Durability of Composite Systems (Woodhead Publishing Series in Composites Science and Engineering) [1 ed.] 0128182601, 9780128182604

Table of contents :
Cover
Durability of Composite Systems
Copyright
Contributors
Introduction
1 - Foundations of modeling of composites for durability analysis
1.1 Introduction to micro–macro modeling with numerical methods
1.1.1 Basic concepts in micro–macro modeling
1.1.2 Historical overview and objectives of this chapter
1.2 Computational grains for particulate composites
1.2.1 Governing equations for 3D elastic heterogeneous materials
1.2.2 Multifield boundary variational principles for 3D computational grain method
1.2.3 Papkovich–Neuber solution
1.2.4 Spherical harmonics
1.2.5 The scaled Trefftz trial functions for the inclusion and matrix
1.2.6 Algorithm for the implementation of computational grains
1.2.7 Validation of computational grains
1.3 Computational grains for fiber composites
1.3.1 Multifield boundary variational principles for fiber composites
1.3.2 Papkovich–Neuber solutions with cylindrical harmonics
1.3.3 Stiffness matrix and algorithmic implementation of computational grains
1.3.4 Validation of computational grains
1.4 Modeling of composites with computational grains
1.4.1 Materials homogenization with computational grains
1.4.2 Parallel computation and direct numerical simulation of composites
1.5 Summary
References
2 - Role of uncertainty in the durability of composite material systems∗
2.1 Durability with uncertainty: engineer's second fundamental problem
2.1.1 What is durability?
2.1.2 What is uncertainty?
2.1.3 What is durability with uncertainty
2.2 Seven types of uncertainty in data and modeling, and eight statistical tools
2.2.1 Three types of data uncertainty, DU1a, DU1b, and DU1c, for a univariate data set
2.2.2 Three types of model uncertainty, MU2a, MU2b, and MU2c, for data modeling
2.2.3 One type of model uncertainty, MU3, for system modeling
2.2.4 Eight statistical tools to estimate seven types of uncertainty
2.3 Data-set uncertainty (DU1a) and data-parameter uncertainty (DU1b)
2.3.1 Data-set uncertainty (DU1a) and Tool-1
2.3.2 Data-parameter uncertainty (DU1b) and Tool-2
2.4 Data-coverage uncertainty (DU1c) from specimen to full-size structures
2.5 Model-function uncertainty (MU2a(f)) for error propagation
2.5.1 Model-function uncertainty (MU2a) and Tool-3 (NUM)
2.5.2 Model-function uncertainty (MU2a) and Tool-4 (error propagation)
2.5.3 Model-function uncertainty (MU2a) and Tool-5 (LLSQ)
2.6 Model-compute uncertainty (MU2b(cf0)) for model verification
2.7 MU2b(cf) and model-physics uncertainty (MU2c(cf))
2.7.1 Model-compute uncertainty (MU2b(cf)) and Tool-7 (DEX)
2.7.2 Model-physics uncertainty (MU2c(cf)) and Tool-7 (DEX)
2.7.3 Model-physics uncertainty (MU2c(cf)) and Tool-8 (ASTM E691)
2.8 Model-system uncertainty (MU3(sf)) for system verification
2.9 Four types of durability with uncertainty for modeling material systems
2.10 Durability with uncertainty (DuU1a) for a smooth simple system under cyclic load without scaling
2.11 Durability with uncertainty (DuU1b) for a cracked simple system under cyclic load without scaling
2.12 Durability with uncertainty (DuU2a) for validating an FRP composite elastic constants database without scaling
2.12.1 A database of elastic constants and thermal expansion coefficients for FRP
2.12.2 Estimation of data-set uncertainty of elastic constants of FRP without scaling
2.12.3 Validation of an FRP composite elastic constants database without scaling
2.13 Durability with uncertainty (DuU2b) for a 2D-holed square composite plate under static load
2.14 DuU3, DuU4, …, DuUn for modeling durability of composite material systems
Appendix A
Statistical analysis software package named DATAPLOT (DP)
What is DATAPLOT?
How to download software and its documentation?
A simple DATAPLOT example
Appendix B
A sample Tool-1 DP code for data uncertainty DU1a
Appendix C
Two sample Tool-2 DP codes for data uncertainty DU1b and DU1c
Appendix D
NIST uncertainty machine (Tool-3) for model uncertainty MU2a
What is NIST Uncertainty Machine?
How to download software and its documentation
A simple NUM example
Instructions
Appendix E
A sample Tool-5 DP code for model uncertainty MU2a
Appendix F
A sample Tool-6 DP code for model uncertainty MU2b
Appendix G
A sample Tool-7 DP code for model uncertainty MU2b, MU2c, and MU3
Disclaimer
References
3 - Durability of aerospace material systems
3.1 Progressive damage analysis by discrete damage modeling
3.1.1 Introduction
3.2 Computational methodology
3.2.1 Static failure analysis
3.2.2 Fatigue failure criterion for MIC insertion
3.2.3 Fatigue cohesive law
3.2.4 Fatigue DDM algorithm
3.2.4.1 Modified fatigue algorithm
3.3 Verification of Rx-FEM coupon-level analysis
3.3.1 DCB analysis
3.3.2 ENF analysis
3.3.3 MMB verification study
3.4 Validation of Rx-FEM subelement-level analysis
3.4.1 Clamped tapered beam—background
3.4.1.1 Static analysis
Experimental observations
Effect of thermal residual stress
Energy dissipation methods
3.4.1.2 Static analysis conclusions
3.4.1.3 Fatigue analysis
CTB fatigue problem statement
Results and comparisons
3.4.1.4 Fatigue analysis conclusions
3.4.2 Three-point bend with flange (3PB-F)
3.4.2.1 Model description
3.4.2.2 Results and discussion
3PB-F Conclusions
3.5 Ply-level constitutive behavior methods
3.6 Machine learning methods
3.7 Chapter conclusions
Acknowledgments
References
4 - Response of composite engineering structures to combined fire and mechanical loading and fatigue durability
4.1 Introduction and background
4.1.1 Marine composites and fire-related design criteria
4.1.2 Wind turbine composites and fatigue-related design criteria
4.2 Mechanistic description of fire and fatigue performance of structural composites
4.2.1 Composites materials subjected to fire conditions
4.2.2 Structural composites materials subjected to combined load and fire conditions
4.3 Fatigue damage of structural composites
4.3.1 Fatigue damage: constant amplitude loading
4.3.2 Fatigue damage: effects of loading frequency
4.3.3 Fatigue damage: spectrum fatigue loading
4.3.4 Fatigue damage: temperature and moisture effects
References
5 - Advanced composite wind turbine blade design and certification based on durability and damage tolerance
5.1 Introduction
5.1.1 Problem statement
5.1.2 Background
5.1.3 Objective
5.2 Methodology
5.2.1 Building block approach (ASTM coupon test standards)
5.2.2 Composite material calibration
5.2.2.1 Continuous fiber
5.2.2.2 Woven fabric composites
5.2.2.3 Nanoenhanced matrix
5.2.3 Multi-scale progressive failure analysis
5.2.4 Fracture mechanics
5.2.4.1 Virtual crack closure technique
5.2.4.2 Discrete cohesive zone modeling
5.2.5 Probabilistic and reliability analysis
5.2.6 Certification approach
5.3 Wind blade design technique and analysis
5.4 Results and discussion
5.4.1 Material modeling calibration and validation
5.4.1.1 Static properties calibration and validation
5.4.2 Tapered blade analysis and results
5.4.2.1 Failure prediction and test validation of tapered composite under static and fatigue loading
Strain energy release rate
Experimentation
Material systems
Simulation results
Static simulation results
5.4.3 Nine meter blade
5.4.3.1 Durability and reliability of wind turbine composite blades using a robust design approach [10]
5.4.3.2 Description of blade FEA model and blade materials
5.4.3.3 Simulation of blade static test
5.4.3.4 Fatigue evaluation of a 9-m wind turbine blade
5.4.3.5 Blade weight analysis
5.4.3.6 Blade durability and damage tolerance probabilistic sensitivity analysis
5.4.3.7 Blade weight reduction with robust design
5.4.3.8 Improving wind blade structural performance with the use of resin enriched with nanoparticles [11]
5.4.3.9 Insertion of silica nanoparticles in a matrix of glass composite
5.4.3.10 D&DTBlade results with glass composite infused with silica nanoparticles
5.4.3.11 Summary
5.4.4 Simulation of a 35-m wind turbine blade under fatigue loading
5.4.5 Conclusion
References
6 - Durability of fiber-reinforced plastics for infrastructure applications
6.1 Introduction
6.2 Application of fiber-reinforced polymer in civil infrastructure
6.2.1 Internal reinforcement (rebar)
6.2.2 External reinforcement
6.3 Environmental conditions
6.4 Durability of composites in aqueous environments
6.5 Durability of composites in subzero and freeze–thaw conditions
6.6 Durability of composites exposed to ultraviolet radiation
6.7 Durability of composites exposed to elevated temperature and fire
6.8 Durability of composites under fatigue loads
6.9 Alkali effects
6.10 Analytical models
6.10.1 Predicting hygrothermal degradation of composites
6.10.2 Prediction of bond strength at elevated temperature
References
7 - Geosynthetics in geo-infrastructure applications
7.1 Introduction and background
7.2 Durability of geosynthetics
7.3 Manufacturing processes
7.4 Infrastructure application areas
7.5 Transportation infrastructure case studies
7.5.1 Case study 1: Geocells for reinforcing base materials
7.5.2 Case study 2: Wicking geotextiles for pavement infrastructure
7.5.3 Case study 3: Geofoam for mitigating bridge approach slab settlements
7.5.4 Case study 4: Slope stability enhancement using fibers
7.6 Summary
Acknowledgments
References
8 - Durability of composite materials for nuclear energy systems
8.1 Introduction
8.2 Mechanical properties of ceramics as pertains to the elemental release
8.3 Corrosion/leaching studies of candidate single-phase and multiphase materials
8.3.1 Corrosion and leaching techniques
8.3.2 PCT product characterization test
8.3.3 MCC-1 monolith test
8.3.4 Vapor hydration test
8.3.5 Corrosion studies of glass waste forms
8.3.6 Corrosion studies of multiphase waste forms
8.3.6.1 SYNROC type waste forms C
8.3.7 Corrosion studies of single-phase waste forms
8.3.7.1 Zirconolite and pyrochlore
8.3.7.2 Perovskite
8.3.7.3 Hollandite
8.3.8 Uranium dioxide
8.4 Radiation effects on surface, mechanical properties, and leaching
8.4.1 Radiation damage processes
8.4.2 Radiation damage process for crystalline structures
8.4.2.1 Frenkel pair defects
8.4.2.2 Electronic defects
8.4.2.3 Volume change
8.4.2.4 Crystalline long-range order amorphization
8.4.3 Radiation induced surface damage
8.4.3.1 Amorphization
8.4.3.2 The effect of volume expansion on the surface
8.4.3.3 How radiation effects mechanical properties
8.4.3.4 How radiation effects leaching
8.5 Morphology drivers for irreversible species diffusion and transport in HeteroFoams
8.5.1 Flux of an included species
References
9 - Work of electrochemical pressurization of a pore in an oxygen ion conducting solid electrolyte and implications concerning ...
9.1 Introduction
9.2 Model
9.2.1 Estimation of strain energy in YSZ
9.2.2 Significance of internal pressurization
9.2.3 The radial displacement
9.2.4 Calculation of strain energy as work done when the pore diameter changes with pressure
9.2.5 Pore pressurization
9.2.6 A simplified derivation assuming a fixed pore radius (applicable at low pressures)
9.2.7 Time required for pressurization
9.2.8 Pore pressurization using a long cylinder and a piston
9.3 Possible experiments
9.4 Summary
Acknowledgments
References
10 - Durability of medical composite systems
10.1 Composites in medical systems
10.1.1 Implantable medical composites
10.1.1.1 Dental composites
10.1.1.2 Composites for organ implant
10.1.1.3 Composites for bone implants
10.1.1.4 Tissue engineered composites
10.1.1.5 Composites for drug delivery
10.1.2 Composites for external medical devices
10.1.2.1 Composite wheelchairs and surgical tools
10.1.2.2 Composites for medical machinery
10.1.2.3 Composites for prosthetic devices
10.1.2.4 Composites for wearable devices
10.2 Properties affecting the durability of medical composite systems
10.2.1 Biocompatibility
10.2.2 Thermal expansion
10.2.3 Elastic modulus and toughness
10.3 Types of failure
10.3.1 Degradation and corrosion
10.3.2 Cavitation
10.3.3 Wear
10.4 Improving durability of medical composite systems
10.5 Closing remarks
References
11- Durability of bonded composite systems
11.1 Introduction
11.2 Types of adhesive bonding
11.3 Theories of adhesive bonding
11.4 Surface treatment method of adhesive bonding
11.5 Surface characterization methods
11.6 Durability and materials state analysis
11.6.1 Environmental/aging
11.6.2 Broadband dielectric spectroscopy for bond material state assessment
11.6.3 Quality assessment of adhesive bonds based on broadband dielectric spectroscopy
11.7 Conclusion
References
12 - Durability of polymer matrix composites fabricated via additive manufacturing
12.1 Introduction
12.2 Approaches to composite additive manufacturing
12.2.1 Short fiber fused filament fabrication
12.2.2 Continuous fiber fused filament fabrication
12.2.2.1 Dual nozzle continuous fiber fused filament fabrication
12.2.2.2 Continuous fused filament fabrication via coextrusion
12.2.2.3 Nonplanar fused filament fabrication
12.3 Techniques for enhancing the durability of composite FFF structures
12.3.1 Voids in printed structures
12.3.1.1 Effect of print parameters on material density
12.3.1.2 Effect of deposition toolpath on material density
12.3.2 Weld strength in composite FFF processing
12.3.3 Wetting and fiber-matrix interfacial bond strength
12.3.4 Effect of printed fiber orientation and distribution
12.3.4.1 Fiber orientation and distribution effects on mechanical properties
12.3.4.2 Fiber orientation around holes
12.3.4.3 Z-pinning
12.4 Engineered composite cellular structures
12.4.1 Open cell composite lattice structures
12.4.2 Closed cell plate lattice structures
12.5 Summary and conclusions
References
Index
A
B
C
D
E
F
G
H
I
L
M
N
O
P
R
S
T
U
V
W
X
Y
Z
Back Cover

Citation preview

Woodhead Publishing Series in Composites Science and Engineering

Durability of Composite Systems

Edited by

Kenneth L. Reifsnider

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2020 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-818260-4 For information on all Woodhead Publishing publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Matthew Deans Acquisitions Editor: Gwen Jones Editorial Project Manager: Joshua Mearns Production Project Manager: Vignesh Tamil Cover Designer: Mark Rogers Typeset by TNQ Technologies

Contributors

Frank Abdi AlphaSTAR Corporation, Long Beach, CA, United States; Imperial College London, London, United Kingdom University of Texas at Arlington (UTA), Fort Worth, TX, United

H.K. Adluru States

Rafal Anay College of Engineering and Computing, University of South Carolina, Columbia, SC, United States Satya N. Atluri

Texas Tech University, Lubbock, TX, United States

AlphaSTAR Corporation, Long Beach, CA, United States

Harsh Baid

Aritra Banerjee Department of Civil and Environmental Engineering, University of Delaware, Newark, DE, United States Kyle S. Brinkman Ye Cao

Clemson University, Clemson, SC, United States

University of Texas Arlington, Arlington, TX, United States

Scott W. Case Reynolds Metals Professor, Via Department of Civil and Environmental Engineering, Virginia Tech, Blacksburg, VA, United States Surya S.C. Congress Zachry Department of Civil and Environmental Engineering, Texas A&M University, College Station, TX, United States Denis Cormier Leiting Dong

Rochester Institute of Technology, Rochester, NY, United States Beihang University, Beijing, China

Amirhossein Eftekharian States

AlphaSTAR Corporation, Long Beach, CA, United

Muthu Ram Prabhu Elenchezhian Mechanical and Aerospace Engineering, University of Texas Arlington (UTA), Arlington, TX, United States James J. Filliben National Institute of Standards and Technology (NIST), Gaithersburg, MD, United States Jeffrey T. Fong National Institute of Standards and Technology (NIST), Gaithersburg, MD, United States Rob Grote

Clemson University, Clemson, SC, United States

x

Contributors

N. Alan Heckert National Institute of Standards and Technology (NIST), Gaithersburg, MD, United States K.H. Hoos

University of Texas at Arlington (UTA), Fort Worth, TX, United States

E.V. Iarve

University of Texas at Arlington (UTA), Fort Worth, TX, United States

John J. Lesko Associate Dean for Research and Graduate Studies and Professor of Mechanical Engineering, Virginia Tech, Blacksburg, VA, United States John J. Myers Missouri University of Science and Technology and the Missouri Center for Transportation Innovation, Rolla, MO, United States Pritam Poddar

Rochester Institute of Technology, Rochester, NY, United States

Anand J. Puppala Zachry Department of Civil and Environmental Engineering, Texas A&M University, College Station, TX, United States Relebohile Qhobosheane Department of Aerospace and Mechanical Engineering, University of Texas Arlington, Arlington, TX, United States Rassel Raihan Mechanical and Aerospace Engineering, University of Texas Arlington (UTA), Arlington, TX, United States Kenneth L. Reifsnider Mechanical and Aerospace Engineering, University of Texas Arlington (UTA), Arlington, TX, United States Wendy Shen Department of Aerospace and Mechanical Engineering, University of Texas Arlington, Arlington, TX, United States Vamsee Vadlamudi University of Texas Arlington Research Institute (UTARI), Fort Worth, TX, United States Anil V. Virkar Department of Materials Science and Engineering, University of Utah, Salt Lake City, UT, United States Paul Ziehl College of Engineering and Computing, University of South Carolina, Columbia, SC, United States

Introduction There are two main elements of any comprehensive statement of a discipline: what are the essential multiple elements of the foundations of the field, and how do they fit together to “give a single heartbeat” to the structure of the discipline. No single volume can hope to provide all of the details of both of those elements, but the substance and structure of this book are designed to serve those two mantras for selected engineering materials and applications. Interestingly, the material in the book is arranged in reverse order. The “heartbeat” is addressed with the most fundamental framework for the subject, the “Foundations for modeling of composites for durability analysis” in Chapter 1 and the “Role of uncertainty in the durability of composite material systems” in Chapter 2. Chapters 3 and 4 follow with discussions of what may be the principle driver of the subject in current society, durability and reliability of aerospace materials and combined mechanical and fire/thermal loading of aerospace, vehicular, and civil structures. Chapter 5 defines the subject for wind turbines, one of the largest free standing composite structures, with growing importance to our society. Chapters 6 and 7 address infrastructure, civil and geoinfrastructure composites that present unique challenges of size, cost, and safety. Chapters 8 and 9 address the broad area of energy conversion and storage, first for the nuclear power industry where the need for reliability and durability is critical for society and then for energy conversion where we are literally at the dawn of a new era. Chapter 10 addresses medical composite systems where the challenges and horizons are remarkably numerous and challengingdperhaps the most multidisciplinary application area of all. Chapter 11 examines bonded systems; it is surprising how many vehicular and other safety critical structures in our lives are “stuck together” with a bond line as a critical element of the component. And Chapter 12 presents what may be the first ever statement of the field for components that are made by additive manufacturing, one of the most rapidly growing areas of technology and science in our society. So what is the “heartbeat” in this complex mixture? What is the “simple takeaway” from our book? Each of us who read (including those of us who wrote) the book will have a different answer to that question. Some candidates include the following. “Durability and Reliability” are defined by “Performance,” i.e., they are defined by what we want/expect a system of materials to do, and by who and what are affected by how well, often, and long the system is able to deliver that performance. So by definition, the subject/discipline must include not only discrete elements of science and math but also many nondiscrete elements

xii

Introduction

of logic, order/disorder, chance, and uncertainty that are just as precisely and rigorously defined and applied. So what is the single theme? Perhaps uniquely, durability and reliability are defined by a unique combination and balance of discrete and nondiscrete rigor that enters every aspect of our economy and our personal livesdas we drive, fly, walk, work, live, and prosper in an ever changing society. On behalf of our group of amazing chapter authors, I extend my welcome to all readers and hope the book helps you to share our fascination and dedication to this subject. Kenneth L. Reifsnider Editor

1

Foundations of modeling of composites for durability analysis

Leiting Dong 1 , Satya N. Atluri 2 1 Beihang University, Beijing, China; 2Texas Tech University, Lubbock, TX, United States

Chapter outline 1.1 Introduction to microemacro modeling with numerical methods

2

1.1.1 Basic concepts in microemacro modeling 2 1.1.2 Historical overview and objectives of this chapter 3

1.2 Computational grains for particulate composites 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7

1.3 Computational grains for fiber composites 1.3.1 1.3.2 1.3.3 1.3.4

5

Governing equations for 3D elastic heterogeneous materials 5 Multifield boundary variational principles for 3D computational grain method 7 PapkovicheNeuber solution 9 Spherical harmonics 9 The scaled Trefftz trial functions for the inclusion and matrix 11 Algorithm for the implementation of computational grains 14 Validation of computational grains 16

18

Multifield boundary variational principles for fiber composites 18 PapkovicheNeuber solutions with cylindrical harmonics 22 Stiffness matrix and algorithmic implementation of computational grains 24 Validation of computational grains 25

1.4 Modeling of composites with computational grains

26

1.4.1 Materials homogenization with computational grains 26 1.4.2 Parallel computation and direct numerical simulation of composites 28

1.5 Summary 33 References 33

Durability of Composite Systems. https://doi.org/10.1016/B978-0-12-818260-4.00001-6 Copyright © 2020 Elsevier Ltd. All rights reserved.

2

Durability of Composite Systems

1.1

Introduction to microemacro modeling with numerical methods

1.1.1

Basic concepts in microemacro modeling

The macroscale properties, macrostress tensor s and macrostrain tensor ε, are defined as the volume average of the microscale stress and strain tensors over the representative volume element (RVE): Z 1 sdU V U Z 1 ε ¼ εdU V U

s ¼

(1.1)

where V is the volume of the RVE. In the absence of body forces, applying the Gauss theorem allows rewriting these two expressions as I 1 t5xdS V vU I 1 1 ðu5n þ n5uÞdS ε ¼ V vU 2 s ¼

(1.2)

The deformation energy at the macroscopic level should be equal to the volume average of microscale stress work. It means that at any equilibrium state of the RVE characterized by the stress field s and the strain field ε, the HilleMandel principle must be satisfied (see Ref. [1]): s: ε ¼

1 V

Z U

s: εdU

(1.3)

From Eqs. (1.1) and (1.2), the Hill-Mandel principle can be rewritten as: I ðt  s$nÞ$ðu  ε$xÞdS ¼ 0

(1.4)

vU

Without loss of generality, the microscopic displacement field on the RVE boundary u can be decomposed into two parts: the mean part u and the zero-mean fluctuation part u: e u ¼ u þ u ¼ ε$x þ u e e

(1.5)

Foundations of modeling of composites for durability analysis

3

These definitions allow one to express the HilleMandel condition in a more convenient form: I vU

t$udS ¼ 0 e

(1.6)

The boundary condition of the RVE for the displacement field and the traction field t should be defined to satisfy Eq. (1.4) or (1.6). Three types of boundary conditions are generally considered: 1. Constant traction boundary condition: the traction field at the boundary of the RVE vV is prescribed in terms of the macroscopic stress:

t ¼ s$n

cx ˛ vU

(1.7)

2. Linear displacement boundary condition: the displacement field at the boundary of the RVE is constrained in terms of the macroscopic strain:

u ¼ u  ε$x ¼ 0 e

cx ˛ vU

(1.8)

3. Periodic boundary conditions: at the boundary of the RVE, vU is decomposed into two parts: a positive part vUþ and a negative part vU with vUþ W vU ¼ vU, vUþ X vU ¼ B, and with the associated outward normal nþ ¼ n at corresponding points xþ ˛ vUþ and x ˛ vU , respectively. The periodic boundary condition represents the periodicity of fluctuations field and antiperiodicity of the traction field on the RVE boundary:

    u xþ ¼ u x cxþ ˛ vUþ and matching x ˛ vU e e     t xþ ¼ t x cxþ ˛ vUþ and matching x ˛ vU

(1.9)

For elastic materials, the average tangent moduli Ceff can be solved by the linear relation of the macrostress tensor s and the macrostrain tensor ε: s ¼ Ceff : ε

1.1.2

(1.10)

Historical overview and objectives of this chapter

There are several widely used analytical tools to predict the overall properties of heterogeneous materials. For example, Hashin and Shtrikman (1963) developed variational principles to estimate the upper and lower bounds of the elasticity or compliance tensor. Hill (1965) developed a self-consistent approach to estimate the homogenized material properties. Analytical methods have their unique value in the study of micromechanics. However, because most of these methods follow the work of Eshelby (1957), namely the elastic field of an ellipsoidal inclusion in an infinite

4

Durability of Composite Systems

medium, these methods can neither consider complex material structures with random distribution of particles, nor give the microscale stress concentrations caused by their interactions. The need for predicting the overall properties of a material with complex geometry, distribution, and arbitrary volume fraction of inclusions promoted the development of computational tools for micromechanics. A popular way of doing this is to use finite elements to model an RVE. By its concept, an RVE is a microscopic material volume, which is statistically representative of the infinitesimal material neighborhood of the macroscopic material point of interest. By modeling simple loading cases of the RVE, the microscopic stress field and strain field in the RVE can be computed by the finite element method. And the homogenized material properties are calculated by relating the macroscopic (average) stress tensor to the macroscopic (average) strain tensor. The finite element method and asymptotic homogenization theory were also combined to perform multiscale modeling of structures composed of heterogeneous materials. However, it is well known that primal finite elements, which involve displacement type of nodal shape functions, are highly inefficient for modeling stress concentration problems. Accurate computation of the fields around a single inclusion or fiber may need many thousands of elements. For example, the mesh of an Al/SiC particulate composite unit cell model typically consists of around 76,000 10-node tetrahedral elements with ABAQUS in the study of Chawla, Ganesh [2]. Moreover, meshing of an RVE, which contains a large number of inclusions/fibers, can be human-labor intensive. For the expensive burden of computation as well as meshing, the aforementioned computational models mostly use a unit cell as the RVE, assuming the microstructure of material is strictly periodic. This obviously cannot account for the complex shapes and distributions of materials of different phases. To reduce the burden of computation and meshing, the authors proposed the idea of computational grains (CGs). Each CG can include an inhomogeneity in it, and the microstructures are discretized with several CGs. Compared with FEM, the CG method needs no fine meshes and can directly model the microstructures. CGs can save several orders of magnitude of computational burden and, in addition, precisely capture the stress concentrations (which enable the prediction of damage precursors at interfaces) around microstructural fibers/inclusions much more accurately than the usual FEM. With CGs, virtual design and virtual testing of composites containing a large amount of inhomogeneities can be carried out, so composites can be designed, tested, and optimized in silico before they are actually manufactured in the laboratory. This chapter is aimed at presenting in detail high-performance computational tools for the modeling of composite materials using CGs, which can directly simulate the micro- and macroscale mechanical behaviors of composite materials efficiently and accurately. It systematically introduces state-of-the-art computational methods to develop a high-performance computational tool with a special focus on the CG method for micromechanical analysis. In the current state of the art, the method of CGs to model the microstructure can be used in conjunction with multiscale modeling, to perform a two-level analysis of the microstructure first to compute

Foundations of modeling of composites for durability analysis

5

the macrostructural properties, and then to perform the macrostructural analysis using structural finite elements. In reverse, a macrostructural analysis may be performed first using structural finite elements, and then a local microstructural analysis using CGs may be performed, to compute the microstructural interfacial stresses accurately, to predict microstructural damage. With advances in petascale and quantum computing, in the near future, it will be possible to model entire structural components using the presently described micromechanical CGs directly rather than using the currently popular structural finite elements. Thus, damaged precursors at the microlevel in a structural component can easily be detected, and the life of a structural component can be more precisely predicted.

1.2

Computational grains for particulate composites

1.2.1

Governing equations for 3D elastic heterogeneous materials

A typical representative material or volume element (RVE) of 3D elastic particulate composites is shown in Fig. 1.1a. Details about algorithms to construct random microstructures and to form the RVE can be found in Ref. [3]. Here, we just use the algorithms to build the RVE and focus on the CG method itself. Using the meshing technology such as Voronoi Diagram, the RVE is discretized into virtual polyhedral CG elements based on the location and size of heterogeneities, as illustrated in Fig. 1.1b. In each CG, a spherical void/elastic inclusion can be included. Fig. 1.1c is a polyhedral CG with a spherical inhomogeneity. The solutions of 3D linear elasticity for the matrix as well as inclusions in each CG should satisfy the following equations of the equilibrium equation, the compatibility equation, as well as the constitutive relations (assuming that both the matrix and the inclusion are linear elastic and undergo only small deformations): V $ sk þ f k ¼ 0

(1.11)

1  k  k T  Vu þ Vu 2

(1.12)

  sk ¼ lk tr εk I2 þ 2mk εk

(1.13)

εk ¼

where the superscript k ¼ m denotes the matrix material, and k ¼ c denotes multiple inclusions. uk , εk , and sk are respectively, the displacements, strains, and stresses in the matrix and in the inclusions. f k is the body force, which can be neglected for micromechanics of composites. V$ and V are the divergence and gradient operators. nk Ek Ek and mk ¼ are Lamé constants for matrix/inclusions, lk ¼ k k ð1  2n Þð1 þ n Þ 2ð1 þ nk Þ where Ek and nk are the Young’s modulus and Poisson’s ratio. I2 is the 3D second order unit tensor.

6

Durability of Composite Systems

(a)

(b)

(c) Set

∂ Ωe ∂ Ωec Ωem

Ωec

Figure 1.1 (a) An illustration of a matrix containing multiple inhomogeneities; (b) the RVE after tessellation into virtual CGs; (c) a representative polyhedral CG with an inhomogeneity. CGs, computational grains; RVE, representative volume element.

Foundations of modeling of composites for durability analysis

7

The interface conditions can be written as: um ¼ uc

at vUec ½matrix=inclusion interface

nm $ sm þ nc $sc ¼ 0

at vUec ½matrix=inclusion interface

(1.14) (1.15)

The boundary conditions can be written as: um ¼ u

at Seu ½b.c on a CG

nm $ s m ¼ t

at Set ½b.c on a CG

(1.16) (1.17)

where u and t are the prescribed boundary displacements and boundary tractions at the displacement boundary Seu and the traction boundary Set of the domain Ue; respectively. In addition to the equations above, displacement continuity and traction reciprocity should be satisfied between neighboring CGs.

1.2.2

Multifield boundary variational principles for 3D computational grain method

In this subsection, multifield boundary variational principles for the CG method are presented in detail. In the CG method, the constitutive equations, compatibility equations, and equilibrium equations are all a priori satisfied by the assumed Trefftz functions for displacement fields in the matrix and the heterogeneity individually, whereas the interface/boundary conditions are to be satisfied in a weak sense from the stationarity conditions of a scalar functional. The detailed discussion is given in the following. To develop CGs, we consider independently assumed displacement fields in the interior of each phase, which is a linear combination of complete Trefftz trial functions as discussed in Section 1.3.2. That is to say, the displacement fields satisfy the Navier’s equation in the matrix/inclusion individually: V2 uk þ

  1 V V $ uk ¼ 0; k ¼ m or c k 1  2n

(1.18)

Another set of displacement e um , which satisfies the inter-CG displacement continuity and essential boundary conditions a priori, is introduced: e um ¼ u

at Seu

(1.19)

Then the inter-CG traction reciprocity as well as the matrix/heterogeneity interface conditions can be satisfied in a weak form by the condition of stationarity of the boundary functional P1 (note that P1 is a boundary-only functional, since the

8

Durability of Composite Systems

equilibrium, compatibility, and constitutive equations are all satisfied by the Trefftz functions uk ; k ¼ m or c): m

u ;u ;u Þ¼ P1 ðe m

c

X

( Z 

vUe

e

þ

( X Z e

1 m m t $ u dS þ e2 þ vUc

Z vUem

Z

1 c c t $ u dS t $ u dS þ vUec vUec 2 m

)

Z t $e u dS  m

m

)

t$e u dS m

Set

c

(1.20) which leads to EulereLagrange equations: tm ¼ t

at Set ½traction b.c. on a CG

um ¼ e um

at vUe ½inter CG compatibilitiy

tmþ þ tm ¼ 0 u ¼u m

c

at

at re ½inter CG traction reciprocity

tm þ tc ¼ 0

(1.21)

½matrix=inclusion interface compatibility

vUec

at vUec ½matrix=inclusion traction reciprocity

ec is merely When the element includes a void instead of an elastic inclusion, u e assumed independently at vUc , and we use the following variational principle: u ; um ; e u Þ¼ P2 ðe m

c

X e

þ

( Z 

vUe

XZ e

vUec

1 m m t $ u dS þ e2 þ vUc

Z vUem

)

Z tm $ e u dS  m

t$e u dS m

Set

tm $ e uc dS (1.22)

which leads to EulereLagrange equations: tm ¼ t

at Set

um ¼ e um

at vUe

tmþ þ tm ¼ 0 um ¼ e uc tm ¼ 0

at vUec at vUec

at re

(1.23)

Foundations of modeling of composites for durability analysis

1.2.3

9

PapkovicheNeuber solution

For 3D isotropic elasticity, the general solutions in the matrix and in the inclusion can be expressed by using the PapkovicheNeuber solution (see Ref. [4]):   u ¼ 4ð1  nÞB  VðR $ B þ B0 Þ = 2m  ¼ 4ð1  nÞB  R $ ðVBÞT  VB0 = 2m

(1.24)

B0 ; B are scalar and vector harmonic functions, which are sometimes called PapkovicheNeuber potentials. The second equation in Eq. (1.24) can be written in the following index form:   ui ¼ ð3  4nÞBi  xk Bk;i  B0;i =2m

(1.25)

An interesting fact of the 3D PapkovicheNeuber solution is that the 3D displacements as in Eq. (1.24) have a very similar form to the displacements in 2D expressed in terms of complex potentials, as shown in Ref. [5]. However, unlike the approach of complex potentials, for a specific displacement field in 3D, the harmonic potentials have high degrees of freedom. That is to say, there may exist many different sets of B0 ; B1 ; B2 ; B3 , which are harmonic potentials of the same specific displacement field. This renders one to think about if it is possible to drop the scalar harmonic function, to express the solution as: u ¼ ½4ð1  nÞB  VR $ B=2m

(1.26)

It was proved in Ref. [4] that Eq. (1.26) is complete for the infinite region, which is external to a closed surface for any v. However, for a simply connected domain, Eq. (1.26) is complete only when n s 0:25. By expressing B0 to be a specific function of B, it is shown that a specific case of the PapkovicheNeuber solution only is: u ¼ ½4ð1  nÞB þ R $ VB  RV $ BÞ=2m

(1.27)

which is complete for a simply connected domain, for any n. For detailed discussions of the completeness of the PapkovicheNeuber solution (see Ref. [4]).

1.2.4

Spherical harmonics

As shown in the previous section, B in Eqs. (1.26) and (1.27) is a vector that should satisfy the Laplace equation. For spherical inclusion/void problems discussed in this chapter, we express B as a linear combination of spherical harmonics. In this subsection, we give a brief introduction to spherical harmonics. For detailed discussions, one can refer to the monograph [6].

10

Durability of Composite Systems

See the spherical coordinates as shown in Fig. 1.2. For the internal problem of a sphere, (see Fig. 1.3a), l, which satisfy the Laplace equation, can be expanded as: lp ¼

N X

( R

n

a00 YC00 ðq; 4Þ þ

n¼0

n  X  m m m am n YCn ðq; 4Þ þ bn YSn ðq; 4Þ

) (1.28)

m¼1

For external problems in a finite domain, a hollow sphere, for instance, see Fig. 1.3b, l can be expanded as: lp þ lk ¼

N X

( c00 YC00 ðq; 4Þ þ

n

R

n¼0

þ

N X

( R

ðnþ1Þ

n¼0

n  X

m m m cm n YCn ðq; 4Þ þ dn YSn ðq; 4Þ

) 

m¼1

e00 YC00 ðq; 4Þ þ

n  X

m m m em n YCn ðq; 4Þ þ fn YSn ðq; 4Þ

) 

m¼1

(1.29)

Figure 1.2 Spherical coordinates.

Figure 1.3 Illustrations of typical domains for the Laplace equation: (a) a sphere; (b) a finite domain external to a sphere.

Foundations of modeling of composites for durability analysis

11

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n þ 1 ðn  mÞ! m P ðcosðqÞÞeim4 Ynm ðq; 4Þ ¼ 4p ðn þ mÞ! n sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n þ 1 ðn  mÞ! m P ðcosðqÞÞ½cosðm4Þ þ i sinðm4Þ ¼ 4p ðn þ mÞ! n

(1.30)

¼ YCnm ðq; 4Þ þ iYSm n ðq; 4Þ Fig. 1.4 gives visual representations of the first few real spherical harmonics. Yellow portions represent regions where the function is positive, and blue portions represent where it is negative. The distance of the surface from the origin indicates the absolute value of Ynm ðq; 4Þ in angular direction.

1.2.5

The scaled Trefftz trial functions for the inclusion and matrix

Now we can express the Trefftz trial functions used in the CG method in terms of the PapkovicheNeuber solution, in which spherical harmonics are employed. For the inclusion, the displacement field uc can be derived by substituting the nonsingular harmonics: Bcp ¼

pc n X R n¼0

Rc

( a00 YC00 ðq; 4Þ þ

n h X

aln YCnl ðq; 4Þ þ bln YSln ðq; 4Þ

) i (1.31)

l¼1

into Eq. (1.27): h i uc ¼ ucp ¼ 4ð1  nc ÞBcp þ R $ VBcp  RV $ Bcp =2mc

Figure 1.4 Illustrations of spherical harmonics.

(1.32)

12

Durability of Composite Systems

where the superscript cdenotes the inclusion, and aln and bln are the unknown coefficients in b (Eq. 1.38) and g (Eq. 1.44). The displacement field um in the matrix is the summation of um p (the nonsingular part) and um (the singular part, with the singularity located at the center of the k inclusion). um can be derived by substituting p Bm p ¼

)

n ( pm n h i X X R 0 0 l l l l cn YCn ðq; 4Þ þ dn YSn ðq; 4Þ c0 YC0 ðq; 4Þ þ Rm n¼0 l¼1

(1.33)

into Eq. (1.27), and um k can be derived by substituting Bm k ¼

km ðnþ1Þ X R n¼0

Rc

( e00 YC00 ðq; 4Þ þ

) n h i X l l l l en YCn ðq; 4Þ þ f n YSn ðq; 4Þ l¼1

(1.34) into Eq. (1.26): m um ¼ um p þ uk

h i m m m um =2mm p ¼ 4ð1  vÞBp þ R$VBp  RV$Bp

(1.35)

  m m m um k ¼ 4ð1  vÞBk  VR$Bk =2m where cln , dln , eln , f ln are the unknown coefficients in a (Eq. 1.37). Rm and Rc are the characteristic lengths introduced to scale the Trefftz trial functions; thereby the difficult algebraic problem of solving ill-conditioned systems of equations can be avoided. For this problem, Rm is the maximum radial distance of points in the matrix, n R and Rc is the radius of the inclusion, see Fig. 1.5. Therefore, and Rm n ðnþ1Þ R R are confined between 0 and 1 for any point in the matrix, and is Rc Rc confined between 0 and 1 for any point in the inclusion. We illustrate the reason why we use characteristic lengths to scale the Trefftz trial functions. See Fig. 1.6 for the geometry of the element. Material properties of the matrix are Em ¼ 1; vm ¼ 0:25. Two kinds of heterogeneities are considered: an elastic inclusion with Ec ¼ 2; vc ¼ 0:3, and a void. Stiffness matrices of the CG are computed, with and without using characteristic lengths to scale Trefftz trial functions. Condition number of coefficient matrices Haa in Eqs. (1.41) and (1.45) used to relate to with/without using characteristic lengths to scale Trefftz trial functions for the element is shown in Table 1.1. We can clearly see that by scaling the Trefftz functions using characteristic lengths, the resulting Haa have significantly smaller condition numbers.

Foundations of modeling of composites for durability analysis

13

Rm Rc

Figure 1.5 Illustrations of the characteristic lengths.

250 200 150 100 50 0 –50 –100 100

100

0

0 –100 –100

Figure 1.6 An element with a spherical inclusion/void used for the condition number test.

Table 1.1 Condition number of coefficient matrices Haa used to relate a; b; g to q. Elastic inclusion Characteristic length

Scaled

Condition number

1.08  10

3

Void

Not scaled

Scaled

3.7  10

2.8  10

35

Not scaled 3

2.6  1035

14

Durability of Composite Systems

1.2.6

Algorithm for the implementation of computational grains

em is assumed at vUe with Wachspress The inter-CG compatible displacement field u coordinates as nodal shape functions. Fig. 1.7 shows the Wachspress coordinates for one node of a regular pentagon, which may be the surface of a polyhedral CG. Using matrix and vector notations, we have: e mq e um ¼ N

at vUe

(1.36)

The displacement field in Uem and its corresponding traction field tm at vUem ; vUec are: um ¼ Nm a

in Uem

tm ¼ Rm a

at vUem ; vUec

(1.37)

It should be noted that Nm and Rm are Trefftz trial function matrices, which were introduced in the previous sections. When an elastic inclusion is to be considered, the Trefftz displacement field in the inclusion is independently assumed. We have: uc ¼ Nc b

in Uec

tc ¼ Rc b

at vUec

(1.38)

1 0.8 0.6 0.4 0.2 0 1 0.5 0 –0.5 –1 –1

–0.5

0

Figure 1.7 Wachspress coordinates as nodal shape functions.

0.5

1

Foundations of modeling of composites for durability analysis

15

Therefore, finite element equations can be derived using the following three-field boundary variational principle (Eq. 1.20), which can be written in the matrix form: P1 ða; b; qÞ ¼

X  aT Haa a þ 1=2aT Gaq q  qT Q þ aT Gab b e

þ 1=2b Hbb b T



(1.39)

This leads to finite element equations: ( )T 2 GT H1 G q aq aa aq 4 d b GTab H1 aa Gaq

GTaq H1 aa Gab GTab H1 aa Gab þ Hbb

3( ) ( )T ( ) Q q q 5 ¼d 0 b b (1.40)

where Z Gaq ¼

vUe

e m dS RTm N

Z Gab ¼

vUe

RTm Nc dS

Z Haa ¼ Z Hbb ¼ Z Q¼

Set

vU

e

vUec

þvUec

RTm Nm dS

(1.41)

RTc Nc dS

e T tdS N m

This equation can be further simplified by static condensation: X

  T k11  k12 k1 22 k12 q  Q ¼ 0

(1.42)

e

where k11 ¼ GTaq H1 aa Gaq k12 ¼ GTaq H1 aa Gab k22 ¼ GTab H1 aa Gab þ Hbb

(1.43)

16

Durability of Composite Systems

ec is merely When the element includes a void instead of an elastic inclusion, u e assumed at vUc . We have: e cg e uc ¼ N

at vUec

(1.44)

We use the following variational principle (Eq. 1.20), which can be written in the matrix form: P2 ða; q; gÞ ¼

X  aT Haa a þ 1=2aT Gaq q  qT Q þ aT Gag g

(1.45)

e

And corresponding finite element equations are: ( )T 2 GT H1 G q aq aa aq 4 d g GTag H1 aa Gaq

3( ) ( )T ( ) GTaq H1 aa Gag q q Q 5 ¼d g g 0 GTag H1 aa Gag

(1.46)

where Z Gag ¼

vUec

e c dS RTm N

(1.47)

Similarly, this equation can be further simplified by static condensation.

1.2.7

Validation of computational grains

To evaluate the overall performances of mathematical CGs for modeling problems with inclusions or voids, we consider the following problems: an infinite medium with a spherical inclusion or void in it. Exact solutions can be obtained using Eshelby’s solution and the equivalent inclusion method. For details of the exact solution, see Ref. [1]. The material properties of the matrix are Em ¼ 1; nm ¼ 0:25. When an elastic inclusion is considered, the material properties of the inclusion are Ec ¼ 2; nc ¼ 0:3. The magnitude of the remote tension P in the direction of x3 is equal to 1. The radius of the inclusion/void is 1. For numerical implementation, the infinite medium is truncated to a finite cube, as illustrated in Fig. 1.8. The length of each side of the truncated cube is equal to 10. For both the two cases with an elastic inclusion or a void, only one CG is used in the computational model. Traction boundary conditions are applied to the outer boundary of the element. For each CG, the displacements of each node on the lower surface are constrained to be the same as the exact solutions. First, we compute the eigenvalues of the stiffness matrices of a single CG. This is done in the original and rotated global Cartesian coordinate system. Numerical results are shown in Table 1.2.

Foundations of modeling of composites for durability analysis

17

Figure 1.8 A spherical elastic inclusion or void under remote tension.

As can clearly be seen, the CG is stable and invariant, because additional zero energy modes do not exist, and eigenvalues do not vary with respect to the change of coordinate systems. We also compare the computed s11 along axis x3 , s33 along axis x1 , to that of the exact solution. As shown in Figs. 1.9 and 1.10, no matter an elastic inclusion or a void is considered, and CGs always give very accurate computed stresses, even though only one CG is used. Table 1.2 Eigenvalues of stiffness matrices of different computational grains when an elastic inclusion/void is considered. Eigenvalues Rotation [ 0 and 45 degrees

Inclusion

Void

1

2.3383

2.2681

2

1.0878

1.0640

3

1.0805

1.0564

4

0.8408

0.8632

5

0.8408

0.8241

6

0.8349

0.8241

7

0.6267

0.6278

8

0.6161

0.6113

9

0.6002

0.5963

10

0.6002

0.5963 Continued

18

Durability of Composite Systems

Table 1.2 Eigenvalues of stiffness matrices of different computational grains when an elastic inclusion/void is considered.dcont’d Eigenvalues Rotation [ 0 and 45 degrees

Inclusion

Void

11

0.5530

0.5707

12

0.5408

0.5707

13

0.5408

0.5602

14

0.5029

0.5029

15

0.2984

0.2994

16

0.2511

0.2518

17

0.2511

0.2518

18

0.1915

0.1978

19

0.1915

0.1978

20

0.1692

0.1726

21

0.1219

0.1257

22

0.0000

0.0000

23

0.0000

0.0000

24

0.0000

0.0000

25

0.0000

0.0000

26

0.0000

0.0000

27

0.0000

0.0000

1.3

Computational grains for fiber composites

A typical RVE of 3D elastic fiber composites is shown in Fig. 1.11a. Details about algorithms to construct random microstructures and to form the RVE can be found in Ref. [3]. Here we just use the algorithms to build the RVE and focus on the CG method itself. Using the meshing technology presented in Ref. [7], the RVE is discretized into virtual polyhedral CG elements by tessellation methods that are based on the location and size of heterogeneities, as illustrated in Fig. 1.11b. In each CG, an elastic fiber can be included. Fig. 1.11c is a polyhedral CG with a fiber.

1.3.1

Multifield boundary variational principles for fiber composites

In this subsection, multifield boundary variational principles for the CG method are presented in detail. In the CG method, the constitutive equations, compatibility

Foundations of modeling of composites for durability analysis

19

0.2

Analytical solution CGM

0.15

σ 11

0.1

0.05

0

–0.05

0

2

4

1.4

x3

6

8

10

Analytical solution CGM

1.3 1.2

σ 33

1.1 1 0.9 0.8 0.7 0.6

0

2

4

x1

6

8

10

Figure 1.9 Computed s11 along axis x3 , s33 along axis x1 for the problem with an elastic inclusion.

equations, and equilibrium equations are all a priori satisfied by the assumed Trefftz functions for displacement fields in the matrix and the fiber individually, whereas the interface/boundary conditions are to be satisfied in a weak sense from the stationarity conditions of a scalar functional. The detailed discussion is given later. Another set of displacement e u, which satisfies the inter-CG displacement continuity and essential boundary conditions a priori, is introduced: e u¼u

at Seu

(1.48)

20

Durability of Composite Systems 0.7

Analytical solution CGM

0.6 0.5

σ 11

0.4 0.3 0.2 0.1 0 –0.1

1

2

3

4

5

x3

6

7

1

8

9

10

Analytical solution CGM

0.8

σ 33

0.6

0.4

0.2

0

–0.2

1

2

3

4

5

x1

6

7

8

9

10

Figure 1.10 Computed s11 along axis x3 , s33 along axis x1 for the problem with a void.

Then the inter-CG traction reciprocity as well as the matrix/heterogeneity interface conditions can be satisfied in a weak form by the condition of stationarity of the boundary functional P3 (note that P3 is a boundary-only functional, since the equilibrium, compatibility, and constitutive equations are all satisfied by the Trefftz functions uk ; k ¼ m or f):   XZ e; um ; e P3 u uf ¼

Z Z 1 m m t $ u dS  tm $ e udS þ t $ e udS vUm St e vUm 2 Z Z 1 f f t $ u dS  þ tf $ e udS vUf 2 vUf

(1.49)

Foundations of modeling of composites for durability analysis

21

Figure 1.11 (a) An illustration of a matrix containing multiple inhomogeneities; (b) the RVE after tessellation into virtual CGs; (c) a representative polyhedral CG with an inhomogeneity. CGs, computational grains; RVE, representative volume element.

22

Durability of Composite Systems

which leads to EulereLagrange equations: tm ¼ t

at Set

tf ¼ t

at Set

um ¼ e u

at vUem

uf ¼ e u

at vUef

tmþ þ tm ¼ 0 tm þ tf ¼ 0

(1.50) at re

at Ge

where Ge is the interface of the matrix and the fiber.

1.3.2

PapkovicheNeuber solutions with cylindrical harmonics

In this subsection, we give a brief introduction to the Trefftz trial functions used in the CG method. For composites containing cylindrical fibers, the Trefftz trial functions are expressed in terms of the PapkovicheNeuber solution, in which cylindrical harmonics are employed. For the fibers, as shown in Fig. 1.11c, the displacement field uf can be derived by substituting the nonsingular harmonics:

Bfp ¼

8 pf1 < a r l cosðmqÞ þ a r l sinðmqÞ X 1l 2l l¼1

þ

:

þ a3l zr l cosðmqÞ þ a4l zr l sinðmqÞ

9 = ;

( ) pf2 b1l coshðnzÞJ0 ðnrÞ þ b2l sinhðnzÞJ0 ðnrÞ X þ b3l cosðnzÞI0 ðnrÞ þ b4l sinðnzÞI0 ðnrÞ 8 9 c1ln cosðnqÞcoshðlzÞJn ðlrÞ þ c2ln sinðnqÞcoshðlzÞJn ðlrÞ > > > > > > > > > > > f f > > > p p < = þ c cosðnqÞsinhðlzÞJ ðlrÞ þ c sinðnqÞsinhðlzÞJ ðlrÞ 3 4 n n X X 3ln 4ln l¼1

þ

> > > > :

l¼1 n¼1 > > > þ c5ln

cosðnqÞcosðlzÞIn ðlrÞ þ c6ln sinðnqÞcosðlzÞIn ðlrÞ

þ c7ln cosðnqÞsinðlzÞIn ðlrÞ þ c8ln sinðnqÞsinðlzÞIn ðlrÞ

> > > > > > > ; (1.51)

into: h  i  uf ¼ ufp ¼ 4 1  nf Bfp þ R $ VBfp  RV $ Bfp =2mf

(1.52)

Foundations of modeling of composites for durability analysis

23

The displacement field um in the matrix is the summation of um p (the nonsingular part) and um (the singular part, with the singularity located at the center of the incluk sion). um can be derived by substituting: p 8 l l pm 1 < e1l r cosðmqÞ þ e2l r sinðmqÞ X

Bm p ¼

l¼1

:

þ e3l zr l cosðmqÞ þ e4l zr l sinðmqÞ

9 = ;

( ) pm f 1l coshðnzÞJ0 ðnrÞ þ f 2l sinhðnzÞJ0 ðnrÞ 2 X

þ

þ f 3l cosðnzÞI0 ðnrÞ þ f 4l sinðnzÞI0 ðnrÞ 8 9 g1ln cosðnqÞcoshðlzÞJn ðlrÞ þ g2ln sinðnqÞcoshðlzÞJn ðlrÞ > > > > > > > > > > > m m > > p p < = þ g3ln cosðnqÞsinhðlzÞJn ðlrÞ þ g4ln sinðnqÞsinhðlzÞJn ðlrÞ > 3 4 X X l¼1

þ

> > > > > :

l¼1 n¼1 > >þ

g5ln cosðnqÞcosðlzÞIn ðlrÞ þ g6ln sinðnqÞcosðlzÞIn ðlrÞ

þ g7ln cosðnqÞsinðlzÞIn ðlrÞ þ g8ln sinðnqÞsinðlzÞIn ðlrÞ

> > > > > > > ; (1.53)

into: h i m m m m m um ¼ 4ð1  v ÞB þ R $ VB  RV $ B p p p p =2m

(1.54)

and ukm can be derived by substituting Bm k ¼

8 l km 1 < h1l r cosðlqÞ þ h2m r l sinðlqÞ X l¼1

þ

:

þ h3l zr l cosðlqÞ þ h4l zr l sinðlqÞ

9 = ;

( ) km i1l coshðlzÞY0 ðlrÞ þ i2l sinhðlzÞY0 ðlrÞ 2 X þ i3l cosðlzÞK0 ðlrÞ þ i4l sinðlzÞK0 ðlrÞ 8 9 j1ln cosðlqÞcoshðnzÞYl ðnrÞ þ j2ln sinðlqÞcoshðnzÞYl ðnrÞ > > > > > > > > > > > m m > > > k k < = þ j cosðlqÞsinhðnzÞY ðnrÞ þ j sinðlqÞsinhðnzÞY ðnrÞ 3 4 X X l l 3ln 4ln l¼1

þ

> > > > :

> cosðlqÞcosðnzÞKl ðnrÞ þ j6ln sinðlqÞcosðnzÞKl ðnrÞ > > > > > > ; þ j7ln cosðlqÞsinðnzÞKl ðnrÞ þ j8ln sinðlqÞsinðnzÞKl ðnrÞ (1.55)

n¼1 l¼1 > > > þ j5ln

24

Durability of Composite Systems

into:   m m m um k ¼ 4ð1  vÞBk  VR $ Bk =2m

(1.56)

Then the displacement field um in the matrix can be expressed as: m um ¼ um p þ uk

(1.57)

where Jn , In , Yn , Kn are Bessel functions.

1.3.3

Stiffness matrix and algorithmic implementation of computational grains

For the CGs containing fibers, the inter-CG compatible displacement field e u is assumed at vUe . Using matrix and vector notations, we have: e e u ¼ Nq

at vUe

(1.58)

The displacement field in Uem and its corresponding traction field tm at vUem as: um ¼ Nm a

in Uem

tm ¼ Rm a

at vUem

(1.59)

It should be noted that Nm and Rm are Trefftz trial function matrices. The Trefftz displacement field in the fiber is independently assumed. We have: uf ¼ Nf b

in Uef

tf ¼ Rf b

at vUef

(1.60)

Therefore, finite element equations can be derived using the following three-field boundary variational principle (Eq. 1.49), which can be written in the matrix form: P3 ða; b; qÞ ¼

X 1 e



1 a Haa a  a Gaq q þ q Q þ bT Hbb b  bT Gbq q 2 2 T

T

T

(1.61) This leads to finite element equations: X e

  T 1 e GTaq H1 aa Gaq þ G bq Hbb Gbq q  Q ¼ 0

(1.62)

Foundations of modeling of composites for durability analysis

25

where Z Gaq ¼ Z Gbq ¼

vUem

vUef

e RTm NdS

e RTf NdS

Z Haa ¼ Z Hbb ¼ Z Q¼

1.3.4

vUem

vUef

RTm Nm dS

(1.63)

RTf Nf dS

e T tdS N

Set

Validation of computational grains

To evaluate the overall performances of mathematical CGs for modeling problems with fibers, we consider the following problems: an infinite medium with a cylindrical fiber in it. Exact solutions can be obtained using Eshelby’s solution and the equivalent inclusion method. For details of the exact solution, see Ref. [1]. The material properties of the matrix and the fiber are Em ¼ 1; nm ¼ 0:25, Ef ¼ 2; nf ¼ 0:3, respectively. The magnitude of the remote tension P in the direction of x2 is equal to 1. The radius of the fiber is 1. For numerical implementation, the infinite medium is truncated to a finite cube, as illustrated in Fig. 1.12. The length of each side of the truncated cube is equal to 100. For this example, only one CG is used in the

Figure 1.12 A cylindrical fiber under remote tension.

26

Durability of Composite Systems 0.25

Analytical solution CGM

σ 11 (MPa)

0.2

0.15

0.1

0.05

0

–0.05

0

2

4

x2

6

1.2

8

10

Analytical solution CGM

1.1 1

σ 22

0.9 0.8 0.7 0.6 0.5

0

2

4

x1

6

8

10

Figure 1.13 Computed s11 along axis x2 , s22 along axis x1 for the problem with a fiber.

computational model. Traction boundary conditions are applied to the outer boundary of the element. For each CG, the displacements of each node on the lower surface are constrained to be the same as the exact solutions. We compare the computed s11 along axis x2 , s22 along axis x1 , to that of the exact solution. As shown in Fig. 1.13, CGs always give very accurate computed stresses, even though only one CG is used.

1.4 1.4.1

Modeling of composites with computational grains Materials homogenization with computational grains

The mesh of an RVE using the CG method is generally nonperiodic so that periodic boundary conditions cannot be directly enforced in a node-to-node fashion.

Foundations of modeling of composites for durability analysis

27

5 4

1 A– 3

2

A+

Figure 1.14 Illustrations of imposing periodic boundary conditions.

Thus, here we illustrate how the periodic boundary conditions are enforced for CGs without periodic meshing. Fig. 1.14 is a schematic diagram for an RVE after tessellation. For each node Aþ at vUþ , there is a corresponding point A at vU by projection. One can locate A in a specific polygonal surface of one CG at vU . And thus, by using the nodal interpolation function of the Wachspress coordinates, the first equation in Eq. (1.19) can be written as:         e m x u x ¼ ε $ xþ  x u xþ  N

(1.64)

e m is the interpolation function. By applying Eq. (1.64) for every node on vUþ where N of the RVE, the following constraint equation is obtained: Mq ¼ g

(1.65)

where q denotes the displacement vector of all nodes. This constraint can be enforced using the penalty method. After solving the global stiffness matrix equation of an RVE together with the constraints periodic boundary conditions, s and ε can be calculated. For elastic materials, the average tangent moduli Ceff can be solved by the linear relation of the macrostress tensor s and the macrostrain tensor ε: s ¼ Ceff : ε

(1.66)

Up to now, we have discussed in detail the entire set of steps to model particulate composites and porous materials. By embedding all these steps in an in-house CG code, an entirely automatic and highly efficient process of micromechanics simulation and homogenization is realized. One should, however, notice that, while taking advantages of analytical methods such as the PapkovicheNeuber solution and harmonic functions, the current methodology is very much different from the analytical micromechanics approaches such as the self-consistent method and variational bounds (see Refs. [1,8]). The current methodology consisted of direct numerical simulations by CGs is indeed a highly accurate and efficient numerical approach, where random distributions of particles and pores can be easily considered, and stress concentrations in

28

Durability of Composite Systems

Figure 1.15 The mesh of an Al/SiC unit cell model using: (a) a large number of 10-node tetrahedral elements with ABAQUS; (b) one computational grain. Table 1.3 Homogenized Young’s modulus using different methods. Method

Young’s modulus (GPa)

CG

103.8

ABAQUS

100.0

CG, computational grain.

and around heterogeneities can be directly captured, as well as obtaining the homogenized material properties simply through computation. Now we give examples of using CGs for materials homogenization. In the first example, a unit cell model of Al/SiC material is considered. The material properties are EAl ¼ 74GPa, nAl ¼ 0:33, ESiC ¼ 410GPa, nSiC ¼ 0:19. The volume fraction of SiC is 20%. This model was studied in Ref. [2], using around 76,000 10-node tetrahedral elements with ABAQUS. However, in this chapter, we use just one CG (see in Fig. 1.15). As shown in Table 1.3, although only one CG is used, the homogenized Young’s modulus is quite close to what is obtained by using round 76,000 10-node tetrahedral elements with ABAQUS. In the second example, the response surface of Eh (the effective Young’s Modulus) with respect to EAl and ESiC , with different volume fractions of SiC is investigated. In this example, random microstructures with multiple inclusions are generated and used as RVEs, and the periodic boundary conditions are applied. Three typical RVEs automatically generated, with different volume fractions, as shown in Fig. 1.16. By performing CG simulations with different values of EAl and ESiC , we can obtain the relation between the effective Young’s Modulus Eh and the modulus of each constituent material. As can be seen from Fig. 1.17, at specific volume fractions of SiC, Eh demonstrates significant variations with respect to EAl and ESiC .

1.4.2

Parallel computation and direct numerical simulation of composites

Although CGs have shown high efficiency in modeling an RVE containing a small number of inclusions [9e12], parallel computation may further reduce computational

Foundations of modeling of composites for durability analysis

(a)

29

(b)

(c)

0.02

0.02

0.02

0.01

0.01

0.01

0

0

0

–0.01

–0.01

–0.01

–0.02 0.02 0.01

0 0 –0.01 –0.01 –0.02 –0.02

0.01

0.02

–0.02 0.02 0.01

0 0 –0.01 –0.01 –0.02 –0.02

0.01

0.02

–0.02 0.02 0.01

0 0 –0.01 –0.01 –0.02 –0.02

0.01

0.02

Figure 1.16 Three typical RVEs generated by the RVE construction algorithm with different volume fractions. RVE, representative volume element.

time when simulating an RVE containing a large number of inclusions. In this section, to accelerate the analysis, parallel computation is implemented. The flowchart of the parallel algorithm is illustrated in Fig. 1.18. The CG analysis starts with input and initialization of the data structure with given CGs, material properties, constraints, and loads. The next step is to determine the number of parallel threads. Suppose there are n CGs after tessellation and k parallel threads for computation. Each of first n-1 parallel threads should compute and assemble the stiffness matrices of ½n=k CGs (½ x rounds x to the nearest integer). The last parallel thread should compute and assemble the stiffness matrices of the rest of the CGs. Finally, we can solve the system of equations with the assembled global stiffness matrix to obtain the nodal displacements. In this subsection, examples of directly computing the microscale stress distributions and concentrations using CGs are given. Firstly, we study an RVE of Al/SiC material, with n randomly distributed spherical SiC particles. The RVE is shown in Fig. 1.19, discretized with n currently discussed CGs. The material properties of Al and SiC are the same as those in the last example. The size of the RVE is 100 mm  100 mm  100 mm. A uniform tensile stress of 100 MPa is applied in the x3 direction. CGs are used to study the microscopic stress distribution in the RVE. The time for modeling 100, 1000, and 10,000 spherical inclusions with/without parallel computation are listed in Table 1.4. Fig. 1.20 shows the computed stress distributions of s33 for these three cases. While the inclusions are presenting a relative uniform stress state, the stress distributions in the matrix show high concentration. To be more specific, high stress concentration is observed near the inclusions, in the direction that is parallel to the direction of loading. On the other hand, at the locations near the inclusions, in the direction that is perpendicular to the direction of loading, very low stress values are observed. This gives us the idea at where damages are more likely to initiate and develop, for materials reinforced by stiffer particles. As seen from Table 1.4, the CPU time needed for the simulation has been significantly reduced after employing parallel computation. Using the CG method with parallel computation, an RVE with 10,000 inclusions can be simulated within 50 min. Considering the procedure is also entirely

30

Durability of Composite Systems

Volume fraction = 0.1

Eh (GPa)

100 95 90 85 80 75 500 450 400 ESi/C (GPa)

350 65

70

75

80

85

EAl (GPa)

Volume fraction = 0.2

Eh (GPa)

115 110 105 100 95 90 500 450 400 ESi/C (GPa)

350 65

70

75

80

85

EAl (GPa)

Volume fraction = 0.3

135

Eh (GPa)

130 125 120 115 110 105 500 450 400 ESi/C (GPa)

350 65

70

75

80

85

EAl (GPa)

Figure 1.17 Response surface of Eh (the effective Young’s modulus) with respect to EAl and ESiC , with different volume fractions of SiC.

Foundations of modeling of composites for durability analysis

31

Figure 1.18 The flowchart of the parallel algorithm for computational grains.

Figure 1.19 An RVE with n spherical inclusions. RVE, representative volume element.

Table 1.4 Time needed for the CG method. n

100

1000

10,000

Without parallel algorithm

360.28s

2886.23s

30,819.18s

38.17s

254.32s

2857.74s

With parallel algorithm CG, computational grain.

32

Durability of Composite Systems

(a)

65 60 55

50 40 30 20 10 0 –10 –20 –30 –40 –50 50

50 45 40 35 30 25

0 –50

–50

–40

–30

–20

–10

0

10

20

30

40

50

20 15

(b)

70

60

50 40 30 20 10 0

50

40

–10 –20 –30 –40 –50 50

30

0 –50

–50

–40

–30

–20

–10

0

10

20

30

40 50

20

10

(c)

80 70

50 40 30 20 10 0 –10 –20 –30 –40 –50 50

60 50 40 30

0 –50 –50

–40

–30

–20

–10

0

10

20

30

40 50

20 10

Figure 1.20 Distribution of s33 by the CGs for the problem of a finite matrix containing (a) 100, (b) 1000, and (c) 10,000 randomly distributed spherical inclusions. CGs, computational grains.

Foundations of modeling of composites for durability analysis

33

automatic, where in the RVE construction, Voronoi tessellation, and CG computation are efficiently executed, a direct numerical simulation of the composite material is indeed achieved. This is probably impossible when the traditional FEM is used, where thousands of simple finite elements are needed to model a single CG [2].

1.5

Summary

This chapter discussed the development of 3D polyhedral CGs for a simple and direct numerical simulation for homogenization and micromechanical stress/strain analysis of particle/fiber composites. In this model, the Trefftz displacement trial functions, which satisfy the Navier’s equations, are constructed from Papkoviche Neuber general solutions. Derivation of these functions in terms of spherical/cylindrical harmonics is a major contribution. Algorithms regarding the development of stiffness matrices of CGs, and the enforcement of periodic boundary conditions, are also discussed in detail. Various numerical examples are given to demonstrate the power of the developed CG model. Compared with analytical methods such as the self-consistent method and variational bounds [1,8], the current methodology featuring direct numerical simulations by CGs can automatically and efficiently model random distributions of particles and fibers, as well as directly and accurately simulate stress concentrations around heterogeneities. These results of interfacial stresses are of paramount importance in studying the initiation of damage near inclusions/fibers.

References [1] S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier, 2013. [2] N. Chawla, V.V. Ganesh, B. Wunsch, Three-dimensional (3D) microstructure visualization and finite element modeling of the mechanical behavior of SiC particle reinforced aluminum composites, Scripta Mater. 51 (2004) 161e165. [3] T.J. Vaughan, C.T. McCarthy, A combined experimentalenumerical approach for generating statistically equivalent fibre distributions for high strength laminated composite materials, Compos. Sci. Technol. 70 (2010) 291e297. [4] A. Lurie, Three-Dimensional Problems in the Theory of Elasticity, Theory of elasticity, Springer, 2005, pp. 243e407. [5] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Springer Science & Business Media, 2013. [6] E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, CUP Archive, 1931. [7] C. Rycroft, Voroþþ: A Three-Dimensional Voronoi Cell Library in Cþþ, Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States), 2009. [8] J.R. Willis, Bounds and self-consistent estimates for the overall properties of anisotropic composites, J. Mech. Phys. Solid. 25 (1977) 185e202.

34

Durability of Composite Systems

[9] G. Wang, L. Dong, J. Wang, S. Atluri, Three-dimensional Trefftz computational grains for the micromechanical modeling of heterogeneous media with coated spherical inclusions, J. Mech. Mater. Struct. 13 (2018) 505e529. [10] L. Dong, S.N. Atluri, Development of 3D T-Trefftz voronoi cell finite elements with/ without spherical voids &/or elastic/rigid inclusions for micromechanical modeling of heterogeneous materials, Cmc -Tech Sci. Press 29 (2012) 169e211. [11] L. Dong, S.N. Atluri, T-Trefftz voronoi cell finite elements with elastic/rigid inclusions or voids for micromechanical analysis of composite and porous materials, Comput. Model. Eng. Sci. 83 (2012) 183e219. [12] L. Dong, S.N. Atluri, Development of 3D Trefftz voronoi cells with ellipsoidal voids &/or elastic/rigid inclusions for micromechanical modeling of heterogeneous materials, Comput. Mater. Continua (CMC) 30 (2012) 39.

2

Role of uncertainty in the durability of composite material systems*

Jeffrey T. Fong, N. Alan Heckert, James J. Filliben National Institute of Standards and Technology (NIST), Gaithersburg, MD, United States

Chapter outline 2.1 Durability with uncertainty: engineer’s second fundamental problem

37

2.1.1 What is durability? 37 2.1.2 What is uncertainty? 38 2.1.3 What is durability with uncertainty 40

2.2 Seven types of uncertainty in data and modeling, and eight statistical tools 40 2.2.1 2.2.2 2.2.3 2.2.4

Three types of data uncertainty, DU1a, DU1b, and DU1c, for a univariate data set 40 Three types of model uncertainty, MU2a, MU2b, and MU2c, for data modeling 43 One type of model uncertainty, MU3, for system modeling 43 Eight statistical tools to estimate seven types of uncertainty 44

2.3 Data-set uncertainty (DU1a) and data-parameter uncertainty (DU1b) 44 2.3.1 Data-set uncertainty (DU1a) and Tool-1 44 2.3.2 Data-parameter uncertainty (DU1b) and Tool-2 46

2.4 Data-coverage uncertainty (DU1c) from specimen to full-size structures 48 2.5 Model-function uncertainty (MU2a(f)) for error propagation 50 2.5.1 Model-function uncertainty (MU2a) and Tool-3 (NUM) 50 2.5.2 Model-function uncertainty (MU2a) and Tool-4 (error propagation) 51 2.5.3 Model-function uncertainty (MU2a) and Tool-5 (LLSQ) 54

2.6 Model-compute uncertainty (MU2b(cf0)) for model verification 2.7 MU2b(cf) and model-physics uncertainty (MU2c(cf)) 59

55

2.7.1 Model-compute uncertainty (MU2b(cf)) and Tool-7 (DEX) 59 2.7.2 Model-physics uncertainty (MU2c(cf)) and Tool-7 (DEX) 62 2.7.3 Model-physics uncertainty (MU2c(cf)) and Tool-8 (ASTM E691) 63

2.8 Model-system uncertainty (MU3(sf)) for system verification

*

63

Contribution of the National Institute of Standards & Technology. Not subject to copyright.

Durability of Composite Systems. https://doi.org/10.1016/B978-0-12-818260-4.00002-8 Copyright © 2020 Elsevier Ltd. All rights reserved.

36

Durability of Composite Systems

2.9 Four types of durability with uncertainty for modeling material systems 64 2.10 Durability with uncertainty (DuU1a) for a smooth simple system under cyclic load without scaling 65 2.11 Durability with uncertainty (DuU1b) for a cracked simple system under cyclic load without scaling 66 2.12 Durability with uncertainty (DuU2a) for validating an FRP composite elastic constants database without scaling 70 2.12.1 A database of elastic constants and thermal expansion coefficients for FRP 70 2.12.2 Estimation of data-set uncertainty of elastic constants of FRP without scaling 72 2.12.3 Validation of an FRP composite elastic constants database without scaling 74

2.13

Durability with uncertainty (DuU2b) for a 2D-holed square composite plate under static load 74 2.14 DuU3, DuU4, ., DuUn for modeling durability of composite material systems 76 Appendix A 79 Statistical analysis software package named DATAPLOT (DP) 79 What is DATAPLOT? 79 How to download software and its documentation? 79 A simple DATAPLOT example 81

Appendix B 83 A sample Tool-1 DP code for data uncertainty DU1a

83

Appendix C 87 Two sample Tool-2 DP codes for data uncertainty DU1b and DU1c

87

Appendix D 89 NIST uncertainty machine (Tool-3) for model uncertainty MU2a 89 What is NIST Uncertainty Machine? 89 How to download software and its documentation 89 A simple NUM example 90

Appendix E 92 A sample Tool-5 DP code for model uncertainty MU2a 92

Appendix F 95 A sample Tool-6 DP code for model uncertainty MU2b 95

Appendix G 100 A sample Tool-7 DP code for model uncertainty MU2b, MU2c, and MU3 100

Disclaimer 102 References 102

Role of uncertainty in the durability of composite material systems

2.1 2.1.1

37

Durability with uncertainty: engineer’s second fundamental problem What is durability?

All material systems, whether manmade or natural, simple1 or composite,2 are useful and important to some segments of the society simply because they answer “yes” to both of the following two questions in engineering [1]3: First fundamental problem (feasibility): Does it work? Second fundamental problem (durability): Does it last, and how long? In general, the first question is easier to answer than the second (durability, or, more specifically, the time to failure) because the amount of knowledge we already have to answer “yes” to the feasibility question is larger than that for answering the second for either a simple or composite material system. In other words, there are more facts and less “uncertainty” in the knowledge base and modeling tools for inferring an answer to feasibility than durability. Owing to the presence of uncertainty, neither of the above two questions can be answered for sure. For example, there is the uncertainty due to sampling, as we conduct a limited number of tests to characterize the strength of a base metal that goes into an I-beam of a building. Another obvious source of uncertainty is due to scaling when we use small-size specimens to predict that their laboratory test results apply to the full-scale structures. A third common source of uncertainty is due to modeling, as we make plausible yet unproven assumptions in the analysis to predict how durable a product will be without knowing the full future loading environment. As a result, the best we could do to answer the durability question is to (1) gather facts and comparable observations, (2) analyze the problem assuming a set of known and hypothetical laws and conditions, and (3) infer an answer on durability with uncertainty using an appropriate set of statistical tools.

The purpose of this chapter is, therefore, twofold: (1) We will define and quantify seven types of uncertainty in data and modeling in general and introduce eight statistical tools to compute. (2) We will define and quantify four types of durability with uncertainty (DuU) and provide for each type an example of how to use one or more of the eight tools to compute.

To accomplish the goal (1), we will first introduce in Section 2.1.2 the fundamental concept of uncertainty. In this chapter, we will first list seven distinct types of data or model uncertainties, and then introduce eight statistical tools to compute the uncertainties. Finally, in Sections 2.3e2.8, we will fully define the seven uncertainties and show examples of how to use the statistical tools to compute.

1

A simple material system is one for which all its failure mechanisms are assumed to be fully understood. A composite material system consists of two or more simple systems with intersystem failure mechanisms either fully or partially understood. 3 The number in square brackets denotes a reference listed in Section 2.15 of this chapter. 2

38

Durability of Composite Systems

To accomplish the goal (2), we will first introduce the concept of component DuU in Section 2.1.3. We will then list four distinct types of DuU0 s in Section 2.9 and provide for each type an example application in Sections 2.10e2.13. A list of references is provided in Section 2.15, followed by seven appendices to assist the readers in using the eight statistical tools described in Sections 2.2e2.8.

2.1.2

What is uncertainty?

The term, uncertainty, when used in general as well as in the context of durability, is usually expressed as a percentage of some mean value, or, in the case of durability, the percent interval around the mean time to failure (mTF) of a material system. For example, a 20% uncertainty of the durability of a composite material system with a mean time to failure equal to mTF denotes a lower and upper bound of its time to failure to be 0.8 mTF and 1.2 mTF, respectively. Unfortunately, this simple-minded concept has given rise to considerable confusion. To illustrate, let us consider a statement that a product has a useful life of, say, 200,000 cycles of some specified loading history with a 20% uncertainty, that is, its life has a lower bound of 160,000 cycles and an upper bound of 240,000 cycles. That statement, however, is incomplete because uncertainty is a probabilistic concept, -and any expression of uncertainty must always be accompanied by a measure of the probability of that expression such as the notion of the so-called “level of confidence.” The correct statement would have to be as follows: A product has a useful life of 200,000 cycles of some specified loading history with a 20% uncertainty at, say, a 95% level of confidence.

A graphical representation of the above statement is given in Fig. 2.1. In other words, it takes two numbers, 20% and 95%, or, 0.20 and 0.95, to quantify uncertainty, and not just one, unless it is understood, as commonly practiced and interpreted in the engineering literature, that the level of confidence is assumed to be 95% for safety considerations. There is another reason why it is a good practice to state explicitly the level of confidence whenever an expression of uncertainty is mentioned. In the field of A normal distribution 0.025

Probability

0.02 0.015 0.01

Mean (m) = 200 Stand. Dev. (s) = 20

200

20% Uncertainty at 95% level of confidence.

160

240 Area (red) under curve between 2 bounds = 0.95, level of confidence.

0.005 0

–100

0

100

200

300

400

500

Figure 2.1 A graphical representation of a correct statement on uncertainty.

Role of uncertainty in the durability of composite material systems

39

metrology and measurement science, uncertainty is rigorously defined in the International Vocabulary of Metrology (VIM) [2] and Guidelines on Uncertainty in Metrology (GUM) [3] as presented in a recent web-based publication [4] by the National Institute of Standards and Technology (NIST) as follows: Measurement is an experimental process that produces a value that can reasonably be attributed to a quantitative property of a phenomenon, body, or substance. . The GUM defines measurement uncertainty as a “parameter, associated with the result of a measurement (i.e., measurand), that characterizes the dispersion of the values that could reasonably be attributed to the measurand.” The VIM defines it as a “non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used.” For scalar measurands, the VIM suggests that this parameter may be, for example, a standard deviation called standard measurement uncertainty (or a specified multiple of it), or the half-width of an interval that includes the measurand with a stated coverage probability. This suggestion follows from the position that measurement uncertainty expresses incomplete knowledge about the measurand and that a probability distribution over the set of possible values for the measurand is used to represent the corresponding state of knowledge about it: in these circumstances, the standard deviation aforementioned is an attribute of this probability distribution that represents its scatter over the range of possible values. . It is this distribution that imparts meaning to the parameter that is chosen to quantify measurement uncertainty.

So, the metrological community leaves the quantitative definition of a measurement uncertainty to the user if a probability distribution is specified and an attribute of the distribution representing the dispersion of the measurand is chosen. For example, if the distribution is Gaussian (or normal), and the attribute chosen is one standard deviation, the level of confidence is, by definition, only 68.3%. If two standard deviations are chosen, the level of confidence goes up and reaches 95.4%. If it is three standard deviations, the level of confidence is higher and equals 99.7%, and so on. In engineering, all quantities of interest have uncertainties but are not always measurable. For example, the ultimate tensile strength (UTS) of a composite laminate under a controlled loading condition is measurable in a laboratory and so is its fatigue life at the scale of the specimen size. However, the UTS of an aircraft wing made of composite material under real flying conditions is not measurable, nor is its durability or fatigue life. For those quantities of interest that are not measurable, we can only infer them and estimate their uncertainties with assumed probability distributions rather than known ones. The need to specify a probability distribution for expressing uncertainty, nevertheless, still exists, and it may not even be the normal distribution as we are accustomed to. It could be a three-parameter Weibull (see, e.g., Ref. [5]) for the UTS of an aircraft wing, as shown in an example of data uncertainty in Section 2.3.

40

Durability of Composite Systems

In summary, we recommend throughout this chapter that all uncertainties be expressed in a three-part format, the first part being a percentage or fraction of a mean value as the size of the uncertainty, the second, a probability distribution describing the nature of the uncertainty, and the third, a percentage or fraction (less than 1) denoting the level of confidence for that uncertainty. In case where the second part is not specified, it will be understood that the probability distribution is the normal distribution, and that the uncertainty is to be expressed in two numbers: the size of the uncertainty and its level of confidence.

2.1.3

What is durability with uncertainty

Having defined durability intuitively as the mTF or mean cycles to failure (mCF) of a material system in Section 2.1.1 and described how to express uncertainty in general in Section 2.1.2, we need to introduce three facts before we could address the role of uncertainty in durability. As shown in Fig. 2.2, failure and failure mechanisms are scale-dependent, and the notion of durability needs to be subdivided into two categories: (1) component durability at Level 3, and (2) structural durability at Levels 4 and above. Secondly, both carry uncertainty and we need to estimate component durability with uncertainty, and structural durability with uncertainty, using statistical tools accessible to a computer-literate engineer. Finally, the space limitation of this chapter simply does not allow a complete exposition of the mathematical basis of those tools to satisfy their user. Consequently, we have adopted a method of presenting this introductory chapter on the role of uncertainty in durability by emphasizing the first category, namely, the component durability with uncertainty having a shorter acronym, DuU, and by taking a computational and numerical approach to equip the reader with the means to learn by computing based on example problems. As mentioned earlier in Section 2.1.1, we present seven uncertainty types and eight tools in Sections 2.2e2.8, and then four types of component durability with uncertainty in Sections 2.9e2.13. In Section 2.14, we conclude with a challenge that with our set of tools, readers will be able to compute as many more DuU’s as they need in the years to come. For more information on the structural durability with uncertainty, readers may consult Refs. [6e9].

2.2

Seven types of uncertainty in data and modeling, and eight statistical tools

2.2.1

Three types of data uncertainty, DU1a, DU1b, and DU1c, for a univariate data set

We begin with a sample data set, A, of n univariate data as follows: Data Set A: fxi g;

i ¼ 1; 2; .; n;

(2.1)

where each xi is real. We also define sample mean, m, and sample standard deviation, s:
X  xi =n; with i ¼ 1; 2; .; n: m¼ (2.2)

Role of uncertainty in the durability of composite material systems

41

Figure 2.2 A schematic representation of the six levels of scale for a suspension bridge, ranging from Level 1 at the microstructural scale to Level 6 at the full scale of the bridge structure.

hX i1=2 ðxi  mÞ2 s ¼ ½1 = ðn  1Þ ;

with i ¼ 1; 2; .; n:

(2.3)

To define an uncertainty of the set A, we need to introduce a probability distribution, which is to be described, in general, by a probability density function as follows: z ¼ f ðx; a1 ; a2 ; .; ah Þ;

(2.4)

42

Durability of Composite Systems

where h is the number of parameters of the density function. For example, a Gaussian (or normal) distribution has the following density function with two parameters, a1 and a2: h
pffiffiffiffiffiffi i    z ¼ 1 = a2 2p exp ðx  a1 Þ = 2a22

(2.5)

A 3-parameter Weibull has the following density function with three parameters, a1, a2, and a3: h  i z ¼ a3 ðx  a1 Þa3 1 = aa23 expf½ðx  a1 Þ=a2 a3 g

(2.6)

To fix ideas on a fundamental concept of uncertainty, let us select the normal density function, Eq. (2.5), as the underlying probability distribution with a1 ¼ m and a2 ¼ s. As explained in Section 2.1.2, any uncertainty needs to be expressed in three parts: (1) a percentage, u 100%, or a fraction, u, of the sample mean, m, denoting the size of the uncertainty, (2) a confidence coefficient, g, representing the g 100% level of confidence for the uncertainty, and (3) the underlying probability distribution, which is assumed to be normal. For a normal distribution, it is well-known in the statistics literature (see, e.g., Refs. [10e12]) that the following three types of intervals can be defined: Confidence interval (basic): m  sk1 ; or; ðm  sk1 ; m þ sk1 Þ with k1 ¼ t½ð1  gÞ = 2; n  1 ð1=nÞ1=2 ;

(2.7)

where n denotes the number of current observations. Note that the confidence interval for the mean quantifies the uncertainty for the mean based on the currently available data. Prediction interval (future): m  sK2 ; or ðm  sK2 ; m þ sK2 Þ with K2 ¼ t½ð1  gÞ = 2; nn  1ð1 þ 1=nnÞ1=2 ;

(2.8)

where nn denotes the number of new observations. Note that the prediction interval for the mean quantifies the uncertainty of the mean of future observations. Tolerance interval : m  sK3 ; with K3 ¼ qðy; nÞrðn; pÞ;

(2.9)

where p is the coverage denoting the proportion of the population one is interested in covering, and q and r are two factors depending on g, n, p as tabulated in Ref. [12].

Role of uncertainty in the durability of composite material systems

43

Note that the tolerance interval quantifies the uncertainty for an interval that contains a certain proportion of the population (the coverage). As a result of the above, we are now ready to introduce the following three types of data uncertainty, to be denoted by DU1a, DU1b, and DU1c, for a univariate data set A of sample size n and an assumed normal distribution: [Data-Set Uncertainty, DU1a], based on the concept of a prediction interval, Eq. (2.8), for set A. [Data-Parameter Uncertainty, DU1b], based on the concept of a prediction interval for m and s. [Data-Coverage Uncertainty, DU1c], based on the concept of a tolerance interval, Eq. (2.9), and the concept of coverage, p, for set A.

Clearly, the above formulation can be generalized to the case when the underlying distribution is not normal. For example, if the distribution is the 3-parameter Weibull, then the Data-Parameter Uncertainty is based on the concept of a prediction interval for a1, a2, a3 of Eq. (2.6). A more complete description of the three types of data uncertainty introduced earlier, and the statistical tools to compute them are given in Sections 2.3 and 2.4.

2.2.2

Three types of model uncertainty, MU2a, MU2b, and MU2c, for data modeling

Let {xi}, i ¼ 1, 2, ., n, be a set of scalar variables, and let a model, M, be defined as a function of the n independent variables, xi, i ¼ 1, 2, ., n, and a single scalar dependent variable, y, such that a computational model exists to compute as follows: Model M : y ¼ gðxi Þ; i ¼ 1; 2; .; n;

(2.10)

where the function g may either be explicitly defined, or implicitly specified through a computer algorithm. Three types of uncertainty associated with the model, M, or model uncertainty for the function g, to be denoted by MU2a, MU2b, and MU2c, are listed below: [Model-Function Uncertainty, MU2a], when the function g is explicitly defined. [Model-Compute Uncertainty, MU2b ], when the mathematical accuracy needs to be verified. [Model-Physics Uncertainty, MU2c ], when the validity of the function, g, representing the correct physics of the model, needs to be confirmed.

A complete description of the three types of model uncertainty introduced above, and the statistical tools to compute them, are given in Sections 2.5e2.7.

2.2.3

One type of model uncertainty, MU3, for system modeling

Let W be a system of k models that are connected either in series or in parallel such that the system output, w, is a function of the k outputs, yj, j ¼ 1, 2, ., k, of the k models

44

Durability of Composite Systems

acting as independent variables, each of which has its own model uncertainty as described in Section 2.2.2: System W : w ¼ g½y1 ðxi1 Þ; y2 ðxi2 Þ; .; yk ðxik Þ

(2.11)

Such a system is very common in engineering and deserves a study of its uncertainty when we are interested in the durability of that system. It turns out that mathematically speaking, model-system uncertainty is no different from model-compute uncertainty, as we will show in Section 2.8. In engineering, a system is much more complicated than a component, so the uncertainty analysis of a system is also more complicated and is beyond the scope of this chapter. Nevertheless, we believe it is important to include it in a list of uncertainties that could be computed using one of our eight statistical tools as shown in Section 2.8. Thus, we define [Mode-System Uncertainty, MU3], when a system is made of models with known uncertainties.

2.2.4

Eight statistical tools to estimate seven types of uncertainty

Recent advances in computer hardware and software technologies have made it easier for engineers to estimate both data uncertainties and model uncertainties by using black-box types of tools. To estimate the seven types of uncertainty described above, we introduce eight tools, seven of which are computer codes written in an open-source and English-command-based language named DATAPLOT [13e15], and the eighth, a web-based application software [4] that runs on a NIST computer and is accessible to the public at no cost to the user. Each of the eight tools will be introduced with examples when it is needed to estimate a specific uncertainty in Sections 2.3e2.8. We wish to add, however, that the seven tools using DATAPLOT and an eighth tool named the NIST Uncertainty Machine are included here for illustrative purposes and that similar tools can be developed in other software programs, since the underlying ideas of these tools are not specific to any one software. We demonstrate them with a specific software like DATAPLOT to make the ideas more concrete.

2.3 2.3.1

Data-set uncertainty (DU1a) and data-parameter uncertainty (DU1b) Data-set uncertainty (DU1a) and Tool-1

We will begin with an example problem. Consider the following data set A, consisting of 31 sample test data for the ultimate tensile strength, in MPa unit, of an aircraft window material, borosilicate crown BK-7 glass [16]: A: f129:83; 143:42; 149:33; 158:79; 160:17; 165:83; 167:69; 175:82; 175:96; 177:89; 184:03; 184:58; 184:65; 186:51; 190:79; 206:16; 214:50; 228:91; 232:57; 232:78; 233:67; 239:67; 246:50; 247:60; 254:98; 255:67; 255:74; 272:90; 303:69; 312:28; 312:90g

Role of uncertainty in the durability of composite material systems

45

Using Eqs. (2.2) and (2.3), we can easily compute for a sample size n of 31, the sample mean m ¼ 212.45, and the sample standard deviation s ¼ 50.01. To estimate the uncertainty of a material testing data set such as A, we note that in general the purpose of the test is to predict how the same material in a full-size component will behave, that is, a future observation of the property of the same material for engineering design purposes. Consequently, the proper vehicle to arrive at an estimate of the uncertainty of A would be the prediction interval given by Eq. (2.8). We now introduce the first of seven uncertainties to be defined in this chapter as follows: [Data-Set Uncertainty], or, symbolically, DU1a(A),

which is defined as the half-width of the prediction interval given by Eq. (2.8) if the underlying distribution is normal. Otherwise, DU1a(A), the uncertainty of the data-set, A, is given by the “standard measurement uncertainty” as defined in the International Vocabulary of Metrology [2] and the Guides to Uncertainty in Metrology [3] and quoted in excerpts in Section 2.1.2 of this chapter. The task of computing DU1a(A) can thus be accomplished in three steps as follows: Step 1: Assuming A is normal, and the confidence coefficient, g, is 0.95, we evaluate the prediction interval of Eq. (2.8) by first looking up from a table such as Table B2 in Ref. [12], the value of the t-distribution for a ¼ (1  g)/2 ¼ 0.025, and n ¼ n  1 ¼ 30, such that t (0.025, 30) ¼ 2.042, and then computing the factor, (1 þ 1/n)1/2 ¼ 1.016. As the sample standard deviation, s is known to be 50.01, we compute from Eq. (2.8) that the half-width of the prediction interval ¼ 50.01  2.042  1.016 ¼ 103.75. Knowing that the sample mean m ¼ 212.45, we conclude that DU1a ðAÞ ¼ 103:75=212:45 ¼ 0:49; or;

(2.12)

the data-set uncertainty of A is 49% at 95% confidence assuming A is normal. Step 2: To find out if the best-fit distribution of the data set A is indeed or close to normal, we introduce tool No 1, or Tool-1, which is a statistical tool in the form of a computer code written in an open source and English-command-based language named DATAPLOT [13e15]. An introduction of DATAPLOT is given in Appendix A. The filename of an example DATAPLOT code for Tool-1 is rd101z.dp, and the name of its input file containing the univariate data set A is fong101x.dat (see Appendix B). As shown in Appendix A, the software, DATAPLOT can be downloaded and installed in many platforms such as Windows 7/8/10, Unix/Linux, and MacOS. As an example of running Tool-1, we ran the code, rd101z.dp, for estimating the parameters of a large number of 2-parameter, 3-parameter, and 4-parameter continuous distributions using two methods of estimation and four goodness-of-fit criteria [17]. A typical output of that tool when applied to the data file of A, fong101x.dat, is given in Table 2.1: We observe from an examination of Table 2.1 that the normal distribution ranks very low among the 18 selected for comparison, and the best-fit distribution for the data set A is the 3-parameter Weibull. This leads us to the next Step 3.

46

Durability of Composite Systems

Table 2.1 A typical output (truncated) showing the ranking of 18 distributions for fitting A.

Step 3: Based on Refs. [2,3], we can specify our definition of DU1a(A) as long as we choose an attribute of an underlying distribution that describes the dispersion of the data. That attribute can be a multiple of the sample standard deviation s. In this case, we choose 2s, and DU1a(A) ¼ 2  50.01/212.45 ¼ 0.47, or, the data-set uncertainty of A is 47% at 95% confidence level assuming a 3-parameter Weibull.

2.3.2

Data-parameter uncertainty (DU1b) and Tool-2

The second of the seven uncertainties to be defined in this chapter concerns the parameters, ai, of a given data set, A, and is named [Data-Parameter Uncertainty], or, symbolically, DU1b(ai),

which is defined as either the half-width of the prediction interval given in Eq. (2.8) or the “standard measurement uncertainty” of any parameter of distribution, following the suggestion in the International Vocabulary of Metrology [2] and the Guides to

Role of uncertainty in the durability of composite material systems

47

Uncertainty in Metrology [3] and quoted in excerpts in Section 2.1.2 of this chapter. In short, we need to calculate the standard deviation, or the so-called “standard error,” of each parameter (see, e.g., Bury [11, p. 129] for a normal distribution), with which we could express DU1b as we did for DU1a in the last subsection. It turns out that for a normally distributed set of data, it is known that the standard error of the sample mean, s.e.(m), and the standard error of the sample standard deviation, s.e.(s), are known as follows [11]: sdðmÞ ¼ sð1=nÞ1=2

(2.13)

sdðmÞ ¼ sf1=½2ðn  1Þg1=2 :

(2.14)

For several commonly used distributions such as normal, 2-parameter Weibull, 3parameter Weibull, 2-parameter Log-normal, and 3-parameter Log-normal, we have developed a tool named tool No. 2 (or, Tool-2), which is also a computer code written in DATAPLOT and listed as an example code in Appendix C with the filename, 3pW_0.05x.dp. We applied this code to the input file, fong101x.dat, for the data set A given in Section 2.3.1, we obtain an output Table 2.2 (truncated for brevity) for a 3-parameter Weibull distribution of A. When we made a change in the DATAPLOT code to make it applied to a normal distribution, we obtain an output Table 2.3 (truncated for brevity) for a normal distribution of A. In either Tables 2.2 or 2.3, we can easily calculate the half-width of the 95% confidence interval underlined in red to obtain the following answers for DU1b(a1) and DU1b(m): DU1b ða1 Þ ¼ ½ð145:316  97:990Þ = 2=121; 653 ¼ 0:20; or; 20% at 95% confid:; 3pW: (2.15)

Table 2.2 A typical output (truncated) from running TooI-2 for 3-parameter Weibull.

48

Durability of Composite Systems

Table 2.3 A typical output (truncated) from running Tool-2 for a normal distribution.

DU1b ðmÞ ¼ ½ð230:790  194:101Þ = 2=212:445 ¼ 0:09; or; 9% at 95% confid:; Normal:

2.4

(2.16)

Data-coverage uncertainty (DU1c) from specimen to full-size structures

In the last section, we introduced two uncertainties associated with a univariate sample data set A of sample size n. The first one, DU1a, concerns the uncertainty of the entire set A itself, and the second, DU1b, the uncertainty of the parameters of a probability distribution that fits A. We also introduced two tools in the form of DATAPLOT codes (see Appendices AeC) to deal with DU1a and DU1b, respectively: Tool-1 (A goodness-of-fit-based probability distribution ranking code for DU1a), and. Tool-2 (A parameter confidence interval estimation code for DU1b and a coveragebased tolerance interval estimation code for a third data uncertainty, DU1c),

Role of uncertainty in the durability of composite material systems

49

where the third data uncertainty is defined as follows: [Data-Coverage Uncertainty], or, symbolically, DU1c(A),

which is defined as the half-width of the tolerance interval given by Eq. (2.9) as implemented in tables given by Natrella [18], Nelson, et al. [12], etc., for coverage p, and a normal probability distribution. Otherwise, DU1c(A) for a few limited non-Gaussian distributions such as 2-parameter Weibull, 3-parameter Weibull, 2-parameter Log-normal, and 3-parameter Log-normal, can be evaluated using Tool-2 (see Appendix C and Ref. [17]). The theory of tolerance interval, developed in the 1940se50s (see Refs. [19e27]) and the uncertainty, DU1c, are important because they allow engineers to extrapolate the information from small-size specimens in the laboratory to full-size components or structures in the field using a concept known as “coverage,” which is denoted by p, (0 < p < 1.0). In the aircraft industry, the concept of tolerance intervals is routinely used for the design (see, e.g., Ref. [28]), where p ¼ .99 is used for the critical component design (so-called A-allowable), and p ¼ .90 for less-critical one (B-allowable). For computational convenience, we use the lack of coverage, 1  p, instead of the coverage, p, as a primary variable when we developed Tool-2 and display the output results in tables with a column headed by (1  p). To illustrate an application of Tool-2 for finding DU1c, we need to provide a data set such as A as well as state a specific probability distribution associated with A. Using the Table 2.4 A typical output (truncated) of TooI-2 for g ¼ 0.95 and a 3-para. Weibull distrib.

50

Durability of Composite Systems

Table 2.5 A typical output (truncated) of TooI-2 for g ¼ 0.95 and a normal distribution.

data file named fong101x.dat, which represents the data in set A, and the computer code named 3pW_0.05x.dp given in Appendix C that works for a 3-parameter Weibull, we obtain an output in a truncated format (see Table 2.4), where we find the upper and lower limits of the tolerance interval for the data set A at 95% confidence level for a range of 10 “lack of coverage,” 1  p, from 0.00000001 to 0.10, which is identical to a range of coverage p from 0.99999999 to 0.90. In Table 2.4, we compute for p ¼ .99, or (1  p)100% ¼ 1% (lack of coverage), the half-width ¼ (166.890  139.537)/ 2 ¼ 13.676, and DU1c(A) ¼ 13.676/153.214 ¼ 0.09, or 9%. After making the necessary changes to the Tool-2 code for a normal distribution, we obtain an output file (truncated) in Table 2.5, where we compute for p ¼ .99, DU1c(A) ¼ 0.169, or, 17%.

2.5 2.5.1

Model-function uncertainty (MU2a(f )) for error propagation Model-function uncertainty (MU2a) and Tool-3 (NUM)

In the previous two sections, we introduced three uncertainties, DU1a, DU1b, and DU1c, each of which describes some dispersion property of a univariate data set A, or the parameters, a1, a2, a3, ., etc., of its “best-fit” or assumed probability distribution with a density function, z ¼ f(x; a1, a2, a3, .), and a random variable, x, associated with the data set A of sample size n.

Role of uncertainty in the durability of composite material systems

51

In this section, we will introduce a random variable, y, which is a function of a finite number of random variables, xi, i ¼ 1, 2, 3, ., k, such that the function, y ¼ f ðxi Þ; i ¼ 1; 2; 3; .; k;

(2.17)

represents a mathematical model of a scalar dependent random variable y and k scalar independent random variables, xi. All xi’s and y are real numbers. Assuming that the model function, f, is continuous and explicitly described as sum, product, fraction, trigonometric, logarithmic, exponential, or powers of the independent variables, xi, and each xi is either normally distributed with known values of sample mean, mi, and sample standard deviation, si, or otherwise, as long as their data-set uncertainties, DU1a(xi), are known, we are ready to define an uncertainty for the model function f as follows: [Model-Function Uncertainty], or, symbolically, MU2a( f ),

which is defined as either the half-width of the prediction interval given in Eq. (2.8) for a normally distributed y, or the “standard measurement uncertainty” of the random variable y, following the suggestion made in the International Vocabulary of Metrology [2] and the Guides to Uncertainty in Metrology [3] and quoted in excerpts in Section 2.1.2 of this chapter. In short, we need to calculate the standard deviation of y, with which we could express MU2a( f ) as we did in Section 2.3 for the data-set uncertainty, DU1a(xi), for each xi, i ¼ 1, 2, ., k. As it turns out, the NIST has recently released a public software [4] named the “NIST Uncertainty Machine (NUM)” that will estimate the model-function uncertainty, MU2a( f ), for any given f as long as the conditions stated above in the definition of MU2a( f ) are met. An introduction of NUM and an example of how to use NUM are given in Appendix D. We therefore recommend that NUM be used as tool No. 3 (Tool-3) for computing the model-function uncertainty, MU2a( f ). For further details on Tool-3, the reader may consult Lafarge and Possolo [29].

2.5.2

Model-function uncertainty (MU2a) and Tool-4 (error propagation)

In many engineering applications, the random independent variables, xi, in the function, f, are either known or assumed to be normally distributed. In that case, we would recommend tool No. 4 (Tool-4), based on the theory of the propagation of errors (see, e.g., Ku [30]), as an alternative to NUM for finding the model-function uncertainty, MU2a( f ). We shall introduce Tool-4 in four steps as follows: Step 1. Let x and y be two independent random variables, each of which is normally distributed. Let w be a dependent random variable that is a function of x and y, w ¼ f(x, y), such that f is continuous and explicitly described as sum, product, fraction, trigonometric, logarithmic, exponential, or powers of the independent variables, x and y.

52

Durability of Composite Systems

Step 2. For x, y, and w, respectively, let mx, my, and mw be their means, sx, sy, and sw be their standard deviations, vx, vy, and vw be their variances (vx ¼ s2x , vy ¼ s2y , and vw ¼ s2w ), and cx, cy, and cw be their coefficients of variations (cx ¼ sx/mx, cy ¼ sy/my, and cw ¼ sw/mw). For x and y, respectively, let nx and ny be their sample sizes. Step 3. Let the model function w ¼ f(x, y) of random variables, x and y, be associated with an identical function f: mw ¼ f ðmx ; my Þ;

(2.18)

where the variables, x, y, and w, are replaced by their means, mx, my, and mw, respectively. Step 4. Then, according to the theory of error propagation [30], the following nine statements are approximately true when nx and ny are large, and x and y are normally distributed: ðSumÞ

if mw ¼ Amx þ Bmy ;

then;

vw ¼ A vx þ B vy : 2

ðProductÞ

2

(2.19)

if mw ¼ mx  my ;

then;

vw ¼ m2y vx þ m2x vy : ðSquareÞ if mw ¼ m2x ; vw ¼

(2.20) then;

4m2x vx :

(2.21)

if mw ¼ mx =my ; then;
 vw ¼ ðmx =my Þ2 vx = m2x þ vy = m2y :

ðQuotientÞ

ðReciprocalÞ vw ¼

if mw ¼ 1=mx ;

(2.23)

ðSquare rootÞ if mw ¼ m1=2 x ; vw ¼ ð1 = 4Þvx =mx :

vw ¼

then;

vw =m4x :

ðLogarithmicÞ

(2.22)

if mw ¼ 1nðmx Þ;

then; (2.24) then;

vx =m2x

ðFractionÞ if mw ¼ mw =ðmx þ my Þ; then;
 vw ¼ vx  m2y þ vy = m2x =ðmx þ my Þ4 : ðPowersÞ if mw ¼ kmax may ; then;
 vw ¼ m2w a2 vw = m2y þ b2 vy = m2y :

(2.25)

(2.26)

(2.27)

Role of uncertainty in the durability of composite material systems

53

Note that in Ku [30, p. 269, Table 2.1], there are five additional statements similar to Eqs. (2.19) through (2.27), but those five are omitted here for brevity, because Eqs. (2.19) through (2.27) are adequate for us to recommend as Tool-4 for finding MU2a( f ). Let us apply Tool-4 to a model function in fracture mechanics as documented by Fong et al. [31], where the function f has two independent random variables, ai and mm, and one dependent random variable, Nif, as shown below: ð1mm=2Þ

Nif ðai ; mmÞ ¼ ai

=fCðHÞmm ðmm = 2  1Þg;

(2.28)

where C and H are constants. Taking the natural logarithm of both sides of Eq. (2.28), we obtain LnðNif Þ ¼ ð1  mm = 2ÞLnðCÞ mmLnðHÞ  Lnðmm = 2  1Þ.

(2.29)

Reorder the five terms on the right-hand side of Eq. (2.29) as follows: LnðNif Þ ¼  LnðCÞ  mmLnðHÞ  Lnðmm = 2  1Þ þ Lnðai Þ  ðmm = 2ÞLnðai Þ; (2.30) where the first term is a constant, the second, a linear term involving the variable mm, the third and fourth terms, two log terms each involving one of two variables, mm and ai, and the fifth term, a product of the variable mm with the log of the other variable, ai. We are now ready to apply the error propagation statements listed in Eqs. (2.19), (2.20), and (2.25) to obtain a relationship between the variance of Nif and the variances of mm and ai. Repeated applications of Eqs. (2.19), (2.20), and (2.25)e(2.30) yield the following result:   fcðNif Þg2 ¼ n1 þ mm2 = 4 fcðai Þg2 o þ ½ðmm=2ÞLnðai Þ2 þ ½mmLnðHÞ2 þ ½mm=ð2  mmÞ2 fcðmmÞg2 ; (2.31) where c(X) is the coefficient of variation of the variable X. It is interesting to observe that Eq. (2.31) resembles a weighted Pythagoras formula with the term, c(Nif), acting as the hypotenuse of a right-angled triangle. We also observe that when we insert engineering constants, C and H, in Eq. (2.31), we obtain a result that a small change of the crack growth exponent, mm, yields a shockingly large change in the remaining fatigue life, Nif. For the example cited in Ref. [31] on an AISI 4030 steel specimen at room temperature in a cyclic loading environment, Eq. (2.31) becomes fcðNif Þg2 ¼ 3:6fcðai Þg2 þ 566fcðmmÞg2 ; or; cðNif Þ z 23:8cðmmÞ;

(2.32)

if we neglect the uncertainty in the measurement of the initial crack length, ai, and focus only on that of the crack growth exponent, mm. Since the coefficient of variation,

54

Durability of Composite Systems

c, equals the standard deviation, s, divided by the mean, m, it is essentially identical to either the data-set uncertainty, DU1a(mm), or the model-function uncertainty, MU2a( f ), each chosen to be one standard deviation at 68% confidence level, normal distribution. Eq. (2.32) can thus be written in the following form: MU2a ð f Þ z 23:8 DU1a ðmmÞ;

(2.33)

where f is given by Eq. (2.28). Note that it is incorrect to equate MU2a( f ) with a fictitious uncertainty quantity related to the dependent random variable, y, or DU1a(y), because y is not an isolated univariate data set, and DU1a(y) simply does not exist.

2.5.3

Model-function uncertainty (MU2a) and Tool-5 (LLSQ)

In the previous two subsections, we considered a multivariable model function, f, given by Eq. (2.17), that is quite general if the function is continuous and expressible in algebraic form, linear or nonlinear. In this subsection, we shall restrict f to be the simplest kind, namely, a linear function of a single random variable, x, with two parameters, a and b, as follows: y ¼ f ðxÞ ¼ a þ bx;

(2.34)

where the function f represents a model, M, for a set of data, {xi, yi}, i ¼ 1, 2, ., n. Eq. (2.34) represents a straight line that fits the data of the model M with a y-intercept equal to a and a slope equal to b. Both parameters, a and b, can be estimated used the method of least squares or regression (see, e.g., Draper and Smith [32], Mosteller and Tukey [33]). To find the model-function uncertainty of f, MU2a( f ), we have written a DATAPLOT code named LLSQ.dp, which is listed in Appendix E, and is offered here as uncertainty tool No. 5 (Tool-5) for a simple linear function of a single random variable and a set of n pairs of data, {xi, yi}. To illustrate an application of Tool-5, let us consider a set of fatigue life, Nf2, versus cyclic stress amplitude, sa, data for AISI 4340 steel (see Dowling [34,35]) as listed in Table 2.6: Assuming that the fatigue life, Nf2, is related to the stress amplitude, sa, by a power law: Nf2 ¼ a*(sa)c, and let xx ¼ log10 sa, and yy ¼ log10 Nf2, then xx and yy are linearly related: yy ¼ f(xx) ¼ A þ C * xx, where A ¼ log10 a, and C ¼ log10 c. An application of Tool-5 yields a plot with all values of the constants and their standard deviations displayed in Fig. 2.3: We now compute the model-function uncertainty, MU2a( f ), for a specific operating stress amplitude such as 398 MPa as shown in the upper left corner of Fig. 2.3. The 95% upper and lower confidence limits were found to be 3.41 and 1.54, and the mean was 2.27 million cycles. The model-function uncertainty, MU2a( f ), thus equals (3.41e1.54)/(2  2.27) ¼ 0.41, or, 41%.

Role of uncertainty in the durability of composite material systems

55

Table 2.6 Fatigue data for AISI 4340 steel. Stress amplitude sa, MPa

Cycles-to-failure Nf2, cycles

948

222

834

992

703

6004

631

14,130

579

45,860

524

132,150

After N.E. Dowling, Fatigue life and inelastic strain response under complex histories for an alloy steel, J. Test. Eval, ASTM 1 (4) (1973) 271e287; N.E. Dowling, Mechanical Behavior of Materials, second ed., Prentice-Hall, 1999.

Linear least squares fit of log (Nf) vs. Log ( sigmaA )

Y = cycles-to-failure, Nf (cycle)

AISI 4340 steel fatigue data (Dowling, N.E., J. Test. Eval. ASTM, Vol. 1, No. 4, pp. 271-281 (1973))

A power-law model (units in natural scale) : Cycles-to-failure Nf (cycle) = L* { sigmaA (MPa) } ** C, where Nf = Y = 10**yy, and sigmaA = X = 10**xx, or, yy = log_10(Y), and xx = log_10 (X).

10E+8 10E+7 10E+6 10E+5

Equivalent to a linear model (in log-log scale):

Operating stress Amp = 398 MPa.

yy = A + C* xx, where A = yy-intercept = 33.869, s.d. of yy-intercept = 0.716, and Lamda = L = 10**A = 7.40 E+33.

10,000 1,000

95% confidence upper limit

100

C = slope = –10.582, and s.d. of C = 0.252.

10

95% confidence lower limit

1 316

398

501

631

794

1000

1259

X = stress amplitude, sigmaA (MPa)

1585 LLNf.dp + 6data_nf.dat

Where blue circles denote fatigue data as reported by dowling (1973).

Figure 2.3 A logelog plot of the fatigue life versus stress amplitude data of Table 2.6 using Tool-5.

2.6

Model-compute uncertainty (MU2b(cf0)) for model verification

In the last section, we introduced a model uncertainty, MU2a( f ), for a class of continuous functions, y ¼ f(xi), i ¼ 1, 2, 3, ., k, with the requirement that f can be explicitly

56

Durability of Composite Systems

expressed in an algebraic form containing terms involving sums, products, powers, fractions, exponentials, trigonometric, logarithmic, quotients, etc. In general, after we applied one of the three tools, Tool-3 (NUM), Tool-4 (by formulas), and Tool-5 (LLSQ), we recommended for three different cases of the function, f, we proceeded to compute the uncertainty for a specific set S of model solution data that could involve a counting parameter, t, such that the set S is defined as follows: S: fxi ðtÞ; yðtÞg ¼ fx1 ðtÞ; x2 ðtÞ; .; xk ðtÞ; f ðxi ðtÞÞg for i ¼ 1; 2; .; k;

(2.35)

where the first k entries of the set were the values of the k independent variables at a pseudotime parameter, t, and the last entry was the predicted solution of the mathematical model based on the function f. The model-function uncertainty, MU2a( f ), in computational terms, becomes an estimate of the uncertainty for the data set, S, or, MU2a(S). For the example problem introduced in Section 2.5.3 involving a linear function of a single variable, the set S becomes a pair of two numbers, namely, S: foperating stress; mean fatigue life at the operating stressg ¼ f398; 2; 270; 000g: (2.36) In this section, we introduce a second type of model uncertainty, to be known as [Model-Compute Uncertainty], or, symbolically, MU2b(cf ),

where cf is again a continuous function of k independent random variables as in Eq. (2.17) but is no longer expressible as an algebraic function. Instead, it is a black-box type of function defined by a computer algorithm with input data files, output solution files, and a number, K, known as the degree of freedom of the solution algorithm. For a mathematically well-posed problem associated with the model function cf, the solution algorithm for cf should yield a series of approximate solutions as K changes, and that series of candidate solutions should converge to the mathematically correct solution as K approaches infinity. Such a function is sometimes called a “computer-function,” so we name MU2b(cf ) a model-compute uncertainty. There are two kinds of computer functions, y ¼ cf(xi), i ¼ 1, 2, 3, ., k, and we offer two different tools to deal with them. The first and simplest kind, cf0, is deterministic in the sense that we assume the data-set uncertainties, DU1a(xi), of all k independent variables, xi, are zero, but the solution variable, y, is approximate and contains a model-compute uncertainty, MU2b(cf0). The second kind, again denoted by cf, is more general in the sense that we do know the data-set uncertainties, DU1a(xi), of all k independent variables, xi. For the computer function of the first kind, cf0, we offer tool No. 6 (Tool-6), which uses a nonlinear least squares method and is named NLLSQ. For that of the second kind, cf, we offer tool No. 7 (Tool-7), named DEX for the method of design of experiments. As DEX is capable of handling both MU2b(cf) as well as a third type of model uncertainty known as “model-physics uncertainty,” or, MU2c(cf) (to be introduced in the next section), we will, for convenience, postpone presenting Tool-7 until next section for both MU2b and MU2c.

Role of uncertainty in the durability of composite material systems

57

2-parameter logistic function : Y=1–1/{1+ exp[–K*(X–a)] } (Reference: fong-Filliben-Heackert-Marcel-Rainsberger-Ma, 2015)

Logistic function variable Y

1.5

Let y1 = upper bound of a 4-parameter logistic function. Let y0 = lower bound, and L = y1 - y0. Let a = mean, and k = shape steepness coeff. y1 = 1.0

1

(a, L/2), or, (10, 0.5)

(a, L/2)

0.5

y0 = 0.0

0

Legend for 2 S-curves : 2-para.with a = 0, k = 1.

–0.5

2-para.with a = 10, k = 1.

–20

–10

0

10

20

Logistic function variable x

Figure 2.4 Plots of two 2-parameter (a, k) logistic functions where the two asymptotes are assumed to be 0 (lower) and 1 (upper), and the two pairs of (a, k) values are (0, 1) and (10, 1).

To present Tool-6 (NLLSQ) for finding MU2b(cf0), we need to first define a logistic function, which was named after Pierre Francois Verhulst [37] for his use in a study of population growth in 1845. A logistic function is an S-curve with two asymptotes and is commonly represented by the following equation with four parameters, y1, L, k, and a: f ðxÞ ¼ y1  L=ð1 þ expð k  ðx  aÞÞÞ;

(2.37)

where y1 is the upper asymptote, L ¼ y1 e y0 with y0 equal to the lower asymptote, k is the S-curve shape steepness coefficient, and a, the x-value of the S-curve midpoint (sometimes denoted by x0). To visualize this 4-parameter function, let us simplify Eq. (2.37) by assigning y1 ¼ 1, and L ¼ 1 (or y0 ¼ 0). Eq. (2.37) thus becomes a 2-parameter logistic function with two sample plots given in Fig. 2.4. The parameter L is, therefore, a scale factor for the difference between the upper and the lower asymptotes. As we mentioned earlier, a black-box function contains a computer algorithm that produces a sequence of solutions as its degree of freedom, K, changes. As K can be quite large, it is more convenient for us to plot that sequence of solutions versus Log10 K, as K increases to a very large number (a practical limit chosen by the user). This makes the logistic function an ideal candidate to model such a sequence.

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Durability of Composite Systems

4-para. Logistic : Y = y1 – L* (exp(–k*(xx – x0)0 / (1 + exp(–k*(xx – X0)))) where xx = Log_10(X) (nonlinear least squares fit, fong-filliben-heackert, 2015)

5-run predicted stress at infinite d.o.f. = 362.49 +/– 11.27 MPa

Y = maximum mises stress (MPa)

390

(uncertainty estimated at 1-billion d.o.f.)

5-run predicted 95% confidence upper limit

380 370 360 350

1-billion degrees of freedom

340 330

Y (MPa)

X (d.o.f.)

322.45 326.76 339.37 355.02 361.40

10,743 31,476 47,022 74,226 118,750

1-million degrees of freedom

320 4

5

6

7

8

9

10

XX = LOG_10 (X) where X = degrees of freedom (d.o.f) of wrench stress analysis (COMSOL) with 5 runs of Tetra-04 mesh design (blue circles)

Figure 2.5 Plot of a nonlinear 4-parameter logistic function least squares fit of a 5-candidate sequence of finite element analysis solution by COMSOL [38]. After J.T. Fong, N.A. Heckert, J.J. Filliben, P.V. Marcal, R. Rainsberger, Uncertainty of FEM Solutions Using a Nonlinear Least Squares Fit Method and a Design of Experiments Approach, Proc. COMSOL Users Conference, Oct. 7e9, 2015, Boston, MA, U.S.A., 2015. http://www. comsol.com/conference-2015; J.T. Fong, J.J. Filliben, N.A. Heckert, P.V. Marcal, R. Rainsberger, L. Ma, Uncertainty quantification of stresses in a cracked pipe elbow weldment using a logistic function fit, a nonlinear least squares algorithm, and a superparametric method, Procedia Eng. 130 (2015) 135e149; J.T. Fong, J.J. Filliben, N.A. Heckert, P.V. Marcal, R. Rainsberger, A new approach to FEM error estimation using a nonlinear least squares logistic fit of candidate solutions, in: Proc. ASME 2016 V&V Symp., May 16-20, 2016, Las Vegas, NV, Paper No. VVS2016-8912, New York, NY: American Society of Mechanical Engineers, 2016; Fong, J. T., Marcal, P. V., Rainsberger, R., Ma, L., Heckert, N. A., and Filliben, J. J., Finite element method solution uncertainty, asymptotic solution, and a new approach to accuracy assessment, Proc. 7th Annual ASME Verification and Validation Symposium, May 16-18, 2018, Minneapolis, MN, U.S.A., Paper VVS2018-9320. New York, NY: American Society of Mechanical Engineers, 2018.

In Fig. 2.5, we show an example of this in the stress analysis of a wrench using a finite element analysis software named COMSOL [38] and a nonlinear least squares fit macro written in DATAPLOT. It is interesting to observe in Fig. 2.5 that the 95% confidence limits of the asymptotic solution broadened indefinitely as Log10 K increases toward infinity. As a practical matter, we chose in a series of papers [39e42] to set a limit to the maximum degree of freedom to the black-box algorithm to one billion, and defined the

Role of uncertainty in the durability of composite material systems

59

model-compute uncertainty, MU2b(cf0), by the half-width of the 95% confidence prediction interval computed for the asymptotic solution at Log10 K ¼ 9.0. For the example displayed in Fig. 2.5, MU2b(cf0) ¼ 11.27/362.49 ¼ 0.03, or 3%. A full listing of an example DATAPLOT code for NLLSQ (Tool-6) is given in Appendix F. The code may easily be modified for cases when the sequence of candidate solutions tends to decrease, and the reader may consult Fong et al. [39e42] for more information.

2.7 2.7.1

MU2b(cf ) and model-physics uncertainty (MU2c(cf)) Model-compute uncertainty (MU2b(cf)) and Tool-7 (DEX)

In the last section, we introduced a nonlinear least squares (NLLSQ) tool (Tool-6) to compute a model-compute uncertainty, MU2b(cf0), for a deterministic black-box-type of computer function, cf0, with degree of freedom, K. That uncertainty is useful as a verification metric (see, e.g., Fong, et al. [42], Oberkampf and Roy [43]) in assessing the mathematical correctness of the model solution, when all the independent variables, xi, in the equation, y ¼ cf(xi), i ¼ 1, 2, 3, ., k, are assumed to be nonrandom with zero uncertainty. In reality, most engineering models are formulated with variables and parameters that are only known within certain physical limits without precise knowledge of their variability. Our Tool-6 (NLLSQ) is no longer adequate to deal with the so-called computer function of the second kind, cf(xi), where all or most of the variables having uncertainties DU1a(xi). One of the most common tools available for MU2b(cf ) is the Monte Carlo method (see, e.g., Robert and Casella [44], Coleman and Steele [9], Oberkampf and Roy [43], etc.). When the degree of freedom, K, of the computer function, cf, is large, as in the case of many finite element computational models, the Monte Carlo method becomes too costly and sometimes impractical to implement. In this chapter, we introduce an alternative method, to be named Tool-7, that is based on the theory of design of experiments (DEX) (see, e.g., Box, Hunter and Hunter [45]). We begin with the broad notion that an experiment can be either physical, as it is carried out in a laboratory, or numerical, as it is done by pencil and paper, or with the help of a calculator or computer. In any such experiment, we change one or more process variables (factors) to observe the effect, the changes have on one or more response variables. DEX is an efficient procedure for planning experiments so that the data obtained can be analyzed to yield valid and objective conclusions. The statistical theory underlying DEX begins with the concept of process models. A process model of the black-box type is formulated with several discrete or continuous input factors that can be controlled, and one or more measured output responses. The output responses are assumed continuous. Both real (results of physical experiments) and virtual (values conceived from experience and judgment of the experimentalist) data are used to derive an empirical (approximate) model linking the outputs and inputs. These empirical models generally contain first-order (linear) and secondorder (quadratic) terms.

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Durability of Composite Systems

Table 2.7 A 5-factor ultrasonic testing (UT) experiment set-up data. Factor

Variable (Unit)

Low

Center

High

X1

Operator’s experience (year)

2.0

4.0

6.0

X2

UT machine age (year)

2.0

5.0

8.0

X3

Cable length (feet)

6.0

8.0

10.0

X4

Transducer probe angle (degree)

42.0

45.0

48.0

X5

Plastic shoe thickness (inch)

0.25

0.50

0.75

After J.T. Fong, N.A. Heckert, J.J. Filliben, S.R. Doctor, Three approaches to quantification of NDE uncertainty and a detailed exposition of the expert panel approach using the sheffield elicitation framework, in: Proc. 2018 ASME PVP Division Conference, July 2018, Prague, Czech Republic, Paper No. PVP2018-84771. New York, NY: American Society of Mechanical Engineers, 2018. http://www.asmeconferences.org/PVP2018.

The most popular experimental designs are two-level fractional factorial designs. For an excellent introduction of that design, we refer our readers to the book by Box, Hunter, and Hunter [45]. Because of space limitations, we will not repeat it here and instead we shall assume that the reader is somewhat familiar with a general idea of experimental design, and proceed to learn how to use our Tool-7 (DEX) by working with an example DATAPLOT code named fong155_5_8.dp, a listing of which is given in Appendix G. The formulation of the example problem and its solution using Tool-7 (DEX) have also appeared in Fong et al. [46]. In Table 2.7 and Fig. 2.6, we see the formulation of a 2-level, 5-factor, 8-run fractional factorial UT experimental design problem. The response variable is the

Ordered response Y (crack length (mm))

70

8-run UT experiment for detecting flaw “L” response = crack length Ordered data plot

k=6 n=8

60

Y X1 X2 X3 X4 X5 Lg(mm) OpExp MachAge CabLen Angle Thick 43.2 54.6 44.5 67.3 57.2 53.3 33.0 63.5

50

40

–1 +1 –1 +1 –1 +1 –1 +1

–1 –1 +1 +1 –1 –1 +1 +1

–1 –1 –1 –1 +1 +1 +1 +1

+1 –1 –1 +1 +1 –1 –1 +1

+1 –1 +1 –1 –1 +1 –1 +1

30 X1: X2 X3: X4: X5:

– + + – –

– – – + +

– + – – +

+ – + – +

+ – – – –

– – + + –

+ + + + +

+ + – + –

1: OperExp 2: MachAge 3: CableLen 4: ProbeAng 5: ShoeThick

Factor settings

Figure 2.6 A 2-level, 5-factor, 8-run fractional factorial design of a UT experiment for finding crack length (mm) [46].

Role of uncertainty in the durability of composite material systems

61

8-run UT experiment for detecting flaw “L” response = crack length main effects plot

Mean response Y (crack length (mm))

60

k=5 n=8

Mean = 52.075

55

50

45

40

15.2 29 95.51 *

–

+

X1 OperExp

0 0 0 –

0.65 1 5.75 +

X2 MachAge

–

+

X3 CableLen

11.45 22 83.37

1.9 4 16.74

–

–

+

X4 ProbeAng

|Effect| Rel.|Effect| (%) Fcdf (%)

+

X5 ShoeThick

Factors (5)

Figure 2.7 Plot of main effects from the analysis of a 2-level, 5-factor, 8-run DEX. After J.T. Fong, N.A. Heckert, J.J. Filliben, S.R. Doctor, Three approaches to quantification of NDE uncertainty and a detailed exposition of the expert panel approach using the sheffield elicitation framework, in: Proc. 2018 ASME PVP Division Conference, July 2018, Prague, Czech Republic, Paper No. PVP2018-84771. New York, NY: American Society of Mechanical Engineers, 2018. http://www.asmeconferences.org/PVP2018.

crack length to be found by the ultrasonic testing (UT) operator using a machine with a cable, a plastic shoe, and a transducer probe. In Fig. 2.7, we display 1 of the 10 plots typically generated by Tool-7 (DEX), where the main effects of the five factors on the crack length response variable are plotted. Acting on the result that X1 and X4 are dominant, the code will follow up with a regression analysis giving the result of an uncertainty estimation in Fig. 2.8, where the uncertainty, MU2b(cf), for the UT experimental determination of crack length is found to be equal to 14.8/52.1 ¼ 0.28, or 28%. In Appendix G, we included the DATAPLOT code, fong155_5_8.dp with an input data file named fong155_5_8.dat, to show how the specific information for the UT experimental design was incorporated into the input file. To illustrate how the same tool can be applied to a different problem, we need to change that input data file, fong155_5_8.dat. In Fong et al. [47], we did exactly that by applying the tool to the analysis of the vibration of a single-crystal silicon cantilever beam in an atomic microscope, of which no analytical solution existed. Using a finite element software package named ANSYS, we found the uncertainty MU2b(cf) to be 5% [47].

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Durability of Composite Systems

Uncertainty quantification of UT measurement of length of a subsurface crack 95% confidence predictive limits based on a design of experiments (DEX) exercise 0.1

Probability

0.075

DEX: K= no. of factors. n = no.of runs. For this exercise, k = 5, n = 8.

Crack length = L = 52.1 (14.8) mm. (based on a linear regression on two dominant factors)

X1 and X4 are dominant.

0.05

66.9 mm. 37.3 0.025

95% confidence

95% conf. Predicitve lower limit

Predictive upper limit

0 20

30

40

50

60

70

80

90

Crack length (L), mm (n = 8, sd = 5.4) Figure 2.8 Crack length distribution plot from the analysis of a 2-level, 5-factor, 8-run DEX with the mean and uncertainty estimated at 95% confidence level using Tool-7. After J.T. Fong, N.A. Heckert, J.J. Filliben, S.R. Doctor, Three approaches to quantification of NDE uncertainty and a detailed exposition of the expert panel approach using the sheffield elicitation framework, in: Proc. 2018 ASME PVP Division Conference, July 2018, Prague, Czech Republic, Paper No. PVP2018-84771. New York, NY: American Society of Mechanical Engineers, 2018. http://www.asmeconferences.org/PVP2018.

2.7.2

Model-physics uncertainty (MU2c(cf)) and Tool-7 (DEX)

When one models a material system with a computer function, one needs to ask two fundamental questions, namely, (1) is the model mathematically correct? and (2) is the model physically correct? As defined by Oberkampf and Roy [43], the process of answering the first question is verification, and that of the second, validation. In Section 2.6, we introduced the model-compute uncertainty, MU2b(cf0), and Tool-6 to answer the first question (verification) for a model with a deterministic computer function, cf0. In the previous Section 2.7.1, we introduced the model-compute uncertainty, MU2b(cf), and Tool-7 (DEX) to answer the same question (verification) for a model with a general computer function, cf. We are now ready to answer the second question (validation) by introducing a third model uncertainty, to be called: [Model-Physics Uncertainty], to be symbolically denoted by MU2c(cf),

where the uncertainty is defined as a “measurement uncertainty” within the framework of the terminology used in the International Vocabulary of Metrology [2] and the Guides to the expression of Uncertainty in Metrology [3]. As a matter of fact, modelphysics uncertainty, MU2c( f ). Also exists and can be defined for any f that satisfies Eq. (2.17).

Role of uncertainty in the durability of composite material systems

63

It turns out that our Tool-7 (DEX) is fully capable of finding either MU2c(cf), or MU2c( f ), as demonstrated in Section 2.7.1, where we found the uncertainty of an 8-team measurement exercise using the ultrasonic testing equipment to measure crack length. So, we conclude this subsection by recommending Tool-7 (DEX) as one of two tools to find either MU2c(cf), or MU2c( f ),

2.7.3

Model-physics uncertainty (MU2c(cf )) and Tool-8 (ASTM E691)

To find the model-physics uncertainty, MU2c(cf), or MU2c( f ), it is common to conduct the same experiment in three or more laboratories to validate a model, that is, to find out if the physics of the model is correct. It turns out that there exists a DATAPLOT code that can handle this task, and the reader is referred to Refs. [48e50] and the following link: https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/ e691.htm to learn how to download and apply this tool, which is named Tool-8 (ASTM E691).

2.8

Model-system uncertainty (MU3(sf)) for system verification

In Section 2.6, we defined for a model M, a computer function, cf, and a modelcompute uncertainty, MU2b(cf), where cf has k random variables, xi, as follows: Model M : y ¼ cf ðxi Þ; i ¼ 1; 2; 3; .; k:

(2.38)

In Section 2.2.3, we introduced a system W of k models that are connected either in series, in parallel, or in any combinations thereof, such that the system output, w, is a function of the k outputs, yj, j ¼ 1, 2, ., k, of the k models acting as independent variables, each of which has its own model uncertainty as described in Section 2.2.2: System W : w ¼ sf ½y1 ðxq1 Þ; y2 ðxq2 Þ; .; yk ðxqk Þ

(2.39)

Where the subscripts, q1, q2, ., qk, can each vary from one to some finite number depending on the make-up of each model, yj, j ¼ 1, 2, ., k. We also introduced a model uncertainty named: [Mode-System Uncertainty], denoted symbolically by, MU3(sf).

By comparing Eq. (2.39) with Eq. (2.38), we observe that mathematically speaking, the two computer functions, sf and cf, are no different, because both have random variables as arguments with known uncertainties. We, therefore, conclude that a modelsystem uncertainty is the same as a model-compute uncertainty, and the recommended tool, Tool-7 (DEX), for i MU2b(cf) is also applicable to MU3(sf).

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Durability of Composite Systems

2.9

Four types of durability with uncertainty for modeling material systems

In Sections 2.3e2.8, we introduced seven types of uncertainty and eight tools to compute as listed in Table 2.8. We are now ready to introduce and define four types of DuU, which are known to occur when we model the fatigue and durability behavior of a material system. The four types of durability uncertainty are: (DuU1a) Durability Uncertainty for Smooth Simple System under Cyclic Load without scaling. (DuU1b) Durability Uncertainty for Cracked Simple System under Cyclic Load without scaling. (DuU2a) Durability Uncertainty for Validating an FRP Composite Elastic Constants Database without scaling. (DuU2b) Durability Uncertainty for 2D-Holed Square Composite Plate under Static Load.

In Section 2.10, we show two examples to compute DuU1a (Fong et al. [36,57]) using Tool-5 and Tool-6. In Section 2.11, we show a third example to compute DuU1b (Fong et al. [31]) using Tool-4. In Section 2.12, we show a fourth example of computing DuU2a as reported by Melo and Fong [60] using again Tool-4. Finally, in Section 2.13, we show the fifth example to compute DuU2b as reported by Shah, Melo, Cimini, and Fong [64] using Tool-7 and Tool-2.

Table 2.8 List of uncertainty types and tools to compute for modeling materials systems. Symbol

Category

Type of uncertainty

Tool(s) to compute

Remark

DU1a

Data

Data-set uncertainty

Tool-1 (goodness-of-fit)

Appendix B

DU1b

Data

Data-parameter uncertainty

Tool-2 (confidence interval)

Appendix C

DU1c

Data

Data-coverage uncertainty

Tool-2 (confidence interval)

Appendix C

MU2a

Model

Model-function uncertainty

Tool-3 (uncertainty machine) Tool-4 (error propagation) Tool-5 (LLSQ)

Appendix D Section 2.5.2 Appendix E

MU2b

Model

Model-compute uncertainty

Tool-6 (NLLSQ) Tool-7 (DEX)

Appendix F Appendix G

MU2c

Model

Model-physics uncertainty

Tool-7 (DEX) Tool-8 (ASTM E691)

Appendix G Section 2.7.3

MU3

System

Model-system uncertainty

Tool-7 (DEX)

Appendix G

Role of uncertainty in the durability of composite material systems

2.10

65

Durability with uncertainty (DuU1a) for a smooth simple system under cyclic load without scaling

A recent sampling of the literature on fatigue and durability of composite materials since 1969 [51e54] indicated tremendous progress in our understanding of the damage mechanisms and life prediction of composite materials. With the exception of material property databases such as the MIL-HDBK-17 [55] and the Updated Composite Materials Handbook (CMH-17) [56], most research results in the composite durability literature were still reported without a proper expression of their uncertainty. In this section, we will show two examples of how to express the simplest type of durability uncertainty, namely, DuU1a. Our first example (steel) has been given in Section 2.5.3, where we have shown in Fig. 2.3 that, at the operating stress level of 398 MPa, the mean cycles to failure were 2.27 million cycles, and the durability uncertainty, DuU1a, was 41% (normal distribution, 95% confidence). Our second example (plain concrete) was reported by Fong et al. [57], using data published by Lee and Barr [58] in 2004. We show in Fig. 2.9 the original plot by Lee and Barr [58] without the confidence limits, and in Fig. 2.10 the same plot with the limits [57]. In Fig. 2.11, we show the analysis results by applying our Tool-5 (LLSQ) to the same data with Log10 (life N) as the y-coordinate to get the proper confidence limits for the life N. At the operating stress level of 0.3 times the ultimate compressive strength, the predicted mean cycle to failure was found to be 1.12 billion cycles,

Figure 2.9 A fatigue SeN curve for plain concrete under compression using a linear least squares fit with the absence of the lower and upper confidence limits. After M.K. Lee, B.I.G. Barr, An overview of the fatigue behaviour of plain and fibre reinforced concrete, Cement Concr. Compos. 26 (2004) 299e305.

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Durability of Composite Systems

3-para. logistic nonlinear LSQ fit : y = U0 –(U0 – y1)/(1+ exp( –K*(xx-X0))) where xx = Log_ 10(X), and X = cycles-to-failure, N (material: plain concrete)

1.0

Linear LSQ semi-log model U0 = static ultimate strength. y = fatigue stress amplitude. y = y / U0 (non-dimensional).

Y = non-dim. stress amplitude

0.9 0.8 0.7 0.6 0.5 0.4 0.3

3-para logistic model predicted (Nonlinear LSQ fit) Endurance stress = 0.567 *U0 ( +/– 0.149 * U0).

Min. endurance at 95% lower confidence limit = 0.418 * U0 (with N> 1.0 E+15 cycles)

Linear model predicted N at Op. stress of 0.3*U0 = 1.78 E+12

0.2 –2

0

2

4

6

8

10

12

14

16

18

20

xx = Log_10 (X), where X = cycles-to-failure, N (cycles) of

plain concrete subjected to an operating stress amp. of 0.3 x static ultimated strength.

Figure 2.10 A nonlinear least squares logistic function fit of an SeN curve of a plain concrete fatigue test superimposed over the result of a linear least squares fit. After J.T. Fong, N.A. Heckert, J.J. Filliben,P.H. Ziehl, A nonlinear least squares logistic fit approach to quantifying uncertainty in fatigue stress-life models and an application to plain concrete, in: Proc. 2018 ASME PVP Division Conference, July 2018, Prague, Czech Republic, Paper No. PVP2018-84739. New York, NY: American Society of Mechanical Engineers, 2018. http://www.asmeconferences.org/PVP2018.

and the 95% upper and lower confidence limits were 62.14 and 0.02 billion cycles, respectively. The DuU1a, was 2770% (normal distribution, 95% confidence).

2.11

Durability with uncertainty (DuU1b) for a cracked simple system under cyclic load without scaling

Consider a textbook exercise problem given by Dowling [35, pp. 522e524, Example 11.4], where a center-cracked plate (half-width ¼ 38 mm, thickness ¼ 6 mm) of the AISI 4340 steel was given an initial crack length, ai ¼ 1 mm, and subjected to a cyclic tensile load between 80 and 240 kN. This problem exists in reality as shown in Fig. 2.12. Two questions were asked: (1) At what crack length af is failure expected? Is the failure due to yielding or brittle fracture? (2) How many cycles can be applied before failure occurs?

The answer to the first question was (a) af ¼ 15.8 mm, and (b) it failed by brittle fracture. The answer to the second question was (c) Nif ¼ 70,600 cycles, where Nif is the remaining service (fatigue) life of a component from an initial state (i) when a crack was first detected with length, ad (¼ ai), to a hypothetical final state ( f ) defined by the existence of a critical crack length, ac (¼ af) when a catastrophic failure occurs.

Role of uncertainty in the durability of composite material systems

67

Linear least square fit of Log_10 ( fatigue life, N) vs. stress level (S) material is plain concrete (after Lee and Barr (2004), Fig. 1(a)) 15

Oper. stress = 0.30

Y = Log_10 (N)

Predicted Mean N = 1.12 E+09 cycles 10

5

Linear least squares Log(N) vs. stress model: Y = C * X + A, where Y = Log_10 (N), and N = No. of Cycles, X = stress level, S.

0

C = Slope = – 11.53 , sd(C) = 0.9322 , and A = Y-intercept = 12.505 , sd(A) = 0.7754. 0

0.25

0.5

0.75

1

X = stress level, S.

Figure 2.11 A linear least squares fit of 82 data points on the fatigue of plain concrete using Tool-5 (LLSQ) and the DATAPLOT code, LLSQ.dp, given in Appendix E. After J.T. Fong, N.A. Heckert, J.J. Filliben,P.H. Ziehl, A nonlinear least squares logistic fit approach to quantifying uncertainty in fatigue stress-life models and an application to plain concrete, in: Proc. 2018 ASME PVP Division Conference, July 2018, Prague, Czech Republic, Paper No. PVP2018-84739. New York, NY: American Society of Mechanical Engineers, 2018. http://www.asmeconferences.org/PVP2018.

÷XN

Failure, N if

a, crack length

ac (a) Inspect

ad 0

(a) Deterministic formulation

ad

Np

ad

2Np

ad

3Np

^ N

N, cycles

Figure 2.12 Growth of a worst-case crack from the minimum detectable length ad to failure critical length ac, and variation of worst-case scenarios with periodic inspections. After N.E. Dowling, Mechanical Behavior of Materials, second ed., Prentice-Hall, 1999.

68

Durability of Composite Systems

Nif is given by the following formula (DS ¼ Smax  Smin) (see Eq. (11.32) in Dowling [35, p. 520]): 1m=L

Nif ¼

1m=L

 ai pffiffiffi m CðFDS p Þ ð1  m=2Þ af

ðm s 2Þ

(2.40)

where the initial crack length, ai, and the crack growth exponent, m, are assumed constant. In reality, as shown in Figs. 2.13e2.16 m can vary considerably with an uncertainty of about 13% (95% confidence). Based on a result in Eq. (2.33) as derived in Section 2.5.1 using our Tool-4 (error propagation), the uncertainty, DuU1b(Nif), equals 23.8 * DU1a(m) ¼ 23.8 * 0.13 ¼ 3.09, or 309% (95% confidence), indicating that the model is unable to predict a lower limit.

? 5

4

For a/W = 0.25

Experimental results of Von Euw et al. a/w = 0.25 a/w = 0.45

predicted da/dN = 3.0 x 10–5

?

da/dN , 10–5 in.

3.35 3

2.7

Equation (8.2-33) for a/w = 0.25

0.9

Equation (8.2-33) for a/w = 0.45

2

2.15

1.8 For a/W = 0.45

1

0

1.2

predicted da/dN = 2.5 x 10–5

5

10

15

20

25

'K, ksi in.1/2

Figure 2.13 Comparison of predicted and experimental fatigue crack growth rate results in 2024-T3 aluminum alloy for R ¼ 0. After E.F.J. von Euw, R.W. Hertzberg, R. Roberts, Delay effects in fatigue-crack propagation, in: Stress Analysis and Growth of Cracks, ASTM STP 513, pp. 230-259. American Society for Testing and Materials, 1972.

Role of uncertainty in the durability of composite material systems

69

LLSQ fit of log(crack growth rate) vs log(delta-K), Y = da/dN (1.0E–05 in/cycle), material is 2024–T3 aluminum. specimen size ratio: a/w = 0.25. A power-law model (units in natural scale) :

Y = da/dN (1.0E–05 in/cycle )

100

Crack growth rate da/dN (1.0E–05/cycle) = L* { delta-K (Ksl–sqrt(in)) } ** m, where da/dN = anti-Log (yy), and delta-K = anti-log (xx). 10

1

Equivalent to a linear model (in log-log scale): 95% confidence upper limit

yy = A + m * xx, where A = yy-intercept = –3.29, s.d. of yy-intercept = 0.194 , and Lamda = L = anti-Log (A) =0.51 E–03.

0.1 95% confidence lower limit

0.01 5.0

6.3

7.9

10

m = slope = 2.86 , and s.d. of m = s = 0.16, cv = s/m = 0.06 12.6

15.8

20

X = delta–K (Ksi-sqrt(in) )

25.1 LLSQ6.dp + 17data_cgr6.dat

Figure 2.14 An LLSQ fit of 2024-T3 aluminum crack growth rate data of von Euw et al. [59] for a/w ¼ 0.25. The data-parameter uncertainty, DU1b(m) ¼ 0.12, or, 12% (95% confidence).

LLSQ fit of log(crack growth rate) vs log(delta-K), Y = da/dN (1.0E–05 in/cycle), material is 2024–T3 aluminum. specimen size ratio: a / w = 0.45. A power-law model (units in natural scale) :

Y = da/dN (1.0E–05 in/cycle )

100

Crack growth rate da/dN (1.0E–05/cycle) = L* { delta-K (Ksl–sqrt(in)) } ** m, where da/dN = anti-Log (yy), and delta-K = anti-log (xx). 10 Equivalent to a linner model (in log-log scale): 1

95% confidence upper limit

yy = A + m * xx, where A = yy-intercept = –2.93, s.d. of yy-intercept = 0.227 , and Lamda = L = anti-Log (A)

0.1

= 1.18 E–03. 95% confidence lower limit

0.01 5.0

6.3

7.9

10

m = slope = 2.51 , and s.d. of m = s = 0.18, cv = s/m = 0.07 12.6

15.8

X = delta-K (Ksi-sqrt(in) )

20

25.1 LLSQ4.dp + 13data_cgr4.dat

Figure 2.15 An LLSQ fit of 2024-T3 aluminum crack growth rate data of von Euw et al. [59] for a/w ¼ 0.45. The data-parameter uncertainty, DU1b(m) ¼ 0.14, or, 14% (95% confidence).

70

Durability of Composite Systems

qNif qac

÷XN

a, crack length

ac

Failure, Nif

(a)

(b) Stochastic formulation

Inspect

qad ad 0

ad

qNp

Np

ad

2Np

ad

3Np

^ N

N, cycles

Figure 2.16 Growth of a worst-case crack from the minimum detectable length ad to failure critical length ac with periodic inspections in a stochastic formulation. After Fong, Heckert, Filliben, and Freiman [36].

2.12

2.12.1

Durability with uncertainty (DuU2a) for validating an FRP composite elastic constants database without scaling A database of elastic constants and thermal expansion coefficients for FRP

To model the durability property of any composite material system, it is imperative that a material property database exists for that system, and that each entry in that database be expressed with uncertainty, properly sourced, and validated. Unfortunately, such is rarely the case, as shown in Fig. 2.17, as shown in an example page from MIL-HDBK17 [55, p. 4e29], where the Poisson ratio, n12, was missing, and there was no evidence that neither of the two moduli was validated. To fulfill our second goal of this chapter, namely, to show the role of uncertainty in the durability of composite material systems, there is no better way to start with an example problem of determining the five elastic constants, E1, E2, n12, G12, and n23, of a specific fiber-reinforced plastic (FRP) tape, expressing them with durability uncertainty, DuU1a, and finding a way to validate them. In 2010, Melo and Fong [60] reported experimental and data analysis results that clearly showed the role of uncertainty in modeling the durability of an FRP material named the APC-2/IM7, PEEK/carbon unidirectional prepreg tape, which was manufactured by Cytec Engineered Materials, with a quoted resin content of 32% and fiber areal weight of 145 g/m2. To present the results of Melo and Fong [60], we begin with the sign convention as shown in Fig. 2.18, where the coordinates (1; 2; 3) refer to the local coordinate system for a single layer of lamina, which corresponds to (1) the fiber direction, (2) the in-plane direction transverse to the fiber, and (3) the through-thickness direction, respectively. The coordinates (x; y; z) correspond to the global or laminate directions. In 2002, Melo and Radford [61] developed a set of equations that relate the coefficients of thermal expansion (CTEs) of a [(þq/q)n]s laminate and the lamina elastic constants. Such equations have also appeared later in Tsai [ [62], Appendix A,

Role of uncertainty in the durability of composite material systems

71

Figure 2.17 A truncated page from MIL-HDBK-17 [55, p. 4e29] for properties of a Celon tape. z-axis or 3-axis (laminating or through-thickness direction)

x-axis 1-axis (fiber direction)

θ

y-axis y-axis x-axis 2-axis (transverse direction)

Figure 2.18 Definition of local (lamina) reference axes (1e2e3) and global (laminate) ones [60].

72

Durability of Composite Systems

Eq. (4.33)]. More specifically, when q ¼ 30 degree, the equation for the modulus, E2, has the following form: E2 ¼

E1 ð4a1  3ax ð30Þ  ay ð30ÞÞ 4v12 ðax ð30Þ þ ay ð30Þ  a1  a2 Þ þ ax ð30Þ þ 3ay ð30Þ  4a2

(2.41)

where a1 and a2 are the CTEs of the unidirectional lamina in the 1 and 2 directions, respectively, and ax(30), ay(30), and az(30) are the CTEs of the [(þq/q)n]s laminate in the x, y, z directions, respectively. By symmetry, a3 ¼ a2, so that a3 does not appear in Eq. (2.41). For brevity, we omit the display of two other equations that relate G12 and n23 to the other variables as shown in Eq. (2.41). When the eight variables on the right-hand-side of Eq. (2.41) are measured, a theoretical value of E2 is obtained and can be used to validate a measured E2.

2.12.2

Estimation of data-set uncertainty of elastic constants of FRP without scaling

As reported by Melo and Fong [60], nine sets of measurements in a laboratory with four specimens in each test were made with the results shown in Table 2.9: In this example, we choose to express uncertainty as one standard deviation (68% confidence, normal). Then, from Table 2.9, we learn that the experimental value of E2 is 9.9 GPa, and the durability uncertainty of E2 ¼ DuU1a(E2) ¼ DU1a(E2) ¼ 0.2/ 9.9 ¼ 0.02, or 2% (68% confidence, normal).

Table 2.9 Experimental results of nine tests with four samples in each test.

After J.D.D. Melo, J.T. Fong, A new approach to creating composite materials elastic property database with uncertainty estimation using a combination of mechanical and thermal expansion tests, in: Proc. ASME 2010 Pressure Vessels & Piping Division/K-PVP Conference, July 18-22, 2010, Bellevue, WA, U.S.A., Paper No. PVP2010-26144. New York, NY: American Society of Mechanical Engineers, 2010.

Role of uncertainty in the durability of composite material systems

73

Table 2.10 Formulas for variances of sum, quotient, and product of two variables. #

Function form

Approx. formula for variance

1

w ¼ Ax þ By

varðwÞ ¼ A2 $varðxÞ þ B2 $varðyÞ

2

w ¼ x=y

varðwÞ ¼ varðxÞ þ varðyÞ$ðxÞ 2 4

3

w ¼ x$y

varðwÞ ¼ ðyÞ $varðxÞ þ ðxÞ2 $varðyÞ

2

ðyÞ 2

ðyÞ

where: x ¼ sample mean of x; y ¼ sample mean of y; sneI ¼ sample standard deviation; var. ¼ square of std. dev.(SneI).2 After H.H. Ku, Notes on the use of propagation of error formulas, J. Res. Natl. Bur. Stands. 70C 4 (1966) 263e273.

To find a theoretical value of E2, we need to apply our Tool-4 (error propagation), as described in Section 2.5.2, to the right-hand-side of Eq. (2.41). As Eq. (2.41) only involves sums, products, and quotient of two random variables, we only need to apply Eqs. (2.19), (2.20), and (2.22), which we reproduce in Table 2.10 for our reader’s convenience: We now apply Tool-8 (error propagation) to Eq. (2.41) as shown in Table 2.11: As variance is the square of standard deviation, and the means and standard deviations of the eight variables in the right-hand-side of Eq. (2.41) are given in Table 2.9, we can compute the theoretical value of E2 from Eq. (2.39), and its variance from Table 2.11 as follows: E2 ¼ 13:2 GPa; varðE2 Þ ¼ 18:18 GPa2 ; and s:d:ðE2 Þ ¼ 4:3 GPa

(2.42)

Table 2.11 Define w1,., w6, and derive var(w1),., var(w6) using formulas in Table 2.10.

74

Durability of Composite Systems

Eq. (2.40) allows us to conclude that the theoretical durability uncertainty of E2 ¼ DuU1a(E2) ¼ DU1a(E2) ¼ 4.3/13.2 ¼ 0.33, or 33% (68% confidence, normal).

2.12.3

Validation of an FRP composite elastic constants database without scaling

A two-sided t-test [12,63] is conducted to determine whether the theoretical value of E2 (¼13.2  4.3 GPa) is acceptable with 95% confidence as compared with the experimental value (¼9.9  0.2 GPa), which was obtained for a sample size of n ¼ 4. The null hypothesis is that E2 ¼ 9.9 GPa is true. We first calculate the t-statistic as t ¼ (13.2  9.9)/(4.3/O4) ¼ 1.555. From a table of t-values, we obtain t (0.025, 3) ¼ 3.183. As 1.555 is not greater than 3.183, the null hypothesis is not rejected, and the theoretical value of E2 is statistically valid (95% confidence).

2.13

Durability with uncertainty (DuU2b) for a 2D-holed square composite plate under static load

In the last section, we mentioned that many handbook-based composite material property databases (see, e.g., Refs. [55,56]) contain incomplete information. Sometimes only the test average values of material properties are given. Failure envelopes generated from these databases are thus deterministic. It would then be desirable to include uncertainties in order to be able to assess the reliability of the design. Computing the so-called A-basis (99% coverage; 95% confidence) and B-basis (90% coverage; 95% confidence) design allowable is a natural way for us to introduce to our readers the durability uncertainty, DuU2b. In this section, a case study by Shah, Melo, Cimini, and Fong [64] is presented, where an average design allowable failure envelope of open-hole specimens was obtained numerically (finite element) for a carbon fiber-epoxy laminate using material property data of smooth specimens by Soutis and Lee [65]. Shah et al. [64] used Tool-7 (design of experiments), to obtain uncertainty estimates for the average design allowable failure envelope. We will go further by adding the results of applying Tool-2 (coverage) to obtain not only the durability uncertainty, DuU2b, but also the A-basis and B-basis design allowable failure envelopes. We begin with the geometry of an open hole square notched plate of 32 mm  32 mm with a central circular opening of diameter 6.35 mm as shown in Fig. 2.19(a). The laminate was made up of quasi-isotropic [45/90/45/0]4s IM7/ 8552 composite, with an overall thickness of 4 mm. The mechanical properties of the material are shown in Table 2.12. To apply Tool-7 (DEX), we need to vary the four ply strengths within feasible ranges as shown in Table 2.13, where a 2-level, full-factorial design for four factors plus a center point has been chosen. This means we need a total of 24 þ 1, or, 17 runs, designated as Runs 1e17. Finite element analysis results for each plate material according to the run number will be considered as input data for DEX runs using a DATAPLOT code, fong_155_4_16.dp, which is a slight modification of the code, fong155_5_8.dp, given in Appendix G. The finite element method (FEM) model was created with conventional shell elements of quadrilateral S4R type.

Role of uncertainty in the durability of composite material systems

(a)

75

(b)

V2

I 6.35 mm 32 mm

V1

V1

32 mm

2 θ

V2

1

Figure 2.19 (a) A 4-mm thick open hole IM7/8552 square plate. (b) A finite element mesh design. Table 2.12 Ply properties of [45/90/e45/0]4s IM7/8552 composite plate. E11 GPa

E22 GPa

G12 GPa

vl2

X/X0

150

11

4.6

0.3

2400/1690

MPa

Y/Y0 MPa

S MPa

Thickness mm

111/250

120

0.125

After C. Soutis, J. Lee, Scaling effects in notched carbon fibre/epoxy composites loaded in compression, J. Mater. Sci. 43 (2008) 6593e6598.

Table 2.13 A 4-factor, 17-run, 2-level full-factorial orthogonal experimental design (Tool-7).

76

Durability of Composite Systems

As delamination was not observed in the experimental study [65] for the considered plate thickness, no delamination was simulated. A structured meshing scheme was implemented with 512 elements and 544 nodes as shown in Fig. 2.19(b). A simultaneous degradation scheme was chosen to compute the tensile or compressive ultimate strength of the specimen. In that scheme, once a failure initiation was noticed as per Tsai-Wu criterion, elastic properties of the whole laminate are degraded to obtain the ultimate ply failure strength. A typical result of the uncertainty analysis using Tool-7 (DEX) consists of a plot of the main effects (Fig. 2.20), a plot of the solution with uncertainty, DuU2b, as shown in Fig. 2.21, and a table of solutions with uncertainty bounds (Table 2.14). Using fong820a.dp, a sample DATAPLOT code given in Appendix C for Tool-2 (Coverage), we obtain new results on the A- and B-basis design allowable failure envelope as shown in Table 2.15, Figs. 2.22 and 2.23.

2.14

DuU3, DuU4, ., DuUn for modeling durability of composite material systems

In Section 2.12 and 2.13, we show in two examples how to apply two of our set of eight tools, namely, Tool-4 (error propagation) and Tool-7 (design of experiments), to estimate laboratory-scale durability uncertainty, DuU, when a model function equation exists either in an algebraic form, as in Eq. (2.39), or in a computer function form as in Section 2.13. To obtain component durability uncertainty, CduU, we leave it to Composite open hole specimen using MicMac/FEA with uncertainty estimation Main effect plot

Mean response Y (Ult. strength, Mpa)

270

k=4 n=16

260 250 240 230 220 210 200

66.44 28%

–

X1 X

+

–

0.00 0% X2 Xp

+

–

0.0425 0% X3 Y

+

0.0075 0%

–

X4 Yp

+

Factors

Figure 2.20 One of the 10 plots by Tool-7 (DEX) showing the main effects of the four factors.

Role of uncertainty in the durability of composite material systems

77

Composite open hole specimen biaxial +1 +0 (ASTM D5766) with MicMac/FEA 95% uncertainty bounds plot with dataplot (Fong-Marcal-Filliben-Shah, 2010) 0.04

Probability

0.03

0.02

“DEX” = Design of Experiments, where k = no. of factors, n = no. of runs.

Ultimate strength ( + U1 ), MPa = 266.25 ( 42.04 ) (for a 17 - run “DEX” with k = 4, n = 16 plus a center point)

95% lower bound = 224.21

95% upper bound = 308.29

0.01

0

Expt. data (n = 5, s = 21.4) due to Camanho, et al. (2007): U = 480.6 (65.08) MPa (95% conf.)

190 232.5 275 317.5 360 Ultimate strength (+ U1), MPa (n = 17, s = 18.88) T300/N520: X = 2400 +/– 14 %, Xp = 1690 +/– 21 %, Y = 110 +/– 11 %, Yp = 250 +/– 20 %

Figure 2.21 Open-hole laminate ultimate tensile strength for Case (a): load ratio s1:s2 ¼ 1:0 [64].

Table 2.14 Open-hole laminate strengths estimated with uncertainty using TooI-7 (DEX). Estimated open-hole laminate strength MPa Load ratio s1: s2

Mean

Standard deviation

Coefficient of variations

95% bounds

1:0

266.25

18.88

0.071

42.04

1:1

415.19

29.41

0.071

65.50

0:1

266.25

18.88

0.071

42.04

1:1

124.69

13.21

0.106

29.43

1:0

169.71

18.00

0.106

40.08

1: 1

264.69

27.99

0.106

62.32

0: l

169.71

18.00

0.106

40.08

1: 1

124.69

13.21

0.106

29.43

our readers to apply Tool-1 (goodness of fit) and Tool-2 (coverage) to obtain the critical result for applications to the design, operation, and maintenance of full-size components. We conclude this chapter with a challenge to our readers that, in the years to come, many more durability uncertainties, DuU3, DuU4, ., DuUn, might be computed with the tools provided in this chapter to advance our understanding, manufacturing, and management of composite material systems.

78

Durability of Composite Systems

Table 2.15 A- and B-basis design allowable estimated from smooth specimen data (TooI-2).

600

V2 (Mpa)

Case (b). 1:1

400

200

–400

–200

0

Case (a). 1:0 V1 (Mpa) 0

200

400

600

Case (c). 1:–1 –200 Failure envelope for run –1 Data points for runs –2 to 17 –400

Case (d). 0:–1

Case (e). –1:–1

Figure 2.22 Five cases of the failure envelope.

Role of uncertainty in the durability of composite material systems

79

σ 2 (Mpa)

σ 1 (Mpa)

A-basis allowable (99% coverage). B-basis allowable (90% coverage). Estimated mean (smooth specimens).

Figure 2.23 Mean values and two design allowable.

Appendix A Statistical analysis software package named DATAPLOT (DP) What is DATAPLOT? DATAPLOT is a free, public-domain, multi-platform (Unix/Linux, MacOS, Windows 7/8/10) software system for scientific visualization, statistical analysis, and non-linear modeling. After the software is downloaded and installed, it can be run either in a traditional command line mode or with a graphical user interface (GUI). A unique feature of DATAPLOT is that all of its commands are English based. For example, some of its most used command lines begin with words like LET, READ, PLOT, FIT, TITLE, CHARACTER, COLOR. END, EXIT, etc. For details about DATAPLOT, see Refs. [10e12].

How to download software and its documentation? To access the DATAPLOT homepage, use the following link: https://www.itl.nist.gov/div898/software/dataplot/homepage.htm. In addition to an introduction of the software, four tables will appear online as follows: Every entry in each of the four tables can be clicked to open a new web page with information on DATAPLOT. For example, in Table A.1, one can click Download to download the software, and Documentation to download documentation, as indicated by two red arrows. In Table A.2, one can click Getting Started, and obtain an introduction on how to get started by clicking on four more links as follows: 1 Getting Into and Out of Dataplot 2 Setting Your Graphics Devices and Printing Graphics 3 A Simple Example

80

Durability of Composite Systems

Table A.1 Dataplot: Introduction.

Table A.2 Dataplot tutorial. Getting started

Language features

GUI mode

Command mode

I/O

Data types

Supported graphics devices

Program control

Customizing dataplot

Command categories

Commands

Distributions

Table A.3 Dataplot resources. Elementary operations

Functions

Random number generation

Experiment designs

Macros

Programs

Data set library

Text file library

Table A.4 Dataplot special features. Lines

Characters

Spikes

Bars

Plot annotation

TEXT subcommands

Colors

Sizing

4 Typical Problems

By clicking A Simple Example, we obtain the problem statement, a DATAPLOT code with a built-in input data list, and the two output plots as shown in the next section.

Role of uncertainty in the durability of composite material systems

A simple DATAPLOT example See Tables A.3 and A.4

81

82

Durability of Composite Systems

Role of uncertainty in the durability of composite material systems

Appendix B A sample Tool-1 DP code for data uncertainty DU1a

. ----------------------------------------------------------------------------------------------------. This is a dataplot code Filename: rd101z.dp . DATA File name: fong101x.dat . Purpose: To Rank many Distributions (RD, or rd) . using two fit methods (Max. Likelihood and PPCC) . and four Goodness-of-Fit (GoF) criteria, i.e., . GoF Criterion-1 = Anderson-Darling (AD) . GoF Criterion-2 = Kolmogorov-Smirnov (KS) . GoF Criterion-4 = Probability Plot Correlation Coefficient (PPCC) . GoF Criterion-5 = Chi-Square (CS) . Date first coded: Oct. 2, 2012 Revision-8: Aug. 4, 2016 . By: N. Alan Heckert, NIST Div. 776, [email protected] . and Jeffrey T. Fong, NIST Div. 771, [email protected] . -------------------------------------------------------------------------------------------------------------------------. ----- Step 1.0 Read input data file (See text of a data file on p. A-7 at end of this code.) dimension 40 columns skip 15 read fong101x.dat y . . Note-1: Bin data for chi-square, use 2 different binning algorithms . set histogram class width normal corrected let y2 x2 = binned y set histogram class width default let y3 x3 = binned y let minsize = 3 let y4 xlow xhigh = combine frequency table y3 x3 skip 0 set write decimals 7 let ymin = min y let ymin2 = 0.05 * ymin . . ----- Step 1.1 Use macro-11 for fit method-1 (Maximum Likelihood) with . Goodness-of-fit criterion-1 (Anderson-Darling), and . output results in a data file named best11.out echo on capture screen on capture best11.out set best fit method ml set best fit criterion ad best distributional fit y end of capture echo off . . ----- Step 1.2 Use macro-12 for fit method-1 (Maximum Likelihood) with . Goodness-of-fit criterion-2 (Komol-Smirnov), and . output results in a data file named best12.out

83

84

Durability of Composite Systems

write "_____" echo on capture screen on capture best12.out set best fit method ml set best fit criterion ks best distributional fit y end of capture echo off . . ----- Step 1.3 Use macro-21 for fit method-2 (PPCC) with . Goodness-of-fit criterion-1 (Anderson-Darling), and . output results in a data file named best21.out write "_____" echo on capture screen on capture best21.out set best fit method ppcc set best fit criterion ad best distributional fit y end of capture echo off . . ----- Step 1.4 Use macro-22 for fit method-2 (PPCC) with . Goodness-of-fit criterion-2 (Komol-Smirnov), and . output results in a data file named best22.out write "_____" echo on capture screen on capture best22.out set best fit method ppcc set best fit criterion ks best distributional fit y end of capture echo off . . ----- Step 1.5 Use macro-24 for fit method-2 (PPCC) with . Goodness-of-fit criterion-4 (PPCC), and . output results in a data file named best24.out write "_____" echo on capture screen on capture best24.out set best fit method ppcc set best fit criterion ppcc best distributional fit y end of capture . . ----- Step 1.6 Use macro-15 for fit method-1 (ML) with . Goodness-of-fit criterion-5 (CS), and . output results in a data file named best15.out . . Note-2: The "best distributional fit" does not currently . support binned data, so manually compute for normal, . 2-para. Weibull, 3-para. Weibull, 2-para. lognormal, and 3-para. lognormal. .

Role of uncertainty in the durability of composite material systems

. Note-3: The ML estimates will be computed for the unbinned data set . and then the chi-square goodness of fit will be applied to the binned data. . echo on capture screen on capture best15.out write "_____" print "Normal Corrected Bins" print x2 y2 print "Default Bins (0.3*s)" print y4 xlow xhigh write "_____" normal mle y let ksloc = xmean let ksscale = xsd normal chi-square goodness of fit y2 x2 normal chi-square goodness of fit y4 xlow xhigh . weibull mle y let ksloc = 0 let ksscale = alphaml let gamma = gammaml weibull chi-square goodness of fit y2 x2 weibull chi-square goodness of fit y4 xlow xhigh . 3-parameter weibull mle y let ksloc = locml let ksscale = scaleml let gamma = shapeml weibull chi-square goodness of fit y2 x2 weibull chi-square goodness of fit y4 xlow xhigh . lognormal mle y let ksloc = 0 let ksscale = scaleml let sigma = sigmaml lognormal chi-square goodness of fit y2 x2 lognormal chi-square goodness of fit y4 xlow xhigh . 3-parameter lognormal mle y let ksloc = locml let ksscale = scaleml let sigma = sigmaml lognormal chi-square goodness of fit y2 x2 lognormal chi-square goodness of fit y4 xlow xhigh end of capture . exit

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. This is dataplot data file . . Filename: fong101x.dat . . ---------- line 5 . . Assoc. dp filename: rd101z.dp . . . ---------- line 10 . . skip 15 . . . ----------- line 15 129.83 143.42 149.33 158.79 160.17 165.83 167.69 175.82 175.96 177.89 184.03 184.58 184.65 186.51 190.79 206.16 214.5 228.91 232.57 232.78 233.67 239.67 246.5 247.6 254.98 255.67 255.74 272.9 303.69 312.28 312.9

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Role of uncertainty in the durability of composite material systems

Appendix C Two sample Tool-2 DP codes for data uncertainty DU1b and DU1c

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Durability of Composite Systems capture 3pW_0.05x___ML.out 3-parameter weibull mle y end of capture exit . ----------------------------------------------------------------------------------------------------------------. Step 5b (for case Normal ): Run Nor-ML analysis by deleting codes below Step 5a . ----------------------------------------------------------------------------------------------------------------capture Nor_0.05x___ML.out normal mle y end of capture exit . ----------------------------------------------------------------------------------------------------------------. Step 5c (for case 2pW ): Run 2pW-ML analysis by deleting codes below Step 5a and Step 5b . ----------------------------------------------------------------------------------------------------------------capture 2pW_0.05x___ML.out weibull mle y end of capture exit

Note-2: The following is the listing of a DATAPLOT cide named, fong820a.dp : , ---------------------------------------------------------------------------------------------------. Date: Aug. 20, 2012 Filename: fong820a.dp . By: Jeffrey T. Fong, Div. 771, NIST, [email protected] . and N. Alan Heckert, Div. 776, NIST, [email protected] . ---------------------------------------------------------------------------------------------------. Purpose: Given mean (xmean), standard deviation (xsd), and sample size (n) . and assume the univariate distribution is normal, . To find the one-sided lower tolerance limits for 95% confidence level, . and 90% (B-basis), 99% (A-basis) coverages . ---------------------------------------------------------------------------------------------------. Step 1: Input sample size (n), mean (xmean), and stand. dev. (xsd) . ---------------------------------------------------------------------------------------------------echo on capture fong820a.out set write decimals 6 let n = 17 let xmean = 266.25 let xsd = 18.88 . ---------------------------------------------------------------------------------------------------. Step 2: Calculate one-sided lower tolerance limits (display table format) . ---------------------------------------------------------------------------------------------------normal summary lower tolerance limits xmean xsd n . --------------------------------------------------------------------------------------------------. Step 3: Calculate and display A-basis (A2) and B-basis (B2) allowables . -------------------------------------------------------------------------------------------------. ----- Note: Input specific confidence level (alpha) and coverage (gamma) let alpha = 0.95 . ------- one-sided tolerance limits for A-basis allowable (gamma = 0.99) let gamma = 0.99 let xmeanvec = data xmean let xsdvec = data xsd let nvec = data n let A2 = summary normal toleranc one sided lower limit xmeanvec xsdvec nvec . ------- one-sided tolerance limits for B-basis allowable (gamma = 0.90) let gamma = 0.90 let B2 = summary normal toleranc one sided lower limit xmeanvec xsdvec nvec end of capture echo off Exit

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Appendix D NIST uncertainty machine (Tool-3) for model uncertainty MU2a What is NIST Uncertainty Machine? The NIST Uncertainty Machine is a web-based software application produced by the NIST to evaluate the measurement uncertainty associated with a scalar or vectoral output quantity that is a known and explicit function of a set of scalar input quantities for which estimates and evaluations of measurement uncertainty are available. The NIST Uncertainty Machine implements the approximate method of uncertainty evaluation described in the “Guide to the expression of uncertainty in measurement (GUM)” [3], and the Monte Carlo method of the Gum Supplements 1 and 2. Input and output quantities are modeled as random variables, and their probability distributions are used to characterize measurement uncertainty. For inputs that are correlated, the NIST Uncertainty Machine offers the means to specify the corresponding correlations, and how they will be considered. The output of the NIST Uncertainty Machine comprises: 1. 2. 3. 4.

An estimate of the output quantity (measurand). Evaluations of the associated standard and expanded uncertainties. Coverage intervals of the true value of the measurand. An uncertainty budget that quantifies the influence that the uncertainties of the inputs have upon the uncertainty of the output.

For details about the NIST Uncertainty Machine, and examples of its application, please refer to its user’s manual, available online [4] and further described in How to download software and its documentation section, and a paper by Lefarge and Possolo [29].

How to download software and its documentation The NIST Uncertainty Machine is a unique computer application software in the sense that it does not require a user to download its software. Instead, all a user needs to do is to click the following link, download a user’s manual, and start running the software on a NIST computer at no cost to the user: https://uncertainty.nist.gov/

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A simple NUM example Instructions • • • • • • •

Select the number of input quantities. Change the quantity names if necessary. For each input quantity choose its distribution and its parameters. Choose and set the correlations if necessary. Choose the number of realizations. Write the definition of the output quantity in a valid R expression. Run the computation.

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Appendix E A sample Tool-5 DP code for model uncertainty MU2a . ---------------------------------------------------------------------------. This is a DATAPLOT program. Filename: LLSQ.dp . Note-1: To run this code, we need the following additional macros: . Data file: 6data_crt.dat . Annotation file: annotate.dp . ---------------------------------------------------------------------------. Purpose: Linear Least Squares Fit of a set of data (X, Y) in a . log-log plot with xx = log(X), yy = log(Y), and model: yy = A + C * xx . ---------------------------------------------------------------------------. Date first coded: December 8, 2005 Revised: Nov. 23, 2016 . by: Jeffrey T. Fong, NIST Applied & Computational Math Div (771) . Gaithersburg, MD 20899-8910 U.S.A. . Contact: [email protected] , or, [email protected] , (301) 975-8217 . ---------------------------------------------------------------------------probe iopsy1 if probeval = 1 device 1 x11 end of if device 2 ps device 2 color on let string header = Linear Least Square Fit of Log(Creep Rupture Time) vs Log(Stress at 600 C) let string subhead = Y = Rupture Time (Hour), yy = Log( Y ). X = Stress (MPa), xx = Log( X ). let string trailer = LLSQ.dp + 6data_crt.dat let string currdate = 11/23/2016 01:40 PST let pagecoun = 0 . -------------------------------------- read input data file skip 15 read 6data_crt.dat y1 x1 skip 0 let x = ln(x1)/ln(10.0) let y = ln(y1)/ln(10.0) let voffset = .4 char circle blank character circle character hw 1.5 1.1 character color blue character thickness 0.3 lines bl dash . --------------------------------------------- specify xlimits ylimits xlimit 1.7 2.4 xtic label size 3 ylimit 0 8 ytic label size 3 y1label displacement 11 . ================ Plot Note-1: x1label varies with application ======= let string xtitle = X = Stress ( MPa ) x1label size 4 x1label ^xtitle x3label . =================================================== End of Plot Note-1 y1label size 4 y1label Y = Creep Rupture Time ( Hour ) . ----------------------------- new lines of code begin x1tic label format alphabetic x1tic label contents 50 63 79 100 126 158 200 251 y1tic label format alphabetic y1tic label contents 1 10 100 1,000 10,000 10E+5 10E+6 10E+7 10E+8 . ----------------------------- new lines of code end plot y x . ---------------------------------------------------- plot no. 1

Role of uncertainty in the durability of composite material systems hw 3 1.5 move 50 92; just center text Plot 1: ^subhead call annotate.dp . ============================================================================ . step 2: fit the data via model (straight line) . ============================================================================ echo on capture screen on capture 6data_crt.out let function g = a + c*x fit y = g end of capture echo off skip 0 read parameter dpst1f.dat int sdint skip 1 read parameter dpst1f.dat slope sdslope print int sdint slope sdslope . pause . ======================================================== . ----- calculate y-intercept a and its anti-log lamda . ======================================================== let a4 = round(a, 4) print a4 . pause let lamda = 10**a let sdint4 = round(sdint, 4) print sdint4 . pause let sdlamda = sdint*lamda . ================================================== . ----- calculate slope c and sdslope . ================================================== let c4 = round(c, 4) print c4 . pause let sdslope4 = round(sdslope, 4) . ---------------------------- least square fit algorithm char circle blank lines blank dash y1label displacement 11 . ================ Plot Note-1: x1label varies with application ======= let string xtitle = X = Stress ( MPa ) x1label size 4 x1label ^xtitle x3label . =================================================== End of Plot Note-1 y1label size 4 y1label Y = Creep Rupture Time ( Hour ) plot y x and plot g for x = 1.7 0.1 2.3 . -------------------------------------------------------- plot no. 2 hw 3 1.5 move 50 92; just center text Plot 2: Material is API 579 Grade 91 Steel at 600 C ( 1112 F ) NRIM-Heat-MgB move 50 66; just left text Equivalent to a Linear Model (in log-log scale) : move 60 61 text yy = A + C * xx , where move 58 56; just left text A = yy-intercept = 24.0685 , move 62 52; just left text s.d. of yy-intercept = 2.285 , and move 70 48; just left text Lamda = L = anti-Log (A) move 76 44 text = 1.17 E+24 . move 72 38; just left text C = Slope = - 9.83 , and

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Durability of Composite Systems move 76 34 text s.d. of C = 1.09 . move 38 86; just left text A Power-law Model (units in natural scale) : move 40 80 text Creep Rupture Time t (hour) = L * { Stress (MPa) } ** C , move 42 76 text where t = anti-Log (yy) , and Stress = anti-Log (xx) . call annotate.dp . pause . =========================================== End of LLSQ Fit plot pre-erase off let yyupper = 8 let xxmax = 2.3 let xxmin = 1.7 ylim 0 ^yyupper let x2 = ^xxmin 0.01 ^xxmax . ------- Note x = log_10(x1) and y = log_10(y1) let xbar = mean x let del2 = x - xbar let del2sq = del2**2 let denom = sum del2sq let num = (x2-xbar)**2 let term3 = num/denom let n = number y let term2 = 1/n let term1 = 1 let sd2 = ressd*sqrt(term1+term2+term3) print sd2 x2 pause let df = n-2 print df . pause let t = tppf(.975,df) let half = t*sd2 print half . pause let function g2 = a4 + c4*x2 let pred2 = g2 print pred2 x2 . pause let lower2 = pred2 - half let upper2 = pred2 + half char circle blank blank blank lines color black blue red red lines thickness 0.1 0.3 0.3 0.3 lines blank solid dash dash print y x . pause print pred2 lower2 upper2 x2 pause plot y x and plot pred2 lower2 upper2 vs x2 hw 3 1.5 move 16 48; just left text 95 % Confidence Upper move 16 44; just left text Limit with no stress error lines solid line thickness 0.1 lines color red drawdddd 2 5 1.87 3.5 move 22 28; just left text 95 % Confidence Lower move 22 24; just left text Limit with no stress error drawdddd 2.05 0 2.23 1.5 pause exit

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Appendix F A sample Tool-6 DP code for model uncertainty MU2b . ====================================================================== . This is an Executable Code named nllsq.dp . ====================================================================== . Note-1: To run this code, install a statistical analysis software package named DATAPLOT . Note-2: DATAPLOT is free online, http://www.itl.nist.gov/div898/software/dataplot.html . -------------------------------------------------------------------------------------------. Note-3: nllsq denotes Non Linear Least SQuares. (also NLLSQ) . Note-4: FEM denotes Finite Element Method. . Note-5: cms denotes COMSOL, a general-purpose FEM analysis software package. . ------------------------------------------------------------------------------------------. Note-6: n denotes the total number of FEM runs. Minimum of n = 5. . Note-7: w = wrench stress analysis problem using element type tetra-10. . Note-8: w27 = wrench stress analysis problem using element type hexa-27 . ====================================================================== . Note-9: To run this code as a template, make changes at nine (9) places as tagged. . WARNING: THIS VERSION USES SKIP 15. CHANGE AS NEEDED. . ====================================================================== . Purpose: To estimate simulation result at infinite degrees of freedom (dof) based on . a sequence of n runs of FEM solutions, using a NLLSQ fit, where the . left asymptote at -infinity is less than the right asymptote at +infinity. . ------------------------------------------------------------------------------------------. Note-10: When the left asymptote at -infinity is expected to be greater than the right . asymptote at +infinity, use an alternative code named nllsq2.dp . ======================================================================= . Name of this DATAPLOT application code : nllsq.dp . -------------------------------------------------------------------------------------------. 1 associated data file: nw_cms.dat . 1 associated annotation file name: annotate.dp . ================================================================================ . Date: Nov. 23, 2016 . By: Jeffrey T. Fong (NIST, Gaithersburg, MD 20899, [email protected]) . -------------------------------------------------------------------------------------------. TABLE OF CONTENTS . Step 0. Preliminaries . Step 1. Naming of String and Numeric Parameters . Step 2. How many runs do you have to run the nonlinear least squares logistic fit ? . Step 3. Give values of string and numeric parameters . ----------------------------------- Plot 1 begins ----------------------------------------. Step 4. Read input data file, nw_cms.dat once for n runs . Step 5. Introducing 4-parameter Logistic Fit and Assign Initial Values of 4 parameters . Step 6. Nonlinear Least Squares (NLLSQ) Fit of data just read, nw_cms.dat . Step 7. Some graphics commands for Plot 1 . Step 8. Commands for Plot 1 (NLLSQ Fit of n data without confidence bounds) . ------------------------------------ Plot 2 begins ---------------------------------------. Step 9. Add commands for Plot 2 (NLLSQ Fit of n data with 95 % confidence bounds) . ------------------------------------ Combined Plots 1 + 2 Graphics -------------------. Step 11. Add annotation for Plot 1 (NLLSQ Fit for n runs), and Plot 2 (Uncertainty) . Step 12. Add local tailor-made graphics commands for Combined Plots 1+2 . ======================================================================== . Step 0. Preliminaries (No change is required.) . ======================================================================= probe iopsy1 if probeval = 1 device 1 x11 end of if . device 2 ps device 2 color on . =====================================================================

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. Step 1. Naming of String and Numeric Parameters (No change is required.) . ---------------------------------------------------------------------------------. Names of string Parameters: header header2 . currdate trailer . ylab . xlab xlab3 . ---------------------------------------------------------------------------------. Names of numeric Parameters: k nk = k+5 . ylm1 ylm2 xlm1 xlm2 . ynk2 halfnk y152 half5 . yheader2 pagecoun . ============================================================== . Step 2. How many FEM runs do you have to run the NLLSQ fit ? . --- TAG-1 ---( No. of FEM runs ) let n=5 . ======================================================================== . Step 3. Give values of string and numeric parameters (keep unchanged except as tagged) . --------------------------------------------------------------------------------------------let string header = 4-para. Logistic : Y = y1 - L*( exp(-k*(xx-x0)) / (1 + exp(-k*(xx-X0)))) let string header2 = where xx = Log_10( X ) (Nonlinear Least Squares Fit, Fong-Filliben-Heckert, 2015) let yheader2 = 91.5 . --- TAG-2 ---( Today's date and time ) let string currdate = 11/24/2016 at 8:33 PST let pagecoun = 0 . --- TAG-3 ---( Change name of data file that was read.) let string trailer = nllsq.dp + 5w_cms.dat . --- TAG-4 ---( Label of Y-axis ) let string ylab = Y = Maximum Mises Stress (MPa) . -------------------------------------------let string xlab = xx = LOG_10 ( X ) where X = degrees of freedom ( d.o.f. ) of . --- TAG-5 ---( Second line lable of X-axis ) let string xlab3 = Wrench Stress Analysis (COMSOL) with ^n runs of Tetra-10 ( circles) . --- TAG-6 ---( Lower and upper limits of Y-axis ) let ylm1 = 320 let ylm2 = 390 . --- TAG-7 ---( Lower and upper limits of X-axis ) let xlm1 = 4 let xlm2 = 10 . ------------------------------------------------------------------------------------. Note-11: Additional numeric parameters, ynk2, halfnk, y152, half5, will be . defined later after 3 parameters, y111, half, y15, are evaluated. . ====================================================================== . Step 4. Read input data file, ^nw_cms.dat , once for n data ( ^ means value of.) . ====================================================================== skip 15 . --- TAG-8 ---( Change name of input file as needed. ) read 5w_cms.dat y11 x11 print y11 x11 pause let ymax = max y11 let ymin = min y11 let n = size y11 print ymax ymin n pause . ----------------------------------------let xln = log(x11) let xlog = xln/log(10) let xaven = mean xlog let xlog = sortc xlog y11 print y11 xlog . pause . ====================================================================== . Step 5. Introducing 4-parameter Logistic Fit and Assign Initial Values of 4 parameters . ====================================================================== . 4-para Logistic Fit: Y = f(xlog | 4 parameters, y1, L, k, x0 ) , i.e., . Y = y1 - L ( exp(-k(xlog - x0)) / (1 + exp(-k(xlog - x0)))), . where an S-curve is obtained between left asymptote (-inf, y0) . and right asymptote (+inf, L), with a midpoint defined . by (x0, y1-L/2). k is the slope or steepness coeff. . Hints for choosing values of L , y1 , x0 , and k .

Role of uncertainty in the durability of composite material systems

. . (1) Let y1 = largest value of data . (2) Let L = largest value of data - smallest value of data . (3) Let x0 = ave. of x-data = xavenk . (4) Let k = 1.0 (an initial guess to be changed later when needed) . ------------------------------------------------------------. ------------------------------------------------------------let y1 = ymax let L = ymax - ymin let x0 = xaven let k=1 print y1 L x0 k pause . ====================================================================== . Step 6. Nonlinear Least Squares (NLLSQ) Fit of data read in Step 4, ^nw_cms.dat . ===================================================================== . --- TAG-9 (Last change) ---( Change name of output file.) capture 5w_cms.out . =============================================== . THIS IS THE LAST TAGGED CHANGE FOR nllsq.dp . =============================================== . =============================================== . =============================================== fit y11 = y1 - L*(exp(-k*(xlog-x0))/(1 + exp(-k*(xlog-x0)))) end of capture . ------------------------------------------------------------let function yy = y1 - L*(exp(-k*(xx-x0))/(1 + exp(-k*(xx-x0)))) let pred1 = pred let ressd1 = ressd let ressd1 = round(ressd1,4) let y111 = y1 let L11 = L let k11 = k let x011 = x0 . --------------------------------------------------- define parameter ynk2 let ynk2 = round(y111, 2) print ynk2 . pause . ------------------------------------------------------------------------print y111 L11 k11 x011 . pause let x11 = xlog print y11 x11 . pause . -------------------------------------------. -----------------------------------------let xx11 = sequence xlm1 0.1 xlm2 let yy11 = y111 - L11*(exp(-k11*(xx11-x011))/(1 + exp(-k11*(xx11-x011)))) print yy11 xx11 . pause . ===================================================================== . Step 7. Some graphics commands for Plot 1. . ===================================================================== character fill off character circle character hw 2.0 1.4 character color red character thickness 0.4 line blank title offset 2 title case asis case asis label case asis lines blank dash lines thickness 0.4 lines color red red x3labels x1label size 4 x1label displacement 9 x1label ^xlab

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Durability of Composite Systems x3label size 3.2 x3label ^xlab3 . --------------------------------------------- add red text command to xlab3 move 79 3 hw 3 1.5 color red text red color black . --------------------------------------------y1label size 4 y1label displacement 9 y1label ^ylab lines blank dash line thickness 0.4 ylimits ylm1 ylm2 ytic label size 3 xlimits xlm1 xlm2 xtic label size 3 . ======================================================================== . Step 8. Commands for Plot 1 (NLLSQ Fit of n data without confidence bounds) . ======================================================================= plot y11 vs xlog and plot yy11 vs xx11 call annotate.dp . pause . ====================================================================== . Step 9. Add commands for Plot 2 (NLLSQ Fit of n data with 95 % confidence bounds) . ===================================================================== pre-erase off . ---------------------------------------------let xxmax = xlm2 let xxmin = xlm1 let x2 = ^xxmin 0.1 ^xxmax print x2 . pause let xbar = mean x11 let del2 = x11 - xbar let del2sq = del2**2 let denom = sum del2sq let num = (x2 - xbar)**2 let term3 = num/denom let n = number y11 let term2 = 1/n let term1 = 1 let sd2 = ressd*sqrt(term1 + term2 + term3) print sd2 ressd term1 term2 term3 . pause . --------------------------------------------let df = n-2 let t = tppf(.975, df) let half = t*sd2 print k11 x011 L11 y111 . pause let function z2 = -^k11*(x2 - ^x011) print z2 x2 . pause let function h2 = exp(z2) let hh2 = h2/(1+h2) let pred2 = ^y111 - ^L11*hh2 print pred2 x2 . pause let lower2 = pred2 - half let upper2 = pred2 + half print lower2 upper2 x2 . pause let low07 = lower2 subset x2 = 9.0 let low07a = lower2 subset x2 = 9.9 let pre07 = pred2 subset x2 = 9.0 let upp07 = upper2 subset x2 = 9.0 print pre07

Role of uncertainty in the durability of composite material systems . pause let low07s = sum low07 let low07ss = sum low07a let pre07s = sum pre07 let upp07s = sum upp07 . --------------------------------------------------- define parameter halfnk let halfnk = pre07s - low07s let halfnk = round(halfnk, 2) print halfnk . pause . ------------------------------------------------------------------------print low07s pre07s upp07s . pause char bl bl bl lines solid dash dash lines thickness 0.4 0.4 0.4 line color red red red plot pred2 lower2 upper2 vs x2 . pause . --------------------------------------------------pre-erase off . --------------------------------------------------char fill on character circle character hw 0.6 0.4 character color red character thickness 0.3 . ============================================ let x2s = 9.0 print low07s pre07s upp07s x2s . pause let low07s = round(low07s, 1) let pre07s = round(pre07s, 1) let upp07s = round(upp07s, 1) let temp1 = pre07s print temp1 . pause let zxx = data x2s x2s let zyy = data low07s upp07s . ================================ rev 3/25/2016 line dash . ================================ end of rev print zxx zyy . pause plot zyy vs zxx . pause hw 3.5 1.75 just left color red move 46 85 text ^n-run predicted = ^ynk2 +/- ^halfnk MPa justification center hw 3.2 1.6 move 90 81 color black text (uncertainty move 90 77 text estimated at move 90 73 text 1-billion d.o.f.) . -------------------------------- add percent uncertainty let pu = 100 * halfnk / ynk2 let pu = round(pu, 2) justification left hw 3.2 1.6 move 82 64 color red text Pu = Percent move 82 60 text uncertainty

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Durability of Composite Systems move 82 56 text = ^pu % . -------------------------------- annotation for 10E+9 d.o.f. just left color black move 45 44 text 1-billion move 45 40 text degrees of move 45 36 text freedom . -------------------------------- add 10E+06 d.o.f. move 45 30 text 1-million move 45 26 text degrees of move 45 22 text freedom . ------------------------------hw 3.5 1.75 move 73 36 color red text ^n-run predicted 95 % move 73 32 text confidence lower limit . =================================================================== . Step 10. Add local tailor-made graphics commands for Combined Plots 1+2 . =================================================================== line solid line thickness 0.15 line color red . ----------------------------------------- pointer mean drawdddd 9.45 389 9.8 ^ynk2 . ---------------------------------------- pointer Lower Limit drawdddd 8.7 334 9.25 334 drawdddd 8.7 334 9.9 ^low07ss line color black . -----------------------------------pointer for 1-million dof drawdddd 6.0 331 6.45 328 . ------------------------- vertical line for 1-million dof drawdddd 6.0 314 6.0 380 . ------------------------------------- pointer 1-billion dof drawdddd 7.5 344 9.0 354 . ------------------------ vertical line for 1-billion dof drawdddd 9.0 314 9.0 370 exit

Appendix G A sample Tool-7 DP code for model uncertainty MU2b, MU2c, and MU3 This is dataplot program file fong155_5_8.dp 3/13/18 Purpose: Sensitivity analysis for NDE Uncertainty Date: 3/13/18

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. -----start point----. -----Step 1: Define header and trailer info----. let string header ¼ Sensitivity Analysis for NDE Uncertainty let string header ¼ 8-run UT Experiment for Detecting Flaw "L" let string header2 ¼ Response ¼ Crack length let string headhw1 ¼ 2.8 1.4 let string headhw2 ¼ 2.8 1.4 let string currdate ¼ 3/13/18 let pagecoun ¼ 0 let string trailer ¼ fong155_5_8.dp let string trailer2 ¼ Datafile: fong155_5_8.dat . -----Step 2: Read in the data----- Comment-1 See next page for text of a data file. . skip 25 read fong155_5_8.daty x1 x2 x3 x4 x5 . -----Step 3: Define Response and Factor info----. let string sty ¼ Crack Length let string stx1 ¼ OperExp let string stx2 ¼ MachAge let string stx3 ¼ CableLen let string stx4 ¼ ProbeAng let string stx5 ¼ ShoeThick . -----Step 4: Plot the data----. let k ¼ 5 call dex10stepanalysis.dp . Exit

************ END OF CODE

************ This is dataplot data file fong155_5_8.dat 3/13/18

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Durability of Composite Systems Contents: Data for NDE Sensitivity Analysis Date: 3/13/18 Number of observations ¼ 8 Number of variables per line image ¼ 6 Order of variables on a line image: 1. Response ¼ Crack Length (inches) 2. Factor 1 ¼ Operator Experience (2 levels: 2 and 6 years) 3. Factor 2 ¼ UT Machine age (2 levels: 2 and 8 years) 4. Factor 3 ¼ Cable Length (2 levels: 6 and 10 feet) 5. Factor 4 ¼ Transducer Probe Angle (2 levels: 42 and 48 degrees) 6. Factor 5 ¼ Plastic Shie Thickness (2 levels: .24 and .75 inches) To read this data file into dataplot: skip 25: read fong155_5_8.dat y x1 x2 x3 x4 x5 x6

Y

X1

X2

X3

X4

X5

CrLen OpExp MachAge CabLen Angle Thick -------------------------------------------1.70 -1 -1 -1 þ1 þ1 2.15 þ1 -1 -1 -1 -1 1.75 -1 þ1 -1 -1 þ1 2.65 þ1 þ1 -1 þ1 -1 2.25 -1 -1 þ1 þ1 -1 2.10 þ1 -1 þ1 -1 þ1 1.30 -1 þ1 þ1 -1 -1 2.50 þ1 þ1 þ1 þ1 þ1

Disclaimer Certain commercial equipment, materials, or software are identified in this paper to specify the computational procedure adequately. Such identification is not intended to imply endorsement by NIST, nor to imply that the equipment, materials, or software identified are necessarily the best available for the purpose.

References [1] J.T. Fong, Fatigue mechanisms – key to the solution of the engineer’s second fundamental problem, in: J.T. Fong (Ed.), Fatigue Mechanisms, Proceedings of an ASTM-NBS-NSF Symposium, Kansas City, Mo., May 1978, American Society for Testing and Materials, Philadelphia, PA 19103, 1979, pp. 3e8. ASTM STP 675.

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[2] JCGM (Joint Committee on Guides in Metrology), International Vocabulary of Metrology - Basic and General Concepts and Associated Terms (VIM), third ed., 2012. Section 2.26, p. 25, https://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf. [3] B.N. Taylor, C.E. Kuyatt, Guidelines for evaluating and expressing the uncertainty of NIST measurement results, in: NIST Technical Note 1297, National Institute of Standards and Technology, Gaithersburg, MD, 1944. [4] NIST, Measurement Uncertainty, 2019. https://www.nist.gov/itl/sed/topic-areas/ measurement-uncertainty. [5] M. Evans, N. Hastings, B. Peacock, Statistical Distributions, third ed., Wiley, 2000. [6] A. Haldar, A. Guran, B.M. Ayyub (Eds.), Uncertainty Modeling in Finite Element, Fatigue and Stability of Systems, World Scientific Publishing Co. Pte. Ltd., Singapore, 1997, p. 912805. [7] B.M. Ayyub (Ed.), Uncertainty Modeling & Analysis in Civil Engineering, CRC, 1998. [8] B.M. Ayyub, G.J. Klir, Uncertainty Modeling and Analysis in Engineering and the Sciences, Chapman and Hall/CRC, 2006. [9] H.W. Coleman, W.G. Steele, Experimentation, Validation, and Uncertainty Analysis for Engineers, third ed., Wiley, 2009. [10] W.Q. Meeker, G.J. Hahn, L.A. Escobar, Statistical Intervals: A Guide for Practitioners and Researchers, second ed., Wiley, 2017. [11] K. Bury, Statistical Distributions in Engineering, Cambridge University Press, 1999. [12] P.R. Nelson, M. Coffin, K.A.F. Copeland, Introductory Statistics: Engineering Experimentation, Elsevier Academic Press, 2003. [13] J.J. Filliben, N.A. Heckert, Dataplot: A Statistical Data Analysis Software System, National Institute of Standards & Technology, Gaithersburg, MD 20899, 2002. http://www. itl.nist.gov/div898/software/dataplot.html. [14] N.A. Heckert, J.J. Filliben, NIST Handbook 148: DATAPLOT Reference Manual, Volume I: Commands, National Institute of Standards and Technology Handbook Series, June 2003. [15] N.A. Heckert, J.J. Filliben, NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions, National Institute of Standards and Technology Handbook Series, June 2003. [16] E.R. Fuller Jr., S.W. Freiman, J.B. Quinn, G.D. Quinn, W.C. Carter, Fracture mechanics approach to the design of glass aircraft windows: a case study, in: Proc. Conf., SPIE e the International Society for Optical Engineering, 26e28 July 1994, San Diego, CA, vol. 2286, 1994, pp. 419e430. [17] J.T. Fong, N.A. Heckert, J.J. Filliben, P.V. Marcal, S.W. Freiman, A distribution selectionand scale uncertainty-based approach to estimating a minimum allowable strength for a full-scale component or structure of engineering materials, in: A Manuscript Submitted to a Technical Journal, 2019. Available at: [email protected]. [18] M.G. Natrella, Experimental Statistics, National Bureau of Standards Handbook 91 (Aug. 1, 1963, reprinted with corrections Oct. 1966), pp. 1-14, 1-15, 2-13, 2-14, and 2-15, Tables A-6 and A-7, Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402, 1966. [19] S.S. Wilks, Determination of sample sizes for setting tolerance limits, Ann. Math. Stat. 12 (1941) 91e96. [20] R.B. Murphy, Nonparametric tolerance limits, Ann. Math. Stat. 19 (1941) 581e589. [21] A. Wald, “An extension of Wilks’ method for setting tolerance limits, Ann. Math. Stat. 14 (1943) 45e55. [22] A. Wald, J. Wolfowitz, Tolerance limits for a normal distribution, Ann. Math. Stat. 17 (1946) 208e215.

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[23] A.H. Bowker, Computation of factors for tolerance limits on a normal distribution when the sample is large, Ann. Math. Stat. 17 (1946) 238e240. [24] J.W. Tukey, Non-parametric estimation, II; statistically equivalent blocks and tolerance regions, the continuous case, Ann. Math. Stat. 18 (1947) 529e539. [25] F. Proschan, Confidence and tolerance intervals for the normal distribution, J. Am. Stat. Assoc. 48 (1953) 550e564. [26] D.A.S. Fraser, Nonparametric tolerance regions, Ann. Math. Stat. 24 (1953) 44e55. [27] P.N. Somerville, Tables for obtaining non-parametric tolerance limits, Ann. Math. Stat. 29 (1958), 599-601. [28] Anon, MIL-HDBK-17/1 Revision F (Volume 1 of 5): Guidelines for Characterization of Structural Materials, Chapter 8 (Statistical Methods), U.S. Department of Defense, 2002. http://mil-17.udel.edu/. [29] T. Lafarge, A. Possolo, The NIST uncertainty machine, NCSLI Meas. J. Meas.Sci. 10 (3 (September)) (2015) 20e27. [30] H.H. Ku, Notes on the use of propagation of error formulas, J. Res. Natl. Bur. Stands. 70C (4) (1966) 263e273. [31] J.T. Fong, N.A. Heckert, J.J. Filliben, Design of an Intelligent PYTHON Code for validating crack growth exponent by monitoring a crack of zig-zag shape in a cracked pipe, in: Proc. 2019 ASME PVP Division Conference, July 14-19, 2019, Minneapolis, MN, United states, Paper No. PVP2019-93502, American Society of Mechanical Engineers, New York, NY, 2019. http://www.asmeconferences.org/PVP2019. [32] N.R. Draper, H. Smith, Applied Regression Analysis, Chap. 1-3, pp. 1-103, and Chap. 10, pp. 263-304, 1966. Wiley (1966). [33] F. Mosteller, J.W. Tukey, Data Analysis and Regression, Addison-Wesley, Boston, 1977. [34] N.E. Dowling, Fatigue life and inelastic strain response under complex histories for an alloy steel, J. Test. Eval, ASTM 1 (4) (1973) 271e287. [35] N.E. Dowling, Mechanical Behavior of Materials, second ed., Prentice-Hall, 1999. [36] J.T. Fong, N.A. Heckert, J.J. Filliben, S.W. Freiman, Uncertainty in multi-scale fatigue life modeling and a new approach to estimating frequency of Inservice inspection of aging components, Strength Fract. Complex. 11 (2018) 195e217. https://doi.org/10.3233/SFC180223. [37] P.F. Verhulst, Recherches mathematiques sur la loi d’accroissement de la population (Mathematical Researches into the Law of Population Growth Increase), in: re-published in Nouveaux Memoires de l’Academie Royaledes Sciences et Belles-Lettres de Bruxelles, vol. 18, 1845, pp. 1e42 (2013). [38] Anon, Introduction to COMSOL Multi-Physics, v.5.1, COMSOL, Inc., Burlington, MA, 2015. http://cdn.comsol.com/documentation/5.1.0.180/IntroductionToCOMSOLMultiphysics.pdf. [39] J.T. Fong, N.A. Heckert, J.J. Filliben, P.V. Marcal, R. Rainsberger, Uncertainty of FEM Solutions Using a Nonlinear Least Squares Fit Method and a Design of Experiments Approach, in: Proc. COMSOL Users Conference, Oct. 7e9, 2015, Boston, MA, United states, 2015. http://www.comsol.com/conference-2015. [40] J.T. Fong, J.J. Filliben, N.A. Heckert, P.V. Marcal, R. Rainsberger, L. Ma, Uncertainty quantification of stresses in a cracked pipe elbow weldment using a logistic function fit, a nonlinear least squares algorithm, and a superparametric method, Procedia Eng. 130 (2015) 135e149. [41] J.T. Fong, J.J. Filliben, N.A. Heckert, P.V. Marcal, R. Rainsberger, A new approach to FEM error estimation using a nonlinear least squares logistic fit of candidate solutions, in: Proc. ASME 2016 V&V Symp., May 16-20, 2016, Las Vegas, NV, Paper No. VVS20168912, American Society of Mechanical Engineers, New York, NY, 2016.

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[42] J.T. Fong, P.V. Marcal, R. Rainsberger, L. Ma, N.A. Heckert, J.J. Filliben, Finite element method solution uncertainty, asymptotic solution, and a new approach to accuracy assessment, in: Proc. 7th Annual ASME Verification and Validation Symposium, May 1618, 2018, Minneapolis, MN, United states, Paper VVS2018-9320, American Society of Mechanical Engineers, New York, NY, 2018. [43] W.L. Oberkampf, C.J. Roy, Verification and Validation in Scientific Computing, Cambridge University Press, 2010. [44] C.P. Robert, G. Casella, Monte Carlo Statistical Methods, second ed., Springer, 2004. [45] G.E. Box, W.G. Hunter, J.S. Hunter, Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building, Wiley, 1978. [46] J.T. Fong, N.A. Heckert, J.J. Filliben, S.R. Doctor, Three approaches to quantification of NDE uncertainty and a detailed exposition of the expert panel approach using the Sheffield Elicitation Framework, in: Proc. 2018 ASME PVP Division Conference, July 2018, Prague, Czech Republic, Paper No. PVP2018-84771, American Society of Mechanical Engineers, New York, NY, 2018. http://www.asmeconferences.org/PVP2018. [47] J.T. Fong, J.J. Filliben, N.A. Heckert, R. deWit, Design of experiments approach to verification and uncertainty estimation of simulations based on finite element method, in: Proc. Conf. Amer. Soc. for Eng. Education, June 22-25, 2008, Pittsburgh, PA, Paper AC2008-2725, 2008. [48] J. Mandel, R. Paule, Interlaboratory evaluation of a material with unequal number of replicates, Anal. Chem. 42 (1970) 1194e1197. [49] ASTM E691-18, Standard Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method, ASTM International, 100 Barr Harbor Drive, PO BOX C700, West Conshohoceken, PA 19428-2959, USA, 2018. [50] J.T. Fong, N.E. Dowling, Analysis of fatigue crack growth rate data from different laboratories, ASTM STP 738 (1981) 171e193. [51] L. Broutman, S. Sahu, Proc. of the 24th society of the plastics industry conference, Feb. 1969, 1969, pp. 1e12. [52] M.J. Owen, R.J. Howe, J. of Phys. D: Appl. Phys. 5 (1972) 1637. [53] W.W. Stinchcomb, K.L. Reifsnider, Fatigue damage mechanisms in composite materials: a review, in: J.T. Fong (Ed.), Fatigue Mechanisms, Proceedings of an ASTM-NBS-NSF Symposium, Kansas City, Mo., May 1978, American Society for Testing and Materials, Philadelphia, PA 19103, 1979, pp. 762e787. ASTM STP 675. [54] P.W.R. Beaumont, Structural integrity and the implementation of engineering composite materials, in: P.W.R. Beaumont, C. Soutis, A. Hodzic (Eds.), Chapter 15 in Structural Integrity and Durability of Advanced Composites, Elsevier Woodhead Publishing, Cambridge, CB22 3HJ, U.K., 2015, pp. 353e398. [55] Anon, Composite Materials Handbook, Volume 2, Polymer Matrix Composites Materials Properties, MIL-HDBK-17-2f, Volume 2 of 5, 24 May 2002, U.S. Department of Defense and the Federal Aviation Administration, 2002. [56] J.D. Kiser, R. Andrulonis, C. Ashforth, K.E. David, C. Davies, Updated composite materials handbook-17 (CMH-17) volume 5 - ceramic matrix composites, in: Y. Kagawa, D. Zhu, R. Darolia, R. Raj (Eds.), Advanced Ceramic Matrix Composites: Science and Technology of Materials, Design, Applications, Performance, and Integration, 2017. ECI Symposium Series (2017), http://dc.engconfintl.org/acmc/46. [57] J.T. Fong, N.A. Heckert, J.J. Filliben, P.H. Ziehl, A nonlinear least squares logistic fit approach to quantifying uncertainty in fatigue stress-life models and an application to plain concrete, in: Proc. 2018 ASME PVP Division Conference, July 2018, Prague, Czech Republic, Paper No. PVP2018-84739, American Society of Mechanical Engineers, New York, NY, 2018. http://www.asmeconferences.org/PVP2018.

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[58] M.K. Lee, B.I.G. Barr, An overview of the fatigue behaviour of plain and fibre reinforced concrete, Cement Concr. Compos. 26 (2004) 299e305. [59] E.F.J. von Euw, R.W. Hertzberg, R. Roberts, Delay effects in fatigue-crack propagation, in: Stress Analysis and Growth of Cracks, ASTM STP 513, American Society for Testing and Materials, 1972, pp. 230e259. [60] J.D.D. Melo, J.T. Fong, A new approach to creating composite materials elastic property database with uncertainty estimation using a combination of mechanical and thermal expansion tests, in: Proc. ASME 2010 Pressure Vessels & Piping Division/K-PVP Conference, July 18-22, 2010, Bellevue, WA, United states, Paper No. PVP2010-26144, American Society of Mechanical Engineers, New York, NY, 2010. [61] J.D.D. Melo, D.W. Redford, Determination of the elastic constants of a transversely isotropic lamina using laminate coefficients of thermal expansion, J. Compos. Mater. 36 (11) (2002) 1321e1330. [62] S.W. Tsai (Ed.), Strength and Life of Composites, Composites Design Group, Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305 USA, 2008. www.compositesdesign.stanford.edu. [63] G. Vining, S.M. Kowalski, Statistical methods for engineers, second ed., Thomson Brooks/ Cole, Belmont, CA 94002 USA, 2006, pp. 213e217. [64] P.D. Shah, J.D.D. Melo, C.A. Cimini Jr., J.T. Fong, Composite material property database using smooth specimens to generate design allowables with uncertainty estimation, in: Proc. ASME 2010 Pressure Vessels & Piping Division/K-PVP Conference, July 18-22, 2010, Bellevue, WA, United states, Paper No. PVP2010-26145, American Society of Mechanical Engineers, New York, NY, 2010. [65] C. Soutis, J. Lee, Scaling effects in notched carbon fibre/epoxy composites loaded in compression, J. Mater. Sci. 43 (2008) 6593e6598.

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3

Kenneth L. Reifsnider 1 , E.V. Iarve 2 , Rassel Raihan 1 , H.K. Adluru 2 , K.H. Hoos 2 1 Mechanical and Aerospace Engineering, University of Texas Arlington (UTA), Arlington, TX, United States; 2University of Texas at Arlington (UTA), Fort Worth, TX, United States

Chapter outline 3.1 Progressive damage analysis by discrete damage modeling 3.1.1 Introduction 108

3.2 Computational methodology 3.2.1 3.2.2 3.2.3 3.2.4

110

Static failure analysis 110 Fatigue failure criterion for MIC insertion 111 Fatigue cohesive law 112 Fatigue DDM algorithm 115 3.2.4.1 Modified fatigue algorithm 117

3.3 Verification of Rx-FEM coupon-level analysis

118

3.3.1 DCB analysis 121 3.3.2 ENF analysis 123 3.3.3 MMB verification study 125

3.4 Validation of Rx-FEM subelement-level analysis 127 3.4.1 Clamped tapered beamdbackground 128 3.4.1.1 Static analysis 128 3.4.1.2 Static analysis conclusions 139 3.4.1.3 Fatigue analysis 140 3.4.1.4 Fatigue analysis conclusions 144 3.4.2 Three-point bend with flange (3PB-F) 144 3.4.2.1 Model description 144 3.4.2.2 Results and discussion 146

3.5 Ply-level constitutive behavior methods 3.6 Machine learning methods 158 3.7 Chapter conclusions 159 Acknowledgments 159 References 160

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3.1 3.1.1

Durability of Composite Systems

Progressive damage analysis by discrete damage modeling Introduction

Fatigue properties of carbon fiber composites were studied intensively in the past and satisfactorily described for most isolated damage modes. The transverse strength properties have been shown to follow the SeN curve methodology based on an underlying static failure criterion [1] and the fracture-related damage modes such as delamination have been shown to follow the Paris law [2,3]. The delamination onset and propagation investigation in composite laminates have been a critical research topic for several decades and are the subjects of many reviews, for example, Refs. [4e6]. Significant achievements in the practical application of the virtual crack closure technique (VCCT) [7] to delamination propagation in laminated composite panels both in static and fatigue regimes were recently reported by Deobald and coauthors [8]. Although the delamination failure mode is of great practical importance, it cannot be considered in isolation from other less critical damage modes, for example, matrix cracking [9]. Depending upon the layup and loading profile, the delamination propagation can be precipitated by matrix crack formation, which can drastically affect its propagation. Several scenarios directly influencing the damage tolerance assessment are possible: matrix cracking can temporarily arrest the delamination; it can divert the delamination to a different interface [10,11]; it may cause an avalanche of multiple delaminations through the thickness of the part. Progressive damage modeling methodologies potentially addresses the damage evolution and interaction phenomena. To date, significant progress has been achieved in developing progressive damage modeling methodologies for the damage evolution and interaction phenomena of laminated composite materials. The continuum damage modeling (CDM) methodology is the most common and widely used approach for progressive damage modeling in laminated composites. The CDM represents all forms of damage as the local volumetric stiffness degradation, which describes matrix and fiber damage by developing local constitutive material models for directionally reducing the stiffness properties of the affected elements. The ease of implementation for this methodology in the conventional finite element framework has greatly proliferated its application [12e14]. However, it has significant difficulties in predicting local effects of interactions between various damage modes and local effects of stress redistribution in the damage area without any special meshing techniques such as fiber aligned meshing [15e17]. This issue has been investigated in detail [18,19] where the CDM did not predict the fiber direction stress relaxation as a function of longitudinal splitting in unidirectional open hole composites accurately. Discrete damage modeling (DDM) represents an approach to progressive damage analysis when multiple individual damage events, such as matrix cracks in plies, delamination between plies and their interaction, are explicitly introduced into the model through the displacement discontinuities. The DDM approach to modeling

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crack-induced displacement discontinuities employs the mesh-independent crack (MIC) technique for crack growth without remeshing, such as the eXtended Finite Element Method (x-FEM) [20], which has been developed over the past decade. In the x-FEM formulation, additional degrees of freedom are added to the element along the crack surface to describe the displacement discontinuities. This crack surface partitions the element into two parts in which the integration schema must be performed separately. Although the x-FEM formulation is not straightforward and much more complicated than CDM, it can solve some of the most significant problems associated with the CDM methodology. The modeling approach reported later represents the DDM method, which is based on the direct simulation of displacement discontinuities associated with individual instances of matrix cracking occurring inside the composite plies, and delaminations at the interfaces between the plies. These methods employ variants of x-FEM [20] and its regularized implementation (Rx-FEM) [21e24] in particular and allow on the fly insertion of the matrix cracks in the direction independent of mesh orientation. These methods were extensively applied to static failure analysis [25e28]. The Rx-FEM allows modeling the displacement discontinuity associated with individual matrix cracks in individual plies of a composite without regard to mesh orientation by inserting additional degrees of freedom in the process of the simulation. The propagation of the mesh independent crack is then performed by using a cohesive zone method. The kinematic aspect of the technique does not require any modification for fatigue loading; however, the constitutive components do. There are two components of the DDM framework, which require modification: the failure criterion for crack insertion and the cohesive law for damage initiation and propagation. It is these developments that will be discussed in detail in this chapter, whereas the Rx-FEM formulation details can be found in Refs. [21e24]. The conceptual difficulty of application of cohesive zone models (CZM) in fatigue lies in different representations of the crack tip in such models as compared to classical crack tip representation, which is used in the definition of the Paris law. This difficulty was recently addressed by Harper and Hallett [29]. It was shown how to extract the fracture mechanics point crack tip energy release rate (ERR) from the cohesive zone model and subsequently directly apply the Paris law for cohesive crack propagation. In this work, we further develop this concept by introducing interpolation procedure for ERR calculation in the cohesive zone model and thus allowing using coarser meshes for analysis. We also complete the fatigue cohesive zone framework by combining the SeN information for the initiation phase with Paris law for the propagation phase for cracking and delamination. The proposed fatigue CZM makes no explicit or implicit assumptions regarding initial crack size for fatigue analysis thus extending this key feature of CZM from static to fatigue. The final development required for completing the DDM fatigue framework is the new crack insertion identification law, which in a static regime is performed by using the LaRC04 criterion [30]. We use Palmgren-Miner’s rule to develop a generic crack insertion methodology for the fatigue loading regime. The propagation of the mesh independent crack is then performed by using the CZM.

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The goal of this section is to present a computational methodology for the DDM framework under static and fatigue loading. The verification and validation of the DDM framework were performed for coupon-level laminated compositesddouble cantilever beam (DCB), end-notched flexure (ENF), and mixed-mode bending (MMB) next. This was followed by the application of DDM to subelement-level compositesdclamped tapered beam (CTB) specimen under quasistatic and monotonic fatigue loading conditions and three-point bend with flange (3 PB-F) under quasistatic loading. The effects of residual thermal stresses and energy dissipationbased load control algorithms for unstable crack propagation for CTB specimen are discussed.

3.2

Computational methodology

In this section, the static DDM framework will be reviewed followed by fatigue failure criterion. The failure criteria in the DDM framework are used to identify the location, crack orientation, and numbers of cycles (in case of fatigue) at which matrix cracks are inserted. The discussion will continue with fatigue CZM development, including damage initiation, propagation, and finally describing the fatigue DDM framework.

3.2.1

Static failure analysis

The DDM approach consists of MIC modeling of transverse cracks in each ply of the laminate and modeling the delamination between the plies using a cohesive formulation at the ply interface. The matrix cracks are modeled by using the regularized formulation [21e24], termed Rx-FEM. The regularized formulation deals with continuous enrichment functions and replaces the Heaviside step function in Ref. [20] with continuous function changing from 0 to 1 over a narrow volume of the so-called gradient zone. The formalism tying the volume integrals in the gradient zone to surface integrals in the limit of mesh refinement was discussed in Ref. [22]. The simulation begins without any initial matrix cracks, which then are inserted based on a failure criterion during the simulation. The LaRC04 failure criterion [30] is chosen in the present work. When the failure criteria are met, a CZM, associated with a matrix crack is inserted, which then begins to open. We will imply the same by saying that we insert an MIC. The propagation of each MIC, that is, opening of the CZM is performed by using formulation [31]. Note that the delamination between the plies is also simulated by CZM; however, the cohesive elements between the interfaces are inserted during initial model preparation. The schematic of the process is shown in Fig. 3.1. After the failure criterion is met at a certain location, an MIC is added by using Rx-FEM. Next, the load is increased and the program enters a nonlinear NewtoneRaphson iteration step at which the inserted crack(s) and/or delaminations are being opened. After convergence is achieved, the failure criterion is checked again and if met a new crack is inserted. This process is coupled with the progressive simulation of fiber failure [23,33e35] and/or fiber criterion, such as a critical failure volume method [34,36] if applicable.

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Model preparation

Nonlinear static analysis (delamination & MIC growth)

Increase load

Insert cracks s Ye

Failure criterion (LaRC03, LaRC04)

Matrix Fiber

Yes

Yes Failure

Critical failure volume

No s Ye

tion

Op

No

Progressive fiber failure No

Figure 3.1 Static solution algorithm [32].

3.2.2

Fatigue failure criterion for MIC insertion

In this section, the fatigue failure criterion is described. The fatigue failure criterion is built upon 3D static failure criterion LaRC04 described in Ref. [30]. The idea of constructing a fatigue failure criterion based upon a static criterion was proposed by Hashin and Rotem [1]. For a given frequency 6 and R ¼ smin=smax ratio, the fatigue failure load s is defined as the failure load amplitude versus the number of cycles to failure and can be expressed as s ¼ ss f ðR; N; 6Þ

(3.1)

where ss is the static strength and f (R,N,6) is the material fatigue function. In the present case, only matrix failure modes are considered. The idea expressed and experimentally verified in Ref. [1] consists of applying a static failure criterion with degraded ply-level strength properties, which are given by two SeN curves, namely normal tension y(R,N,6) and shear s(R,N,6), such that YðNÞ=Yt ¼ 1  s1 LogðNÞ and SðNÞ=S ¼ 1  s2 LogðNÞ

(3.2)

where Y(N) and S(N) are reduced strength values as the number of cycles for transverse normal and shear strength and Yt and S are their static values, respectively; s1 and s2 are material parameters. According to Ref. [1] the fatigue strength can be predicted by applying static failure criterion with the respective ply-level strength values

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provided by the SeN curves. Two modifications are required to this methodology [1] for application in the DDM framework. First, a 3D failure criterion capable of predicting the transverse cracking angle relative to the normal direction to the ply interface needs to be employed. Second, owing to constantly changing loading amplitude during progressive damage simulation, it is needed to generalize this approach to variable amplitude loading. To address the first requirement, we employ the LaRC04 criterion [30]. In the tension-shear quadrant, it represents a quadratic criterion similar to Ref. [1]; however, it is applied to the so-called failure plane where the failure index attains its maximum value. In the case of transverse tension loading, this plane is normally perpendicular to the ply midsurface; however, in the presence of transverse shear stresses and under transverse compression, it can form an angle to the ply interface. The second step required to complete the fatigue failure criterion development for DDM is a generalization to variable amplitude loading. It is accomplished by applying the Palmgren-Miner linear damage accumulation hypothesis. For implementation purposes, let us define a material point loading history parameter dI as q

dI ¼

q X DN k k¼1

Nfk

(3.3)

where the summation is carried out over all previous load cycle steps 1 . q and Nkf is the limit number of cycles the specimen can survive within a given block of loading alone. The failure, according to the Palmgren-Miner hypothesis, occurs when dI attains a value between 0.7 and 2.2 depending on the material, R, and loading frequency. Without restricting generality, we will assume that fatigue failure corresponds to dI ¼ 1. Based on this hypothesis and Eq. (3.3), we can compute at each load cycle step q the number of cycles until dI ¼ 1, which in our context corresponds to the insertion of a matrix crack CZM as h i q  qþ1 DNc ¼ min 1  dI Nf x

(3.4)

Sometimes, the SeN relationship for transverse tension and shear strength is provided in the form sðNÞ ¼ Aðlog NÞB sstatic

(3.5)

whereas the Rx-FEM standalone package (BSAM) uses the simple log-linear relationship as in Eq. (3.2). When constants provided are in the form of Eq. (3.5), least squares regression analysis method is applied to determine constants s1 and s2 parameters for tension and shear strength.

3.2.3

Fatigue cohesive law

As seen in the previous section, the extension of static failure criterion in the tensionshear quadrant for matrix failure mode to fatigue is conceptually transparent.

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However, a significantly more complicated situation arises with the CZM. Based on previous work [29,37e39], two problem areas can be identified. One area is the simulation of the propagation of existing delamination according to Paris law. The difficulty here is that Paris law is defined within the classical fracture mechanics framework and uses ERR or stress intensity factor magnitude associated with ideal crack tip singularity. On the other hand, the CZM introduces a process zone concept instead of the ideal crack tip. Two different approaches have been proposed to address this issue. One is based on explicitly bringing in the process zone length into the propagation mechanism as described in Ref. [39]. The other approach proposed in Refs. [29e31,34,36,37] extracts the ERR value of the classical crack tip from the process zone information and uses it to propagate a fatigue crack similar to VCCT. The first methodology allows for the continuous evolution of the damage variable and maintains the traditional implementation of the CZM method; however, it requires estimation of the length of the process zone to conduct the analysis. This presents a problem for variable mode mixity and therefore the second approach for the delamination propagation phase is adopted. The second problem area of the cohesive zone model extension to fatigue analysis is the damage initiation phase and its transition into the propagation phase. It is this feature of the cohesive zone method, which has earned its popularity in static analysis. Consider bilinear static cohesive law shown in Fig. 3.2 with a dashed line. In the process of static deformation, material points move along the traction displacement jump curve continuously from left to right, assuring the transition from undamaged to damaged portion and eventually to the fully separated state under load or stored energy. In the fatigue regime, the situation is completely different in both the initiation and propagation regime in that the material points do not follow continuously the static cohesive traction law through the initiation and propagation phases, and in some situations are not even located on the static cohesive law curve at all. First, consider the initiation stage and a material point A on the ascending portion of the cohesive law (see Fig. 3.2). The point A has a ratio of stress amplitude to the static strength of approximately 0.5. According to SeN curves in Eq. (3.2) this point will fail 2

Cohesive traction

1.5

Static

Static strength 1

Fatigue

S-N strength

A

0.5

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Displacement jump (∆u)

Figure 3.2 Schematics of a cohesive zone model for static and fatigue simulation [32].

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after DNI ¼ exp(0.5/s1) cycles, assuming that it was loaded in mode I. However, what does it mean in terms of the cohesive law? In Ref. [29], it is proposed to assign to this element the damage variable of d ¼ 1 moving it to the fully separated condition. The rationale behind it is that a highly loaded crack tip zone will be formed, which will be in the propagation regime and can be treated by the VCCT-like approach. This damage initiation approach is, however, similar to assuming a certain size of the initial flaw and that this size is also tied to the mesh density. Alternatively, point A can be moved to the damage side of the cohesive law by a parallel shift to the right thus maintaining the cohesive traction at this point. There could be other reasonable scenarios proposed with a certain rationale behind them. We propose instead to modify the cohesive law in each point depending upon the stress level at which the SeN curve failure condition was achieved. Namely, when a given point A fails at stress below static failure stress we propose to reduce the initiation strength of the cohesive law to that at which the failure condition was reached under cyclic loading as is illustrated in Fig. 3.2, where a new cohesive law for point A is shown by a solid line. The slope of the cohesive zone model is also changed to maintain the static propagation characteristics, that is, Gc. According to this hypothesis, each material point may have its cohesive law. The proposed approach eliminates any ambiguity or need for initial damage size or presence of any cracks or delaminations in the structure. Such an approach, while also completely arbitrary, maintains the overall consistency of the CZM, where there is a continuous path for all material points to progress from fully undamaged to completely separated state. To keep track of the loading history of each material point, a point loading history parameter dI defined in Eq. (3.3) is introduced. The value of this parameter starts accumulating before crack CZM is inserted and continues accumulating after CZM is inserted. Only when the condition dI ¼ 1 is achieved, the material point A appears at the onset of the propagation stage and the cohesive law is modified as shown in Fig. 3.2. As mentioned previously, the approach by Harper and Hallett [29] to propagate matrix cracks and delaminations will be used. In this method, the cohesive law is used to extract the value of ERR and subsequently propagate the damage in a way similar to VCCT. In the propagation regime, the modified Paris law   da Gmax m ¼ C dN Gc

(3.6)

was used where Gc is the static critical value of the ERR, Gmax is the maximum value of ERR during the cycle, and C and m are material fatigue constants. Analogs to VCCT, the crack, and/or delamination was propagated one element at a time so that the number of cycles DNp for propagation was DNp ¼

le  Gmax m C Gc 

(3.7)

Durability of aerospace material systems

115

where the length value for crack extension le is equal to one element size. Eqs. (3.6) and (3.7) are applicable provided that the R ratio of the input data and applied loading are the same. Several approaches to include the effect of R ratio in the composite material application are discussed in Ref. [40]. An approach based on the original formulation of the Paris law in the power form of the change of the crack stress intensity factor during a loading cycle has been adopted in this work. pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2m da ¼ D Gmax  Gmin dN pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Introducing ¼ Gmin = Gmax , Eq. (3.8) can be rearranged as da ¼ Dð1  RÞ2m ðGmax Þm dN

(3.8)

(3.9)

In composite applications, the Paris law constants for both modes I and II are often provided in the form da ¼ Ap ðGmax Þm dN

(3.10)

for certain R ratio R ¼ R0. For our application, we use the normalized form of the Paris law   da Gmax m ¼ Cð1  RÞ2m dN Gc

(3.11)

where C ¼ Ap ðGc Þm =ð1  R0 Þ2m Please refer to Ref. [41] for 2D validation examples illustrating the ERR extraction technique and important modification proposed to the integration point extraction procedure [29]. Namely, we propose to use the maximum extrapolated nodal values of ERR as the crack tip ERR instead of the integration point values. Such modification was shown to allow using 1.27 mm linear size cohesive elements for mode II delamination prediction with the ERR values of 0.25GIIc [41]. In the process of computation, all matrix cracks and delaminations propagate in a mixed-mode regime. The experimental data to validate mixed-mode fatigue crack propagation are not readily available. In this work, a BeK approximation adopted in Refs. [29,31,37,38] was used for all parameter approximation including Paris law parameters with h ¼ 2.25. This value was established based on experimental data for static analysis in Ref. [39].

3.2.4

Fatigue DDM algorithm

Recall the main events described in the previous two sections and notation associated with the number of cycles for each of them taking place: DNc d insertion of new MIC,

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DNI d initiation of CZM damage progression (CZM modification), DNp d CZM damage propagation of one interval, Note that we are not distinguishing between CZM evolution stages in matrix cracking and delaminations; instead, in each case, the number of intervals is counted to the first occurrence of this event over the entire structure. Two types of algorithms can be envisioned for modeling progressive failure in fatigue. Cycle-based algorithm (CBA): In this algorithm, one predefines several fatigue cycles on each solution step and simulates the damage, which occurs during these cycles. Event-based algorithm (EBA): In this algorithm, one defines an increment of damage or damage event, such as new crack insertion or delamination extension for one element, and computes the number of cycles required to advance to this event. The number of cycles for each increment DNmax for an EBA is DN ¼ minðDNc ; DNI ; DNp Þ

(3.12)

In a complex DDM simulation, a combination of CBA and EBA is employed. Namely, the simulation starts in the EBA regime. As the simulation progresses, the CZM propagation cycle count DNp becomes smaller and smaller. After it reaches a certain predefined minimum value, DNmin, the propagation events determine the number of cycles per increment that is, even if DNc and DNI are DN ¼ DNp smaller and the cracks are inserted not exactly when their number of cycles is reached but at the end of the step determined by delamination propagation. This may delay a crack insertion somewhat but allows the computation to proceed. The same is true for CZM modification, which as a result will happen in more than one point at a time. As evident, the CZM propagation is selected as a dominant driver for fatigue simulation with understanding that it is this event, which has an immediate impact on the deformed state and requires reequilibration. The fatigue DDM algorithm is based on the static algorithm shown in Fig. 3.1 with some changes or additions. Fig. 3.3 shows the fatigue simulation algorithm. The modifications and additions are shown in crisp red. Consider a fixed amplitude cyclic loading. The maximum cycle load has a special meaning in progressive fatigue simulation because complex multisite damage evolution involves both static equilibrium and cycle driven fatigue damage evolution based on Paris law and S/N curves. The static equilibrium is evaluated at maximum cycle loads for obvious reasons. Thus the simulation starts with ramping up to the cycle maximum load in the static regime. It is possible that in this process the MIC insertion and opening will already begin and therefore the ramp-up should be performed in more than one step. Note that in here fatigue failure criterion becomes identical to static if the number of cycles in Eq. (3.2) is N ¼ 0. Without restricting generality, assume that we ramped up the load and no MIC insertion nor delamination damage took place. At this initial step, the EBA will be controlled by several cycles until the first crack insertion

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117

Model preparation

Nonlinear static analysis (delamination & MIC growth)

Increase load Insert cracks

Find ∆Np and update d I

d=

Ye

s

1

CZM

Find ∆NI and update dI

Matrix

No

Fatigue criterion (find ∆Nc, update dI)

Fiber tion Yes

Ye

s

Op

Yes Failure

Critical failure volume

No Progressive fiber failure

No

Figure 3.3 Fatigue simulation algorithm [42].

DNc given by Eq. (3.4). The first step cycle count as DN1 ¼ DNc is recorded and the process continued to equilibrate the solution. This time the equilibrium will result in no changes to the stressestrain state because we have not modified any CZM and the load is below static initiation load. On the next step, we modify the CZM in all points with dI ¼ 1 and compute the cycles to new modification DNI, and to new crack insertion DNc, normally DNp is still large. We determine the new cycle count as DN 2 ¼ min(DNc, DNI) and update dI everywhere. The next reequilibrium will have effects from the CZM opening and will bring down DNp. The process continues in EBA mode until DNp < DNmin after which we do not use DNI and DNc even though we continue inserting cracks and updating CZM dI with some delay less than DNmin. The sequence of cycle counts DNi, i ¼ 1,2,3, . is recorded in a file and the damage distribution on each step forms the fatigue loading history. After a given number of cycles, one can revert to static loading for residual strength evaluation; however, in this case, we save damage distributions and infuse them into a separate static simulation.

3.2.4.1

Modified fatigue algorithm

Ref. [39] outlines the cycle-based and event-based fatigue algorithm methods. The event-based algorithm predicts the cycle increment required until an elemental failure event such as MIC insertion (DNc) or delamination and/or MIC propagation (DNp) by a given increment such as one mesh size. In addition to these more apparent physical events, an additional procedural event had to be included, which consisted of transformation from the undamaged to damaged state in the fatigue CZM formulation (DNI).

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Durability of Composite Systems

These cycle increments are evaluated based on the examination of the equilibrium state of stress, which in the case of the implicit solution technique is achieved as a result of NewtoneRaphson (N-R) iterations. Next, in the context of strict event-based formulation, a new cycle increment is determined as DN ¼ minðDNc ; DNp ; DNI Þ

(3.13)

and the solution updated depending upon which of the events occur first. After the solution is updated, the state of stress is equilibrated and the procedure repeated. The P total number of cycles after n equilibration steps will be N ¼ DN. Such procedure n

can be very efficient in the initial stages of simulation where the event cycle increments are very large, however, with an accumulation of damage when multiple cracks and delaminations propagate at the same time, the subject increments can become on the order of a single cycle and significantly burden the implicit solution algorithm. At present, a modified algorithm, partially addressing this problem is proposed. The CZM transformation cycle increment, DNI is excluded from Eq. (3.13) and the new cycle increment is computed as DN ¼ minðDNc ; DNp Þ

(3.14)

In the modified algorithm, the CZM transformation is performed within the N-R iterations and not after the equilibrium state is achieved. In this case, the cycle count is incremented by DN and either MIC is inserted or the delamination is propagated or both and N-R iterations performed to equilibrate the solution. The difference is that during these iterations the initiation stress in CZM is not kept constant but constantly recalculated to be equal to the residual strength at a given location, which is evaluated by using SeN curves and fatigue accumulation law, that is, Palmgren-Miners rule in the present formulation. The approach proposed in Ref. [29] for propagation modeling evaluates the ERR of a crack simulated by CZM by looking at the maximum value of ERR in all elements in the process zone. As this implementation uses explicit time-cycle integration, the damage variable is increased based on Paris Law. In the implicit implementation [43], the same concept for ERR evaluation is used and cycle increment DNp for the delamination to propagate by 1 interval length is computed and the damage variable in the respective element is set to 1 analogous to VCCT method. The ERR in Ref. [29] was evaluated in the integration points; however, in Ref. [43] it was interpolated into the nodes. This approach appeared more accurate in benchmark problems with coarse meshes.

3.3

Verification of Rx-FEM coupon-level analysis

In this section, the results for verification and partial validation of DDM methodology are presented. The experimental reports [2,3,44] for mode I, mode II, and mixed-mode delamination propagation are presented for validation. The verification of these

Durability of aerospace material systems

119

coupon models, preceding the validation, is accomplished by comparing available analytical solutions for DCB, ENF, and MMB configurations. These solutions were applied for stiffness and static strength calculations. The computational model predicted static results close to the analytical model in all cases including mixedmode static loading. In DCB and ENF, the stiffness and static strength were also in close agreement to the experimental data. These cases were further processed with validation including cyclic loading. In MMB case, verification with benchmark case [45] was performed for MMB coupon analysis in both static and fatigue loading and the results were discussed. Finite element models were developed for DCB, the ENF, and the MMB specimens. These geometries and the finite element meshes for these specimens were developed using ABAQUS CAE. The model geometries and boundary conditions for the three test specimens are shown in Fig. 3.4. The green lines in these figures represent the initial delamination length (Teflon inserts) specified in the experimental investigations from Ref. [2,3,44]. All of the specimens were modeled using 3D solid elements similar to C3D8 in Abaqus. Half-width specimens were modeled using symmetry through the width of the specimen and fixtures. An approximately same stiffness beam as the experiment was developed for the MMB tests. A single node line connection was considered to attach the loading arm to the composite specimen. The center arm is connected (tied together) in vertical displacement with respective nodes of the arm and the plate and the lateral (end) arm connection are tied together in all displacements. IM7/8552 tape laminate based on 60% f.v.f properties were considered for the

(a)

(b)

P

(c)

Pm

Pe

Figure 3.4 Model geometries and boundary conditions for (a) DCB, (b) ENF, and (c) MMB.

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Durability of Composite Systems

composite and the fixture in the MMB specimen was considered as aluminum. The critical ERR based on [2,3,44] is considered as well. These properties for the analysis are shown in Tables 3.1e3.3. To verify each model, specific test parameters were chosen to compare simulations with either experimental data or closed-form solutions from the literature.

Table 3.1 Static material properties for DCB, ENF, and MMB. IM7/8552

Steel

Aluminum

E11 (GPa)

158

207

69

E22 (GPa)

8.9

207

69

E33 (GPa)

8.9

207

69

G13 (GPa)

5.17

79.6

25.6

G23 (GPa)

3.98

79.6

25.6

G12 (GPa)

5.17

79.6

25.6

n13

0.32

0.3

0.35

n23

0.44

0.3

0.35

n12

0.32

0.3

0.35

6 

0.0

N/A

N/A

6 

a2 (10 / C)

30.1

N/A

N/A

XT (MPa)

2501

N/A

N/A

XC (MPa)

1716

N/A

N/A

YT (MPa)

64.1

N/A

N/A

YC (MPa)

285

N/A

N/A

S (MPa)

91.1

N/A

N/A

GIC (N/mm)

0.2

N/A

N/A

GIIC (N/mm)

1.0

N/A

N/A

a1 (10 / C)

Table 3.2 Fracture toughness properties considered in simulations. Study [2,43,45]

Benchmark

ACC-CRT

GIc (kJ/m )

0.240

0.212

0.240

GIIc (kJ/m )

*0.744

0.774

*0.739

h

2.1

2.1

2.1

2

2

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121

Table 3.3 Fracture material properties considered in simulations. Study [2,43,45]

Benchmark

S1

0.03

0.03

S2

0.075

0.075

m1

7.1

Properties from Ref. [45] for each individual

m2

5.44

Mode ratio

C1 (mm)

0.00762

As described

C2 (mm)

0.044

Below

3.3.1

DCB analysis

For mode-I DCB analysis, nominal dimensions of 24 ply IM7/8552 specimen in Ref. [2], which are 24.5 mm (1 in.) wide and 4.5 mm (0.18 in.) thick, are considered. The initial delamination length was a0 ¼ 50.8 mm (2 in.) from the load application point to the tip of the delamination. Loads are applied for a static model as per ASTM Standard D5528. Only the half-width model with symmetry boundary conditions are considered. The stiffness from the simulation is compared with the solution from classical beam theory for the cantilever beam as shown in Fig. 3.5. The result from classical beam theory is slightly stiffer than the numerical analysis, as the latter takes into account transverse shear displacements. Overall, the beam theory analysis supports the BSAM prediction. A series of analysis with increased crack length were performed and the compliance C was found from this analysis, based on ASTM standard D5528. Delamination compliance (C1/3) versus delamination length (a) are plotted in Fig. 3.6 and compared with experimental results [2] and classical Mode-I

25

Load ( lbs )

20 BSAM-simulation 15 Classical beam theory 10

5

0 0

0.1

0.2

0.3

0.4

Displacement ( in. )

Figure 3.5 Load displacement curves for DCB configuration.

0.5

0.6

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Durability of Composite Systems

Compliance calibration DCB 0.4 0.35

C1/3 (in\lb)

0.3 0.25 0.2 Classical beam theory

0.15

Experimental

0.1

BSAM 0.05 0

0

1

2

3

4

5

Delamination length (in.)

Figure 3.6 DCB compliance calibration curves for static analysis.

beam theory results. The experiment results, BSAM predictions, and analytical solutions predict very similar compliance. Fatigue analysis was performed as per ASTM Standard D6115. The analysis was performed under displacement controls to be consistent with experimental data [2]. Three progressive damage analysis (PDA) simulations were performed under constantly applied displacement amplitude corresponding to initial delamination ERR of 65%, 50%, and 35% of the GIC values. The ERR is calculated by direct differentiation dW/dA and compared to the ERR from CZM. Crack length versus ERRs is plotted in Fig. 3.7 for different G/GC ratios. The crack growth rate is then calculated by postprocessing the delamination length versus cycle’s data by direct numerical differentiation. These results are presented in Mode-I ERR different loads 0.8 0.7 dW/dA

0.6

65% G/Gc

CZM ERR

G/Gc

0.5 0.4 0.3 50%

0.2 0.1

35%

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Crack length (in.)

Figure 3.7 DCBdERR plots for different loads.

0.7

0.8

0.9

1

Durability of aerospace material systems Delamination growth rate

1.E–04

65% load

1.00E–05

da/dN (in/cycle)

(b)

1.00E–04

1.E–05

50% load 35% load Experiment

1.00E–06 1.00E–07

da/dN, in/cycle

(a)

123

1.00E–08

y = 0.0003X

1.00E–09

7.1188

1.00E–10

1.E–06

1.E–07

1.E–08

a

1.00E–11 1.E–09

1.00E–12

0.01

0.1

1

G/Gc

Figure 3.8 Delamination growth rate versus normalized GImax (a) and experimental data [2] (b).

the normalized fashion as da=dN versus GImax=GIC in Fig. 3.8(a). These results are directly compared with experimental data corrected for crack bridging R-curve and presented in a similar normalized fashion as da/dN versus GImax=GIR in Ref. [2] and Fig. 3.8(b). The log-log trend line to the analysis data in Fig. 3.8(b) recovers the Paris law fit to experimental data precisely. Note that the analysis da=dN begins with significantly lower crack growth rates than the experiment data. It is attributed to neglecting the growth initiation thresholds presently not included in the model.

3.3.2

ENF analysis

For the verification of mode-II ENF delamination propagation, a report [3] was considered. Two types of specimens from two sources were considered in Ref. [3]: nonprecracked (NPC) specimen, where the delamination growth started from a Teflon film inserted during the manufacturing and precracked (PC) specimens where an approximately 1 in. long crack was statically induced before fatigue testing. The initial critical ERR GIIC averages of 4.22 and 6.5 lb/in were obtained for PC and NPC, respectively, for the first source. It was shown that the NPC and PC condition had a significant effect on the fatigue onset curves whereas the source of the material did not affect Paris law crack growth parameters. Three specimens with an initial crack length of 1.5  0.4 in. were considered in Ref. [3] for compliance calibration and are used for model verification. The loade deflection curves for NPC specimens from Figs. 3.5 and 3.6 of [3], as well as the predicted responses, are shown in Fig. 3.9. The predicted stiffness for all three specimens as well as the failure load for the 1.5 in. precracked specimen is in close agreement with experimental values. Note the shift of the experimental data from the origin and thus the parallel instead of the coincident location of the predicted and experimental response curves. The stiffness of the 1.1 and 1.9 in. crack length compliance calibration specimens is also well within the variation expected from the thickness variation of the individual samples. Only the specimen with 1.5 in. crack length was tested to failure and the prediction matched the measured strength accurately using the higher GIIC ¼ 6.5 lb/in. value of the critical ERR for NPC specimens.

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Durability of Composite Systems

Non pre-cracked ENF loaddisplacement curves

Load (Ibs)

400

a0 = 1.1 in.

350

BSAM

300

Experimental

a0 = 1.5 in.

250 200

a0 = 1.9 in.

150 100 50 0 0

0.02

0.04

0.06

0.1

0.08

Displacement ( in. )

Figure 3.9 Loadedisplacement curves for ENF configuration. Mode II delamination growth 1.00E–03

1.00E–04

da/dN (in/cycle)

50% 1.00E–05

1.00E–06

35%

Expt. = 0.0017x5.45

Experiment

1.00E–07

y = 0.0025x5.5914 1.00E–08

1.00E–09 0.01

0.1

1

G/Gc

Figure 3.10 Normalized delamination growth results by initial GIImax.

The crack growth validations were performed in fatigue loading. PDA under two different load levels corresponding to 50% and 35% of the GIIC initially were performed and the crack growth rate computed postprocess by direct numerical differentiation of the delamination length by cycle count. The results of data reduction are presented in Fig. 3.10. A slight deviation from the predicted crack growth rate manifested by the difference in the trend line and the Paris law are observed and are both shown in Fig. 3.10.

Durability of aerospace material systems

3.3.3

125

MMB verification study

This benchmark study [45] employed three mixed mode ratios 0.2, 0.5, and 0.8 with the size of the initial delamination as a0 ¼ 25.4 mm. The results in here are presented in SI units following the benchmark study. The static loadedeflection curves are shown in Fig. 3.11e3.13 for mode mixity ratios of 0.8, 0.5, and 0.2, respectively. In all cases, the BSAM analysis is less stiff than the benchmark cases. The difference in stiffness increased with a decrease in the mode mixity ratio B. One possible explanation would be that due to relatively small aspect ratio of the loading arm, approximately 1:10, which in combination with finer subdivision of the BSAM model, showed higher compliance than the benchmark solution. Fatigue crack propagation under constant amplitude displacement loading and nominal initial 60% Gc load level were considered for fatigue analysis. For the verification of 900 800

Benchmark 700

Load (N)

600

BSAM

500 400 300

GII/GT = 0.8

200 100 0

0

1

2

3

4

Displacement (mm)

Figure 3.11 MMB specimen analysis with 0.8 mode ratio.

450 400

Benchmark

350

BSAM

Load (N)

300 250 200

GII/GT = 0.5

150 100 50 0

0

1

2

Displacement (mm)

Figure 3.12 MMB specimen analysis with 0.5 mode ratio.

3

4

126

Durability of Composite Systems 160 140

Benchmark

Load ( N )

120

BSAM

100 80

GII/GT = 0.2

60 40 20 0

0

1

2

3

4

Displacement ( mm )

Figure 3.13 MMB specimen analysis with 0.2 mode ratio.

benchmark cases in fatigue analysis, minor adjustments were made. As the mixed-mode Paris law parameters used in Ref. [45] do not fit the BeK law interpolation parameters considered in BSAM, all mixed-mode ratio-dependent parameters were fixed at the values provided in Table 3.1 of [45] for each mode ratio analysis. For mode mixity ratio of 0.8 and 0.5, exact maximum displacement amplitude values of 1.28 and 1.04 mm are considered for fatigue loading. Owing to compliance differences in the 0.2 mode mixity ratio case, the benchmark value of 1.27 mm resulted pffiffiffiffiffiffiin ffi only 50% Gc loading. The loading amplitude was recalculated as wmax ¼ uf 0:6 resulting in a 1.44 mm load amplitude, where uf ¼ 1.86 mm displacement was applied at static failure. The results of the crack growth prediction as a function of the number of cycles are shown in Fig. 3.14e3.16 for mode mixity ratios of 0.8, 0.5, and 0.2, respectively. In all three cases, the predicted crack growth with respect to cycles is in good agreement with benchmark solutions. The faster crack growth rate in the case of B ¼ 0.8 is partially explained by slightly higher 0.62Gc loading than the nominal value of 0.6Gc in the benchmark solution. One significant difference in the predictions is the gradual crack growth initiation at a lower number of cycles in BSAM compared to benchmark Crack length increase ( in mm)

35 30 25 Benchmark 20 BSAM 15 10 5 0 1E+0

1E+1

1E+2

1E+3

1E+4

1E+5

Number of cycles

Figure 3.14 Fatigue MMB specimen [45] analysis with 0.8 mode mixity ratio.

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127

Crack length increase (in mm)

35 30 25

Benchmark BSAM

20 15 10 5 0 1.E+0

1.E+2

1.E+4

1.E+6

Number of cycles

Figure 3.15 Fatigue MMB specimen [45] analysis with 0.5 mode mixity ratio.

Crack length increase (in mm)

25 20

Benchmark 15

BSAM

10 5 0 1.E+0

1.E+2

1.E+4

1.E+6

1.E+8

Number of cycles

Figure 3.16 Fatigue MMB specimen [45] analysis with 0.2 mode mixity ratio.

solution. The difference is due to the lack of onset threshold in BSAM cohesive zone models. Overall, the results are in reasonable agreement between different mode mixity ratios.

3.4

Validation of Rx-FEM subelement-level analysis

In this section, the development and validation of Rx-FEM analysis tools to predict the performance and durability of structural elements (subelement level) made from composite materials are presented. A representative subelement, CTB, which was designed and tested at NASA Langley Research Center, is considered, as shown in Fig. 3.17. This specimen was designed to study both the failure initiation, propagation, and delamination migration in both quasistatic and monotonic fatigue loading.

128

Durability of Composite Systems

Figure 3.17 End and edge view of clamped tapered beam (CTB) [25].

3.4.1

Clamped tapered beamdbackground

Previously, a single cantilever beam specimen [11] was studied to reproduce the delamination migration from one interface to another in a controlled setting. The specimen was designed to grow delamination that will migrate from one ply interface to another ply interface at a repeatable location. This location could be varied by changing the load application point, thus providing a spectrum of information required for validation in the numerical analysis tools. The stacking sequence of the specimen was [0/90/02/906/02/90/0/PTFE/904/010/9012/04], and the initial delamination was in the form of a Teflon insert (PTFE) positioned at the interface of a single 0-degree ply and a stack of four 90-degree plies. This stacking sequence (highlighted earlier), serves as a fracture path to enable the initial delamination to migrate from its original interface to the interface between this stack and a 0-degree ply. This method was extended to multidirectional laminates in Ref. [46]. In these studies, the specimens contained initial delamination, which propagated in the stable regime and subsequently migrated to the neighboring interface. Several discrete damage modeling approaches were applied to the simulation of migration and accurately predicted the loads and the delamination length before migration [10,41,47e50]. For the present study, a more representative subelement, termed CTB specimen shown in Fig. 3.18, was designed at NASA Langley Research Center. Even though CTB is conceptually similar to short beam cantilever, this specimen does not contain Teflon insert and the specimen was tapered to form a configuration consistent with stiffener termination areas. This pristine specimen was designed to study both the failure initiation and propagation including delamination migration. CTB also has a cross-ply stacking sequence like a previous short-beam cantilever specimen, which is not representative of a structural element; however, the overall skin-flange configuration realistically represents the stiffener termination region and provides a testbed for model validation.

3.4.1.1

Static analysis

Two configurations differing only in the load application location were manufactured and tested to provide a broader basis for validation of the analysis methods. The predictions were conducted in two phases with the first blind phase completed before the

Durability of aerospace material systems

129

Load

Z 2.26” (57.5 mm) X Skin

1.76” (44.7 mm)

Configuration A Hinge

Ref: left BC

Width: 0.5” (12.7 mm)

Flange

1.581” (40.16 mm)

Ref: right BC

Span 4.520” (115.mm)

1.240” (31.5 mm)

1.240” (31.5 mm)

7.000” (177.8 mm)

Load

Configuration B

2.76” (70.2 mm) Hinge

1.76” (44.7 mm)

Width: 0.5” (12.7 mm)

1.581” (40.16 mm)

Ref: left BC 1.240” (31.5 mm)

Span 4.520” (115. mm)

Ref: right BC 1.240” (31.5 mm)

7.000” (177.8 mm)

Figure 3.18 CTB geometry with loading information for two configurations [25].

experimental program and the second phase performed after the experimental data were known. The parameters of interest included the peak load as well as delamination migration location and associated load. The DDM method was employed for this analysis. The initial static results and follow-up analysis are discussed in detail in Ref. [25]. Some basic details, like BC’s, mesh details are discussed here. The boundary conditions were applied in the edge regions marked in Fig. 3.18 and consisted of constraining vertical uz displacement on the left side and both ux and uz on the right side edge. The unidirectional IM7/8552 material properties given in Table 3.4 are considered for static analysis. As only 3D hexahedral elements are available for Rx-FEM simulation in BSAM, a 3D model was created. Only one element in the y-direction was modeled taking advantage of the 2D nature of the problem and the uy ¼ 0 condition was imposed at the y ¼ 0 plane, thus providing a half-width symmetric model. Three different meshes were initially created in ABAQUS CAE 2016 and imported to BSAM software [51], where the Rx-FEM is implemented for fracture modeling. • • •

Coarse meshdsingle ply thickness size mesh model (single element per ply) has an approximate element size of 0.18 mm. Fine meshd1/2 ply thickness size mesh model has an approximate element size of 0.09 mm. Finer meshd1/4 ply thickness size mesh model has an approximate element size of 0.045 mm.

Experimental observations The documented experimental data included loadedisplacement curves, delamination migration locations, and X-ray CT as well as optical imaging of the postfailure

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Durability of Composite Systems

Table 3.4 Unidirectional IM7/8552 ply stiffness and strength properties. Property

Value

E11 (GPa)

157.2

E22 (GPa)

8.96

E33 (GPa)

8.96

G13 (GPa)

5.08

G23 (GPa)

2.99

G12 (GPa)

5.08

n13

0.32

n23

0.5

n12

0.32 6 

0.0

6 

a2 (10 / C)

30.1

XT (MPa)

2723

XC (MPa)

1200

YT (MPa)

127

YC (MPa)

200

S (MPa)

95.8

GIC (N/mm)

0.24

GIIC (N/mm)

0.739

a1 (10 / C)

specimens, which were performed at NASA Langley Research Center. In this section, experimental details observed by means of X-ray CT and optical imaging are discussed, which will drive the potential to further investigate numerical studies. The schematics shown in Fig. 3.18 represent the design of an ideal specimen. A closer look at the manufactured samples showed unintended curvature in plies neighboring the ply drop region, as shown in Fig. 3.17. From the optical imaging after postfailure, it was observed that the crack originated at the free edges inside the 90-degree cluster as shown in Figs. 3.19 and 3.20 but not at the interface of 0-degree ply as expected. In general, the delamination propagation including migration events occurred in the unstable regime. It means that the experimental peak loads obtained in the initial phase for Configuration A and B [26] correspond to the initiation and opening of the matrix crack, which leads to delamination and its migration and continuation on a different surface without an increase of the load. An exception is the Test 4 in Fig. 3.8 in Ref. [26] showing the Configuration B. In this case, load increase was required for the delamination to continue to grow after migration. There was no difference in the specimen and/or load application. It was worth noting that the delamination energy

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131

Figure 3.19 (a) coarse mesh w0.18 mm (b) fine mesh w0.09 mm (c) finer mesh w0.045 mm [25].

Figure 3.20 Crack insertion and propagation for Configuration A [25].

release rate is a decreasing function of length beyond load application point and it is therefore considered possible that within experimental scatter a specimen might require additional loading to grow the delamination beyond the migration event. Another experimental observation was the presence of three-dimensionality and variation of the delamination front through the width of the specimen. The X-ray CT images of longitudinal cross-sections for the two configurations at selected locations are shown in Figs. 3.21 and 3.22. The specimen shown for Configuration B is the Test 4 specimen where load increase for continued postmigration delamination

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Durability of Composite Systems

Figure 3.21 Delamination migration scenario when the original delamination has extended beyond the migration location (shown with blue line) [25].

Figure 3.22 Multiple crack delamination migration pattern for Configuration B (shown with blue line) [25].

growth was required. As one can see, the migration location contains several matrix cracks and even a region of delamination overlap on the two surfaces. This observation is similar to that predicted for Configuration B earlier and shown in Figs. 3.8 and 3.9(b) in Ref. [26]. The loadedisplacement curve here has a knee-type behavior, which is correlating with the migration pattern in Fig. 3.9(b) of [26] when the delamination is seen to have an overlapping region where it grows on both surfaces before full migration. Only after the additional load is applied, the delamination growth continues on the new surface. Only 1 out of 3 specimens showed such behavior indicates that it is not consistent. As will be seen later the numerical results also show that this behavior may not be consistent, namely we did not see similar step like force-deflection behavior after the thermal residual stresses were included.

Effect of thermal residual stress The mesh refinement studies discussed in Ref. [25] did not take into account the effect of thermal residual stress. The CTB subelement combines both damage initiation and propagation-controlled phenomena. It was therefore interesting to evaluate the effect of residual processing stress, which may affect the damage initiation stage in the failure process. The Yt values considered in the analysis were obtained on unidirectional specimens and thus do not implicitly account for the residual stress present in the laminate. An estimate of the processing residual stresses was performed by taking into account the thermal mismatch effect only. The coefficients of thermal expansion for unidirectional ply, which was used in the analysis, are shown in Table 3.4. A uniform temperature change of DT ¼ 165 C, approximately equal to the difference between cure and room temperature, was applied. A two-stage load application procedure was utilized, where the first stage is a pure thermal loading with boundary conditions restricting rigid body motion only. On the second stage, the mechanical loading was

Durability of aerospace material systems

(a)

133

700 Test 1

600 Test 2

Force (N)

500

Test 3

400

1 ply mesh

300

1/2 ply mesh 1/4M ply mesh

200 100 0

(b)

0.00

0.50

1.00

2.00

1000 Test 4

900

Test 5

800

Test 6

700

Force (N)

1.50

Displacement (mm)

1 ply mesh

600

1/2 ply mesh

500

1/4M ply mesh

400 300 200 100 0

0.00

0.50

1.00

1.50

2.00

Displacement (mm)

Figure 3.23 Forceedisplacement curves with thermal residual stress effect (a) Configuration A and (b) Configuration B [25].

superimposed with the boundary conditions shown in Fig. 3.18. The loade displacement responses obtained with the three mesh refinements for Configurations A and B are shown in Fig. 3.23. The experimental data are also shown for comparison. Two observations are noteworthy: (1) the results are practically mesh independent and (2) the predicted strength was 4.7% and 3.6% higher for Configurations A and B, respectively, than that predicted without taking into account the residual stresses. However, the more interesting observation was that the matrix crack initiation location on the surface of the tapered region shifted away from the interface with 0 degree when the residual stresses were included. The close-ups of the initial crack region with and without taking into account the residual stress are shown in Fig. 3.24. Here the 1/4 ply size mesh results are shown; however, the initial crack location was identical for all meshes. Location A corresponds to the beginning of the tapered region; location B is the MIC insertion location when residual stresses were taken into account; and the location C was the MIC insertion location predicted without residual stress, which coincided with the 0-degree ply interface and free edge intersection. To understand the shift of the failure initiation site, the distribution of the thermal residual stress was examined. The distribution of the thermal residual sxx stress along the free edge

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Durability of Composite Systems

Config-A A

Config-B

B C

B

A

C

Without residual

A

B

C A

B

C

With residual stress

Figure 3.24 Matrix crack insertion and delamination initiation location with and without residual stress [26].

boundary (path along points AeBeC in Fig. 3.24) of the tapered region in the 90-degrees ply cluster is shown in Fig. 3.25. This stress component represents the transverse normal stress in the 90-degrees ply and is indicative of matrix cracking. The stress distributions shown here are from the 1/4M mesh model, that is, modified 1/4 ply mesh. This additional mesh refinement was not dictated by convergence issues but rather by a desire to rule out the possibility of underestimating the role of the weak stress singularity at location C, where the crack inserted and opened before the residual stress was taken 150

Stress (Vxx in MPa)

100

Location of crack initiation 50

0

A 70.5

B 71

71.5

72

72.5

C

73

Uz = 0, residual –50 Uz = 1.05 mm Uz = 1.17 mm –100

X coordinate value (in mm)

Figure 3.25 Axial sxx stress distribution along the free edge of the 90-degree cluster. Displacement levels 0, 1.05 and 1.17 mm correspond to thermal residual stress only; MIC cohesive zone insertion and crack opening, respectively [26].

Durability of aerospace material systems

135

into account. The stress is shown along the free edge as a function of the axial coordinate x along path ABC for three applied displacements 0, 1.05, and 1.17 mm for Configuration A. The sxx stress before mechanical loading, which corresponds to the zero displacement, is negative at the location C. Note that all interfaces are modeled as cohesive zones and therefore the stress singularity is strictly speaking not present; however, a sharp stress concentration can be seen. The applied displacements of 1.05 and 1.17 mm correspond to MIC cohesive zone insertion and opening (i.e., crack propagation) loads, respectively. Under mechanical load, the sxx stress becomes positive including location C, and the crack inserts when the applied displacement reaches 1.05 mm. After the crack opens, the mechanical stress is relieved and the 0-degree interface (location C) sees compressive stress, sxx, consistent with the zero applied displacement case. Note that the crack insertion location does not coincide with the location of maximum sxx because the MIC insertion is based on the LaRc04 failure indicator and not just on a single stress component value. The sxx, however, clearly illustrates the phenomenon and is easier to output in the analysis. The matrix crack location predicted after the residual stress was taken into account was more consistent with the experimental observations shown in Fig. 3.20. Although several other factors can contribute to the fact that the matrix crack in the actual specimens does not originate at the point where the mathematical stress singularity would be predicted, it is the presence of residual stress that is most likely responsible for crack initiation locations. In this particular type of specimen, the precise location of the initial matrix crack affects the predicted failure load value less than 5% leading to a slight increase. The peak forces are summarized in Fig. 3.26. The predicted peak force increased when the residual stress is taken into account, which is initially counterintuitive, but consistent with the finding that the residual stress relieves the tensile transverse stress in the critical location and there is more energy required to propagate the crack in the new location. 1000

w/o residual 900

w residual 800

Peak load (N)

Experiment 700 600 500 400 300 200 100 0

1 ply mesh

1/2 ply mesh

Config A

1/4 M ply mesh

1 ply mesh

1/2 ply mesh

1/4 M ply mesh

Config B

Figure 3.26 Peak load predictions with and without account for the residual thermalmechanical stress [26].

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Durability of Composite Systems

Energy dissipation methods In all experiments, the delamination migrated during the period of unstable crack growth. The sequence of modeling events leading up to delamination migration described previously required insertion of matrix crack to facilitate the migration. Such insertion in the present algorithm is only possible at the end of the loading step. Thus the algorithm is likely to over predict the migration distance if it occurs during the unstable propagation. Several methods can be used to improve the predictions of delamination migration in the unstable propagation regime within a static framework. One approach consists of reducing the loading step size. All previous analyses were performed with a loading step of 0.015 mm/step. Reducing the step size, however, has only limited effect because it does not prevent the unstable propagation and it does not control the extent of delamination propagation at each step. Two methods were applied to address the problem of unstable propagation. They are based on the energy dissipation arc-length technique [52], which were extended to problems including thermal residual stress as well as shear nonlinearity by van der Meer et al. [53]. The essence of this methodology is to rescale the applied load to grow the damage in predefined increments, whereas, in the experiment, it would grow unstably under constant load. Though such an assumption changes the physics of the problem, it is necessary, as the unstable propagation is a dynamic process and cannot be modeled adequately in a static formulation. The quintessential component of the dissipation-based arc-length method is the on-the-fly calculation of the energy dissipated during each loading step. Consider a load step at the beginning of which there is a solution vector of U0 corresponding to applied load vector of lF0, which is a generalized load vector resulting from either applied displacements or pressure loading or both. In this method, proportional loading is required and l corresponds to the amplitude of the load at the beginning of the step. During the new load step, a solution U0 þ DU0 under load (l þ Dl)F0 is sought. Consider the following quantity DE ¼

1 T F ðlDU 0  DlU0 Þ 2 0

(3.15)

where DE is the measure of energy dissipation. If the response of the structure is linear and no dissipation takes place, then DU0 ¼ DlðU0 =lÞ and DE ¼ 0; otherwise, DE is the measure of dissipation. Ref. [53] discusses modifications required to Eq. (3.15) in the presence of nonlinear response and thermal-mechanical load. In the present work, Eq. (3.15) is applied to the energy dissipation calculation directly without additional corrections. The dissipation energy, DE, is an indirect measure of damage extent and is found to track the damage growth adequately in the presence of residual stress. The solution algorithm shown in Fig. 3.1 was minimally modified to implement the dissipation-based arc-length control method. First, a threshold value of the energy dissipation, which is allowed on a given load step was defined and denoted as DElimit. Next, an appropriate load step size was chosen, which in the present case was Dl ¼ 0.015 mm/step and the standard NewtoneRaphson method based solution algorithm initiated by augmenting each iteration with the energy dissipation DE evaluation

Durability of aerospace material systems

137

by using Eq. (3.15). If the solution converged before DE ¼ DElimit, the load was increased and a new load step was performed. If, however, DE > DElimit and convergence was not achieved, then the load update loop was activated by calculating a revised load increment as follows: Dl ¼ Dl þ

FT0 ðlDU0  DlU 0 Þ  DElimit FT0 U 0

(3.16)

Next, a new NewtoneRaphson iteration was performed with an updated load (l þ Dl*)F0. Thus, there was no modification of the original NewtoneRaphson algorithm except for changing the applied load magnitude at each step. As Eq. (3.15) is linear, one could always calculate the value of Dl , which would yield DE ¼ DElimit precisely; however, a more robust solution convergence was achieved by using an iterative Newton method correction Eq. (3.16). The iterations were repeated until convergence was achieved. The solutions obtained by this method are called as the Load Update (LU) solutions. The updated load can change dramatically and Dl often are on the order of l so that the applied load can be nearly 0 to achieve convergence. The physics of such a method is not clear as the applied load or displacement in the experimental setting is not reducing during the unstable propagation. A simple alternative to the LU method is the following dissipation-based iteration exit approach. Here, DE is monitored in the same manner used in the former method; however, when DE ¼ DElimit, the iteration step was terminated without achieving equilibrium. A new step starts with Dl ¼ 0 without any load increase and DE continued to be monitored. The second term of Eq. (3.15) is equal to zero; however, the first term will start growing if the damage continues to grow without adding any load, that is, in an unstable manner. Such steps were repeated until convergence was finally achieved while DE  DElimit, after which the applied load was increased by a nominal load step value. At the end of each step, the stress fields were examined and new matrix crack cohesive zones are inserted to facilitate the delamination migration during unstable crack propagation. Results obtained by this method are called as the load hold (LH) solution. In this case, more realistic loading conditions are achieved and at the same time, nonequilibrium deformation fields are dealt with during LH steps. The results of analysis by using the LU and LH methods were obtained for a range of DElimit values between 5 and 20 mJ and will be shown for DElimit ¼ 10 mJ. This threshold value resulted in approximately a 4 mm growth of delamination per load step. Fig. 3.27 shows the forceedisplacement curves for Configurations A and B obtained with 1/2 ply size mesh. The LU method results have characteristically nonmonotonic curves with a very sharp load retraction after the initial peak load when the matrix crack opens and the delamination begins propagating. On the other hand, in the case of the LH solution, the applied displacement is constant during unstable propagation. By comparing the LH and LU solution, it is reasonable to assume that the delamination propagates in an unstable fashion until the force drops to 60 N and the two solutions coincide. The delamination migration event, which is marked on each curve, occurred in the unstable regime in all cases. The migration distance

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Durability of Composite Systems

(a)

600 500

LH

Force (N)

400

LU 300

Migration 200 100 0 0.00

0.50

1.00

1.50

2.00

2.50

Displacement (mm)

(b)

1000 900 800

LH

Force (N)

700

LU

600 500 400

Migration

300 200 100 0

0.00

0.50

1.00

1.50

2.00

2.50

Displacement (mm)

Figure 3.27 Forceedisplacement curves predicted using the load update (LU) and load hold (LH) methods for configurations A (a) and B (b) [26].

predictions from the LU and LH methods for both configurations and all three-mesh densities are summarized in Fig. 3.28. Horizontal lines show the range of the experimental data and the calculated migration distance without residual stress is indicated by the black circles. The first observation is that the inclusion of the residual stress led to a higher overprediction of the delamination migration distance than without including residual stress. This observation is unsurprising as higher peak stress was predicted by including the residual stress and that led to propagating the delamination further with the standard method, where no intermediate steps are allowed until the delamination stops. The application of the LU and LH methods provides increased accuracy during the unstable propagation regime. As seen in Fig. 3.28, the predicted migration distance with all three mesh densities for Configuration A are within the experimental range for both the LH and LU methods and slightly underpredicted for Configuration B. For both Configurations A and B, the predicted migration distances with the load control solution techniques LH and LU are closer to the lower bounds of experimental data. This can be explained by the inertia of the dynamic unstable propagation, which is not accounted for in the present model. The results predicted by both

Durability of aerospace material systems

Migration distance from load point (mm)

30.0

139

Migration distance without residual stress

Standard LH

25.0

LU 20.0

Experiment

15.0

10.0

5.0

0.0 1 ply mesh

1/2 ply mesh

1/4 M ply mesh

Config A

1 ply mesh

1/2 ply mesh

1/4 M ply mesh

Config B

Figure 3.28 Delamination migration distance predictions for Configurations A and B with three mesh densities using the three different convergence approaches. The bars correspond to predictions with residual stresses included [26].

methods are close to each other and avoid over predictions of the migration distance due to unstable crack growth.

3.4.1.2

Static analysis conclusions

The regularized extended finite element method was used to predict matrix cracks, delamination initiation, and propagation, including migration from one interface to another. The simulations were performed simultaneously with experimental measurements conducted at NASA Langley Research Center and showed good quantitative agreement with experimental data in most cases, based on several key metrics such as peak loads and delamination migration distances. The addition of thermal residual stress yielded slightly higher values of the peak load, which were practically mesh independent for single, 1/2 and 1/4 ply thickness mesh size densities. The initial matrix crack location predicted was in good agreement with experimental data, while the predictions without the residual stress placed the initial crack at the location of a stress singularity. Delamination migration represents an energy dissipation mechanism, which allows for delamination to grow on a different interface if it becomes energetically unfavorable to continue growing on the original interface. In the composite tapered beam specimen considered, the delamination migration took place during the unstable propagation regime in all experiments and the distance between the load application point and the migration location was overpredicted for both specimen configurations with and without residual stresses. Two methods based on the energy dissipation arc-length control technique were implemented and predicted the migration distances within the experimental range for all three mesh densities for Configuration A and slightly underpredicted for Configuration B.

140

3.4.1.3

Durability of Composite Systems

Fatigue analysis

The CTB discussed in earlier sections is further analyzed in monotonic fatigue loading. Similar to static analysis, these studies were performed in two stagesdinitial blind predictions where the simulations were performed simultaneously along with the experimentation conducted at NASA LaRC [54]. The first phase of simulations was performed before knowing the experimental data and the second round of predictions was done after experimental data was revealed. The parameters of interest in fatigue analysis include several cycles to matrix crack onset; delamination progression w.r.t cycle count including migration and delamination migration distance. In this section, the CTB fatigue problem statement followed by results and conclusions are discussed.

CTB fatigue problem statement The CTB specimen considered for fatigue analysis is the same as the CTB specimen discussed in the static analysis, with the same material, same thickness, ply layup and the clamping. In this work, monotonic cycling loading with maximum arm displacements of 0.91 mm (high load) and 0.825 mm (low load), which are approximately 80% and 70% of peak static loads are studied. Three different meshes developed earlier for static analysis with the needed node sets were considered for these studies. From the mesh convergences studies in the static analysis, it was concluded that the fine model (mesh size of approximately 1/2 ply thickness) is appropriate considering both accuracy and computational time. Hence, the results reported are obtained using a fine (1/2 ply) model. The material properties required for static analysis are listed in Table 3.4. The additional properties required for analyses under fatigue loading are discussed in this section. Table 3.5 summarizes the fatigue parameters used for analysis. All fatigue input data was provided for R ¼ 0.1; however, the experimental setup was later changed to R ¼ 0.2 to prevent setup issues arising from unloading the specimen. This change in R ratio was only taken into account at the correction stage of these analyses. Only Table 3.5 Material properties for fatigue analysis. Property

Value

SeN constants

(R ¼ 0.1)

S1 (normal)

0.0765

S2 (shear)

0.045

Paris law

(R [ 0.1)

CI (mm)

0.0051

CII (mm)

0.0443

mI

6.710

mII

5.450

h

2.1

Durability of aerospace material systems

141

Table 3.6 Rescaled material properties for fatigue analysis. Property

Value

Paris law

(R ¼ 0.2)

CI (mm)

0.0010

CII (mm)

0.012

mI

6.710

mII

5.450

h

2.1

the Paris law parameters are rescaled based on Eqs. (3.6), (3.9), and (3.10) and are shown in Table 3.6.

Results and comparisons The predictions were performed in two stages before the experimental data was known and after. The initial predictions were performed by using the methodology in Ref. [43] at R ¼ 0.1 and at high load amplitude of 80% and low load amplitude of 70% of static failure load without taking into account the processing residual stress. The failure process is generally similar to that observed under static loading and consists of matrix crack formation in the 90 degrees 4 ply cluster at the fillet, stage (1)e(2) in Fig. 3.29(a) with subsequent delamination initiation and propagation, stage (2)e(3) followed by migration, stage (3)e(4) and continued propagation (5). The main experimentally observed difference between the damage evolution under static and fatigue loading was the difference between unstable propagation in static regime on all stages including part of the stage (5) whereas in the fatigue regime the propagation was stable on all stages for both load amplitudes. The analysis, however, predicted unstable delamination propagation at the initial stages of delamination propagation and in the case of 80% loading in the migration stage as well. The failure patterns predicted in the blind prediction stage for 80% and 70% loading on Configuration A are shown in Fig. 3.29. The delamination migration distance in the case of the 80% loading is overpredicted, which is consistent with the observations during static propagation and is a result of unstable propagation of the delamination. In the case of 70% loading, the migration distance is predicted more accurately due to the stable propagation of the delamination near the migration. The revised predictions took into account R ¼ 0.2 load ratio, thermal residual stress and modified fatigue CZM formulation as described in previous sections. The respective delamination and matrix cracking paths are shown in Fig. 3.30. The R ¼ 0.2 resulted in slower rates of delamination growth and resulted in a more accurate prediction of the delamination migration distance in both cases. Note that the matrix cracking initiation location was predicted on the tapered edge at the interface of the 0-degree and 90-degree plies before the residual stress was taken into account (Fig. 3.29) and in the middle of the 90 degrees 4 ply cluster after it was taken into account in Fig. 3.30.

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Durability of Composite Systems

(a) am ~ 15.3 mm

(1)

(4) (5) (3)

(2)

69 70 71 72 73 74 75 76 77 78 79 00 01 02 03 04 05 06 07 08 09 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

Delamination length ~ 32 mm

(b) am ~ 10.4 mm

Delamination length ~ 26.9 mm

Figure 3.29 Damage events during blind predictiond(1) crack initiation, (2) delamination growth (3), (4) delamination migration, and (5) growth past migration of CTB specimen at high load amplitude. (a) w80% and (b) w70% loading [42].

(a) am ~ 12.5 mm

69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

Delamination length ~ 29.5 mm

(b) am ~ 7.5 mm

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100 101 102

Delamination length ~ 24.5 mm

Figure 3.30 Delamination and matrix cracking propagation path predicted during the correction stage: (a) w80% loading and (b) w70% loading [42].

Durability of aerospace material systems

143

50

Delamination length (mm)

45 40

Experiment, ~80% loading

Migration location

Blind prediction, R=0.1

35 30

Revised prediction, R=0.2

25 20 15 10 5 0 10

100

1000

10000

100000

1000000

Log N (cycles)

Figure 3.31 Delamination length verses cycle count at 80% loading [42].

Delamination length (mm)

40 35

Experiment, ~ 70% loading

30

Blind predictions, R=0.1

25

Revised predictions, R=0.2

Migration location

20 15 10 5 0 100

1000

10000

100000

1000000

Log N (cumulative cycles)

Figure 3.32 Delamination length verses cycle count at 70% loading [42].

The total delamination length as a function of cycles including experimental data is displayed in Fig. 3.31 for the 80% load amplitude and in Fig. 3.32 for 70% load amplitude. In both cases, the rate of delamination growth was significantly overestimated in the blind prediction phase. As seen in Figs. 3.31 and 3.32, the rate of propagation of the mode I crack is approximately five times lower for R ¼ 0.2 than R ¼ 0.1 and has a strong effect on the rate of propagation. It is also observed that including the residual strength calculation in the N-R iteration loop has affected the matrix cracking initiation stage reducing the cycle count for delamination initiation and early propagation stage at both load levels. The results appear to correlate better with the experimental data, especially for the 70% loading case. However, in both the blind prediction and correction phases the methodology predicted partial unstable delamination growth for the 80% loading, which was not observed in the experiments. This difference could be a result of bridging and delving effects not accounted for in the computations.

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Durability of Composite Systems

3.4.1.4

Fatigue analysis conclusions

Rx-FEM methodology was applied to predict failure in laminated composite CTB under monotonic fatigue loading at amplitudes of 80% and 70% of the static failure load. Matrix crack initiation leading to delamination initiation and propagation including its migration from one interface to another and number of cycles to these damage events under monotonic cyclic loading were predicted. The initial prediction performed at R ¼ 0.1 significantly overpredicted the rate of delamination growth as compared with experiments at R ¼ 0.2, which were conducted simultaneously by NASA team at NASA LaRC. The corrected prediction was performed at R ¼ 0.2, taking into account the residual thermal processing stress and modified algorithm. The modified algorithm included, calculating residual strength directly in the NewtoneRaphson iteration loop. A reasonable agreement with experimental observations including the delamination migration distances was achieved in the correction stage. However, in both the blind prediction and correction phases, the methodology predicted partial unstable delamination growth for the 80% loading, which was not observed in the experiments. This could be a result of bridging and delving effects not accounted for in the computations.

Three-point bend with flange (3PB-F)

3.4.2 3.4.2.1

Model description

Tape25 plies damage enriched

Layered C3D8 tape part – 22 plies no damage

Flange

Skin

For more complex structures, it was determined that using the standard ply-by-ply DDM with damage enabled in all plies was both excessive and computationally intractable. To reduce the computation cost, layered 8-node brick (LC3D8) elements were implemented for the first time in DDM simulation [27] of common feature test component (CFTC). The full 3PB-F model with a close-up of the region near the load application is shown in Fig. 3.33. The flange is modeled with nine plies of carbon/epoxy fabric to match the flange region of the hat. All damage is disabled in the fabric plies.

Fabric9 plies no damage

Load washer

Figure 3.33 Three-point bend with Flange model with close-up of the model construction [27].

Durability of aerospace material systems

145

w

Arrow color

Description of w = 0 boundary condition

z

Span 1, span length = 2.6 in Span 2, span length = 3.5 in Span 3, span length = 4.15 in

y

Span 4, span length = 8.6 in

Figure 3.34 3PB-F boundary conditions for all four spans [27].

Near the load application region, the skin is modeled ply-by-ply with full damage enabled throughout. The three skin plies closest to the flange are modeled as full plies throughout the length of the specimen. Away from the damage region, the rest of the skin plies are modeled using the LC3D8 part to model the stiffness of these 22 plies. The load washer and countersunk fastener are modeled as steel. 3PB-F simulations were set up to match experimental loading conditions. The boundary conditions for the model are shown in Fig. 3.34. Displacement in-plane with the specimen was restricted at the center node point of the fastener on the load application side of the specimen. The specimen was tested at four different span lengths in static loading, and one in fatigue loading. The span lengths for static loading range from 2.6 to 8.6 in. This wide variation of spans produces different stiffnesses, failure loads and failure modes. Three different finite element mesh refinements were considered for the 3PB-F analysis. These three meshes with information are shown in Table 3.7. In the mesh images, the blue elements correspond to the skin, while the orange elements correspond to the flange. It is anticipated that the damage initiation and growth is likely to occur in the left and right regions of the flange. The first mesh developed was TPB32, which used a parametrically defined mesh, developed in Sigma. This mesh was used in an attempt to capture both first peak failure modes as well as any secondary pull-through failure modes that may occur in the specimen. This was used for static modeling for all four spans. However, during the static simulation of Span 4, it became apparent that the model would not simulate failure in a reasonable amount of time due to model size, even in static loading. As fatigue loading takes significantly longer than static loading in DDM, a new mesh was developed to reduce the size while maintaining the first peak strength. Thus, a mesh was developed in ABAQUS/CAE using unstructured meshes in the damage regions with large elements. This mesh is named ABQ2 and is approximately one-third of the size of TPB32. Following the development of the first common feature test component (CFTC) model [27], a 3 PB-F mesh that matched element sizes in the left and right areas of the flange mesh was developed named ABQ3. This further reduced the size of the simulation to a little over one-fifth of the TPB32 model. The elastic, strength, fracture, and fatigue properties for the carbon/epoxy tape are provided in Table 3.8, which are similar to Ref. [27]. Table 3.9 contains the carbon/ epoxy fabric properties as well as the interface properties related to the tape/fabric

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Table 3.7 3PB-F model names, meshes, and model sizes [27]. Name

Mesh

Starting DOF

TPB32

2026k

ABQ2

676k

ABQ3

451k

interface that is critical in 3PB-F specimens. As no Paris law data were generated for the tape/fabric interface of this material system to date, an initial assumption was made that the tape/fabric interface would have the same parameters as a tape/tape interface. For fasteners and load washers, steel elastic properties were used. No failure was simulated in the fabric plies, resin, or steel parts.

3.4.2.2

Results and discussion

The static simulations for all spans along with the experiments are plotted in Fig. 3.35. In all cases, the simulation surprisingly predicted a less stiff response compared to the experiment. This was consistent with simulations using other finite element codes. The TPB32 mesh was used for all four spans and produced accurate first peak response predictions in all cases. ABQ2 and ABQ3 meshes were also simulated for span 4 as a

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147

Table 3.8 Carbon/epoxy tape material properties for DDM analysis [27]. Carbon/epoxy plain weave fabric elastic Properties

Tape/fabric interface Properties

E11av

10.809

msi

Fracture

E22av

10.769

msi

F33t

8

ksi

E33av

1.251

msi

F33c

217.6

ksi

n12

0.039

S

11

ksi

n13

0.35

GIc

2.7

lbf/in

n23

0.35

GIIc

5.98

lbf/in

G13

0.605

msi

G12

0.591

msi

SeN

G23

0.591

msi

S1

0.09852

K12

43.7

ksi

S2

0.05427

K13

45.6

ksi

H

2.25

n12

0.297

n13

0.293

a11

0.0

a22

0.0

Fatigue

Paris law C1

0.006934

m-in/in/F

m1

7.322

m-in/in/F

C2

0.035554

m2

6.35

h

2.25

comparison to TPB32. All three meshes produced a very similar first peak response, which was the basis for the fatigue loading. Following the first peak, the response diverges significantly from the experiment. The postpeak behavior in the experiment is driven by backface buckling in the skin, which cannot be simulated due to the large elements in the damage region. It should be noted that TPB32 did not reach 1 in. of displacement during simulation due to the size of the simulation and was stopped. Fig. 3.36 shows the post-peak damage state of ABQ2 compared with the experiment. Similar to the experiment, delamination migration was predicted between the flange/skin through a matrix crack in the top skin ply. The back-face buckling mentioned earlier is shown in the experiment image in Fig. 3.36. Delamination was restricted near the fastener due to the large element size. The delamination (or crack) propagation is shown in Fig. 3.36. The offset between the simulation ABQ2 and experiment seems related to the higher predicted first peak load as well as the element size in the delamination region.

3PB-F Conclusions DDM was applied successfully to 3PB-F in static loading at four different span lengths. To accommodate computationally intractable size of this subelement,

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Table 3.9 Carbon/epoxy fabric properties and tape/fabric interface properties for DDM analysis [27]. Carbon/epoxy plain weave fabric elastic properties

Tape/fabric interface properties

E11av

10.809

msi

Fracture

E22av

10.769

msi

F33t

8

ksi

E33av

1.251

msi

F33c

217.6

ksi

n12

0.039

S

11

ksi

n13

0.35

GIc

2.7

lbf/in

n23

0.35

GIIc

5.98

lbf/in

G13

0.605

msi

G12

0.591

msi

SeN

G23

0.591

msi

S1

0.09852

K12

43.7

ksi

S2

0.05427

K13

45.6

ksi

H

2.25

n12

0.297

n13

0.293

C1

0.006934

a11

0.0

m-in/in/F

m1

7.322

a22

0.0

m-in/in/F

C2

0.035554

m2

6.35

h

2.25

Fatigue

Paris law

LC3D8 elements were implemented successfully. This allowed a significant reduction in the overall computation size for this and several other subelement components. It has been demonstrated that DDM can accurately predict peak loads and damage locations in all four configurations.

3.5

Ply-level constitutive behavior methods

There are in fact many composite material systems and applications for which PDA is not yet useful or adequate at the present time. Applications of composite materials to hypersonic and other high-speed airframes, to jet engine components such as combustor liners and thrust deflectors, and to structures subjected to impact and high energy rate loading are immediate examples. For these and other situations we will follow arguments presented by Reifsnider et al. [55] for damage accumulation in composites by postulating that composite strength at the structural level may be used as a damage metric for the prognosis of remaining strength and life under continued loading. We assume that the remaining strength at

Durability of aerospace material systems

(a)

149

(b)

3500

900 800

3000

700

Load (Ibf)

Load (Ibf)

2500 2000 1500 1000

0

0.05

0.1

400 300

Rx-FEM TPB32 3PBF-7 3PBF-8 3PBF-9

100

3PBF-1 0

500

200

Rx-FEM TPB32

500

600

0

0.15

0

Displacement (in)

0.1

(d) 500 450

450

3PBF-2

400

3PBF-3

0.2

3PBF-4

350

350

300

Rx-FEM ABQ2

250

Rx-FEM ABQ3

Load (Ibf)

400

300 250 200

Rx-FEM TPB32

200 150

150

Rx-FEM TPB32 3PBF-10

100 50 0

0.15

Displacement (in)

(c)

Load (Ibf)

0.05

3PBF-11

100 50 0

0

0.05

0.1

0.15

0.2

1

0

2

3

Displacement (in)

Displacement (in)

Figure 3.35 3PB-F static simulation and experiment results for (a) span 1, (b) span 2, (c) span 3 and (d) span 4 [27].

Ply 23/ply 24 interface

Hat/skin interface

Ply 24/ply 25 interface

Rx-FEM cracks

Skin ply

Skin ply

Figure 3.36 Postpeak load experiment damage (left) compared to ABQ2 post first peak load damage (right) [27].

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Durability of Composite Systems

Normalized stress level

Remaining strength curves

1 1

Sr

1

Sa

t1

2 Sa

Life curve

t2

Time

Figure 3.37 Equivalent damage defined as equivalent change in remaining strength at the structural level.

the structural level may be determined (or predicted) as a function of load level and some form of generalized time and history of loading. For a given load level, a particular faction of life corresponds to a certain reduction in remaining strength. We claim that a particular fraction of life at a second load level is “equivalent” to the first if and only if it produces the same reduction in remaining strength, as illustrated in Fig. 3.37. In the case of Fig. 3.37, time t1 at an applied stress level S1 is equivalent to time t2 at stress level S2 because it gives the same remaining strength. In addition, the remaining life at the second load level is given by the amount of generalized time required to reduce the remaining strength to the applied load level. In this way, the effect of several increments of loading may be accounted for by adding their respective reductions in remaining strength. For the general case, the strength reduction curves may be nonlinear, so the remaining strength and life calculations are path dependent, not simple ‘ratios of time to life or cycles to life as in classical damage summation methods. To enable this approach, we postulate that normalized remaining laminate strength (our damage metric) is an internal state variable for a damaged material system. This normalized remaining strength is based upon the selection of an appropriate failure criterion (such as maximum stress or maximum strain) in the critical elements of our structure that control failure. In this way, we can consider a single scalar quantity rather than the individual components of the strength tensor, and most importantly, we can consider the specific failure modes that control our composite laminate and structure for a given laminate and application. We denote this failure function by Fa. We next construct a second state variable, the continuity, which we shall define to be (1  Fa). Then we define our normalized remaining strength in terms of that failure criterion and represent the remaining strength as the variable Fr so that Z Fr ¼ 1  DFa ¼ 1 

ð1  FaðsÞÞds

(3.17)

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151

where, s is a generalized variable that marks progress in the damage process, such as time, cycles, or some process variable. In such a form, the failure condition is the point at which the remaining strength equals the applied load written in terms of our failure function (Fr ¼ Fa), using the simple idea that failure occurs at the point in loading history when the structural strength is reduced to the level of applied loading. To support the idea of nonlinear changes in remaining strength with history (like the “sudden death” behavior so often observed wherein the change of strength is very little early in life and accelerates as damage collects), we introduce the idea that change in strength may depend on some power of the generalized time variable and use the form Z Fr ¼ 1  DFa ¼ 1  ð1  FaðsÞÞjsj1 ds (3.18) Which has been found to fit many cases of observed behavior by the authors [55]. The generalized time may be something as simple as the current number of cycles (n) divided by the fatigue life (N) in the local material element that controls structural failure for a specific laminate in a given applied condition (such as the “zero degree ply”) in a laminate loaded in simple tension. We use the usual representation of the fatigue life in the local element with the form FaðtÞ ¼ S=Sult ðin local controlling elementÞ ¼ An þ Bn logðNÞ

(3.19)

These are local quantities, and as we have seen from the discrete defect (PDA) results earlier, as damage develops in the other plies of the laminate (or delamination, etc. occurs) the stress in the local ply controlling failure changes so when we solve for life N in our controlling local element at a given point in the loading history, it is a variable with cycles, time, local stress S, and current material strength in that element Sult (which may be changing with temperature, time, etc.). Entering s ¼ n/N in Eq. (3.19), we see that dðn=NÞ ¼ 1=N dn  n=N2 dN

(3.20)

Finally, if cycle frequency is introduced by setting n ¼ f * t where f is the frequency in cycles per unit of time and t is the time unit, then the most general form of the strength change equation becomes     Z  S t j1 Fr ¼ 1  1  Fa  j j f Su N

f f t e dN = dt NðtÞ NðtÞ2

! dt (3.21)

This approach has the disadvantage of being an empirical formulation based on available experience, so it is not truly predictive in the strict fundamental sense. At the same time is has the advantage of making direct use of discrete defect analysis

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Durability of Composite Systems

to use one set of composite (as manufactured) strength data in a laminate and structural code to predict laminate and structural response for many structural failure modes and applied environments including temperature and rate effects for example. So it provides a critical step in the predictive design and analysis of composite laminates and structures as it allows the designer to compare different materials, laminates, and structural configurations for actual applied conditions to sort out the optimum design. Further, as the integral sums up the effects of specific histories of loading, it can be used for the time-resolved prognosis of an operating structure for structural integrity, remaining life and strength for example. Those capabilities have been demonstrated in the literature, and we will look at some examples later. In the remainder of this chapter, we will use Eqs. (3.17)e(3.21) to explain how each physical variation effects remaining strength and life, beginning with how a bilinear fatigue curve of the baseline material influences the results. To close the chapter, we will add only a short discussion of the “next phase” of this approach which is to incorporate data analysis methods (including AI-assisted analysis to assist when missing physics can be augmented by continuing experience), which shows considerable promise. We select first an example in which the baseline fatigue data for a unidirectional composite material is bilinear as a function of applied cyclic load level. The material used in this example is the aromatic polymer composite (APC-2), which is PolyEtherEtherKetone (PEEK) matrix reinforced with AS-4 carbon fibers. PEEK is a semicrystalline polymer with a glass transition temperature of 144 C and a melting point of 335 C. Typical unidirectional properties of this material system are given in Tables 3.10 and 3.11. This material system was chosen because it has been well characterized experimentally. In particular, Picasso and Priolo [56] have characterized the unidirectional (0 degree) fatigue behavior of APC-2. Those data were found to be well represented by a bilinear form as follows: Table 3.10 Unidirectional properties for APC-2 considered. Quantity

Value

E1

19.4 msi (134 GPa)

E2

1.29 msi (8.89 GPa)

G12

0.74 msi (5.10 GPa)

n12

0.35

Xt

303 ksi (2090 MPa)

Xc

176 ksi (1210 MPa)

Yt

11.6 ksi (80 MPa)

Yc

40.6 ksi (280 MPa)

S

11.6 ksi (80 MPa)

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Table 3.11 Stress redistribution caused by ply cracking. s Sx

s Sy

sxy

Fa

Undamaged

2295

65.9

0

0.94

One cracked 45-degree ply

2536

90.2

0

0.92

Two cracked 45-degree plies

2873

126.9

0

0.89

0.9 0.7

0.8

Data from picasso and prido

0.6

Fitted curve

0.4

0.5

Normalized stress level

1.0

1.1

S/Sult ¼ An þ Bn log(Nf) where An ¼ 1.312, Bn ¼ 1.818 for N  5982 cycles and An ¼ 0.786, Bn ¼ 0.0425 for N > 5982 cycles. Fig. 3.38 shows this representation. This bilinear representation is used at the ply level for all laminates made from that material and loaded in whatever manner is required for a given application. To illustrate the method, consider the data of Simonds and Stinchcomb [57] wherein a quasiisotropic laminate made from this material was loaded in cyclic tension/compression. The strength and life predictions are shown in Fig. 3.39. Other load levels can be found in Ref. [55]. Although there is some displacement of the data from the predicted results, it can be seen that the “sudden death” effect (especially for the 22 ksi case) and general accuracy of the method are evident. A second example demonstrates the influence of the damage in other plies in the laminate on the stresses in the last ply to fail, and thereby on the predicted/observed failure load through the ply-level applied stress level Fa(S/Sult) in Eq. (3.21). The material, in this case, is a T300-5208 graphite-epoxy laminate with a stacking sequence of [(0,þ45)s]4s. We consider uniaxial tensile loading and using the standard laminate

10

100

1000

10000

100000

1000000

Cycles

Figure 3.38 Comparison between measured and fitted fatigue lives for unidirectional APC-2 based on the data of Picasso and Priolo [56].

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Durability of Composite Systems

Maximum applied stress (Ksi)

45 40 35 30 25

Data from simonds and stinchcomb 22 Ksi 27 Ksi Predicted

20 15 10 1

10

100

1000

10000

100000 1000000 10000000

Cycles

Figure 3.39 Comparison between predicted and measured remaining strength for quasiisotropic APC-2 based on the data of Simonds and Stinchcomb [57].

analysis to determine that the initial stress in the 0-degree plies (the direction of loading) divided by the strength of the material in that direction has a value of 0.94. Then cracking of one and then both of the 45-degree plies is considered, so that all stress transverse to the fibers in those plies is set to zero, simulating the “real” situation. The results on the value of the failure function, Fa (S/Sult), in the zero degree ply as used in the remaining strength Eq. (3.22) are shown later [55]. In this case, the “scissors effect” of the 45-degree plies increases the axial stress in the 0-degree plies until cracking reduces that constraint and reduces S/Sult and Fa as damage develops. The result is that the predicted life and predicted remaining strength increases as damage develops using the model equations earlier, and the predicted behavior aligns well with observed data, as seen in Fig. 3.40. It should be noted that predictions of both remaining strength and life are shown in that figure and that both are well aligned with observations despite a great deal of nonlinear behavior including matrix cracking and large local stress changes due to that damage development. The effect of this change in the “last ply failure stress” is substantial, in the model and in reality, as shown by Fig. 3.40. Our final example, considers the silicon carbide matrix composite processed by a chemical vapor infiltration technique [58,59]. This material system is simply called as Nicalon/E-SiC. The emphasis is mainly on the cyclic fatigue behavior of Nicalon/E-SiC at elevated temperatures. Thermomechanical behavior as well as potential damage modes and failure mechanisms of these materials were examined under various load levels, stacking sequences, specimen geometries, and testing temperatures. Stiffness changes due to damage (as considered earlier), cycle-dependent failure (in which our generalized time is cycles) was included, and finally, time-dependent failure was considered in the modeling of the timedtemperaturedcycles changes in strength in our local controlling element Sult and included in the changes that were used in Eq. (3.21).

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F3–1 F5–5

1.0

Residual strength ratio

0.8 F2–2 * 0.6 * Predicted life Observed life Observed residual strength Predicted residual strength with biaxial correction Predicted residual strength without correction

0.4

0.2

* F1–9

* F4–6

F5–5: 71.7 ksi, 10 kn, RS 78 ksi F2–2: 71 ksi, 21.3 kn life F3–1: 65 ksi, 200 kn, RS = 87 ksi F1–9: 64 ksi, 290 kn life F4–6: 57 ksi, 330 kn, RS = 81.8 ksi

0 3.0

4.0

5.0

6.0

Log cycles

Figure 3.40 Comparison between predicted and measured remaining strength for T300-5208 laminates based on the data of Case and Reifsnider [55].

Damage evolution due to cyclic loads was represented by a stiffness degradation curve. For a [(0,90)/(0,90)]2s laminate (cross-ply), it is assumed that each (0,90) woven ply may be represented by a 0-degree fiber ply and a 90-degrees matrix ply. We have also assumed any reduction in laminate stiffness is due to degradation of matrix stiffness only. Using results of data considered, stiffness degradation for 10, 18, and 15 ksi stress-level tests was best represented by EðnÞ ¼ b þ m ðlog nÞ Eo

(3.22)

where (b,m) ¼ (0.926489,0.000000) for 10 ksi. ¼ (0.683933,d0.00347) for 13 ksi. ¼ (0.680945,d0.01962) for 15 ksi Where E is the initial stiffness of laminate (undamaged) and E(n) is the laminate stiffness (damaged) which is a function of cycles. The failure function used was a maximum strain-failure criteria as follows: Fa ¼ ε=εc

(3.23)

where ε and εc, represent strain due to applied stress, so, and strain to failure of composite, respectively. We can write the strain to failure of the composite as

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Durability of Composite Systems

  εc ¼ Xt = vf  Ef þ bð1  vfÞE0m

(3.24)

Where Xt is UTS of laminate (35.8 ksi for our example). Parameter b is used as a scaling factor because upon the application of first cycles of tensile loading, the matrix cracks and its stiffness drops from E0m to b* E0m. The remaining cyclic dependent input should represent the life of the local controlling element under constant stress state. A suitable representation is the SeN relationship for a unidirectional material in the controlling local ply under a constant cyclic loading in the fiber directions having the form SaðnÞ ¼ An þ Bnðlog nÞp SuðnÞ

(3.25)

where Sa(n) is the maximum cyclic stress value and Su(n) is the material strength in that loading direction at the local level. For our next example, we include the timedependent failure of the Nicalon/E-SiC high-temperature material, which is controlled by temperature, exposure to oxidation at 1800 F in air, and creep-rupture failure. For our example, the rupture time tr can be expressed in terms of parameters such as stress level s, and test temperature T with the following equation tr ¼



   A þ Bs þ Cs2 þ D þ E Tr þ F Tr2

(3.26)

in which tr ¼ TTpest Tpest with Tpest equal to the “pest temperature” for which oxidation is severe (with much less effect below and above that temperature as often observed at such high temperatures), and A ¼ 17.95, B ¼ 18.98, C ¼ 0, D ¼ 0.611, E ¼ 0.733, F ¼ 18.94, and Tpest ¼ 1550 F. With these values, Eq. (3.21) produces the following predicted remaining strength and life for this complex set of material changes caused by time, temperature, and cyclic loading. Our final example considers a polymer membrane in a polymer electrolyte membrane (PEM) fuel cell, as used for hydrogen propulsion in some commercial automobiles and trucks, and for auxiliary electrical power supplies in industrial and housing applications. The membrane in these fuel cells separates the anode from the cathode, allowing only the flow of one species of ions through the cell to create a source of electricity driven by electrochemical potentials. The membranes are mounted in a rigid frame, so that there is a temperature and hydration cycle as the power from the cell increases and decreases during use, which creates cyclic mechanical stress and strain. However, there is also a chemical degradation element, which is a function of the concentration of hydrogen peroxide, which is an electrochemical byproduct of the fuel cell during operation, and the time of exposure to those local chemical acids. εf ðtÞ ¼ ð1  0:02 tÞC=CoððT  TgÞ = ToÞ

(3.27)

where t is time, C is the concentration of the peroxide that collects in and around the membrane, T is temperature absolute, and To is a reference temperature.

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157

So, for this situation, our local failure function that represents the effect of the combined degradation effects on the strain to failure can be stated as Fa ¼

RHmax DεðhydrationÞ ¼ A  B logðNðtÞÞ 120 εf ðtÞ

(3.28)

where RHmax is the maximum relative humidity in the membrane, normalized by the saturation value, A and B are material constants, and N(t) is the number of cycles of hydration to failure calculated at time t. Then solving for N(t) in Eqs. (3.27) and (3.28) and using that form in our integral Eq. (3.22) produces the form Z FðtÞ ¼ 1 

t

ð1  FðaÞÞ  j  a j1 f q j h

i 1 j B  logðFaðaÞÞA (3.29)

d ! FaðaÞ da da 1 A FaðaÞlogFaðaÞ a

1.2

Data

0.6

0.8

1.0

Remaining strenght prediction

Value of failure function

0.4

Remaining strengh and local failure function as ratios of current strengh in the controlling local element

which, calculates remaining strength at time t under after exposure to the history of temperature and concentration changes caused by the operation of the fuel cell (variations in output power required) for that calculation period. For a typical operating sequence, including the chemical degradation in the model decreases the predicted and observed life by as much as 50% [60]. Figs. 3.41 and 3.42 shows an example of such a calculation.

1

10 E2 E3 E4 Cycles of loading at one cycle per second in powers of 10

E5

Figure 3.41 Comparison between experimental and predicted remaining strengths of [(0/90) (0/ 90)]2s Nicalon/E-SiC laminates for R ¼ 1, f ¼ 1 Hz, at 1800 F [59].

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Durability of Composite Systems 1

Normalized strength

Remaining strength, Fr

Applied condition (failure function), Fa

0.5

Data point

0

0

5

10

15

20

Time, t(h)

Figure 3.42 Predicted and observed life of a PEM membrane using Eq. (3.18). If chemical degradation is eliminated the life of the membrane doubles [55].

It is seen that the failure function that carries the time/history-dependent information about hydration and chemical concentration increases quite a bit during the lifetime, “accelerating” the degradation and significantly reducing the predicted and observed life.

3.6

Machine learning methods

The last subject to be mentioned in this chapter is the role of artificial intelligence and machine learning to the reliability of the aerospace system in operation. This is, in fact, a broad subject worthy of extensive discussion and presentation, but alas, the consensus on the scope and details of this field is still somewhat uncertain. The general concept looks like Fig. 3.43. We have discussed this approach in Ref. [60], for situations in which the real-time input data during service are dielectric material properties. We only mention here that random forest methods of classification produced an excellent agreement between Real time performance prediction and control methodologies using artificial intelligence

Real time sensor data collection

Al predicts the performance

Figure 3.43 Approach to incorporation of physics-based analysis into real time prediction of reliability for aeronautical systems during operation [60].

Durability of aerospace material systems

159

predicted reliability using data-based machine learning and observations, so the foundations of this field are certainly firm. There is a myriad of other topics that deserve attention, but we defer those discussions to a later opportunity.

3.7

Chapter conclusions

Advanced numerical methods are successfully applied for verification and validation of laminated composites, modeling damage initiation and progression. Rx-FEM modeling approach, which models displacement discontinuities associated with matrix cracks without regard to mesh orientation are applied to complex laminated composites. Verification of coupon-level components like DCB, ENF, and MMB is successfully demonstrated for both quasistatic and monotonic fatigue loading. During the subelement-level validation process, several new features were implemented and discussed such as thermal residual stresses, energy dissipation arc-length methods, layered 8-node brick elements and R ratio effects. There are many composite material systems and applications for which DDM has not yet been developed at this time. Applications of composite materials to hypersonic and other high-speed airframes, to jet engine components such as combustor liners and thrust deflectors, and to structures subjected to impact and high energy rate loading are some current examples. However, we have shown that ply-level constitutive-based models, with empirical models of strength reduction caused by cyclic loading and various combinations of applied conditions including elevated temperature, chemical degradation environments, and variable rate/level loading can be used to construct life prediction models of composite materials and structures for those extreme conditions. We have also shown that there is very much more to be done in this general area. Beyond the obvious limitations of the details of the approaches, we cannot yet answer some basic questions, like “what is a fundamental canonical data set for life prediction of composite materials and structures?” We can guide structural design, but can we guide material system design in any fundamental way? Moreover, finally, in application worlds that often and more frequently involve unknown histories of applied conditions and unknown material properties (e.g., after some history of use), how can we use machine learning and other AI algorithms to improve the performance reliability and safety of aerospace composite material systems? These horizons beacon us for exploration and continued development.

Acknowledgments This work was partially funded under NASA contract number NNL16AA02C with the University of Texas at Arlington, Arlington, TX. The authors are grateful to Drs. Nelson Vieira de Carvalho, Ronald Krueger, T. Kevin O’Brien and the Advanced Composite Consortium members for encouragement and fruitful discussions. We also acknowledge continuous support and collaborations with Air Force team Drs. David Mollenhauer, Lauren Fugerson, Eric Zhou; Boeing team Drs. Joe Schaefer, Haozhoung Gu.

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References [1] Z. Hashin, A. Rotem, A fatigue failure criterion for fiber reinforced materials, J. Compos. Mater. (1973). [2] G.B. Murri, Evaluation of delamination onset and growth characterization methods under mode i fatigue loading, in: NASA STI Program, 2012. [3] T.K. O’Brien, W.M. Johnston, G.J. Toland, Mode II Interlaminar Fracture Toughness and Fatigue Characterization of a Graphite Epoxy Composite Material, 2010. Nasa/Tme2010216838. [4] T. O’Brien, Towards a damage tolerance philosophy for composite materials and structures, in: Composite Materials: Testing and Design, Ninth volume, 2009. [5] T.E. Tay, Characterization and analysis of delamination fracture in composites: an overview of developments from 1990 to 2001, Appl. Mech. Rev. (2003). [6] N.J. Pagano, G.A. Schoeppner, Delamination of polymer matrix composites: problems and assessment, in: Comprehensive Composite Materials, 2000. [7] R. Krueger, Virtual crack closure technique: history, approach, and applications, Appl. Mech. Rev. (2004). [8] L.R. Deobald, G.E. Mabson, B. Dopker, D.M. Hoyt, J. Baylor, D. Graesser, Interlaminar fatigue elements for crack growth based on virtual crack closure technique, in: Collection of Technical Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2007. [9] S.R. Hallett, W.G. Jiang, B. Khan, M.R. Wisnom, Modelling the interaction between matrix cracks and delamination damage in scaled quasi-isotropic specimens, Compos. Sci. Technol. (2008). [10] J. Ratcliffe, N. Carvalho, Investigating Delamination Migration in Composite Tape Laminates, 2014. NASA/TMe2014-218289. [11] J.G. Ratcliffe, M.W. Czabaj, T.K. O’brien, Characterizing delamination migration in carbon/epoxy tape laminates, in: 27th Annual Technical Conference of the American Society for Composites 2012, Held Jointly with 15th Joint US-Japan Conference on Composite Materials and ASTM-D30 Meeting, 2012. [12] J.W. Lee, D.H. Allen, C.E. Harris, Internal state variable approach for predicting stiffness reductions in fibrous laminated composites with matrix cracks, J. Compos. Mater. (1989). [13] B.N. Nguyen, Three-dimensional modeling of damage in laminated composites containing a central hole, J. Compos. Mater. (1997). [14] R. Talreja, Continuum mechanics characterization of damage in composite materials, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. (1985). [15] T.E. Tay, G. Liu, V.B.C. Tan, X.S. Sun, D.C. Pham, Progressive failure analysis of composites, J. Compos. Mater. (2008). [16] P. Maimí, P.P. Camanho, J.A. Mayugo, C.G. Davila, A continuum damage model for composite laminates: part I - constitutive model, Mech. Mater. (2007). [17] P. Maimí, P.P. Camanho, J.A. Mayugo, C.G. Davila, A continuum damage model for composite laminates: part II - computational implementation and validation, Mech. Mater. (2007). [18] F.P. van der Meer, L.J. Sluys, A phantom node formulation with mixed mode cohesive law for splitting in laminates, Int. J. Fract. (2009). [19] F.P. van der Meer, Mesolevel modeling of failure in composite laminates: constitutive, kinematic and algorithmic aspects, Arch. Comput. Methods Eng. (2012).

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[20] N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, Int. J. Numer. Methods Eng. (1999). [21] E.V. Iarve, Mesh independent modelling of cracks by using higher order shape functions, Int. J. Numer. Methods Eng. (2003). [22] E.V. Iarve, M.R. Gurvich, D.H. Mollenhauer, C.A. Rose, C.G. Davila, Mesh-independent matrix cracking and delamination modeling in laminated composites, Int. J. Numer. Methods Eng. 88 (8) (2011) 749e773. [23] D. Mollenhauer, et al., Simulation of discrete damage in composite overheight compact tension specimens, Compos. A Appl. Sci. Manuf. (2012). [24] M.J. Swindeman, E.V. Iarve, R.A. Brockman, D.H. Mollenhauer, S.R. Hallett, Strength prediction in open hole composite laminates by using discrete damage modeling, AIAA J. (2013). [25] H.K. Adluru, K.H. Hoos, E.V. Iarve, Discrete damage modelling of delamination migration in clamped tapered laminated beam specimens, in: 32nd Technical Conference of the American Society for Composites 2017 2, 2017, pp. 917e930. [26] H.K. Adluru, K.H. Hoos, E.V. Iarve, J.G. Ratcliffe, Delamination initiation and migration modeling in clamped tapered laminated beam specimens under static loading, Compos. A Appl. Sci. Manuf. (2019). [27] H. Kevin H., M.J. Scott, A. Hari K., S. Joseph D., G. Haozhong, I. Endel V., Modeling complex sub-elements by using discrete damage modeling, in: Proceedings of the American Society for Composites - 34th Technical Conference, 2019. [28] N.V. De Carvalho, B.R. Seshadri, J.G. Ratcliffe, G.E. Mabson, L.R. Deobald, Simulating matrix crack and delamination interaction in a clamped tapered beam, in: 32nd Technical Conference of the American Society for Composites 2017, 2017. [29] P.W. Harper, S.R. Hallett, A fatigue degradation law for cohesive interface elements development and application to composite materials, Int. J. Fatig. (2010). [30] S.T. Pinho, C.G. Davila, P.P. Camanho, L. Iannucci, P. Robinson, Failure Models and Criteria for FRP under In-Plane or Three-Dimensional Stress States Including Shear Nonlinearity, 2005. Nasa/Tm-2005-213530. [31] A. Turon, P.P. Camanho, J. Costa, C.G. Davila, A damage model for the simulation of delamination in advanced composites under variable-mode loading, Mech. Mater. (2006). [32] E.V. Iarve, D.H. Mollenhauer, Mesh-independent matrix cracking and delamination modeling in advanced composite materials, in: Numerical Modelling of Failure in Advanced Composite Materials, 2015. [33] E.V. Iarve, K. Hoos, M. Braginsky, E. Zhou, D.H. Mollenhauer, Tensile and compressive strength prediction in laminated composites by using discrete damage modeling, in: 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2015. [34] E.V. Iarve, D. Mollenhauer, T.J. Whitney, R. Kim, Strength prediction in composites with stress concentrations: classical Weibull and critical failure volume methods with micromechanical considerations, J. Mater. Sci. (2006). [35] K. Hoos, E.V. Iarve, M. Braginsky, E. Zhou, D.H. Mollenhauer, Static strength prediction in laminated composites by using discrete damage modeling, J. Compos. Mater. (2017). [36] E.V. Iarve, R. Kim, D. Mollenhauer, Three-dimensional stress analysis and Weibull statistics based strength prediction in open hole composites, Compos. A Appl. Sci. Manuf. (2007). [37] M. May, S.R. Hallett, A combined model for initiation and propagation of damage under fatigue loading for cohesive interface elements, Compos. A Appl. Sci. Manuf. (2010).

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[38] L.F. Kawashita, S.R. Hallett, A crack tip tracking algorithm for cohesive interface element analysis of fatigue delamination propagation in composite materials, Int. J. Solid Struct. (2012). [39] A. Turon, J. Costa, P.P. Camanho, C.G. Davila, Simulation of delamination in composites under high-cycle fatigue, Compos. A Appl. Sci. Manuf. (2007). [40] L.R. Deobald, et al., Guidelines for VCCT-Based Interlaminar Fatigue and Progressive Failure Finite Element Analysis, 2017. [41] E.V. Iarve, K.H. Hoos, D.H. Mollenhauer, Damage initiation and propagation modeling in laminated composites under fatigue loading, in: Proceedings of the American Society for Composites - 31st Technical Conference, ASC 2016, 2016. [42] H.K. Adluru, E.V. Iarve, K.H. Hoos, Discrete damage modelling of clamped tapered beam specimen under fatigue loading, in: 33rd Technical Conference of the American Society for Composites, vol. 3, 2018, pp. 1816e1827. [43] E.V. Iarve, K. Hoos, M. Braginsky, E. Zhou, D.H. Mollenhauer, Progressive failure simulation in laminated composites under fatigue loading by using discrete damage modeling, J. Compos. Mater. 51 (15) (2017) 2143e2161. [44] J.G. Ratcliffe, W.M. Johnston, Influence of mixed mode I-mode II loading on fatigue delamination growth characteristics of a graphite epoxy tape laminate, in: Proceedings of the American Society for Composites - 29th Technical Conference, ASC 2014; 16th USJapan Conference on Composite Materials; ASTM-D30 Meeting, 2014. [45] R. Krueger, N. Carvalho, In search of a time efficient approach to crack and delamination growth predictions in composites, in: Proceedings of the American Society for Composites 31st Technical Conference, ASC 2016, 2016. [46] M.F. Pernice, N.V. De Carvalho, J.G. Ratcliffe, S.R. Hallett, Experimental study on delamination migration in composite laminates, Compos. A Appl. Sci. Manuf. (2015). [47] X.J. Fang, Q.D. Yang, B.N. Cox, Z.Q. Zhou, An augmented cohesive zone element for arbitrary crack coalescence and bifurcation in heterogeneous materials, Int. J. Numer. Methods Eng. (2011). [48] X.F. Hu, B.Y. Chen, M. Tirvaudey, V.B.C. Tan, T.E. Tay, Integrated XFEM-CE analysis of delamination migration in multi-directional composite laminates, Compos. pA Appl. Sci. Manuf. (2016). [49] B.Y. Chen, T.E. Tay, S.T. Pinho, V.B.C. Tan, Modelling delamination migration in angleply laminates, Compos. Sci. Technol. (2017). [50] Y. Gong, B. Zhang, S.R. Hallett, Delamination migration in multidirectional composite laminates under mode I quasi-static and fatigue loading, Compos. Struct. (2018). [51] T. Hoos, Kevin, M. Swindeman, Whitney, B-spline Analysis Method (BSAM)-FE, User Manual, University of Dayton Research Institute, Dayton, 2014. [52] C.V. Verhoose, J.J.C. Remmers, M.A. Gutiérrez, A dissipation-based arc-length method for robust simulation of brittle and ductile failure, Int. J. Numer. Methods Eng. (2009). [53] F.P. van der Meer, C. Oliver, L.J. Sluys, Computational analysis of progressive failure in a notched laminate including shear nonlinearity and fiber failure, Compos. Sci. Technol. (2010). [54] B.R. Seshadri, N.V. de Carvalho, J.G. Ratcliffe, G.E. Mabson, L. Deobald, Simulating the clamped tapered beam specimen under quasi-static and fatigue loading using floating node method, in: AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2018, 2018. [55] S.W. Case, K.L. Reifsnider, Damage Tolerance, Durability, and Life Prediction for HighTemperature Polymer Composite Structures, John Wiley, 2002.

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[56] B. Picasso, P. Priolo, Damage assessment and life prediction for graphite-peek quasi isotropic composites, in: American Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication) PVP, 1988. [57] R. Simonds, W. Stinchcomb, Response of notched AS4/PEEK laminates to tension/ compression loading, in: Advances in Thermoplastic Matrix Composite Materials, 2008. [58] K.L. Reifsnider, et al., Multiphysics design and development of heterogeneous functional materials for renewable energy devices: the heterofoam story, J. Electrochem. Soc. (2013). [59] S.W. Case, K.L. Reifsnider, Mechanical characterization and investigation of damage accumulation of EPM model CMC, in: Damage Tolerance and Durability of Material Systems, 1997. [60] P. Elenchezhian, R. Raihan, K.L. Reifsnider, The role of uncertainty in machine learning as an element of control for material systems and structures, in: Procedings of ASME 2018 Pressure Vessels and Piping Conference, 2018.

Response of composite engineering structures to combined fire and mechanical loading and fatigue durability

4

Scott W. Case 1 , John J. Lesko 2 1 Reynolds Metals Professor, Via Department of Civil and Environmental Engineering, Virginia Tech, Blacksburg, VA, United States; 2Associate Dean for Research and Graduate Studies and Professor of Mechanical Engineering, Virginia Tech, Blacksburg, VA, United States

Chapter outline 4.1 Introduction and background

165

4.1.1 Marine composites and fire-related design criteria 166 4.1.2 Wind turbine composites and fatigue-related design criteria 167

4.2 Mechanistic description of fire and fatigue performance of structural composites 168 4.2.1 Composites materials subjected to fire conditions 168 4.2.2 Structural composites materials subjected to combined load and fire conditions 170

4.3 Fatigue damage of structural composites 4.3.1 4.3.2 4.3.3 4.3.4

References

4.1

181

Fatigue damage: constant amplitude loading 181 Fatigue damage: effects of loading frequency 187 Fatigue damage: spectrum fatigue loading 188 Fatigue damage: temperature and moisture effects 194

199

Introduction and background

Polymeric composite material systems have found regular and mainstream use in a number of structural applications including transportation, marine, wind energy, aerospace, and infrastructure. Automotive and transportation makes up w20% of the global market for composites, with aerospace and defense, wind energy, electrical and electronics, pipes and tanks, and construction and infrastructures contributing to w10%e15% each. Marine applications make up w5% of the overall market [1].

Durability of Composite Systems. https://doi.org/10.1016/B978-0-12-818260-4.00004-1 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Deployment of polymer composite systems in these applications, with either glass or carbon fiber reinforcement, is driven by several characteristics including • • •

High material strength-to-weight ratio allowing for reduced system weight, increasing efficiency and allowing for increases in payload capacity Durability and damage tolerance (e.g., damage, fatigue, corrosion resistance) allowing for extended operation, and reductions in maintenance and life cycle cost Design flexibility and part consolidation that allows for achieving performance that is difficult to be achieved by traditional materials, and the reduction in part and assembly costs

While the advantages of composites have increasingly driven their adoption in structures traditionally constructed from metals, concrete, or timber, limitations still remain for their use. Moreover, shortcomings exist in our ability to predict their performance with a level of confidence suitable for regular deployment. Certification and standardization bodies present limitations to advancing the use of composites in various applications due to uncertainty in their performance or cost. Thus, the need remains to improve our fundamental understanding of composite material performance. In this chapter, we focus on the fundamentals of structural fire response for composite materials in ship applications and mechanics of composite fatigue performance in marine and wind energy. We review the state of our understanding of our ability to predict composite material system performance and comment on future directions for research. First, we provide context to the design criteria as they are related to composites and fire in marine applications and fatigue performance of composite wind turbine blades.

4.1.1

Marine composites and fire-related design criteria

Composite materials usage in the boat-building industry reached w6% (c.2017); for ships less than 50 m in length, composite materials have w70% share [2]. As might be obvious, reducing the ship’s structural load through the use of composites [3] allows for increased payload, thus increasing cargo transport and profits or weapons carrying capacity with associated increases in lethality. Because of the increased corrosion resistance of composites in comparison to metals, maintenance and life cycle costs are reduced and longer service lives can be achieved [4]. Incorporating composite superstructures (e.g., deskhouses and masts) can lower the center of gravity, thus increasing the seaway stability of the ship. Composites also enable the construction of ships that have reduced magnetic and acoustic signatures and that have been designed for operation at various speeds to avoid frequencies that are uncomfortable to passengers [5]. While the advantages of composites in marine vessel applications exist, restrictions on material combustibility limit the use of composites for ships >50 m [6]. In particular, the International Marine Organization (IMO) Safety of Life at Sea regulations embody these restrictions. Chapter II-2 of the IMO regulations on fire protection requires structural materials to be noncombustible, effectively excluding composites for ships operating in international waters and carrying more than 12 people or cargo ships exceeding a gross tonnage of 300 tons. However, a recent regulation (Chapter II2/17) that came into effect in 2002 allows alternative systems that meet a risk-based

Response of composite engineering structures to combined fire

167

design approach and demonstrate equivalent safety performance. While the new regulation has been successfully used to go beyond the prescriptive code, polymer composites’ structural applications still have not been approved due to their combustibility. Likewise, EU passenger ships and ferries make no allowances for structural polymer composites (EU Directive 2009/45/EC) [7], requiring that all materials to be of “steel or another equivalent noncombustible material.” In contrast, the IMO International Code of Safety for High-Speed Craft [8] resolution MSC. 36 (63) does allow for the use of polymer composite materials in craft capable of certain speeds based on stringent criteria that must be confirmed through fire testing. Composites material systems are allowed in hulls, bulkheads, walls, and decks (with restriction on areas of high fire risk) for EU inland waterway vessels and are covered by the 2006/87/EC directive [9]. As it is evident by the codes, time and distance are primary factor in the acceptance of fire risk for composite structures ships. Limitations on the ability to evacuate the ship (speed or distance) drive the nature of the standards. Thus, a better understanding of fire-related mechanisms that drive structural polymer composite failure in fire conditions is needed to provide greater confidence.

4.1.2

Wind turbine composites and fatigue-related design criteria

Composite wind turbine blades make it possible to achieve larger wind turbines taking advantage of the material strength to weight ratio due to the ability to tailor the local stiffness of the blade and durability and damage tolerance. However, worldwide estimate (c.2015) of “wind turbine rotor blades failure rate is around 3800 a year, 0.54% of w700,000 or so blades that are in operation worldwide.” Blade failures “are the primary cause of insurance claims in the US onshore market” and “account for over 40% of claims, ahead of gearboxes (35%) and generators (10%).” [10] Blade failures are costly given the loss revenue due to repair downtime as well as the cost of the repair time and materials [11]. These blade faults are primarily due to structural causes, such as cracks, delamination, debonding, and overall fatigue failures. If too much damage accumulates within the blade, excessive tip deflectionsda structural failure of the bladedincrease drag near the end of the blade [12]. Certification of wind turbines and their blades is essential to distinguish manufacturers’ products. These certifications are important for securing public clean energy funds and provide the organizations who procure wind turbines a means to ensure that their investments are prudent with limited risk. The certifications by independent third parties are subject to various country and regional codes, whether onshore or offshore. For example, in the United States, turbines are certified to the American Wind Energy Association Small Wind Turbine Performance and Safety Standard (AWEA 9.1-2009), which is an American National Standards Institute accredited standards development organization. AWEA 9.1-2009 incorporates three international standards including IEC 61400-2, 61400-11, and 61400-12. Other international standards (e.g., DNV-DS-J102 [13] and CSA PLUS 61400) rely on the IEC 61400

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standard. Composite blade designs are validated by full-scale testing under extreme and fatigue loads to ensure that the blade will withstand the target design loads for 20 years lifetime. The DNV-DS-J102 standard outlines the design, manufacturing, and testing of onshore and offshore wind turbine blades and provides insight into blade design processes in compliance with IEC requirements [14]. The IEC 61400-23 [15] outlines the full-scale testing of rotor blades, and IEC 61400-13 “Wind turbines e Part 13: Measurement of mechanical loads” describes the measurement of structural loads on wind turbines. These certifications depend on various fatigue design methods and criteria. These design methods rely on the well-known Palmgren-Miner theory, which has limitations in describing the damage accumulation associated with spectrum loading. As blades become larger, improved fatigue analysis is needed. While fatigue design in marine composites is still limited, the importance of robust fatigue design, fatigue resistance of wind turbine blades, is more critical to ensuring profitable wind turbine projects.

4.2

Mechanistic description of fire and fatigue performance of structural composites

The ability to achieve the potential performance benefits noted above is dependent upon the capability to move associated regulatory bodies to adopt composite material systems over traditional materials. Fire and fatigue performance figure prominently into these factors, thus advancing our understanding of the mechanisms, their mechanics, and the ability to deploy them in predictive codes to inform both prescriptive and performance-based design standards. Thus, this chapter addresses the current state of the art for fire and fatigue performance that inform the design and acceptance of critical structural composite material systems.

4.2.1

Composites materials subjected to fire conditions

When polymer composite material structures are subjected to fire, or a high heat flux, the composite goes through a complex set of reversible and irreversible changes that are dependent upon each other. Mouritz et al. [16] outline these processes and classify them into four categories: thermal, chemical, physical, and the progression of damage leading to final failure. In the case of one-sided composite exposure to fire (Fig. 4.1), the composite experiences a through the thickness evolution of material properties and state changes (e.g., volatile gas flow and porous material development) with a front of stiffness changes and damage. Failure modes exhibited are highly dependent on the coupled stress state (i.e., tension, compression, shear), heat flux intensity, duration, and the resulting through the thickness property changes leading to the loss of load carrying capacity. Under fire or high heat flux conditions, the matrix presents a major concern to the composite structure’s integrity and survivability. Polymers are combustible when

Response of composite engineering structures to combined fire

169

Heat flux

Fiber + char Porous decomposition Matrix cracking + delamination Undamaged

Figure 4.1 Generalized depiction of damage, state, and physical property changes in polymer composite materials subjected to fire. Adapted from A.P. Mouritz, S. Feih, E. Kandare, Z. Mathys, A.G. Gibson, P.E. Des Jardin, S.W. Case, B.Y. Lattimer, Review of fire structural modelling of polymer composites, Compos. Appl. Sci. Manuf. 40 (12) (2009) 1800e1814.

exposed to fire or high heat at temperatures well above the glass transition temperature, Tg. The process of combustion, the reactions that take place, the by-products of those reactions (e.g., char and released gases) produce fundamental changes in the polymer composites in physical and mechanical properties [17]. During the initial stages of temperature increase within the structure (200
C (well below the annealing temperatures of 657 and 736
C for boron-free E-glass and E-glass, respectively [21]), glass fibers exhibit substantial tensile strength degradation for both single fibers and fiber bundles. They found that boron-free E-glass exhibits higher softening temperature (920
C for boron-free Eglass and 830e860
C for E-glass) and increased resistance to corrosion and stress rupture.

170

4.2.2

Durability of Composite Systems

Structural composites materials subjected to combined load and fire conditions

While polymer composites under fire conditions exhibit a host of damage mechanisms and loss in strength, a reliable understanding on how long these composite structures tolerate sustained load and fire is important to broader adoption. It is impractical to test every structure to be put into service under a multitude of fire scenarios, and thus we require confidence in our predictive capabilities for the response of composite structures under combined load and fire conditions. Below we examine investigations of residual composite strength and time to failure for fire and sustained loads. We restrict our review to the compression mechanics of composites given their greater susceptibility to sudden collapse under load and fire conditions. Significant progress has been made in the last 15 years to understand compression failure mechanics under combine fire and load. This work spans several thermal conditions and physical scales from coupons to subcomponents, and ultimately to full-scale elements. Our discussion will be limited in this chapter to small and intermediate composite laminate tests under load and controlled heat flux levels. Both experimental and modeling have focused on sustained axially compressionloaded (in load control) laminates where the vertical edges are unconstrained. A representative set of studies are presented in Table 4.1 for glass and carbon composites with thermosetting resins. These compression tests were performed with effective pinnedpinned boundary conditions, where the effective length, Le, is equal to the unsupported length, Lu. The exception to this is the studies undertaken by Bausano [22] and Boyd [23] where the unsupported length is short and assumed to present fixed-fixed boundary conditions with the effective length Le ¼ 0.5Lu. The slenderness ratios have been calculated for comparison. The heat flux exposures range from 10 to 75 kW/m2 with nearly all the maximum surface temperatures exceeding the glass transition temperatures of the resin systems. If we consider the A-60 fire rating, the door or bulkhead must not allow the passage of flame and smoke for 60 min when subjected to the standard fire test [28]. The standard cellulosic fire reaches a radiation level of 50 kW/m2 after 5 min, with surface temperatures reaching 500
C and rising to 945
C over time. In cases where a 50 kW/m2 heat flux is used, none of the composites come close to meeting the A60 standard. Thus, the use of a composite laminate alone is impractical for in-service structures that must bear load under fire conditions and survive for 60 min. This suggests the use of sandwich constructions. For naval applications, balsa wood is used as the core in part serving as an insulator to the composite face sheet on the back side of the sandwich. The failure modes from the tests summarized in Table 4.1 include forms of global laminate buckling, microbuckling, and kinking, and forced-response deflection. While these terminologies are not always used consistently in the literature, we review each and provide a more descriptive discussion of the failure modes where available. In the study by Grigoriou and Mouritz [24], laminates were constructed from a layup of carbon fiber prepreg of a relatively high Tg resin in various quasiisotropic ply sequences. Here the fibers are nominally without the undulations present in woven or stitched engineering fabrics. In all cases, the failures were due to global buckling as

Table 4.1 Summary of coupon level tests conducted under sustained load and incident heat fluxes.

Authors

Fiber and volume fraction Vf

Matrix

Tg (8C) Laminate

Specimen Slenderness dimension ratio Le/r (mm)

Test Q (kW/ m2)

wTMax surface (8C)

XB (Xc) (MPa) at RT

Applied load as % of RT XB

Range of w tfailure at (XB %) (seconds)

Grigoriou [24]

Hexcel AS4 Tows Vf ¼ 65%

3501-6 (prepreg)

210

Quasiisotopic variations

h ¼ 6.6 w ¼ 50 Lu ¼ 430

226

50

700

24 ()a 13 ()b 37 ()c 20 ()d

80% to 10 %

110 to 260 120 to 270 60 to 220 75 to 210

Burns [25]

24 K PAN Woven Vf ¼ 55%

Kinetix R118 (infusion)

65

0/90

h ¼ 9.5 w ¼ 50 Lu ¼ 560

204

10 25 50

270 480 650

65 ()

60% to 5%

25 to 250

Summers [26]

E-glass fabric 830 g/m2 Woven Vf ¼ NR

Derakane 411-350 (infusion)

120

0/90

h ¼ 6, 9, 12 w ¼ 203 Lu ¼ 737

213, h ¼ 12 mm 284, h ¼ 9 mm 426, h ¼ 6 mm

8e 11.8 19.3 38

240f 300 360 520

7.5g 25.2 59.8 (w300)

53% to 27% e 53% to 15% 53% to 11%

350 to 2000f e 150 to 1300 50 to 510

Feih [27]

E-glass fabric 800 g/m2 Woven Vf ¼ 55%

Derakane 411350 (infusion)

120

0/90

h¼9 w ¼ 50 Lu ¼ 560

216

10 25 50 75

280 420 600 700

21.5 (435)

90% to 10%

290 to run out 150 to 360 75 to 700 50 to 460 Continued

Table 4.1 Summary of coupon level tests conducted under sustained load and incident heat fluxes. (cont’d)

Authors

a

Fiber and volume fraction Vf

Matrix

Tg (8C) Laminate

Specimen Slenderness dimension ratio Le/r (mm)

c

wTMax surface (8C)

XB (Xc) (MPa) at RT

Applied load as % of RT XB

Range of w tfailure at (XB %) (seconds)

Bausano [22]

E-glass 630 g/m2 0/90 stitched tows Vf ¼ 55%

Derakane momentum 411-350 (pultrusion)

120

Quasiisotropich

h¼4 w ¼ 25 Lu ¼ 50

43.3i

5 10 15 20 30

95 145 180 215 e

(161)j

26% to 6%

400 to run out 150 to run out 70 to run out 60 to run out 40 to 2000

Boyd [23]

E-glass fabric 810 g/m2 Woven Vf ¼ 55%

Derakane 510Ae40 (infusion)

105

[ 45 ]2S and quasi isotropick

h ¼ 6.1 w ¼ 25 Lu ¼ 50

28.4i

5 10 15 20

130 150 170 190

(300)j

37% to 18% 40% to 5% 29% to 10% 30% to 3%

360 to 2960 120 to 600 110 to run out 90 to 600

[0/þ45/90/45/0/þ45/90/45/0/þ45/90/45]s. [þ45/90/45/0/þ45/90/45/0/þ45/90/45/0]s. [0/0/0/þ45/90/45/þ45/90/45/þ45/90/45]s. d [þ45/90/5/þ45/90/45/þ45/90/45/0/0/0]s. e Heat flux measured at the test surface and not the incident of the source. f Temperatures measured on thickness, h ¼ 12 mm. g Buckling stresses, from top to bottom, for the thickness h ¼ 6, 9, and 12 mm at LB ¼ 203 mm. h [0/90 / 45/CSM]s. i FixeddFixed boundary condition assumed. j Compression strength and not buckling stress due to short gauge length. k [0/þ45/90/45/0]S. b

Test Q (kW/ m2)

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a result of the high slenderness ratio. Stacking sequence had no effect on the structural failure mode after heating times of w100e150 s and within the period of time needed to reach the glass transition temperature. Burns et al. [25] studied a woven plain weave carbon fiber cross-ply fabric infused with a very low Tg resin. Microbuckling of the near-surface plies and delamination cracking between the plies were observed at failure. Note that the times to failure for these carboneepoxy composites are short in comparison to the other composite laminates in Table 4.1, which have E-glass as the reinforcement, a result that is likely due to the high thermal conductivity of the carbon fibers. Examining the E-glass vinyl ester composites failure modes summarized in Table 4.1, Summers noted three general posttest failure modes (shown in Fig. 4.2(a) e(c)) for these large slenderness ratio test coupons, or “intermediate-scale tests”: (a) kinking, (b) localized kinking, and (c) forced-response deflection. Kinking was observed when the applied loads were 15% of the Euler buckling load and transitioned to localized kinking at higher sustained loads. This failure mode included some out-of-plane deflection with kinking through the thickness. Forced-response deflection was observed at loads 50% of the Euler buckling load with significant out-of-plane deflection and little to no through the thickness kinking of the laminate. With a very similar resin infused E-glass vinyl ester composite and comparable slenderness ratios to the Summers h ¼ 12 mm thick samples, the failures observed by Feih (see Fig. 4.2(e)) appear typical of the kinking observed by Summers (Fig. 4.2 (a)). The failures were noted to occur suddenly and with “plastic tow kinking” with the presence of delamination that may be a postfailure process result of the kink band formation. A kink angle, f, of w60 degrees was observed with the width of the kink band w4 mm for the h ¼ 9 mm thick. It was noted that the room temperature kink bands are typically under 0.2 mm in width. Similarly, Bausano et al. observed a single kink band for these pultruded quasiisotropic E-glass/vinyl ester composites with some coupons exhibiting a double kink, as shown in Fig. 4.2(e) and (f). Due to the presence of the 0
ply on the outside face of the [0/90/ 45/CSM]S, surface kinks on the exposed face were observed (see Fig. 4.2(g)). This local surface damage likely preceded the final collapse of the laminate leading to the kink band formation. Boyd et al. studied a [0/þ45/90/45/0]S woven E-glass laminates with a Derakane 510A-40 vinyl ester resin (Derakane 510A-40 is brominated bisphenol-Aebased vinyl ester that is similar in mechanical performance to the Derakane 411-350). The failure modedkink formationdis similar to all the other E-glass/vinyl ester composites discussed by Bausano. Furthermore, kinking phenomena were observed for both cross-ply and quasiisotropic E-glass vinyl ester laminates independent of slenderness ratios and fabrication methods (infusion vs. pultrusion). The commonality in these failure modes presents a common basis for exploring approaches to modeling and is discussed next. Modeling Compression Failure of Composites Under Combined Mechanical Load and Fire Conditions.

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Durability of Composite Systems

(a)

(b)

(d)

(e)

(c)

(f)

(g)

90°

Lamp side

ply

0°

ϕ

ply

Matrix crack = 60°

Figure 4.2 Failure modes observed for “intermediate scale” tests of Summers [26]: (a) kinking, (b) localized kinking, and (c) forced-response deflection; (d) observed “plastic tow kinking” of Feih [27] with a kink angle, f, of w60 degrees and kink bandwidth w 4 mm for the h ¼ 9 mm thick laminate; typical compression failure modes of Bausano [22] and Boyd [23] showing (e) single and (f) double kink, and (g) cross section of the front/exposed face (right side) shows a 0 degree ply buckle prior with cracking of the matrix prior to final failure of the quasiisotropic laminate.

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The studies of Table 4.1 show that • • • •

With increasing applied stress and heat flux, the time to failure is reduced There are combinations of heat flux and applied load where the coupon does not fail in test times approaching 60 min Carbon fiber composites, whether of woven or stitched fabrics, have relatively lower times to failure at nearly equal specimen slenderness ratios compared to E-glass fiber composites Most failure modes appear to be initiated by the kinking of the fiber or tows and aided by initial imperfections due to the crossing of fibers within the weave or stitched fabrics

Exploring other potential relationships between the various parameters investigated in this type of testing, Summers [29] noted a relationship between the applied stress and the average through the thickness temperature. A reasonable trend was noted between the average through the thickness temperature and the normalized applied stress to the Euler buckling stress for the spans of thicknesses investigated. Here, the through the average thickness temperature was determined by using a numerical integration scheme. Summers [29] further calculated values for the nondimensionalized stress, s; and the slenderness ratio, l; s¼ l¼

sapp : sc ðTÞ l l ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffi lo EðTÞ : p scr ðTÞ

(4.1)

(4.2)

and sought to develop buckling design curves employing the average through thickness temperature and slenderness ratio. Incorporating data from three other studies (Boyd et al. [30], Feih et al. [27], and larger-scale test by Hugi [31]), Summers shows most of the data collapse to a standard critical compressive stress versus slenderness ratio, l, as shown in Fig. 4.3. The analysis relies on the isothermal properties of axial modulus and buckling stress as a function of temperature. It is noted that the relationship holds with the exception of the tests by Boyd et al. [30] (noted in the legend by VT) where the slenderness ratio (see Table 4.1) is low and the failure mode is not representative of Euler buckling. Assessing an average through the thickness temperature and using isothermal characterized strength with temperature provides heuristic guidelines for design. However, a better understanding of the through the thickness temperature distribution is required to describe ply-wise residual properties including time- and temperature-dependent physical and mechanical material properties. This understanding must include the description of the associated irreversible damage (e.g., porous resin, matrix cracking, and delamination) due to decomposition and is addressed next in this chapter. Time-Evolved Thermal Modeling of Composite Laminates at High Temperatures. Henderson et al. [33] developed a one-dimensional model that is often used directly, or slightly modified, to describe the through thickness temperature

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Durability of Composite Systems

1

12 mm 9 mm 6 mm RMIT VT EMPA

σapp/σc

0.8

0.6

0.4

0.2

0 0

5

10

15

20

λ/λ0

Figure 4.3 Nondimensionalized critical buckling stress versus slenderness ratio based on the average thickness temperature for one-sided fire and sustained load tests taken from Summers [32]. Data by S. Feih, Z. Mathys, A.G. Gibson, A.P. Mouritz, Modelling the compression strength of polymer laminates in fire, Compos. Appl. Sci. Manuf. 38 (11) (2007) 2354e2365 and E. Hugi, Personal Communication, in: P. Summers (Ed.) (2010) are noted as RMIT and EMPA, respectively, in the legend.

distribution in composites subjected to fire. The governing equation for energy flow is given by rCp

 vT v2 T vk vT vT vr  ¼k 2 þ  m_ g Cpg  Q p þ hc  hg vt vx vx vx vx vt

(4.3)

The four terms on the right hand side of the equation are, respectively, (1) the effective heat conduction through the laminate thickness, (2) an additional heat conduction term that accounts for the change in through the thickness conductivity as a function of temperature, (3) the internal convection of thermal energy due to the flow of hot decomposition gases toward the heated surface, and (4) the temperature change due to heat generation or consumption from polymer resin decomposition. Variations on Henderson’s equation have been made by Gibson et al. [34] to include the polymer decomposition reaction rate described by a single reaction with Arrhenius temperature dependence, where reaction rate constants are as determined from thermogravimetric analysis. Based on studies by Sullivan [35,36], Loohey [37] developed a modified differential equation for energy flow including a decomposing, expanding, porous composite. The expansion of the composite is driven by the thermochemical changes that produce

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decomposition gases, and the storage of these gases was first considered by Henderson and Wiecek [38]. 

   vT rg gCpg vP vT v vT ð1  fÞrCp þ frg Cpg ¼ k  vt vx vx vx vx mg  vrr   Q p þ hc  hg vt

(4.4)

The rate of change in internal energy (left-hand side of the equation) includes the porosity and the contributions of the gas to the internal energy, with f describing the porosity volume. The first term of the right-hand side is similar to the first two terms of Eq. (4.3) with the k ¼ ð1 fÞk þ fkg ; where k and kg are, respectively, the thermal conductivity of the polymer and the decomposition gas in the pores. The second term of Eq. (4.4) is the energy transfer due to the gas mass diffusion evaluated in terms of the pressure gradient and assumed to conform to Darcy’s law. The governing equation (Eq. 4.4) for the energy flow is complemented with two additional governing expressions for decomposition gases through the decomposing, expanding, porous composite: gas mass diffusion equation rg vT v rg g vP ¼ vx mg vx M vt

!  rg a

 vT  v vux vrr þ þ rg fbg þ ab vt vx vt vt

(4.5)

and the through-thickness deformation equation E

v2 ux vT vP þa þ Fx ¼ 0:  bE 2 vx vx vx

(4.6)

The three equations are combined into a coupled matrix and solved for each time interval using the BubnoveGalerkin finite element formulation. The predicted temperature versus time curves at the front and back as well as several internal locations show very reasonably agreement with experiments for a glass woven roving/polyester resin composite having a fiber volume fraction of 46.5% [35]. Thermomechanical Modeling of the Time to Failure Under Combined Load and One-Sided Fire Exposure. With the means to describe the through-thickness temperature distribution using the energy balance, we now turn our attention to approaches for describing the thermomechanical response of a composite under combine load and fire conditions. In reviewing the modeling approaches, we summarize the thermomechanical mechanisms and failure criteria for several representative studies in Table 4.2. The summary of models includes analysis attributes for the description of through-thickness temperature distribution, how the matrix property evolves, and the stress analysis (both analytical and numerical) and failure criteria.

Through the Thickness temperature profile

Laminate material properties

Analysis attributes

Bausano et al. [22]

Gibson et al. [34]

Boyd et al. [23]

Feih et al. [27]

Zhang et al. [39]

Summers et al. [26]

Heat conduction

U

U

U

U

U

U

Decomposition

U

U

U

U

Flow of gases

U

U

U

U

Mass loss

U

U

U

U

Thermal expansion

U

U

Pressure rise

U

Linear elastic (T)

U

U

U

U

U

Linear viscoelastic (T, t)

U

U

Nonlinear viscoelastic (s, T, t)

U

U

Viscoplastic (s, T, t)

U U

U U

Decomposed material/char

U U

U

U

Euler buckling Kinking

U

Average stress

U

U

U

U

Ply stresses

U

U

U

U

Ply discount

U

U

U

U U

U

Durability of Composite Systems

Nonlinear elastic (s, T)

Yielding

Stress analysis and timeto-failure criteria

U

178

Table 4.2 Summary of various compression modeling schemes for combined compression loading and one-sided fire exposure of glass/thermoset composite laminates.

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These studies employ a multitude of approaches to deal with pertinent timedependent evolution of thermal and mechanical properties, and the means to describe time-dependent failure. All studies except that of Zhang assume constant heat fluxes representative of those presented in Table 4.1. Through-thickness Temperature Profiles: Most of the modeling approaches to predict the through-thickness temperature distribution are based on Henderson’s model [33] or variations to the model. In contrast, the models by Bausano and Boyd do not include the effects of thermal decomposition. Both studies employed a finite difference solution of the energy equation, including temperature-dependent specific heat and thermal conductivity but assuming constant density. Through-thickness thermal expansion is considered in Zhang and Summers, while Zhang also considers the pressure rise in the material due to the evolution of decomposition gases. The porosity and permeability are calculated in Zhang’s approach using the rule of mixtures as a function of decomposition factor. Utilizing an Abaqus analysis, a user-defined heat transfer subroutine, UMATHT, is used to implement the decomposition equation, the heat transfer equation, and implement the gas diffusion equation. The analysis shows little influence of pressure on temperature, and the temperature profiles are similar to the temperature prediction where no accumulation of decomposition gases is assumed. An acknowledged shortcoming of each of the models is that the insulating characteristics of delaminated regions are not considered. Of these, Zhang’s is the most comprehensive in that it deployed Loohey’s model including thermal expansion and pressure rise as well as the viscoelastic behavior of the composite. Laminate Materials Properties: Turning our attention to the evolution of the laminate material properties, several approaches are deployed including linear and nonlinear effects, and time dependence (viscoelasticity). Temperature-dependent yielding of the polymer is considered by Bausano and Boyd, where the yielding characteristics of the laminates are measured under isothermal condition and used in the determination of failure. Gibson describes mechanical property evolution through heuristic means and the empirical function PðTÞ ¼

     n PU þ PR PU  PR  tanh k T  T ' R : 2 2

(4.7)

based in part on the work of Mahieux and Reifsnider [40]. The property P(T) is related to the measured PU and PR, which are the unrelaxed “low temperature” and relaxed “high temperature” values of the properties to be described (strength or stiffness). The overall expression is modified by a power law factor, Rn, describing the residual resin content. The constant k describes the breadth of the loss modulus, and T0 is the mechanically determined Tg. The works of Bausano, Boyd, and Zhang consider time- and stress-dependent laminate material properties in the stress analysis and subsequent determination of failure. Boyd characterized the nonlinear elastic and viscoelastic, including viscoplastic, properties to describe the shear creep compliance master-curve using a Prony series of the generalized KelvineVoigt [41].

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Durability of Composite Systems

Stress Analysis and Time-to-failure Criteria: When considering failure modes, several approaches developed considered buckling, and kink band formation. At small slenderness ratios, Bassano, Boyd, and Zhang determined failure by comparing the strength of the 0
plies to the stress level computed from temperature-dependent classical laminate theory or anisotropic finite element modeling. When both the front and back face 0
plies fail, total failure of the laminated is noted at the given time and thermal gradient. The compression strength of the 0
ply was computed using a modified Budiansky and Fleck [42] model to describe kink band formation following initial microbuckling.  1  n1 #1 3 n f=gY n sc ðt; T; sÞ ¼ G12 ðt; T; sÞ 1 þ n 7 n1 "

(4.8)

DMA data were used in Bausano to describe the temperature-dependent G12(T), while Boyd implemented the Prony series for the time, temperature, and shear stress dependence for G12(t,T, s). Gibson and Feih determine ply stresses from lamination theory, where the temperature-dependent compliances were determined by Eq. (4.7). Ply level stresses were compared to the experimentally determined isothermal strength reduction also described by Eq. (4.7), and a ply discount scheme was used when the predicted ply stress reached the strength. The time-to-failure criterion is determined when residual compressive strength of the laminate is less than the applied compressive stress. Euler buckling was not observed in the experiments. Kink band formation was observed (see Fig. 4.2(d)) and thus the model appears to capture this failure mode with fidelity. Summers observed a range of failure modes (see Fig. 4.2(aec)) for a range of slenderness ratios >200, and the model developed addressed the bending of axially loaded columns with a consideration of the thermal moment and the shift in the neutral axis due to time-dependent thermal changes in stiffness. The thermomechanical stress analysis predicts with good fidelity the movement of the laminate toward or away from the heat flux source depending upon the load and heat flux conditions. The thermostructural model uses a localized failure criterion to determine laminate failure. Failure is predicted when the maximum combined compressive stress at a point is equal to the compressive strength at the same location. The failure criteria assume a very conservative condition; when one of the plies fails, the entire laminate fails. The model does not include progressive failure or ply discount methods. Elastic and compression strength properties as a function of temperature are described by Eq. (4.7). Future Research Directions on Combined Fire and Load of Composites: This review of modeling of E-glass fiber thermosetting matrix composite laminates subjected to sustaining one-sided heat flux exposure and compressive load demonstrates a reasonable ability to predict the time to failure. However, describing the development and presence of matrix cracking and delamination was not addressed. The thermal models including decomposition and the ability to evolve lamina properties demonstrate strong fidelity for describing the through the thickness temperature profile with the ability to reasonably predict the development of porous material evolution as a

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result of decomposition. Much of the modeling is dependent on the measurement of constitutive properties at temperature with some heuristic descriptions of strength and irreversible properties. Ideally, we would like to minimize the number of needed characterizations for the purposes of predicting these fundamental mechanisms. However, once we understand the evolution of the material properties under fire conditions, the use of finite element user-defined property subroutine can be leveraged to describe, at a first order, larger and more complex critical load-bearing structures under various fire conditions.

4.3

Fatigue damage of structural composites

The degradation and failure of many mechanical or structural systems is associated with dynamic cyclic (or fatigue) loading. Fatigue life is usually measured as the number of cycles to failure for a given applied load level. In a general sense, applied loads include mechanical and environmental loads. Available literature has suggested that over 50% of all mechanical or structural failures are related to fatigue, and most of them are unexpected fractures [43]. Therefore, design decisions involving fatigue are required for many mechanical or structural systems (e.g., wind turbines and composite naval structures). Fatigue has been recognized as a material and engineering problem [44]. Early studies on fatigue were almost always related to metallic fatigue. However, the topic of fatigue of composites has received extensive investigation over the past several decades [45e51]. A comparison of the fatigue of metallic materials and FRP composites is summarized in Table 4.1. Although the analytical and experimental tools for composites are based on the knowledge, we have accumulated from studying the fatigue of metals and alloys; the fatigue behavior and factors influencing the material’s fatigue behavior are quite different from metallic materials to composites. One significant difference is that for composites, there is little available direct experimental evidence to support the assumption of a fatigue (or endurance) limit under which failure does not occur for a large number of loading cycles [52]. Therefore, it is questionable to use the infinite life fatigue criterion for composites (Table 4.3).

4.3.1

Fatigue damage: constant amplitude loading

For multidirectional continuous fiber composites subjected to constant amplitude tensile fatigue loading conditions, the general sequence of damage development, summarized in Fig. 4.5, is well accepted [47]. The first damage mode to develop is matrix crackingdcracks that develop in directions perpendicular to the fibers (in the transverse direction) in plies that experience tensile stresses. These cracks are driven by stress concentrations resulting from the stiffness mismatch between the fiber and matrix and may result in significant laminate stiffness changes shown in Region I of Fig. 4.4. Additionally, at these matrix cracks, localized fiber breakage may occur and is associated with the high local stresses at the crack tips. Along with these fiber fractures, debonds often occur near the broken fiber ends. Moderate amounts of matrix

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Durability of Composite Systems

Table 4.3 Comparisons of fatigue behavior of metals and fiber-reinforced composites. Items

Materials (steel, Aluminum)

Composites

Basic material behavior

Isotropic, ductile; corrosion fatigue, fretting fatigue, and thermal fatigue significant

Anisotropic, brittle; due to anisotropy, a small strain in the main fiber direction may cause strains normal to the fibers or the fibereresin interface that are high enough to cause deterioration and fatigue damage

Fatigue process

Fatigue damage accumulation is localized; fatigue is often preceded by the initiation of a dominant crack and the propagation of that crack to a critical size

Fatigue damage accumulation is in a general rather than a localized fashion; failure is not controlled by a dominant crack; interaction of damage modes determines failure

High-cycle (lowstress) fatigue

Fatigue at low stress will harden the material slightly

No hardening effect

Fatigue limit

A fatigue limit often exists, and, generally, a stronger material will have a higher fatigue resistance and the fatigue ratio (fatigue limit/tensile strength) is roughly constant

Little direct experimental evidence is available to support the assumption that a fatigue limit exists

Fatigue design criteria

Infinite life design, safe-life design, fail-life design, damage-tolerant design, probabilistic design

Damage-tolerant design, probabilistic design

Factors that influence fatigue

Processing, residual stress, surface finish, loading conditions, environmental conditions

Fiber types and stacking sequence, resin types, fiber sizing and fiber matrix interphase, manufacturing technique, loading conditions, environmental conditions

cracking do no significantly affect the strength of a laminate. This surprising observation is the result of the simple fact that the initial strength of the material is measured by straining it to the ultimate load; during that experiment, matrix cracks develop so that the presence of microcracks is included in the measurement of the ultimate strength [45]. Indeed, the saturation density of the microcracks in static tests and in fatigue tests is often quite similar. This damage state is referred to as the characteristic damage state (CDS) of the laminate. The third major type of damage that occurs during fatigue is that of secondary matrix cracks, as shown in Fig. 4.6. While these cracks may sometimes extend for

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1.10

Normalized stiffness

1.00

0.90

0.80

0.70

I

II

III

0.60

0.50

0

100000 200000 300000 400000 500000 600000 700000 800000 Cycles

Figure 4.4 Experimentally determined stiffness loss, normalized to the first-cycle stiffness, for a [þ45/0/45/90]2s graphite fiberereinforced composite laminate subjected to tensile fatigue. The three regions of material behavior are evident [45].

3-delamination fiber breaking

1-matrix cracking fiber breaking

Damage

0°

CDS

0°

0° 0°

0°

2-crack coupling, interfacial debonding, fiber breaking 0

Percent of life

5-fracture

0°

0° 0°

0°

0°

4-delamination growth, fiber breaking (localized) 100

Figure 4.5 Stages of damage development in the lifetime of a composite material [47].

great distances, most often they extend only a short distance away from the primary matrix crack. Delaminations often initiate at the crossover between primary and secondary matrix cracks as also shown in Fig. 4.6. Once these delaminations initiate, their subsequent growth is driven by the fact that local regions of the composite (such as different plies) have different Poisson’s ratio and coefficients of mutual influence and therefore would deform differently in

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Durability of Composite Systems

Applied load

Figure 4.6 X-ray radiograph illustrating secondary matrix cracks (parallel to applied loading direction) in addition to primary matrix cracks. A small delamination at a crossover point is illustrated by the arrow [45].

response to the local loads if they were not bonded together. The energy associated with this mutual constraint is released if the regions separate; that energy drives the subsequent delamination growth. Radiographic images of typical delaminations are shown in Fig. 4.7. These delaminations are typical of results from plates in that they start and spread from the edges of the laminate. In laminates with internal free surfaces, such as edges of holes, delaminations may also initiate and grow in a similar fashion. The stiffness change associated with delamination growth, shown in Region II of Fig. 4.4, can be quite large and is directly proportional to the change in stored energy during a given deformation of the laminate for the case of two strip delaminations on either side of the laminate [53]. Such delamination growth is one of the few examples of self-similar crack growth in composite materials and hence is one of the few cases in which fracture mechanics techniques may be employed. The importance of the stiffness changes related to delaminations is discussed in Ref. [45]. Not only can the delamination affect structural response (such as by changing the natural frequency

Delaminations

Figure 4.7 Development of free-edge delaminations in a laminated composite material [45].

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185

of the system) but they can also greatly change the stress distribution within the undelaminated as well as the delaminated portions of the composite. When this happens, some of the material in the interior that has not changed in stiffness must carry more of the load when other material unloads. If there is a ply with fiber orientations close to the principal loading direction, for example, then the stress in those plies typically increases with the growth of the delamination, sometimes by substantial amounts. Hence, unlike matrix cracking, delamination may change the strength of the laminate. This leads us to an important distinction that we must make in the study of the subject at hand. Delamination is rarely, if ever, a failure mode; instead, it is a typical damage mode. This process of damage accumulation and associated stress redistribution within the laminate continues until significant stiffness changes with respect to cycles are observed to occur (Region III of Fig. 4.4). Within this region, significant localized fiber fracture occurs until the point at which the composite as a whole can no longer sustain the load, leading to fatigue failure. A distinguishing feature of this failure is that, unlike fatigue failure in metals, it is rare to identify a dominate crack to leads to failure. Rather, it is this interactive, distributed damage that results in failure. While such interactive, distributed damage is desirable for long fatigue lifetimes, it greatly complicates predictions of the fatigue behavior. These discussions have focused on tensile fatigue behavior. Experimental observations [54e56] show that tensionecompression fatigue is more detrimental than tensionetension fatigue in multidirectional laminates. The sequence of damage development appears to be quite similar to that observed in tensionetension fatigue loading: initial matrix cracking (perhaps near a stress concentration), secondary matrix cracking, and delamination development. This similarity in sequence is summarized in Fig. 4.8 from the work of Gamstedt and Sj€ ogren [56]. Because failures of these laminates are often controlled by compression rather than tension, the ultimate failure is by fiber microbuckling [54,57]. This microbuckling is the direct result of an instability that occurs at the local level. Consequently, having outer ply orientations at angles other than 0
can delay the failure [54]. The damage development processes above have been discussed for a laminated composite material. Fujii et al. [58] examined tensile fatigue damage mechanisms in a plain-weave graphite fiberereinforced composite material. They found the fatigue damage development in the composite to have three stages: (1) an initial region where rapid modulus decay with cycles occurred, (2) an intermediate region consisting of gradual modulus decay with cycles, and (3) a final region where decay occurred for the last few cycles. Thus, the stiffness reduction during fatigue of the woven material appeared to have the regions of damage shown in Fig. 4.5. However, the underlying physical mechanisms relevant to this case were found to be different. Based on their experimental observations, Fujii et al. hypothesized that the damage developed as follows. The initial rapid modulus decay was caused by the initiation and accumulation of debonds in the weft and matrix cracks in the warp in regions surrounding a fiber undulation. Furthermore, this initial cracking ceased after approximately 10% of the fatigue lifetime. Unlike laminated composites, this damage state occurs at nearly all of the crossover points. The authors termed this damage state to be the “meta-CDS.”

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Durability of Composite Systems

Tension-compression

Tension-tension

Crack tip opening in compression

Contact zone in tension

High growth rate

Low growth rate

Accelerated initiation of transverse cracks

Retarded transverse crack formation

Debond profile

Debond propagation

Transverse cracking

Transverse crack density

0° 90° 0°

Few cracks

Many cracks

Fatigue behaviour

0° 90° 0°

S

T.T T.C

Log N Shorter fatigue lives and more rapid degradation for fatigue with compressive load excursions

Figure 4.8 Schematic diagram of link from micromechanisms to macroscopic fatigue behavior [56].

In the second fatigue stage, the gradual modulus decay is attributed to the development of “metadelaminations” (small delaminations between warp and weft fiber bundles) occurring at the fabric crossover points. From experimental results, the authors observed weave pitch of the fabric to be a parameter of metadelamination

Response of composite engineering structures to combined fire

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because the metadelamination and weft debonds occur at each crossover point. This is somewhat similar to angle-ply laminates in which interior delaminations tend to occur where cracks intersect each other and where cracks meet fibers from adjacent plies. Finally, Fujii et al. claim that woven composites have unit cells of damage accumulation, which can be used as parameters describing fatigue accumulation. The authors go on to consider the possibility that many damaged unit cells are produced everywhere within the specimen in the middle stage where the gradual modulus decay represents the sum of fatigue damage of unit cells having different damage accumulations. As Case and Reifsnider [45] note, this type of damage development was observed in the work of Patel [59] on PR500 epoxy reinforced with woven (five harness satin weave) AS-4 carbon fiber. Given the nature of composites, the variability of their fatigue behavior is much greater than that associated with metallic materials. This variability comes from not only the statistical nature of the progressive damage and the stochastic nature of loading and environment but is also more due to the variable quality of commercial fibers, resin systems, and the fabrication of composite materials. Because of this, the experimental characterization of composite fatigue lifetimes is still challenging. The most common practice is to test replicate samples at different stress levels. The experimental data provide statistical information at each stress level and provide probability of failure through stresselife curves in addition to mean life or median life. This practice, however, brings the problem of how many replicates should be tested at each stress level considering both of statistical requirements and economic issues. The ASTM standard suggests at least six replicates at each stress level, while from the statistical point of view, at least 20 replicates at each stress level may be necessary for a confident statistical analysis of the experimental results [52]. The resulting lifetimes are then characterized using two- or three-parameter Weibull distributions [44,52].

4.3.2

Fatigue damage: effects of loading frequency

Zhou et al. [51] have suggested that the main mechanism of loading frequency’s effects on fatigue life is due to the internal heating of the composite during fatigue. In general, as the frequency of fatigue loading increases, internal heating of the composite increases and fatigue life of a composite decreases. They further suggested than from various published data, frequencies below 5 Hz have been shown to produce negligible internal heating in glass fiberereinforced composites. The fatigue lifetime results from our own study on the behavior of a plain-woven E-glass/Derakane 8084 composite tested at frequencies from 0.1 to 40 Hz under R ¼ 0.1 loading are summarized in Fig. 4.9. As the frequency of loading increases from 0.1 Hz to approximately 10 Hz, the fatigue lifetimes increase slightly. However, for higher frequencies, the lifetimes decrease for higher frequencies. A potential explanation is that there is a small creep rupture effect at the lower frequency range (samples tested at higher frequencies experience less time under tensile load for a given number of cycles to failure than those tested at the higher frequencies). Fig. 4.9 shows the temperature

188

Durability of Composite Systems 100000

Cycles to failure

10000

1000

100 0.1

1

10

100

Frequency (Hz) 148 MPa

110 MPa

193 MPa

252 MPa

Figure 4.9 Fatigue lifetime data for plain-woven E-glass/Derakane 8084 composites tested a various loading frequencies and maximum stresses for a fatigue R-ratio of 0.1.

as a function of cycles for the fatigue of woven E-glass/8084 matrix composites over a range of frequencies. The temperature increase is below 80
C (postcuring temperature) for frequencies lower than 22 Hz (Fig. 4.10).

4.3.3

Fatigue damage: spectrum fatigue loading

Ideally, one would be able to develop fatigue lifetime predictions without the benefit of an underlying stresselifetime (SeN) curve. Because of the complicated nature of the damage and failure processes in composites, there is presently no validated method to Frequency (Hz)

120

Temperature (°C)

100 80 60 40 20 0 0

10000

20000

30000 40000 Fatigue cycles

50000

60000

0.2 1 1 5 5 10 10 10 15 15 20 20 25 25 30 30 35.4 39

Figure 4.10 Temperature as a function of cycles for composites of Fig. 4.9 over a range of frequencies (R ¼ 0.1, maximum applied stress ¼ 110 MPa).

Response of composite engineering structures to combined fire

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do so. Consequently, SeN curves based upon constant amplitude data collected at different fatigue R-ratios (such as that shown in Fig. 4.11) most often are used as the basis for such predictions. To apply those data to practical engineering applications, fits to the constant lifetime data are employed to develop constant lifetime plots such as that shown in Fig. 4.12. Prior studies have shown that a typical constant lifetime diagram for composite materials is shifted so that the mean stress corresponding to a specified lifetime is to the right of zero mean stress [60]. It is common to use interpolation to connect the segments, although Vassilopoulos et al. [61] have suggested a piecewise nonlinear constant life diagram developed in the stresseratio/stresse amplitude plane. A review on the fatigue studies of composite materials by Van Paepegem and Degrieck [62] suggested that for nonconstant amplitude fatigue data, there is an inconsistency with regard to published results. In some cases, high stress amplitude followed by low stress amplitude is more damaging; in other cases, the opposite behavior is true. For example, fatigue testing of cross-ply E-glass/epoxy specimens indicated that low stress followed by high stress was the more damaging case [63]. These data have been used by several authors to validate their models. Other studies have found that the reverse effect can be true, where high stress followed by low stress amplitude was more damaging [64]. In some ways, this is an academic question, as realistic loading conditions most often include more than just two load levels. Post et al. [65] conducted an evaluation of 12 different approaches to predict lifetimes for composites subjected 700

600 Absolute peak applied stress (MPa)

R = 0.10 500 R = –0.37 400

300 R = 7.00 200 R = –1.00 100

0 1

10

100

1000

10000

100000

1000000 10000000

Cycles to failure (Nf)

Constant amplitude stresselifetime data at a frequency of 10 Hz for Figure 4.11  ð  45Þ2 02 0 s laminates fabricated using Toray fibers in an epoxy matrix material. T60024k fibers were used for the 0
layers, and T700-12k were used for the 45 degrees layers. Tests in which fatigue failure did not occur are shown as open symbols.

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Durability of Composite Systems 500

R=–0.37 450

Stress amplitude (MPa)

400 1

10

350

R=0.10 2

10

300 R=–1.00

3

10

250

4

10 5

10

200

6

R=7.00

10 150 100 50

0

–400

–200

0

200

400

600

800

1000

Mean stress (MPa)

Figure 4.12 Constant lifetime plots based upon constant amplitude stresselifetime data of Fig. 4.11. Fa 1 0.9

90% confidence interval

0.8 0.7 0.6 0.5 0.4 0.3 0.2

Least squares regression curve fit log(N) = –8.7 log(Fa) + 1.6

0.1 0 100

1000

10000

100000

1000000

10000000

log(N)

Figure 4.13 Fatigue lifetimes for E-glass vinyl ester composites tested at R ¼ 0.1 [66].

to multiple (spectrum) loading conditions. One of their interesting conclusions was that “considering the poor reputation of [Palgrem-Miner] based on two block fatigue loading the model performs well under the repeated spectrum loading considered here with the worst prediction being about 4 times the experimental fatigue life in the unidirectional UD2 material and otherwise giving predictions within 2.5 times the experimental life.” They also found that a “simple residual strength model introduced

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191

by Broutman and Sahu [63] gave more conservative prediction of fatigue life than [Palmgren-Miner] in all cases due to failure occurring at the highest stress present in the spectrum. Since [Palmgren-Miner] generally over predicted fatigue life, the [Broutman-Sahu] results were generally better.” To demonstrate the ability of such a residual strength models to address spectrum loading conditions in which amplitude and/or mean stress in fatigue loading vary, we examine the work of Ref. [66] who characterized the response of E-glass/vinyl ester composites consisting of 10 layers of E-glass (Vetrotex 324) with a [0/þ45/90/45/ 0]s stacking sequence (denoted by the warp direction) in a vinyl ester matrix (Ashland Derakane 810A). Static strength measurements were conducted on 20 samples to determine the distribution of the initial static strength. Subsequently, constant amplitude fatigue tests were conducted using 88 samples to determine the SeN response. This SeN response is summarized in Fig. 4.13 and Table 4.4, where the normalized applied fatigue stress is Fa ¼

sa sult

(4.9)

(the maximum fatigue stress in a cycle divided by the ultimate strength). Additionally, residual strength tests were conducted at three different stress levels and five life fractions after constant amplitude fatigue loading to determine the shape of the residual strength curve with fatigue cycles. In total, 186 such tests were conducted. These results are summarized in Fig. 4.14, where the normalized residual strength is given by Fr ¼

sr sult

(4.10)

where sr is the measured residual strength. For the initial attempt at modeling spectrum loading fatigue, a 99,999,998 cycle spectrum with 30 stress levels based on a 30 year lifetime was selected so that the distribution of loads would be somewhat realistic. The spectrum loads were scaled based Table 4.4 Summary of fatigue lifetime data for E-glass vinyl ester composites [66].   s Fa applied Xt

Number of replicates

Median

Weibull a

Weibull b

0.71

16

917

3.8

1064

0.52

16

11,162

4.3

12,517

0.44

16

44,010

3.7

50,422

0.40

16

109,397

4.2

116,877

0.36

16

377,012

1.4

471,390

0.32

3

1,542,553

e

e

Lifetime, N (cycles)

192

Durability of Composite Systems 1.0 0.9

Total of 186 tests Error bars = 1 st. dev. Least squares fit: j = 1.16

0.8 0.7

n

Fr = 1– (1 – Fa) ( N ) j

0.6 0.5

Fa=0.52

0.4

Fa=0.44 Fa=0.36

0.3

RS fit at Fa=0.52 RS fit at Fa=0.44

0.2

Initial median strength = 334 MPa

RS fit at Fa=0.36

0.1 0.0 1

10

100

1000

10000

100000

1000000

Figure 4.14 Residual strength measurements for E-glass vinyl ester composites.

on the residual strength analysis to predict failure at the maximum applied stress level after exactly one time through the spectrum. To obtain reasonable testing times, the spectrum was truncated to the highest 22 stress levels leaving a total of 735,641 cycles after which the residual strength could be measured and compared to model predictions. These fatigue stress levels and corresponding numbers of applied cycles are given in Table 4.4. During the spectrum fatigue tests, an R-ratio of 0.1 was maintained for each valley following a corresponding peak. The spectrum was applied in three ways: first, as block loads in ascending order from lowest stress to highest, second, as block loads in descending order from highest stress to lowest, and third, the entire spectrum was randomized so that each individual cycle had an equal probability of occurring at any point during the fatigue. Because there are so many more cycles of the lowest two stress levels than any of the others, the randomized spectrum ends up looking like a low-level fatigue with various higher peaks interspersed [66]. An illustration of the stress levels applied for the descending order arrangement is given in Fig. 4.15. The residual strength analysis was applied to the actual stress peaks applied experimentally (rather than the desired peaks in the spectrum). Because the test equipment was not able to hit each peak perfectly, there is a small difference in the residual strength at the end of the descending block and ascending block spectrum, which in theory should produce identical results because the model is not load order dependent. Comparisons between samples produced very little variation so the predicted result reported is the average result for the loading of all specimens tested for each spectrum block configuration. Predicted and experimentally determined failure lifetime distributions are shown in Fig. 4.16. This agreement looks quite good and provides an indication that the measured residual strengths are independent of the block ordering (as expected based on simple residual strength theories). However, when the stresses were randomized (rather than

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250 200 250

150

200

100 50

150

0 100

0

10

50

20

•

30

40

Applied as: – Descending ordered blocks

0 0

200000

400000

600000

800000

– Ascending ordered blocks

n

– “Completely randomized”

Figure 4.15 Illustration of stresses (in MPa) applied during spectrum loading for E-glass vinyl ester composites (stresses in MPa). (b)

(a)

1.0

1.0

0.9

0.9 Measured: α = 19.1 β = 295 MPa

0.8 0.7

Pf

0.7 0.6

0.6 0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 245

265

285

305

σ failure (MPa) Ascending order block loading spectrum

Predicted: α = 13.0 β = 295 MPa

0.5

Predicted: α = 13.8 β = 298 MPa

0.5

0.0 225

Measured: α = 19.5 β = 293 MPa

0.8

325

0.0 225

245

265

285

305

325

σ failure (MPa) Descending order block loading spectrum

Figure 4.16 Predicted and measured residual strength distributions for (a) ascending and (b) descending ordered blocks of fatigue loading [66].

applied in blocks), all 12 of the samples tested under this loading condition failed at a median of approximately 280,000 cycles (none of the samples subjected to the block loading failed prematurely). Remembering that the initial stresses had been selected so that failure occurred after approximately 100,000,000 cycles, this was a surprising (and disturbing) occurrence. This type of behavior where frequent changes in the fatigue loading level lead to a greater reduction in strength than the same cycles applied in a more ordered fashion has been termed the “cycle mix effect” [67]. To further investigate this effect, additional data were collected from material consisting of 10 layers of the same woven roving E-glass (Vetrotex 324) with a [0/þ45/90/ 45/0]s stacking sequence (denoted by the warp direction) in a rubber toughened vinyl ester matrix (Ashland Derakane 8084)da matrix similar to that used in the study above, except for the absence of halogens [65]. The variable amplitude fatigue loading

194

Durability of Composite Systems

data for this material system include Rayleigh-distributed loading with different degrees of autocorrelation (a measure of the degree of load ordering, see Ref. [68]) with a nominal value of the fatigue R-ratio, R ¼ 1 (Fig. 4.17). The resulting fatigue lifetimes are shown in Fig. 4.18. In this case, there is no significant difference in the fatigue lifetimes (indicating no “cycle mix” [67] effect). It appears that this cycle mix effect depends upon the R-ratio of the loading conditions. In fact, these spectrum fatigue results are well predicted by a residual strength model (an example of such predictions is shown in Fig. 4.19).

4.3.4

Fatigue damage: temperature and moisture effects

Given the complicated nature of the fatigue damage development process, it is unsurprising that the role of environmental factors is complicated and often material system dependent. The fiber reinforcements are used to control stiffness and strength and to limit time-dependent behavior of the material. Glass fibers exhibit excellent resistance against temperatures (>650
C) well above the decomposition temperature of the polymer resin (typically ranges from 250e350
C). However, even though the fiber reinforcements are not directly affected by temperature, the fiber direction stiffness, strength, and fatigue life of a composite can be strongly dependent on temperature, especially in temperature ranges above the glass transition temperature (Tg) of the polymer matrix. This is mainly due to the contribution of strong temperature4

Autcorrelation = 0%

Normalized stress

Normalized stress

4 3 2 1 0 –1 –2

3

Autcorrelation = 95%

2

4 3

0 –1 –2 –4

–2 –3

Time (s)

Autcorrelation = 99.99%

1

–3

0 –1

Time (s)

Time (s)

Normalized stress

Normalized stress

4

2 1

–4

–3 –4

Autcorrelation = 50%

3

2 1 Peaks 0 0 1000 –1 Valleys –2

2000

3000

–3 –4

Figure 4.17 Illustration of different degrees of autocorrelation studied [68].

4000

5000 cycle

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45.0 40.0 RMS stress (MPa)

35.0 30.0 25.0 20.0 15.0

Rayleigh1.0 Rayleigh0.95 Rayleigh0.5 Rayleigh0.0

10.0 5.0 0.0 1000

10000 100000 1000000 Cycle to failure, N (log scale)

10000000

Figure 4.18 Cycles to failure results for fatigue loading conditions with different degrees of autocorrelation [68]. 50 45 40

RMS(σ ) (MPa)

35 30 25 20 15 10

Experimental data TC Model

5 0 100

1000

10000

100000

1000000

10000000

Cycles, N (log scale)

Figure 4.19 Predicted and measured fatigue lifetimes for Rayleigh-distributed loading with autocorrelation of 0.95 [68].

dependent behavior of polymer matrix above its Tg, as discussed above in the composites and fire section above. The degradation of stiffness, strength, and life of the composite is the result of the complex internal and local processes that control degradation and the strong interactions between fibers and the viscoelastic matrix. Temperature can have a significant effect on the fatigue of composites. A study on the fatigue of a pultruded E-glass/vinyl ester composite is shown in Fig. 4.20. It is seen that the strength decaying ratio increases as temperature increases (but below the Tg of the matrix). The effects of moisture-induced stress states on the fatigue behavior of the material were shown to be important by Smith and Weitsman [69]. They conducted fatigue tests

196

Durability of Composite Systems

20000

90 80

15000

70 4 °C

60 50

10000

40 30 °C

65 °C

30 20

5000

R = 0.1 Freq = 10 Hz

10

0 1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

Max stress (psi)

% Ultimate tensile strength

100

0 1.0E+07

Fatigue cycles (N)

Figure 4.20 Fatigue lifetimes of an E-glass/vinyl ester composite as a function of temperature (R ¼ 0.1).

on AS4/3501-6 ([0/903]s, 8-ply]) samples fatigued dry in air, previously saturated samples fatigued in air, and saturated samples tested while immersed. SeN curves for the three different cases are shown in Fig. 4.21. The differences in behavior were attributed the relief of curing-induced shrinkage provided by sorption-induced swelling. The fact that previously saturated specimens fatigued while immersed in water exhibited the lowest fatigue lives was attributed to the presence of water draw by capillary action into the mechanically-induced fatigue cracks. In contrast, Pfeiffer et al. [70] evaluated the durability of E-glass fiberereinforced vinyl ester (Derakane 411-350) composites. Fatigue tests were conducted on these composites under two different conditions: in the as-manufactured state and after 95

σmax/σult (%)

90

85

80

75

(c) 70 1x102

1x103

1x104

1x105

(a)

(b)

1x106

1x107

Nf

Figure 4.21 SeN curves for (a) dry specimens fatigued in air, (b) saturated specimens fatigued in dry air, and (c) saturated specimens fatigued while immersed in seawater [69].

Response of composite engineering structures to combined fire

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180 160 140

Applied stress (MPa)

120 100 80 60 40 20 0 100

1,000

10,000

100,000

1,000,000

Cycles to failure As fabricated

Aged for 14 days

Unaged model

Aged 14 days predictions

Figure 4.22 Experimental fatigue lifetime predictions for aged and unaged materials, compared with model results. Reproduced with permission of the Society of Plastics Engineers E. Pfeiffer, S.W. Case, J. Lesko, J. Moore, N. Verghese, J. Bicerano, H. Pham, Lifetime Prediction of Glass Fiber/ DERAKANE 411-350 Composites, vol. 3, 2004.

immersion for 14 days in 80
C water. Samples fatigued after being aged for 14 days demonstrated significantly reduced fatigue lifetimes, as shown in Fig. 4.22. They hypothesized that these differences were due to the degradation of the tensile strength of the glass fibers as a result of the immersion. Further, they used a residual strengthe based fatigue lifetime model to attempt to predict the fatigue lifetimes of the previously immersed samples based on the dry fatigue data and strength versus aging time data from a long-term study (summarized in Fig. 4.23). For other material systems, the effect of moisture and temperature may be much less pronounced. For example, Patel and her coworkers [59,71e73] evaluated the effects of moisture, elevated temperature, and temperature and moisture cycling on the fatigue durability PR500 epoxy reinforced with woven (five-harness satin, [0/90]4s) AS-4 carbon fiber. Residual strength measurements were conducted for samples tested at room temperature dry, for samples tested at 30
C and 85% relative humidity, samples tested at 120
C, and for samples that were hygrothermally cycled. These results, summarized in Fig. 4.24, indicate no significant differences due to the environmental exposure. In addition, the authors conducted tests on samples that had been previously hygrothermally cycled. They found that changes in physical appearance, thermal analysis results, fracture surfaces, and moisture diffusion behavior all indicated that the material was affected by the aging process. However, experimental testing also

198

Durability of Composite Systems 1.2 80°C-(0/90°)s 65°C-(0/90°)s

1.0

45°C-(0/90°)s 25°C prediction

Xt/Xto

0.8

0.6

0.4

0.2

0.0 0

100

200

300

400

500

600

700

Time (Days)

Figure 4.23 Normalized strength (aged strength/initial strength) versus immersed aging time for glass/vinyl ester composites.

Normalized residual strength

1.1

Calculated residual strength curves corresponding to specified fatigue stress levels

1.0 70% UTS 0.9

0.8

0.7

0.6 100

85% UTS

75% UTS

Life curve (S-N curve) 25°C data 30°C and 85% RH data 120°C data Hygrothermal data 1000

10000

100000

1000000

10000000

Fatigue cycles

Figure 4.24 Residual strength data and predictions based on room temperature data.

showed that the initial and residual tensile properties of the aged material were virtually unaffected by the imposed environmental aging. These results, and others from the literature, highlight the difficulty in drawing general conclusions about factors that influence the fatigue resistance of composites. While the general sequence of damage events is well understood, the manner in which that damage interacts to lead to failure is complex.

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[61] A.P. Vassilopoulos, B.D. Manshadi, T. Keller, Piecewise non-linear constant life diagram formulation for FRP composite materials, Int. J. Fatig. 32 (10) (2010) 1731e1738. [62] W.V. Paepegem, J. Degrieck, Effects of load sequence and block loading on the fatigue response of fiber-reinforced composites, Mech. Adv. Mater. Struct. 9 (1) (2002) 19e35. [63] L.J. Broutman, S. Sahu, A new theory to predict cumulative fatigue damage in fiberglass reinforced plastics, in: Composite Materials: Testing and Design (Second Conference), ASTM International, West Conshohocken, PA, 1972, pp. 170e188. [64] W. Hwang, K.S. Han, Cumulative damage models and multi-stress fatigue life prediction, J. Compos. Mater. 20 (2) (1986) 125e153. [65] N.L. Post, S.W. Case, J.J. Lesko, Modeling the variable amplitude fatigue of composite materials: a review and evaluation of the state of the art for spectrum loading, Int. J. Fatig. 30 (12) (2008) 2064e2086. [66] N.L. Post, J. Cain, K.J. McDonald, S.W. Case, J.J. Lesko, Residual strength prediction of composite materials: random spectrum loading, Eng. Fract. Mech. 75 (2008) 2707e2724. [67] J.R. Schaff, B.D. Davidson, Life prediction methodology for composite structures. Part 1 constant amplitude and two-stress level fatigue, J. Compos. Mater. 31 (2) (1997) 128e157. [68] N.L. Post, Reliability Based Design Methodology Incorporating Residual Strength Prediction of Structural Fiber Reinforced Polymer Composites under Stochastic Variable Amplitude Fatigue Loading in Engineering Mechanics, Virginia Tech, Blacksburg, VA, 2008. [69] L.V. Smith, Y.J. Weitsman, The immersed fatigue response of polymer composites, Int. J. Fract. 82 (1) (1996) 31e42. [70] E. Pfeiffer, S.W. Case, J. Lesko, J. Moore, N. Verghese, J. Bicerano, H. Pham, Lifetime Prediction of Glass Fiber/Derakane 411-350 Composites, vol. 3, 2004. [71] S.P. Davison, Enviro-mechanical durability of graphite/epoxy composite materials, in: Engineering Mechanics, Virginia Tech, Blacksburg, VA, 2003. [72] S.R. Patel, Durability of advanced woven composites in aerospace applications, in: Engineering Mechanics, Virginia Tech, Blacksburg, VA, 1999. [73] S.R. Patel, S.W. Case, Durability of hygrothermally aged graphite/epoxy woven composite under combined hygrothermal conditions, Int. J. Fatig. 24 (12) (2002) 1295e1301.

Advanced composite wind turbine blade design and certification based on durability and damage tolerance

5

Frank Abdi 1,2 , Harsh Baid 1 , Amirhossein Eftekharian 1 1 AlphaSTAR Corporation, Long Beach, CA, United States; 2Imperial College London, London, United Kingdom

Chapter outline 5.1 Introduction

204

5.1.1 Problem statement 205 5.1.2 Background 205 5.1.3 Objective 206

5.2 Methodology

208

5.2.1 Building block approach (ASTM coupon test standards) 208 5.2.2 Composite material calibration 208 5.2.2.1 Continuous fiber 208 5.2.2.2 Woven fabric composites 208 5.2.2.3 Nanoenhanced matrix 212 5.2.3 Multi-scale progressive failure analysis 213 5.2.4 Fracture mechanics 214 5.2.4.1 Virtual crack closure technique 214 5.2.4.2 Discrete cohesive zone modeling 216 5.2.5 Probabilistic and reliability analysis 217 5.2.6 Certification approach 218

5.3 Wind blade design technique and analysis 5.4 Results and discussion 226

222

5.4.1 Material modeling calibration and validation 226 5.4.1.1 Static properties calibration and validation 226 5.4.2 Tapered blade analysis and results 226 5.4.2.1 Failure prediction and test validation of tapered composite under static and fatigue loading 226 5.4.3 Nine meter blade 241 5.4.3.1 Durability and reliability of wind turbine composite blades using a robust design approach [] 241 5.4.3.2 Description of blade FEA model and blade materials 242 5.4.3.3 Simulation of blade static test 242 5.4.3.4 Fatigue evaluation of a 9-m wind turbine blade 245 5.4.3.5 Blade weight analysis 247

Durability of Composite Systems. https://doi.org/10.1016/B978-0-12-818260-4.00005-3 Copyright © 2020 Elsevier Ltd. All rights reserved.

204

Durability of Composite Systems 5.4.3.6 Blade durability and damage tolerance probabilistic sensitivity analysis 249 5.4.3.7 Blade weight reduction with robust design 252 5.4.3.8 Improving wind blade structural performance with the use of resin enriched with nanoparticles [] 254 5.4.3.9 Insertion of silica nanoparticles in a matrix of glass composite 256 5.4.3.10 D&DTBlade results with glass composite infused with silica nanoparticles 257 5.4.3.11 Summary 259 5.4.4 Simulation of a 35-m wind turbine blade under fatigue loading 260 5.4.5 Conclusion 265

References

5.1

269

Introduction

Recently, the electricity generated by wind power is increasing dramatically. The larger the turbine, the greater the amount of electricity produced. The need for large commercial wind turbines gives rise to challenges in materials and manufacturing technologies. At present, turbine blades are usually made of E-glass fiber-reinforced polyester composites. The use of advanced composites in wind blades is becoming more attractive due to its advantageous weight-to- stiffness and weight-to-strength ratios. In addition, composite structures are usually used to subject severe combined environments and are expected to survive for long periods. Therefore, it is necessary to assess the damage mechanism in large wind blades. Current wind turbine blade design with advanced composites is based on high factors of safety and traditional design/stress analysis practices to ensure the target static strength levels and service life lengths. To achieve low production costs, material systems such as resin-infused woven and stitched fiberglass are utilized to try to hit an approximate $5/lb. pound target product cost. To reduce operational costs, realtime structural health monitoring is not used to assess the condition of the blades. The design process can be described as one that focuses on service life rather than damage tolerance. In addition, composite materials have considerable scatter in nature due to voids, fiber waviness, and manufacturing anomalies. Minimizing the scatter would involve considerable costly testing. A combination of these design constraints can significantly impact the turbine blade weight and performance. A design process that uses advanced damage modeling approaches for composites will lead to blades that are optimized to be damage resistant and tolerant while being light and inexpensive. The use of advanced composites in product design is becoming increasingly more attractive due to their advantageous weight-to-stiffness and weight-to-strength ratios. Increasingly, composite structures are being subjected to severe combined environments and are expected to survive for long periods. There is neither an adequate test database for composite structures nor significant long-life service experience to aid in risk assessment. To ensure safe designs, aerospace companies spend many millions of dollars per year on testing. Owing to the difficulty and cost in assessing and

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205

managing risk for new and untried systems, the general method of risk mitigation consists of applying multiple conservative factors of safety and significant inspection requirements to already conservative designs instead of costly full system tests. Unfortunately, this approach can lead to excessively conservative designs and the full potential of composite systems is often not fully realized. Determination of allowable properties is a time-consuming and expensive process, as a large amount of testing is required. To reduce costs and product lead-time, certification by analysis (CBA) can be used to reduce necessary physical tests both for certification and for determining allowables. Whereas current advanced composite industrial practice tends to rely on expensive test-intensive empirical methods to establish design allowables for sizing advanced composite structures, the proposed CBA methodology relies on physics-based failure criteria to reduce its dependence on such empirical-based procedures.

5.1.1

Problem statement

Wind turbine blades are low-cost composite structures that are subjected to highly demanding, and often unexpected, environmental dynamic loads during their operational life that can lead to catastrophic failure. Wind turbines are subjected to high numbers of cyclic fluctuations due to wind intensity in various directions, therefore, resulting in a severe strain on wind turbine blades if the wind intensity is high, which can ultimately lead to failure. Wind turbine failure can have detrimental impacts on the forms of safety, downtime, and public exposure. Mechanical components, such as structural parts/housing, are estimated to account for 4% of wind turbine failures. Rotor blades are estimated to account for 7% of wind turbine failures. One aspect to account for the relatively high percentage of rotor blade failures is manufacturing defects. One such defects is delamination of the layered composite, which results in a severe loss of its structural integrity. An interesting fact is that the fracture process in layered composite materials usually initiates, and propagates, while under low loading conditions. This differentiates composite materials from metallic materials, which raise the question of whether or not traditional stress/strain analysis is sufficient for evaluating the failure of composite materials. Many theories for the mechanics of composite materials exist; therefore, there are also many tools available to evaluate the failure mechanisms of fiber-reinforced layered composite materials. In theory, wind turbine blades are composed of joints, noodle areas, critical junctions, and others. Traditionally, big blades are manufactured in parts and assembled together. The traditional way of solving composite problems is based on the classical laminate theory that is very conservative. The traditional way of analyzing composite is not enough, and it requires rigorous numerical-based simulation to analyze structural integrity.

5.1.2

Background

Tapered laminated structures, as are found in some areas of wind turbine blades, are formed by dropping off some of the plies at discrete positions over the laminate. Structural details like this and others have received much attention from researchers because

206

Durability of Composite Systems

Figure 5.1 Simulated blade structure with material thickness transition [7]. [+/– 45]4

Drop off zone : 7 mm

Depth : 25.4 mm

[08]

[06]

11.6 mm

14.2 mm

Resin rich area

[+/– 45] Gage length : 101.6 mm

Figure 5.2 Layout of a two plies drop-off specimen [7].

of their structural tailoring capabilities, damage tolerance, and their potential for creating significant weight savings in engineering applications. The inherent weakness of this construction is the presence of material and geometric discontinuities at the ply drop region that induces premature interlaminar failure at interfaces between dropped and continuous plies. A review of recent developments in the analysis of tapered laminated composite structures with an emphasis on interlaminar stress analysis, delamination analysis, and crack growth analysis applied to a blade structure (Fig. 5.1) is presented in this paper. A 2 plies drop-off, as shown in Fig. 5.1, is illustrated in Fig. 5.2. The gage is 101.6 mm long and the drop off zone is 7 mm wide.

5.1.3

Objective

The objective of this chapter is to demonstrate and verify CBA capability for wind turbine blades made from advanced lightweight composite materials. The approach integrated durability and damage tolerance analysis with robust design and virtual testing capabilities to deliver superior, durable, low weight, low cost, long life, and reliable wind blade design. Durability and life prediction approach was used as the primary simulation tool. First, a micromechanics-based computational approach was used to assess the durability of composite laminates with ply drop features commonly used in wind turbine applications. Ply drops occur in composite joints and closures of wind turbine blades to reduce skin thicknesses along the blade span. They increase localized stress concentration, which may cause premature delamination failure in composite and

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reduced fatigue service life. Durability and damage tolerance (D&DT) were evaluated utilizing a multiscale microemacro progressive failure analysis (PFA) technique. PFA is a finite element based and is capable of detecting all stages of material damage including initiation and propagation of delamination. It assesses multiple failure criteria and includes the effects of manufacturing anomalies (i.e., void, fiber waviness). Two different approaches have been used within PFA. The first approach is the virtual crack closure technique (VCCT) PFA while the second one is strength-based. Constituent fiber and matrix stiffness and strength properties for glass and carbonbased material systems were reverse engineered for use in D&DT evaluation of coupons with ply drops under static loading. The root cause of the problems can be analyzed using a micromechanics approach including the effect of defects by implementing independent fiber and matrix constituent properties. Lamina and laminate properties calculated using manufacturing and composite architecture details matched closely published test data. Similarly, resin properties were determined for fatigue life calculation. The simulation not only reproduced static strength and fatigue life as observed in the test, it also showed composite damage and fracture modes that resemble those reported in the tests. The results show that computational simulation can be relied on to enhance the design of tapered composite structures such as the ones used in turbine wind blades. A computational simulation for D&DT and reliability of composite wind turbine blade structures in the presence of uncertainties in material properties was performed. A composite turbine blade was first assessed with finite element-based multiscale progressive failure analysis (MS-PFA) to determine failure modes and locations as well as the fracture load. D&DT analyses were then validated with the static test performed at Sandia National Laboratories. The work was followed by a detailed weight analysis to identify the contribution of various materials to the overall weight of the blade. The methodology ensured that certain types of failure modes, such as delamination progression, are contained to reduce the risk to the structure. Probabilistic analysis indicated that composite shear strength has a great influence on the blade ultimate load under static loading. Weight was reduced by 12% with robust design without loss in reliability or D&DT. Structural benefits obtained with the use of enhanced matrix properties through nanoparticle infusion were also assessed. Thin unidirectional fiberglass layers enriched with silica nanoparticles were applied to the outer surfaces of a wind blade to improve its overall structural performance and durability. The wind blade was a 9-m prototype structure manufactured and tested subject to three saddle static loading at Sandia National Laboratory. The blade manufacturing did not include the use of any nanomaterial. With silica nanoparticles in glass composite applied to the exterior surfaces of the blade, the D&DT results from MS-PFA showed an increase in the ultimate load of the blade by 9.2% as compared to baseline structural performance (without nano). The use of nanoparticles leads to a delay in the onset of delamination. Loadedisplacement relationships obtained from testing of the blade with baseline neat material were compared to the ones from analytical simulation using neat resin and using silica nanoparticles in the resin. MS-PFA results for the neat material construction matched closely those from the test for both loadedisplacement and location and type of damage and failure.

208

5.2

Durability of Composite Systems

Methodology

5.2.1

Building block approach (ASTM coupon test standards)

Fig. 5.3 shows the ASTM-based static and fatigue coupon calibration and validation for composite materials. Composite ASTM Standard static tests will be defined such as (1) unnotched: longitudinal/transverse tension (D3039, D638), longitudinal/ transverse compression (D695, D3410), in-plane off-axial tension at
45 degrees (D3518); (2) off-axial tension at 45 degrees; (3) in-plane sheardIosipescu (ASTM D5379); (4) 4-point bending (ASTM D6272); (5) in-plane biaxial bending; (6) open hole (ASTM D5766, D6484); and (7) center crack, inclined crack, etc. ASTM Standard fatigue test plan will be defined such as (a) longitudinal tension, stress life fatigue (E 466); and (b) compact tension: da/dN versus DK (E 647). It is envisioned that virtual testing guides/reduces testing at each level. The detail of both static and fatigue test matrix for unidirectional carbon fiber, fabric, and chopped composite will be presented in Section 5.4.

5.2.2 5.2.2.1

Composite material calibration Continuous fiber

Fiber and matrix properties calibration is performed using a reverse-optimization process to determine the matrix-stiffness/strength (stressestrain curves), and the fiber stiffness/strength to match the unnotched (longitudinal/transverse tensile, longitudinal/ transverse compression, and shear) composite coupon tests at the lamina and laminate levels (Fig. 5.4 and Table 5.1). Using reverse-engineering and durability and damage tolerance analysis, stiffness, strength, Poisson’s ratio, and strength of the lamina and laminates are predicted and verified against the test data. Material uncertainty analysis is also performed to identify the effects of composite fiber/matrix material property and manufacturing uncertainties on laminate response. To obtain the in-plane material properties, five physical tests are required. These tests include tension and compression tests in the weft and warp direction and an in-plane shear test. The types of tests needed for the calibration processes are not limited to certain ASTM or other standards. The main issue is to create a good virtual counterpart of the physical tests. In other words, the material buildup, boundary conditions, loading, and test conditions should be included in the model as accurately as possible. For the calibration process, the stressestrain information is used as a comparison parameter between the virtual model and the physical test. To get good results, a complete experimental stressestrain curve is desired. This means that the stressestrain (loadedisplacement) curve has to be recorded and documented through the entire test until the test structure collapses. It is recommended to do this for the verification of the simulation results with the tests as well.

5.2.2.2

Woven fabric composites

Woven fabric composites for engineering structures draw on many traditional textile forms and processes. These textiles are generally those that most effectively translate

Test description

ASTM

Validation Description

Test description

ASTM D3846

Longitudinal/transver se tension

D3039 D638

Double notched compression (interlaminate shear)

Longitudinal/transver se compression

D695 D3410

Four point bending

D6272

In-plane shear (r45 in tension)

D3518

Short beam bending

D2344

Iosipescu (in-plane shear) or in plane shear

D5379 or D3518 or D4255

Flatwise tension/cring (interlaminar tension)

C297

Fatigue: stress life

E 466

Open-hole tension open-hole compression

D5766 D6484

Fatigue: compact tension: da/dN versus DK

E 647

Compact tension

E1802

Description

Advanced composite wind turbine blade design

Calibration

Figure 5.3 ASTM based static and fatigue coupon calibration and verification [25].

209

210

Durability of Composite Systems

(a)

Some of the textile forms available for high performance composite structures

(b)

Tape layup laminate

Plain weave fabric

Triaxial fabric

z Filler Stuffer Filler Plies

Layer-to-layer angle interlock woven Body warp weaver Surface warp weaver

Orthogonal interlock woven

Through-the-thickness angle interlock

Different composite architectures considered

(c)

L

Q3D

Q3D03

2D

Q3DO5

Various weaves: L (laminated), Q3D (quasi-three-dimensionalwoven), 2D (two-dimensional woven), Q3DO3 (quasi-three-dimensional woven with three harnesses) and Q3DO5 (quasi-three-dimensional with five harnesses).

Figure 5.4 Fiber architecture.

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211

Table 5.1 Fiber and matrix failure criteria applied at the micromechanics scale of the composite [20]. Mode of failure

Description

Longitudinal tensile (S11T)

Fiber tensile strength and fiber volume ratio.

Longitudinal compressive (S11C)

(1) Rule of mixtures based on fiber compressive strength and fiber volume ratio (2) Fiber microbuckling based on matrix shear modulus and fiber volume ratio (3) Compressive shear failure or kink band formation, which is mainly based on ply intra-laminar shear strength and matrix tensile strength

Transverse tensile (S22T)

Matrix modulus, matrix tensile strength, and fiber volume ratio

Transverse compressive (S22C)

Matrix compressive strength, matrix modulus, and fiber volume ratio.

Normal tensile (S33T)

Plies are separating due to normal tension

Normal compressive (S33C)

Due to very high surface pressure, i.e., crushing of laminate

In-plane shear (þ) (S12s)

Failure due to positive in-plane shear with reference to laminate coordinates

In-plane shear () (S12s)

Failure due to negative in-plane shear with reference to laminate coordinates

Transverse normal shear (þ) (S23s)

Shear failure due to shear stress acting on the transverse crosssection that is taken on transverse cross-section oriented in the normal direction of ply

Transverse normal shear () (S23s)

Shear failure due to shear stress acting on the transverse crosssection that is taken on negative transverse cross-section oriented in a direction of ply

Longitudinal normal shear (þ) (S13s)

Shear failure due to shear stress acting on the longitudinal cross-section that is taken on a positive longitudinal crosssection oriented in the normal direction of ply

Longitudinal normal shear () (S13s)

Shear failure due to shear stress acting on the longitudinal cross-section that is taken on negative longitudinal crosssection oriented in the normal direction of ply

Relative rotation criterion

Considers failure if adjacent plies rotate excessively with respect to one another

stiff, strong yarns into stiff, strong composites. A textile composite has an internal structure on several scales. At the molecular scale, both the polymer matrix and the fibers exhibit structural details that profoundly affect strength and stiffness. Matrix properties are determined by chain morphology and cross-linking, among other things.

212

Durability of Composite Systems

Carbon fibers, which are often the preferred choice in aerospace materials, owe their axial stiffness and strength to the arrangement of carbon atoms in oriented graphitic sheets. On a coarser scale, typically w1 mm, lots of fibers are bundled into yarns or tows. Within the finished composite, each tow behaves as a highly anisotropic solid entity, with far greater stiffness and strength along its axis than in transverse directions. Because tows are rarely packed in straight, parallel arrays, stresses and strains often possess strong variations from tow to tow. Thus, composite mechanical properties such as elasticity can only be considered approximately uniform on scales that are even larger still, say w10 mm or higher, where the effects of the heterogeneous structure at the tow level are averaged out. Finally, the textile forms part of an engineering structure, perhaps the stiffened skin of a wing or fuselage. As the engineering structure itself usually has some dimensions as small as w 10 mm, the fabrication of the composite material and the fabrication of the engineering structure may no longer be considered distinct operations. To fabricate the textile composite is to fabricate the structure. Fig. 5.4 introduces the most important groups of textile forms that are candidates for airframes, many of which were investigated by NASA. The left column (weaving, braiding, etc.) categorizes textiles according to the machines and processes used in creating them. High-performance composite structures have been created using all the processes listed, conventional textile machinery having been modified in many cases to handle the high modulus fibers needed in airframes and to reduce costs through automation. The unit cells of different weave types are shown in Fig. 5.4(c).

5.2.2.3

Nanoenhanced matrix

In the suggested approach, the effective nanocomposite (or enhanced matrix) material properties, where silica nanoparticles are analytically infused in the matrix. The analysis approach uses well-known MorieTanaka formulation for calculating the anisotropic nanocomposite properties from isotropic matrix and nanoparticle properties (stiffness, aspect ratio, and volume fraction). For a composite material reinforced with aligned fiber-like particles, the Tandon and Weng (1984) prediction of the moduli E11 (aligned particle direction), E22 (transverse to the aligned particle direction), the inplane shear modulus G12, and the out-of-plane shear modulus G23 of the composite are as follows: E11 ¼ Em=ð1 þ f ðA1 þ 2y A2Þ=AÞ p m

(5.1)

   E22 ¼ Em 1 þ f  2y A3 þ ð1  y ÞA4 þ 1ð1 þ y ÞA5A =ð2AÞ
p m m m

(5.2)


G12 ¼ Gm 1 þ f m = m  m

þ 2ð1  f ÞH p m p 1212 p m

(5.3)


G23 ¼ Gm 1 þ f m = m  m

þ 2ð1  f ÞH p m p 2323 p m

(5.4)

Advanced composite wind turbine blade design

Neat matrix

Chopped fiber-like nano-particles

Conventional fibers

213

Multi-scale composite

Close-up view of the enhanced matrix

Figure 5.5 Micrographs of enhanced matrix [26].

where A and Ai are constants depending on the components of the Eshelby tensor and the matrix/nanoparticles properties, and Hijkl are the Cartesian components of the Eshelby tensor. A closed-form analytical solution for the complete set of anisotropic elastic properties of the composite derived by Tandon and Weng (1984) by combining the Eshelby theory and the MorieTanaka model, as shown in Eqs. (5.1)e(5.4) is used to obtain the stiffness properties of the play with nanoparticles infused in the matrix. The analytical approach discussed earlier is used to calculate the effective stiffness for the lamina after infusing its matrix with silica nanoparticles. Fig. 5.5 shows how the neat matrix is infused with nanoparticles to enhance its structural properties. Once the lamina or laminate properties are updated, MS-PFA is then used to determine the strength of the composite ply with nanoparticles. This is done by assessing the failure mechanisms derived by Chamis. The ply is loaded to failure and the analysis detects laminate loading that produces damage matrix cracking and fiber failure.

5.2.3

Multi-scale progressive failure analysis

The evaluation of local damage due to cyclic loading is embedded in the composite mechanics module. The fundamental assumptions for cyclic fatigue are the following. Fatigue degrades all ply strengths at approximately the same rate. Fatigue degradation may be due to the following: (1) (2) (3) (4)

mechanical loading (tension, compression, shear, and bending), thermal stresses (elevated to cryogenic temperature), hygral stresses (moisture), and combined effects (mechanical, thermal, and hygral).

Laminated composites generally exhibit linear behavior to initial damage under uniaxial and combined loading. All ply stresses (mechanical, thermal, and hygral) are predictable by using linear laminate theory. The composite mechanics module with cyclic load analysis capability evaluates the local composite response at each node subjected to fluctuating stress resultants. The number of cycles required to induce local structural damage is evaluated at each node. After damage initiation, composite properties are reevaluated based on degraded ply properties and the overall structural response parameters are recomputed. Iterative application of this computational procedure results in the tracking of progressive damage in the composite structure subjected to cyclic load increments. The number of cycles for damage initiation and the number of cycles for structural fracture are identified in each simulation. After damage initiation, when the number of load cycles

214

Durability of Composite Systems

As-build/as-is

Residual stress distortion

Manufacturing defects • Matrix void shape/size/distribution • Thickness effect • Residual stress • Fiber waviness • Resin rich

Manufacturing curing and distortion • Matrix shrinkage vs. Temp/pressure • Modulus/CTE vs.temp • Viscosity vs.temp • Fiber volume vs.temp • Thickness vs.temp

Damage evolution Trans laminar failure • Matrix crack density • Matrix failure (L/T) •Tension •Compression •Shear • Inter phase • Fiber failure (L/T) • Long compression •F/matrix delamination • Fiber micro buckling • Fiber compression • Shear kink band • Ply failure • Tension • Compression • Shear

Fracture evolution Interlaminar failure

Propagation Fracture initiation Fracture propagation

• Interlaminar shear • Crack path • .005 inch • Interlaminar tension •2-d • Relative rotation •3-d • Edge declamination

• Gic • Gllc • Mixed mode

Residual strength

Figure 5.6 Multiscale damage (translaminar and interlaminar), and fracture (modes IeII) in MS-PFA.

reaches a critical level, damage begins to propagate rapidly in the composite structure. After the critical damage propagation stage is reached, the composite structure experiences excessive damage or fracture that causes its collapse. Iterative application of this computational procedure results in the tracking of progressive damage in the composite structure subjected to cyclic load increments. The damage and fracture tracking are decomposed from the global structural level to the microscale level. The stresses and strain at the microlevel are calculated using a mechanics-of-material approach from the finite element analysis (FEA) results of the macro-mechanical analysis at each load increment. Displacements, stresses, and strains derived from the structural scale FEA solution at a node or element of the finite element model are passed to the laminate and lamina scales using laminate theory. Stresses and strains at the microscale are derived from the lamina scale using microstress theory. The latter are interrogated for damage and fracture using a set of failure criteria components of which are listed in Figs. 5.6 and 5.7. This analysis is performed progressively, enabling the analysis of damage initiation and progression including fracture initiation on a microlevel.

5.2.4 5.2.4.1

Fracture mechanics Virtual crack closure technique

To further access the crack propagation or delamination into the ply drop coupon, the VCCT was introduced to this study. VCCT is a fracture mechanics-based approach to study crack propagation, which involves computing strain energy release rates ðGI ; GII ; GIII Þ and comparing these values to their corresponding critical values

Advanced composite wind turbine blade design

215

3 1

4

5

2

Y

X

Figure 5.7 Schematic for VCCT

ðGIC ; GIIC ; GIIIC Þ, where I, II, and III correspond to mode I, mode II, and mode III crack propagation modes, respectively. From a finite element perspective, VCCT determines the strain energy release rates from the nodal forces and displacements, thus not adding any complexity to the finite element formulation. The VCCT has been performed using the local coordinate system, based on the geometric relationships among the nodes surrounding the crack and the tip of the crack itself, to facilitate the separation of the different fracture models. Fig. 5.1 illustrates the scheme behind VCCT. The basis behind VCCT is an interface element based on the modified crack closure integral. The nodes for this element are numbered in a manner such that nodes 3 and 4 are located behind the crack, nodes 1 and 2 are located at the crack tip, and node 5 is ahead of the crack. To determine the nodal forces at the tip of the crack, a stiff spring is essentially placed between nodes 1 and 2. Nodes 3e5 do not contribute to the stiffness matrix used to calculate the nodal forces; however, nodes 3 and 4 are used to determine information concerning the opening of the crack behind its tip while node 5 carries information about the jump length in front of the crack tip. All this information combined is used to calculate the strain energy release rates. For a 2-D model, the mode I and mode II strain energy release rates can be expressed as follows: GI ¼

Y12 v34 2BDa

GII ¼

X12 m34 2BDa

where GI and GII are the mode I and mode II strain energy release rates, respectively, Y12 and X12 are the nodal forces in the X and Y directions for nodes 1 and 2, m34 and v34 correspond to the X and Y displacement, respectively, between nodes 3 and 4, Da is the crack extension, and B is the thickness of the model. The fracture criteria used to determine crack initiation and propagation based on the computed strain energy release rates is 

GI f¼ GIC

a



GII þ GIIC

b 1

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Durability of Composite Systems

where f represents the crack growth parameter. According to Ref. [1], the exponents a and b are assumed to be 1. Once the crack growth parameter f  1, the stiffness matrix associated with the interface element is set equal to zero and crack initiation or propagation occurs. For additional information concerning the numerical methods of VCCT, please see the references. VCCT can be used with MS-PFA providing some knowledge of the location for crack initiation, and the path of crack propagation is provided. This information can be obtained experimentally, through a preliminary MS-PFA, or based on user experience. As VCCT does not add any complexity to the finite element formulation, the need for extensive mesh preparation is eliminated.

5.2.4.2

Discrete cohesive zone modeling

An additional method to potentially access the crack propagation or delamination into the ply drop coupon is known as discrete cohesive zone modeling (DCZM). DCZM, like VCCT, is also a fracture mechanics-based approach to study crack propagation. This particular method is noted for its ability to simulate crack initiation and propagation even when various material nonlinearities are present, where VCCT is mostly used when linear elastic materials are present. DCZM essentially implements a discrete spring foundation at the process zone that is attached to the interfacial node pairs of the surfaces to be separated. In other words, a nonlinear spring type interface element is placed between interfacial nodes to model the cohesive effects between the surfaces to be separated or decohered. Fig. 5.8 illustrates this concept. As can be seen in Fig. 5.9 DCZM uses a triangular cohesive law for mixed-mode failure analysis. The triangular form of the cohesive law is dependent on the corresponding cohesive strength and stiffness. Cohesive strength is the strength that causes the virtual spring elements’ stiffness to decrease to a point where they begin to simulate nonlinear responses of adhesives. The cohesive stiffness is the initial stiffness of these spring elements before reaching this nonlinear state. In Fig. 5.9, s1c ; s2c ; and s3c correspond to the tensile (mode I fracture), shear (mode II fracture), and twisting (mode III fracture) cohesive strengths respectively, d1m ; g2m ; and g3m correspond to the maximum crack tip separation for a corresponding fracture mode, and d1c ; g2c ; and g3c correspond to

σ σG K IF

G

δ

Figure 5.8 DCZM virtual spring elements.

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217

τ2(MPa)

σ1(MPa) τ2c

σ1c

τ3c

GII

GI δ1(mm)

GIC δ1c

τ3(MPa)

δ1m

GIIC γ2c

γ2(mm) γ2m

GIII

γ3(mm)

GIIIC γ3c

γ3m

Figure 5.9 DCZM triangular cohesive law for mixed-mode failure analysis.

the crack tip separation at the associated cohesive strength for a corresponding fracture mode. For a detailed explanation concerning the interface element, equations for the cohesive stiffness, cohesive strength, and the overall construction and implementation of the cohesive law for a 2-D case please see reference. Crack propagation is controlled through the sequential releasing of nodes along a user-defined crack path. This takes place when the strain energy release rates ðGI ; GII ; GIII Þ exceed their corresponding critical values ðGIC ; GIIC ; GIIIC Þ. The comparison between the strain energy release rates and their associated critical values is performed using either the BeK (BenzeggagheKenana) or Power Law, which is also user-defined. As with VCCT, DCZM can be used with MS-PFA, therefore providing some knowledge of the crack propagation path is known. Once again, this information can be obtained experimentally, through a preliminary MS-PFA, or based on user experience.

5.2.5

Probabilistic and reliability analysis

With the direct coupling of composite micro- and macromechanics, structural analysis, and probabilistic methods, it is possible to simulate uncertainties in all inherent scales of composites, from constituent materials to the whole structure and its loading conditions. The evaluation process starts with the identification of the primitive variables at the micro- and macrocomposite scales including fabrication. These variables are selectively perturbed to generate a database for determining the relationships between the desired materials’ behavior and/or structural response and the primitive variables. The approach for probabilistic simulation is shown in Fig. 5.10. Composite micromechanics is used to carry over the scatter in the primitive variables to the ply and laminate scales (Fig. 5.10). The laminate theory is then used to determine the scatter in the material behavior at the laminate scale. This step leads to the perturbed resultant force/moment-displacement/curvature relationships used in the structural analysis. Next, the finite element analysis is performed to determine the perturbed structural responses corresponding to the selectively perturbed primitive variables. This completes the description of the hierarchical composite material/structure synthesis shown on the left side of Fig. 5.10. The multiscale progressive decomposition of the structural response to the laminate, ply, and fiber-matrix constituent

218

Durability of Composite Systems

Component Finite element

Structural analysis

D

E Loads, geometry boundary conditions

Laminate theory

C

F Probabilistic composite mechanics

B

P

σ

A

Laminate theory

Multiscale progressive decomposition

Ply response

Ply properties

Composite micromechanics theory

Structural analysis

Laminate response

Probabilistic structural analysis

Laminate properties

Composite structure synthesis

Finite element

G

Composite micromechanics theory

T Micromechanics t unit cell

M

P = f(σ, T, M, t)

Nonlinear multifactor interaction model for constituent properties

Figure 5.10 Technical approach for probabilistic evaluation of wind blade composite structures.

scales is shown on the right side of Fig. 5.10. After the decomposition, the perturbed fiber, matrix, and ply stresses can be determined. MS-PFA can be coupled with optimization and probabilistic methods [2] to deliver a design that is affordable, durable, and reliable. However, relying on traditional computational simulations to perform robust design can be impractical due to the level of computation involved. Designers can use effectively the sensitivity analysis to identify influential material and fabrication variables that produce scatter in the blade failure load. For the present case, MS-PFA was validated for static test simulation of the blade. Then, the code evaluated the weight and D&DT contribution of key materials used in the blade. Probabilistic sensitivity analysis identified the material and properties that influence the failure load. Weight was finally reduced by iterating on the percent of foam volume that can replace some of the materials without affecting the durability of the blade.

5.2.6

Certification approach

With extensive modeling and simulation approaches, analysis tools are capable of detecting aircraft certification categories of damage for primary aircraft structures. Damage on commercial aircraft can fall into one of the five Federal Aviation Administration (FAA)-designated five categories (Table 5.2): (1) BVID (barely visible impact damage); (2) VID (visible impact damage); (3) obvious damage seen in walk-around

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219

Table 5.2 FAA categories of damage/defects for primary composite aircraft structures. Category

Examples

Safety considerations

Category 1: Damage that may go undetected by field inspection methods (or allowable defects)

BVID, minor environmental degradation. Scratches, gouges. Allowable mfg. defects

Demonstrate reliable service life; retain ultimate load capability; design-driven safety

Category 2: Damage detected by field inspection methods @ specified intervals (repair scenario)

VID (ranging small to large), mfg, defects/ mistakes, major environmental degradation

Demonstrate reliable inspection; retain limit load capability; design, maintenance. mfg,

Category 3: Obvious damage detected within a few flights by operations focal (repair scenario)

Damage obvious to operations in a “walkaround” inspection or due to loss of form/fit function

Demonstrate quick detection; retain limit load capability; design, maintenance, operations

Category 4: Discrete source damage known by pilot to limit flight maneuvers (repair scenario)

Damage in flight from events that are obvious to pilot (rotor burst birdstrike, lightning)

Defined discrete-source events; retain “get home” capability; design, operations, maintenance

Category 5: Severe damage created by anomalous ground or flight events (repair scenario)

Damage occurring due to rare service events or to an extent beyond that considered in design

Requires new substantiation requires operations awareness for safety (immediate reporting)

Courtesy of FAA. AMTAS spring 2006 meetind (April 11, 2006).

inspection, critical in-flight damage known to pilot such as (4) discrete source damage and (5) severe damage resulting in a get-home-safe scenario and damage due to rare unexpected event. In composite commercial aircraft panels, damage may or may not be visible on the surface but will involve delamination, fiber breakage/buckling, matrix cracking, and join-up of damaged areas. NDE inspection methods are sometimes used. The aerospace industry has attempted to predict the five categories of damage by progressive damage analysis to reduce test at every step and to obtain type certificate by analysis. Fig. 5.11 shows the certification on analysis supported by coupon and component test evidence in compliance with guidelines issued by the FAA [3]. Fig. 5.12 shows the product test sequence including the test conditions and damage types: (A) Apply small damages 1. 60% design limit strain surveydsix conditions; 2. Fight spectrumdone lifetime;

220

Durability of Composite Systems

Topic

Reg

Subject

Material and process

25.603

• Control of materials

specifications

25.605

• Fabrication methods

Material properties

25.613

• Material strength properties and design values

Proof strength

Damage tolerance

25.615

• Design properties

25.619

• Special factors

25.305

• Strength and deformation

25.307

• Proof of structure

25.571

• Fatigue evaluation • Residual strength • Disccrete source damage

Other

25.581

• Lightning protection

25.609

• Protection of structure

Figure 5.11 FAA test rules/recommendations.

“Visible” damages “Element” damages

“Small” damages

1

2

1

2

3

4

5

6

3

Repair visible & element damages

7 8 9 10 11

Figure 5.12 Product test sequence. 3. 60% design limit strain surveydthree conditions; 4. Fatigue spectrumdone lifetime including load enhancement factor; 5. Design limit strain surveydsix conditions; 6. Design ultimate loadsdthree conditions); (B) Apply visible impact damages 7. Fatigue spectrumdincluding load enhancement factor; 8. Failure safe (limit) loadsdthree conditions); and (C) Apply element damages 9. Get home loads (approximately 70% limit)dthree conditions; 10. Design ultimate loadsdthree conditions; 11. Destruction test

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221

Loads for residual strength substantiation, FAR25.571: The residual strength evaluation must show that the remaining structure can withstand loads (considered as static ultimate loads) corresponding to the following conditions: 1. The limit symmetrical maneuvering conditions specified in Sec. 25.337 at all speeds up to Vc and in Sec. 25.345. 2. The limit gust conditions specified in Sec. 25.341 at the specified speeds up to Vc and in Sec. 25.345. 3. The limit rolling conditions specified in Sec. 2. 4. 5.349 and the limit unsymmetrical conditions specified in Secs. 25.367 and 25.427(a) through (c), at speeds up to Vc. 5. Limit yaw maneuvering conditions specified in Sec. 25.351 (a) at specified speeds up to Vc. 6. For pressurized cabins, the conditions are as follows: (i) The normal operating differential pressure combined with the expected external aerodynamic pressures applied simultaneously with the flight loading conditions specified in paragraphs (b) (1) through (4) of this section, if they have a significant effect. (ii) The maximum value of normal operating differential pressure (including the expected external aerodynamic pressures during 1-g level flight) multiplied by a factor of 1.15, omitting other loads.

A typical building block approach required for composite structures certification is shown in Fig. 5.13. The certification process is defined in five steps. The traditional approach to the component, part, and assembly development has been to test materials to obtain the necessary “material allowable,” which then serves as the input to design and analysis software that calculates the ultimate analytical performance of the final structure. This methodology is commonly called the “building block testing approach” and has proven to be successful and reliable when moving from simple material coupon to the final assembled part (Fig. 5.13). Fig. 5.13 shows the role of analysis by virtual testing to establish a building-block testing strategy to (1) guide the MS-PFA analysis processes

Subcomponents

- Non-generic

Structural features

sam

ple

Requires minimum validation

Configuration

eas

Generic specimens

Incr

Calibration process

ing

Verification process

Details

Elements Coupons

n esig of d tion ses gra ces Inte & pro

specimens

size

Defines risk mitigation

Components

Data base

Figure 5.13 Type certification strategy of five steps of building-block testing approach: (a) coupon; (b) elements; (c) details; (d) subcomponents; (e) components.

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Durability of Composite Systems

integration and design processes; (2) identify causes when failure to meet performance requirements occurs; (3) benefit certification process by establishing reduced test plan at each level (use reversed Taguchi: regression analysis); and (4) consider scattering in geometry, manufacturing, and material levels. The experimental approach has also proven to be expensive and time consuming. This cost may be tempered by the integration of the building block approach with “virtual testing.” More importantly, together these two techniques can introduce new materials in structures earlier in the development process. Note also that there are usually many replicates, many test orientations (for composite materials), many batches of processed materials, various environmental effects (temperature, humidity, fluid exposure, moisture content, etc.), and other factors for even a few test configurations. As the sample configurations get larger, taking on more “structural features” associated with ultimate geometry and design aspects, the samples get larger geometrically. The tests get more sophisticated in terms of loading and load levels induced. Finally, as one moves up to the ultimate test “sample”dfor example, the full structural component configurationdthe number of test samples decreases, but the test cost (fabrication of the test part, testing costs, reporting, etc.) increases exponentially. Consequently, the cost of testing per the “building block” approach is costly if one desires to develop and qualify new advanced materials technologies into the mainstream of future applications in shorter time spans. It is certainly important to develop appropriate “material allowables” for design applications as well as to qualify final structural components and full-scale structures for their intended ultimate use. In recent years, “virtual testing” has increased in use to try new materials in structures earlier in an application timeline. Virtual testing is made possible by conducting progressive failure analysis and combining those results to predict structure/component safety based on the physics and micro/macromechanics of materials, manufacturing processes, available data, and service environments. The approach takes progressive damage and fracture processes into account and accurately assesses reliability and durability by predicting failure initiation and progression based on constituent material properties. An example is shown in Fig. 5.3. Such approaches are becoming more widespread and economically advantageous in some applications.

5.3

Wind blade design technique and analysis

Current polymer matrix composite (PMC) wind turbine PMC blade design is driven by high factors of safety. These cover unknowns in material properties and strengths, analysis methods simplifications, manufacturing tolerances, and anomalies as well as uncertainties in the design load envelope. Cost and time constraints have limited the material and structural testing as well as nondestructive inspection (NDI). High design factors of safety are used instead to ensure adequate wind turbine blade

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223

performance. Fall-outs of high factors of safety are higher weight and thus larger gravitational loads, as well as possibly more expensive structures. An alternative design approach is to utilize a CBA method. This involves a building-block approach, integrating materials, and structural testing with advanced strength and life prediction analysis methods, to determine an optimum weight/cost turbine polymer matrix composite blade design that is driven by D&DT requirements. A CBA approach minimizes testing, NDI, and active structural health monitoring through the use of sophisticated D&DT analysis methods. This chapter will demonstrate the ability of MS-PFA advanced structural residual strength and life analysis approach to predict the static and fatigue load response of a current Sandia wind turbine blade design to its design loads/environment envelope. Employing advanced D&DT methodology, time-dependent reliability analysis, and micromechanics-based progressive failure analysis will be used to validate the current Sandia wind turbine blade design against laboratory and system dynamics modeling data. The Sandia wind turbine blade concept will then be reoptimized with the validated proposed methodology to achieve a light-weight, low-cost robust design (maximum durability, reliability, and longevity) that has an optimum stiffness distribution for aeroelastic and load requirements. The design approach emphasized analytic approaches to reduce the current high design-to factors of safety and minimize nondestructive testing and real-time structural health monitoring. A blade design from Sandia National Lab is selected to demonstrate certification by analysis capability. The Sandia Blade System Design Study (BSDS) blade is a subscale research blade that was developed to examine several design innovations that had the potential to increase the structural efficiency of utility-scale blades. The blade is 9 m in length and was designed nominally as a 100-kW blade [4]. AlphaSTAR demonstrated under this grant certification by analysis capability using geometry, load, and test data from Sandia National Lab obtained through the BSDS program. Other design features of the BSDS blade include a carbon fiber spar cap, embedded root studs, and high-performance outboard airfoils. A schematic of the major blade laminate regions is shown in Fig. 5.14. The blade is predominately glass/epoxy with unidirectional glass in the root and a biaxial glass/balsa sandwich structure throughout most of the outboard region. The narrow carbon/glass hybrid spar cap is seen to extend for the entire length of the blade. The blade was manufactured by first laying up dry fiber and core in skin and shear web molds. The dry fiber was then infused with epoxy using a vacuum-assisted resin transfer mold (VARTM) process and cured at elevated temperature and pressure. The shear web was then glued to the low-pressure skin as bucking occurs on this surface and thus is the most critical bond. Finally, the high-pressure skin was glued to the low-pressure skin at the leading and trailing edge, along with the shear web using a blind adhesive joint. The laminate construction and adhesive joints are shown in Fig. 5.15. The materials used are described in Table 5.3. Fig. 5.16 shows the blade finishing from Ref. [4].

224

White gelcoat (0.381-0.505 mm thick) (0.15-.02 in thick)

Tape off 1/2’ at the root Before spraying gelcoat

3/4oz mat Leading edge

LEADING EDGE

Trailing edge TRAILING EDGE 0000 0000

1000

2000

3000

4000

5000

6000

7000

8000

1000

2000

3000

5000

6000

7000

8000

9000

Layer 3-- ¾ 0z material

Layer 1- white gelcat

Dbm-1708 (Mat up)

At-prime (About 0.127 mm thick) (about 0.005 in thick)

Tape off 1/2’ at the root Before applying at-prime

4000

9000

Leading edge

Leading edge Trailing edge Trailing edge 0000

0000

1000

2000

3000

4000

5000

6000

7000

8000

1000

2000

9000

3000

4000

5000

6000

7000

8000

Layer 4 - DBM 1708 material

Layer 2 - AT-prime adhesion

Le flanges

1 layer of c260 Root filler on each Side of the carbon

The two c260 Layers would meet At station 1200

1 layer of triax

Sta 7875

Leading edge

Leading edge

Insert the 1/2’ plywood Root dam before lay-up

0000

1000

2000

3000

4000

5000

6000

7000

8000

0000

1000

2000

3000

4000

5000

6000

7000

9000

Layers 5 & 6 - seartex triax and C20 Te flanges

Install return flanges/plywood root DAM

Figure 5.14 BSDS blade planform with major laminate regions [4].

8000

9000

Durability of Composite Systems

Trailing edge Trailing edge

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225

Figure 5.15 BSDS blade assembly fixture [4].

Table 5.3 Materials used in BSDS construction. Material

Description

Area of use

DBM-1708/ DBMe1208


45 degrees stitched glass with chopped glass mat backing

Blade skins, shear web, and leading edge

C520/C260/ ELT5500

96% 0 degrees, 4% 90 degrees stitched glass

Blade skin root

Woven rug


45 degrees woven glass

Blade skin root

Carbon triax

0 degrees carbon stitched with 45 degrees and 45 degrees glass facings

Spar cap

Balsa

e

Outboard blade skin panels and shear web

Figure 5.16 BSDS blade finishing [4].

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5.4

Durability of Composite Systems

Results and discussion

5.4.1 5.4.1.1

Material modeling calibration and validation Static properties calibration and validation

As a first step in simulating the behavior of laminated composites with complex geometric features, the fiber and matrix mechanical properties were calibrated using test data available from Refs. [5,6]. The referenced data pertain to lamina and laminate properties for glass or carbon-based composite material systems. The calibration process is essential for determining root cause problem for composite damage and failure. Effective properties, calculated using the technical approach described in the previous section, are listed in Table 5.4 for the glass-epoxy material system. Note that the test data available in the reference were provided for [0/90] laminate. The in situ fiber and matrix properties were reverse engineered as the [0/90] laminate properties were reproduced. Table 5.5(a) shows the characterization of carbon epoxy (NCT307-D1-34-600) for [0]10 carbon/epoxy prepreg with 47% fiber and 2% voids. Table 5.5(b) shows the characterization for carbon epoxy (NCT307-D1-34-600) for [0]10 carbon/epoxy prepreg laminate with 53% fiber and 2% voids. For each material, the in situ material properties were derived using the approach described earlier in this chapter.

5.4.2 5.4.2.1

Tapered blade analysis and results Failure prediction and test validation of tapered composite under static and fatigue loading

Strain energy release rate Characterization of delamination growth was performed using the strain energy release rate, which is the energy dissipated per unit area of delamination growth. The energy that must be supplied to a crack tip for it to grow must be balanced by the amount of energy dissipated due to the formation of new surfaces and other dissipative processes such as plasticity. For problems involving cracks that move in a straight path, the stress intensity factor (K) is related to the energy release rate (G). Stress intensity (K) in any mode situation is directly proportional to the applied load on the material. These load types are categorized as modes I, II, and III (Fig. 5.17). In the blade structure, the mode II is the prevailing one. mode II is sliding or an in-plane shear mode where the crack surfaces slide over one another in a direction perpendicular to the leading edge of the crack.

Experimentation The experimental work [7] was carried out by the Department of Chemical and Biological Engineering, Montana State University as part of the DOE/MSU Composite Material database [4]. The database, maintained in cooperation with Sandia National Laboratories [5], is a collection of static and fatigue tests of a wide variety of materials used in wind turbine blades (Table 5.6).

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227

Table 5.4 Calibrated glass epoxy with fiber content of 39% and void content of 2% material: NB307-D1 7781 497A E-glass with epoxy. Property

Unit

E-glass calibrated

Epoxy calibrated

Lamina calibrated

Laminate [0/90] references [5,6]

E11

[GPa]

62.27

4.95

19.19

19.2

E22

[GPa]

62.27

d

19.19

19.2

E33

[GPa]

d

d

11.76

d

v12

[e]

0.22

0.215

0.13

0.13

v23

[e]

0.22

d

0.26

d

v13

[e]

d

d

0.26

d

G12

[GPa]

9.4

2.037

3.95

3.95

G23

[GPa]

9.4

d

3.43

d

G13

[GPa]

d

d

3.43

d

Tensile strength (11)

[MPa]

1408

d

337.01

337

Compressive strength (11)

[MPa]

1408

d

496.89

497

Tensile strength (22)

[MPa]

d

d

337.01

337

Compressive strength (22)

[MPa]

d

d

496.89

497

Shear strength (12)

[MPa]

d

d

115.1

115

Matrix tensile strength

[MPa]

d

80

d

d

Matrix compressive strength

[MPa]

d

357

d

d

Matrix shear strength

[MPa]

d

179

d

d

Material systems Panels containing ply drops were infused under vacuum through two flow medium layers and one peel ply layer on the top and the bottom surfaces of the laminate. Table 5.7 gives the main properties of the four studied laminates and Fig. 5.18 illustrates the stress/strain curves of these specimens. The nominal fiber volume fraction for the ply drop panels was 54%, giving a thin-side and thick-side panel thickness of 13.7 and 11.5 mm, respectively.

228

Table 5.5 Comparative evaluation of analysis simulation with MS-PFA and test mechanical properties for various laminates without ply drops. (a) E-glass with epoxy (NCT 307-D1-E300) ply property for [0]10 E-glass/epoxy prepreg with 47% fiber and 2% voids Longitudinal direction Elastic constants NCT307-D1-E300 E-glass with epoxy

EL

ET

GPa

GPa

Test

35.5

8.33

Transverse direction

Tension

Compression

Shear

Tension

GLT

UTSL

εmax

UCSL

εmax

s

vLT

GPa

MPa

%

MPa

%

0.33

4.12

1005

2.83

788

2.22

Compression

UTST

εU

UCST

εU

MPa

MPa

%

MPa

%

112

51.2

0.74

TU

2.02 168

GENOA

35.5

8.33

0.33

4.12

1001.21

2.82

791.86

2.23

112.03

51.05

0.61

2.02 168.58

(b) Carbon epoxy (NCT307-D1-34-600) ply property for [0]10 carbon/epoxy prepreg with 53% fiber and 2% voids Longitudinal direction

NCT307-D1-34-600 carbon with epoxy

EL

ET

GPa

GPa

Test

123

8.2

GENOA

123

8.2

vLT

Tension

Compression

Shear

Tension

Compression

GLT

UTSL

εmax

UCSL

εmax

sTU

UTST

εU

UCST

εU

GPa

MPa

%

MPa

%

MPa

MPa

%

MPa

%

0.31

4.71

1979

1.32

1000

0.9

103

59.9

0.76

223

2.72

0.31

4.709

1968.32

1.6

1009.2

0.82

103.56

59.91

0.73

222.93

2.72

Durability of Composite Systems

Elastic constants

Transverse direction

Advanced composite wind turbine blade design

Mode Ι: Opening

229

Mode ΙΙ: In-plane shear

Mode ΙΙΙ: Out-of-plane shear

Figure 5.17 Crack opening modes. Table 5.6 Higher order ASTM-based coupon verification. Verification Test description

ASTM

Double notched compression (interlaminate shear)

D3846

Four-point bending

D6272

Short beam bending

D2344

Flatwise tension

C297

Open-hole tension/compression

D766 (tension) D6484 (compression)

Compact tension

E1802

Iosipescu (in-plane shear)

D5379

Table 5.7 Properties of four laminate types. Resin

Fiber content (%)

Thickness (mm)

UTS (MPa)

Strain at UTS (%)

Initial E (GPa)

EP-1

44

4.57

168

2.4

13.4

UP-1

44

4.52

175

2.4

14.3

VE-1

46

4.21

160

3.1

17.0

VE-2

44

4.54

156

2.5

15.2

Longitudinal tensile test [7]: The complex coupon with ply drops employs an unsymmetrical geometry shown in Figs. 5.1 and 5.2. This test method required significant test development to arrive at a lay-up and dimensions that would have minimal bending, be compatible with testing machine (250 kN) capacity and grip capacity while representing blade materials and structure of current interest (Fig. 5.19). The lay-up chosen allows convenient infusion with a variety of resins of interest for blades and features failure modes including delamination at the ply drops, damage in the
45 degrees surface layers (which represent blade skin materials) and load

230

Durability of Composite Systems 200 180

Stress (MPa)

160 140 120 100

EP–1

80

UP–1

60

VE–1

40

VE–2

20 0 0

0,005

0,01

0,02 0,015 Strain

0,025

0,03

0,035

Figure 5.18 Stress/strain curves of four laminate types.

Figure 5.19 Longitudinal tensile test.

redistribution between the surface skins and primary structural 0 degree plies as damage develops and extends. The finite elements model (Fig. 5.20(a)) contains 24,204 elements and 30,563 nodes. The applied loads and boundary conditions (Fig. 5.20(b)) simulate a simple longitudinal tensile test.

(a)

(b)

Geometry

Figure 5.20 Finite element model.

Loads and boundary conditions

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231

Simulation results The methodology and its various failure criteria and material-degradation submodels were compared and assessed by performing analyses for four material (fiber/matrix) systems: EP-1, UP-1, VE-1, and VE-2 combined with E-glass. Constituent properties of the composite laminates are derived by modeling the actual coupon tests and comparing simulation results with measured results. An optimization process is used to select the constituent properties that best fit the test data. Table 5.8 describes the resin details for polyester, vinyl ester, epoxy, and toughened vinyl ester [7], all used with glass fabrics. The calibrated SeS curves are shown in Fig. 5.21, and the SeN curves for fatigue tests are shown in Fig. 5.22. The purpose of this effort is to compare composite failure predictions of MS-PFA against tests. Test data were collected for four tapered laminates corresponding to longitudinal tension. The material (fiber/matrix) constituent properties were calibrated using a material modeling approach. Note that calibration is not required if actual fiber and matrix properties are known. Four tests were simulated MS-PFA capabilities corresponding to each material system: EP-1, UP-1, VE-1, and VE-2. An initial crack was modeled in the resin-rich area. Test results [7] and Fig. 5.23 show that damage and crack initiate at the bottom of the ply drop. This is caused by a stress concentration at this location due to a higher displacement of the continuous plies compared to cut-off plies: Delamination mode II. Then, the crack propagates along the delamination path between continuous plies and cut-off plies. Fig. 5.24 shows a comparison between simulated crack opening and test results. The most vulnerable lay-up to delamination is the one where the crack is located at the interface between two 0-degree plies (Fig. 5.24, green circle), although another crack is to be considered, at the transition between 0-degree plies and
45 degrees, on the inner side of the wrapping. The length of the crack was investigated according to the applied load on the test specimen. Fig. 5.25 illustrates the good correlation between simulation and tests for the four material systems. In most cases, it appears that the propagation rate is slow in the beginning and rapidly growing until catastrophic failure of the laminate.

Static simulation results The finite element models of flat specimen and ply drop off specimen are shown in Figs. 5.27 and 5.28. The FEM contains 24,204 elements and 30,563 nodes. The applied loads and boundary conditions (Fig. 5.28) simulate a simple longitudinal tensile test. Table 5.8 Description of various resin systems [7]. Resin

Resin details

EP-1

Hexion MGS RIMR 135/MGC RIMH 1366

UP-1

Hexion/uPICA TR-1 with 1.5% MEKP

VE-1

Ashland Derakane Momentum 411 with 0.1% CoNap, 1.0% MEKP and 0.02 phr 2,4-Pentanedione

VE-2

Ashland Derakane 8084 with 0.3% CoNap and 1.5% MEKP

232

(b)

180

180

160

160

140

140

120

EP–1 UP–1 VE–1 VE–2

100 80 60

Stress (MPa)

Stress (MPa)

(a)

120 100

Genoa

60

40

40

20

20

0

Test

80

0 0

0.02

0.04

0.06 0.08 Strain

0.10

0.12

Figure 5.21 Simulation of EP-1 matrix.

0

0.01

0.02

0.03

Strain

Test vs. simulation

Durability of Composite Systems

Calibrated matrix S-S curves

0.14

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233

180 160

Stress (MPa)

140 120

EP–1

100

UP–1

80

VE–1 VE–2

60 40 20

7 1, E+ 0

6

5

1, E+ 0

E+ 1,

1, E+ 0

04

3 1, E+ 0

2 1, E+ 0

01 E+ 1,

1,

E+

00

0

Cycles

Figure 5.22 Calibrated matrix SeN curves.

(a)

(b)

Von misses stresses

Damage initiation locations

Figure 5.23 Crack initiation.

(a)

(b) Crack propagation

Delamination path

Simulation

Test [1]

Figure 5.24 Crack propagation.

Static tests using building block validation strategy: D&DT of flat laminate coupons (without ply drops): PFA is used to predict failure loads and modes for flat coupons made from two material systems. The 0-degree plies are made from carbon fibers and epoxy matrix while the
45 degree plies are made from the E-glass epoxy 2-D weave system. The finite element model used in this evaluation is shown in Fig. 5.27. Table 5.9 lists the stiffness (modulus) and strength prediction of the [
45/08/
45] laminates under tensile and compressive loading conditions.

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Durability of Composite Systems

(b)

EP–1

UP–1

30

35

25

30

20 Test GENOA

15 10 5

Crack length (mm)

Crack length (mm)

(a)

25 20 10

0

5 0

0

25

50

100

75

125

150

0

25

50

75

Load (kN)

Load (kN)

Material EP–1

Material UP–1

(c)

(d)

VE–1

35

30

30

25 Test Test2 GENOA

20 15 10 5 0

100

125

VE–2

35 Crack length (mm)

Crack length (mm)

Test Test2 GENOA

15

25 Test Test2 GENOA

20 15 10 5 0

0

25

50

75

100

125

150

0

25

50

75

100

Load (kN)

Load (kN)

Material VE–1

Material VE–2

125

150

Figure 5.25 Crack length versus loaddsimulations versus experimental tests [7]. Failure durability of tapered composite laminates under static and fatigue loading The complex coupon with ply drops employs an unsymmetrical geometry shown in Fig. 4.10. This test method required significant test development to arrive at a lay-up and dimensions which would have minimal bending, be compatible with testing machine (250 kN) capacity and grip capacity, while representing blade materials and structure of current interest (Fig. 5.26). The lay-up chosen allows convenient infusion with a variety of resins of interest for blades, and features failure modes including delamination at the ply drops, damage in the
45 surface layers (which represent blade skin materials) and load redistribution between the surface skins and primary structural 0 plies as damage develops and extends. The same figure shows the finite element model used in the progressive failure analysis to determine load magnitude that produce damage initiation (matrix cracking and or delamination), damage progression, fracture initiation, and fracture propagation to failure.

The prediction results matched closely those of the test (difference ranged from 2.88% to 5.56%). The analysis was able to detect a test anomaly for the compression case as explained in Table 5.10. The strength reported for the laminate with mixed layups under compression is higher than that of the 0-degree ply. This is inconsistent with the physical behavior of composites. As layups other than 0 degrees are introduced in the laminate, the laminate strength is reduced to values lower than that of the 0-degree ply consistent with the physics, as reported by the PFA predictions. Evaluating coupon behavior without a ply drop is essential to establish a baseline. It will allow the assessment of loss in strength as the ply drops are introduced. D&DT of specimens with ply drops (complex laminate configuration) under tensile and compressive load: With the successful simulation of laminates for flat coupons, a specimen with complex geometric features such as ply drops is

0° layers 25mm

Plane of symmetry

Test specimen schematic [3] 203 mm

Advanced composite wind turbine blade design

FEA model generated for progressive failure analysis ±45° layers

25 mm Fiberglass tabs

89 mm

Layup details: [±45/02/09/02/±45] E-glass epoxy in ±45 piles (2 D weave) Carbon epoxy in 0 deg plies

25 mm

Figure 5.26 Schematic of test specimen with typical ply drops at the surface of 0-degree plies and finite element model generated for use in progressive failure analysis.

235

236

Durability of Composite Systems

±45° layers 0° layers

Figure 5.27 Finite element model used in laminate [
45/02/09/02/
45] progressive failure analysis (mixed layup without ply drop). r45 layer

resin area

0 layer

Figure 5.28 Shell finite element model for laminates with ply drops. Table 5.9 Static stiffness and strength (tension/compression) from simulation compared to test [5] without ply drop-[
45/08/
45]dcarbon epoxy 0-degree plies and E-glass epoxy 45 degrees plies. Laminate without ply drop lay-up: [±45/08/±45]

Tension modulus

Tension strength

Tension strain

Compressive strengtha

Compressive strain

GPa

MPa

%

MPa

%

Test

101

1496

1.4

1070a

1.04

MS-PFA

106.62

1511.5

1.426

780.19

1.01

Difference from test

5.56%

1.04%

1.86%

27.09%

2.88%

a

Test anomaly (reported test value is incorrect as the 0-degree unidirectional strength is 1000 MPa); With mixed layups, the laminate strength must be lower than that of the unidirectional lamina strength; MS-PFA predictions are consistent with the physics.

Table 5.10 Static stiffness and strength (tension/compression) from simulation compared to test [5] with ply dropdcarbon epoxy 0-degree plies and E-glass epoxy 45-degree plies. Laminate with ply drop

[
45/02a/09/02a/
45] Laminate with ply drop

[
45/02a/09/02a/
45] a

Dropped plies.

Compressive strength (MPa)

Compressive strain (%)

Test

MS-PFA

% Error

Test

MS-PFA

% Error

617

588.46

4.63

0.64

0.57

10.94

Tensile strength (MPa)

Tensile strain (%)

Test

MS-PFA

% Error

Test

MS-PFA

% Error

827

852.77

3.12

0.85

0.85

0

Advanced composite wind turbine blade design

(a)

Damage initiation stress: 161.52 MPa

(b)

237

(c)

Final failure stress: 852.77 MPa

Damage propagation stress: 161.52 MPa

Figure 5.29 Damage evolution process from PFA of specimen with ply drop under static tensile loading. (Red color identifies damaged elements, damage and failure modes: longitudinal tensile, compression and in-plane shear).

(a)

Damage initiation stress: 138.75 MPa

(b)

(c)

Damage propagation stress: 294 MPa

Final failure stress: 588.46 MPa

Figure 5.30 Damage evolution process from PFA of specimen with ply drop under static compressive loading. (Red color identifies damaged elements, damage and failure modes: longitudinal compression and in-plane shear).

evaluated under incremental static load. Ply drops effect at the surface of 0-degree plies (on top and bottom surfaces of the structure) was evaluated by progressive failure analysis for both tension and compression static loading. The finite element model generated for PFA analysis and depicting the section with the ply drops is presented in Fig. 5.28. The ply drop region had to be very carefully meshed to ensure converged FEA solution. The 45-degree plies are made from the 2-D weave E-glass epoxy material while the 0-degree plies are made from the carbon/epoxy material. Fig. 5.28 shows the schematic of a test specimen with typical ply drops. Figs. 5.29 and 5.30 show the damage evolution process and associated failure mechanisms of ply drop complex laminate configuration under tensile and compressive load. Compression and tension [
45/02/09/02/
45] laminate strain and strength results were obtained from the durability evaluation by PFA strength-based approach. The MS-PFA predictions ranged from 10.94% to 3.12% as compared to the test. Fatigue tests using building block validation strategy: In addition to determining failure modes, location, and loads under static loading, we evaluated the fatigue life of these laminates under service loading. Wind turbine blades must be designed for adequate handling of static loads as well as very high numbers of fatigue cycles. Fatigue is the progressive and localized structural damage that occurs when a material is subjected to cyclic (repetitive) loading. As a first step in the prediction process, it is important to characterize the fatigue properties of the constituents of the composite. Similar to the static application, we used reverse engineering principles to derive fatigue properties of the epoxy resin that can be used in simulating fatigue behavior of composite laminates.

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Durability of Composite Systems

Stress (MPa)

200 Reverse engineered fatigue property for use in GENOA

150 100 50 0 1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

Cycle, N

Figure 5.31 Epoxy resin SeN curve obtained by reverse engineering for use as input to PFA fatigue analysis. 2000

Stress (MPa)

1500

1000

Test [3] 500

GENOA PFA simulation

0 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Cycle, N

Figure 5.32 Fatigue life prediction compared to test for a [
45/04]s laminate without ply drop (carbon epoxy in 0-degree plies and e-glass epoxy for 45-degree plies and 0.1 stress ratio).

Fig. 5.31 shows the room temperature derived fatigue properties for the epoxy resin. It is assumed that the fiber undergoes little degradation while the resin experiences significant degradation under fatigue loading. The fatigue life of the laminated coupon was then evaluated (without ply drop) subjected to a tensile loading under a stress ratio of 0.1. The stress ratio is the ratio of minimum stress to maximum stress applied during cycling. The prediction results for various applied stress levels are compared to those from tests [5] in Fig. 5.32. As illustrated in the same Fig. 5.32, the predictions are in remarkable agreement with the reported test results. This indicates that the reverseengineered properties can be used with confidence in fatigue evaluation of any structure where the same material constituents are used including those with tapered laminates. VCCT/PFA of specimens with ply drops (complex laminate configuration) under compressive loading: For this study, VCCT/PFA was used to simulate crack initiation and propagation, for the same ply drop composite coupon previously discussed, subject to static compressive loading. Of interest were the compressive strength and compressive strain determined through the use of VCCT/PFA compared to both the previous PFA and experiment. To assure accurate results, it was paramount that the proper crack initiation point and propagation path were specified.

Advanced composite wind turbine blade design

239

Crack initiation point

Figure 5.33 Crack propagation path for compression test simulation using VCCT/PFA.

Pictured in Fig. 5.24(b) from Ref. [5] is the delamination pattern observed during a compression fatigue test of a similar ply drop composite coupon as the one used in this study. Pictured in Fig. 5.33 is the ply drop region of the finite element model used for the VCCT/PFA compression test simulation. Using the information gained from the previous PFA in Fig. 5.30 and that pictured in Fig. 5.24(b), the proper crack initiation point and propagation path were chosen accordingly as shown in Fig. 5.34. The fracture toughness values used in the VCCT simulations were G1c ¼ 364 J/m2 and G2c ¼ 1829 J/m2. Both values were obtained from Ref. [5]. The numerical results of the VCCT/PFA compression simulation are listed in Table 5.11 and compared to the experimental results obtained in Ref. [5]. Pictured in Fig. 5.34 is the damage evolution and crack propagation process at various stages throughout the VCCT/PFA compression simulation. Lastly, Figs. 5.35 and 5.36 display load versus deflection and load versus crack length, respectively, between Table 5.11 Static stiffness and strength (compression) from VCCT/PFA simulation compared to test from Refs. [5] with ply dropdcarbon epoxy 0-degree plies and E-glass epoxy 45-degree plies. Compressive strength (MPa) Laminate with ply drop [
45/02*/09/ 02*
45]

Compressive strain (%)

Test

GENOA VCCT/PFA

% Error

Test

GENOA VCCT/PFA

% Error

617

611.52

0.89

0.64

0.60

6.25

(a)

Crack initiation stress: 82.08 MPa

(b)

Crack and damage propagation stress: 328.08 MPa

(c)

Final failure stress: 611.52 MPa

Figure 5.34 Crack and Damage evolution process from VCCT/PFA of specimen with ply drop under static compressive loading. (Red color identifies damaged elements, damage and failure modes: longitudinal compression and in-plane shear).

240

Durability of Composite Systems 3500 3000

Load (N)

2500 2000 1500 1000

PFA VCCT/PFA

500

Max. experimental load

0 0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

Displacement (mm)

Figure 5.35 Comparison of PFA and VCCT/PFA load versus displacement. 3500 3000 PFA

2500

Load (N)

VCCT/PFA

2000

Experimental max. load

1500 1000 500 0 0

2

4

6

8

10

12

Crack length (mm)

Figure 5.36 Comparison of crack growth due to increasing load.

the previous PFA and the VCCT/PFA. In addition, displayed in these figures is the maximum load achieved experimentally. The results of the VCCT/PFA are in agreement with not only the experimental results but also the PFA strength-based approach. Table 4.8 indicates less than 1% error for the compressive strength between the VCCT/PFA and experiment and an associated 6.25% error for the percentage compressive strain. Figs. 5.35 and 5.36 also indicate closer results to the actual experiment than the previous PFA. In support of [8,9] Fig. 5.36 illustrates that crack initiation and propagation occurred under low loading conditions. These results prove that GENOA VCCT/PFA is a reliable and useful tool for predicting the behavior of composite structures comprising ply drop features subject to compressive loads. GENOA VCCT/PFA and DCZM/PFA can also be used to predict the behavior of composite structures comprising ply drop features subject to tensile loads. The results of this work will be presented in a future publication.

Advanced composite wind turbine blade design

5.4.3 5.4.3.1

241

Nine meter blade Durability and reliability of wind turbine composite blades using a robust design approach [10]

The use of advanced composites in product design is attractive due to advantageous weight-to-stiffness and weight-to-strength ratios. Composite structures are being subjected to severe combined environments and are expected to survive for long periods. Although large amounts of coupon test data exist, there is neither an adequate test database for composite structures nor significant long-life service experience to aid in risk assessment. Owing to the difficulty and cost in assessing and managing risk for new and untried systems, the general method of risk mitigation consists of applying multiple conservative factors of safety and significant inspection requirements to already conservative designs instead of costly full system tests. Unfortunately, this approach can lead to excessively conservative designs. The full potential of composite systems is often not fully realized. Current wind turbine blade design with advanced composites is based on high factors of safety and traditional design/stress analysis practices to ensure the target static strength levels and service life lengths. To achieve low production costs, material systems such as resin-infused woven and stitched fiberglass are utilized to attempt to achieve an approximate $5 per pound or lower target product cost. In addition, real-time structural health monitoring is not used to assess the condition of the blades. Finally, the design process can be described as one that focuses on service life rather than damage tolerance. The combination of these design constraints can significantly impact the turbine blade weight and performance. A design process that uses a composite damage modeling approach can lead to blades that are optimized to be damage resistant and tolerant while being light and inexpensive. The objective of this section is to perform virtual testing (VT) process that involves an accurate simulation of physical tests using MS-PFA including the scatter in physical tests by using probabilistic analysis. The multiscale analysis is based on a hierarchical analysis, where a combination of micromechanics and macromechanics is used to analyze material and structures in great detail. Certification required predictions are important for reducing risk in structural designs. Moreover, the determination of allowable properties is a time-consuming and expensive process, as a large amount of testing is required. To reduce costs and product lead time, VT can be used to reduce necessary physical tests both for certification and for determining allowables. In summary, whereas industry tends to rely on expensive test-intensive empirical methods to establish design allowables for sizing advanced composite structures, the developed VT methodology relies on physics-based failure criteria to reduce its dependence on such empirical-based procedures. This is more than a simple mix of analysis and test because of the following: (1) The root cause of failure at the microscale is modeled for accurate failure and life prediction, (2) VT is incorporated into each stage of the building-block process and the certification categories of damage tolerance and (3) Natural material and manufacturing data scatter is created giving rise to the unique capability to estimate strength allowables.

242

5.4.3.2

Durability of Composite Systems

Description of blade FEA model and blade materials

The blade considered was designed and built through a research partnership between TPI composites and Sandia National Laboratories. The testing of the blade was completed by Sandia and the National Renewable Energy Laboratory. Fig. 5.37 shows a finite element model of the wind blade [4]. The model was used by MS-PFA to simulate analytically the static test. The blade was 9-m long. It was scaled down from the initial 44-m planform from the first phase of Sandia’s Blade System Design Studies project [4]. The different colors in Fig. 5.37 indicate regions with different materials. These regions are called as ply schedules correlating distinct physical regions in the model with specific material or number of materials, layup information, thickness, and other physical properties. The blade was modeled with 22,774 shell elements and 67,608 nodes. Fig. 5.38 shows the loading profile at three cross-sections of the blade. The loads were applied at 3, 4.8, and 6.6 m from the root to approximate the loading of the ultimate load test. Table 5.12 lists the various materials used in the construction of the blade. Table 5.13 lists the type of material used in different ply schedules of the blade. More details on blade design can be found in Ref. [4]. The list of the blade materials was supplied by Sandia.

5.4.3.3

Simulation of blade static test

During testing, blade bending loads were applied at three blade stations. These loads were increased until the failure of the test article. Simulation results were obtained numerically using MS-PFA software. The analytical predictions were validated not only for the ultimate failure load but also for load-deflection results. Table 5.14 lists ply mechanical properties for ELT5500 glass epoxy composite. Properties for other 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2y 1

z x

Figure 5.37 Sandia 9 m blade finite element model with correspondent ply schedules.

Advanced composite wind turbine blade design

243

Blade tip

Blade root

Figure 5.38 Blade bending loads applied at three stations. Table 5.12 List of materials used in the blade model. Material #

Material

1

Stitched double bias glass (DBM1708)

2

Balsa

3

Steel

4

Gelcoat

5

Chopped glass mat (0.75 oz)

6

Uni Glass (ELT5500)

7

Woven double bias glass (6 oz)

8

Rod filler (smeared stud/glass)

9

Stitched double bias glass (DBM1208)

10

Carbon/glass triax

11a

Chopped glass mat (0.75 oz) 10

12a

Stitched double bias glass (DBM1708) 10

a

13 a

Balsa 10

Material with 10 stiffness and density, 0.1 ply thickness.

materials can be found in Ref. [4,5]. MS-PFA predicted damage initiation and propagation, fracture initiation and propagation, and the final residual strength in the blade. Compared with test results, analytical simulation gives a good prediction as well as the damage progression sequence and structural response characteristics during different degradation stages. Fig. 5.39 shows the progressive damage in the blade under increased static loading. The initial damage started near the root of the blade at a

244

Durability of Composite Systems

Table 5.13 Materials and ply schedule correlation. Ply schedule

Material #

1

3

2

4,5,1,6,7,6,6,8,6,9,6, 1

3

4,5,1,6,7,6,6,8,6,9,6, 1

4

4,5,1,6,7,6,6,9,6,1

5

4,5,1,6,7,6,6,9,6,1

6

4,5,1,10,7,10,6,9,6,1

7

4,5,1,6,7,6,6,1

8

4,5,1,6,7,6,6,1

9

4,5,1,10,7,10,6,1

10

4,5,1,6,7,6,1

11

4,5,1,6,7,6,1

12

4,5,1,10,7,10,1

13

4,5,1,2,1

14

4,5,1,6,7,6,1

15

4,5,1,6,7,6,1

16

4,5,1,2,1,1,1

17

4,5,1

18

4,5,1,10,7,1 0,1

19

4,5,1,10,1

20

4,5,1

a

a

21

4,11,12,13,1 2

22

4,5,1

23

1,2,1

Used at the leading edge to avoid shell radius/thickness violation.

load of 8.7 kN due to delamination in composite layers (Fig. 5.39(a)). When the load reached 24.36 kN, additional damage criteria were activated: transverse tensile, longitudinal compressive, shear, and delamination (Fig. 5.39(b)). When the load applied reached the ultimate peak value of 48.29 kN, damage propagated to most zones of the blade (Fig. 5.39(c)). At ultimate, the damage modes were longitudinal and transverse tensile failure, longitudinal and transverse compressive failure, and shear failure and delamination. The final damage of the blade from the test is shown in Fig. 5.39(d). As indicated in the figure, the test indicated extensive damage near the root consistent with the findings of the simulation. The final position of blade obtained from simulation and showing excessive deformation is presented in Fig. 5.39(e).

Advanced composite wind turbine blade design

245

Table 5.14 Lamina properties used for ELT5500 (unidirectional glass epoxy material). Property

Symbol

Value

Measurement

Longitudinal modulus

El11

4.180Eþ04

N/(mm2)

Transverse modulus

El22

1.403Eþ04

N/(mm2)

Transverse modulus

El33

1.416Eþ04

N/(mm2)

In-plane shear modulus

Gl12

2.630Eþ03

N/(mm2)

Transverse shear modulus

Gl23

2.323Eþ03

N/(mm2)

Shear modulus

Gl13

2.630Eþ03

N/(mm2)

Poisson’s ratio

Nul12

2.806Ee01

Poisson’s ratio

Nul23

4.534Ee01

Poisson’s ratio

Nul13

2.759Ee01

Longitudinal tensile strength

Sl11T

1.125Eþ03

N/(mm2)

Longitudinal comp. strength

Sl11C

7.492Eþ02

N/(mm2)

Transverse tensile strength

Sl22T

6.083Eþ01

N/(mm2)

Transverse comp. strength

Sl22C

2.041Eþ02

N/(mm2)

In-plane shear strength

Sl12S

3.020Eþ01

N/(mm2)

Normal tensile strength

Sl33T

6.083Eþ01

N/(mm2)

Normal comp. strength

Sl33C

2.041Eþ02

N/(mm2)

Fig. 5.39(f) shows the structural response under increased static loading from simulation compared to the test. The load deflection measured by string potentiometers located at three blade stations is plotted in the figure. Station SP1 is for blade station 3 m away from the root of the blade. Stations 2 and 3 (SP2 and SP3) are located 4.8 and 6.6 m away from the blade root, respectively. As noted in Fig. 5.39(f), the results from station SP3 indicate a potential anomaly with the data collected from the test. This might be attributed to a problem with the strain gage. The loads at the three stations of the blade were increased proportionally until structural failure took place. The results presented here illustrate the effectiveness of the MS-PFA approach to virtually predict the behavior of wind blade structures. It provides a reliable means to assess structural response by predicting test behavior and not replicating it.

5.4.3.4

Fatigue evaluation of a 9-m wind turbine blade

MS-PFA is used to analyze the durability and damage tolerance of the 9-m blade Sandia blade subject to 17 blocks of fatigue loading. Table 5.15 lists the load sequence applied to the blade in the test. Table 5.16 lists the validation of displacement from analysis with that from the test for the first block of loading at the point of load application. This established confidence that the structural response under fatigue loading was captured properly with analysis. Fig. 5.40 shows the airfoil station where

246

Durability of Composite Systems

(a)

(b) Damage (red color)

Damage propagation at load of 24.36 KN

Damage initiation at load of 8.7 KN

(d)

(c) Damage (red color)

Damage at test peak load of 48.612 KN (2m from root of the blade)

Damage at peak load of 48.29 KM

(e)

(f) 2.622E+00

1.577E+00

Load (N)

2.099E+00

Final Position

1.055E+00

5.323E-01

Original Position 9.918E-03

Total displacement in meter at peak load of 48.29 KN

Out-of-plane deflection (meters)

Test and predicted load-deflection for sandia’s BSDS blade under static loading

Figure 5.39 Progressive failure analysis and test validation of Sandia’s 9 m blade under static loading.

the load was applied. Displacement measurement was taken from the actuator that was located at 4.8 m. During testing, Sandia’s blade developed a crack running from the spar cap to the leading edge at the 1200 mm station. This shuts the test down automatically. This was after 97k cycles in load block 17 were completed. Using MS-PFA, structural fracture occurred at the end of block 16. Typical damage observed during cycling is shown in Fig. 5.41 for block 5.

Advanced composite wind turbine blade design

247

Table 5.15 Sandia’s 9 m blade fatigue loading sequence. Load block

Load type

1

Flap

2

End cycle

Number of cycles

1

1,045,762

1,045,761

3797

Edge

1,045,762

2,071,869

1,026,107

797

3

Flap

2,071,869

2,572,033

500,164

4271

4

Flap

2,572,033

3,075,051

503,018

4659

5

Flap

3,075,051

3,576,257

501,206

4929

6

Flap

3,576,257

4,078,930

502,673

5489

7

Flap

4,078,930

4,329,294

250,364

5962

8

Flap

4,329,294

4,582,725

253,431

6388

9

Flap

4,532,725

4,833,015

250,290

6989

10

Flap

4,833,015

5,085,684

252,669

7783

11

Flap

5,085,684

5,336,174

250,490

8166

12

Flap

5,336,174

5,588,916

252,742

9094

13

Flap

5,588,916

5,838,815

249,899

9941

14

Flap

5,838,815

6,101,283

262,468

10,748

15

Flap

6,101,283

6,188,069

86,786

14,252

16

Flap

6,188,069

6,439,067

250,998

11,873

17

Flap

6,439,067

6,686,541

247,474

13,026

5.4.3.5

Start cycle

Mean load (N)

Blade weight analysis

Building block strategy for robust design requires a detailed understanding of the role that each material of the blade plays with respect to contribution to weight and durability. To determine accurately the weight of the blade and identify the contribution of the various materials, weight analysis was performed. Table 5.17 shows the volume and weight contribution from each material used in the blade construction, which was previously listed in Table 5.12. As indicated in Table 5.17, the stitched double bias glass (or DBM1708) material system constituted about 31% of the total blade weight. The DBM1708 material and unidirectional glass (ELT5500) undergo damage and fracture at increased loading conditions as evidenced by progressive failure analysis results. Working with these materials will change the damage characteristics of the blade. Enhancing the properties of materials #1 and #6 combined with lower density can effectively reduce the weight of the blade without reducing its performance. However, this has to be done in a way that the overall shape and stiffness of the blade remain unaltered. The total weight predicted by the analysis is lower by

248

Durability of Composite Systems

Table 5.16 Validation of displacement from analysis.

# Of fatigue cycles

Analysisdflapwise displacement (m)

Testdflapwise displacement (m)

50,000

0.0954

0.093

100,000

0.0954

0.093

150,000

0.0954

0.093

200,000

0.0954

0.093

250,000

0.0956

0.093

500,000

0.0956

0.093

550,000

0.0957

0.093

600,000

0.0958

0.093

650,000

0.0958

0.093

700,000

0.0958

0.093

750,000

0.0958

0.093

800,000

0.0958

0.093

850,000

0.0958

0.093

900,000

0.0958

0.091

950,000

0.0958

0.091

1,000,000

0.0958

0.091

1,050,000

0.0958

0.091

Figure 5.40 Sandia fatigue test point of load application.

16 kg as compared to the one reported in Ref. [4] (113 kg from analysis vs. 129 kg from the lab). The difference could be attributed to the fact that the analytical model did not include the weight of steel studs extending from the blade and adhesive included in the blade when it was weighed in the lab.

Advanced composite wind turbine blade design

249

Damage modes

Figure 5.41 Structural damage in red obtained from analysis after applying 3.5 million cycles (end of block 5).

5.4.3.6

Blade durability and damage tolerance probabilistic sensitivity analysis

It is prudent to identify the influential material with respect to D&DT and weight. Results from D&DT and weight analysis guide the weight reduction strategy. First, a parametric study was performed by varying the mechanical properties of various material systems, one at a time, to assess the influence on the final failure load. The static load applied is a three-saddle load similar to the one applied in the test performed by Sandia (bending load at three locations along the span of the blade). Table 5.18 shows the results from a total of 21 progressive failure analysis (MS-PFA) runs. A description of the perturbed properties is given here: S11t ply longitudinal tensile strength; S11c

250

Durability of Composite Systems

Table 5.17 Volume and weight contribution for various materials used in the blade design. Material system #

Total volume m3

Density kg/m3

Total mass kg

% of total mass

Material

1

1.89297E02

1814

34.338

30.76%

Stitched double bias glass (DBM1708)

2

5.29436E02

230

12.177

10.91%

Balsa

3

6.80245E04

7850

5.340

4.78%

Steel

4

4.88732E03

1235

6.036

5.41%

Gelcoat

5

2.86724E03

1678

4.811

4.31%

Chopped glass mat (0.75 oz)

6

1.24760E02

1874

23.380

20.94%

7

1.65826E03

2076

3.443

3.08%

8

4.02312E03

3006

12.093

10.83%

9

3.52024E04

1814

0.639

0.57%

Stitched double bias glass (DBM1208)

10

3.44028E03

1685

5.797

5.19%

Carbon/glass triax

11

2.48228E05

16,780

0.417

0.37%

Chopped glass mat (0.75 oz) 10

12

1.15840E04

18,140

2.101

1.88%

Stitched double bias glass (1708) 10

13

4.63358E04

2300

1.066

0.95%

Balsa 10

Total

1.02862E01

Uni glass (ELT5500) Woven double bias glass (6 oz) Rod filler (smeared stud/ glass)

111.64

ply longitudinal compressive strength; S22t ply transverse tensile strength; S22c ply transverse compressive strength; S12 ply shear strength; E11 ply longitudinal modulus; E22 ply transverse modulus. The table shows the normalized variable considered in each analysis run and the predicted failure load from the analysis. Three material systems (DBM1708, ELT5500, and carbon/glass triax) were considered in the parametric study for D&DT sustainment based on initial progressive failure analysis of the blade. ELT5500 was a mixed material system that included plies with carbon fibers

Advanced composite wind turbine blade design

251

Table 5.18 Summary of a parametric study showing the influence of mechanical properties from three material systems on blade failure load. Material system

DBM1708

ELT5500

Saertex

Mechanical property (normalized value)

Failure load (N)

S11t

S11c

S22t

S22c

S12

E11

E22

Analysis

1.1

1.0

1.0

1.0

1.0

1.0

1.0

48,285

1.0

1.1

1.0

1.0

1.0

1.0

1.0

48,285

1.0

1.0

1.1

1.0

1.0

1.0

1.0

48,285

1.0

1.0

1.0

1.1

1.0

1.0

1.0

48,720

1.0

1.0

1.0

1.0

1.1

1.0

1.0

51,330

1.0

1.0

1.0

1.0

1.0

1.1

1.0

51,330

1.0

1.0

1.0

1.0

1.0

1.0

1.1

48,285

1.1

1.0

1.0

1.0

1.0

1.0

1.0

48,285

1.0

1.1

1.0

1.0

1.0

1.0

1.0

48,285

1.0

1.0

1.1

1.0

1.0

1.0

1.0

48,285

1.0

1.0

1.0

1.1

1.0

1.0

1.0

48,285

1.0

1.0

1.0

1.0

1.1

1.0

1.0

48,285

1.0

1.0

1.0

1.0

1.0

1.1

1.0

48,720

1.0

1.0

1.0

1.0

1.0

1.0

1.1

47,850

1.1

1.0

1.0

1.0

1.0

1.0

1.0

48,285

1.0

1.1

1.0

1.0

1.0

1.0

1.0

48,285

1.0

1.0

1.1

1.0

1.0

1.0

1.0

48,285

1.0

1.0

1.0

1.1

1.0

1.0

1.0

48,285

1.0

1.0

1.0

1.0

1.1

1.0

1.0

48,285

1.0

1.0

1.0

1.0

1.0

1.1

1.0

48,285

1.0

1.0

1.0

1.0

1.0

1.0

1.1

48,285

Test

48,612

and others with glass fiber, while the carbon/glass triax is a carbon fiber-based material system. From the results presented in the referenced table, it is clear that the DBM1708 material and ELT5500 play key roles with respect to the blade D&DT as their mechanical properties influence the final failure load of the blade. Fig. 5.42 shows the scatter in the ultimate failure load of the blade obtained from MS-PFA as a result of 5% coefficient of variation in the DBM1708 material random variables. The data shown in the referenced figure are of great importance for obtaining reliable design. For example, if the applied load is kept under 36,700 N, a reliability of 0.999 can be attained (highly desired outcome). If we produce 1000 blades, very few

Durability of Composite Systems

Probability

252 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 30000

Tesr failure load: 48,612

40000

50000

60000

70000

Ultimate load (N)

Figure 5.42 Probabilistic scatter in blade failure load subject to static loading due to uncertainties in properties of DBM1708 material. 1

Sensitvity (max = 1.0)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 S11T S11C S22T S22C S12

E11

E22

Random variable

Figure 5.43 Probabilistic sensitivity of random variables due to uncertainties in the properties of DMB1708 material.

will fail when the load applied is lower than 36,700 N. Most will fail when the load applied is close to the mean load (about 48,000 N). If the load applied is increased beyond 61,000 N, most blades would have failed by then. The blade failure load exhibited a scatter of 24,000 N based on the assumed uncertainties. Fig. 5.43 shows the results obtained from probabilistic sensitivity analysis assuming variation in the mechanical properties of the DBM1708 material. The longitudinal stiffness and ply shear strength are very influential when it comes to affecting the blade ultimate load. Controlling scatter in these two random variables reduces variability in the blade ultimate load. With the data presented here, it is established that the DBM1708 material is of great importance when it comes to the blade failure followed by ELT5500 material. Results obtained from weight reduction are discussed next.

5.4.3.7

Blade weight reduction with robust design

Weight reduction is an important area to address as it is directly proportional to manufacturing, transportation, and installation costs. Weight reduction should be done

Advanced composite wind turbine blade design

253

60000 SP1

SP3

SP2

50000

Load (N)

40000 SP1-test

30000

SP3-test SP2-test

20000

SP1-GENOA SP2-GENOA SP3-GENOA

10000 0 –0.4

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Blade deflection (m)

Figure 5.44 Loadedisplacement relationship comparing blade response under static load from the test with analysis results. 20% of ELT5500 material was replaced with foam reducing weight from 112.7 to 103.7 kg.

without loss in stiffness and strength or the life of the blade. The authors assessed the potential for weight reduction by replacing a volume of the blade material with foam. In the study performed, the blade was subjected to a static load similar to the one used by Sandia during testing (at three locations on the blade). This work presented an opportunity to reduce weight without loss in stiffness or strength. The study was based on the following independent evaluations: (1) Replace 20% of ELT5500 material system # 6 (carbon and glass system) with foam and (2) Replace 15% of the DBM1708 material system # 1 with foam. The volume of the blade remained unchanged in order not to alter the aerodynamic performance.

Fig. 5.44 shows the loadedisplacement from three locations on the blade compared to the test for ELT5500 that includes 20% foam. The weight, in this case, was reduced by 8.7% as it decreased from 109.4 to 100.4 kg with minimal loss in performance as compared to the original design of the blade. The original layup and properties of ELT5500 material system were used; only the ply thickness of ELT5500 was scaled down to reflect a 20% reduction in the volume of ELT5500. Test results presented in Fig. 5.44 pertain to the test under the static loading of the blade with the original design performed at Sandia National Laboratories. The foam inserts in ELT5500 were used by the analytical simulation. The performance with reduced weight is comparable to that of the original design, especially from the aspect of not affecting the overall stiffness of the blade or ultimate load as the ELT5500 material is mainly used in the root area. Fig. 5.45 shows the loadedisplacement from three locations on the blade from test compared to analysis when 15% of DBM1708 material is replaced with foam.

254

Durability of Composite Systems

60000 SP1

SP2

SP3

50000

Load (N)

40000 SP1-test

30000

SP3-test SP2-test

20000

SP1-GENOA SP2-GENOA SP3-GENOA

10000

0 –0.5

0.0

0.5

1.0

1.5

2.0

Blade deflection (m)

Figure 5.45 Loadedisplacement relationship comparing blade response under static load from the test with analysis results. 15% of DBM1708 was replaced with foam reducing weight from 112.7 to 100.2 kg.

The weight, in this case, was reduced by 12.5% as it decreased from 109.4 to 96.9 kg with no loss in performance as compared to the original design of the blade. Using more than 15% foam could affect the D&DT of the blade by reducing its ultimate load as compared to that of the original design. One other concern for this case is the reduction in stiffness as compared to that of the original design (Fig. 5.45). This was expected as the stiffness of the foam was lower than that of the DBM1708 material. The new design is more flexible than the original one. This can impact the frequency and buckling resistance of the blade and require further investigation. Further studies need to be carried out to determine the combination of ply angles that can maintain the same original stiffness. Note that for this case study, one material ply property was modified. The shear strength of DBM1708 material was increased by 20% that can be obtained from commercially available enhanced resin properties. Other design variables should be considered in future work, especially those dealing with manufacturing of the composite such as ply thickness and ply orientation.

5.4.3.8

Improving wind blade structural performance with the use of resin enriched with nanoparticles [11]

Laminated fiber-reinforced polymer (FRP) composites have found wide use for a range of structural and functional applications in industries such as aerospace, automobile, marine, and infrastructure including wind blade applications. FRPs are high-performance materials, having high stiffness and strength-to-weight ratios compared to metals. FRPs exhibit high corrosion resistance, a significant reduction in maintenance

Advanced composite wind turbine blade design

255

costs, and, most importantly, expand a designer’s horizons to design a material to meet the limit load requirements. The fibers in FRP composites reinforce mostly the in-plane properties. The weaker matrix interface dominates the out-of-plane properties, such as delamination resistance. Delamination along with ply interfaces (interlaminar shear strength) due to fiber/matrix debonding is one of the major failure modes in laminated FRPs when subjected to transverse and compressive loading. Mandell et., al. [7] concluded that delamination between plies is the root cause of many failures of composite materials structures such as wind turbine blades. Mandell also elaborated that design methodologies to prevent such failures have not been widely available for the materials and processes used in blades. The use of reinforcement in the matrix to enhance resistance to delamination is a benefit that would be desired for the wind blade industry. The work described here assesses benefits resulting from the use of nanoparticles in wind blade applications under static loading. Durability and damage tolerance results from the use of nanomaterial are evaluated analytically and compared to results from a baseline static test of a neat blade (without nano). The benefits of fatigue life are not addressed in the work presented in this paper. Any insertion of nanoparticles in the wind blade structure must take into account the manufacturability and cost aspects. For ease of applications, the analysis assumed that strips of thin unidirectional glass composites are applied to the blade after the blade is manufactured. Reinforcement of the matrix of laminated composites via stitching, weaving, braiding fibers, or addition of toughening particles is a widely used approach to improve delamination resistance. These approaches, however, can degrade other FRP properties. For example, using toughening particles with lower stiffness than the matrix can reduce the overall stiffness of a laminate. Improving interlaminar shear strength (ILSS) by reinforcing the laminate interface with nanoparticles such as double-walled [12] or multiwalled [13] carbon nanotubes (CNT) and nanoclay [14] have recently been considered [15]. The nanoparticles used had high aspect ratios, high stiffnesses/strengths, and were nanosized. A low-weight ( > > > 1 þ n 1 1 1 þ n > > > > ln  patm > >

2 þ 3  ln  > > > > 3 2Y ð1 þ nÞ 1þn > > > > > > 2 1  3 1  > > > > 3 3 > > > > > > > > > > > >" # > > > > > > 1 1 > > > >  > > þ

  > > 2 3 > > 1þn 1þn > > > > > > p p 2 1  3 1  atm atm > > > > 2Y 2Y > > > > > > > > > > > >  > > > > 1 þ n > > ð1 þ nÞ " " # # > > > > p 1  atm > > 1  > > 1 1 1 1 2Y > > 3 > >þ



  ln ln  þ > > > > > > ð1 þ nÞ ð1 þ nÞ 1þn 1þn 2 2 > > > > 2 1 p p 2 1  atm atm > > > >  3 3 2Y 2Y = 4pr3 2Y < 3 II ¼   > 1þn > 3 > > 1þn > > ð1 þ nÞ#2 " ð1 þ nÞ # " " "2  1 þ n p #2 > > > patm # > atm > > 2  > > 1 1 1 1 2Y 2Y > > 3 3 > >   ln ln þ  þ  > > > > > > ð1 þ nÞ ð1 þ nÞ 1þn 1þn 6 6 3 3 > > > > 1  1  p p 1  1  atm atm > > > > 3 3 2Y 2Y > > > > > > > > > > > > > >  > > > > > > 1þn 1 1 > > > > p þ þ 



 atm > > 2 2 > > 2Y > > ð1 þ nÞ 1 þ n > > > > p 2 1  2 1  > > atm > > > > 3 2Y > > > > > > > > > > ! > > > > > > > > 1 1 > > > > þ  > >



 > > 3 3 > > ð1 þ nÞ 1 þ n > > > > > > p 3 1  3 1  atm ; : 3 2Y (9.43)

Work of electrochemical pressurization of a pore in an oxygen ion

355

with Y ¼ 210 GPa, n ¼ 0.25 and patm ¼ 21,278 Pa, the values of the two terms are as follows: I¼

4pr3  5894:4  109 m3 Pa ðJoulesÞ 3

(9.44)

and II ¼ 

4pr 3  447:45  109 m3 Pa ðJoulesÞ 3

(9.45)

Thus, the net electrochemical work of pressurization at the condition of criticality is given by Welectrochem ¼

4pr 3  5446:95  109 m3 Pa ðJoulesÞ 3

(9.46)

The above includes the energy in the gas phase and the strain energy in the YSZ sample. Note that this value is much greater than the strain energy in YSZ (Eq. 9.29 by more than a factor of 10). This is entirely to be expected as the gas is compressible. Thus, most of the energy of pressurization is stored in the gas phase, not in YSZ. This result has major significance insofar as how the failure of electrochemical devices occurs under electrolytic operation as shown previously for a penny-shaped crack in a solid electrolyte [17]. At the condition of criticality for cracking to occur, the stored energy in the gas is much higher than that in the solid. It is the stored energy in the solid that determines when fracture begins (stress intensity factor exceeding fracture toughness). However, as soon as the crack grows, the volume of the pore/crack increases. I This leads to a decrease in the local stress intensity factor. That is, dK dc < 0, where c is the crack (possibly emanating from the pore). Thus, a crack arrest occurs. As the loading is due to gas pressurization, more gas must be precipitated, which depends upon the net current. Thus, operating conditions determine how the fracture propagates. By controlling electrolytic conditions, one can control the rate of crack growth.

9.2.6

A simplified derivation assuming a fixed pore radius (applicable at low pressures)

The following is a simplified derivation assuming that the pore diameter remains unchanged, which is applicable at low pressures. The number of moles of oxygen gas in the pore is given by nðpÞ ¼

4pr 3 p 3RT

(9.47)

356

Durability of Composite Systems

The differential work of pressurization is given by dWðpÞ ¼

 4pr 3 p ln dp patm 3

(9.48)

Eq. (9.47) integrates to WðpÞ ¼

4pr 3 ½plnp  p  plnpatm þ patm  3

(9.49)

Although this equation is applicable at low pressures, we will still evaluate numerical value at criticality for the sake of comparison. The work of pressurization is given by   

2Y 4pr 3 2Y 2Y 2Y 2Y ln W ¼   lnpatm þ patm 3 3 3 3 3 3

(9.50)

or W

 2Y 4pr 3  2; 058  109 m3 Pa ðJoulesÞ ¼ 3 3

(9.51)

Note that this considerably lower than when a change in the pore radius is included. Note, however, that this includes the strain energy in the solid and the energy stored in the gas phase, and thus is much larger than the value in Eq. (9.6).

9.2.7

Time required for pressurization

Under the applied voltage EA , the transport of oxygen ions occurs from the porous surface electrode toward the buried electrode. A Nernst voltage is developed between the pore and the outside atmosphere, EðpÞ, which opposes the flow of current. The net current is given by IðpÞ ¼

dqðpÞ ðEA  EðpÞÞ ¼ dt Ri

(9.52)

where Ri is the ionic resistance between the porous electrode and the buried electrode tip. This also includes the electrode polarization resistance at both electrodes. Thus, dt ¼

Ri dqðpÞ ðEA  EðpÞÞ

(9.53)

Work of electrochemical pressurization of a pore in an oxygen ion

357

9 8  > > > > 1þn > > > 3 p > = < 3 4pr Ri 1 2Y dqðpÞ ¼ 4FdnðpÞ ¼ 4F 3 þ 4 dp   > 3RT > 1þn 1þn > > > > > p > 1 ; : 1  2Y p 2Y

Now

(9.54) Thus, the time required for pressurization is given by 88 99  > > >> 1þn > > > > > > 3 p > > > < => > > 1 2Y > > > > > > þ



  > 3 4 >> > > > > > 1 þ n 1 þ n > > > > p > > > Z Z : 1 p p ;> 1 < = 3 4pr Ri 2Y 2Y   dp t¼ dt ¼ 4F > > RT p 3RT > > > > E ln  patm > A > > > 4F patm > > > > > > > > > > > > > > > > : ; (9.55) Note that the time required to pressurize to a given pressure is proportional to the initial pore volume. Eq. (9.55) will need to be evaluated numerically. The maximum Nernst voltage (corresponding to criticality) is given by  RT 2Y EN ðmaxÞ ¼ ln 4F 3patm

(9.56)

Its magnitude at 800 C is w0.35 V. This means an applied voltage greater than 0.35 V, regardless of the size of the pore and the sample, will eventually lead to internal pressurization to cause failure. Its rate, however, depends upon the applied voltage (the higher the applied voltage the faster the rate) and the resistance (the smaller the resistance, the faster the rate). In addition, as to be expected, the smaller the pore, the faster is the kinetics of pressurization. In reality, the time required could be much larger if the geometry of the pore is not spherical. This is because, at high pressures, the stoichiometry of YSZ itself can change, due to the pressure dependence of the oxygen chemical potential [7]. More oxygen can be accommodated in the lattice by filling up oxygen vacancies and even creating oxygen interstitials. In the case of a spherical pore, it is to be noted that the hydrostatic component of the stress, given by sxx þ s44 þ sqq is zero [18]. Thus, if in an experimental arrangement of the type 3 shown in Fig. 9.3 can be realized, where the pore is spherical (and not of some other geometry or a crack), the net hydrostatic stress (pressure) in the YSZ surrounding the pore, will be zero. This means the oxygen chemical potential in the YSZ near the pore

358

Durability of Composite Systems

will be the same as far away from the pore. In such a case, the time to pressurization given in Eq. (9.54) may be reasonably accurate. In experiments reported by Lim and Virkar [20], the electrochemical generation of pressure was investigated through the measurement of the Nernst voltage as a function of time. However, the geometry of the crack (or possibly many cracks) was unknown. In such a case, it is not possible to quantitatively estimate the kinetics of internal precipitation and pressurization. In the present case, in principle, it should be possible to investigate the kinetics of pressurization without the complication of changes in local stoichiometry. If  the applied voltage is much greater than the Nernst voltage, that is if EA [ RT p ln , which would be the case in the early stages of the experiment, 4F patm Eq. (9.55) approximately reduces to tz



 4pr 3 Ri 3 1þn  2 p  p2atm ðp  patm Þ þ 2 Y 3RTEA

(9.57)

Thus, in the initial stages, it should be possible to compare the theoretical calculations presented here with experimental results.

9.2.8

Pore pressurization using a long cylinder and a piston

So far, we have used an electrochemical approach to analyze the pressurization of a pore. From thermodynamics, however, we know that the final state for a given pressure in the gas in the pore and the corresponding strain energy in the solid, the net reversible work done must be the same regardless of how the process is achieved. To illustrate this, we consider a hypothetical arrangement given in Fig. 9.4, which shows a long cylinder of cross-sectional area A and length lo such that the total volume corresponds to gas at patm but containing the same number of moles in the pore-cylinder 2Y . We corresponding to the condition of criticality; that is corresponding to p ¼ 3 know that the number of moles in the pore corresponding to pressure p is given by nðpÞ ¼

4pr 3 3RT

 1

p

3 1þn p 2Y

A l

Figure 9.4 Hypothetical poreecylinderepiston assembly.

(9.58)

Work of electrochemical pressurization of a pore in an oxygen ion

359

2Y , the number of moles is Thus, at the condition of criticality, that is when ¼ 3 given by  2Y r3 Y n p¼ (9.59) ¼ 12p 3 RT ð2  nÞ3 So, the initial volume of the pore-cylinder assembly is given by  2Y n p¼ 12pr 3 Y RT 12pr 3 Y 3 Vð0Þ ¼ RT ¼ ¼ 3 patm RT ð2  nÞ patm ð2  nÞ3 patm

(9.60)

Thus, the initial volume of the cylinder is given by Alo ¼

12pr 3 Y 3

ð2  nÞ patm



4pr 3 3

(9.61)

where A is the cross-sectional area of the cylinder. Thus, lo ¼

12pr 3 Y Að2  nÞ3 patm



4pr 3 3A

At some pressure p, during pressurization as the piston is moved in  2Y n p¼ 3 Al ¼ RT p

(9.62)

(9.63)

or l¼

12pr 3 Y Að2  nÞ3 p



4pr 3 3A

(9.64)

In the above, we have neglected the change in pore volume as the change in pore radius is small compared to the change in l. Thus dl ¼ 

12pr 3 Y dp Að2  nÞ3 p

(9.65)

The corresponding work done is dW ¼  Apdl ¼

12pr 3 Y dp ð2  nÞ3 p

(9.66)

360

Durability of Composite Systems

Thus, the work done is 2Y

Z3 W¼

dW

(9.67)

patm

which integrates to  4pr 3 9Y 2Y W¼ ln 3patm 3 ð2  nÞ3

(9.68)

In addition, the work is proportional to the initial pore volume. The numerical value is  2Y 4pr 3  5536  109 m3 Pa ðJoulesÞ W p¼ ¼ 3 3

(9.69)

This value is remarkably similar to Eq. (9.46). It is slightly different (higher) as we neglected the change in the second term in Eq. (9.64) with the change of pressure.

9.3

Possible experiments

As stated earlier, investigation of internal pressurization and cracking in actual solid oxide electrolyzers is complicated because of geometry, poorly defined geometry of possible cracks or voids into which oxygen precipitation occurs, and complicated operating conditions. Lim and Virkar [19] have previously measured Nernst potential at an embedded probe in a bilayer sample consisting of one layer of YSZ doped with ceria and one undoped. The maximum Nernst voltage of 0.22 V was measured, which corresponded to local oxygen pressure of w715 atm. At this pressure, local cracking occurred, as determined by a decrease in Nernst voltage. Although the local pressure could be determined as a function of time by measuring the Nernst voltage as a function of time, it was not possible to determine the kinetics of pressurization and cracking as no details were known about the exact nature of defects at the interface. The analysis presented here allows for possible experimental verification. A possible experiment (Fig. 9.3) consists of fabricating a YSZ sample with a well-defined pore, made by inserting a carbon bead during the powder pressing stage. A platinum wire, with an insulating coating (such as alumina) deposited, is placed in contact with the bead (during the powder pressing stage). The carbon bead is burnt off during slow heating. The compact is then sintered to fully densify the material. Porous platinum (or some other) electrodes are applied on the two sides of the disc. At elevated temperature, a DC voltage (EA ) is applied as shown in Fig. 9.3. The current flowing through the sample is measured, which will be a function of time. In addition,

Work of electrochemical pressurization of a pore in an oxygen ion

361

measured across the embedded Pt wire and the other porous electrode is the Nernst voltage, EðpÞ, which is a measure of the pressure in the pore at any time. This is also expected to be time dependent. The net resistance (which includes the polarization resistances), Ri , can be measured by electrochemical impedance spectroscopy. This may also be time dependent. In concept, this information can be used to approximately determine the kinetics of pressurization using Eq. (9.56) or the simplified Eq. (9.57).

9.4

Summary

A model is presented for the pressurization of a pore in an oxygen ion conductor such as YSZ, a material commonly used as an electrolyte in solid oxide cells (SOC). In electrolyzer mode, the precipitation of oxygen inside the electrolyte (in addition to electrode delamination) is known to occur. For a typical geometry of an SOC, if transport properties and electrode polarizations are known, it is generally possible to estimate the conditions under which degradation can occur. The nature of the degradation is generally complex and it is not easily possible to quantitatively estimate conditions under which degradation can occur. In the present work, a model is presented to estimate the pressurization of a spherical pore. In concept, it should be possible to design experiments that may be used to investigate pressurization quantitatively.

Acknowledgments This work was supported in part by the National Science Foundation under Grant No. DMR1742696 and in part by the US Department of Energy, Office of Basic Energy Sciences under grant number DE-FG02-03ER46086. The author thanks Prof. Dinesh Shetty for useful comments on the chapter.

References [1] A.V. Virkar, J. Nachlas, A.V. Joshi, J. Diamond, Internal precipitation of molecular oxygen and electrochemical failure of zirconia solid electrolytes, J. Am. Ceram. Soc. 73 (11) (1990) 3382e3390. [2] K. Reifsnider, G. Ju, X. Huang, Y. Du, Time dependent properties and performance of a tubular solid oxide fuel cell, ASME J Fuel Cell Sci Technol 1 (2004) 35e42. [3] M.A. Laguna-Bercero, Recent advances in high temperature electrolysis using solid oxide fuel cells: a Review, J. Power Sources 203 (2013) 4e16. [4] G. Schiller, A. Ansar, M. Lang, O. Patz, High temperature water electrolysis using metal supported solid oxide electrolzer cells, J. Appl. Electrochem. 39 (2009) 293e301. [5] A. Momma, T. Kato, Y. Kaga, S. Nagata, Polarization behavior of high temperature solid oxide electrolysis cells (SOEC), J. Ceram. Soc. Jpn. 105 (1997) 369e373.

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[6] J.R. Madwsley, J.D. Carter, A.J. Kropf, B. Yildiz, V.A. Maroni, Post-test evaluation of oxygen electrodes from solid oxide electrolysis stacks, Int. J. Hydrogen Energy 34 (9) (2009) 4198e4207. [7] M.A. Laguna-Bercero, R. Campana, A. Larrea, J.A. Kilner, V.M. Orera, Electrolyte degradation in anode-supported microtubular yttria-stabilized zirconia-based solid oxide steam electrolysis cells at high voltages of operation, J. Power Sources 196 (21) (2011) 8942e8947. [8] A. Hauch, S.H. Jensen, S. Ramousse, M. Mogensen, Performance and durability of solid oxide electrolysis cells, J. Electrochem. Soc. 153 (9) (2006) A1741eA1747. [9] R. Knibbe, M.L. Traulsen, A. Hauch, S.D. Ebbesen, M. Mogenson, Solid oxide electrolysis cells: degradation at high current densities, J. Electrochem. Soc. (2010). B1209-B1217. [10] J. Schefold, A. Brisse, M. Zahid, Electronic conduction of yttria-stabilized zirconia electrolyte in solid oxide cells operated in high temperature water electrolysis, J. Electrochem. Soc. 156 (2009) B897eB904. [11] A.V. Virkar, Mechanism of oxygen electrode delamination in solid oxide electrolyzer cells, Int. J. Hydrogen Energy 35 (2010) 9527e9543. [12] K.J. Harry, D.T. Hallinan, D.Y. Parkinson, A.A. MacDowell, N.P. Balsara, Detection of subsurface structures underneath dendrites formed on cycled lithium metal dendrites, Nat. Mater. 13 (2014) 69e73. [13] J.M. Tarascon, M. Armand, Issues and challenges facing rechargeable lithium batteries, Nature 44 (2001) 359e367. [14] J. Wang, J. Yang, Y. Nuli, R. Holze, Room temperature Na/S batteries with sulfur composite cathode materials, Electrochem. Commun. 9 (2007) 31e34. [15] W. Xu, J. Wang, F. Ding, X. Chen, E. Nasybulin, Y. Zhang, J.-G. Zhang, Lithium metal anodes for rechargeable batteries, Energy Environ. Sci. 7 (2014) 513e537. [16] C. Brissot, M. Rosso, J.-N. Chazalviel, S. Lascaud, Dendritic growth mechanisms in lithium/polymer cells, J. Power Sources 81e82 (1999) 925e929. [17] A.V. Virkar, ‘Failure of ion-conducting materials by internal precipitation under electrolytic conditions’, 59-76, in: T. Ohji, M. Singh (Eds.), Engineered Ceramics: Current Status and Future Prospects, Wiley-American Ceramic Society, 2016. [18] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, third ed., McGraw-Hill, New York, 1970. [19] S.M. Selby (Ed.), CRC Standard Mathematical Tables, fifteenth ed., CRC Publications, 1967. [20] H.-T. Lim, A.V. Virkar, A study of solid oxide fuel cell stack failure by inducing abnormal behavior in a single cell test, J. Power Sources 185 (2008) 79e800.

Durability of medical composite systems

10

Wendy Shen, Relebohile Qhobosheane Department of Aerospace and Mechanical Engineering, University of Texas Arlington, Arlington, TX, United States

Chapter outline 10.1

Composites in medical systems

363

10.1.1 Implantable medical composites 364 10.1.1.1 Dental composites 365 10.1.1.2 Composites for organ implant 366 10.1.1.3 Composites for bone implants 367 10.1.1.4 Tissue engineered composites 368 10.1.1.5 Composites for drug delivery 369 10.1.2 Composites for external medical devices 369 10.1.2.1 Composite wheelchairs and surgical tools 369 10.1.2.2 Composites for medical machinery 370 10.1.2.3 Composites for prosthetic devices 371 10.1.2.4 Composites for wearable devices 371

10.2

Properties affecting the durability of medical composite systems

372

10.2.1 Biocompatibility 372 10.2.2 Thermal expansion 375 10.2.3 Elastic modulus and toughness 376

10.3

Types of failure

377

10.3.1 Degradation and corrosion 377 10.3.2 Cavitation 378 10.3.3 Wear 379

10.4 Improving durability of medical composite systems 10.5 Closing remarks 380 References 380

10.1

379

Composites in medical systems

Recent developments in composites have contributed to the durability of medical composite systems. These are systems made of different constituents with improved strength, corrosion resistance, shape memory, and biocompatibility. The durability of medical composite systems is directly linked to their area of application. Durability of Composite Systems. https://doi.org/10.1016/B978-0-12-818260-4.00010-7 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Composites in medical systems have different applications including the development of medical devices such as biosensors and artificial organs. The usage of materials in these medical devices such as ligaments, bone plates, joint replacements, and dental implants has contributed to the studies of medical composite systems durability. Fig. 10.1 shows some of the common applications of composites in medical systems. There are different combinations of constituents in medical composites, and these are based on the required property for a specific application. These combinations are based on the materials’ required characteristics such as hardness, shear modulus, flexural modulus, stiffness or fracture toughness, geometry coefficient of thermal expansion or electrical conductivity, adhesion strengths, impact strength, and reproducibility [1]. These properties of medical composites systems are specific to each application and will be detailed in the next section.

10.1.1

Implantable medical composites

Implantable medical composites in this chapter are composites used in medical applications that only have to do with the inside of a human being. These are the types of composites designed with properties to sustain the moist and varying temperature environment of the human internal system. This chapter focuses on four internal applications of different properties that are dental, organ implant, bone replacement, and tissue engineering composites. It should be noted that there are more composite internal applications with properties similar to those covered in these sections based on their area of application. Medical composites Implantable medical composites

Dental composites Composites for organ implant Composites for bone implants Tissue engineering composites

Composites for external medical devices

Composite wheel chairs

Composites for prosthetic devices

Surgical tools composites Composites for medical

Composites for drug delivery

Figure 10.1 An overview of medical composite systems.

Wearable devices composites

Composites for medical research

Durability of medical composite systems

365 Titanium

Osseointegration

Bone

Implant

Space

Zirconia Biointegration

Implant

Bone

Corrosion layer

Figure 10.2 Dental composite made of Ti and an osseointegration constituent at the bottom [4].

10.1.1.1 Dental composites Composites used in dental restoration applications directly or indirectly are lightly cured. Initial developments in dental composites were boded on enamel. New methods of bonding composites to dentin evolved these bonding methods and therefore the use of composites for dentistry restoration became common [2]. These composites used in dental applications entail three segments: • • •

Fillers such as particles, fibers, or coupling agents form the dispersed segment The matrix forms the organic segment The combination of the two forms the interfacial region

The polymers in dental composites are fabricated in interlinking, network formation, and copolymerization to archive the preferred high conversion rates of double bonds on completion of polymerization. The double bonds in these polymers within dental composites that are unreacted may lead to materials with higher water absorption and softening. This is not a preferred property in dentistry as the composites may change in color, which leads to degradation of mechanical properties and leaching of toxins. Unbonded monomers are therefore kept at minimum because of their potential to cause allergic reactions and toxicology [3]. The main aim is to have replicas as similar to the real tooth as possible, and this is an advantage in composites as different shades can be blended to create a specific color. Dental composites also can bond to an existing tooth structure therefore helping protecting it from excessive temperature variations and breakage. As developed as dental composites are, patients still experience certain levels of discomfort after implantation. The composite may be extra sensitive to changes in the environment around it. They may also change in color due to different food, drinks, or tea. In most cases the dental composite is coated with a clear polymer to protect the color. As well as composites perform in small cavities, dental composites tend to wear out quick in large cavities compared with other fillings such as silver. Preferred properties in dental composites include a high compressive strength, less toxins release, and others, which will be discussed in the next sections.

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Recent developments in dental composite research aims to achieve a complete regeneration of bone and migration of epithelial cells to the implant and fibrous encapsulations. There are numerous combinations of constituents to formulate dental composites including common ones such as siliconecarbon composites and carbon finer composites. These types of composites have proved to have a great deal of advantages over alloys and dental ceramics [5]. Composites exhibit good long-term response from the host because of their low elastic modulus and sufficient stiffness, which is like that of natural teeth therefore minimizing the variation of the stress field. Composites formed of constituents that include Ti and HA display even better properties than most dental composites. These types of composites maintain the bioactive and osteoconductive requirement on the part of the teeth implanted within the host made of HA and the required mechanical properties from the top part made of Ti as shown in Fig. 10.2. Dental composite properties and further developments in a combination of constituents are detailed in the subsequent sections.

10.1.1.2 Composites for organ implant There are different types of composite applications in organ transplant. This involves different environments demanding high biocompatibility. These include the development of heart valves, which even though displayed longevity in their operation have a substantial risk of thromboembolic complications [6]. This is the formation of clots within the bloods cells soft enough to be distributed to another vessel. This has therefore led to the development of biomaterials for valves using glutaraldehydepreserved porcine aortic could combine with the use of bovine pericardial valves. A substantial fraction of prosthetic heart valves implanted annually in the United States are mechanical, and although durable, they are associated with a substantial risk of thromboembolic complications. Hence, bioprosthetic implants such as glutaraldehyde-preserved porcine aortic valves and bovine pericardial valves have become increasingly popular. These valves may go through structural failure therefore bringing forth the need for reoperation due to cuspal calcification. The use of tissue and organ implants also involves xenografts which have proper hemodynamics and the more compatible allografts. These still have an issue of calcification coupled with a limitation of supply. This type of system has numerous disadvantages including not being able to adjust for patient growth which results in repeated operations to improve vascular flow as the patient grows. The use of composites for organ implant is an option tissue engineers are considering to improve on these issues for most organ implants. Numerous experiments are run on this work to develop a valve that can mimic properties of actual valves using biocompatible composite materials. These are made of different constituents, which include PGA, PLA, or copolymers. These are proven to also have limitations of being bulky for blood valves, too stiff, and very degradable in human systems environments. Further developments were carried out in the past. The development of composites entailing nonwoven PGA mesh coated with poly-4-hydroxybutyrate (PH4B) to fabricate a trileaflet valve were carried out [7]. This showed an improvement in biocompatibility properties in an environment with autologous myofibroblasts and

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367

endothelial cells. Even in this condition, this type of composite valves displayed longer degradation time while maintaining the valve mechanical strength. These are some of the developments in composite organ implants and a detailed description of different types in relation to their properties will be elaborated on in the next sections.

10.1.1.3 Composites for bone implants Numerous developments in composites for bone replacements or implants have been made in recent years. It is evident that the lack of donors for supply of tissue used in autografts, increased risk of infection, additional surgeries, and donor site morbidity have affected the popularity of the use of autografts for treatment of bone defects and fractures [8]. This has therefore led to the use of metals and ceramics which have displayed a brief success and were halted due to their indifference in properties to bone tissue. Another issue with metals was that proteins could not be embedded with the only option being coating, which is less preferred. Bone replacement composites require a material that can mimic the properties of an actual bone, cable of assimilating into the bio environment tissue eliminating infection potential, differentiation of osteogenic and progenitor cells, and able to stimulate their proliferation [9]. This therefore led to a combination of two or more constituents to formulate a material with all the required properties. Bone replacement composites have different continents to mimic the structure of an actual bone as shown in Fig. 10.3. These include the inorganic mineral component made of hydroxyapatite, the main organic component made of collagen, other organic components in small amounts and water. This has led to improvements in the regeneration of bone structure using composite materials. Another material of interest to composite bone replacement students is ceramic polymer composites used as fillers in defects within the bone structure. There are several ceramic composites used in

Pelvis Acetabulum Femoral head Cu

t

Insert Femur

Articulating surfaces Native hip

Acetabular cup

Textured stem

Total-hip components

Figure 10.3 Joint and bone replacement implant of metal and polymer constituents’ composites [11].

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Durability of Composite Systems

this type of application with properties of biocompatibility, osteointegration, and osteoconduction. These include hydroxyapatite, tricalcium phosphate, and calcium phosphate ceramics [10]. Some composite materials such as bioceramic glasses illustrated preferable even vascularization and bonding with bone properties, their only limitation being their stiffness and brittle nature. Therefore, these types of materials are added to a polymers to develop a material with properties from individual constituents. Composites that have good binding properties for osteogenic proteins and mineral deposition are the most preferred, hence the popularity of collagen type I. Similar to discussed polymers, collage is combined with ceramics to formulate a better material for bone regeneration. This also brought forth the use of biocompatible composites such as one made of two constituents of bioceramics and polyglycolic acid (PLA). The fabrication process of this composites follows the submergence of mSBF and PLG in a slurry of bioglass for a few minutes resulting in a composite with smooth pores similar to an actual bone structure [12]. These properties and materials used in bone replacement composites will be discussed in the next sections.

10.1.1.4 Tissue engineered composites The process of defected or damaged tissue restoration has brought forth the study of tissue engineered therapies. This involves different methods, but this chapter focused on the use of implantable composite materials to promote repair of defected tissues. A majority of tissue engineering composites have not made it to practical applications because of their unstable biological properties, mechanical properties, and in other cases methods of fabrication [13]. Studies have been developed to resolve these issues by the use of constituents involving new biomaterials, inductive factors, or cell sources to contribute to the improvement of damaged tissue growth. This is done by adding characterized biomaterial with required tissue repair properties to an otherwise constituent to develop a composite with better properties. Other applications of these types of composites have played a great role in the central nervous system (CNS) research. The CNS is a complex system and any cell damage after injury within is not replaced leading to the neuroglia-restricted course. This, therefore, leads to the growth of axons in the defect terminating at the lesion site limiting the regeneration of neuronal circuitry [14]. This process results in the formation of a barrier to axon generation known as the glial scar. This is a challenge that tissue engineers addressed with the development of a composite that can allow proliferation and neuronal infiltration without affecting the blood brain barrier in such a sensitive environment. There are different materials that contribute the composite constituents used in tissue engineering. For the same application in CNS environments, the use of composites entailing single-walled carbon nanotubes has become prevalent because of their corrosion resistant, electrical resistant, and mechanical stability properties. There are different combinations with examples used in neuroblastoma and glioma hybrid cultured cells such as composites made from

Durability of medical composite systems

369

polydiallyldimethylammonium chloride (PDDA) and layer-by-layer assembly of SWNTs and polyacrylic acid (PAA) [15]. These types of materials showed more compatible properties with the bioenvironment.

10.1.1.5 Composites for drug delivery The use of drug delivery systems that deliver the required treatment to the desired location while reducing the side effects to the overall human has led to numerous materials research. Biodegradable composite systems when triggered by a lase are making a greater impact in these applications. This drug delivery systems research has gained popularity because of their safety factor, convenient for the host and very good for the system’s efficacy [15]. This type of devices has the advantage of delivering controlled doses of drugs either for a long period of time or short periods of dosages and contribute to the reduction of any side effects as the number of drug administrations is also reduced. Polymer composites development for this purpose have therefore been of interest especial for the development of colloidal systems that can just be injected in to the human body and carry drugs to the required location. Hydrogels (HGs) have, therefore, become one of the common constituents used in these system composites. This is because of their properties of softness, thermodynamically compatible with water, elastic, and ability to carry cells and drugs [15]. Other properties that makes these materials more preferable for this application are as follows: • • • • • •

Mucoadhesive Bioadhesive characteristics Enhancing drug residence time Tissue permeability Relatively deformable Conform to the shape to which they are confined

These materials have limitations too that lead to the addition of polymeric nanoparticles (NPs), which include the inability to control the release of low steric hindrance molecules, incapacity to carry and release hydrophobic drugs and finalize the selection of cells therapies [16]. NPs bring the advantage of size flexibility, hydrophilic turntable nature, and functionalization of the surface.

10.1.2 Composites for external medical devices This section gives details of composites that are used in different medical applications that are not placed inside a human body. These composites include wheelchairs, hospital tools, medical machinery, and wearable devices. Composites in this realm have similar properties to most composites used in different industries as the environment of operation is not as complex as that of internal composites.

10.1.2.1 Composite wheelchairs and surgical tools There are different developments that have been made in the wheelchair and other patient-assisting composite tools industry. Composites in this application focus

370

Durability of Composite Systems

more on the ability to be designed into required shapes and the fatigue performance of the material. This has therefore led to the use of carbon fiber reinforced polymer composites. Coating constituents are added to these types of materials to improve their biocompatibility properties. Traditional wheelchair models were made of purely metals, and this in addition to other limitations had a lower fatigue performance. Comparison studies have been made for materials such as carbon fiber reinforced polymer composite with titanium and stainless steel. During various dynamic loading tests metal components showed earlier fatigue failure in the range of 40,000 cycles for titanium, 440,000 cycles for steel, and 4 million cycles for components with carbon fiber. This improvement in fatigue performance is an advantage in material durability extending the life span of the device. Some parts of the wheelchair and certain patient devices may require softer material and therefore softer configurations of PEEK constituent to metal are fabricated and interconnected with available structure to archive patient comfortability. There are numerous applications of composites in medical tools. These vary from application to application. The design and constituents of the composite are determined based on the medical environment the tool is used in. These composite tools could be very small parts in minor but sensitive applications such as bolts used for bone support, to components of medical machinery such as X-ray equipment or any other large-scale component. Applications also include complex designs of state-of-the-art carbon fiber exoskeletons and mammogram plates. In medical tools and other machinery, carbon prepreg with a designed specific matrix is used because of its ability to maintain its transparency and still has the mechanical properties of carbon fibers [17]. Further properties and their relation to the environment of application will be discussed in next sections. The advantage of composites in this application is that they have good structural strength compared with metals and still maintain greater flexibility. Composites also have a property of radiolucency, hence their application in X-rays, with an added advantage of limited tissue adhesion.

10.1.2.2 Composites for medical machinery Composites in this application are very similar and interconnected to the application in medical tools. The improvement of the quality of life and deployment of a health and life-saving processes in the main objective of medical machinery. A majority of medical machinery use material with high strength to weight ratio property, materials that are inert to the human body environment and nonferromagnetic. This has therefore contributed to the increased use of metals for these applications such as gold, steel, platinum, niobium, silver, and tantalum. The main disadvantages of these materials include the difficulty to find them and flexible manufacturability. This has therefore led to the need to develop new materials as replacements at good cost while still maintaining the required properties. Composite materials have therefore been a viable option to replace the use of metals in these applications. Composites can be used in different environments without drastic variations that may affect the host, which include magnetic exposure; hence their use in postoperative magnetic resonance imaging (MRI) equipment. Throughout the development of composites materials, various

Durability of medical composite systems

371

biocompatible resins have been developed, which are noncarcinogenic and nontoxic and therefore used with carbon fiber constituents to develop a composite with the best properties at a lower cost. Further students in medical machinery composites have contributed to the development of cryogenic ablation systems and tools, delivery for mobile respiratory systems therapies, MRI safe medical equipment and light-weight oxygen storage systems. The use of composites in these applications have improved the machinery electrical performance and extended the life span of the devices. Most developed resins for medical systems can provide ultrahigh magnetic fields that are very useful in the design of new MRI machines coupled with advanced resin insulations, which are more affordable. In all medical machine industry research, a high focus is paid to the development of resin and composite materials that improve the magnetic field generation and proper insulation to guarantee smooth operation of medical machinery with composites systems.

10.1.2.3 Composites for prosthetic devices This section details composites that used in prosthetic applications. These are devices used to replace missing components of the human body lost due to diseases or accidents. The design of prosthetic is of different types be it shank, socket or if lower limb, then foot. The attachment of a lower limb can vary based on the cutoff section. It may be just a foot or include joints, and all this contributes to the materials constituents. All these vary in their object hence also contributing to their design. All these prosthetics have different names depending on how they are attached to the body, which include ankle disarticulation, lisfranc, symes, shopart, and others [18]. Prosthetics are also applied in cutoff parts of the body such as a finger. This is more popular, and different materials for this application have been developed. Composites used in this type of prosthetics require biocompatibility to impose the interaction of vessels with the prosthetic. The composite constituent used in this type also have both elastic properties and biocompatibility properties [19]. The stated application of cutoff or broken finger prosthetic requires grafted composites that can be attached to the tip of the target host. This are nonimplantable options for prosthetics and therefore have different properties. There are two processes involved while the composite is attached to the target, first being the nourishment of the tip by diffusions followed by neovascularization. The use of composite grafting is cost-effective and a very simple process. This process has the capability of sensory restoration, glabrous soft tissue coverage, near normal nail complex using the host or patient’s tissue interlinked with that actual composite structure.

10.1.2.4 Composites for wearable devices Composite materials have gained momentum in the field of devices that are attached to the human body for collection of useful data. These devices can monitor different functions within the body including host blood flow, diseases, heartbeat, and others. There are two classification of wearable devices that are invasive and continuous invasive

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Durability of Composite Systems

devices [20]. These types of devices provide a better solution compared with the traditional noninvasive devices. Wearable devices promote continuous monitoring of bodily functions and can perform certain routine examinations. This further promotes awareness on certain health-related issue and gives hosts an opportunity to look in their body performance. These devices are therefore designed for both hush handling by nonprofessionals and still have the sensitivity to detect even the lightest pulse within the host’s body, Fig. 10.4 shows an example of a wearable device design. Developments in this study have led to the use of flexible composite materials. The application of composite constituents in applications such as pressure, strain, heartbeat rate, temperature, skin odor, and breath sensing has become the main objective in this area [22]. The composite type selected for each application is designed and prepared to sense a specific property without being altered by its surroundings. The constituents of these composites vary for devices that physical markers such as pressure pulse, body temperature, or devices that detect chemical species on human skin such as minerals, trace elements, lactic acid, and other organic compounds [23]. Composites for wearable devices should therefore be compatible with the human skin, that is, they ought to be flexible, stretchable, and limited irritation to the host [24]. The use of flexible elastomers as constituents has prevailed in the development of wearable devices for strain responsive devices. Polymers with soft fibers and matrix that can adapt the host shape and still able to maintain sensing properties are used.

10.2

Properties affecting the durability of medical composite systems

This section gives the properties of medical composite systems contributing to their durability. Each property is related to a specific application in medical systems. A review of the developments in composite properties is done highlighting biocompatibility of composites, thermal and electrical expansion, adhesion strength, shear modulus, and impact strength.

10.2.1

Biocompatibility

The application of composites in biomedical research requires materials that do not contain any toxins and diffusible substance that could have any side effects on any organ they get in contact with. Biocompatibility of composite systems mainly focuses on the protection of humans and minimizes the number and exposure of test animals [25]. The property of biocompatibility is observed in different medical composite systems and contributes to the measure of composite systems durability. An example of this would be biocomposites dental devices. Dental composites are composed of a resin matrix and filler materials. Coupling agents are used to improve adherence of resin to filler surfaces in presence of activation systems including heat, chemical, and photochemical initiate polymerization. Plasticizers are solvents that

Nanoparticle (NP)

(b)

(i) Nano-structured plastic base (NPB)

(ii) Coplanar electrode patterning

(iii) nano-composite insulator (NCI) coating

B A’

5 cm

(d)

A

(c)

B’

c=

A

(iv) UV-curing

NCI

Nanograting PUA PET substrate

NPB A’

(e)

εA d

Coplanar electrode

(v) Hierarchical nano-composite (HNC) nano-force touch sensor (unit sensor)

(f)

Durability of medical composite systems

(a)

B

B’

(Percolation effect in HNC film)

Nanocomposite insulator

Press

Press

Δε 1 cm

Nanograting PUA

Sparse NPs

Release

Dense NPs

373

Figure 10.4 Wearable device design using different material constituents from a more durable functioning composite [21].

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Durability of Composite Systems

contain catalysts for mixture into resin. Monomer, a single molecule, is joined together to form a polymer, a long chain of monomers. More relevant materials in composites related to medical applications and their biocompatibility are given in Table 10.1. Table 10.1 Biocompatible materials used in composite applications and properties. Material

Application

Biocompatibility

Glasseionomer cements

Dental composites

Resin-modified glass ionomer and resin composite despite showing some degree of biocompatibility interfered with the development of new bone and the connective tissue attachment process

BiseGMA/TEGDMA-based composite resins

Dental and bone composites

Evaluated immunochemistry, morphologic, total-etching adhesive systems confirm biocompatibility of self-etching or total-etching adhesive systems when placed directly on dentin

Polyvinyl alcohol (PVA) and magnesium

Biomedical devices and implants

Material of biodegradable and biocompatible nature with good drug release properties

Polymethyl methacrylate (PMMA) denture base material

Dental composites

Denture base materials have been developed for patients with allergy to PMMA. These materials contain Significantly lower residual monomer than PMMA

Bisphenol-epoxy/carbon-fiberreinforced composite

Bone replacement

Stimulated osseointegration inside the tibia bone marrow. High-strength biomaterials with a density close to bone for better stress transfer and electrical properties that enhance tissue formation

Collagen-based composite with calcium phosphate (CaP) bioceramics and calcium silicate (CaSi) bioceramics

Bone structure and other implants

Possess great biocompatibility, osteoconductivity, and bone-bonding ability

Single-walled carbon nanotubes (SWCNTs)/poly (lactic-co-glycolic acid) (PLAGA) composites

Bone and tissue regeneration

Composites exhibit biocompatibility like the food and drug administration approved biocompatible polymer. No elicit a localized or general overt toxicity. An acceptable biocompatibility to warrant further long-term and more invasive in vivo studies.

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Physical characteristics improve by combining more than one type of monomer and those materials are referred to as a copolymer. Cross-linking monomers join long chain polymers together along the chain and improve strength [26]. Unbound monomers may result in the reduction of biocompatibility of a polymer composite. Organic polymerized matrix in dental composites also improves the composite biocompatibility coupled with inorganic quartz fillers, which are more stable in an aqueous environment.

10.2.2 Thermal expansion Medical system applications include environments of different temperatures. The developments in composite structures have improved the durability of such materials in these environments. This includes contributions from composite with a high coefficient of thermal expansion and coefficient of moisture expansion. The difference in thermal or moisture expansion between the constituents of a composite or a coated medical system or material can induce large stresses or strains and can eventually lead to failures. The thermal mismatch between constituents is minimized to such a degree that stresses generated in the experienced temperature domain are acceptable. The coefficient of thermal expansion (CTE) of composite materials intended for high stability structural applications in sensitive medical applications is a factor to be determined and controlled. A sensitivity analysis is performed before material application in different medical systems in relation with the inaccuracies due to the manufacturing process to validate the required CTE. A few examples of some of the composite materials used in medical composites applications are given in Table 10.2.

Table 10.2 Thermal expansion of materials used in medical composite applications and properties. Material

Application

Thermal expansion

Polymer/carbon nanotube (CNT) composites

Organ implants

High thermal conductivity on the order of 1000 Wm
1 K
1 at room temperature. Some efforts have been made in fabricating polymer/CNT composites which have a better thermal transport property than bulk polymers

Silk fibroin base composites

Drug delivery and other applications

High thermal conductivity, up to 416 W/ m$K. Outstanding mechanical durability, stable chemical properties and good biocompatibility Continued

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Durability of Composite Systems

Table 10.2 Continued Material

Application

Collagen-based composites

Implants and drug delivery

Polyvinyl alcohol (PVA) incorporated with boron nitride nanotubes

Organ restoration

Higher thermal conductivity than that of pure PVA mats. The k value of the film increases as the volume fraction and alignment degree of nanofiller increases.

Graphene nanoplatelets composites

Implantable devices and organs

The thermal conductivity is improved according to the filler size; the highest bulk and in-plane thermal conductivity values of the composites are at the highest filled GNPs composite, respectively.

10.2.3

Thermal expansion

Elastic modulus and toughness

The property of elastic modulus in medical composite systems is applicable in all medical environments entailing various loads. At the atomic level, elastic deformation takes place by the deformation of bonds in a composite system during loading. For composites used in medical systems involving varied loads, a property of high impact strength, and fracture toughness is necessary to archive high durability. Fracture toughness is a measure of the damage tolerance of a material containing initial flaws or cracks. The fracture toughness in materials is described by the plain strain value of the critical stress intensity factor. The fracture toughness depends on the environment of application. Dental composites should have various properties which include compressive strength. This is the resistance of a material to breaking under compression. This is especially important in posterior restorations. Composites should also have adaptation and handling ability. It is defined as how well it adapts to the walls of the preparation. Handling is related to personal preference, how well one is able to work with the material to achieve the desired results. Translucency is another property. Bulk fills must be translucent in order to cure in one layer to greater depths. This can necessitate opaquer and liners to block stains and mimic the natural tooth. Depending on the esthetic demands and the location in the mouth, this can affect material choice. The limiting factor in the long-term effectiveness and prognosis of a composite is shrinkage during polymerization. Shrinkage leads to poor marginal seal, higher potential for recurrent caries, and marginal staining. Microfilled composites offer high elastic modulus. Hybrid composites also have high elastic modulus. The newer nanofilled composites have even greater elastic modulus than hybrids.

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377

Types of failure

Failure in medical composite systems is due to different parameters that hinder the defined required properties. This may be due to the environment a composite is placed in or the loads and forces it goes through. In general, this is the event that damages the composite structure or the point of structural collapse in medical composite systems. Current failure modes analysis studies reveal the importance of the individual constituents’ properties contributing to the overall composite properties. This section only focuses on failures in medical composite systems that affect the system’s durability. These failures include composite degradation, wear and tear, composite fracture, cavitation, and chemical corrosion.

10.3.1 Degradation and corrosion The degradation of composites in medical is systems is due to different factors including moisture, temperature, and environmental chemical reactions. Different mechanisms of degradation are involved which include time dependent deformation and resultant damage accumulation, environmental attack, and accelerating effects of elevated temperatures and microstructural and compositional changes. One of the commonly used composites in medical systems is the fiber-reinforced polymer composites. Attention is brought to this material due to its desirable properties of low friction coefficient, wear resistance, toughness, damage tolerance, chemical resistance, and corrosion resistance [27]. The use of this type of composites in medical systems has brought about unforeseen failures such as degradation. Studies show that composites may suffer corrosion and biofouling in the presence of microorganisms such as bacteria. The presence of moisture in other biomedical systems such as implantable devices lead to degradation coupled with osmotic blistering due to the difference in working environments. Materials such as hybrid resin composites with photo initiating substances benzil (BL) and dimethoxybenzoine (DMBZ) and triethylene glycol dimethacrylate (TEGDMA)ebased composites used in dental applications have displayed a lower monomer leaching value, which in turn improves composite degradation and durability. Medical systems also involved composites that are coupled with metals. This may be in bone replacement organs or body sensors. This type of composite systems also goes through degradation and may be due to similar reasons coupled with other factors. Studies showed that carbon or polymer composites galvanically coupled with metals are degraded by cathodic reactions in oxygenated seawater. This type of degradation is also present in humans in the presence or oxygenated bodily fluids. Other factors proved in different studies are of steel elements coated with epoxy and nylon to form a hybrid with metal composite degradation due to the presence of bacteria. All sections of a fiber reinforced polymer composite, fibers, resin, and fibereresin interface are susceptible to some form of microbial degradation affecting the overall medical composite system durability. Possible mechanisms for microbial degradation of polymeric composites include direct attack of the resin by acids or enzymes, blistering due to gas evolution, enhanced cracking due to calcareous deposits and gas evolution, and polymer destabilization by concentrated chlorides and sulfides. Any attack may result in loss of strength due to fracture, debonding, or delamination, and ultimate failure.

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The chemical environment to which a material is subjected in its life span may cause changes in the material properties. Corrosion is the reaction of the engineering material with its environment with a consequent deterioration in properties of the material. Corrosion will include the reaction of composites with environments that embrace liquid metal, gases, nonaqueous electrolytes and other nonaqueous solutions, coating systems, and adhesion systems in medical applications. Corrosion in medicals systems is one of the major failures contributing to the reduction of material durability. This may be in different environments with heat, moisture, chemicals, radiation, electricity, biological substances, or combination in applications such as organ implants, wound support composites systems to dental composite systems with different corrosions rates and durability. Chemical environments in medical composite systems may also lead to corrosion fatigue. Corrosion fatigue indicates crack formation and propagation caused by the effect of alternating loading in the presence of a corrosion process. Because of the time dependence of corrosion, the number of cycles to failure depends on frequency. Since chemical attack requires time to take effect, its influence is greater as the frequency becomes lower.

10.3.2

Cavitation

Medical composite systems are subjected to environmental effects of different properties and have varied effects on the composite material. Environmental effects on composite materials depend on the properties of the individual componentsdfiber, matrix, and the interface between the fiber and the matrix. Liquid environments for medical systems may lead to cavitation and bubbles formation in composite materials resulting in the composite lifespan reduction. Cavitation may be defined as a phenomenon, which creates damage to a structural material in contact with a liquid in which the pressure fluctuates because of the flow pattern or vibrations in the system [28]. As cavitation occurs, a composite structure is subjected to repeated local shock loading and experience very high strain rate. There are varied deformations during cavitation which include localized surface and subsurface effects and the deformation of the whole structure. Major effects are from the localized impact loadings from bubble collapse in liquid environments embedded medical composites. The major concern in cavitation studies is on the localized impact loading by the bubble collapse. Fiber-reinforced metal matrix composites will have a superior resistance to cavitation erosion compared to that of the matrix metals and alloys. In metal matrix composites, choices can be made between two classes of reinforcements, namely, ductile fibers such as stainless steel, tantalum or beryllium, and brittle fibers such as graphite, silicon carbide and alumina. In ductile matrix or ductile fiber composites, both constituents can contribute to the resistance to cavitation damages such as composite indentations, pits, troughs and valleys, craters, micro- and macrofracture. The limiting factors for this contribution will be fiber diameters and orientations, fiber volume fraction and the characteristic of fiber and matrix interface. Cavitation energy absorption by the composites in medical systems can be maximized by the proper selection of matrix and fiber. This selection should include composites with low stacking fault energy, low strain-rate sensitivity, relatively large diameter fiber which

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may undergo plastic deformation without fracturing, moderate fiber volume fraction to allow larger interfiber distance, fibers normal to the surface, and good interfacial bonding to allow subsurface microbuckling of fibers.

10.3.3 Wear There are several mechanisms of composite wear including adhesive wear, abrasive wear, fatigue, and chemical wear. Adhesive wear is created by extremely small contacts and therefore extremely high forces, of two opposing surfaces. When small forces release, material is removed. All surfaces have microscopic roughness which is where extremely small contacts occur between opposing surfaces. Abrasive wear is when a rough material gouges out material on an opposing surface. A harder surface gouges a softer surface [29]. Materials are not uniform so hard materials in a soft matrix, such as filler in resin, gouge resin and opposing surfaces. Fatigue causes wear. Constant repeated force causes substructure deterioration and eventual loss of surface material. Chemical wear occurs when environmental materials such as saliva, acids or like affect a surface.

10.4

Improving durability of medical composite systems

For years, composites have been considered as the advanced materials in most medical systems applications. Studies on their reliability and lifespan have been conducted to better understand their durability and accurately determine their lifespan. Various experiments under accelerated aging environments have been conducted in controlled laboratory conditions and acquired data compared with in-service data to formulate data on medical composites durability. Medical composite systems that suffer degradation due to moisture in the application environment required specific steps to improve their durability. i. Certain manufacturing process for composites used in medical tools or machinery such as hot-pressing fabrication method has proved to yield composites that can withstand moisture. ii. Composite treatment, coating with polymers that protections from certain chemical reactions reduces decay time and therefore improves durability. iii. Use of nanoscale fillers in composite materials improves the material mechanical properties hence improving durability. iv. Curing temperature of composites varies based on the type of composite and area of application and it also affects the composite strength. v. The use of biocompatible constituent in the fabrication of implanted composites improve the medical composite system durability.

Failures such as this including composite blistering can be eliminated by proper manufacturing and maintenance procedures [30]. Composite systems have varying degrees of durability and this depends on their properties and application environment. Decay of composites will generally occur where the moisture content is above tolerance specific to a composite material for a prolonged period. Where durability

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against the risk of decay or chemical attack is required, chemical preservative treatments may be applied. Preservatives are often used as a second line insurance against design or construction failures that could lead to moisture contents above limit. Alternatively, specific combinations of composite constituents may be adopted for applications where durability is required. Liquid environments have been shown to have a higher effect on composites durability in medical applications. This is due to different factors such as surface swelling caused by osmosis. Osmosis can be defined as the equalization of solution strength by passage of a liquid through a semipermeable membrane. Normally the fluid will pass through the material without affecting it but, there may be soluble materials. This results in composite swelling leading to blisters with associated surface undulation reducing the life span of a composite material. Numerous composites in medical applications contain large proportions of polymers derived from petrochemicals and have durability costs associated with their application. These materials are highly resistant to biological degradation but susceptible to chemical decomposition.

10.5

Closing remarks

Medical composite systems are applied broadly from traditional medical machinery and surgical tools to emerging fields of tissue engineered platforms and implantable devices. These composites are designed to exhibit properties, such as improved strength, corrosion resistance and biocompatibility, which are uniquely appropriate for each application. The durability of the medical composite systems often depends on a complex interaction between the composite systems and their application environment. Understanding their failure processes supports the efforts of improving the durability of the medical composite systems, which can be achieved through materials designs and the design of their manufacturing processes. Ensuring the durability of medical composite systems are crucial in many clinical applications where repeated or chronic usage of such composite systems are required.

References [1] S.I. Sadaq, N. Seetharamaiah, J. Dhanraj Pamar, A. Mehar, Characterization and mechanical behavior of composite material using FEA, Int. J. Eng. Res. 2 (2) (2013) 125e131. [2] E. Wintermantel, J. Mayer, T.N. Goehring, Composites for Biomedical Applications,” in Encyclopedia of Materials: Science and Technology, Elsevier, 2001, pp. 1371e1376. [3] Composite Fillings e Tooth-Colored Fillings e American Dental Association. [Online]. https://www.mouthhealthy.org/en/az-topics/c/composite-fillings. (Accessed 27 November 2019). [4] S.P. Victor, C.K.S. Pillai, C.P. Sharma, “Biointegration,” in Biointegration of Medical Implant Materials, Elsevier, 2020, pp. 1e16.

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[5] Dental Composites: What to Look For and What to Get e Spear Education. [Online]. https://www.speareducation.com/spear-review/2017/01/dental-composites-in-2017-whatto-look-for-and-what-to-get. (Accessed 27 November 2019). [6] S. Iyer, Vascularised composite allotransplants: transplant of upper extremities and face, Indian J. Plast. Surg. 48 (2) (May 2015) 111e118. [7] P.E. Herbert, A.N. Warrens, Solid organ and composite organ transplantation - the dynamic equilibrium of rejection and immunosuppression, J. Surg. 2 (1) (2004) 12e14. Surgical Associates Ltd. [8] (PDF) Application of Bioactive Composite Green Polymer for the Development of Artificial Organs. [Online]. https://www.researchgate.net/publication/310892760_ Application_of_Bioactive_Composite_Green_Polymer_for_the_Development_of_ Artificial_Organs. (Accessed 27 November 2019). [9] N. Sikich, A. Mitchell, A. Ali, C. Soulodre, Health Technology Assessment at Health Quality Ontario, 2016. [10] H. Davis, J. Leach, J. Kent Leach, Hybrid and Composite Biomaterials in Tissue Engineering Hybrid and Composite Biomaterials for Tissue Engineering, 2008. [11] New Joint Implant Coating to Prevent Bacterial Infections j Medgadget. [Online]. https:// www.medgadget.com/2017/07/new-joint-implant-coating-prevent-bacterial-infections. html. (Accessed 28 November 2019). [12] S. Eo, G. Doh, S. Lim, K.Y. Hong, Analysis of the risk factors that determine composite graft survival for fingertip amputation, J. Hand Surg. Eur. Vol. 43 (10) (December 2018) 1030e1035. [13] F. Idone, A. Sisti, J. Tassinari, G. Nisi, Cooling composite graft for distal finger amputation: a reliable alternative to microsurgery implantation, In Vivo 30 (4) (2016) 501e505. [14] D. Younge, O. Dafniotis, A Composite Bone Flap To Lengthen A Below-Knee Amputation Stump, 1993. [15] E. Mauri, A. Negri, E. Rebellato, M. Masi, G. Perale, F. Rossi, Hydrogel-nanoparticles composite system for controlled drug delivery, Gels 4 (3) (September 2018) 74. [16] C.Y. Yu, B.C. Yin, W. Zhang, S.X. Cheng, X.Z. Zhang, R.X. Zhuo, Composite microparticle drug delivery systems based on chitosan, alginate and pectin with improved pHsensitive drug release property, Colloids Surf. B Biointerfaces 68 (2) (February 2009) 245e249. [17] A. Leto Barone, A. Arun, N. Abt, S. Tuffaha, G. Brandacher, Nerve regeneration in vascularized composite allotransplantation: current strategies and future directions, Plast. Aesthetic Res. 2 (4) (2015) 226. [18] J. Choo, B. Sparks, M. Kasdan, B. Wilhelmi, Composite grafting of a distal thumb amputation: a case report and review of literature, Eplasty 15 (2015) e5. [19] A. Foroohar, L.S. Levin, Quadrimembral Amputation: A Review and Perspective on the Role of Composite Tissue Allotransplantation, 2011. [20] Y. Shi, C. Wang, Y. Yin, Y. Li, Y. Xing, J. Song, Functional soft composites as thermal protecting substrates for wearable electronics, Adv. Funct. Mater. 29 (45) (November 2019), 1905470. [21] J.Y. Yoo, M.H. Seo, J.S. Lee, K.W. Choi, M.S. Jo, J.B. Yoon, Industrial grade, bendinginsensitive, transparent nanoforce touch sensor via enhanced Percolation effect in a hierarchical nanocomposite film, Adv. Funct. Mater. 28 (42) (October 2018). [22] M. Zhao, D. Li, J. Huang, D. Wang, A. Mensah, Q. Wei, A multifunctional and highly stretchable electronic device based on silver nanowire/wrap yarn composite for a wearable strain sensor and heater, J. Mater. Chem. C 7 (43) (2019) 13468e13476.

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[23] B. Song, F. Wu, K.S. Moon, R. Bahr, M. Tentzeris, C.P. Wong, Stretchable, printable and electrically conductive composites for wearable RF antennas, in: In Proceedings e Electronic Components and Technology Conference, vol. 2018, May 2018, pp. 9e14. [24] Y.Q. Li, et al., Multifunctional wearable device based on flexible and conductive carbon sponge/polydimethylsiloxane composite, ACS Appl. Mater. Interfaces 8 (48) (December 2016) 33189e33196. [25] P.A. Wagner, B.J. Little, K.R. Hart, R.I. Ray, Biodegradation of Composite Materials, 1996. [26] Biocompatibility Testing At Pacific Biolabs. [27] A.K. Mohanty, M. Misra, G. Hinrichsen, Biofibres, biodegradable polymers and biocomposites: an overview, Macromol. Mater. Eng. vols. 276e277 (2000) 1e24. [28] D.A. Hammond, M.F. Amateau, R.A. Queeney, Cavitation erosion performance of fiber reinforced composites, J. Comp. Mater. (1993). [29] M. Raji, H. Abdellaoui, H. Essabir, C.-A. Kakou, R. Bouhfid, A. el kacem Qaiss, Prediction of the cyclic durability of woven-hybrid composites, in: In Durability and Life Prediction in Biocomposites, Fibre-Reinforced Composites and Hybrid Composites, Elsevier, 2019, pp. 27e62. [30] E. Bari, J.J. Morrell, A. Sistani, Durability of natural/synthetic/biomass fiberebased polymeric composites, in: In Durability and Life Prediction in Biocomposites, FibreReinforced Composites and Hybrid Composites, Elsevier, 2019, pp. 15e26.

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Rassel Raihan 1 , Muthu Ram Prabhu Elenchezhian 1 , Vamsee Vadlamudi 2 1 Mechanical and Aerospace Engineering, University of Texas Arlington (UTA), Arlington, TX, United States; 2University of Texas Arlington Research Institute (UTARI), Fort Worth, TX, United States

Chapter outline 11.1 11.2 11.3 11.4 11.5 11.6

Introduction 383 Types of adhesive bonding 386 Theories of adhesive bonding 387 Surface treatment method of adhesive bonding Surface characterization methods 390 Durability and materials state analysis 392

389

11.6.1 Environmental/aging 394 11.6.2 Broadband dielectric spectroscopy for bond material state assessment 395 11.6.3 Quality assessment of adhesive bonds based on broadband dielectric spectroscopy 396

11.7 Conclusion References 399

11.1

398

Introduction

Adhesive bonding is a material joining process where an adhesive solidifies between the adherend surfaces. In the past decades, there has been an increased use of adhesively bonded joints as an alternative to mechanical joints in engineering application across many industries such as aerospace, automotive, sport, biomedical, electronics, marine, and oil The history of using adhesive to bond different materials is as old as humankind. Early humans learned different techniques from the best and that is Nature, in which they observed how insects and birds were building and repairing their nest with mud and clay. They developed the first composites with straw and mud that they used to make their habitat to protect themselves from increment weather and ferocious animals. Prehistoric people used natural glue to fix broken potteries. Babylonians utilized bituminous cements to attach ivory eyeballs to the statues for their temples. Ancient Egyptians were familiar with the production of bonded abrasives; they created laminated wood for bows and furniture by using crude animal and casein glues. Durability of Composite Systems. https://doi.org/10.1016/B978-0-12-818260-4.00011-9 Copyright © 2020 Elsevier Ltd. All rights reserved.

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In modern time, first plant-based glue commercialized in 1690 in Holland and in 1840 for postage stamps starch adhesives was used [1]. Thomas Hancock have used hard rubber (Ebonite) for bonding to metals. In 1927, sulfuric acid used to treat rubber to make solvent-based thermoplastic rubber cements for metal-to-rubber bonding. Thermosetting solvent-based rubber cements first became available between World Wars I and II [2]. Polyester resins (alkyds) were commercialized in 1926 to use in coatings. Fiber glass composites were made by using unsaturated polyesters as matrix resins in the 1940s. The United States typically uses expensive epoxy adhesives where most of the countries in the world for similar applications use unsaturated polyesters as adhesives for thermoset plastics bonding. After these, many new types of adhesive entered in the commercial market, i.e., nitrile-phenolic, nylon-phenolic, epoxyphenolic, nylon-epoxy, modified epoxy-phenolic, polyimide, bismaleimide, and many more, with better favorable adhesion properties for different operating conditions. Nylon epoxy adhesives, extremely “tough” adhesives, were used to laminate helicopter rotor blades in 1960. Fig. 11.1 is indicating the areas of Fokker F-100 aircraft that are adhesively bonded [3]. Fuselage, wing structure, and engine housing were adhesively bonded. Automotive industries use adhesive bonding for automobile hoods where the top panel is bonded with the stiffener, automobile doors and hem-flange bonding are used to bond and seal, and adhesives are used to bond friction surfaces in brakes and clutches. Construction industries and furniture industries use adhesive bonding routinely. Microelectronics industry uses adhesives in the manufacturing of integrated circuits.

Adhesively bonded laminate and stringers Adhesively bonded laminate Adhesively bonded metal sandwich Aramid fibre composite Carbon fibre composite

Figure 11.1 Adhesively bonded section of Fokker F-100 Aircraft.

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The use of adhesive bonding is increasing because of numerous advantages; a few are mentioned in the following [2,4,5]. ❖ Adhesively bonded joints distribute stress evenly across the bond line, whereas mechanical fasteners create stress concentration points, which lead to premature failure (Fig. 11.2) and can therefore support larger stress-bearing areas. ❖ Adhesive bonding enables joining of thin or thick materials of any shape, similar or dissimilar materials. ❖ Polymeric adhesives between the substrate minimize or prevent electrochemical (galvanic) corrosion between dissimilar materials, i.e., metaleCFRP, steelealuminum. ❖ They have high tolerance to fatigue and cyclic loads. ❖ Unlike mechanical joints, adhesive bonds ensure smooth contours and are also faster and inexpensive to form than mechanical fastening. ❖ Adhesive layers act as a sealant against a variety of environments and also act as an insulative layer against heat transfer and electrical conductance and dampen vibration and absorb shock. ❖ Unlike welded joints, adhesively bonded joints require less heat to make the bond, which is too low to reduce the strength of the metal parts. ❖ Adhesively bonded joints have high strength/weight ratio.

Tough adhesively bonded joints have many advantages, but still they have some weaknesses that hinder wide use of these bonding systems for critical structural bonding [4,5].

Mechanically bonded joint

Stress

❖ Bond strength depends most of the time on the condition of the substrate. Durability of the bond in extreme environmental conditions is affected by the surface condition. ❖ Sometimes, adhesive bonding is more expensive than mechanical bonding and requires tremendous process control, which increases production time. Some adhesives require long cure times. Most of the methods require extra fixtures, presses, oven, and/or autoclaves for bonding process. ❖ Some adhesives are subject to attack by bacteria, mold, rodents, and vermin. ❖ Some adhesives can be hazardous to health.

Bolted joint

Stress

Adhesively bonded joint

Adhesive bondline

Figure 11.2 Stress distribution in mechanically bonded joint versus adhesively bonded joint.

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Adhesive properties Adherend/substrate properties

Bonding process

Materials

Performance and durability of adhesively bonded joints

Loading condition Joint shape and dimension

Bond design

Environment

Surface treatment of the adherend Preparation and application of adhesive Adherend assembly Curing adhesive layer

Temperature Moisture Hygrothermal Chemical attach Other factors (UV radiation, chemical and hygrothermal togather)

Figure 11.3 Factors impacting durability of the adhesively bonded joints. ❖ There is a limitation on the shelf life of the adhesive. ❖ Some adhesives fail catastrophically because of poor resistance to crack propagation. ❖ Visual examination of the bond line is not possible unless the substrate is transparent.

The performance and durability of adhesively bonded joints depends on the bonding process, bonding type, applied loading conditions and application, and environment. Fig. 11.3 is a schematic representation of the factors that influence the longterm performance and durability of the bond. The main objective of this chapter is to discuss the durability and materials state analysis of the adhesively bonded joint using broadband dielectric spectroscopy (BbDS), but before that a brief discussion of different types of bonding, theories of adhesion, role of surface treatment on durability, common surface treatment methods, and characterization techniques will be presented.

11.2

Types of adhesive bonding

A wide variety of adhesive joints are available. The joint strength and long-term durability are affected by the stress concentration at the end of overlap. Many researchers have modified the joint design and geometry to reduce the stress concentration. Fig. 11.4 is a schematic of common joint designs that have been analyzed in the literature. The strength and performance of an adhesively bonded joint depends on (1) type of load, (2) stress distribution within the joint/geometry of the joint, and (3) the mechanical properties of adhesive and adherend [6].

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(a) (d)

(b)

(e)

(c)

(f)

(g)

(h)

(i)

(j) Figure 11.4 Different types of bonded joints. (a) single-lap share joint (b) double-lap share joint (c) tapered single-lap joint (d) joggle-lab joint (e) symmetrical single-lap joint (f) single -strap joint (g) double-strap joint (h) tapered-double-strap joint (i) Recessed-double-strap-joint (j) scarf joint.

11.3

Theories of adhesive bonding

Although adhesiveeadherend interactions take place at the molecular level, there are many theories of adhesion that exist to explain the mechanism, and it is difficult to fully credit an individual theory for adhesive bonding [5,7]. Table 11.1 is a list of common theories that are used to explain adhesive bonding to the substrate and their scale of action. A combination of these mechanisms is most likely responsible for bonding of substrate/adherend and adhesives. In mechanical interlocking, the contact surface between adherend and adhesive is the microscopic parameter of interest (Fig. 11.5(a)). Based on this theory, adhesive

Table 11.1 Theories of adhesion and scale of action. Common theories of adhesion

Scale of action

Mechanical interlocking

Microscopic

Electrostatic

Microscopic

Diffusion

Molecular

Adsorption/surface reaction/wettability

Molecular

Chemical bonding

Atomic

Weak boundary layer

Molecular

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Adherend atoms or molecules Adhesive

Adhesive

Adherend Adherend

(a)

Adherend atoms or molecules

Adhesive

(d) Adhesive

Adherend

(b)

Adherend-highly wettable

Adhesive Adhesive

Adherend-moderately wettable

Adherend

(e)

(c) Adhesive Weak boundary layer Adherend

(f) Figure 11.5 Adhesion theories (a) mechanical interlocking (b) electrostatic (c) diffusion (d) chemical bonding (e) wettability (f) weak boundary layer.

penetrates into the surface irregularities (i.e., pores, cavities) and displaces the trapped air at the interface to make an interlock between the substrate and adhesive. The electrostatic theory is established on the basis of the electrostatic effects between the adhesive and the adherend (Fig. 11.5(b)), which gained support from the observation of electronic discharge during the peel of an adhesive from a substrate. Unlike electronic band, structures between the adhesive and the adherend cause an electron transfer that results an electrical double layer at the adhesivedadherend interface that opposes the separation of adhesive and adherend. Diffusion theory proposes that adhesion is developed through the interdiffusion of molecules in between the adhesive and the adherend (Fig. 11.5(c)). The nature of the adherend and adhesive materials with the bonding condition influences the extent of diffusion. When molecules of two materials come in contact with each other, surface forces develop at the interface, and this is the driving force of adhesion between adherend and adhesive based on wetting theory (Fig. 11.5(e)). Wetting is the continuous contact between substrate and adhesive. To wet a surface and construct bond, adhesive should have lower surface tension than the critical surface tension of the adherend. Chemical bonding mechanism addresses the chemical forces between the adhesive and adherend that takes place during bonding (Fig. 11.5(d)). The chemical composition

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of the interface dictates the exact nature of the interactions for a given adhesive bond. The most common interactions are covalent bonds, hydrogen bonds, Lifshitzevan der Waals forces, and acidebase interactions. Impurities or contamination (i.e., oil, oxide layers) on the substrate or the adhesive layers can result in the formation of weak boundary layers near the bonding surface (Fig. 11.5(f)). Because of the contamination or trapped air on the surface, the adhesive does not wet the adherend properly and adequately to form a good adhesive bond.

11.4

Surface treatment method of adhesive bonding

Surface preparation of the adherend or substrate plays a crucial role in the durability and long-term performance of adhesively bonded joints. The main objective of the treatment of the surface is to promote construction of strong physical and chemical bonding between adhesives and adherend [8,9]. The surface treatments are carried out to achieve the following: ❖ Remove the contamination (i.e., dust, oxide layers, grease, and undesirable coatings): avoid weak boundary layer formation. ❖ Improve texture of the surface: control the surface roughness. ❖ Improve chemical affinity of the surface: increase the surface free energy. ❖ To protect adherend surface before bonding.

The selection of surface treatment method depends on various factors: ❖ Materials related: properties of the adherend/substrate and adhesive materials, surface chemical state. ❖ Design related: shape, dimension, and positioning of the adherend. ❖ Environment of the workplace.

Fig. 11.6 is an example of overall surface preparation methods for formation of a durable adhesive bonds. At the primary stage of surface treatment, mechanical treatment is done to remove loose particles from the adherend surface. Degreasing is a cleansing process that is performed to remove all sorts of contamination, i.e., dust, debris, organic or nonorganic substances of the adherend surface. Edge deburring is done to make sure of intimate contact between adherend surfaces. The aim of the fundamental stage of surface pretreatment is to improve surface free energy and roughness. Mechanical treatment at this step is done to produce a favorable roughness on the adherend surface for increased surface free energy and also to help in interlocking of the adhesive with the adherend surface. To achieve corrosion protection and high physicochemical activity, chemical treatment is done by etching of the surface. On some surfaces, electrochemical treatment is carried out in the form of anodic oxidation techniques [9]. Special treatments methods, i.e., corona discharge, laser, plasma, flame, electromagnetic radiation, and ozone oxidation, are also carried out to ensure increased surface free energy to promote better and durable bonding. Some additional surface treatment methods, i.e., priming, rinsing, drying, and heating, are also performed on the adherend surface for a strong bonding in some cases.

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Surface treatment method

Primary

Fundamental

Mechanical treatment (remove loose particle)

Mechanical treatment (produce favorable roughness)

Degreasing

Chemical treatment

Edge deburring

Electrochemical treatment

Corona discharge

Leaser

Additional

Primming Immersion

Plasma Rinsing Flame

Special treatment

Drying Electromagnetic radiation based Ozone oxidation

Spray

Combined

Atmospheric Heating Hot-air Compresse d-air Vacuum

Sorption

Figure 11.6 Surface treatment methods.

11.5

Surface characterization methods

For the long-term durability and performance of adhesively boned joints, the knowledge of the composition, topological features, and structure of the adhered/substrate surface is essential. The assessment of the pretreated surface helps in design, failure analysis, and performance prediction of the bond. The following are a list of some techniques that are commonly used for surface and materials characterization of the adherend of the bonded joint prior to bonding [10e12]. ❖ Determination of adhesive properties of the adherend surface: Researchers have found that materials with better surface wetting properties result in better adherends for adhesive bonding [13e16].

The following characteristics of the surface can be determined by the contact angle measurements.

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(i) Wettability (ii) Surface Free energy (iii) Surface work function methods

Contact angle is an angle that a liquid makes with another liquid or solid surface. The attraction and repulsion forces between all of the phases (i.e., liquidevapor, solideliquid, and solidevapor) determine the nature of the angle. When there is an attraction between similar molecules such as between the liquid molecules (i.e., hydrogen bonds and van der Waals forces), it is called cohesive force, and when the attraction forces are between dissimilar forces, they are called adhesion force. The balance between the cohesive forces and the adhesive forces will determine the contact angle created in the solid and liquid interface. Fig. 11.7 is a schematic of the contact angle of a liquid drop that is in equilibrium on a solid surface, where g represents the relevant surface energy at the three-phase contact point (i.e., solidevapor [sv], solide liquid [sl], and liquid-vapor [lv]) and q is the equilibrium contact angle. ❖ Determination of surface roughness measurements: Optical profilometer is an interference microscope, with which one can measure surface texture, finish, and roughness in seconds. Optical interference profiling is a very well-known technique of obtaining precise surface measurements. Advantages of optical profilometers are noncontact method, speed, reliability, and small spot size. ❖ Fourier transform infrared spectroscopy (FTIR): FTIR is a form of vibrational spectroscopy that utilizes the absorbance, transmittance, or reflectance of infrared light. The wavelength of the incident electromagnetic radiation that is absorbed by the sample surface depends of the chemical nature of the surface. With this technique, contaminants on the adherend surface can be detected. ❖ Raman spectroscopy: Raman spectroscopy is vibrational spectroscopy technique where an inelastic scattering phenomenon probes the molecular vibrations to provide a molecular fingerprint of materials. This is a good technique for qualitative analysis and is used for identification of organic and/or inorganic compounds that present on the adherend surface. ❖ Ion scattering spectroscopy (ISS) or low-energy ion scattering (LEIS): When a beam of ions hits a solid surface, part of the ions are scattered. Information (i.e., mass, kinetic energy) of these backscatter ions are utilized in ISS to understand the composition of the surfaces. Depending on the energy of the incident ion beams, ISS can be classified in three categories. (i) LEIS where primary energy level is between 100 ev and 10 keV. (ii) Medium-energy ion scattering where primary energy level is between 100 and 200 keV.

γ1v Vapor

γsv

θ Liquid γs1 Solid

Figure 11.7 Surface tension on a solid surface.

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(iii) High-energy ion scattering (HEIS) where primary energy level is between 1 Mev to several MeV. HEIS is also known as Rutherford back scattering (RBS). RBS is suited for determining the concentration of the trace elements heavier than major constituent of the adhered ❖ Time-of-flight secondary ion mass spectroscopy (TOF-SIMS): A finely focused and pulsed particle excites the sample surface, which causes secondary ions and ion clusters to be emitted from the sample surface. The identity of an element or molecular fragments on the sample surface depends on the exact mass and intensity of the SIMS peak, which is measured by the TOF analyzer in TOF-SIMS. This techniques provides elemental, isotopic, and molecular information at extremely high surface sensitivity [7,17,18]. The molecular fingerprint information of the adherend surface allows insight into the adhesive reactions. ❖ Scanning electron microscopy (SEM): This is an electron beam technique where an electron beam scans the sample surface and electron scattering is used to get the topological information of the surface under investigation. This method allows greater understanding of the change in surface topology after different surface treatment methods. ❖ Energy-dispersive X-ray (EDS or EDAX): Properly equipped SEM can perform EDS analysis, which is used for identifying and quantifying elemental compositions of the sample. ❖ Electron spectroscopy for chemical analysis (ESCA) or X-ray photoelectron apectroscopy (XPS): By directing X-ray irradiation onto a solid surface and simultaneously measuring the kinetic energy and electrons that are emitted from the top, XPS spectra are obtained. ESCA/XPS is utilized for analyzing the surface chemistry of a material, which can measure the elemental composition, empirical formula, chemical state, and electronic state of the elements within a material. ❖ Auger electron spectroscopy (AES): AES provides quantitative elemental and chemical state information of the substrate surface. In this technique, a sample’s surface is excited with a finely focused electron beam, which causes Auger electrons to be emitted from the surface. An electron energy analyzer and counter are used to measure the energy and number of the emitted Auger electrons. The kinetic energy and intensity or number of the emitted Auger electrons provide the elemental identity and concentration of the elements of the surface. ❖ Atomic force microscopy (AFM) scanning probe microscopy (SPM): A finely pointed probe attached to a cantilever scans the surface under investigation and gets the probe surface information. The probe tip deflects because of the interaction with the surface features. This pointwise deflection information can be utilized to get the surface roughness, surface electrical, and chemical interaction. This probe can be arranged and/or utilized differently to achieve required surface information.

11.6

Durability and materials state analysis

The performance of adhesively bonded joints relies on many factors such as how the bonds are made, how the surface was prepared, material properties of both adherend and adhesive, and design of the bond (adhesive thickness, overlap length, stacking sequence, ply angle, fillet etc.) [19e23]. A proper selection of the manufacturing bonding method is very important. The performance dependency on the bonding process has been widely researched. Song et al. [24] and Mohan et al. [25] found that cocured bonding joints yield lesser performance than secondary bonding. Fig. 11.8 shows the schematic of different bonding

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Co-cured without adhesive layer Uncured laminate No adhesive layer

Secondary bonding Cured laminate Adhesive layer Cured laminate Co-curing Uncured laminate Adhesive layer Uncured laminate

Performence based on bonding methods

Cured laminate

Co-bonding Cured laminate Adhesive layer Uncured laminate

Figure 11.8 Bonding methods and performance.

methods and performance ranking based on the methods. For the cocured bonding, moisture present in the prepreg spread over the adhesive layer during the cocuring process, which leads to a weaken interface and causes lower performance of the bond [19,26]. Failure modes of the bond vary with prebond moisture content. With the increase of the moisture content, multiple cracks are converted into a single crack growth [27]. Prebond moisture also has a substantial impact on the fatigue threshold behavior of the crack growth rate curves. The fatigue threshold energy decreases with the increasing prebond moisture level in the adherend for both film joints [27]. Castagnetti el al. studied the effect of adhesive thickness of single lap joints (SLJs) and found reduction in joint strength and performance, which was attributed to the fact that thicker bond line contains more defects such as voids, microcracks, and higher interface stresses [28]. Clean surface is one of the necessary requirements for a durable bond. For secondary bonding, a peel ply is used to ensure a clean surface, but it was observed that the resin system interacts with the material of peel ply and creates a weak interphase between adherend and adhesive. This weak interphase reduces the shear strength or fracture toughness and also reduces long-term durability [29,30].

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In the presence of oxygen, plasma treatments can oxidize the surface of the polymer. It can thus remove organic contaminants from the surface. Early research on the plasma treatment of adherends has concluded that the cross-linking of lowmolecular-weight surface species is the mechanism for eliminating a weak boundary layer [31]. The durability of adhesively bonded joints is very sensitive to the existence of hydrophobic contaminants on the adherend surface prior to bonding [32]. Plasma treatment of the adherend surface increases the formation of chemical bonds across the interface of adherendeadhesive by surface cleaning, ablation of surface polymer chains, surface cross-linking of polymer chains, and introduction of polar functional groups that also result in increased surface energy and can significantly improve the strength and durability of bonded materials but also highly dependent on the substrate type [33,34]. The degradation of the fracture toughness and long-term durability of adhesively bonded joints by the humid environment is highly impacted by the surface roughness of the adherend [35e37]. Fig. 11.9 presents the elastic energy release rate at slow-crack velocities in a humid environment. It is seen that GIc depended strongly on the adherend surface profile angle (a) [35]. This test is done on ultramilled aluminum adherends by a wedge test.

11.6.1

Environmental/aging

The effect of moisture on the strength and long-term durability of adhesively bonded joint and structures has been comprehensively investigated by many researches [34,38,39]. Diffusion of water plasticizes the epoxy resin and reduces Tg by about 10 C for every 1% of water intake. Fig. 11.10(a) represents glass transition temperature versus water uptake for different samples of resin. The loss of strength of the adhesive due to the ingression of the moisture is caused by hydrodynamic displacement

α =60°

GIc (J/m2)

1000

Grit-blast 100

α =30°

10

α =0°

0

0.5

1.0

1.5

2.0

tan α (°)

Figure 11.9 GIc determined from wedge tests conducted in 95% RH at 50 C shows the effect of surface roughness profile angle (a) [35].

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(b) 5 400

Water uptake (%)

Glass transition temperature (K)

(a)

375

350

4 3

B

2

C

1

A 325

0 0

1

2

3

Water uptake (%)

4

0

1000

2000

3000

4000

5000

Time (h)

Figure 11.10 (a) Glass transition temperature versus water uptake. b) Water uptake at 50 C versus time for three different samples of the resin [39].

of adhesive from the substrates, adhesive plasticization, and corrosion at the adhesiveeadherend interface and for the case of bonding between dissimilar materials can lead to cathodic corrosion [39]. The effect of absorption followed by desorption is illustrated in Fig. 11.10(b). Resin samples were exposed to water for different durations and then dried again and then reexposed to water. The process of water intake alters the materials internal structure that allows further cure of the material and the creation within the matrix of microscopic voids, which can retain water [39].

11.6.2 Broadband dielectric spectroscopy for bond material state assessment BbDS is a very well-known tool for material characterization. BbDS is used to study the interaction of electromagnetic waves with matter, usually in the frequency range from 10 6 to 1012 Hz [40]. This dynamic range generally can provide information about the molecular and collective dipolar fluctuations, as well as about charge transport and polarization effects that occur at inner and outer boundaries in the form of different dielectric properties of the material under study. Fig. 11.11 shows the effect of different charge displacement mechanisms on dielectric response and their corresponding effective frequency ranges. Hence, broadband dielectric spectroscopy can also be used as an effective tool to detect and interpret the damage state of heterogeneous material systems such as adhesively bonded joints [41e43]. Extensive studies have been undertaken to monitor the way in which the dielectric properties of adhesive joints change with time to understand the interaction of water with the adhesive [39,45,46]. It is possible to identify at least two types of water in the system: water that is bound to the epoxy matrix and water that exists as essentially water droplets within voids in the matrix by the study of temperature dependent dielectric relaxation in epoxy resins [47e49]. Fig. 11.12 represents the changes that happen to the dielectric spectrum. Because of the large dipole moment of water, BbDS can easily detect the presence of water inside the bonded joints; Fig. 11.12(a) shows an

396

Durability of Composite Systems E=0 E≠0 E=0 E≠0

Inerfacial & space charge

E=0 E=0

E≠0

E≠0

Orientational ε’

lonic Electronic

ε’’

10–2

1

102

104

106 108 Log (frequency)

1010

1012

1014

1016

Figure 11.11 Dielectric response to different polarization mechanisms in different frequency regimes [44].

increase in the real part of the permittivity with the increase of the water intake in the epoxy resin but depending on the water molecules interaction with the molecules of the resin to determine the shape of the curve change [48,50]. The location on the frequency domain will change depending on exactly how the water molecules are interacting with the epoxy matrix. The presence of pendant OH groups in the matrix is responsible for the initial dielectric spectrum, which exhibits a frequency-dependent permittivity. There also a small amount of water segregated into voids created during the cure process. As the water is absorbed, there is an increase in the dielectric loss at around 105 and 109 Hz. The permittivity changes with time in a very controlled manner for a joint in which plasticization of the resin occurs.

11.6.3

Quality assessment of adhesive bonds based on broadband dielectric spectroscopy

The “kissing bond” or weak adhesion bond that is a weak interfacial bond between the two adherent surfaces is a typical occurrence due to the presence of contamination and unwanted surface features that are the result of surface pretreatment on the adherend during manufacturing process, which can lead to the catastrophic failure of the structure. This weak interfacial region reduces the strength and durability of the bonded material significantly [51]. One of the major difficulties faced in this type of defective bonding is their detectability using all the state-of-the-art nondestructive testing (NDT) methods. Most of the NDT methods can detect a flaw or defect if there is a distinct change of properties between material interfaces but “kissing bonds” do not have any properties/feature of that kind. So it is difficult to assess the quality of a bond with a “kissing bond”. Elenchezhian et al. [52] created adhesive bonds with different grades of contamination and used BbDS to assess the material state. In their study, they consider “dielectric relaxation strength (DRS)” to assess the material state. DRS is the algebraic difference between the low-frequency real part of the permittivity

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(a) 5.0

5.0

4.5

4.0 4.0 3.5 3.5

Dielectric permittivit y

Dielectric permittivity

4.5

3.0 3.0

Lo

2.5 5.50 5.75 6.00 6.25 6.50 6.75 7.00

g

fre

qu

en

7.25

cy

7.50

50 25

75

Age

0

100

ing t

ime

125

150

175

2.5 200

s) (day

(b) 1.4

1.4

1.2

1.0 1.0 0.8 0.8 0.6

Dielectric loss

Dielectric loss

1.2

0.6 0.4 0.4

Lo

7.50 7.25 7.00 6.75 6.50 6.25 6.00

g

fre

qu

100

en

5.75

cy

5.50

25 0

50

75

Age

ime ing t

125

(day

150

175

200

s)

Figure 11.12 Variation of dielectric permittivity (a) and dielectric loss (b) as a function of exposure time [50].

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Durability of Composite Systems

Normalized DRS 45

Normalized DRS value based

40 35 30 25 20 15 10 5 0

A

B

C

D

E

F

Specimens

Figure 11.13 Normalized DRS values of all specimens. Normalization was done based on the least value of the sample (which turns out to be the specimen that was sanded on both surfaces of the joint) [52].

and high-frequency real part of the permittivity. Six different types of specimen were prepared to simulate the weak adhesion bond and study interfacial polarization that arises because of the presence of different contamination. Specimen AdThe specimen was sanded on both surfaces of the joint (ASTM D 2093 was followed for surface treatment). Specimen BdThe specimen was sanded on only one surface. Specimen CdThe specimen was not sanded on any surface. Specimen DdThe specimen was prepared with a water drop on an unsanded surface. Specimen EdThe specimen was completely immersed in water before bonding. Specimen FdThe specimen was prepared with oil on one surface. Fig. 11.13 shows that the DRS can be used as an indicator of the contamination level present on an adherend surface, which can create unwanted “kissing bonds.” Higher contamination levels showed higher measured DRS values than the baseline samples.

11.7

Conclusion

There are many factors that impact the durability of an adhesive bond. The very first thing that should be in consideration is the proper selection of the adhesive material to increase the fatigue life of the bonded joints. Then, a careful material-specific surface

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preparation followed by a controlled surface characterization and quality control scheme to certify the bond is mandatory to ensure a durable bond. Environmental effects pose a threat to the long-term performance and durability and restrict the wide application of adhesively composite-bonded joints in structural applications. Prebond moisture, postbond moisture, and temperature cycle history should be well documented and discussed before and after the bonding method, as they directly influence durability of the joints.

References [1] J. Delmonte, The Technology of Adhesives, Reinhold, New York, 1947. [2] A. Pizzi, K.L. Mittal, Handbook of Adhesive Technology, CRC press, 2017. [3] A. Kwakernaak, J. Hofstede, J. Poulis, R. Benedictus, 8 - improvements in bonding metals for aerospace and other applications, in: M.C. Chaturvedi (Ed.), Welding and Joining of Aerospace Materials, Woodhead Publishing, 2012, pp. 235e287. [4] A. V Pocius, Adhesion and Adhesives Technology an Introduction, Carl Hanser Verlag GmbH Co KG, 2012. [5] S. Ebnesajjad, A.H. Landrock, Adhesives Technology Handbook, William Andrew, 2014. [6] M.D. Banea, L.F.M.M. Da Silva, Adhesively bonded joints in composite materials: an overview, Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl. 223 (1) (2009) 1e18. [7] F. Awaja, M. Gilbert, G. Kelly, B. Fox, P.J. Pigram, Adhesion of polymers, Prog. Polym. Sci. 34 (9) (2009) 948e968. [8] S. Ebnesajjad, C. Ebnesajjad, Surface Treatment of Materials for Adhesive Bonding, second ed., 2013. [9] A. Rudawska, Surface treatment methods, Surf. Treat. Bond. Technol. (Jan. 2019) 47e62. [10] S. Ebnesajjad, Handbook of Adhesives and Surface Preparation, 2011. [11] C.M. Chan, T.M. Ko, H. Hiraoka, Polymer surface modification by plasmas and photons, Surf. Sci. Rep. 24 (1-2) (1996) 1e54. [12] R. Jones, N. Matthews, A.A. Baker, V. Champagne, Aircraft Sustainment and Repair, 2017. [13] A. Rudawska, Assessment of surface preparation for the bonding/adhesive technology, In: Surface Treatment in Bonding Technology, Academic Press, 2019, pp. 227e275. [14] H.M.S. Iqbal, S. Bhowmik, R. Benedictus, Study on the effect of surface morphology on adhesion properties of polybenzimidazole adhesive bonded composite joints, Int. J. Adhesion Adhes. 72 (January 2017) 43e50. [15] W. Eckert, Improvement of Adhesion on Polymer Film, Foil and Paperboard by Flame Treatment, In: TAPPI European PLACE Conference, 2003. [16] M. T., R.M.B. Duncan, R. Mera, D. Leatherdale, Techniques for characterising the wetting, coating and spreading of adhesives on surfaces, In: National Physical Laboratory Report, NPL Report DEPC-MPR-020 (2006) 1e48. [17] D. Briggs, J.C. Vickerman, In: ToF-SIMS SurfaceAnalysis by Mass Spectrometry, IM Publ. Surf. SpectraLtd., Chichester, 2001. [18] J.C. Vickerman, I.S. Gilmore, Surface Analysis - the Principal Techniques, second ed., 2009.

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[19] S. Budhe, M.D. Banea, S. de Barros, L.F.M.M. da Silva, An updated review of adhesively bonded joints in composite materials, Int. J. Adhesion Adhes. 72 (October 2016) (2017) 30e42. [20] L.F.M. da Silva, R.J.C. Carbas, G.W. Critchlow, M.A.V. Figueiredo, K. Brown, Effect of material, geometry, surface treatment and environment on the shear strength of single lap joints, Int. J. Adhesion Adhes. 29 (6) (2009) 621e632. [21] G. Meneghetti, M. Quaresimin, M. Ricotta, Influence of the interface ply orientation on the fatigue behaviour of bonded joints in composite materials, Int. J. Fatig. 32 (1) (2010) 82e93. [22] L. Liao, C. Huang, T. Sawa, Effect of adhesive thickness, adhesive type and scarf angle on the mechanical properties of scarf adhesive joints, Int. J. Solid Struct. 50 (25-26) (2013) 4333e4340. [23] M. Mokhtari, K. Madani, M. Belhouari, S. Touzain, X. Feaugas, M. Ratwani, Effects of composite adherend properties on stresses in double lap bonded joints, Mater. Des. 44 (2013) 633e639. [24] M.G. Song, et al., Effect of manufacturing methods on the shear strength of composite single-lap bonded joints, Compos. Struct. 92 (9) (2010) 2194e2202. [25] J. Mohan, A. Ivankovi
c, N. Murphy, Mode i fracture toughness of co-cured and secondary bonded composite joints, Int. J. Adhesion Adhes. 51 (2014) 13e22. [26] J. Mohan, A. Ivankovi
c, N. Murphy, Effect of prepreg storage humidity on the mixedmode fracture toughness of a co-cured composite joint, Compos. Part A Appl. Sci. Manuf. 45 (2013) 23e34. [27] S.R. Budhe, Effect of Pre-Bond Moisture on the Static and Fatigue Behaviour of Bonded Joints Between CFRP Laminates for Structures Repairs, University of Girona, 2014. [28] D. Castagnetti, A. Spaggiari, E. Dragoni, Effect of bondline thickness on the static strength of structural adhesives under nearly-homogeneous shear stresses, J. Adhes. 87 (7-8) (2011) 780e803. [29] M. Kanerva, O. Saarela, The peel ply surface treatment for adhesive bonding of composites: a review, Int. J. Adhesion Adhes. 43 (2013) 60e69. [30] M. Kanerva, E. Sarlin, M. Hoikkanen, K. R€am€o, O. Saarela, J. Vuorinen, Interface modification of glass fibre-polyester composite-composite joints using peel plies, Int. J. Adhesion Adhes. 59 (2015) 40e52. [31] H. Schonhorn, R.H. Hansen, Surface treatment of polymers for adhesive bonding, J. Appl. Polym. Sci. 11 (8) (1967) 1461e1474. [32] T. Pribanic, et al., Effect of surface contamination on composite bond integrity and durability, FAA JAMS 2014 Tech. Rev. Meet. Paper 6, March 25-26 (2014) 1e24. [33] T.S. Williams, H. Yu, P.C. Yeh, J.M. Yang, R.F. Hicks, Atmospheric pressure plasma effects on the adhesive bonding properties of stainless steel and epoxy composites, J. Compos. Mater. 48 (2) (2014) 219e233. [34] A. Baldan, Adhesively-bonded joints and repairs in metallic alloys, polymers and composite materials: adhesives, adhesion theories and surface pretreatment, J. Mater. Sci. 39 (1) (2004) 1e49. [35] A.N. Rider, D.R. Arnott, J.J. Mazza, Surface treatment and repair bonding, In: Aircraft Sustainment and Repair, Elsevier Ltd., 2018, pp. 253e323. [36] A.N. Rider, D.R. Arnott, The influence of adherend topography on the fracture toughness of aluminium-epoxy adhesive joints in humid environments, J. Adhes. 75 (2) (2001) 203e228. [37] A. Rider, Surface Properties Influencing the Fracture Toughness of Aluminium-Epoxy Joints, University of New South Wales, 1998.

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[38] A.J. Kinloch, The durability of adhesive joints, In: Adhesion Science and Enginereing 1, 2002, pp. 661e698. [39] R.A. Pethrick, Design and ageing of adhesives for structural adhesive bonding-A review, Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl. 229 (5) (2015) 349e379. [40] F. Kremer, A. Sch€onhals, Broadband Dielectric Spectroscopy, Springer Science & Business Media, 2002. [41] R. Raihan, J.-M. Adkins, J. Baker, F. Rabbi, K. Reifsnider, Relationship of dielectric property change to composite material state degradation, Compos. Sci. Technol. 105 (2014). [42] R. Raihan, J.-M. Adkins, F. Rabbi, J. Baker, Q. Liu, K. Reifsnider, A dielectric spectroscopy method of directly assessing composite micro-state evolution during manufacturing and deformation, In: International SAMPE Technical Conference, 2013. [43] V. Vadlamudi, R. Raihan, and K. Reifsnider, “Dielectric assessment of composite damage states”, in International SAMPE Technical Conference, 2016, Vol. 2016-January. [44] V. Vadlamudi, others, Assessment of Material State in Composites Using Global Dielectric State Variable, 2019. [45] J.M. Zhou, J.P. Lucas, Hygrothermal effects of epoxy resin. Part I: the nature of water in epoxy, Polymer 40 (20) (1999) 5505e5512. [46] P. Boinard, W.M. Banks, R.A. Pethrick, Use of dielectric spectroscopy to assess adhesively bonded composite structures, Part I: water permeation in epoxy adhesive, J. Adhes. 78 (12) (2002) 1001e1014. [47] D. Hayward, E. Hollins, P. Johncock, I. McEwan, R.A. Pethrick, E.A. Pollock, The cure and diffusion of water in halogen containing epoxy/amine thermosets, Polymer 38 (5) (1997) 1151e1168. [48] R.A. Pethrick, G.S. Armstrong, W.M. Banks, R.L. Crane, D. Hayward, Dielectric and mechanical studies of the durability of adhesively bonded aluminium structures subjected to temperature cycling. Part 1: examination of moisture absorption, Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl. 218 (3) (2004) 169e182. [49] R.A. Pethrick, E.A. Hollins, I. McEwan, E.A. Pollock, D. Hayward, P. Johncock, Effect of cure temperature on the structure and water absorption of epoxy/amine thermosets, Polym. Int. 39 (4) (1996) 275e288. [50] R.A. Pethrick, Non-destructive Evaluation (NDE) of Composites: Dielectric Methods for Testing Adhesive Bonds in Composites, In: Non-destructive Evaluation (NDE) of Polymer Matrix Composites, Elsevier, 2013, pp. 185e219. [51] P.N. Marty, N. Desai, J. Andersson, NDT of kissing bond in aeronautical structures, In: 16th World Conference on NDT, 2004. [52] M.R.P. Elenchezhian, et al., Quality assessment of adhesive bond based on dielectric properties, Int. SAMPE Tech. Conf. (2017) 1336e1349.

Durability of polymer matrix composites fabricated via additive manufacturing

12

Denis Cormier, Pritam Poddar Rochester Institute of Technology, Rochester, NY, United States

Chapter outline 12.1 12.2

Introduction 403 Approaches to composite additive manufacturing

404

12.2.1 Short fiber fused filament fabrication 405 12.2.2 Continuous fiber fused filament fabrication 407 12.2.2.1 Dual nozzle continuous fiber fused filament fabrication 408 12.2.2.2 Continuous fused filament fabrication via coextrusion 410 12.2.2.3 Nonplanar fused filament fabrication 412

12.3

Techniques for enhancing the durability of composite FFF structures 414 12.3.1 Voids in printed structures 415 12.3.1.1 Effect of print parameters on material density 415 12.3.1.2 Effect of deposition toolpath on material density 418 12.3.2 Weld strength in composite FFF processing 419 12.3.3 Wetting and fiber-matrix interfacial bond strength 421 12.3.4 Effect of printed fiber orientation and distribution 424 12.3.4.1 Fiber orientation and distribution effects on mechanical properties 424 12.3.4.2 Fiber orientation around holes 429 12.3.4.3 Z-pinning 430

12.4

Engineered composite cellular structures 432 12.4.1 Open cell composite lattice structures 433 12.4.2 Closed cell plate lattice structures 434

12.5 Summary and conclusions References 435

12.1

435

Introduction

Additive manufacturing (AM) is a class of manufacturing technologies in which components are fabricated by selectively depositing one layer of material on top of another until the three-dimensional shape is complete. ISO/ASTM 52900:2015 [1] defines seven categories of AM technologies that include material extrusion, vat Durability of Composite Systems. https://doi.org/10.1016/B978-0-12-818260-4.00012-0 Copyright © 2020 Elsevier Ltd. All rights reserved.

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polymerization, sheet lamination, powder bed fusion, directed energy deposition, binder jetting, and material jetting. The foundational patents for most of these processes date to the late 1980s and early 1990s. Although each category of AM technologies uses a fundamentally different way of depositing material, all of them share the same fundamental advantages over traditional manufacturing processes. The first advantage is the freedom to make extremely complex geometric shapes that would be virtually impossible to produce via other manufacturing technologies. This can, for example, give designers the ability to reduce weight in aerospace or automotive components, thus increasing fuel economy and/or travel range. A second fundamental advantage is the ability to manufacture parts without the need for tooling. The elimination of tooling costs is a significant benefit when only low or medium quantities of parts are needed. There are also cases where multipart assemblies can be redesigned and additively manufactured as one single component because the engineers no longer have to design components that are machinable or removable from molds or dies. The reduction in tooling and labor costs in these cases can be enormous. Despite the rapid growth of interest in AM technologies, AM with composite materials has been a relatively recent development. Markforged is widely credited with bringing the first continuous fiber composite AM machine to the market. The company’s original Mark One machine made its commercial debut in 2014. As the Mark One’s introduction, several other polymer matrix composite AM companies such as EnvisionTEC, Anisoprint, Impossible Objects, Arevo Labs, and Desktop Metal have entered the market. Many of these companies promote the idea that their machines can be used to fabricate stronger and lighter polymer matrix composite alternatives to parts that were previously manufactured out of metal. Research involving metal and ceramic matrix composite materials can be found in the literature. However, the overwhelming majority of published composite AM research focuses on thermoplastic polymer matrix composite materials with either short or continuous fiber reinforcement. Thermoplastic polymer matrix fiberreinforced composite additive manufacturing is the focus of this chapter.

12.2

Approaches to composite additive manufacturing

To understand how the properties of additively manufactured composites potentially differ from those of composites manufactured using more traditional approaches, it is useful to first review the different types of composite AM processes. The interested reader is directed to review papers on composite AM technologies by Parandoush and Lin [2] and Brenken et al. [3]. Among the seven categories of AM technologies, material extrusion and sheet lamination dominate polymer matrix composite AM landscape. Fused filament fabrication (FFF) is a type of material extrusion AM in which a bead of thermoplastic material is heated above its melting point and extruded through a nozzle while the nozzle traverses a predefined material toolpath that prints one cross-sectional slice of material on top of another. Each cross-sectional slice is composed of one or more boundary contours that surround an infill region. The printed contours form the outer surface (i.e., skin) of the part, while the printed infill forms the

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inner core beneath the part’s surface. The infill may be solid, or it may consist of hollow triangular, rectangular, or hexagonal patterns. As extruded thermoplastic material exits the nozzle, it bonds with previously deposited tracks of material as it cools down and solidifies. The degree of bonding, or weld strength, has a major influence on durability of additively manufactured thermoplastic composite parts. FFF processes have been adapted to work with both short/chopped fiber and continuous fiber-reinforced thermoplastic matrix composites as illustrated in Fig. 12.1 [4]. In the simplest case involving short fiber composite FFF, the extruded feedstock material consists of chopped fibers of carbon, glass, Kevlar, or other reinforcement that have been compounded with a thermoplastic such as polyamide (PA), acrylonitrile butadiene styrene (ABS), polylactic acid (PLA), or polyether ether ketone (PEEK) as seen in Fig. 12.1(a). In the case of continuous fiber FFF, two approaches have been used. In Fig. 12.1(b), a thermoplastic filament and a continuous fiber are coextruded through a single nozzle. The thermoplastic matrix material, in this case, may be a neat polymer or a chopped fiber composite into which the continuous fiber is fed during coextrusion. Fig. 12.1(c) illustrates the case where two separate extrusion nozzles are useddone for the thermoplastic matrix material and one for the continuous fiber. In both approaches to continuous fiber FFF, the extrusion subsystem includes a cutting mechanism to cut the continuous fiber when it is necessary to lift the deposition head to move to a new location elsewhere on the build platform. From a durability perspective, a key aspect of composite FFF is that the material deposition toolpath that the extrusion head follows dictates the fiber orientation at any location. This theoretically allows one to intelligently engineer local fiber orientation within the part according to expected mechanical loading conditions. In practice, composite AM processes impose limitations on the degree to which fiber orientation can be controlled. This is discussed in subsequent sections.

12.2.1 Short fiber fused filament fabrication In short fiber FFF, a chopped fiber reinforcing phase material such as carbon or glass is compounded with a thermoplastic matrix (e.g., PA, PEEK, ABS) and formed into a filament having an appropriate diameter for use with the FFF machinedtypically 1.75 or 2.85 mm. The composite feedstock filament is fed into a material extrusion AM machine in which the tapered nozzle’s exit diameter is typically in the 0.4e0.8 mm range. As the composite mixture of chopped fibers in a thermoplastic matrix flows through the tapered nozzle, the high degree of wall shear results in the substantial alignment of fibers as seen in Fig. 12.2 [4]. Although the shear alignment of fibers is desirable, the likelihood of fiber entanglement and the resulting clogged nozzle increases as the length of fibers and fiber loading fraction increase. Chopped fibers used in composite FFF feedstock materials typically have lengths and diameters on the order of 100e200 mm and 1e10 mm, respectively. The fiber loading fraction is typically in the 10e20 wt% range. This relatively low fiber loading fraction can be an important limitation of the process with respect to mechanical properties. Furthermore, the use of short chopped fibers has implications with respect to the critical fiber length and fiber pull-out.

406

(a)

(b)

Chopped material spool

Conventional FDM print head Print head movements Platform movements

(c)

Reinforce material spool

Single nozzle print head Print head movements

Matrix material spool

Platform movements

Part

Part Print bed

Reinforce material spool

Dual print head Print head movements

Matrix material spool

Platform movements

Part Print bed

Print bed

Figure 12.1 (a) FFF with short fiber thermoplastic filament; (b) FFF via coextrusion of continuous fiber and thermoplastic matrix through a single nozzle; (c) FFF of continuous fiber and thermoplastic matrix through separate nozzles [4]. Durability of Composite Systems

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Figure 12.2 Shear alignment of short fibers as they flow through an extrusion nozzle [5].

In the context of the durability of additively manufactured chopped fiber composites, the tendency of chopped fibers to shear align as they flow through a tapered nozzle is significant [5,6]. The fact that reinforcing fibers align as they exit the nozzle opens up the possibility of generating material deposition toolpaths that allow local control over fiber orientation. For example, one can intelligently generate material deposition toolpaths that result in fiber orientations that optimally withstand expected mechanical loading conditions of the component in service. Fig. 12.3 demonstrates a spiral extrusion toolpath for an epoxy carbon fiber blend in which micrographs taken at various points around the spiral path reveal an excellent correlation between the direction of print head travel and carbon fiber orientation [7].

12.2.2 Continuous fiber fused filament fabrication The origins of continuous fiber additive manufacturing can be traced back to conventional automated fiber placement (AFP) technologies that have been in use for decades. Traditional AFP typically involves the robotic deposition of one or more spools of continuous epoxy prepreg tape onto a mandrel using a desired fiber layup strategy. Bands of tape may be deposited side-by-side during AFP to cover the area needed for a given application. The tape may also be laid down one layer upon another, in which case the orientation of each layer is typically rotated 45 degrees with respect to the previous layer. AFP can be considered a precursor to composite AM in the sense that it lays down one layer of material on top of another.

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Figure 12.3 Correlation between material deposition toolpath and chopped fiber orientation [7].

An important limitation of AFP is that it is difficult to lay down tape in paths that are not essentially linear. If the deposition toolpath deviates substantially from a nonlinear path, the tape tends to lift up, or pucker, on the inner edge relative to the turning direction of travel. Furthermore, AFP is normally performed on a mandrel that may or may not be removed after fabrication is complete. Additive manufacturing is done without tooling.

12.2.2.1 Dual nozzle continuous fiber fused filament fabrication With dual nozzle continuous fiber FFF, one nozzle extrudes the matrix material while a second nozzle extrudes continuous fiber. By separating the placement of the matrix and continuous fiber materials, it is possible to route fiber only in those areas of a part requiring reinforcement. This is advantageous from a cost perspective given that the price of carbon fiber can exceed that of the thermoplastic matrix by a considerable amount. Markforged (Watertown, MA) is widely credited with being the first company to introduce continuous fiber composite additive manufacturing to the market. Matrix materials include either neat PA6 (polyamide) copolymer or a PA6 copolymer blend with approximately 15 wt% chopped carbon fibers. The second nozzle extrudes continuous fiber such as 1K carbon fiber tow, Kevlar, or glass fiber. The heated fiber exiting the nozzle remelts the previously extruded matrix material. Pressure from the face of the nozzle spreads the fiber tow to a width of approximately 1 mm and helps embed the fiber into the matrix. When it is necessary to stop depositing fiber to move to another location, a fiber cutter within the extrusion head is engaged. Carbon fiber is

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typically used for cases requiring the greatest strength and stiffness with minimum mass. Kevlar has good flexibility and is well suited for impact resistance. Fiberglass is used as a relatively inexpensive alternative to carbon fiber in applications where stiffness and weight are not as critical. Fig. 12.4 illustrates two typical material deposition strategies for dual nozzle composite FFF. The black lines represent pathing for the thermoplastic matrix material, whereas the blue lines represent pathing for the continuous fiber. Fig. 12.4 (left) shows a case where the outermost wall consists of two concentric rings of thermoplastic matrix material that surround two concentric rings of continuous fiber. The number of concentric fiber rings can be increased or decreased as needed. The hollow hexagonal matrix material infill used in this case greatly reduces mass and printing time for the part. The infill density in this example is just 27%. Infill density may also be increased or decreased depending on the demands of the application. For applications where an engineer wishes to enhance resistance to impact events, this type of configuration could be used with Kevlar as the reinforcing phase material. Fig. 12.4 (right) has the same outer skin configuration as Fig. 12.4 (left), however, the infill consists of densely printed continuous fiber at an angle of 45 degrees. It is possible to print all fiber at a specific desired angle, and the fiber angle may be rotated from one layer to the next when the fiber is printed in multiple layers. The fiber content in this type of printing strategy is much greater than that of Fig. 12.4 (left). Furthermore, fiber can be oriented to provide maximum strength and stiffness in the desired direction based on expected loading conditions. As previously mentioned, a significant potential advantage of composite AM involves the ability to route continuous fiber in nonlinear paths. Fig. 12.5 shows a tensile coupon printed using PA6 copolymer matrix material and a 1k carbon fiber tow. This sample was printed using five concentric carbon fiber rings on the top and

Thermoplastic matrix Long fiber reinforcement

Figure 12.4 Dual nozzle continuous fiber FFF depositions strategies employing (left) concentric continuous fiber contours and hollow hexagonal matrix material infill, and (right) concentric continuous fiber contours and isotropic continuous fiber infill.

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Figure 12.5 Demonstration of continuous carbon fiber routing using dual nozzle FFF.

bottom face layers. Between the face layers, a 35% dense triangular infill was used. Between the left and right ends of the coupon, the gradual curvature of the fiber deposition path can be seen. At the ends of the coupon, the fiber’s innermost corner turning radius is approximately 2 mm. This example also shows that a very high fiber loading fraction can be achieved in desired regions with continuous fiber pathing using two extrusion nozzles. Fig. 12.6 illustrates the pronounced effect that fiber loading has on strength and elastic modulus as the volumetric fiber loading fraction is increased from approximately 30% to just over 90% through the addition of concentric fiber rings [8]. The choice between using an isotropic fiber routing strategy versus a concentric fiber routing strategy is discussed in Section 12.3.4.1.

12.2.2.2 Continuous fused filament fabrication via coextrusion An alternative to dual nozzle continuous fiber FFF is a coextrusion of both the thermoplastic matrix and continuous fiber through a single nozzle [9e11], and [12]. This approach is schematically shown in Fig. 12.7.

100

350

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300

70 60

250

50 40

200

30

Elastic modulus (GPa)

90

20

150

10 100

0

20

40

60

80

0 100

νCF (%)

Figure 12.6 Strength and elastic modulus as a function of fiber loading using a concentric fiber routing strategy [8].

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Spool Thermoplastic filament Spool Reinforcing fiber Feeder Cutter Melting chamber Extruder Heater Nozzle

Feeder Channel of remote feed Bed Composite part

Figure 12.7 Coextrusion of matrix and continuous fiber through a single nozzle [12].

Adumitroaie et al. [9] suggest that a fundamental limitation of thermoplastic matrix continuous fiber AM is poor wetting between viscous thermoplastic matrix materials and the continuous fiber as they are coextruded through a nozzle. A dual-matrix approach is therefore advocated in which the continuous fiber is first impregnated with a thermoset that provides excellent wetting and interfacial bond strength with the fiber tow. This prepreg fiber is then coextruded through the same nozzle with a thermoplastic material of choice, such as PA6, during the FFF process. The thermoset used for impregnation of the fiber before coextrusion is chosen such that its glass transition temperature (Tg) is below the melting temperature (Tm) of the thermoplastic matrix. The prepreg fiber, therefore, softens during coextrusion and does not suffer a brittle fracture. When using high temperature engineering thermoplastics such as PEEK whose FFF extrusion temperatures can reach 400
C or higher, a thermoset prepreg that can withstand those temperatures is necessary. Although relatively little published data on mechanical properties achieved through bimatrix coextrusion FFF processing is available, Azarov et al. [12] provide the specific modulus and strength of a bimatrix carbon fiber composite in comparison with PLA, 1030 steel, 2024-T351 aluminum, and traditional carbon epoxy composite. As can be seen in Fig. 12.8, the specific modulus and strength of the bimatrix coextruded composite exceeds values for the steel and aluminum alloys listed. In contrast to the assertions of Adumitroaie et al. Liu et al. [13] caution that care must be taken when considering interfacial bond strength between fiber, thermoset sizing, and a thermoplastic matrix. In this study, carbon fiber with epoxy sizing was coextruded in a PA6 matrix via the FFF process. For comparison, the epoxy sizing was removed in a solvent bath, rinsed, and then resized in an aqueous thermoplastic PA6 dispersion. The carbon fiber with thermoplastic PA6 sizing was then coextruded with the same FFF process. This allowed for a direct comparison of all thermoplastic coextrusion with a bimatrix thermoset þ thermoplastic coextrusion approach. In this study, the interlaminar shear strength of samples produced with all-thermoplastic

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Specific modulus, km

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PLA

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1030 steel

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2024-T351

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2000 3D printed composite

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120 1030 steel

100 80

2024-T351

60 38.9

40 17.2

20 0

3.1

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Carbon-epoxy

3D printed composite

Figure 12.8 Specific modulus and strength of a bimatrix coextruded composite compared with other materials of interest [12].

coextrusion was approximately 40% higher than that of the samples produced via the bimatrix thermoset þ thermoplastic coextrusion. Suffice it to say that interfacial bond strength between sizing and fibers and sizing and the surrounding matrix are both important factors that must be taken into consideration when selecting material systems for composite FFF.

12.2.2.3 Nonplanar fused filament fabrication Practical consideration of composite FFF as described thus far is the fact that each layer is printed in the XeY plane (i.e., one planar layer of material on top of another). The orientation of fibers may therefore only be controlled within the XeY plane. It is not possible to print fibers that are vertically inclined in the Z-axis direction. Given that the load paths in a large percentage of applications are not planar (e.g., body panels, wing structures, etc.), it would be advantageous to pursue AM fiber placement

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strategies that allow true 3D fiber orientation. Toward that aim, several research groups and companies have been developing nonplanar deposition techniques that employ robotic or five-axis CNC motion stages [14e17]. Although the precise steps for nonplanar FFF material deposition vary from implementation to implementation, the steps of one representative approach may be seen in Fig. 12.9 [15]. Fig. 12.9(e) and (f) illustrates the final additively manufactured component and its corresponding CAD model. For cases such as this where a nonplanar sandwich panel construction is desired, the first step is to completely print a 3D support structure for the downward-facing surface of the panel as shown in Fig. 12.9(a). This can be printed at low density with hexagonal or other print strategy using the standard layerwise FFF deposition approach. The next step is to print a flexible thermoplastic elastomer (TPE) that spans, or bridges, over the gaps in the open-cell support base. It is in this step that the FFF extrusion nozzle follows a path that follows the undulating surface of the support base. In other words, the print head can simultaneously move in the X-, Y-, and Z-axis directions rather than just the XeY plane. This is shown in Fig. 12.9(b). Finally, the bottom skin of the sandwich panel structure is printed using the desired panel material. For example, the lower skin of the panel

Figure 12.9 Nonplanar fused filament fabrication [15].

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Figure 12.10 Nonplanar fused filament fabrication of an airfoil geometry [17].

structure can be printed using the same chopped or continuous carbon fiber composite materials that have been previously discussed. Fig. 12.9(d) then shows a hollow honeycomb structure that is printed to fill the space between opposing faces of the sandwich panel. Lastly, the top surface of the panel is printeddagain using a nonplanar toolpath with whichever material the application calls for. After printing, the soft and flexible TPE layer (Fig. 12.9(b)) allows the completed sandwich panel to be separated from the underlying support base. The key distinction between planar material deposition and nonplanar material deposition is that the nonplanar deposition strategies make it possible to route chopped or continuous reinforcing fibers in a 3D conformal path that flows with the outer surface of the structure. This is much closer to the case of conventional composite manufacturing where woven prepreg is vacuum-formed to mold such that the woven fibers flow with the 3D mold surface. Having said that, nonplanar FFF toolpath generation algorithms are still in their infancy. Much research is needed to intelligently generate nonplanar conformal FFF toolpaths for composite materials. Although most published research on nonplanar FFF has made use of three-axis motion stages (i.e., linear motion in X, Y, and Z axes), there are limits to the degree of curvature in the printed part that can be achieved before the extrusion nozzle collides with previously deposited material. In general, this technique is best suited for curved panels with a gentle curvature (i.e., automotive body panel, aerofoils on a UAV, etc.) as illustrated in Fig. 12.10.

12.3

Techniques for enhancing the durability of composite FFF structures

Nearly all AM technologies involve layerwise laminations of material. The durability of composite AM parts is largely influenced by (1) voids, or lack thereof in the printed material; (2) polymer bond strength between and within layers; (3) wetting and interfacial bond strength between the polymer matrix and the fibers; and (4) fiber orientation relative to mechanical loading conditions.

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12.3.1 Voids in printed structures It is intuitively obvious that the presence or absence of voids in an additively manufactured structure will affect mechanical properties. There are generally two potential causes of voids in composite FFF structures that can affect durability. One cause is voids that result from printing 3D structures via extrusion through a circular orifice. The second cause is voids resulting from the printing toolpaths in relation to the cross-sectional area that must be covered in each layer.

12.3.1.1 Effect of print parameters on material density Achieving high density in material fabricated via FFF is a challenge given that parts are printed via extrusion through nozzles having circular orifices. Fig. 12.11 shows a fracture surface of an ABS sample printed via FFF in which the print orientation between successive layers was alternated between 0 and 90 degrees [18]. It is readily apparent that the cross-sectional profile of the extruded material tracks results in voids and incomplete bonding. Although complete elimination of voids in FFF components is extremely difficult, steps can be taken to minimize voids. Material exiting the vertical nozzle bends 90 degrees as it lays down on the horizontal surface of the previously deposited material. When a material having a circular profile bends, the profile deforms to an elliptical shape. It is standard practice in FFF material deposition to select a layer thickness (i.e., distance between the tip of the nozzle and the substrate) such that the face of

Figure 12.11 Fracture surface of ABS plastic printed via FFF [18].

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the nozzle will compress the extruded material to a degree. This flattens the top surface and causes lateral spreading at the sides. A general rule of thumb is that the thickness of each printed layer should be between 20% and 50% of the extrusion nozzle’s inner diameter. For an extrusion nozzle having an inner diameter of 500 mm, the recommended layer thickness would, therefore, be in the range of 100e250 mm. If the layer thickness is too small, there will not be enough room for the material to exit the nozzle. If the layer thickness is too large, then there will be an insufficient flattening of the extruded bead to produce sufficiently high material density. Although the aforementioned rule of thumb for selecting layer thickness based on the nozzle diameter is important, Hebda et al. [19] observed that the height (H) of a printed track of material is larger than the nominal layer thickness (i.e., gap distance between the nozzle and substrate during printing) due to die swell as the polymer exits the nozzle. Die swell also affects the actual width (W) of a printed track of material. From Ref. [19], the prediction of H and W is based on the nozzle radius (RN ), the radially averaged velocity of polymer exiting the nozzle (UN), the velocity of the nozzle over the print bed during printing (UP), and an experimentally derived prefactor (a). Note that UN and UP need not be the same. If the velocity of the print head across the print bed exceeds the velocity of polymer exiting the nozzle (i.e., UP > UN), then tension on the soft extruded bead of material will elongate it and reduce the printed track width. If, on the other hand, material flows out of the nozzle at a greater velocity than the print head is moving (i.e., UN > UP), then the extruded material will spread out laterally from beneath the nozzle orifice during printing, and the printed track width will increase. Eqs. (12.1) and (12.2) estimate the actual width and height of a printed bead of material [19]. rffiffiffiffiffiffiffi UN W ¼ 2RN a UP H¼

2RN a

rffiffiffiffiffiffiffi UN UP

(12.1)

(12.2)

The swelling prefactor a varies from polymer to polymer and is also influenced by the temperature of the polymer during extrusion and the temperature of the substrate. Hebda et al. experimentally determined a values of 1.750 and 1.252 for PLA and ABS, respectively, with extrusion temperatures of 220
C and substrate temperatures of 50
C. Fig. 12.12 shows good agreement between the model and experimental data. An accurate estimate of the printed width of material for a given nozzle size is extremely important for reducing void space in a printed structure. Specifically, the width of printed material is needed to determine the stepover distance between adjacent printed tracks of material that will minimize void space. Fig. 12.13 illustrates the combined effect of layer thickness and track spacing on material density in an FFF process. In Fig. 12.13(a), the road spacing is large enough that there is a gap between adjacent tracks of material. No bonding between adjacent tracks takes place in this scenario. Fig. 12.13(b) illustrates the case where the spacing results in extruded

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RN

–R

Z (mm)

Experiment model

0.5 GN

0

–0.8

–0.4

0

0.4

0.8

X (mm)

Figure 12.12 Predicted and experimental FFF printed track width and height [19].

Figure 12.13 (a) Excessive track spacing leading to gaps between adjacent tracks; (b) track spacing leading to contact between adjacent tracks with void space; (c) proper track spacing that eliminates void space between adjacent tracks.

tracks that make tangential contact without merging. The void space within the printed material is evident. Fig. 12.13(c) shows the case where a proper selection of track spacing based on accurate estimates of printed track width and thickness results in essentially complete densification of the material. Koch et al. [20] introduce the notion of a solidity ratio (SR), which is the ratio of cross-sectional area in an extruded bead to the rectangular bounding box area of the bead as illustrated in Fig. 12.14. An extruded bead with a perfectly rectangular cross-section will produce 100% dense material and will have an SR of 1. In this study that related solidity ration to mechanical properties of FFF samples, ASTM D638 Type 1 dog-bone tensile specimens were fabricated in two orientations. Specifically, the axis of printed tracks in one batch was parallel to the tensile axis, and it was perpendicular to the print orientation in the second set. As the SR value of coupons pulled perpendicular to the print direction increased from 0.63 to 1.00, the tensile strength increased

(a)

(b)

hlayer

hlayer Wbead

Figure 12.14 Concept of the solidity ratio in FFF parts [20].

Wbead

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from approximately 3e28 MPa. This underscores the importance of eliminating void space in FFF printed samples. Note, however, that even with an SR value of 1.00 (i.e., essentially fully dense parts), the tensile strength of the FFF samples in which the track orientation was perpendicular to the tensile axis was only 69.86% that of the injection-molded ABS samples and 77.4% that of the samples whose tracks were printed parallel to the tensile axis. This underscores the fact that density is just one of several factors that contribute to material performance in these components.

12.3.1.2 Effect of deposition toolpath on material density Although void space in printed parts can be minimized through proper selection of layer thickness and track spacing in relation to the nozzle diameter, voids that result from the deposition toolpath in relation to the area to be printed are also a concern. This is seen in Fig. 12.15, which shows chopped carbon fiber in a PA6 matrix printed around a hole. As previously discussed, FFF machines typically print outer contours that form the surface, or skin, of a part. They then fill in the space inside of the boundary contours. Given that the extruded bead of material has a width as discussed in Section 12.3.3.1, there will be locations where the width of the printed bead of material is simply too large to fit. The red arrows in Fig. 12.15 highlight two voids beneath a printed hole where the curvature of the printed contours around the holes results in a wedge-shaped void where the printed track cannot fit. This is largely unavoidable. In some cases, the part geometry and/or printing parameters may be adjusted such that voids resulting from deposition toolpaths are located in regions of the part that are not critically loaded.

2000 μm

Figure 12.15 Voids resulting from areas that are too small for the printed track of material to fit into.

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12.3.2 Weld strength in composite FFF processing Although reducing void content through proper control of printing parameters is critical to achieving the desired material properties, high density alone is not sufficient to ensure that additively manufactured material properties will approach those of the bulk material. Poor inter and/or intralayer fusion in FFF is often attributed to inadequate entanglement of polymer chains across the interface between an extruded bead of polymer and previously deposited material. Reptation theory [21] is often used as the basis for predicting inter/intralayer bond strength in FFF processing. Under appropriate processing conditions, a sufficient quantity of polymer chains will penetrate far enough across the interface between newly and previously extruded material that the bond strength at the interface (s) will essentially match the strength of the bulk polymer (sN ). The time needed to reach bulk strength at a weld interface having a given temperature is called as the reptation time (tr ). The complicating factor for composite FFF is that after material exits the nozzle and the extrusion head moves away, the temperature of the weld interface rapidly drops below the polymer’s glass transition (Tg) or crystallization (Tc) temperature. Excellent treatments of the science behind filament weld formation may be found in Refs. [22,23,24]. The composite polymer weld strength and resulting durability of the additively manufactured composite part are dependent upon many factors including the extrusion temperature, molecular weight and melt viscosity of the polymer, and extrusion head velocity. Extrusion temperature is perhaps the easiest and most direct factor to control during composite FFF. Each polymer has a recommended range of extrusion temperatures. If the temperature is too low, then very poor interlayer bond strength is obtained. If the temperature is excessively high, potential problems include thermal degradation of the polymer, unwanted oozing of the polymer from the nozzle, loss of geometric accuracy from slumping, poor surface finish, and warping. The selection of the extrusion temperature involves making tradeoffs between mechanical properties and geometric accuracy. It is well understood that the rate of molecular diffusion and polymer entanglement across the weld interface drops sharply when the temperature falls below the polymer’s glass transition (Tg) or crystallization (Tc) temperature. McIlroy and Olmsted [23] suggest that the temperature typically drops below Tg on the order of 100 ms or less. Ignoring potential problems with warping, slumping, or other print defects, Fig. 12.16 fairly clearly shows just how strongly extrusion temperature affects fracture toughness. The vertical axis plots fracture toughness normalized by the equilibrium fracture toughness of the bulk material. The horizontal axis represents the entanglement number that relates to the molecular weight. McIlroy and Olmsted found that the number and depth of polymer entanglements (and hence mechanical properties) are inversely related to molecular weight. Fig. 12.16 shows that at the highest printing temperature of 300
C, the weld fracture toughness essentially matches that of the bulk material regardless of molecular weight. Fracture toughness drops sharply though as the extrusion temperature decreases to 250 and then 200
C. As previously mentioned, increasing the melt extrusion temperature to raise the temperature of previously deposited material above Tg or Tc to enhance weld strength

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1.6

Bulk strength

GCweld/GC

0.8 0.4 0.2

TN = 300 TN = 250 TN = 200

0.1 5

10

20

Eq.40 Eq.40 Eq.40

40

80

Zeq

Figure 12.16 Plot of normalized fracture toughness as a function of different printing temperatures [23].

presents risks for geometric accuracy and surface finish. Another approach to increasing weld strength is to instead focus an external heat source such as a laser just ahead of the advancing extrusion nozzle to raise the temperature above Tg or Tc. Fig. 12.17(a) from Luo et al. [25] conceptually illustrates the limited degree of polymer entanglement in a carbon fiber PEEK composite due to the temperature of the previously printed layer falling below Tg when no laser heating is used. Fig. 12.16(b) conceptually illustrates how the temperature rises above Tg using a 10W fiber laser results in substantially greater polymer entanglement. Care must be taken to avoid overheating the polymer with the laser to the point where it begins to degrade though. Luo et al. found that a laser power of 10 W and a printer scanning

(a)

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100°C

Printed layer

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Printing layer Laser Printed layer

400°C

Figure 12.17 (a) Relatively weak polymer entanglement and interpenetration depth at the weld interface without laser heating; (b) much higher polymer entanglement and interpenetration depth at the weld interface with laser heating [25].

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speed of 120 mm/min increased the interlaminar shear strength (ILSS) by slightly over 250% versus the ILSS obtained with conventional FFF printing and no laser heating. This laser preheating technique is used commercially by Arevo Labs (Milpitas, CA). Although heating the material is an effective way to increase weld strength, there are additional methods for increasing the mobility of polymer chains to improve mechanical properties. For example, the rate of diffusion can be increased by decreasing the molecular weight of the polymer. This approach is not typically used in practice, as lowering the molecular weight may adversely affect the mechanical properties of the polymer. Another way to increase the rate of diffusion is to decrease the melt viscosity through the addition of plasticizing agents. For example, Ko et al. [24] add triphenyl phosphate (TPP) plasticizer in concentrations of 5 and 10 wt% to polycarbonate-acrylonitrile butadiene styrene (PC-ABS) to lower the melt viscosity. This had the desirable effect of lowering Tg from 107.9
C for the neat PC-ABS to 99.3 and 91.8
C for the PC-ABS with 5 and 10 wt% TPP respectively. Under a given set of processing conditions, the welding time (tw) is the time that the interface between newly printed material and previously printed material must maintain a given temperature to produce a bond whose strength essentially matches that of the bulk material. Ko et al. found that tw decreased by nearly an order of magnitude with the addition of just 5 wt% TPP. As expected, tw also decreased as the temperature increased, thus reinforcing the notion that increasing temperature and/or decreasing viscosity of the polymer melt through the addition of plasticizing agents are both effective ways to promote weld strength in FFF processing.

12.3.3 Wetting and fiber-matrix interfacial bond strength A key consideration affecting the durability of additively manufactured fiberreinforced composites is the interfacial bond strength between the fibers and the surrounding matrix. Thermoset matrix materials such as epoxy have historically been favored by many in traditional composite manufacturing due to their low viscosity, relative ease of fiber wetting, and high interfacial bond strength with fibers. However, FFF with epoxy-fiber composite mixtures has not been widely employed due in large part to difficulties with the extremely high curing speeds of the thermoset that would be needed for the extruded material to retain its shape. Composite FFF with thermoplastic matrix materials, in comparison, is relatively straightforward to implement. Achieving good wetting of the fibers and good distribution of fibers across the cross-sectional area of the extruded bead is a significant challenge. There are several ways that wetting and dispersion are being tackled. One approach to enhancing wetting is to raise the extrusion temperature during printing. As the thermoplastic extrusion temperature increases, viscosity decreases, melt flow increases, and wetting of fibers improves. It is worth noting, however, that a carbon fiber composite blend has higher thermal conductivity and greater strength and stiffness than that of the neat thermoplastic matrix material. It is therefore often possible to extrude carbon fiber composite blends at significantly higher temperatures than the maximum recommended temperature of the neat thermoplastic matrix without sagging or other complications. Tian et al. [26] demonstrated this with a study

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of the effect of various printing conditions on flexural strength and modulus when a 1K continuous carbon fiber tow was coextruded with thermoplastic PLA under a variety of printing conditions using a customized FFF machine. A key finding of the study was that as extrusion temperature increased from 180 to 240
C, impregnation of PLA into the fiber bundle was enhanced due to lower viscosity and increased melt flowability of the heated PLA. Fig. 12.18(a)e(f) show flexural fracture surfaces of specimens printed at extrusion temperatures of 180 and 240
C, respectively. Referring to Fig. 12.18(b), there is essentially no evidence of PLA impregnation into the 1K fiber tow at an extrusion temperature of 180
C. Fig. 12.18(e), however, shows better impregnation of PLA into the fiber bundle at the higher extrusion temperature of 240
C. Note that the typical extrusion temperature for PLA is approximately 185e205
C; hence, the coextrusion of carbon fiber and PLA was possible at a much higher temperature than is typically used. As the extrusion temperature was increased from 180 to 230
C, the flexural strength increased approximately linearly from 111 to 145 MPa (w30% increase). Similarly, the flexural modulus also showed an approximately linear increase from 5.1 to 8.6 GPa (w68% increase). However, Fig. 12.18 does exhibit some degree of fiber pull-out at the fracture surface. This underscores the need to achieve both proper wetting and high interfacial bond strength. Fig. 12.18 underscores a fundamental challenge associated with coextrusion where a polymer filament and fiber bundle are simultaneously fed into a heated extrusion nozzle. The fiber bundle does not easily spread out, thus achieving good impregnation of the fiber bundle during composite FFF is difficult. Garofolo and Walczyk [27] instead advocate impregnating carbon fiber tows before the FFF process. Their approach is to use barrel-shaped rollers to fan out a carbon fiber tow into a wide tape profile rather than a circular bundle. The fanned-out tow is then fed through a

(a)

(b)

(c)

10 mm

(d)

(e)

(f)

10 mm

Figure 12.18 (a)e(c) Fracture surfaces of carbon fiber PLA composite coextruded at 180
C; (d) e(f) Fracture surfaces of carbon fiber PLA composite coextruded at 240
C [26].

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thermoplastic slot die coater. Because the individual strands of fiber have been spread out to an essentially linear array, each fiber is readily coated with a thermoplastic material. The coated fiber tow then progresses through a circular die that reshapes the coated fiber tow back into a cylindrical profile that can be spooled for use with a composite FFF machine. In contrast to the challenges with achieving proper fiber tow impregnation with the coextrusion FFF approach, achieving good dispersion and impregnation with chopped fibers is relatively straightforward. Liao et al. [28] use twin-screw extrusion to compound long carbon fibers 15e20 mm in length with a PA12 matrix at loading fractions of 2, 4, 6, 8, and 10 wt%. The chopped carbon fiber PA12 composite is extruded into feedstock filament that is ready for printing via the single nozzle composite FFF approach. Fig. 12.19 shows fibers at 0 and 90 degrees to the fracture surfaces of tensile coupons in which the fiber loading was 10 wt%. Here, the fibers are well dispersed, the impregnation of PA12 between fibers is excellent, and PA12 coating on the fibers at the fracture surface is evident thus providing evidence of good interfacial adhesion. Liao et al. also assessed the impact strength of printed samples with fiber contents ranging from 0 to 10 wt%. The impact strengths of the 2% and 4% carbon fiber content samples were nearly 50% lower than that of the neat PA12 material as seen in Fig. 12.20. The impact strength then increased with higher fiber loading fractions to the point where the impact strength of 24.8 kJ/m2 at 10 wt% fiber loading exceeded the 22.5 kJ/m2 impact strength of the neat PA12. The authors hypothesized that at low fiber loading levels, stresses at the end of each fiber during impact resulted in cracks that easily propagated due to the low fiber loading in the material. As the modulus increased with the addition of more fiber, the material was better able to resist the propagation of cracks. The flexural strength and modulus of the composite FFF material at 10 wt.% fiber were 251.1% higher and 346% higher, respectively, than the neat PA12 samples.

Figure 12.19 Tensile coupon fracture surfaces for a chopped carbon fiber PA12 composite with 10 wt% loading at (a) 0 degrees and (b) 90 degrees to the tensile axis [28].

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Impact strength (KJ/2)

Impact strength (KJ/m2)

25

20

15

10

5

0 0

2

4

6

8

10

Carbon fiber contents (% )

Figure 12.20 Impact strength for chopped carbon fiber PA12 composite coupons at fiber loading fractions ranging from 0 to 10 wt% [28].

12.3.4

Effect of printed fiber orientation and distribution

As previously described, the path that the extrusion head follows during material deposition determines the orientation of fibers in the printed sample. This holds true even when the reinforcing phase is in the form of chopped fibers due to the shear alignment of fibers as they flow through the nozzle. In this section, we discuss the influence of fiber routing strategies and the distribution of fibers through the thickness of a sample on the composite mechanical properties.

12.3.4.1 Fiber orientation and distribution effects on mechanical properties The effect of reinforcement fiber type and print configuration for samples produced via continuous fiber composite FFF has been well studied [8,29e32]. Although carbon fiber reinforcement has received the greatest attention, glass, and Kevlar fiber reinforcement has also been studied [33]. Different from conventional composite processes that employ woven sheets, continuous composite FFF processes lay down one fiber bundle in the desired path at a time such that the overall fiber content and fiber orientation may be easily controlled. Most studies consider fiber orientation in terms of isotropic fiber that is printed parallel, perpendicular, or staggered with respect to the mechanical testing axis. Fig. 12.21(a) shows an ASTM D638 Type I tensile bar in which the fiber (blue lines) is routed in concentric rings. Note that because each ring of fiber is routed in a complete loop, there is always an even number of fibers (i.e., 10 fibers through the gauge region in this case). In addition, note the red circle in Fig. 12.21(a) in which the start and end locations of fiber deposition are located

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Figure 12.21 (a) Concentric fiber pathing; (b) isotropic fiber pathing parallel to tensile axis without any concentric rings; (c) isotropic fiber pathing perpendicular to tensile axis without any concentric rings; (d) isotropic fiber pathing parallel to the tensile axis with two concentric rings.

in the gauge region of the sample. This is detrimental to the mechanical properties of the sample and should be avoided. Fig. 12.21(b) shows the same coupon in which fiber has been routed using an isotropic fiber orientation that is parallel to the tensile axis. Note that the number of fibers need not be even with isotropic fiber routing. Fig. 12.21(b) shows 11 fibers rather than 10 passing through the gauge region. Fig. 12.21(c) employs isotropic fiber routing with fibers oriented perpendicular to the tensile axis. Lastly, Fig. 12.21(d) shows a combination of two concentric perimeters of fibers that surround isotropic fibers fitted in the remaining space . In addition to considering fiber routing within each layer, it is possible to control the number of layers containing fiber. Fig. 12.22 shows three scenarios in which fiber is only located in the opposing faces of a panel (case B), it is located in the opposing faces plus the center layer of a panel (case B þ C), or it is evenly distributed through the thickness of the panel (case E) [30]. Starting with the behavior of composite FFF structures printed using the concentric fiber routing strategy, van de Werken et al. conducted a finite element study of ASTM D3039 Type I samples printed with differing numbers of concentric perimeter shells [8]. Fig. 12.23(a) illustrates the fiber routing with four contour shells, while

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Levels

Factors Reinforcement type

Isitropic (I)

Concentris (C)

Reinforcement distribution

Borders (B)

Borders + Center (B+C)

Equidistant (E)

PA6 matrix 100% infill rectangular pattern 0° –45° 90° +45 Isotropic CF reinforcement x

(B)

x

y

(B+C)

z

x

y

(E)

z

Concentric

y z

Figure 12.22 Strategies for distributing fiber only in opposing faces (B), in the opposing faces plus the center (B þ C), or evenly distributed through the thickness of the sample (E) [30].

(a) Nylon

3D printed carbon fibre reinforced filament

(b)

0

50

100

150

200

250

300

350

400

450

(c)

–50

–40

–30

–20

–10

0

10

20

30

40

50

Figure 12.23 (a) Concentric fiber routing in ASTM D3039 Type I coupon; (b) FEA normal stress at failure (MPa); (c) FEA shear stress at failure (MPa) [8].

Fig. 12.23(b) and (c) shows the normal stress at failure and the shear stress at failure, respectively, from the FEA study for samples with six contour shells. The peak normal and shear stresses occur where the fiber follows the curved radius between the grip region and the gauge region. The fiber in these regions is not oriented parallel to the tensile axis. Not surprisingly, it was noted that this is where samples tended to fail. In light of van de Werken’s findings, it is natural to study the isotropic fiber routing strategy in which all fibers may be printed parallel to the axis of tension. Pyl et al. conducted tensile tests on ASTM D638-14 coupons containing 18 layers of carbon fiber printed using the isotropic and concentric fiber routing strategies [31]. The tensile

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strengths of the isotropic and concentric samples were 422 and 380 MPa, respectively. The elastic modulus of the isotropic and concentric samples were 48.75 and 43.77 GPa, respectively. Failure analysis suggested that samples printed using the concentric fiber routing strategy failed at the neck region where fiber curves into the gauge region of the sample. This is consistent with the findings of van de Werken and reinforces the notion that fiber routing should be as parallel to the axis of tension as possible. Pyl et al. also studied the effect that the distribution of fibers through the thickness of a sample has when the sample is placed in tension. Two sets of ASTM D638-14 tensile test coupons were prepared. All samples contained eight layers of carbon fiber. However, one set of samples was printed with all eight layers of carbon fiber printed one on top of another. In the other set of samples, each layer of carbon was separated by a printed layer of PA6 (nylon). When the fiber layers were separated by a layer of PA6, the modulus of elasticity dropped considerably from 57.09 to 31.65 GPa. As previously mentioned, there are price-performance tradeoffs between carbon, Kevlar, and glass fiber-reinforced AM parts. Al Abadi et al. compare the tensile performance of coupons fabricated using each of these three reinforcing fiber types using equivalent fiber loading fractions [33]. As seen in Fig. 12.24, the strength and stiffness of the carbon fiber-reinforced samples is significantly higher than that of Kevlar or glass. Glass fiber’s performance is slightly below that of Kevlar; however, glass is typically much less expensive than Kevlar.

400 350

40% caron - 60% nylon E = 37000 MPa

Axial strains

300

Stress (MPa)

250 200 40% kevlar - 60% nylon E = 8700 MPa

150

40% glass - 60% nylon E = 6400 MPa

100 50 0 –0.01 –0.005 0 Lateral strain (mm/mm)

0.005

0.01

0.02 0.015 Axial strain (mm/mm)

0.025

0.03

Figure 12.24 Stressestrain curves for tensile coupons printed with continuous carbon, Kevlar, and glass fiber reinforcements at 40% fiber loading fractions [33].

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Although properties of composite FFF structures in tension and flexure have been well studied, data and design recommendations for fatigue and fracture toughness with composite FFF is still sparse. Agarwal et al. compared the fatigue life of composite FFF samples printed using isotropic and concentric fiber routing strategies with a PA6 thermoplastic matrix at 25 and 50 vol% glass fiber loading [32]. ASTM D3479 tensionetension composite fatigue test coupons were prepared for testing. Samples printed using the isotropic fiber routing strategy outperformed samples printed using concentric fiber routing as seen in Fig. 12.25. It is also interesting to note that the fatigue performance of the printed glass-PA6 composite samples outperformed conventionally manufactured vacuum-bagged epoxy fiberglass with uniaxial fiber orientation parallel to the tensile axis. Similarly, Imeri et al. studied fatigue life of carbon, glass, and Kevlar-reinforced samples having a range of fiber loading fractions [34]. Coupons were printed using both concentric and isotropic fiber routing strategies. Results from this study were consistent with the findings of Agarwal et al. and suggested that isotropic fiber routing was preferable over concentric fiber routing for fatigue life. It is noted that in the flexural loading of sandwich panel structures, fiber is normally routed only in the opposing faces of the panel, and the space between the reinforced faces is filled with a sparse infill. This is important for AM processes, as the sparse infill printing greatly reduces print time. However, the user still has the choice of which infill geometry pattern to print, and how dense that infill pattern should be. NaranjoLozada et al. found that when one wishes to print hollow infill patterns beneath a solid-skinned part, as in the case of sandwich panels, the effect of infill percentage had a very modest impact on elastic modulus and tensile strength at yield within the range of 10%e70% infill density. They did, however, conclude that the shape of the hollow infill pattern matters. The use of triangular infill patterns yielded significantly better results than when rectangular infill patterns were used.

Stress amplitude (MPa)

250.00

50 % volume fraction, 0 degree orientation Isotropic

200.00

Concentric, 4 layers

150.00

Concentric, 8 layers Concentric, 12 layers

100.00

Wet lay up

50.00 0.00 1.E+0

Vacuum bagging 1.E+1

1.E+2

1.E+3

1.E+4

1.E+5

1.E+6

1.E+7

Log N N=number of cycles

Figure 12.25 Fatigue life data for glass fiber composites produced via isotropic fiber routing FFF, concentric fiber routing FFF, and conventional composite techniques [32].

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12.3.4.2 Fiber orientation around holes The overwhelming majority of composite AM research published to date has focused on ASTM standard tensile, flex, and/or compression coupon geometries. Structures containing holes present a special challenge in composite manufacturing due to stress concentrations. It is therefore not surprising that several researchers have studied the potential advantages of using AM technologies to route fibers around holes such that stress concentrations may be reduced. Pyl et al. hypothesized that by printing concentric rings around open holes, similar to what is seen in Fig. 12.15, more uniform stress concentrations would be observed in comparison with the case where a hole of the same diameter is drilled in a solid printed panel [35]. Samples were printed with and without holes in a carbon fiber PA6 composite. Holes of an equivalent diameter were then drilled in the samples printed without holes. Digital image correlation (DIC) was used to observe engineering strain distributions around the printed and drilled holes as the samples were loaded in tension. Fig. 12.26 shows DIC results, which indicate that, although the load distribution is somewhat more distributed with the printed and fiber-reinforced holes, the magnitude of strain is higher around the printed holes. The maximum load for the sample with the drilled hole was 22 kN versus just 14 kN for the sample with the printed hole. Similarly, the tensile stiffness for the sample with the drilled hole was 46 versus 37 GPa for the sample with the printed hole. Although the reasons for this are not immediately clear, the previous discussion in Section 12.3.1.2 pertaining to unavoidable voids and discontinuities in the printed material around holes may at least partially explain this result. Up to this point in the discussion, concentric and isotropic fiber routing strategies have been discussed. The majority of published research to date employs commercially available continuous fiber FFF machines that employ a closed architecture

(a)

eyy 0.0105

0.0052

0.015

(b)

eyy 0.00548

0.00348

0.00148

Figure 12.26 Longitudinal engineering strain distribution in (a) sample with printed hole and reinforcing rings, and (b) sample with drilled hole [35].

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that does not allow the user to generate custom toolpaths. Consequently, very little research has been focused on the generation of novel material routing strategies that go beyond isotropic and concentric fiber placement. A noteworthy exception is the work of researchers working with custom-built open-architecture FFF machines to develop printing strategies that emulate (to a certain degree) behavior of woven textiles [36e38]. Fig. 12.27 shows a tailored fiber placement strategy used to print woven structures via the continuous composite FFF technique [36]. This unique serpentine space-filling approach results in fibers that pass over and under each other at specific locations. Fig. 12.28(a) shows a special case where the pathing has been modified to permit fiber to flow around a hole from multiple directions [37]. In a loading condition where a bolt is placed in the hole, and the sample is placed in tension, this fiber routing strategy helps transmit the load around the hole. This is in contrast to Fig. 12.28(b) where a drilled hole cuts fibers that then terminate at the periphery of the hole. Fig. 12.29 shows experimental and simulation results indicating that the woven hole performs substantially better than the drilled hole and nearly as well as an intact sample of material that has no hole at all [38].

12.3.4.3 Z-pinning In recognition of the fact that laminations of sheets are susceptible to delamination, the use of reinforcing fiber pins through the thickness of lamination has been used with traditional composite manufacturing processes for decades. Mouritz provides a survey of z-pinned composite laminates [39]. Fig. 12.30 shows top-down and transverse views of pins that have been inserted downward through multiple plies of the composite to “stitch” them together to better resist delamination [39]. This same concept has been adapted by researchers at Oak Ridge National Laboratories for use with composite additive manufacturing processes [40,41]. The basic process steps are illustrated in Fig. 12.27 [41]. Namely, a component is printed up

Figure 12.27 A tailored fiber placement strategy used to print woven structures via the continuous composite FFF technique [36].

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Tailor woven

431

Drilled

350 300 250

Tensile strain

Tensile strength (MPa)

Figure 12.28 (a) Hole printed into sample via tailor woven fiber deposition strategy, and (b) hole drilled into sample printed using tailor woven fiber deposition strategy [37].

200 150 100 50 0

Drilled

Woven

Experiment result

Intact

Model result

2.0% 1.8% 1.6% 1.4% 1.2% 1.0% 0.8% 0.6% 0.4% 0.2% 0.0%

Woven

Drilled

Experiment result

Intact

Model result

Figure 12.29 Tensile strength and strain of tailor woven samples having a woven hole, a drilled hole, and no hole (intact) [38].

(a)

(b) Pin 2θ

Resin zone

z-pins

Wavy fibres

Figure 12.30 Top-down (a) and transverse (b) views of z-pinning [39].

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(a)

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(b)

(e)

(c)

(f)

Figure 12.31 Z-pinning strategy for [41].

to some height using any suitable composite FFF process. As seen in Fig. 12.31(a), a series of vertical holes of varying depths are intentionally printed into the component. At regular intervals, the extrusion nozzle moves to these hole locations and injects material into the hole as in Fig. 12.31(b). The material could be a chopped fiber polymer composite, or it could even be strands of continuous fiber. In either case, the expectation is that the fiber(s) will be vertically aligned inside the hole. This process of printing and then filling staggered holes at different locations and depths continues until the component is complete as seen in Fig. 12.31(c)e(f). Tensile bars printed in PLA with and without z-pinning were tested [40]. After normalizing results based on the mass of material printed, the specific ultimate tensile strength of z-pinned samples was 67.7 kPa/g compared with a specific Ultimate Tensile Strength (UTS) of 22.5 kPa/g for the nonpinned samples. The toughness of samples printed with and without z-pinning showed similar improvement. Although this technique must still be investigated further for effectiveness with composite AM materials, it shows considerable potential.

12.4

Engineered composite cellular structures

As previously described, the infill patterns used in FFF deposition are typically hollow 2D patterns of triangles, rectangles, hexagons, or other shapes. However, the geometric freedom provided by FFF processes suggests that the infill geometry may be an additional area of opportunity for composite additive manufacturing techniques. Specifically, there is growing interest in engineered lattice structures whose geometric shapes can be tailored to accommodate the specific loading conditions of a given application. Very little work has been done with composite AM of engineered lattice structures; however, this section of the chapter will provide a forward-looking glimpse of emerging areas of opportunity.

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12.4.1 Open cell composite lattice structures Broadly speaking, an engineered lattice structure can be thought of as a cellular structure in which a repeating unit cell has been designed for specific loading conditions. Fig. 12.32(a) shows an open-cell engineered lattice structure. Note that with conventional layerwise FFF deposition, the diagonal strut resembles a lamination of printed “dots.” The sample shown here is printed in ABS plastic to highlight the layerwise laminations, but a chopped carbon fiber material would have the same result. From a composite material perspective, the ideal diagonal strut would be smooth and continuous, and the fibers within the polymer matrix would be axially aligned with the axis of the strut. Axial lattice extrusion (ALE) is an approach whereby the FFF extrusion nozzle is raised up and down along the z-axis during printing such that smooth and continuous struts are produced as seen in Fig. 12.32(b). To prevent slumping of the composite polymer as it exits the nozzle, cool gas is blown on the strut. The closeup view of the strut shown in Fig. 12.32(b) also highlights the fact that the chopped carbon fibers are, in fact, axially aligned with the axis of the strut. This strut, therefore, represents an ideal contrast to the laminated strut with no fiber alignment in Fig. 12.32(a).

(a)

(b)

2 mm

2 mm

Figure 12.32 Open-cell engineering lattice structure printed via (a) layerwise FFF and (b) the axial lattice extrusion technique.

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Closed cell plate lattice structures

Although most AM research involving engineered lattice structures has focused on open-cell geometries, the ALE process briefly described in Section 12.4.1 highlighted the fact that it is possible to extrude material in short distances (e.g., several mm) without any support beneath the material being extruded. This opens up the possibility of fabricating closed-cell engineered lattice structures such as the one shown in Fig. 12.33. This sample consists of a series of intersecting flat plates; hence, we refer to it as a plate lattice structure. The internal structure of this particular geometry is such that the plates encase closed pyramidal volumes. The samples shown in Fig. 12.33 have been printed using a chopped carbon fiber PA6 polymer matrix composite (Markforged Onyx). Mechanical properties will be published at a later date; however, preliminary results from high strain rate impact testing indicate that this combination of a thermoplastic carbon composite and plate lattice geometry has extremely impressive energy absorption capabilities. The sample shown in Fig. 12.33(d) has been subjected to five drop tests in which a 50 kg plate has been dropped from a height of 0.6 m onto the 75 mm  75 mm  75 mm (original size) carbon composite plate lattice sample. The structure exhibits an unexpected viscoelastic recovery behavior following each impact event. This suggests that this combination of composite material and plate lattice geometry holds considerable promise for use as a damage tolerant structure.

Figure 12.33 (a) Drop testing apparatus, (b) and (c) close-up views of the drop mass approaching and landing upon the plate lattice structure, (d) plate lattice structure following five drop tests.

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Summary and conclusions

This chapter has focused on thermoplastic polymer matrix additive manufacturing with both discontinuous and continuous fiber. Key takeaways can be summarized as follows. The majority of composite AM research involves thermoplastic matrix materials rather than thermosets. Both amorphous and semicrystalline thermoplastics are used; however, semicrystalline matrix materials such as PA and PEEK are used much more often than the amorphous thermoplastics. Additive manufacturing via fused filament fabrication provides engineers with the ability to locally control fiber orientation based on expected load paths. Intelligent fiber routing based on FEA results is expected to be a growing area of active research. As an extension of this point, it is only necessary to place fiber where it is structurally needed. This can greatly reduce cost. As FFF extrudes a single fiber tow at a time, getting a high fiber loading fraction can be a challenge. When routing fiber, most tensile, flexural, and fatigue testing students have consistently concluded that isotropic fiber routing generally produces better mechanical properties than concentric fiber routing. Inter- and intralayer bond strength is a major challenge with FFF material deposition. External heating of the bond area above Tg or Tc has been shown to enhance bond quality. Lowering of melt viscosity through the addition of plasticizers has also helped. Higher extrusion temperatures are possible due to the ability of fiber loaded melts to conduct heat and resist deformation. Looking ahead, composite additive manufacturing of open- and closed-cell engineered lattice structures holds considerable promise for light weighting and for applications requiring damage tolerance. Nonplanar printing of panel structures is a promising new area of research as well that allows fiber routing to conform to the shape of curved panels.

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Durability of polymer matrix composites fabricated via additive manufacturing

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[23] C. McIlroy, P.D. Olmsted, Disentanglement effects on welding behaviour of polymer melts during the fused-filament-fabrication method for additive manufacturing, Polymer 123 (2017) 376e391. [24] Y.S. Ko, D. Herrmann, O. Tolar, W.J. Elspass, C. Br€andli, Improving the filament weldstrength of fused filament fabrication products through improved interdiffusion, Addit. Manuf. 29 (2019) 100815. [25] M. Luo, X. Tian, J. Shang, W. Zhu, D. Li, Y. Qin, Impregnation and interlayer bonding behaviours of 3D-printed continuous carbon-fiber-reinforced poly-ether-ether-ketone composites, Compos. Appl. Sci. Manuf. 121 (2019) 130e138. [26] X. Tian, T. Liu, C. Yang, Q. Wang, D. Li, Interface and performance of 3D printed continuous carbon fiber reinforced PLA composites, Compos. Appl. Sci. Manuf. 88 (2016) 198e205. [27] J. Garofolo, D. Walczyk, In-situ Co-extrusion: additive manufacturing of continuous reinforced thermoplastic composites, in: Proceedings of the American Society for CompositesdThirty-Third Technical Conference, 2018. [28] G. Liao, Z. Li, Y. Cheng, D. Xu, D. Zhu, S. Jiang, J. Guo, X. Chen, G. Xu, Y. Zhu, Properties of oriented carbon fiber/polyamide 12 composite parts fabricated by fused deposition modeling, Mater. Des. 139 (2018) 283e292. [29] A.N. Dickson, J.N. Barry, K.A. McDonnell, D.P. Dowling, Fabrication of continuous carbon, glass and Kevlar fibre reinforced polymer composites using additive manufacturing, Addit. Manuf. 16 (2017) 146e152. [30] M. Araya-Calvo, I. Lopez-Gomez, N. Chamberlain-Simon, J.L. Le on-Salazar, T. GuillénGiron, J.S. Corrales-Cordero, O. Sanchez-Brenes, Evaluation of compressive and flexural properties of continuous fiber fabrication additive manufacturing technology, Addit. Manuf. 22 (2018) 157e164. [31] L. Pyl, K.A. Kalteremidou, D. Van Hemelrijck, Exploration of specimen geometry and tab configuration for tensile testing exploiting the potential of 3D printing freeform shape continuous carbon fibre-reinforced nylon matrix composites, Polym. Test. 71 (2018) 318e328. [32] K. Agarwal, S.K. Kuchipudi, B. Girard, M. Houser, Mechanical properties of fiber reinforced polymer composites: a comparative study of conventional and additive manufacturing methods, J. Compos. Mater. 52 (23) (2018) 3173e3181. [33] H. Al Abadi, H.T. Thai, V. Paton-Cole, V.I. Patel, Elastic properties of 3D printed fibrereinforced structures, Compos. Struct. 193 (2018) 8e18. [34] A. Imeri, I. Fidan, M. Allen, D.A. Wilson, S. Canfield, Fatigue analysis of the fiber reinforced additively manufactured objects, Int. J. Adv. Manuf. Technol. 98 (9e12) (2018) 2717e2724. [35] L. Pyl, K.A. Kalteremidou, D. Van Hemelrijck, Exploration of the design freedom of 3D printed continuous fibre-reinforced polymers in open-hole tensile strength tests, Compos. Sci. Technol. 171 (2019) 135e151. [36] A.N. Dickson, K.A. Ross, D.P. Dowling, Additive manufacturing of woven carbon fibre polymer composites, Compos. Struct. 206 (2018) 637e643. [37] A.N. Dickson, D.P. Dowling, Enhancing the bearing strength of woven carbon fibre thermoplastic composites through additive manufacturing, Compos. Struct. 212 (2019) 381e388.

438

Durability of Composite Systems

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Index Note: Page numbers followed by “f” indicate figures and “t” indicate tables. A Additive manufacturing (AM) cellular structures closed cell plate lattice structures, 434, 434f infill patterns, 432 open cell composite lattice structures, 433, 433f components, 403e404 fiber orientation and distribution effects, 426f concentric fiber pathing, 424e425, 425fe426f failure analysis, 426e427 fatigue life data, glass fiber composites, 428, 428f flexural loading, 428 isotropic fiber pathing, 424e425, 425f space-filling approach, 429e430, 431f strain distribution, 429, 429f stressestrain curves, 425, 427f tailored fiber placement strategy, 429e430, 430f tensile strength, 429e430, 431f fracture surfaces, 421e422, 422f fused filament fabrication (FFF), 404e405. See also Fused filament fabrication (FFF) impact strength, chopped carbon fiber PA12 composite coupons, 423, 424f interfacial bond strength, 421 metal and ceramic matrix composite materials, 404 tensile coupon fracture surfaces, 423, 423f thermoplastic polymer matrix, 404 types, 404 wetting, 421e422 Z-pinning, 430e432, 431f Adhesive bonding

advantages, 385 applications, 384 broadband dielectric spectroscopy (BbDS) dielectric permittivity and loss, 395e396, 397f polarization mechanisms, 395, 396f quality assessment, 396e398, 398f durability factors, 386, 386f elastic energy release rate, 394, 394f environmental/aging, 394e395, 395f fiber glass composites, 384 manufacturing bonding method, 392e393, 393f nylon epoxy adhesives, 384, 384f plant-based glue, 384 plasma treatments, 394 polyester resins, 384 single lap joints (SLJs), 393 stress distribution, 385, 385f structural bonding, 385e386 surface characterization methods contact angle measurements, 390e391 long-term durability and performance, 390 on solid surface, 391e392, 391f surface treatment method, 389, 390f theories, 387e389, 387t, 388f types, 386, 387f Aerospace material systems machine learning methods, 158e159, 158f ply-level constitutive behavior methods. See Ply-level constitutive behavior methods progressive damage analysis (PDA). See Progressive damage analysis (PDA) regularized extended finite element method (Rx-FEM). See Regularized extended finite element method (Rx-FEM) AM. See Additive manufacturing (AM)

440

Aromatic polymer composite (APC-2), 152, 152t, 153f, 153t ASTM coupon test standards, 208, 209f Asymptotic homogenization theory, 4 Atomic force microscopy (AFM), 392 Auger electron spectroscopy (AES), 392 Automated fiber placement (AFP) technologies, 407 B Barely visible impact damage (BVID), 218e219 Black-box function, 57e58 Bonded composite systems. See Adhesive bonding Broadband dielectric spectroscopy (BbDS) dielectric permittivity and loss, 395e396, 397f polarization mechanisms, 395, 396f quality assessment, 396e398, 398f Brominated bisphenol-Aebased vinyl ester, 173 BSDS blade. See Sandia Blade System Design Study (BSDS) blade BubnoveGalerkin finite element formulation, 177 Building block validation strategy fatigue tests, 237e238, 238f static tests, 233e234, 236t C Carbon fiber composites, 108 Carbon fiberereinforced polymer (CFRP), 272f, 273, 279 Carbon nanotubes (CNTs), 255e256 Cartesian coordinate system, 16 CDM methodology. See Continuum damage modeling (CDM) methodology Ceramics, 314 fracture toughness, 315e316, 316t mechanical properties, 316, 317f perovskites, 316 pyrochlore, 316 radiation damage, 316 Vickers hardness testing, 315, 315f, 316t Young’s modulus, 316, 316t Certification by analysis (CBA), 205, 223 building-block testing approach, 221e222, 221f

Index

Federal Aviation Administration (FAA). See Federal Aviation Administration (FAA) structural features, 222 CGs. See Computational grains (CGs) Characteristic Damage State (CDS) Chemical bonding adhesion theory, 387e388, 388f Chemical vapor infiltration technique, 154 Clamped tapered beam (CTB) fatigue analysis, 144 damage events, 141, 142f delamination and matrix cracking propagation, 141, 142f delamination length vs. cycle count, 143, 143f material properties, 140e141, 140t mesh convergences, 140 rescaled material properties, 140e141, 141t stages, 140e141 geometry and configurations, 128, 129f static analysis boundary conditions, 129 coarse mesh, 129e130, 131f crack insertion and propagation, 129e130, 131f delamination migration, 130e132, 132f, 139 energy dissipation methods, 136e139, 138fe139f fine mesh, 129e130, 131f finer mesh, 129e130, 131f loadedisplacement curve, 130e132 multiple crack delamination migration pattern, 130e132, 132f thermal residual stress, 132e135, 133fe135f, 139 unidirectional IM7/8552 ply stiffness and strength properties, 129, 130t Cohesive zone models (CZM), 109 crack stress intensity factor, 114e115 delamination propagation phase, 112e113 energy release rate (ERR) extraction technique, 115 evolution stages, 116 mesh density, 113e114 modified fatigue algorithm, 117e118 Paris law, 112e113, 115

Index

static and fatigue simulation, 112e113, 113f static failure analysis, 110 Computational grains (CGs) fiber composites meshing technology, 18 multifield boundary variational principles, 18e22 multiple inhomogeneities, 18, 21f PapkovicheNeuber solutions, cylindrical harmonics, 22e24 representative volume element (RVE), 18, 21f stiffness matrix, 24e25 validation of, 25e26, 25fe26f macrostructural properties and analysis, 4e5 materials homogenization Al/SiC unit cell model mesh, 28, 28f interpolation function, 26e27 macrostress tensor, 27 periodic boundary conditions, 26e27, 27f representative volume element (RVE), 28, 29f stiffness matrix equation, 27 Wachspress coordinates, 26e27 Young’s modulus, 28, 28t, 30f Matlab Parallel Computing Toolbox, 28e29 microscale stress distributions, 29e33 multiscale modeling, 4e5 parallel computation, 28e29, 31f computed stress distributions, 29e33, 32f simulation time, 29e33, 31t particulate composites 3D elastic heterogeneous materials, 5e7, 6f elastic inclusion, 16 finite element equations, 15e16 multifield boundary variational principles, 7e8, 15 PapkovicheNeuber solution, 9 scaled Trefftz trial functions, inclusion and matrix, 11e12, 11f, 13f, 13t spherical harmonics, 9e11, 10f static condensation, 15e16 Trefftz trial function matrices, 14

441

validation, 16e17, 17f, 17te18t, 19fe20f Wachspress coordinates, 14, 14f Continuous fiber fused filament fabrication automated fiber placement (AFP) technologies, 407 coextrusion, 410e412, 411fe412f dual nozzle carbon fiber, 408e409 material deposition, 409, 409f strength and elastic modulus, 410, 410f tensile coupon, 409e410, 410f thermoplastic matrix, 408 nonplanar fused filament fabrication, 412e414, 413fe414f Continuum damage modeling (CDM) methodology, 108e109 CTB. See Clamped tapered beam (CTB) Cycle-based algorithm (CBA), 116 CZM. See Cohesive zone models (CZM) D DATAPLOT (DP) data uncertainty, 83e88 definition, 79 example, 80t, 81e82 model uncertainty, 92e102 software and documentation, 79e80, 80t Data-set uncertainty, 44e46, 46t DCB. See Double cantilever beam (DCB) D&DT. See Durability and damage tolerance (D&DT) Dielectric relaxation strength (DRS), 396e398, 398f Diffusion adhesion theory, 387e388, 388f Discrete cohesive zone modeling (DCZM), 216e217, 216fe217f Discrete damage modeling (DDM) carbon fiber composites, 108 cohesive zone models (CZM), 109 crack stress intensity factor, 114e115 delamination propagation phase, 112e113 energy release rate (ERR) extraction technique, 115 mesh density, 113e114 Paris law, 112e113, 115 static and fatigue simulation, 112e113, 113f

442

Discrete damage modeling (DDM) (Continued) continuum damage modeling (CDM) methodology, 108e109 delamination failure mode, 108 energy release rate (ERR), 109 extended finite element method (x-FEM), 108e109 fatigue algorithm cohesive zone models (CZM) evolution stages, 116 cycle-based algorithm (CBA), 116 event-based algorithm (EBA), 116 modified fatigue algorithm, 117e118 residual strength evaluation, 117 static algorithm, 116, 117f types, 116 matrix crack formation, 108 mesh-independent crack (MIC) technique, 108e109 fatigue failure criterion, 111e112 regularized extended finite element method (Rx-FEM), 109 static failure analysis, 110, 111f verification and validation, 110 Double cantilever beam (DCB) compliance calibration curves, 121e122, 122f energy release rate (ERR), 122, 122f load displacement curves, 121e122, 121f DP. See DATAPLOT (DP) DRS. See Dielectric relaxation strength (DRS) Durability and damage tolerance (D&DT), 206e207, 223 computational simulation, 207 glass composite, silica nanoparticles, 257e259, 258fe259f multiscale progressive failure analysis (MS-PFA), 207, 218 nanomaterial, 255 nine meter blade, 249e252, 251t, 252f, 255, 257e259, 258fe259f probabilistic sensitivity analysis, 249e252, 251t, 252f Durability with uncertainty (DuU) cyclic load without scaling cracked simple system, 66e68, 67fe70f smooth simple system, 65e66, 65fe67f

Index

failure and failure mechanisms, 40, 41f fiber-reinforced plastic (FRP) data-set uncertainty estimation, 72e74, 72te73t elastic constants and thermal expansion coefficients, 70e72, 71f, 74 2D-holed square composite plate A-basis and B-basis design, 74, 76, 78f, 78t, 79f effects, 76, 76f finite element method (FEM) model, 74e76 IM7/8552 composite plate, 74, 75f 2-level full-factorial orthogonal experimental design, 74e76, 75t mechanical properties, 74, 75t open-hole laminate strengths, 76, 77f, 77t mean cycles to failure (mCF), 40 modeling material systems, 64, 64t types, 37 E EBA. See Event-based algorithm (EBA) E-glass vinyl ester composite, 170e173, 190f, 191, 191t, 192fe193f Eigenvalues, 16, 17te18t Elastic modulus, 376 Electrochemical impedance spectroscopy, 360e361 Electron spectroscopy for chemical analysis (ESCA), 392 Electrostatic adhesion theory, 387e388, 388f End-notched flexure (ENF) analysis, 123e124, 124f Energy-dispersive X-ray (EDAX), 392 Energy dissipation methods arc-length technique, 136 delamination migration distance predictions, 137e139, 139f delamination propagation, 136 forceedisplacement curves, 137e139, 138f load hold (LH) solutions, 137 load update (LU) solutions, 137 NewtoneRaphson method, 136e137 nonlinear response, 136 thermal-mechanical load, 136 Energy release rate (ERR), 109, 215e217

Index

double cantilever beam (DCB), 122, 122f Eshelby’s solution, 16, 25, 212e213 EulereLagrange equations, 8, 20e22 Event-based algorithm (EBA), 116 Expanded polystyrene (EPS), 293e294 External medical composite systems magnetic resonance imaging (MRI) equipment, 370e371 medical machinery, 371 prosthetic devices, 371 surgical tools, 369e370 wearable devices, 371e372, 373f wheelchairs, 369e370 F Fatigue damage, structural composites constant amplitude loading characteristic damage state (CDS), 181e182 damage development stages, 181e182, 183f, 185 free-edge delaminations, 183e185, 184f laminate stiffness changes, 181e182, 183f micromechanisms to macroscopic fatigue behavior, 185, 186f Poisson’s ratio, 183e185 secondary matrix cracks, 182e183, 184f stresselife curves, 187 tensile fatigue behavior, 185 loading frequency effects, 187e188, 188f metals and fiber-reinforced composites, 181, 182t spectrum fatigue loading constant lifetime plots and amplitude stresselifetime data, 188e189, 190f cycle mix effect, 192e193 degrees of autocorrelation, 193e194, 194fe195f E-glass vinyl ester composites, 190f, 191, 191t, 192fe193f Rayleigh-distributed loading, 194, 195f residual strength models, 191e192, 193f stresselifetime (SeN) curves, 188e189, 189f temperature and moisture effects, 194e198, 196fe198f Federal Aviation Administration (FAA)

443

damage/defects, primary composite aircraft structures, 218e219, 219t product test sequence, 219e220, 220f residual strength evaluation, 221 test rules/recommendations, 219e220, 220f FFF. See Fused filament fabrication (FFF) Fiber composites meshing technology, 18 multifield boundary variational principles, 18e22 multiple inhomogeneities, 18, 21f PapkovicheNeuber solutions, cylindrical harmonics, 22e24 representative volume element (RVE), 18, 21f stiffness matrix, 24e25 validation of, 25e26, 25fe26f Fiber-reinforced polymer (FRP) composites, 254e255 alkali effects, 279e280 aqueous environments, 277 bond strength, 283e286, 284fe285f data-set uncertainty estimation, 72e74, 72te73t elastic constants and thermal expansion coefficients, 70e72, 71f validation, 74 environmental conditions, 275e276 external reinforcement, 271e275, 281f fatigue loads, 279 hygrothermal degradation, 280e282, 283f internal reinforcement (rebar), 272f, 273, 274fe276f long-term durability, 272 mechanical advantages, 272 steel reinforcement, 273 subzero and freezeethaw conditions, 277e278 temperature and fire, 278e279 ultraviolet radiation, 278 Finite element equations, 24 Finite element method (FEM) model, 4, 74e76, 119e120 35-m blade, fatigue loading, 260, 260fe261f nine meter blade, 242, 242fe243f, 243te244t

444

Finite element method (FEM) model (Continued) tapered blade analysis and results, 230e231, 230f, 236f Fourier transform infrared spectroscopy (FTIR), 391 Fused filament fabrication (FFF) continuous fiber, 405, 406f. See also Continuous fiber fused filament fabrication extruded thermoplastic material, 404e405 printed structures voids deposition toolpath effect, 418, 418f mechanical properties, 405 parameters, 415e418, 415f, 417f short fiber, 405e407, 406fe408f weld strength, 419e421, 420f G Gas mass diffusion equation, 177 Gaussian distribution, 41e42 Gauss theorem, 2 Geocells, 293 base materials reinforcement, 295e298, 297fe298f Geofoam, 293e294 bridge approach slab settlements, 302e304, 303fe304f Geogrid/geotextile layers, 290 Geonets, 292e293 Geosynthetic clay liner (GCL), 293 Geosynthetic-reinforced soil (GRS), 295 Geosynthetics characteristics, 291 definition, 289e290 durability of, 292 function, 294e295, 294t geogrid/geotextile layers, 290 geosynthetic-reinforced soil (GRS), 295 laboratory model tests, 290 manufacturing processes additives, 292 expanded polystyrene (EPS), 293e294 geocells, 293 geofoam, 293e294 geonets, 292e293 geosynthetic clay liner (GCL), 293 multi-ply geomembranes, 293 planar reinforcement, 293

Index

polypropylene and polyethylene, 292e293 planar reinforcements, 290 roadway construction, 290 settlement ratios, 290 structural integrity, 294e295 synthetic polymers, 289 transportation infrastructure geocells, base materials reinforcement, 295e298, 297fe298f geofoam, bridge approach slab settlements, 302e304, 303fe304f slope stability enhancement, 305e307, 306f wicking geotextiles, 298e302, 299fe301f types, 291, 291f, 294e295, 294t Glass fiberereinforced polymer (GFRP), 273, 276f, 279 Guidelines on Uncertainty in Metrology (GUM), 38e39 H Heterofoams cesium ions, 332 flux, 332e333, 333f hollandite structure, 331e332, 332f NernstePlank relationship, 332 surface-to-volume ratio, 333 Voronoi tessellation, 333, 334f waste storage phase, 335 Heterogeneous materials, 3e4 High-energy ion scattering (HEIS), 392 HilleMandel principle, 2e3 Hydrogels (HGs), 369 Hygrothermal degradation, 280e282, 283f I Implantable medical composites. See Medical composite systems Interlaminar shear strength (ILSS), 255 Internal medical composite systems bone implants, 367e368, 367f dental composites, 363e366, 365f drug delivery systems, 369 organ implant, 366e367 tissue engineered composites, 368e369 Ion scattering spectroscopy (ISS), 391

Index

L Laplace equation, 9e10, 10f Least squares regression analysis method, 112 Linear least squares fit, 65e66, 65f, 67f Logistic function least squares fit, 58e59, 58f Low-energy ion scattering (LEIS), 391 M Machine learning methods, 158e159, 158f Macrostrain tensor, 2 Macrostress tensor, 2, 27 Marine composites and fire-related design criteria, 166e167 Matlab Parallel Computing Toolbox, 28e29 Matrix/heterogeneity interface, 7e8 Mechanical interlocking adhesion theory, 387e388, 388f Medical composite systems, 363e364, 364f biocompatible materials, 372e374, 374t chemical preservative treatments, 379e380 compressive strength, 376 elastic modulus, 376 external applications magnetic resonance imaging (MRI) equipment, 370e371 medical machinery, 371 prosthetic devices, 371 surgical tools, 369e370 wearable devices, 371e372, 373f wheelchairs, 369e370 failure types cavitation, 378e379 corrosion, 377e378 degradation, 377e378 parameters, 377 wear, 379 fracture toughness, 376 internal applications bone implants, 367e368, 367f dental composites, 363e366, 365f drug delivery systems, 369 organ implant, 366e367 tissue engineered composites, 368e369 thermal expansion, 375, 375te376t Medium-energy ion scattering, 391

445

Mesh-independent crack (MIC) technique, 108e110 fatigue failure criterion, 111e112 Microemacro modeling average tangent moduli, 3 constant traction boundary condition, 3 deformation energy, 2 linear displacement boundary condition, 3 macrostrain tensor, 2 macrostress tensor, 2 periodic boundary conditions, 3 representative volume element (RVE), 2 Mixed-mode bending (MMB) verification, 125e127, 125fe127f Mixed-mode failure analysis, 216e217, 217f Model-compute uncertainty crack length distribution, 61, 62f DEX, 59, 61, 61f 5-factor ultrasonic testing (UT) experiment, 60e61, 60f, 60t model verification, 55e59, 57fe58f nonlinear least squares (NLLSQ) tool, 59 two-level fractional factorial designs, 60 Model-function uncertainty, error propagation dispersion property, 50 fracture mechanics, 53 LLSQ, 54, 55f, 55t NIST Uncertainty Machine (NUM), 51 prediction interval, 51 Pythagoras formula, 53e54 standard deviations, 52 Model-physics uncertainty, 62e63 Model-system uncertainty, system verification, 63 Modified fatigue algorithm, 117e118 Molecular dynamics simulation techniques, 255 MorieTanaka formulation, 212e213 Multiscale progressive failure analysis (MS-PFA) 35-m blade, fatigue loading, 260 durability and damage tolerance (D&DT), 207, 218 nine meter blade, 256

446

N Navier’s equation, 7 NewtoneRaphson (N-R) iterations, 110, 117e118, 136e137 Nicalon/E-SiC, 154, 157, 157f NIST Uncertainty Machine (NUM), 44, 51, 89e91 Nodal shape functions, 4 Nondestructive inspection (NDI), 222e223 Nondestructive testing (NDT) methods, 396e398 Nonlinear least squares (NLLSQ), 57e59, 65e66, 66f Nuclear energy systems candidate single-phase and multiphase materials corrosion and leaching techniques, 317, 318f glass waste forms, 319e320 monolithic leach test (MCC-1), 319 product consistency test (PCT), 317e319, 318f vapor hydration test, 319 ceramics, 314e316, 315f, 316t, 317f crystalline systems, 314 glass vitrification, 314 heterofoams cesium ions, 332 flux, 332e333, 333f hollandite structure, 331e332, 332f NernstePlank relationship, 332 surface-to-volume ratio, 333 Voronoi tessellation, 333, 334f waste storage phase, 335 multiphase waste forms, SYNROC type waste forms C, 320e321 radiation damage processes. See Radiation damage processes radiation effects surface amorphization, 329 leaching, 330e331, 331f mechanical properties, 330, 330f volume expansion, 329 X-ray diffraction, 329 radioactive waste elements, 314 separation process, 314 single-phase waste forms hollandite, 322e323, 323f perovskite, 321e322, 322f

Index

zirconolite and pyrochlore, 321 uranium dioxide, 323e324 zirconolite, 314 Nylon epoxy adhesives, 384, 384f O Optical profilometer, 391 P PapkovicheNeuber solutions cylindrical harmonics, fiber composites, 22e24 particulate composites, 9 scaled Trefftz trial functions, inclusion and matrix, 11 Particulate composites 3D elastic heterogeneous materials, 5e7, 6f elastic inclusion, 16 finite element equations, 15e16 multifield boundary variational principles, 7e8, 15 PapkovicheNeuber solution, 9 scaled Trefftz trial functions, inclusion and matrix, 11e12, 11f, 13f, 13t spherical harmonics, 9e11, 10f static condensation, 15e16 Trefftz trial function matrices, 14 validation, 16e17, 17f, 17te18t, 19fe20f Wachspress coordinates, 14, 14f 3 PB-F model. See Three-point bend with flange (3 PB-F) model PDA. See Progressive damage analysis (PDA) Plant-based glue, 384 Ply-level constitutive behavior methods aromatic polymer composite (APC-2), 152, 152t, 153f, 153t chemical vapor infiltration technique, 154 constant cyclic loading, 156 damage evolution, 155 discrete defect, 151 equivalent damage, 148e150, 150f failure function, 150e151, 154, 157 fatigue life, 151 Nicalon/E-SiC, 154, 157, 157f pest temperature, 156 polymer electrolyte membrane (PEM) fuel cell, 156e157, 158f

Index

strength and life predictions, 153, 154fe155f strength change equation, 151 structural integrity, 151e152 Poisson’s ratio, 183e185, 345 Polyester resins, 384 Polymer electrolyte membrane (PEM) fuel cell, 156e157, 158f Polymeric composite material systems advantages, 166 characteristics, 166 combined load and fire conditions brominated bisphenol-Aebased vinyl ester, 173 BubnoveGalerkin finite element formulation, 177 compression failure mechanics, 170 coupon level tests, 170, 171te172t E-glass vinyl ester composite, 170e173 failure modes, 170, 173 gas mass diffusion equation, 177 heat flux exposures, 170, 180e181 intermediate scale tests, 173, 174f laminate materials properties, 179 nondimensionalized stress, 175, 176f ply-wise residual properties, 175 residual composite strength, 170 slenderness ratio, 175, 176f stress analysis and time-to-failure criteria, 180 thermomechanical modeling, 177, 178t through-thickness temperature profiles, 179 time-evolved thermal modeling, 175 fatigue damage. See Fatigue damage, structural composites fire conditions, 168e169, 169f marine composites and fire-related design criteria, 166e167 wind turbine composites and fatigue-related design criteria, 167e168 Product consistency test (PCT), 317, 318f product characterization test, 317e319 Progressive damage analysis (PDA) carbon fiber composites, 108 cohesive zone models (CZM), 109 continuum damage modeling (CDM) methodology, 108e109 delamination failure mode, 108

447

energy release rate (ERR), 109 extended finite element method (x-FEM), 108e109 matrix crack formation, 108 mesh-independent crack (MIC) technique, 108e109 regularized extended finite element method (Rx-FEM), 109 Progressive failure analysis (PFA), 206e207. See also Virtual crack closure technique (VCCT) R Radiation damage processes crystalline structures amorphization, 326e329, 328f electronic defects, 326 Frenkel pair defects, 324e326 volume change, 326, 327f melting and epitaxial recrystallization, 324, 325f radioactive bearing phases, 324 Raman spectroscopy, 391 RBS. See Rutherford back scattering (RBS) Regularized extended finite element method (Rx-FEM), 109 validation, subelement-level analysis clamped tapered beam (CTB), 127, 128f. See also Clamped tapered beam (CTB) three-point bend with flange (3 PB-F) model, 144e148, 144fe145f, 146te148t, 149f verification, coupon-level analysis crack growth rate, 122e123, 123f double cantilever beam (DCB), 121e122, 121fe122f end-notched flexure (ENF) analysis, 123e124, 124f finite element models, 119e120 fracture material properties, 119e120, 121t fracture toughness properties, 119e120, 120t geometries and boundary conditions, 119e120, 119f mixed-mode bending (MMB) verification, 125e127, 125fe127f static failure analysis, 110 static material properties, 119e120, 120t

448

Representative volume element (RVE), 2 Al/SiC unit cell model mesh, 4 3D elastic particulate composites, 5, 6f 3D elastic fiber composites, 18, 21f finite elements, 4 materials homogenization, 28, 29f spherical inclusions/voids, 29e33, 31f Residual strength models, 191e192, 193f Rutherford back scattering (RBS), 392 S Sandia Blade System Design Study (BSDS) blade, 223 assembly fixture, 223, 225f design features, 223 finishing stage, 223, 225f laminate regions, 223, 224f materials construction, 223, 225t vacuum-assisted resin transfer mold (VARTM) process, 223 Scanning electron microscopy (SEM), 392 Scanning probe microscopy (SPM), 392 Shape array accelerometers (SAAs), 297e298, 297f Short fiber fused filament fabrication, 405e407, 406fe408f Silica nanoparticles, glass composite, 256e257, 256te257t, 258f Slenderness ratio, 175, 176f Solid oxide electrolyzers electrochemical impedance spectroscopy, 360e361 electrochemical process, 341e343 electrolysis, 341e343 geometry of, 343e344, 343f internal pressurization crack growth/pore pressurization, 347 elastic strain energy, 347 electrochemical devices, 346 failure criteria, 345e346 stress intensity factor, 346 thermally induced stress, 345e346 ionic current, 343e344 oxygen gas moles, 355 pressurization electrochemical, 341e344, 344f internal, 357e358, 360 Nernst voltage, 356, 358

Index

net current, 356 net hydrostatic stress, 357e358 pore, 351e355 poreecylinderepiston assembly, 358e359, 358f time, 357 work, 356, 359e360 radial displacement integration constant, 348 linear elasticity, 347e348 pore surface, 349 Young’s modulus, 347 strain energy, 349e350 yttria-stabilized zirconia (YSZ) electrolyte, 341e343, 360e361 pores, 341e343, 342f strain energy, 344e345 Spherical harmonics coordinates, 10, 10f finite domain, 10e11 Laplace equation, 9e10, 10f SPM. See Scanning probe microscopy (SPM) Stiffness matrices, 16, 17te18t, 27e29 Surface characterization methods contact angle measurements, 390e391 long-term durability and performance, 390 on solid surface, 391e392, 391f Surface treatment method, 389, 390f T Thermal residual stress axial stress distribution, 133e135, 134f forceedisplacement curves, 132e133, 133f matrix crack insertion and delamination, 133e135, 134f peak load predictions, 135, 135f Three-point bend with flange (3 PB-F) model, 147e148 boundary conditions, 145, 145f carbon/epoxy fabric properties, 145e146, 148t carbon/epoxy tape material properties, 145e146, 147t meshes, 145, 146t model construction, 144e145, 144f postpeak load experiment damage, 147, 149f

Index

static simulation and experiment results, 146e147, 149f tape/fabric interface properties, 145e146, 148t Time-of-flight secondary ion mass spectroscopy (TOF-SIMS), 392 Titanate ceramics, 314 Trefftz trial functions, 22, 24 displacement fields, 7, 12 scaled functions, inclusion and matrix characteristic lengths, 12, 13f coefficient matrices, 12, 13t condition number test, 12, 13f PapkovicheNeuber solution, 11 spherical harmonics, 11f, 12 U Uncertainty data modeling, 43 data types, univariate data set confidence interval, 42 density function, 42 Gaussian distribution, 41e42 prediction interval, 42e43 probability distribution, 41e42 standard deviation, 40e41 tolerance interval, 43 definition, 38 level of confidence, 38e39, 38f measurement, 39 statistical tools data-coverage uncertainty, 48e50, 49te50t data-parameter uncertainty, 46e48, 47te48t DATAPLOT (DP), 44. See also DATAPLOT (DP) data-set uncertainty, 44e46, 46t model-compute uncertainty. See Model-compute uncertainty model-function uncertainty. See Model-function uncertainty model-physics uncertainty, 62e63 model-system uncertainty, system verification, 63 NIST Uncertainty Machine (NUM), 44, 89e91 system modeling, 43e44

449

ultimate tensile strength (UTS), 39 V Vacuum-assisted resin transfer mold (VARTM) process, 223 Vapor hydration test, 319 Vickers hardness testing, 315, 315f, 316t Virtual crack closure technique (VCCT), 207, 214e216 progressive failure analysis (PFA) crack and damage evolution process, 239e240, 239f crack growth, 239e240, 240f crack propagation path, 239, 239f load vs. displacement, 239e240, 240f ply drop features, 240 static stiffness and strength, 239e240, 239t Visible impact damage (VID), 218e219 Vocabulary of Metrology (VIM), 38e39 Voronoi meshing, 27e28 W Wachspress coordinates, 14, 14f, 24, 26e27 Weak boundary layer adhesion theory, 387e388, 388f Wetting adhesion theory, 387e388, 388f Wind turbine blade ASTM coupon test standards, 208, 209f 35-m blade, fatigue loading boundary conditions, 260, 261f fatigue life, 260, 264f finite element model, 260, 260fe261f material information, 260, 261t material properties, 260, 263t multiscale progressive failure analysis (MS-PFA), 260 ply schedule, 260, 262t, 264f Sandia blade, 261e262, 265, 265fe268f certification by analysis (CBA), 205, 223 building-block testing approach, 221e222, 221f Federal Aviation Administration (FAA). See Federal Aviation Administration (FAA) structural features, 222 composite material calibration fiber and matrix failure criteria, 208, 211t fiber architecture, 208, 210f

450

Wind turbine blade (Continued) material properties and physical tests, 208 nanoenhanced matrix, 212e213, 213f woven fabric composites, 208e212 damage modeling, 204 durability and damage tolerance (D&DT), 206e207, 223 computational simulation, 207 multiscale progressive failure analysis (MS-PFA), 207, 218 fracture mechanics discrete cohesive zone modeling (DCZM), 216e217, 216fe217f virtual crack closure technique (VCCT), 214e216 layout, 206, 206f material modeling calibration and validation, 226, 227te228t mechanical components, 205 micro- and macrocomposite scales, 217 micromechanics-based computational approach, 206e207 multi-scale progressive failure analysis, 213e214, 214fe215f nine meter blade carbon nanotubes (CNTs), 255e256 computational simulation approach, 259e260 durability and damage tolerance (D&DT), 249e252, 251t, 252f, 255, 257e259, 258fe259f durability and reliability, 241 fatigue evaluation, 245e246, 247t, 248f, 248t, 249f fiber-reinforced polymer (FRP) composites, 254e255 finite element model, 242, 242fe243f, 243te244t interlaminar shear strength (ILSS), 255 molecular dynamics simulation techniques, 255 multiscale progressive failure analysis (MS-PFA), 256 silica nanoparticles, glass composite, 256e257, 256te257t, 258f static test simulation, 242e245, 245t, 246f

Index

weight analysis, 247e248, 250t weight reduction, 252e254, 253fe254f nondestructive inspection (NDI), 222e223 probabilistic simulation, 217, 218f progressive failure analysis (PFA), 206e207 Sandia Blade System Design Study (BSDS) blade, 223 assembly fixture, 223, 225f design features, 223 finishing stage, 223, 225f laminate regions, 223, 224f materials construction, 223, 225t vacuum-assisted resin transfer mold (VARTM) process, 223 stress/strain analysis, 205 structural benefits, 207 structural integrity, 205 structure, 206, 206f tapered blade analysis and results ASTM-based coupon verification, 226, 229t building block validation strategy, 233e234, 236t, 237e238, 238f calibrated matrix SeN curves, 231, 233f crack initiation, 231, 233f crack length vs. loaddsimulations vs. experimental tests, 231, 234f crack propagation, 231, 233f durability and damage tolerance (D&DT), 234e237, 237f EP-1 matrix simulation, 231, 232f finite element model, 230e231, 230f, 236f laminate types, 227, 229t longitudinal tensile test, 229, 230f with ply drops, 231, 235f resin systems, 227, 231t shell finite element model, 231, 236f strain energy release rate, 226, 229f stress/strain curves, 227, 230f virtual crack closure technique/ progressive failure analysis (VCCT/ PFA), 238e240, 239f, 239t, 240f weight-to-stiffness ratio, 204 weight-to-strength ratio, 204

Index

Wind turbine composites and fatigue-related design criteria, 167e168 X X-ray photoelectron apectroscopy (XPS), 392 Y Young’s modulus, 28, 28t, 30f, 316, 316t, 345, 347

451

Yttria-stabilized zirconia (YSZ) electrolyte, 341e343, 360e361 pores, 341e343, 342f strain energy, 344e345 Z Z-pinning, 430e432, 431f