessment of marine structures. The effects of artificially induced disbonds on the strain distribution were determined using the finite element method, and the results were verified experimentally. Kesavan et al. (2005) determined the size and location of damage in a T‐joint subject to pull‐off load based on strain distribution. Later, Kesavan et al. (2006a, 2006b) used the strain variation across a GFRP structure for damage detection. They considered the strain distribution of a GFRP T‐joint structure under tensile pullout loads. A real‐time system was developed for detecting the presence, location, and extent of damage from the  longitudinal strains obtained from a set of sensors placed on the surface of the structure. The resulting strain variation across the surface of the structure was verified experimentally. The effect of disbond at the T‐joint subjected to the pull‐off‐loading was examined by Dharmawan et al. (2008) with reference to the horizontal disbond; see Figure 5.11(a), with the initial sizes of 30, 60, and 90 mm. This disbond was chosen since at this damage configuration, the major fracture mode is the mode‐I fracture mode (opening fracture mode) and it is the most critical fracture mode compared with the other modes. It was used to determine the total strain energy release rate, (due to all three modes, I, II, and III, the opening, shearing, and tearing fracture modes, respectively). The distributions of the total strain energy release rate, according to the crack tip element analysis along the depth (y direction) of the structure and its component for the thin and thick T‐joints are shown in Figures 5.11(b) and 5.11(c), respectively. The plots show the energy release rate for three different values of disbond length, 30 mm, 60 mm, and 90 mm. It is seen that the strain energy release distribution is symmetric across the T‐joint depth. It was reported that the mode‐III strain energy release rate is larger than for mode‐II, though

295

(a)

110 mm

230 mm

400 mm

Bulkhead (15 mm thick)

Overlaminate (10 mm thick) Horizontal initial delamination Filler 110 mm

20 mm

15 mm

45° 45 mm

290 mm

Hall (43.5 mm thick)

y x z

(b) 3000 2500

GT (J/m2)

2000 1500 1000 500 0 0

10

20

30

40

50

60

70

80

90

100

70

80

90

100

y location (mm)

(c)

5000 4500 4000

GT (J/m2)

3500 3000 2500 2000 1500 1000 500 0

0

10

20

30

40 50 60 y location (mm)

Figure 5.11  (a) Schematic diagram of half symmetrical T‐joint and the total strain energy release rate distribution along the depth (y) of T‐joint according to the crack tip element analysis results for (b) thin T‐joint and (c) thick T‐joint for three different values of disbond length: ♦ 30 mm disbond, ▪ 60 mm disbond, ▴90 mm disbond. (Dharmawan et al., 2008)

Statistical Pattern Recognition and Vibration-Based Techniques

it was still much smaller than mode‐I. Furthermore, it was found that the percentage differences increase as the disbond length increases. This was attributed to the fact that the T‐joint finite element model used thicker laminates for longer disbond length. The mechanical behavior of transverse stitched T‐joints using a fiber insertion process and PR520 toughened epoxy resin under bending and tensile loading was examined by Stickler and Ramulu (2001). Experimental tests were conducted to determine the modes of failure and ultimate failure strength for each load condition. The results indicated that the flexural specimens fail in part from unsymmetrical loading of the fiber insertions and in part from high stress concentration in the “resin‐rich” fillet region. Tensile specimens have symmetric loading of both sets of fiber insertions and initially fail due to matrix cracking at the web‐to‐flange interface. Later, Stickler and Ramulu (2002) examined the effects of key parameters of T‐joints with transverse stitching under flexure, tension, and shear loading using finite element analysis. These parameters include fiber insertion tow modulus, fiber insertion filament count, fiber insertion depth, and resin‐rich interface zone thickness on T‐joint displacement and damage initiation load. It was found that under flexural loading, increasing the fiber insertion tow modulus and tow filament count increases the T‐joint damage initiation load. On the other hand, increasing the fiber insertion depth reduces T‐joint deflection. Furthermore, reducing the web‐to‐flange interface thickness reduces the T‐joint deflection. The fiber insertion tow filament count and modulus were found to have a negligible effect on T‐joint deflection under tension and initial damage load under shear. The progressive damage of a new type of composite T‐joint with transverse stitching using a fiber insertion process was modeled using numerical methods by Stickler and Ramulu (2006). They also conducted a series of experiments to determine the load–displacement and strain–load history under flexure and tensile loading. Experimental observations of initial failure were manifested by a discrete drop in the load displacement behavior and the initiation and propagation of an interfacial matrix crack at the web‐to‐flange interface. Fiber insertion bridging and fiber insertion breakage were observed at T‐joint ultimate failure. In view of the relatively low modulus of fiber‐reinforced plastic (FRP) material, Dodkins et al. (1994) discussed the problems with forming efficient joints between the major structural components of FRP ships and boats with stiffened single‐skin construction ships. They considered T‐joints between watertight bulkheads and shell plating and attachment of top‐hat stiffeners to plating. The design of top‐hat stiffener to shell plating joints was presented by Shenoi and Hawkins (1995). Finite element modeling was used to identify key variables that control and govern the transfer of load from the panel to the stiffener. T‐joints and top‐hat stiffeners create a potential zone of weakness in ships. The mechanical behavior of these structures was documented in the literature by Phillips and Shenoi (1998) and Phillips et al. (1999). It was reported that delamination‐induced damage in the root of the T‐joint is a potential source of catastrophic failure. The load transfer mechanisms in single‐skin T‐joints were examined by Phillips and Shenoi (1998) in terms of the strain energy release rate and the J‐integral. The fracture mechanics revealed that deep cracks give greater values of the J‐integral than surface cracks, and long straight cracks give greater values of the J‐integral than short straight cracks. Later, the damage tolerance of a top‐hat stiffener to plate connection in FRP marine structures was examined by Phillips et al. (1999), who used stress‐based and

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Handbook of Structural Life Assessment

fracture‐dependent criteria. Numerical modeling was used to determine the internal load transfer characteristics and failure mechanisms in top‐hat stiffeners under static loading. The estimated fracture parameters with regard to delamination in the top‐hat over‐laminates revealed that delaminations propagate under a straight pull‐off load. Kumari and Sinha (2002) studied the static behavior of composite T‐joints made of carbon‐fiber composite materials. A pullout force was applied to the top of the web to determine the strength of the T‐joints. Parametric variations account for two different stacking sequences and variations in filler area and radius of curvature at the web/skin interface. The optimization of FRP top‐hat stiffened single‐skin and monocoque sandwich plates subject to a lateral pressure load was studied by Maneepan et al. (2005). They considered plates made of E‐glass/epoxy, high‐strength carbon/epoxy and ultrahigh‐ modulus carbon/epoxy. Eksik et  al. (2007a, 2007b) experimentally, and numerically examined the behavior of a top‐hat‐stiffened panel under uniform lateral pressure. They developed a finite element model based on an ANSYS three‐dimensional solid element (SOLID45) to determine the static response of a top‐hat‐stiffened panel under uniform lateral pressure. The numerical modeling results were compared to the experimental findings for validation and to further understand an internal stress pattern within the different constituents of the panel for explaining the likely causes of panel failure. 5.3.6.3  Case Study: Crack Formation in T‐Joint Structures

This section deals with the analytical modeling of a T‐joint experiencing crack formation based on representing the main sandwich panel by two outer skins, with the core between them represented by a set of parallel springs (Pilipchuk and Ibrahim, 2011). The two legs of the T‐joint are also represented by an equal number of springs with one cracked side. For perfect and ideal adhesion of the joint, the springs are in complete contact with the main panel. However, the initiation of a crack is manifested by the disconnection of some springs from one side, as shown in Figure 5.12. It will be shown that the disconnection of springs makes the model physically nonlinear. The main objective of this study is to find out how the linear normal modes (under the no‐crack condition) will be affected by the crack as it develops, and therefore that w b–a

d

k kf

x 0

Figure 5.12  Joint representation by an elastic spring system model showing the crack initiation by springs over a width d.

Statistical Pattern Recognition and Vibration-Based Techniques

the  dynamics becomes increasingly nonlinear. The analytical modeling is based on representing cracks by these elastic elements that work normally in compression phases but show no resistance to tension. Note that such an approach is quite common in the reduced order modeling of cracked elastic structures (Pilipchuk, 2009). However, as will be shown later, the geometrical specifics of T‐joints require the additional ­analytical tools of dynamic modeling. For instance, a cracked T‐joint is reduced to a specific nonlinear element embedded in an elastic beam and is localized at the center of the T‐joint. In addition, the crack length is assumed to grow by a smoothed step‐ wise law with a slow rate as compared to any other temporal scale associated with the model eigen‐frequencies and external loading. The model reduction is based on the first three modes of a beam with no joint. As a result, the corresponding differential equations of motion are linearly coupled due to the presence of the joint even though no crack/nonlinearity develops. Consider a simplified elastic model of a T‐joint initially cracked, as schematically shown in Figure 5.12. The main structure is a sandwich beam whose core is modeled by a set of elastic springs with a distributed stiffness of kf per unit length. The upper and lower skins are represented by two thin beams each of length L, density ρ, and bending stiffness EI, where E is the Young’s modulus and I is the area moment of inertia of the skin cross‐section. The reaction of the T‐joint attached to the horizontal beam is represented by a continuous set of linearly elastic springs whose distributed stiffness per unit length is k and occupy the region b a on each leg of the T‐joint. In order to simplify preliminary derivations, it is assumed that the perfect (no crack) joint is symmetric with respect to the origin, x 0. However, when a small area of separation occurs, the elastic reaction of the joint within the area of separation becomes essentially nonlinear and may be described by the following functions kw f x ,w

b, a d

x

kH w w 0

x x

a ,b

a d, a b, a

(5.12)

a ,b

where H(w) is the unit‐step Heaviside function. Note that the deflection function, w(t, x), is associated with the centerline of the sandwich or the upper skin beam as follows. In the case of global long‐wave modes, when the upper and lower beams vibrate in phase, the function w(t, x) effectively describes the position of the entire sandwich. However, in this case, the bending stiffness of the upper beam must be replaced by the effective bending stiffness of the sandwich. According to equations (5.12), there is no interaction between the beam and joint in the area x [ a d , a] whenever w(t ,x ) 0. Therefore, for negative deflections of the upper beam, the joint reaction becomes weak and non‐symmetric with respect to the origin. On the other hand, when the beam deflection is positive everywhere in the area of joint, then the mechanical properties are equivalent to those of the undamaged joint. Note that, on one hand, such kinds of dynamic behavior bring the model into a class of distributed vibrating systems with non‐smooth nonlinearities, which are quite difficult to analyze. But on the other hand, it will be shown shortly that the corresponding characteristics of a nonlinear dynamic response can be employed for damage detection in joints. Finally, based on the joint modeling given by equation (5.12), the total transverse

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Handbook of Structural Life Assessment

force, Qj(t) and bending moment, Mj(t), applied from the joint to the beam are determined as follows:



Qj t

b

f x , w t ,x dx (5.13)

b

Mj t

b

xf x , w t ,x dx (5.14)

b

At this stage, equations (5.13) and (5.14) cannot be used because the function w(t, x) is a priori unknown. Nevertheless, the above expressions may help to facilitate the problem formulation under certain assumptions regarding the function w(t, x) in the area of joint interaction, as will be described in the next section. In view of the relative structural complexity of the T‐joint, its dynamic behavior may significantly depend upon spatiotemporal characteristics of external loads. The idea of the model reduction may be contemplated a priori to account for possible dynamic modes that may occur under given loading conditions. For non‐destructive diagnostic purposes, we can apply necessary dynamic excitations that are capable of generating such modes that exhibit sufficient sensitivity to crack formation. In particular, we can assume that the elastic dynamic state of the upper beam at any time t is slowly varying with respect to the spatial coordinate, x, as compared to the area of contact with the joint. Such a dynamic state can occur provided that the horizontal beam is long enough. In this case, the transverse force, given by equation (5.13), and bending moment, given by equation (5.14), applied from the joint to the beam can be considered localized at the joint center x 0. In addition, the integrals of equations (5.13) and (5.14) admit explicit calculations by using the truncated power series for w(t, x) about the center of the joint, x 0, i.e.

w(t ,x ) w(t ,0) wx (t ,0)x O( x 2 )

(5.15)

Equation (5.15) implies that the function w(t, x) is smooth enough with respect to x. Note that the integrands in both equations (5.13) and (5.14) include the non‐smooth function H(w)w, which can be approximated, nevertheless, by the generalized power series (e.g. Richtmyer, 1985),

H w t ,x w t ,x

H w t ,0

w t ,0

wx t ,0 x

O x 2 (5.16)

where the following property of distributions has been taken into account

zH z

z

z

0 (5.17)

Substituting equation (5.15) into equations (5.13) and (5.14), taking into account equation (5.12), gives where

Qj t Mj t

2k b a w t ,0

kd

2 k b3 a3 wx t ,0 3

( z1 ,z2 ) [1 H ( z1 )]( z1 z2 ).

w t ,0 , wx t ,0 (5.18) kda

w t ,0 , wx t ,0 (5.19)

Statistical Pattern Recognition and Vibration-Based Techniques W

M + dM Q M Mj (x)

Qj (x)

Q + dQ –kfw + p(t, x)

x

0

Figure 5.13  A small segment of the beam including localized joint.

In order to incorporate the localized force and moment, given by equations (5.13) and (5.14), into the model, consider a small segment of the beam as shown in Figure 5.13. Following the classic approach, and ignoring the angular inertia force, gives the following balance of forces and moments Adx



0

2

w t2

M

Q

Q dQ

M dM

Mj

p t ,x x dx

k f w Qj

x dx (5.20)

1 1 Qdx Q dQ dx (5.21) 2 2

where ρ is the mass density of the beam, A is the cross‐sectional area, and kf characterizes the distributed elastic properties of the foundation represented by a continuous set of linearly elastic springs. Eliminating the high‐order term dQdx from equation (5.21), and dividing both sides of the equations by dx, gives A



2

w t2 M x

Q

Q Qj x

x

kfw

p t ,x (5.22)

x (5.23)

Mj 2

w . x2 Figure 5.14 shows the upper skin represented by a simply supported beam of length L by assuming that the joint is localized at the middle of the beam, x L/2. Substituting equation (5.23) into equation (5.22), gives the partial differential equation of motion for the beam’s centerline in the form where M



A

EI

2

w t2

EI

4

w x4

k f w Qj

x

L 2

Mj

x

L 2

p t ,x (5.24)

301

302

Handbook of Structural Life Assessment w Mj [w(t, L/2), W′(t, L/2)]

x Qj [w(t, L/2), W′x(t, L/2)]

0

L

Figure 5.14  Equivalent model for the beam with localized joint.

where

Qj t

2k b a w x, L / 2

2 k b3 a3 w x x , L / 2 3

Mj t

w t , L / 2 , wx t , L / 2 (5.25a)

kd

kda

w t , L / 2 , wx t , L / 2 (5.25b)

Equation (5.24) is subject to the boundary conditions

w t ,x |x

2

0,

0 ,L

w t ,x x

2

|x

0 ,L

0

(5.26a,b)

Note that as a crack develops (i.e. d 0), the model described by equations (5.24)– (5.26) becomes nonlinear such that the strength of nonlinearity is proportional to the length of the crack d. As stated at the beginning of this section, one of the main objectives of this study is to find out how the linear normal modes (under the no‐crack condition d 0) will be affected by the crack as its size d increases. Using the Bubnov–Galerkin method, the beam centerline and external loading functions are represented as a linear combination of the first three mode shapes of the beam with no joint attached

3

w t ,x

i 1

p t ,x

wi t sin i x / L (5.27a)

A

3 i 1

pi t sin i x / L (5.27b)

where ρA is the beam mass per unit length, which is introduced as a scaling factor of the external loading function in order to simplify further notations. Substituting equations (5.27) into equation (5.24) and applying the Bubnov–Galerkin procedure gives the following three equations of motion in terms of the generalized coordinates wi(t), i 1, 2, 3, 1 w 2 w

2 2

2 1 w1

2

2 0

w1 w3

8 2 ra ra rb rb2 02 w2 2 ra 3 3 32 w3 2 02 w1 w3 w

2 0F

w1 w3 , w2

p1 t

2 0F

w1 w3 , w2

p2 t

2 0F

w1 w3 , w2

p3 t



(5.28)

Statistical Pattern Recognition and Vibration-Based Techniques 2

[k f EI j / L ]/ ( A) are the eigenvalues of the beam setting on an elaswhere j 2k (b a)/(AL ) is the natural frequency of an tic foundations with no joints. 0 oscillator whose mass is equal to the mass of the beam, where the stiffness is equal to b/L are non‐dimensional geometrical the total stiffness of the joint, ra a / L and rb parameters characterizing the size of the joint, and d /(b a) is a parameter characterizing the size of the crack. The nonlinearity due to the crack may be described by the non‐smooth function

F w1 w3 , w2

F z1 ,z2

1 H z1

z1 2ra z2 (5.29)

where z1 w1 w3 , and z2 w2 . In the absence of a joint we have 0 0, thus equations ((5.28)) exactly describe the behavior of the first three linear normal modes of the beam. 0. In this In the presence of the joint with no crack, however, we have 0 0, but case, equations ((5.28)) are still linear, but w1(t) and w3(t) are not the principal coordinates any more. To this end it is convenient to introduce a transformation of the princi0) of pal coordinates, qi; i 1, 2, 3, of the linearized perfect system (with no crack, equations ((5.28)), i.e., 2 r12 r22 r2 q1 t 2 q2 t ,

w1 w2 t

r12 r22 r2 q3 t , (5.30)

q1 t

w3 t

1



1 r2

q3 t 2

/ r12

1

1 r2

2

/ r12

where the second mode remains the same since it is already decoupled from the other two modes of the linearized model ( 0), and the parameters of transformation 2 are  expressed through the natural frequencies ω1 and ω3 as r1 4 02 /( 32 1) 2 2 2 2 and r2 ( 3 1 )/( 3 1 ), and the entire spectrum of the linear model with joint is given by



2 1

1 4 2

2 2

2 2

2 3

1 4 2

2 0

2 1

8 2 ra 3 2 0

2 3

rb2

ra rb 2 1

16

2 3

2 0

2 1

2 2 3

2 0

16

(5.31) 2 0

2 1

2 2 3

Substituting the transformation (5.30) into equations (5.28) and adding an effective damping in a phenomenological way, gives q1 2 q2 2

q3 2

1 1 1q

2 1 q1

1

q1 ,q2 ,q3

2 2q

2 2 q2

2

q1 ,q2 ,q3

P2 t

3 3 3q

2 3 q3

3

q1 ,q2 ,q3

P3 t

2

P1 t

(5.32)

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Handbook of Structural Life Assessment

where i , i 1, 2, 3 are modal damping ratios, that will be assumed equal in the numerical simulations,



1

q1 ,q2 ,q3

2

q1 ,q2 ,q3

3

q1 ,q2 ,q3

1 2 2ra 1 2

2 0 2 0 2 0

1 r1 r2

1

1 r2

2

/ r12

q1 ,q2 ,q3

q1 ,q2 ,q3 1 r1 r2

(5.33) 1

1 r2

2

/ r12

q1 ,q2 ,q3

and Φ(q1, q2, q3) is given by substituting expression (5.30) in F (w1 w2 , w3 ) as defined by equation (5.29), and the following notations for the forcing functions have been introduced P1 t P2 t

P3 t

1 r1 p1 t 2 p2 t 1 r1 p1 t 2

1 r2 p3 t

1

1 r2

2

/ r12 (5.34)

1 r2 p3 t

2 / 1 r2

The system of equations (5.32) becomes completely decoupled in terms of the modal 0. Therefore, the effect of modal coupling, coordinates in the absence of any crack, as well as spectral deviations from the frequencies ω1, ω2, and ω3 can be used for the purpose of crack detection. Note that the modal coordinates q1, q2, and q3 cannot provide an exact description of the beam mode shapes with the middle joint, even though no crack developed ( 0). This is clear from the fact that the Bubnov–Galerkin procedure for the boundary value problem (5.24)–(5.26) was based on the linear combination of mode shapes of the beam without a middle joint as given by equation (5.27). Under fixed material mass densities and geometry, the relationship between the natural frequencies of the model is determined by the beam’s bending stiffness, elastic foundation, and joint stiffness. The previous definition of the frequency, ω0, can be physically viewed as a characteristic parameter representing the strength of the T‐joint. For example, Figure 5.15 illustrates the dependence of the modal frequencies on the joint strength parameter, ω0. These plots are obtained for the system parameters: L , ra 0.02 , rb 0.06, 0.01 and 1 0 . 02 . In order to select a convenient time unit 2 3 for simulations, the unit frequency is defined by imposing the condition 1 1.0, which is equivalent to the following constraint on the parameters



EI A

L

4

1 K f (5.35)

2 K f is the frequency of an effecwhere K f k f /( A) k f L/( AL) f and thus f tive oscillator whose mass is the total mass of the beam where its linear stiffness is the total stiffness of the foundation. At this stage, consider the case of no foundation (k f 0) so that EI /( A) 1.0. Figure 5.15 reveals that 1:1 internal resonance 1 2 is reached at a critical value of * 3 . 13202 . Above this critical value, the frequency of the first mode is larger 0 0 than the frequency of the second mode.

Statistical Pattern Recognition and Vibration-Based Techniques 30 25 Weak joint

Strong joint

ωi (rad/s)

20 15 10

ω3

5

ω2 ω1

ω*0 = 3.14202

0 0

2

4

6

8

10

12

14

ω0(rad/s)

Figure 5.15  Dependence of modal frequencies on the joint strength parameter, ϖ0 showing the occurrence of 1:1 internal resonance between the first two modes, 0 2k (b a)/(AL ) . Pilipchuk and Ibrahim (2011)

Figure 5.16 shows two sets of mode shapes belonging to weak (Figure 5.16(a)) or strong (Figure 5.16(b)) joints. Figure 5.16(a) illustrates the first three mode shapes for the case of a weak joint ( 0 2.12, 1 3.0, 2 4.0, 3 9.5), while Figure 5.16(b) shows the mode shapes for the case of a strong joint ( 0 10.61, 1 6.12, 2 4.19, 3 22.24 ). The influence of the joint is manifested in the first mode shape in both weak and strong joints. For the case of a weak joint, the first mode shape is flattened over a wide region near the joint. For the case of a strong joint, on the other hand, the first mode shape exhibits a * wrinkle near the T‐joint. Thus, the point 0 0 can be viewed as a physical boundary between the areas of relatively weak and strong joints. Further, Figure 5.17 illustrates the dependence of the spectrum on the foundation stiffness parameter, Kf. The point *2 K f K *f f , at which the internal resonance 1 2 occurs, can be viewed as a boundary between the areas of weak and strong foundations. Note that, due to the constraint equation (5.35), the increase of foundation stiffness is accompanied by the decrease of bending stiffness of the beam. Eventually, at the point K f 1.0, the beam’s bending rigidity vanishes. This is another point of the internal resonance 1 2, which is possibly beyond the applicability of the model. Indeed, the three‐term expansion of equation (5.27a) may not be enough to capture the behavior of mode shapes near the joint as the beam becomes very flexible. Finally, Figure  5.18 shows the dependence of the beam modal frequencies on the foundation stiffness Kf for the case of a strong joint. Compared with the case of a weak joint represented by Figure 5.17, the area of weak foundations is wider. Also, the relationship ω1 holds on the left side of the internal resonance 1 2, so that stronger foundations bring the order of frequencies back to “normal.” In order to clarify the evolution of the dynamic response due to crack formation in the joint, the following phenomenological time dependence for the fracture length is assumed:



t

0

0

0

1 tanh 1 tanh

c

t tc c tc

1

(5.36)

305

(a)

(b) 1.5

1.5

1

1

0.5

0.5

w

w

0

0

–0.5

–0.5

–1

–1 0

0.5

1

1.5 x

2

2.5

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Figure 5.16  First three mode shapes of the beam with (a) weak joint, (b) strong joint. Pilipchuk and Ibrahim (2011)

ω3

8

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4

Weak foundation

Strong foundation

ω1

2

K*f = 0.54418 0.0

0.2

0.4

0.6

0.8

1.0

Kf

Figure 5.17  Dependence of natural frequencies on the foundation stiffness in the weak joint case: 2.1213, K f kf /( A). (Pilipchuk and Ibrahim, 2011) 1 1.0 , 0

Statistical Pattern Recognition and Vibration-Based Techniques

ω3

20

ωi

15

Weak foundation

10 ω1 ω2

5 0

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K*f = 0.93704

0.2

0.4

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Figure 5.18  Dependence of natural frequencies on the foundation stiffness in the strong joint case: kf /( A). (Pilipchuk and Ibrahim, 2011) 1 1.0 , 0 10.6066 , K f 1.0

ε = d/(b – a)

0.8 0.6 0.4 0.2 0.0 0

50

100 t

150

200

Figure 5.19  Time evolution of the relative length of a crack based on the proposed model. (Pilipchuk and Ibrahim, 2011)

where ε0 and ε1 are the initial and final relative fracture lengths as t 0 and t , respectively; λc and tc are parameters characterizing the temporal scale and phase of crack formation. Figure  5.19 shows a typical plot of the evolution of the fracture length parameter e with time, which also corresponds to the number of cycles. In particular, under specific numerical values of the parameters, these kinds of dependencies can fit experimental data obtained for deterioration of joints under the condition of cyclic loading (e.g. Ibrahim and Pettit, 2005). This seems to be a very general smooth approximation for any step‐wise behavior. In the present study, the step duration is about 50 time units, which is long enough to cover multiple vibration cycles. At earlier stages, the crack propagation is very slow starting from, initially, a very small relative length parameter 0 0.01. Then most of the crack formation develops within the time interval 100, after which the left side of the joint becomes almost completely separated from the beam. In real applications, however, the duration of crack formation depends on the

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number of cycles and the level of vibration amplitude. The corresponding modeling is rather outside the scope of the present study. The proposed model given by equation (5.36) is applied for the purpose of illustrating how the system’s dynamic response is affected by crack formation in the time and frequency domains. First, a series of numerical simulations dealing with different values of stiffness parameters of joints under the no‐elastic‐foundation condition (k f 0) is performed. In all cases, the model is excited by the harmonic loading in equation (5.34) with the first mode frequency as follows: p1 (t ) 0.1 sin 1t , p2 (t ) 0.0, and p3 (t ) 0.0. Therefore, the spatial shape of loading is given by one half of a sine wave that corresponds to the first mode of the beam with no joint. It is seen from equations (5.27b), (5.32), and (5.34) that this loading will directly excite the first and third modes of the beam with joint. However, the first mode will be predominantly excited due to the above choice of input frequency. In addition, there is no direct excitation of the second mode, which may only oscillate as a result of nonlinear modal interaction after the initiation of a crack. The results of simulations are illustrated in Figure 5.20–5.23. In particular, Figure 5.20 displays a series of system trajectory projections on different configuration planes for three different cases of joint strength. The diagrams clearly demonstrate that the system changes its attractor (from A to B) as the crack develops, but in a different way for different joint strengths. In the resonance case (b) where 1 2, the second mode has eventually the largest amplitude with a specific phase shift π/2 that is reflected by the elliptic shape of the attractor’s projection on the q1q2 plane as shown by the top diagram of Figure 5.20(b). Below and above resonance, the projections of the final attractor on the same configuration plane are rather close to straight lines, but directed in a different way: in phase and out of phase for weak and strong joints, respectively. This resembles a typical oscillator behavior passing through the resonance. In our case, the “oscillator” is represented by the second mode, whereas the excitation comes mostly from the first mode subjected to the direct resonance loading. The second row in Figure 5.20 displays projections on the q1q3 plane. Since the second mode does not participate in the projection, then the resonance with the second mode has a minor effect on the projection shapes as one compares cases (a) and (b) in the second row. Case (c) slightly differs though because of effective structural changes in the model due to the stiffer joint. Finally, the third row shows q2q3 projections. The transition to the final (developed crack) attractors has a quite chaotic character in cases (a) and (b). Still, it is seen that the path is more regular in the case of a stiffer joint (c). Related to Figure 5.20 are response time histories and corresponding spectrograms for individual modes. These are shown in Figure 5.21–5.23 for the three different cases of Figure 5.20. The short windowed Fourier spectrograms are generated by the “specgram” procedure built into MATLAB‐7. In both temporal and spectral representations, the developing fracture is clearly seen through the growing amplitudes of secondary modes as well as the spectral widening. Once again, more regularity is observed in the case of the stronger joint. In all three cases shown in Figure  5.21–5.23, it is seen that near t 100 an energy transfer takes place from the first mode, which is directly excited, to the other two modes. The energy transfer is attributed to the nonlinearity created by crack formation. Figure 5.24 shows the projection of the system trajectory on configuration planes for the case of strong joint under the developing fracture condition with the loading of the same spatial mode shape, but with an excitation frequency which is equal to the second

(a)

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q2

Figure 5.20  Projection of the system trajectory on the configuration planes for three different cases of the joint stiffness: (a) weak joint, (b) 1:1 resonance, (c) strong joint, under the developing fracture condition and loading: p1 0.1sin 1t, p2 p3 0: A‐initial crack, and B‐developed crack. (Pilipchuk and Ibrahim, 2011)

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t(s)

Figure 5.21  Time history records and the corresponding spectrograms for the modal coordinates in the case of a weak joint under the developing fracture condition and first linear mode resonance excitation (relates to Figure 5.20(a)). (Pilipchuk and Ibrahim, 2011)

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Figure 5.22  Time history records and the corresponding spectrograms for the modal coordinates in the case of 1:1 resonance under the developing fracture condition and first linear mode resonance excitation (relates to Figure 5.20(b)). (Pilipchuk and Ibrahim, 2011)

0.1 0.05 q1

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t(s)

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Frequency (Hz)

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Figure 5.23  Time history records and the corresponding spectrograms for the modal coordinates in the case of a strong joint under the developing fracture condition and first linear mode resonance excitation (relates to Figure 5.20(c)). (Pilipchuk and Ibrahim, 2011)

Statistical Pattern Recognition and Vibration-Based Techniques No foundation, Kf = 0

(a) 0.004

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Figure 5.24  Projections of the system trajectory on configuration planes in the strong joint case with stiffness under the developing fracture condition and loading; p1 0.1sin 2t , p2 p3 0. (Pilipchuk and Ibrahim, 2011)

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Handbook of Structural Life Assessment 1:1 resonance foundation, Kf = 0.937

(b) 0.075 0.05

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–0.075 –0.4

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Figure 5.24  (Continued )

linear mode frequency. In this case, the loading components are: p1 (t ) 0.1 sin 2t , p2 (t ) 0.0, and p3 (t ) 0.0. In other words, this case deals with the indirect excitation to the second mode through the symmetric modes. In contrast to the previous case, due to the resonance with the second mode, there is no significant energy exchange with the

Statistical Pattern Recognition and Vibration-Based Techniques

first mode. In addition, the influence of a foundation is considered in this case. The 2, leads to quite significant qualitative changes in trajectory internal resonance, 1 projections as follows from the comparison of columns (a) and (b) in Figure 5.24. Also, the internal resonance results in a stronger dynamic complexity in both temporal and spectral characteristics of motion. Note that the third mode q3(t) has a zero initial condition, but it is directly excited as reflected by equations (5.27b), (5.32), and (5.34). The adopted model for a crack exhibits asymmetric nonlinear stiffness characteristics. As a result, the asymmetry observed in time history records and configuration planes can also be viewed as an indication of fracture formation in the joint. Using the coordinate transformation, (5.30), gives the corresponding configuration plane and time history representations of the dynamics in terms of the original geometrical modes as shown in Figures 5.25 and 5.26, respectively. It follows from expressions (5.30) that the second mode is not affected by the transformation, whereas the first and third modes are coupled due to the presence of a joint localized at the middle of the beam. Nevertheless, in most cases, except that shown in Figure 5.24, the first mode appears to be the major energy receiver from the external loading, regardless of the type of coordinates. As a result, the diagrams represented on planes q1, q2 and w1, w2 in Figures 5.20 and 5.25, respectively, look qualitatively similar, but all other planes are quite different. Note that the spatiotemporal modeling of loading given by equation (5.27b) has been specifically chosen for the purpose of crack detection. This can be experimentally implemented in the lab by appropriate actuators. Furthermore, the present study focused on specific features of nonlinear dynamics based on three degrees of freedom. Had the analysis been extended to a more comprehensive structural model, it could result in describing other features unrelated to the crack detection methodology described in this work. From the standpoint of the strength of materials and design issues, finite element software packages have already been developed as outlined in the introduction. Based on the analytical modeling and numerical simulations, different qualitative changes of the dynamics due to crack formation have been predicted in the configuration plane, temporal and spectral representations. The normal mode frequencies of the main structure have been found to be affected by the presence of the crack. Under external dynamic loading with a frequency close to the first mode frequency, the development of the crack has been revealed by the evolution of the configuration plots on the planes. In the absence of a crack the attractor is essentially a straight line and then rotates with time as the crack develops. Another tool for detecting crack formation is through the frequency content evolution of each mode represented by spectrogram plots. Therefore, multiple signal processing tools could be employed for the non‐ destructive diagnostics of joints, where damage may be hidden under layers of material. 5.3.6.4  Damage Detection in Bridges

Bridges are generally subjected to harsh weather conditions, earthquakes, and dynamic loading due heavy ground vehicles. Thus monitoring bridges is challenging and vital in order to maintain thier structural integrity and safety. The state‐of‐the‐art of SHM of bridges was presented by Dong et al. (2010). Bridges have been instrumented with SHM systems of various sizes and complexities that vary in the quality of information they provide. Grimmelsman et al. (2007) discussed the uncertainty related to ambient vibration testing of a long‐span steel arch bridge. The consistency of the identified

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0

w2

Figure 5.25  Projections of the system trajectory on the original generalized coordinate planes under the developing fracture condition and loading p1 0.1sin 1t, p2 p3 0. No elastic foundation assumed. (Pilipchuk and Ibrahim, 2011)

0.0005

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Figure 5.26  Original generalized coordinate time histories under the conditions of Figure 5.25. (Pilipchuk and Ibrahim, 2011)

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parameters was examined through statistical analyses, and the effects of bandwidth and stationarity on the identified parameters were discussed. Collins et al. (2014) reported case studies highlighting the merits of using embedded gages and low‐cost data collection systems to provide increased quality assurance during construction as well as a means of monitoring the health of bridge foundations while in service. A two‐part study to evaluate and develop existing and proposed damage detection schemes for highway bridges was presented by Whelan et al. (2008). The first utilized a laboratory bridge model to investigate the vibration response characteristics induced through introduction of changes to structural members, connections, and support conditions. The second validated the damage detection methods developed from the laboratory testing with progressive damage testing of an in‐service highway bridge scheduled for replacement. A wireless sensor network consisting of 56 accelerometers accommodated by 28 local nodes facilitates simultaneous, real‐time and high‐rate acquisition of the vibrations throughout the bridge structure. The study presented the laboratory model and damage scenarios, a brief description of the developed wireless sensor network platform together with an overview of available test and measurement instrumentation within the laboratory, and baseline measurements of dynamic response of the laboratory bridge model. A large‐scale field deployment of high‐density, real‐time wireless sensors networks for the acquisition of local acceleration measurements across a medium length, multi‐span highway bridge was presented by Whelan et al. (2011). The deployment consisted of 30 dual‐axis accelerometers installed across the girders of the bridge and interfaced with 30 wireless acquisition and transceiver nodes operating in two star topology networks. Real‐time wireless acquisition per channel sampling rate of 128 samples per second was maintained across both networks for the specified test durations of 3 min with insignificant data loss. Output‐only system identification of the structure from the experimental data is presented to provide estimates of natural frequencies, damping ratios, and operational mode shapes for 19 modes. The analysis of the structure provided the measured response of a multiple‐span skewed bridge supported by elastomeric bearings. Different types of excitation applied to an existing bridge were studied by Alwash et al. (2005). Of particular interest are the vibration characteristics produced by realistic truck loading compared to those produced by ambient excitation, as well as the suitability of both for use in vibration‐based damage detection. Cruz and Salgado (2008) evaluated six damage detection methods based on vibration monitoring with two case studies. In the first study, the dynamic simulation and modal parameters of a cracked composite bridge were obtained. The damage detection methods were evaluated under different crack depth, extension of the damage, and noise level. In the second study, damage was identified in a reinforced concrete bridge. This bridge was deliberately damaged in two phases. In the first case study, evaluated damage detection methods could detect damage for all the damage scenarios, but their performance was notably affected when noise was introduced to the vibration parameters. In the second case study, the evaluated methods could successfully localize the damage induced to the bridge. Ratcliffe et al. (2004, 2008) presented a broadband vibration‐based structural irregularity and damage evaluation routine that uses features in complex curvature operating shapes to locate damage and other areas with structural stiffness variations. Experimental results from a composite road bridge and a composite ship hull revealed that structural variability and some manufacturing defects can be located. Damage

Statistical Pattern Recognition and Vibration-Based Techniques

caused over time can be detected by comparing results of an inspection with the results obtained from the structure before it is put into service. The application of vibration‐based damage detection techniques based on changes of mode shapes in a two‐span, slab‐on‐girder, integral abutment bridge in Saskatoon, Canada was reported by Siddique et  al. (2005). The bridge dynamic response under ambient traffic loading was measured periodically using temporarily installed accelerometers over a range of ambient temperatures. A finite element model was developed and calibrated to match the first three measured natural frequencies and mode shapes. It was shown that the ambient temperature significantly influences measured natural frequencies. The performance of the techniques was found to be influenced by the number of sensors used to characterize mode shapes, as well as by the procedures used to normalize the mode shapes. The spatial patterns of changes produced in the fundamental mode shape of a bridge deck were used to determine whether the presence of damage can be reliably detected by the variation in the fundamental mode shape along three longitudinally oriented lines (Siddique et al., 2006). The study identified patterns that would allow a reliable determination of whether damage is present and in what region of the bridge it might be located. It was shown that normalizing mode shapes along individual lines separately emphasized localized changes caused by damage, but also magnifies the influence of random measurement noise, making it more difficult to recognize the global spatial patterns indicative of damage. The identification of damage region and location in the Tsing Ma Suspension Bridge deck using modal data was examined by Ni, Y.Q. et al. (2000). A two‐stage identification method was proposed and implemented through numerical simulation for damage detection of the bridge deck. In the first stage, the main span deck of 1377 m length was divided into 76 segments with the purpose of identifying the deck segment that contains damaged member(s). An index vector derived from mode shape curvatures in both intact and damaged states was developed to identify the damage region. In the second stage, the specific damaged member(s) within the damage region was identified by means of a neural network technique. The simulation results revealed that the developed method can still locate the damage at longitudinal structural members such as bottom chords, top chords, and diagonal members. An approach to identifying the location and the extent of the damage introduced into the steel frame by pseudo‐ dynamic seismic loads was presented by Görl and Link (2001, 2003). The measured dynamic response of the original undamaged structure was used to generate a reference finite element model of the structure. This was followed by using the experimental modal data of the damaged structure to identify the location and the extent of the damage. This was based on comparing the changes of stiffness parameters obtained from the undamaged and the damaged structure. After selecting the parameters best representing the possible stiffness degradations, the numerical values of the parameters were identified by minimizing the test/analysis differences of eigen‐frequencies and mode shapes. With the identified parameters the finite element model reproduced the experimental results. The damage location of complex structures using a multi‐step identification method was adopted by Zhang, Y. et al. (2009). The damaged substructure was identified after a structure is decomposed into several substructures, then the damaged element in the substructure can be recognized. Three decomposition modes were adopted by taking a suspension truss structure as an example. It was shown that the decomposition mode of substructures based on a clustering analysis can make it easier

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to train neural networks, and compared with the other two decomposition modes, its correct identification rate increases by 1% to 5%. The Infante Dom Henrique Bridge over the River Douro in Oporto, Portugal, has been instrumented with 12 force balance accelerometers installed in the box girder, as well as a number of strain and temperature sensors that were embedded in the concrete during construction (Magalhães et  al., 2012). The acceleration sensors were programmed to take data every 30 minutes. The data was processed using different system identification techniques, followed by post‐processing to eliminate environmental factors, and then the data stored to a database. The results of the autonomous operational modal analysis were linked to a graphical user interface that could display the results. Analytical studies showed that should damage occur, the change in natural frequencies should be detected by the system. A framework for the long‐term monitoring was applied to the Rock Island Arsenal Government Bridge by Giles and Spencer (2015). The US Army Corps of Engineers had a system of fiber optic strain gages installed on the bridge. This system was supplemented with a wireless sensor network that measured accelerations on the bridge. The multi‐metric data from the sensor systems was collected and analyzed. Metrics were developed that could be used to determine the health of the structure and the sensor networks themselves. Statistical process control methods were established to detect anomalous behavior in the short and long term. The approach utilized methods to locate and quantify the damage that occurred in the structure once an anomaly was detected. A series of dynamic tests was conducted on a truss bridge model for modal analysis using both wired and wireless sensor systems by Jang and Spencer (2015). For model‐ based damage identification approach, a finite element model was developed and updated based on a visual estimate of the corrosion. The updated model was used to generate baseline information for damage detection. The wireless smart sensor network based autonomous SHM system using the decentralized damage detection application was deployed on the historic steel truss Mahomet Bridge, in Illinois. The damage detection results using the decentralized comprehensive application were compared with those from the centralized approach using wireless smart sensor network. The modal properties matched reasonably in the degrees of corrosion between the model and the visual corrosion estimation. A methodology for damage quantification was introduced to quantify the actual condition of a bridge structure by Wenzel et al. (2015). The proposed approach was based on the spectral analysis applied to on‐site measurement data at different bridges. Detecting the development of structural damage was manifested by the change of the spectral pattern due to the shift of dynamic energy towards a higher frequency range. In China and Hong Kong, representative sophisticated SHM systems for long‐span bridges were introduced by Li, H. et al. (2015). These SHM systems were used to serve as a tool to develop the methods of lifecycle performance design, evaluation, maintenance, and management of bridges. The monitored variables were categorized into three groups: (1) loads and environmental actions, (2) local response, and (3) global response. The load and environmental measurements monitored vehicular weight, the temperature and humidity, the wind, earthquake ground motion, the vessel collision, and rainfall. Local response includes the strain in key elements, cable tension and damage, corrosion, reaction force of supports/anchorages, and scour. The global response includes the acceleration and displacement. Bridge acceleration is induced mainly by

Acceleration (cm/s2)

Statistical Pattern Recognition and Vibration-Based Techniques 2000 1000 0 –1000 –2000 0

1000

2000

3000

Time (sec)

Vertical acceleration (cm/s2)

Figure 5.27  Time history record of wind induced vibration of bridge hanger. (Li et al., 2015) 40 1/2 side span

fo = 0.1831 Hz

20 0 –20 –40

0

100

200

300

Time (minutes)

Figure 5.28  Time history record of the vertical acceleration due to vortex induced vibration of the bridge girder. (Li et al., 2015)

wind and heavy trucks. For suspension bridges, the girder and the hangers close to the towers were found to be excited to dramatic oscillations, as shown in Figure 5.27. For cable‐stayed bridges, stay cables, particularly the longer ones, may oscillate under wind. Vortex induced vibration was observed at the girder of the suspension bridge, as shown in Figure 5.28. 5.3.6.5  Damage Detection in Concrete Structures

Concrete structures have been used extensively in infrastructural systems and their non‐ destructive evaluation is important to secure their safety. Automated non‐destructive evaluation techniques have been developed for real‐time health monitoring of concrete structures. The automated non‐destructive evaluation techniques that enable continuous health monitoring of concrete structures while in operation require the development of a built‐in diagnostic system. Wahab and De (1999) examined the application of the change in modal curvatures to detect damage in a pre‐stressed concrete bridge. A damage indicator called “curvature damage factor” was introduced, in which the difference in curvature mode shape for all modes can be summarized in one number for each measured point. The technique was further applied to a real structure. Damage assessment in a two‐story steel frame and steel‐concrete composite floors structure based on a multi‐layer perceptron was presented by Zapico et al. (2001). A simplified finite element model was used to generate the training data. This model was

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previously updated through another multi‐layer perceptron using two natural frequencies as inputs and the stiffness of the beams and masses as updating parameters. The different combinations of damage at the ends of the longitudinal beams were used as damage scenarios. Two natural frequencies and mode shapes were used as inputs, and three different definitions of damage were tried as outputs. A vibration based damage detection method applied to the monitoring of two large‐scale pre‐bended beams in laboratory conditions was proposed by Siegert et al. (2006). Pre‐bended steel beam consists of a type of composite structure in which very high performance concrete was cast around the lower flange of a pre‐bended steel girder. The beams were loaded quasi‐ statically up to serviceability limit state, ultimate limit state, and failure. Transient acceleration signals were periodically recorded for subsequent modal analysis. Output‐only data was processed in order to extract the modal characteristics. The loss of stiffness induced by loading the beam over the serviceability limit state was detected from deflection measurements. However, no significant shift of the resonant frequencies occurred, because the compressive stresses were high enough to close the cracks. An experimental‐analytical investigation on the small vibrations of steel‐concrete composite beams with damaged connection was presented by Morassi (2002). Damage was induced by removing concrete around some elements connecting the steel beam and the reinforced concrete slab and consequently causing a lack of structural solidarity between the two beams. Experimental observations suggested the formulation of a one‐ dimensional model of a composite beam, where the strain energy of the connection includes an energy term associated to the occurrence of relative transversal displacements between the slab and the steel beam. A damage detection technique based on the measurement of variation in the first flexural frequencies was then applied to the model. Experimental and analytical investigations on damage‐induced changes in modal parameters of steel‐concrete composite beams subjected to small vibrations was presented by Morassi and Rocchetto (2003) and Dilena and Morassi (2003b). Damage was induced by removing concrete around some elements connecting the steel beam and the reinforced concrete slab and consequently causing a lack of structural solidarity between the two beams. Experiments results showed that flexural frequencies exhibit a rather high sensitivity to damage and can be considered as a valid indicator after a diagnostic analysis. Induced damage causes the nodes of flexural vibration modes to displace towards the damaged area. The elements connecting the slab and the metallic beam play a key role in reducing transversal motions between the two beams. Dilena and Morassi (2009) presented an Euler–Bernoulli model of composite beams, describing the dynamic response measured on composite beams with either severe or intermediate levels of damage. A diagnostic technique based on frequency measurements was applied. A Timoshenko model of composite beams was also derived and used for diagnostic purposes. The impact resonance method was used by Gheorghiu et al. (2005) for damage detection in carbon fiber reinforced polymer (CFRP) beams under fatigue loading. The beams were cyclically loaded for two million cycles. The impact resonance method was performed intermittently throughout the cyclic loading, i.e. cycles 0, 1, 50, 100, 200, 1000, 9000, 50,000, and 100,000, and approximately after every 250,000 cycles thereafter. One beam was cycled at a low stress level and the other at a high stress level. It was found that the fundamental frequency, percentage of damping, and FFT secondary peaks changed with increasing number of cycles. A decreasing trend of P‐wave

Statistical Pattern Recognition and Vibration-Based Techniques

velocities with increasing fatigue was reported. Later, Ward et al. (2008) considered the use of the impact resonance method for evaluating the structural health of thermal‐ cycled reinforced concrete beams with and without externally strengthened CFRP pultruded plates. The 1.2 m long specimens were subjected to 55 thermal cycles ranging from 23 °C to −18 °C. Fatigue loading consisting of up to two million cycles at high and low stress levels was performed. It was found that the longitudinal mode of vibration is the most sensitive to damage detection as the majority of the damage was oriented perpendicular to this mode’s vibration direction. A feasibility study for practical application of an impedance‐based real‐time health monitoring technique applying PZT (lead zirconate titanate) patches to concrete structures was presented by Park et al. (2006). Progressive surface damage inflicted artificially on the plain concrete beam was assessed by using both lateral and thickness modes of the PZT patches. It was verified that an impedance‐based damage detection method using both lateral (frequency range >20 kHz) and thickness (frequency range >1 MHz) modes of PZT patches is reliable for real‐time health monitoring and multiple (shear and flexural) crack detection in concrete structures. A root mean square deviation (RMSD) in the impedance signatures of the PZT patches is used as a damage indicator. In‐situ PZT‐based health monitoring approach for a large‐scale reinforced concrete structure was presented by Song et al. (2007). A full‐size reinforced concrete bent cap was used as the testing object for health monitoring purposes. Piezoceramic patches were embedded in the reinforced concrete bent cap prior to casting. The piezoceramic patches were used as both actuator and sensors to detect possible internal cracks inside the reinforced concrete bent cap. The detection of the existence of the cracks and the growth of cracks was monitored. The experimental results showed that the transmission energy between the actuator and sensor would drop dramatically when a crack happens inside. The proposed transmission energy‐based damage index was found to be a good damage index for detecting the existence and severity of the internal cracks. The wavelet packet analysis was applied to form the energy vector to reveal the energy content in the frequency band. Kim, J.K. et al. (2008) proposed a two‐ point elastic wave excitation method to detect concrete structure damages (crack and deterioration). This method does not require any baseline signal to distinguish the damages. Two models including 2D plane and 3D solid models were constructed, and non‐reflecting boundary conditions were applied to the 3D model to simulate a massive concrete structure. A vibration‐based damage identification study applied to a steel‐concrete composite frame structure was presented by Chellini et al. (2008). The structure was subjected to a series of pseudo‐dynamic and cyclic tests with increasing peak ground acceleration. The damaging phenomena, caused by pseudo‐dynamic tests, were assessed and quantified by means of a multi‐level vibration‐based approach suitably designed in order to evaluate the changes in the global dynamic structural response and to experimentally estimate the reduction in joint stiffness. Vibration‐based damage detection techniques were employed by Xu, B. and Gong (2010) to detect damage in reinforced concrete column after different levels of quasi‐static cyclic loadings. Four main stages were carefully examined, including healthy state, concrete cracking, partial yielding, and final yielding. Modal test and analysis were carried out for the RC column in original condition. Zhu and Hao (2009) studied the signatures of nonlinear vibration characteristics of damaged reinforced concrete structures using the wavelet transform. The vibration

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frequencies, mode shapes, and damping ratios at each loading stage were extracted and analyzed. It was found that the vibration frequencies are not sensitive to small damages, but are good indicators when damage is severe. It was found that the skeleton curves are good indicators of damage in the reinforced concrete structures because they are more sensitive to small damages than vibration frequencies. 5.3.7  Operational Deflection Shapes/Vibration Deflection Shapes

An operational deflection shape is defined as the deflection of a structure at a particular frequency and offers rapid visual feedback on the deformation of a structure in both time and frequency domain. It can be defined from any forced motion, either at a moment in time, or at a specific frequency. It can be obtained from different types of time domain responses. It can also be obtained from many different types of frequency domain measurements, including linear spectra, frequency response functions, and transmissibilities. Operational deflection shapes depend on the forces applied to a structure. They have units, typically displacement, velocity, or acceleration, or perhaps displacement per unit of excitation force. They can be defined for nonlinear and non‐ stationary structural motion, while mode shapes are only defined for linear stationary motion. They can also be defined for structures not in resonance while mode shapes are only used to characterize resonant vibration. The term “vibration deflection shape” was used instead of the more conventional “operational deflection shape” because the latter can be confused to mean the deflection shape while the structure is under service loads or in operation. The term vibration deflection shape was used to designate the vibration displacement shape or vibration velocity shape of the structure as measured by the scanning laser Doppler vibrometer. Continuous scanning laser Doppler vibrometer technique was used to obtain the operational deflection shape of a structure within a very short period of time. The ability to predict and to measure the operational deflection shape of a vibrating structure suggests its use to increase the potential for structural damage detection, localization and severity assessment. Operational deflection shapes provide very useful information for understanding and evaluating the absolute structural dynamic behavior (Døssing, 1988). Mode shapes and operating “deflection” shapes are related to one another. In fact, one is always measured in order to obtain the other. Yet, they are quite different from one another in a number of ways. Richardson (1997) discussed the relationships between modal testing, modal analysis, and operating deflection shape measurements. Modes are associated with structural resonances and are inherent properties of a structure. Each mode is defined by a modal frequency, modal damping, and a mode shape. Modes will change if the mass, stiffness, and damping properties, or boundary conditions of the structure change. Mode shapes don’t have unique values, and hence don’t have units associated with them. However, the motion of one point relative to another at resonance is unique. In structural torsional vibrations, the vibration deflection shape is referred to as torsional operational deflection shape, which is defined similar to the operational deflection shape, with the exception that torsional operational deflection shape designates the operational deflection shape of structures vibrating in a rotational, or angular degree of freedom. They can be applied to rotating shafts. Their evaluation requires the measurement and analysis of structural response signals from the vibrating structure.

Statistical Pattern Recognition and Vibration-Based Techniques

Gade et  al. (1994) described the basic concept of measuring torsional operational deflection shapes using a laser‐based torsional vibrometer, a dual‐channel FFT analyzer, and operational deflection shapes software. The direct use of operating deflection shapes was considered by Pascual et al. (1999) for on‐line damage assessment based on an approach of two levels for damage assessment. These levels are referred to as the reactive level and the proactive level. In the reactive level, current experimental operational deflection shapes are compared to the healthy measured operational deflection shapes. The frequency domain assurance criterion is used to track a global evolution and the shifted residual operational deflection shape technique is used to obtain a first damage localization. If changes in the operational deflection shape are significant, the proactive level of damage assessment is activated. The estimation of the operating deflection shapes of a system in an unsteady state was considered by Matsumura et al. (1999). An adaptive identification scheme of the operational deflection shapes for non‐stationary signals was considered. A subspace method was presented to identify well‐excited frequencies and the operational deflection shapes corresponding to those frequencies. The interpretation and accuracy of vibratory responses in identifying damage is dependent on the numbers and types of sensors and actuators. Vibration deflection shapes are used to detect damage (Pandey et al., 1991; Doebling et al., 1996; Richardson, 1997; Schulz et  al., 2003; Schwarz and Richardson, 1999; Sundaresan et  al., 1999a, 1999b, 2003; Vold et  al., 2000a, 2000b; Waldron et  al., 2000, 2002). The literature reported different applications of vibration deflection shapes for damage detection in bridges, wind turbines, and other structures. The use of vibration deflection shapes for structural damage detection provides a visual interpretation of the vibration patterns of the structure, thus allowing anomalies to be detected in terms of the geometric features of the structure. Sundaresan et al. (1999b, 1999c) used a scanning laser vibrometer and piezoceramic actuators for detecting damage on a section of a wind turbine blade. Piezoceramic patches were used to generate the vibration without mass‐loading the structure. Three different methods were used for detecting damage based on changes in transmittance functions, frequency response functions, and operational deflection shapes. Schulz et al. (2003) showed that the frequency range and number of simultaneous excitation forces significantly affect the sensitivity of damage detection. A model independent method of damage detection using measured constrained vibration deflection shapes and pre‐damage data was presented. The constrained vibration deflection shapes occur when the forcing vector contains elements that cause the structure to vibrate in a desired pattern. Yang, S.H. et  al. (1993) outlined the process of estimating the operating deflection shape from sound intensity measurement through a series of experiments, and explained the choice of measuring parameter. The application of operating deflection shape analysis for the resolution of structural vibration problems was described by Alavi et al. (1997) with the purpose of evaluating maintenance history to avoid unnecessary maintenance costs. Vold et  al. (2000a, 2000b) presented a methodology for post‐processing non‐­ stationary operating data as a prerequisite for displaying operating deflection shapes on a three‐dimensional spatial model of a test machine or structure. Waldron et al. (2000, 2002) used operational deflection shapes for structural damage detection. Generation of experimental operating deflection shape was performed using a scanning laser Doppler vibrometer and PZT actuators. The working principle of laser Doppler vibrometers was

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employed by Frank (2005). These vibrometers are capable of measuring the operational deflection shapes or to determine the modal parameters of a dynamic system. A scanning laser Doppler vibrometer can be used to measure the vibration response in a dense spatial pattern to generate vibration deflection shapes and detect small damage. For this purpose, the structure can be excited at high frequencies necessary to detect small damage by using piezoceramic (PZT) patches or a PZT inertial actuator bonded to the structure. Analytical and experimental modal analyses were performed on a plate by Di Maio et al. (2006) to obtain eigenvalues and eigenvectors and to update the finite element model. Four damaged plates were modeled in finite element software upon the position of the damage. The simulation and testing of the operational deflection shapes and the mean square error of four plates in the damaged and the undamaged cases were performed based on continuous scanning laser Doppler vibrometer measurements and compared, respectively. Results showed the effect of vibration excitation location in the damage detection. A flexible tracking system for measuring a whole grid of measurement points on arbitrarily moving structures with a focus on rotating objects using a scanning laser Doppler vibrometer was developed by Bendel (2004). A laser Doppler vibrometer was used by Bail (2008) to determine the natural frequencies and mode shapes of the composite radome structure, a structural enclosure to protect an antenna from aerodynamic loading. The vibration testing was carried out using a few different boundary conditions, which include approximately free‐free boundary conditions and fixed boundary conditions. Delamination was found to possibly cause unpredicted response in the vibration or pressure tests. The operational deflection shape analysis using multi‐channel time history response was considered by Sumimoto et al. (1993). They used random noise reduction in operational deflection shape using principal component analysis and the response‐ratio method. Since structural damage generally causes local dissipation of energy, structural intensity plots (representing spatial flow distribution of the active part of the vibration energy) can be used to detect damage. This motivated Dos Santos et al. (1999) to propose a technique based on the computation of structural intensity using spatially dense vibration data measurements to calculate energy flow maps in plates as a diagnostic tool to locate structural damage. Operational deflection shapes measured over the surface of the plate at particular frequencies were curve‐fitted using a two‐dimensional discrete Fourier series approximation, which minimizes the effects of spatial leakage. The concept of complex modes of dynamic structures was used by Ahmida and Arruda (2007). Complex modes can be interpreted in terms of wave propagation phenomena caused by either localized damping or propagation to the surrounding media. The input/output transfer relations of the structure were obtained using a spectral formulation known as the spectral element method. It was shown that mode complexity characterizes a behavior that is half‐way between purely modal and purely propagative. The mode complexity coefficient consists of the correlation coefficient between the real and imaginary parts of the eigenvector, or of the operational deflection shape. It was shown that, far from discontinuities, this coefficient is zero in the case of pure wave propagation, in which case the plot of the operational deflection shapes in the complex plane is a perfect circle. The wavelet transform was utilized by Baik and Kim (1997) to analyze transient signals to capture time‐varying characteristics in transient operational deflection shape for source identification of transient noise. A mathematical model of estimating the structure transient excitation response swept by a laser transducer during a period of

Statistical Pattern Recognition and Vibration-Based Techniques

transient free response caused by an impact was developed by Ribichini et al. (2008). The combination of a broadband excitation with a full‐field measurement resulted in a highly complex signal, modulated both by the mode shapes of the structure and by the exponential decays due to damping effects. Operational deflection shape analysis technique is an effective tool for troubleshooting of various types of noise and vibration problems in which the measured signals are assumed stationary. Masuda and Sone (2003) presented a multi‐scale operating deflection shape analysis of vibrating structures under operating conditions. A wavelet transform on the vibration data measured at the significant points on the structure was employed to separate the multiple vibration components on the timescale domain. The amplitude and the phase of the wavelet transform at a particular point on the timescale plane give the deflection shapes of each component. The extracted shapes may indicate the changes of mode shapes, or the sources of the transient responses due to some abnormal events. D’Cruz and Herszberg (2009) outlined analytical details of two approaches, which may be used to generate operational curvature shapes information for composite structures solely from structural responses. In particular, these approaches are the operational deflection shape frequency response function technique and the random decrement technique. The use of transmissibility functions in the field of operational modal analysis was introduced by Devriendt et  al. (2010). A post‐processing method based on transmissibility measurements was introduced that allows the estimation of the modal parameters and in particular the un‐scaled mode shapes. A boundary effect detection method for locating defects in plates using operational deflection shapes measured by a scanning laser vibrometer was outlined by Pai and Jin (2000) and Jin and Pai (2001). An operational deflection shape consists of central and boundary solutions. Central solutions are periodic functions, and boundary solutions are exponentially decaying functions due to boundary constraints. Because defects introduce new boundaries to a structure, boundary solutions exist around structural boundaries and defects. The boundary effect detection method uses a sliding‐window surface‐fitting technique to extract boundary solutions from a measured operational deflection shape to reveal locations of defects. Numerical results showed that high‐ order spatial derivatives of operational deflection shapes are sensitive defect indicators for locating small defects. Experimental studies on the 579 × 762 × 3.1 mm aluminum plate with four different defects showed that this defect detection method is capable of pinpointing small defects. A study dealing with a boundary effect detection method for pinpointing locations of small damages in beams using operational deflection shapes measured by a scanning laser vibrometer was presented by Pai and Young (2001) and Pai et al. (2001a, 2001b). The boundary effect detection method requires no model or historical data for locating structural damage. It works by decomposing a measured operational deflection shape into central and boundary layer solutions using a sliding window least squares curve fitting technique. For high‐order operational deflection shapes of an intact beam, boundary layer solutions are non‐zero only at structural boundaries. At damage location, the boundary layer solution of slope changes sign, and the boundary layer solution of displacement peaks up or dimples down. Experiments were performed on several different beams with different types of damage, including surface slots, edge slots, surface holes, internal holes, and fatigue cracks. Experimental results showed that this damage detection method is sensitive and reliable for locating small damages in beams.

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A local strain energy method was derived for estimating crack sizes. The crack‐induced strain energy extracted from an operational deflection shape was compared with that calculated using fracture mechanics to estimate the crack size. The results were verified experimentally on six 22″ × 1″ × 0.25″ 2024‐T4 aluminum beams each having a through‐ the‐width mode‐I crack at its midpoint. Pai et al. (2004a, 2004b, 2006) developed a signal decomposition method for extracting boundary effects from an operational deflection shape of a structure under harmonic excitation. For numerical simulations, exact mode shapes, and operating deflection shapes of beams with damage were obtained by spectral element analysis. 5.3.7.1  Applications of Operational Deflection Shapes

A vibration‐based damage detection method using operating deflection shapes has been employed in many studies to detect the locations and extent of damage in different structural systems such as beams, composite structures, bridges, and machines. For example, Yoon et al. (2010) extended their work (Yoon et al., 2009) and obtained the operational deflection shapes derived from experimental frequency response function data to detect the locations and extent of damage in steel beams, composite beams, and plate‐like structures. Both the analytical and measured operational deflection shapes were differentiated twice to yield the curvature operating shapes. The structural irregularity index at each grid point on the structures (the difference between the analytic and measured curvature operating shapes) was averaged over the selected frequencies to obtain the frequency‐averaged structural irregularity index. The method was originally developed for one‐dimensional beam models. The experimental frequency response function identified the locations and extents of the notches in steel beams, delaminations in the composite beams and plates, and dry spots in a composite hull structure. Arruda and Mas (1998) presented an experimental method adapted for the computation of structural power flow using spatially dense vibration data measured with scanning laser Doppler vibrometers. The operational deflection shapes measured over the surface of a plate curve‐fitted using a two‐dimensional discrete Fourier series approximation that minimizes the effects of spatial leakage. From the wave number frequency domain data, the spatial derivatives necessary to determine the structural power flow were computed. Bae et al. (2011) formulated operational deflection shape frequency response function to normalize the effects from variable excitation force on a beam and plate under the excitation of varying forces. Baz et al. (1993) and Baz and Poh (1996) presented a new class of distributed sensors, which can measure both the modal and physical displacements of vibrating composite beams. The sensor relies in its operation on a set of super‐elastic shape memory alloy wires embedded off the neutral axes of the vibrating beams. The wires were arranged in a special manner which allows continuous monitoring of the deflection curve of the beam. Comparisons between the experimental performance of the shape memory alloy distributed sensor and that of conventional laser sensors were obtained to demonstrate the accuracy and merits of the distributed shape memory alloy sensor. The results suggested the potential of these sensors as a viable means of monitoring the static and dynamic deflections of flexible composite smart beams and plates. The suppression of steady‐state vibrations of a cantilevered skew aluminum plate using nonlinear saturation phenomena and PZT patches was examined by Pai et  al. (1999, 2000). Finite‐element analysis and measurement of operational deflection shapes

Statistical Pattern Recognition and Vibration-Based Techniques

using a scanning laser vibrometer were performed to study the bending‐torsional dynamic characteristics of the plate due to non‐rectangular geometry. The control method uses linear second‐order controllers coupled to the plate via quadratic terms to establish energy bridges between the plate and controllers. Each linear second‐order controller was designed to have a 1:2 internal resonance with one of the plate vibration modes and was able to exchange energy with the plate around the corresponding modal frequency. Because of quadratic nonlinearities and 1:2 internal resonances, saturation phenomena exist and are used to suppress modal vibrations. It was concluded that the saturation controllers can be used to regulate dynamics of structures to prevent resonant vibrations by designing each controller to control one vibration mode. The mode shapes, natural frequencies, and defect detection of circular plates using a scanning laser vibrometer were studied by Pai et al. (2002a). Their study considered the exact dynamic characteristics of a circular aluminum plate having a clamped inner rim and a free outer rim, obtained using Bessel functions and a multiple shooting method. Their numerical results showed that high‐frequency operational deflection shapes are needed to locate small defects. Experimental results revealed that small defects in circular plates can be pinpointed by these approaches. Pai and Lee (2003) demonstrated the use of a scanning laser vibrometer and a signal decomposition method to characterize nonlinear dynamics of highly flexible structures. A scanning laser vibrometer was used to measure transverse velocities of points on a structure subjected to a harmonic excitation. Experimental results revealed the existence of 1:3 and 1:2:3 external and internal resonances, energy transfer from high‐frequency modes to the first mode, and amplitude‐ and phase‐modulation among several modes. Moreover, the existence of nonlinear normal modes was found to be questionable. Some results of damage identification studies performed for the Swiss Z‐24 Bridge, a three‐span post‐tensioned concrete box girder structure, were presented by Catbas and Aktan (2002). Modal properties obtained from the ambient vibration tests of the bridge were utilized for damage identification purposes. The deflected shapes of measurement lines under virtual uniformly distributed load, which is termed as the “bridge girder condition indicator” were presented. A total of 14 damage cases were analyzed for the Z‐24 Bridge using the pseudo‐modal flexibility based on bridge girder condition indicator. The impact of each damage scenario was evaluated by examining the deflection profiles generated for the box girders. Often machinery failure is associated with the vibrational motion of individual machinery components, or of combinations of components. Sometimes the locations, depth, and orientation of wear scars give telling clues concerning the nature of problems, and how to fix them. More often, the wear patterns or component fatigue cracks are able to be interpreted only when combined with other quantitative information. One of the most useful types of information is vibration measurements taken in various key locations. Marscher and Jen (1999) used operational deflection shape and mode shapes for machinery diagnostics. When the vibration measurements include phase (relative direction of motion of one location versus others) as well as amplitude, they can be combined with the wear measurements to facilitate understanding in many vibration related failures. Kromulski and Hojan (1996) presented two methods for the determination of operational deflection shapes. The first method determined the operational deflection shapes by measuring mechanical vibrations at test points of the operating machine (created by the forces occurring in the actual work cycle). The second

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method was based on the modal model of the machine, including the forces which occur in the operation process, and yields the operational modes of vibration. The determination of the forces involved is necessary for the definition of the paths of energy transfer and the analysis of the noise emitted by the mechanical system (machine). Castellini and Revel (1988) used a scanning laser Doppler vibrometer to detect, localize, and characterize delaminations in composite materials. An experimental investigation by scanning laser Doppler vibrometer was performed on panels with known detachments and the results were compared with those obtained by thermal tomography. Later, damage detection and characterization technique based on a laser Doppler vibrometer were presented by Castellini and Revel (2000a, 2000b). A damage detection algorithm based on the principle of curvature changes was developed by Chen, S.E. et al. (1998, 1999). To assess the suitability of using a scanning laser vibrometer for damage detection, a beam specimen was tested and the results confirmed that damage detection using vibration measurements from scanning laser vibrometer is successful. Due to more spatial density, the scanning laser Doppler vibrometer data was shown to be more sensitive than the contact sensor test. Another experimental study was conducted on aluminum and composite plates with different damage scenarios to investigate the feasibility of the approach. A two‐dimensional strain energy distribution was computed for damage detection. Chen, S.S. (2000) and Chen, S.E. et al. (2000) showed that a derivative of vibration deflection shapes, strain energy distribution, is a more sensitive parameter than fundamental vibration parameters for detecting damage such as cracking in the members. Since the strain energy distribution method requires significantly more spatial data points than typical vibration tests, a scanning laser vibrometer capable of providing adequate spatial density is used. The frequencies, displacements, and shapes of the vibrational modes within the structure were measured using equipment such as scanning laser Doppler vibrometers, and any change in the modal properties compared to the defect‐free structure can be used to indicate the presence and location of damage (e.g. Ghoshal et al., 2000, 2003; Waldron et  al., 2002; Qiao et  al., 2007) used scanning laser Doppler vibrometers to detect delaminations in a curved fiberglass composite plate and honeycomb panel. Ghoshal et al. (2000) examined four different algorithms for detecting damage on wind turbine blades made of fiberglass material. These algorithms were the transmittance function, resonant comparison, operational deflection shape, and wave propagation methods. The methods were all based on measuring the vibration response of the blade when it is excited using piezoceramic actuator patches bonded to the blade. The vibration response of the blade was measured using either piezoceramic sensor patches bonded to the blade, or a scanning laser Doppler vibrometer. An experimental investigation to detect embedded delamination and other forms of damage in heterogeneous structures using smart materials and a laser vibrometer was presented by Ghoshal et al. (2003). Experiments were conducted on composite structures with piezoelectric actuator patches and embedded delaminations, using the vibration deflection shape method and a scanning laser Doppler vibrometer. The vibration deflection shapes were shown to be sensitive to structural parameter variations, and hence can be used to detect and locate damage in large composite structures, including a woven fiberglass curved plate and a honeycomb inter‐tank panel. Measurement of structural vibration under operating conditions and a validated finite element analysis were combined to identify detrimental excitation sources and design

Statistical Pattern Recognition and Vibration-Based Techniques

imperfections by Ebi (2001). Operational deflection shape analysis was performed by Powell (2003) on a large fan foundation using normal background turbulence as the only excitation. Ganeriwala et al. (2009) demonstrated the use of the operational deflection shape of a rotating machine as a means of detecting unbalance in its rotating components. The detection of defects in composite T‐stiffened panels using vibration modal analysis was numerically and experimentally studied by Herman et  al. (2013). The analysis was performed on carbon fiber/epoxy laminate panels containing delamination cracks or porosity. Experimental testing revealed that vibrational excitation of the defective panels altered the vibration mode response measured by scanning laser vibrometer. The experimental results revealed that changes to the mode shape curvature of the lower‐order vibrational modes was the most reliable indicator of damage within the stiffened panel, whereas high order modes could not be used to detect damage. The mode shape displacements for the T‐stiffened panels were measured using the scanning laser vibrometer, and Figure 5.29 shows the normalized displacement along the length of the undamaged panel and the panels containing a delamination or porous region along the skin/flange bond‐line. The normalization was taken with reference to the distance from zero to the first local maximum measured in the vertical axis. It is seen that there is no significant difference between the three types of panel, including in the damage region. Accordingly it was concluded that the mode shape displacement response is not a reliable indicator of damage for these specimens. Alternatively, Herman et al. (2013) adopted the mode shape curvature profiles as a better measure of damage detection. Figure 5.30 shows typical mode shape curvature difference profiles along the length of the delaminated and porous panels when measured using a scanning laser vibrometer from the skin side. The profiles were normalized to the peak mode shape curvature difference value, and they were measured for the first mode defined as  one half of a sine wave. The location of the delamination damage was accurately detected by the large spike in the mode shape curvature difference profile, which was due to damping of the first mode response by the large loss in local stiffness over the region of the crack. The mode shape curvature difference method was also capable of detecting the porosity, but the central zone of the porous region was predicted to be about 30 mm from the actual central location. This inaccuracy was attributed to the weaker vibration damping effect of the porous region compared to that of the delamination crack. This was supported by the fact that the maximum mode shape curvature difference value in the damaged region was about four times lower in the porous panel compared to the delaminated panel. The sensitivity of the mode shape curvature difference method for the detection of delamination damage and porosity in the T‐stiffened panels decreased rapidly with higher‐order vibrational modes. It was found that the accuracy of the mode shape curvature difference method in the detection and location of damage within T‐stiffened panels is the most sensitive and accurate when the lowest vibration mode is analyzed. 5.3.7.2  Case Study of T‐Joints

Stress–strain elastic states of sandwich beams may significantly differ from those predicted by the classical Euler–Bernoulli model for thin beams made of homogeneous materials. The classical beam model assumes that a normal to the beam’s centerline remains normal so that elastic energy of bending is accumulated due to tension‐­ compression deformations parallel to the centerline layers. In contrast, during

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(a) Stiffener

Delamination and porosity damage region

Flange Skin

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Figure 5.29  (a) Schematic diagram of the T‐joint showing the region of defect, (b) mode shape displacement profiles along the length of the stiffened panel. (Herman et al., 2013): _______ undamaged panel, – ‐ – delaminated panel, and …. porous panel

bending of hard skin sandwich beams, the upper and lower skins try to preserve their lengths by imposing significant shear deformations on a relatively soft core. In the general case, the elastic energy will be due to both shear and tension‐compression deformations whose relative contribution is determined by elastic properties of skin and core materials. Therefore, a sandwich beam model must include at least two different stiffness parameters in order to describe both components of the elastic energy. The influence of cracking damage on the dynamic response of T‐joint structures was studied analytically and experimentally by Pilipchuk et al. (2013a).

Statistical Pattern Recognition and Vibration-Based Techniques

Normalized mode shape curvature difference

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Figure 5.30  Normalized mode shape difference profiles for the first mode along the length of the stiffened panels containing (a) delamination, (b) porous region. (Herman et al., 2013): ______ profiles measured experimentally, – ‐ – and estimated using the finite element method

An accurate modeling that captures local details of the stress–strain distributions can be developed within two‐ or three‐dimensional elasticity theories. Such approaches are usually taken when dealing with static problems. The common purpose of dynamic studies is to find the dynamic response of the entire beam in terms of its modal amplitudes, natural frequencies, and related parameters. For that reason, it is convenient to

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deal with a single descriptive function of the beam position, such as the coordinate of centerline, say w w(t ,x ), where 0 x L, and L is the length of the beam. All the information regarding the stress–strain distributions through the beam thickness is incorporated via coefficients of the corresponding differential equation of motion with respect to the function w(t, x). Analytical expressions for such coefficients can be derived theoretically by reducing two‐dimensional elastic models. Some complexity of such approaches is caused in part by the necessity of dealing with three different structural subcomponents of sandwich beams. However, taking into account the final goal of the corresponding considerations suggests a phenomenological way of modeling the sandwich beam dynamics as follows. The inertial forces are assumed to be mainly due to transverse motion, and the model is linear. The differential equation of motion for the beam centerline may be written in the following general form: A



2

w t ,x t2

P

x

w t ,x

q t ,x

(5.37)



where ρA be the beam’s mass per unit length, q(t, x) is a distributed external loading per unit length of the beam, and P is a general polynomial of the differential operator / x with as yet unknown coefficients

P

p0

x

p1

x

2

p2

x2

p3

3

x3

p4

4

x4

(5.38)



Although no certain rule exists on how many terms must be kept in expression (5.38), it is reasonable to keep as few terms as possible, while achieving adequate description of the real object based on the comparison with test data. Polynomial (5.38) is truncated such that the classic beam model can be included as some limit case in which the coefficient p4 becomes the bending stiffness. The coefficients in (5.38) may be specified by using physical interpretations and ideas of symmetry. For instance, the odd coefficients, p1 and p3, must be set to zero to prevent equation (5.37) from changing its form when x. Further, a non‐zero p0 would indicate the changing direction of the coordinate x presence of linearly elastic foundation of the stiffness p0. This parameter may be included whenever it complies with the physical problem formulation. In the present case, however, it is taken as zero, p0 0. As mentioned earlier, the parameter p4 should provide the possibility of transition to the limit case of classical Euler–Bernoulli beam model. Finally, the parameter p2 is left to capture the effect of shear deformations. The non‐zero parameters may be represented in the form

2

p2

EI ,

p4

EI (5.39)

where the notation EI means the classic bending stiffness only in the corresponding limit case, when the elastic energy of shear deformation is negligible, which corresponds 0. For finite α, indicating the contribution of shear deformations, the standard to expression for stiffness parameter EI must be modified. Substituting (5.39) into (5.38), brings equation (5.37) to the form

A

2

w t ,x t

2

2

EI

2

w t ,x x

2

EI

4

w t ,x x4

q t ,x



(5.40)

Statistical Pattern Recognition and Vibration-Based Techniques

Note that the second term is similar to tension in a string, but, in our case, it describes the effect of shear deformations rather than tension. However, this term can still account for the longitudinal tension whenever it is applied to the beam. The form of equation (5.40) is similar to that derived by Berdichevsky (2010a) in quite a different way based on the asymptotic analysis of elasticity equations that provides explicit expressions for the coefficients of second and third terms of equation (5.40). Within our phenomenological approach, however, the parameters α and EI will be determined from dynamic characteristics, such as resonance frequencies. Assuming that the vibration mode shapes are described by sine waves, gives the following expression for natural frequencies of the beam

fi

EI 2 2 i i AL4

2

2 1

, i 1, 2 .

(5.41)

L/ . For instance, dynamic tests on the beam, whose length and mass per where 1 unit length are L 0.635 m and A 0.113122 kg/m, respectively, give the following estimates for the first two resonance frequencies: f1 78.13 Hz (490.9 rad/s) and f2 302.7 Hz (1901.92 rad/s). Based on these results, expression (5.41) gives the following parameter estimates EI 41.75 Nm2 and 1 0.299815 m−1. Note that physical and geometrical properties of the specimen used with the shaker characteristics did not allow for a reliable detection of high resonance modes. In cases where high resonance frequencies are known, a least‐square fit over more frequency sets has to be applied. Although this may give somewhat different result, it is assumed this is exactly what may happen practically – some parameter uncertainties are always present. It must be noted also that the current model serves only for illustration purposes. A general view of T‐joint structure with dimensions currently used in tests and modeling is shown in Figure 5.31(a). For modeling purposes, it is assumed that the effect of the transverse beam is similar to that created by a combination of two springs (one linear and the other is torsional) and localized mass shown in Figure 5.31(b). There are also two torsion springs at the ends of the horizontal spring. Such a model can be described by the Lagrange density function in the form Ldens

1 2

A m 2

L x 2

1 w k0 2 x 1 KJ y w 2

x 2

dy dt kL

kJ

w t

2

1 EI 2

2

w x

2

2

w x2

(5.42)

x L w x

2

2

x

L 2

Vdamage

where w w(t ,x ) is the coordinate of centerline of the horizontal beam measured in the moving system coordinates, as shown in Figure 5.31(b), y y(t ) is the vertical support displacement of the beam’s base which is mounted on the shaker’s platform (it is therefore assumed that the system coordinates are attached to the base), denotes the Dirac function,5 ρA is the beam mass per unit length, m is the localized mass of the T‐joint 5  Here the Dirac delta with overbar is used to avoid confusion with the variation symbol.

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Handbook of Structural Life Assessment

(a)

379 mm 48 mm 4 mm

Width–26 mm 48 mm 14.2 mm 663 mm

(b) KJ

w(t,x) y(t)

m

k0

kL x

kJ x=L

x=0

Figure 5.31  (a) The geometry and (b) modeling of T‐joint structure.

materials, k0 and kL are torsion springs at the beam’s right and left ends, respectively, KJ and kJ are linear and torsion spring stiffness parameters, simulating the dynamic influence of the T‐joint on the horizontal spring, and the elastic energy adjustments due to a damage of the T‐joint are introduced through an effective negative stiffness as follows

w 1 KJ y w d 2 x

Vdamage

2

1

y w d

w x

x

L (5.43) 2

0 is a dimensionless small parameter characterizing the damage (or crack) where size, θ denotes the unit‐step Heaviside function, and d is a parameter characterizing the damage shift from the geometrical center of T‐joint on the horizontal beam. The presence of delta function in (5.43) should be interpreted in terms of integral identities, in L other words, the operation Vdamage dx simply substitutes into the “factor” of the delta 0 function in (5.43) under the condition that the factor is continuous in x at x = L/2. In case d 0, equation (5.43) takes the form

Vdamage

/2 KJ y w

2

1

y w

x L / 2 (5.44)

The Lagrangian of the entire structure is



Ltotal

L 0

Ldens dx

(5.45)

Statistical Pattern Recognition and Vibration-Based Techniques

The variation of energy dissipation is

U diss

2

AEI

L 0

3

w wdx x t

(5.46)

2

AEI in expression (5.46) is chosen in such way that ζ would where the factor 2 further have a common meaning of damping ratio. Note that the external loading on the horizontal beam is due to the homogeneously distributed vertical inertia force and the localized transverse force due to the reaction of the vertical beam. Both these loading components are already incorporated into Lagrange function (5.42) through the base displacement y y(t ) and velocity dy(t)/dt. Further, the beam centerline will be described by the truncated Fourier series, including the first three terms:

w t ,x

3 i 1

wi t sin

i x (5.47) L

Note that representation (5.47) cannot provide a detailed description of the vibration mode shapes near the beam ends and its center, where the localized elastic elements are placed to model the effect of the supports and the influence of the transverse beam. This study, however, is to find how distributed mode shapes are affected by localized damages, in order to utilize this information for damage detection purposes. From this standpoint, the elastic energy accumulated by the springs shown in Figure 5.31(b) will be spread among the modal components of representation (5.47) due to substituting equation (5.47) in equation (5.44). The corresponding differential equations of motion are obtained using the Lagrange equation

d dt

Ltotal w i

Ltotal wi

U diss , i 1, 2, 3 (5.48) wi

where w i dwi (t )/ dt , and δ indicates variation. The resulting differential equations are given in matrix form as



1 w 2 M w 3 w 2 J

2

w 1 D w 2 w 3

w1 K w2 w3

6 3 1 2 0 0 y 3 2 3 1

2 J

f y w1 2

d w2 w3 R L (5.49)

 y

where

M

1 2 0 2

0 2 1 0 , K 0 1 2

K 11 K 12 K 13

K 12 K 22 K 23

K 13 K 23 K 33

D 2

1 0 0 0 4 0, R 0 0 9

1 d 2 L 1



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Handbook of Structural Life Assessment

where m/( AL) is the mass ratio of the T‐joint materials to the total mass of the horizontal beam, f ( z ) [1 ( z )]z z |z| / 2 is the piecewise‐linear function such that f ( z ) z if z 0 and f ( z ) 0 if z 0, M is a dimensionless mass matrix, matrices D and R characterize the distribution of damping, and nonlinearity factors among the modes, respectively, the elements of the stiffness matrix K are as follows:



2 1

2

2 0

K 11

1

K 22

4 4

2 1

2

9 9

2 1

2

K 33

2 J

2 L,

4

2 0

2 J

9

2 0

2 J

K 12 4 9

2 0

2

2 L,

K 23

2 L

6

, K 13 2 0

2 L

3

2 0

2 J

,

3

2 L,

(5.50)

2 L

where 1 ( L/ ) is a dimensionless parameter characterizing the intensity of shear deformations in the sandwich beam corresponding to that introduced in equation (5.39), and the following parameters have frequency units associated with different stiffness elements participating in the model versus the total mass of the beam itself: /L

0

4

EI / 2

/ L k0 /

A ,

2K J /

J

AL ,

L

AL , 2

/ L kL /

8

j

AL

2

/ L kJ /

AL , (5.51)

These parameters are introduced in a form that allows for estimating contributions of different stiffness components to the frequency content of the dynamics. For parameter identification purposes, it is useful to estimate sensitivity of the dynamic response to different parameter variations, for instance, in terms of the model spectrum and frequency response curves. For that reason, consider the linear case of undamped 0) described by vibrations ( 0 and

y

A cos t , wi

Bi cos t ; i 1, 2, 3 (5.52)

where A and Bi are constants. Equation (5.49) then gives the following solution for the response amplitudes:



B1 B2 B3

A K

2

M

1

2 J

1 0 1

2 3

2

6 3 0 2 3

(5.53)

Figure  5.32 illustrates amplitude frequency response plots of the horizontal beam alone so that K J 0, k J 0, and m 0. It is assumed that the boundary torsion springs have non‐zero stiffness k0 93.7676 and k L 40.6086. As a result, solution (5.53) possesses non‐zero asymmetric component B2 0. Note that the inverse matrix in (5.53) has a singularity whenever the input frequency Ω is very close to one of the natural frequencies of the beam, as reflected by the presence of discontinuities of the diagrams in Figure 5.32. Figure 5.33 illustrates the dependence of the horizontal beam spectrum on the torsion spring at the left end, while the right end is simply supported (k L 0). Note that Figures 5.32 and 5.33 may be used for parameter identification and validation purposes from dynamic shaker excitation tests. For instance, the amplitude–frequency response of Figure 5.32 shows the modal amplitude signatures at different ranges of the excitation frequency, Ω.

Statistical Pattern Recognition and Vibration-Based Techniques

0.0004

B1 (m)

0.0002 0.0000 –0.0002 –0.0004 0

1000

2000

3000

4000

5000

6000

3000 Ω (rad/s)

4000

5000

6000

3000

4000

5000

6000

Ω (rad/s) 0.0004

B2 (m)

0.0002 0.0000 –0.0002 –0.0004 0

1000

2000

0.0002

B3 (m)

0.0001

0.0000

–0.0001

–0.0002

0

1000

2000

Ω (rad/s)

Figure 5.32  Amplitude–frequency response of the horizontal beam alone. (Pilipchuk et al., 2013a)

The presence of the transverse beam will change the system’s natural frequencies, as shown in Figures 5.34 and 5.35. In particular, Figure 5.34 shows the dependence of the modal frequencies on the tension‐compression stiffness KJ. It is seen that the second frequency is constant because the asymmetric mode does not contribute to tension‐ compression of the vertical beam. In contrast, the torsion spring, kJ, leaves the frequencies of symmetric modes constant, whereas the frequency of asymmetric mode is growing as the torsional stiffness increases, as shown in Figure 5.35.

339

6000

ωf (rad/s)

5000

ω3

4000 3000 ω2

2000 1000 0

ω1 0

1000

2000

3000

4000

5000

k0 (Nm)

Figure 5.33  Eigen‐frequencies of the beam alone versus torsion stiffness at the left boundary. (Pilipchuk et al., 2013a) 5000

ω3

ωf (rad/s)

4000 3000 ω2

2000

A

1000 0

ω1 0

50,000

100,000 kf (N/m)

150,000

200,000

Figure 5.34  Eigen‐frequencies of the model (Figure 5.31(b)) with no boundary springs versus stiffness of the tension‐compression spring; Point A – 1:1 internal resonance between the first two modes (Pilipchuk et al., 2013a)

5000

ω3

A

ωf (rad/s)

4000 3000

ω2

2000 1000

ω1

0 0

2000

4000 6000 kf (N/m)

8000

10,000

Figure 5.35  Eigen‐frequencies of the model with no boundary springs versus stiffness of the central torsion spring; Point A – 1:1 internal resonance between the second and third modes. (Pilipchuk et al., 2013a)

Statistical Pattern Recognition and Vibration-Based Techniques

Consider now the nonlinear case ( 0). In particular, we seek steady‐state periodic solutions in the following form of truncated Fourier series:

wi

Bi 0

Bi

Bi1 cos t

Bi 2 cos 2 t

Bi 4 cos 4 t , i 1, 2, 3 (5.54)

where the coefficients Bi are given by equation (5.53), whereas the rest of the coefficients are to balance the corresponding harmonics generated by the nonlinear term of order ε in equations (5.49). These coefficients therefore must be of order ε. Furthermore, we employ the following truncated Fourier series for the nonlinear term calculated on the harmonic input 1 1 1 2 Y cos t Y cos t Y Y cos t Y cos 2 t 2 2 3 (5.55) 2 Y cos 4 t 15 d where, in our case, Y A B1 2 B2 B3. L Note that the term Y cos t generates only even harmonics in expansion (5.55). Figure 5.36 reveals that expansion (5.55) captures main qualitative features of the non‐ smooth function f(Y cosΩt) and therefore can be used for solution estimates. Substituting equations (5.54) and (5.55) in equation (5.37), taking into account (5.53) and equating terms of order ε in the basis {1,  cosΩt,  cos 2Ωt,  cos 4Ωt}, gives f Y cos t

B10 B20 B30



2 J

B12 B22 B32

2 J

Y

B11 B21 B31

1

K R,

2Y K 4 3

2

M

2 J

1

R

Y K 2

2

M

1

R, (5.56)

B14 B24 B34

2 J

2Y K 16 15

2

M

1

R

0.0

f(Y cosΩt)

–0.2 –0.4 –0.6 –0.8 –1.0 –10

–5

0

5

10

Ωt

Figure 5.36  The temporal mode shape of non‐smooth perturbation: exact (thick line) and the four‐term trigonometric expansion (thin line), the input amplitude is Y = 1. (Pilipchuk et al., 2013a)

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Handbook of Structural Life Assessment

Equations (5.53), (5.54), and (5.56) determine the steady‐state first‐order asymptotic solution whose period is given by the period of external loading in the absence of damp0. In particular, Figure 5.37 illustrates trajectories of this solution in different ing, i.e. configuration planes for no damage ( 0) and damaged structure ( 0.1, d 0.01), shown by curves 1 and 2, respectively. The T‐joint “lumped” parameters are: K J 50000, k J 1000, and m 0.03. Curves 3 show trajectories of numerical simulation of equations (5.49) obtained for the damping parameter 0.02 on the interval 0.9 t 1.1. The presence of damping provides the possibility of reaching the steady state in numerical simulations. Regarding the analytical procedure, the damping term would technically complicate solution (5.53), (5.54) and (5.56), but derivations would still be possible. The damping effect is seen from the diagrams in Figure  5.37. Since the influence of damping on different modes is different, then some phase shift between the modes must occur. As a result, the periodic attractors take loop‐wise shapes on configuration planes indicated by curves 3 in Figure 5.37. This effect is confirmed also by test results that will be described later. The present idea of damage detection is to find such visualizations of the modal content of dynamics that would be sensitive to minor (local) stiffness variations and typical nonlinear effects in modal interactions induced by local damages. Therefore, real‐time information about mode shape characteristics is required for practical implementation of such approaches. The vibration mode shapes were recovered from the measurements of local strain deformations by means of strain gauges installed at three different points on the upper skin of the horizontal beam. Generally speaking, this is a type of inverse problem that may have no unique solution. However, under some additional assumptions, the problem becomes easily and uniquely solvable. First, let us assume that the longitudinal strain of the skin’s upper layer εs is expressed through the beam’s centerline w w(t ,x ) as

s

t ,x

h 2

2

w t ,x x2

(5.57)

where h is the beam’s thickness and β is a dimensionless coefficient characterizing the structural non‐homogeneity of the sandwich beam. In particular, when 1, expression (5.57) is reduced to the classical expression for strain deformation of the Euler– Bernoulli beam model. The reason for assuming expression (5.57) is that strain gauges are installed on the outer side of the upper skin of sandwich, which is a thin homogeneous beam whose centerline is similar to the centerline of the entire sandwich. The inner side is under the influence of contact with the relatively soft core that may result in some modification of the dependence (5.57). For our practical goal, however, an exact knowledge on the mode shapes is of little importance because practically it is hardly possible to exactly recover the mode shapes of real structures even with very detailed relationships (5.57). However, it is important to detect variations of the mode shapes caused by growing damages. Another condition is imposed, namely, the number of strain gauges at different locations must be greater than, or at least equal to, the number of modes in expansion (5.47), which is currently three modes. In this case, substituting (5.47) in (5.57) gives w1 t

w2 t w3 t

1

T x1 ,x2 ,x3

2 3

t t (5.58) t

6 × 10–7 2

4 × 10–7

1

2 × 10–7 W3

0

3

– 2 × 10–7 – 4 × 10–7 – 6 × 10–7

–0.00006 –0.00004 –0.00002

0

0.00002 0.00004 0.00006

W1 0.000015

0.00001

5 × 10–6 W2

0 –5 × 10–6

–0.00001

–0.000015 –0.00006 –0.00004 –0.00002

0

0.00002 0.00004 0.00006

W1 0.000015

0.00001

5 × 10–6 W2

0 – 5 × 10–6

–0.00001

10 – 7 6×

10 – 7 4×

10 – 7

0

2×

10 – 7 ×

×

10 – 7

–2

–4

–6

×

10 – 7

–0.000015

W2

Figure 5.37  Configuration plane trajectories under the input frequency 313.6 Hz: 1 – no damage, zero damping (analytical solution), 2 – damaged structure, zero damping (analytical solution), 3 – damaged structure, non‐zero damping (numerical steady‐state solution). (Pilipchuk et al., 2013a)

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Handbook of Structural Life Assessment

where the transfer matrix is given by x1 L x sin 2 L x sin 3 L sin



T x1 ,x2 ,x3

2 h L

2

x1 L x 4 sin n 2 L x 4 sin 3 L 4 sin

x1 L x 9 sin 2 L x 9 sin 3 L 9 sin

1

(5.59)

In our case, x1 0.062 m, x2 0.262 m and x3 0.381 m, and direct calculation shows that the transfer matrix T(x1, x2, x3) does exist. Taking into account (5.47) and (5.58), brings (5.57) to the form

s

t ,x

h 2 L

2

x 2 x 3 x sin ,4 sin ,9 sin T x1 ,x2 ,x3 L L L

1

t

2

t (5.60)

3

t

Note that the factor (βh/2)(π/L)2 will be cancelled by its inverse, which is present in the expression T(x1, x2x3) given by equation (5.59). Expression (5.60) allows for estimation of skin strain at any location x, based on the strains measured at three different points x1,x2, and x3. Therefore, expression (5.60) can be used for determining approximate damage location by analyzing the distribution of strain in case the damage location is unknown. Based on the present series of tests, however, the problem of damage location would be solved in quite a trivial way due to the specimen symmetry. Note also that the present approach considers T‐joints as localized at some points. As a result, the damage location will always coincide with locations of T‐joints. The problem of exact damage location on a given T‐joint must be addressed rather by using other tools. In the experimental model, both ends of the horizontal sandwich beam are supported by two rollers on the upper and lower skin sides. These rollers provide boundary conditions close to elastic hinges with some torsional stiffness. The horizontal beam is mounted on the aluminum platform attached to the shaker. The upper end of the vertical beam is attached to the immovable frame through some bushes. The core is bounded by two skins made of woven glass fibers of the type 3‐ply 7781 style E‐glass fiber/epoxy fabric laminate with weave pattern of 8 HS (Harness Satin) of which the warp and fill yarns running at 0° and 90° with the length, respectively. Each beam is 1 inch (25.4 mm) wide and 0.556 inch (14.1 mm) thick. The core is 0.5 inch (12.7 mm) thick, while each skin is 0.028 inch (0.7 mm) thick. The T‐joint is made of two metallic angles with six bolted joints. The reason for using the bolted joints is to provide the possibility of creating and removing different structural “damages” by relaxing or tightening some bolts of the T‐joint. The test setup deals therefore with artificially introduced expected effects of damage on stiffness properties instead of real damages. Such “intermediary” phase of experimental validation seems to be useful for dealing with a technique, which is under development, since bolted joints provide reproducible “damages” for testing under different conditions. After different technical issues have been resolved, the eventual goal would be validation of the tool by using specimens with real progressive cracks or other damages. The time history records of the dynamic response of both healthy and damage structure under low excitation frequency loading of 24 Hz are shown in Figure 5.38. Note the

Statistical Pattern Recognition and Vibration-Based Techniques

Figure 5.38  Dynamic response of the T‐joint structure on the low frequency load (24 Hz): left column – healthy structure, right column – ‘damaged’ structure with no bolts on the horizontal beam; first row – strain records. (Pilipchuk et al., 2013a)

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Handbook of Structural Life Assessment

Figure 5.38  (Continued )

Statistical Pattern Recognition and Vibration-Based Techniques

Figure 5.39  Trajectories of the T‐joint structure under the low frequency load (24 Hz): left column – healthy structure, right column – ‘damaged’ structure with no bolts on the horizontal beam. (Pilipchuk et al., 2013a)

top plot on each column of Figure 5.38 represents the output signals of the three strain gauges (ei, i 1, 2, 3). These signals were processed to recover the three modal amplitudes wi, i 1, 2, 3, shown on the same figure. Figures 5.39 and 5.40 show the corresponding configuration plots and FFT, respectively. The response of the healthy structure is shown on the left columns, whereas the right columns illustrate the case with no bolts

347

FT[w1(L)]

0.0020 0.0015 0.0010 0.0005 0.0000

0

10

20

30

40

50

60

40

50

60

60

ω (Hz)

FT[w2(L)]

0.00005 0.00004 0.00003 0.00002 0.00001 0 0

10

20

30 ω (Hz)

0.0001

FT[w3(L)]

0.00008 0.00006 0.00004 0.00002 0.0000 0

10

20

30 40 ω (Hz)

50

0

10

20

30 40 ω (Hz)

50

0.0006

FT[w3(L)]

0.0005 0.0004 0.0003 0.0002 0.0001 0.0000 60

Figure 5.40  Spectral response characteristics of the T‐joint structure under the low frequency load (24 Hz): left column – healthy structure, right column – ‘damaged’ structure with no bolts on the horizontal beam. (Pilipchuk et al., 2013a)

Statistical Pattern Recognition and Vibration-Based Techniques 0.000035

FT(w2(L))

0.00003 0.000025 0.00002 0.000015 0.00001 5 × 10–6 0 0

10

20

30 40 ω(Hz)

50

60

10

20

30 40 ω(Hz)

50

60

FT(w3(L))

0.00004 0.00003 0.00002 0.00001 0 0

Figure 5.40  (Continued )

on the horizontal beam. Even though the principal frequency of the loading is quasi‐ static, the dynamic features are developed in both responses in the form of beats and chaotic components. In particular, the chaotic features occur in the case of damaged structure due to impact interaction between the horizontal and vertical beams. In this case, both configuration plane (Figure  5.39) and spectral (Figure  5.40) representations appear to give sufficient information for damage detection. However, in this case, the structural damage cannot be qualified as local because complete separation of the T‐joint creates a significant variation of the distributed stiffness characteristic. Under high excitation frequency of 313.6 Hz, Figures 5.41–5.44 show transient structural dynamics of the T‐joint with the developing damage under high frequency loading. Note that this excitation frequency slightly exceeds the frequency of the second mode of the horizontal beam alone. In this case, tension‐compression deformations of the vertical beam do not accumulate any significant amount of elastic energy during the vibration process, because the T‐joint location coincides with the node of second mode. However, the asymmetry of the second mode involves torsion stiffness of the T‐joint. As a result any damage even on one side of the T‐joint should have some effect on the dynamics near the second vibration mode. In order to verify this prediction, both bolts on the side close to the first strain gauge were relaxed before the test. During the test run, the bolt relaxations continued to grow with some stick‐slip effect between the bolts and the metallic angles. To some extent this process can be viewed as a model of growing cracking damage of the T‐joint. Figure 5.41 shows the dynamic response of modal

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Handbook of Structural Life Assessment

Figure 5.41  Dynamic response of the T‐joint structure on the high‐frequency load (313.6 Hz): left column – healthy structure, right column – progressive damage with bolt ‘cracking relaxation’ on one side of the T‐joint; first row – strain records. (Pilipchuk et al., 2013a)

coordinates of the horizontal beam in two cases: healthy structure (on the left), and progressive damage (on the right). A noticeable effect of the growing damage is the occurrence of non‐zero mean modal components with step‐wise transitions; see the right column of Figure 5.41. A short‐term fragment of the dynamics of the damaged case is shown in Figure 5.42 for a time duration of 0.08 s. In particular, it is seen that the amplitude of the asymmetric mode is most sensitive to damage variations. More information regarding the above dynamic specifics follows from the configuration plane diagrams in Figures 5.43 and 5.44. In particular, the trajectory shapes look like patterns

Statistical Pattern Recognition and Vibration-Based Techniques

Figure 5.42  Short‐term fragment of the high‐frequency dynamic response from the previous series of diagrams. (Pilipchuk et al., 2013a)

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Handbook of Structural Life Assessment

2 × 10–6 2 × 10–6 0

0 w3

w3 – 2 × 10–6 – 2 × 10–6

–0.00005

0 w1

0.00005

–0.00005 0 0.00005 w1

0.00001

w2

w2

0

0.00001 0

–0.00001

–0.00001 –0.00005

0 w1

0.00005

–0.00005 0 0.00005 w1

0.00001

w2

0.00001

0

w2 0

–0.00001

–0.00001 – 2 × 10–6

0 w3

2 × 10–6

– 2 × 10–6 0 2 × 10–6 w3

Figure 5.43  Trajectories of the T‐joint structure under the high‐frequency load (313.6 Hz): left column – healthy structure, right column – ‘damaged’ structure with no bolts on the horizontal beam; frame ticks are preserved for comparison reason. (Pilipchuk et al., 2013a)

Figure 5.44  Short‐term fragment of the trajectories from the previous series of diagrams. (Pilipchuk et al., 2013a)

Handbook of Structural Life Assessment

(a) 0.0004

w

0.0002

0.0000

–0.0002

–0.0004

0.0

0.1

0.2

0.3 x

0.4

0.5

0.6

0.0

0.1

0.2

0.3 x

0.4

0.5

0.6

(b) 0.0004

0.0002

w

354

0.0000

–0.0002

–0.0004

Figure 5.45  Snapshots of the horizontal beam’s centerline during 5 seconds (4.0