Creep and fatigue in polymer matrix composites [Second edition] 9780081026014, 9780081026021, 2152152182, 0081026013

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Creep and Fatigue in Polymer Matrix Composites

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Woodhead Publishing Series in Composites Science and Engineering

Creep and Fatigue in Polymer Matrix Composites Second Edition Edited by

Rui Miranda Guedes

An imprint of Elsevier

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 © 2019 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: 978-0-08-102601-4 (print) ISBN: 978-0-08-102602-1 (online) For information on all Woodhead publications visit our website at https://www.elsevier.com/books-and-journals

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Contents

Contributors Preface

Part One 1

2

xi xiii

Viscoelastic and viscoplastic modeling

Viscoelastic constitutive modeling of creep and stress relaxation in polymers and polymer matrix composites G.C. Papanicolaou and S.P. Zaoutsos 1.1 Introduction 1.2 Types of polymers 1.3 Viscoelastic behavior 1.4 Properties degradation 1.5 Physical properties of polymers 1.6 Viscoelastic behavior of polymers 1.7 Short-term behavior 1.8 Long-term behavior 1.9 Isochronous and isometric diagrams 1.10 Creep-recovery and stress relaxation analysis 1.11 Linearity 1.12 The time-temperature superposition principle 1.13 The time stress superposition principle 1.14 The time-temperature-stress superposition principle 1.15 Linear viscoelastic models 1.16 Nonlinear viscoelastic behavior of polymers 1.17 Applications to different materials References Further reading Time-temperature-age superposition principle for predicting long-term response of linear viscoelastic materials E.J. Barbero 2.1 Correlation of short-term data 2.2 Time-temperature superposition 2.3 Time-age superposition 2.4 Effective time theory 2.5 Summary

3 3 4 6 10 12 14 15 16 17 19 22 25 26 27 28 43 51 57 59

61 61 63 71 77 79

vi

Contents

2.6 Temperature compensation 2.7 Conclusions References 3

4

5

Effect of moisture on elastic and viscoelastic properties of fiber reinforced plastics: Retrospective and current trends Andrey Aniskevich and Tatjana Glaskova-Kuzmina 3.1 Introduction 3.2 Moisture absorption in polymers 3.3 Moisture absorption in fiber reinforced composites 3.4 Swelling 3.5 Effect of moisture on elastic properties and strength of polymers and composites 3.6 Viscoelastic behavior of polymers and composites 3.7 Moisture in nanocomposites 3.8 Conclusions Acknowledgment References Micromechanics modeling of hysteretic responses of piezoelectric composites Chien-hong Lin and Anastasia Muliana 4.1 Introduction 4.2 Constitutive models 4.3 Fiber- and particle-unit-cell models 4.4 Hybrid piezocomposite model 4.5 Conclusions Acknowledgment References Further reading Predicting the viscoelastic behavior of polymer composites and nanocomposites A. Beyle 5.1 Preface 5.2 Specific features of constituents 5.3 Distinctive characteristics of behavior of heterogeneous materials with polymeric matrix 5.4 Viscoelasticity of matrix 5.5 Other factors contributing to viscoelastic behavior 5.6 Poisson ratio change during creep 5.7 Viscoelasticity of particulate composites: spherical inclusions 5.8 Viscoelasticity of particulate composites: elongated inclusions

80 80 81

83 83 83 89 96 97 102 112 117 118 118

121 121 125 131 142 151 152 152 155

157 157 157 162 165 167 176 179 195

Contents

vii

5.9

Viscoelasticity of particulate composites: oblate and platelet inclusions 5.10 Viscoelasticity of fibrous composites 5.11 Hollow fillers: buckyballs, microballoons 5.12 Hollow fibers and nanotubes 5.13 Viscoelasticity of foams and nanoporous materials 5.14 More complicated cases 5.15 Concluding remarks Acknowledgments References Further reading

6

7

Creep analysis of polymer matrix composites using viscoplastic models E. Kontou 6.1 Introduction 6.2 Viscoplastic creep modeling for polymer composites 6.3 Concluding remarks 6.4 Future trends References Further reading Polymer matrix composites: Update Jacob Aboudi

Part Two 8

196 201 203 203 205 209 210 211 211 214

215 215 218 242 243 244 248 249

Creep rupture

Time and temperature dependence of transverse tensile failure of unidirectional carbon fiber-reinforced polymer matrix composites Jun Koyanagi and Mio Sato 8.1 Introduction 8.2 Failure morphology 8.3 Simple FEA and critical-point stress 8.4 Time and temperature dependence on interface strength 8.5 RVE modeling 8.6 Periodic boundary condition 8.7 Matrix modeling 8.8 Interface modeling 8.9 Numerical results and discussion 8.10 Summary References

253 253 253 254 256 258 259 261 262 264 266 266

viii

9

10

Contents

Lifetime prediction of polymers and polymer matrix composite structures: Failure criteria and accelerated characterization Rui Miranda Guedes 9.1 Introduction 9.2 Time-dependent failure criteria for viscoelastic homogeneous materials 9.3 Time-dependent failure criteria: extension to polymer-based matrix composites 9.4 Long-term failure: accelerated experimental methodologies 9.5 Cumulative damage models: multiple step creep loading 9.6 Micromechanical model 9.7 Conclusions References Time-dependent damage evolution in unidirectional and multidirectional polymer composite laminates Raghavan Jayaraman 10.1 Introduction 10.2 Damage modes in composites 10.3 Damage characterization methods 10.4 Time-dependent evolution of damage modes 10.5 Criterion to predict time-dependent damage initiation and evolution 10.6 Concluding remarks References

269 269 270 278 282 287 292 297 297

303 303 304 306 307 313 317 317

Part Three Fatigue modeling, characterization, and monitoring 11

Accelerated testing methodology for long-term creep and fatigue strengths of polymer composites Masayuki Nakada 11.1 Introduction 11.2 Accelerated testing methodology 11.3 Experimental verification for ATM 11.4 Applicability of ATM 11.5 Theoretical verification of ATM 11.6 Prediction of statistical creep life 11.7 Future trends and research 11.8 Conclusions References

325 325 326 331 336 339 341 344 346 346

Contents

12

13

14

A combined creep and fatigue prediction methodology for fiber-reinforced polymer composites based on the kinetic theory of fracture Faisal H. Bhuiyan and Ray S. Fertig III 12.1 Introduction 12.2 A KTF-based methodology to predict creep and fatigue in the polymer matrix 12.3 Finite element implementation of the fatigue methodology 12.4 Finite element implementation of the creep rupture and strain methodology 12.5 Review of literature on KTF-based durability modeling of FRP composites 12.6 Conclusions References Fatigue testing for polymer matrix composites Ruben Dirk Bram Sevenois and Wim Van Paepegem 13.1 Introduction 13.2 Fatigue testing methods 13.3 Effect of boundary conditions and specimen geometry 13.4 Advanced instrumentation methods 13.5 Future trends and challenges 13.6 Sources of further information and advice References Further reading S–N curve and fatigue damage for practicality Ho Sung Kim 14.1 Introduction 14.2 Fatigue damage 14.3 S–N curve model overview 14.4 The S–N curve model proposed by Kim and Zhang 14.5 Damage validity conditions at R ¼ 0 14.6 Damage function and validity 14.7 Isodamage point b 14.8 Numerical determination of n value 14.9 Examples for predicting the remaining fatigue life 14.10 Concluding remarks References

ix

349 349 352 366 385 394 397 398 403 403 403 415 418 427 428 429 437 439 439 441 442 445 450 454 454 455 456 460 461

x

15

16

17

Contents

Modeling, analysis, and testing of viscoelastic properties of shape memory polymer composites and a brief review of their space engineering applications Wessam Al Azzawi, Madhubhashitha Herath and Jayantha Epaarachchi 15.1 Introduction 15.2 Modeling the viscoelastic behavior of SMP and SMPC 15.3 Finite element simulation procedure for modeling of viscoelastic properties of SMPC 15.4 Finite element simulation results 15.5 Validation of the FEA technique 15.6 Competence of SMPCs for space engineering applications 15.7 Conclusion References Characterization of viscoelasticity, viscoplasticity, and damage in composites Janis Varna and Liva Pupure 16.1 Introduction 16.2 Material model 16.3 Microdamage effect on stiffness 16.4 Viscoplasticity 16.5 Nonlinear viscoelasticity 16.6 Conclusions Appendix Time dependence of VP-strain in one creep test References Further reading Structural health monitoring of composite structures for durability Sreenivas Alampalli 17.1 Introduction 17.2 FRP structures in bridge industry 17.3 Structural health monitoring 17.4 FRP structures and SHM 17.5 Case studies 17.6 Summary References

Index

465 465 467 472 473 477 482 492 493

497 497 498 501 508 518 526 526 528 530 531 531 532 537 538 539 553 555 559

Contributors

Jacob Aboudi Faculty of Engineering, Tel Aviv University, Ramat Aviv, Israel Wessam Al Azzawi School of Mechanical and Electrical Engineering; Centre for Future Materials, University of Southern Queensland, Toowoomba, QLD, Australia Sreenivas Alampalli New York State Department of Transportation, Albany, NY, United States Andrey Aniskevich Institute for Mechanics of Materials, University of Latvia, Riga, Latvia E.J. Barbero West Virginia University, Morgantown, WV, United States A. Beyle University of Texas at Arlington, Arlington, TX, United States Faisal H. Bhuiyan Department of Mechanical Engineering, University of Wyoming, Laramie, WY, United States Jayantha Epaarachchi School of Mechanical and Electrical Engineering; Centre for Future Materials, University of Southern Queensland, Toowoomba, QLD, Australia Ray S. Fertig III Department of Mechanical Engineering, University of Wyoming, Laramie, WY, United States Tatjana Glaskova-Kuzmina Institute for Mechanics of Materials, University of Latvia, Riga, Latvia Rui Miranda Guedes Department of Mechanical Engineering, Faculty of Engineering of the University of Porto, Porto, Portugal Madhubhashitha Herath School of Mechanical and Electrical Engineering; Centre for Future Materials, University of Southern Queensland, Toowoomba, QLD, Australia Raghavan Jayaraman Department of Mechanical Engineering, Composite Materials and Structures Research Group, University of Manitoba, Winnipeg, MB, Canada

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Contributors

Ho Sung Kim The University of Newcastle, School of Engineering, Callaghan, NSW, Australia E. Kontou Department of Mechanics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece Jun Koyanagi Department of Materials Science and Technology, Tokyo University of Science, Tokyo, Japan Chien-hong Lin Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan Anastasia Muliana Department of Mechanical Engineering, Texas A&M University, College Station, TX, United States Masayuki Nakada Materials System Research Laboratory, Kanazawa Institute of Technology, Hakusan, Japan G.C. Papanicolaou Department of Mechanical and Aeronautics Engineering, University of Patras, Patras, Greece Liva Pupure Lulea˚ University of Technology, Lulea˚, Sweden Mio Sato Department of Materials Science and Technology, Tokyo University of Science, Tokyo, Japan Ruben Dirk Bram Sevenois Department of Materials, Textiles and Chemical Engineering (MaTCH), Ghent University, Ghent, Belgium Wim Van Paepegem Department of Materials, Textiles and Chemical Engineering (MaTCH), Ghent University, Ghent, Belgium Janis Varna Lulea˚ University of Technology, Lulea˚, Sweden S.P. Zaoutsos Faculty of Applied Sciences, Technological Educational Institute of Thessaly, Larissa, Greece

Preface

The structural application of polymer matrix composites in civil construction is becoming more important than ever. The success of these applications has enhanced further development of new solutions based on fiber-reinforced polymers. The viscoelastic/viscoplastic nature of the matrix, damage accumulation and propagation within the matrix, and fiber breaking progressively degrade the strength and stiffness of fiber-reinforced polymers, even if they often exhibit initially high mechanical performance. Catastrophic premature failure is one of the consequences of these features, when many structural components are supposed to operate for 50 years. The full spectrum of internal material changes from the microscopic scale up to the full structure length is far from being completely known. The interactions between different mechanisms acting at different scale levels are extremely complex and not yet fully understood. Under typical service loads and environmental conditions, many phenomena like cracking, oxidation, chemical degradation, delamination, and wear, among others, occur concomitantly and eventually interact to evolve as a complex time-dependent degradation process. Additionally, the viscoelastic/ viscoplastic behavior of the matrix may increase the load transfer to the intact fibers, leading to more fiber breaking. In brief, disorder plays an important role in strength of materials. Multiscale models seem to be able to capture these complexities through microscale models that incorporate the material inhomogeneity. However, global and homogeneous analyses, which incorporate damage as a diffuse phenomenon and are simple to formulate and solve, are still convenient for practical applications. Fatigue characterization, in general, is produced through constant load amplitude and frequency tests. Based on this data the structural designers must perform lifetime predictions, assuming the load spectrum and environmental history expected in real service conditions. Several empirical or phenomenological cumulative damage models, with different numbers of parameters to be determined from fatigue data, have been proposed. Despite these efforts, the lifetime predictability performance under general loading conditions is as yet unknown. This second edition of Creep and Fatigue in Polymer Matrix Composites reviews some of the latest experimental and theoretical approaches dealing with creep and fatigue of polymer matrix composites in a durability context. Each chapter reflects on a portion of the issues previously discussed. A few of them attempt a broader vision of the questions, as well as answers, that have arisen in this field of research. The book is divided into three parts, which correspond to different major topics concerning the creep and fatigue of polymer matrix composites. Part I is entirely devoted to the time-dependent models that capture the viscous nature of the polymer matrix. Part II deals with the mechanical failure caused by creep or static fatigue. Part III discusses some aspects of crucial importance in the context of civil engineering

xiv

Preface

applications. The interaction between damage and creep and the creep effects on fatigue are discussed. Finally, the monitoring feasibility of structures in service is discussed in order to assure structural integrity in a durability framework. This updated edition would not have been possible without the contributions of leading researchers from different parts of the world. I am privileged to know some of them personally. These good friends are Evi Kontou, George Papanicolaou and Stefanos Zaoustsos from Greece, Tatjana Glaskova-Kuzmina and Andrey Aniskevich from Latvia, Masayuki Nakada from Japan, Wim Van Paepegem from Belgium, Janis Varna from Sweden, and Anastasia Muliana from the United States. Although I am not privileged to know the others personally, I consider them friends: they are Jayaraman Raghavan from Canada, Ever Barbero, Andrey Beyle, Ray S. Fertig III, and Sreenivas Alampalli from the United States, Jacob Aboudi from Israel, Jun Koyanagi from Japan, and Ho Sung Kim and Jayantha Ananda Epaarachchi from Australia. Above all, I have followed and admired their continuous efforts to push forward scientific knowledge. I want to thank all the contributors for their efforts to duly complete the assigned tasks. Their excellent contributions are the backbone of the book and I expect they enjoy it as much as I do. I extend my gratitude to all, including the Elsevier editorial managers, Sabrina Webber and Peter Llewellyn, for their estimable support. Above all I must acknowledge the kind and caring support of my wife Cristina throughout all these countless years, and my two children, Helena and Matias, who teach me to feel and see more things than I could have imagined. Without them, I would be lost! Rui Miranda Guedes

Part One Viscoelastic and viscoplastic modeling

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Viscoelastic constitutive modeling of creep and stress relaxation in polymers and polymer matrix composites

1

G.C. Papanicolaou*, S.P. Zaoutsos† *Department of Mechanical and Aeronautics Engineering, University of Patras, Patras, Greece, †Faculty of Applied Sciences, Technological Educational Institute of Thessaly, Larissa, Greece

1.1

Introduction

It is difficult to imagine today’s world without the presence of plastics. Today plastics are an integral part of everyone’s life, with applications ranging from simple everyday objects to the most complex industrial constructions. Today, designers/engineers turn to plastics because they offer unique properties that are not found in traditional materials. Plastics offer advantages such as light weight, flexibility, corrosion resistance, transparency, and easy construction and, even though all these properties have their limits, the limits of their application depend only on the designer’s imagination. It is usual to consider plastics as a relatively recent discovery, but in fact, as part of a wider family called polymers, they are basic derivatives of the animal and plant kingdom. Polymers differ from metals in their structure because the former consist of long molecules, the macromolecules, which are chain-shaped. Synthetic “macromolecules” are synthesized by combining thousands of small molecule units called monomers. The chemical process in which two or more monomers combine to form larger molecules that contain repeating structural units (macromolecules) is called polymerization, and the number of monomers that make up the polymeric chain is called degree of polymerization. The terms polymers and plastics are usually considered to be synonymous. In reality, however, there is a difference between them. A polymer is a pure material resulting from the polymerization process and usually represents the family of materials characterized by a macromolecular structure (including elastomers). Pure polymers are rarely used in applications. Typically, polymers contain various additives and are then called plastics. Additives are distinguished in the following categories: antistatic agents, coupling agents, fillers, extenders, flame retardants, lubricants, pigments, plasticizers, reinforcements, and stabilizers. Polymer and polymer-based composite behavior is strongly influenced by the application of mechanical loading, deformation, applied strain/stress rate, temperature, Creep and Fatigue in Polymer Matrix Composites. https://doi.org/10.1016/B978-0-08-102601-4.00001-1 © 2019 Elsevier Ltd. All rights reserved.

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Creep and Fatigue in Polymer Matrix Composites

humidity, and time. The main characteristic of their behavior is the so-called viscoelastic response. This refers to the simultaneous elastic and viscous response, which is more pronounced during creep, relaxation, and dynamic mechanical loading of these materials. The problem of predicting/describing the viscoelastic behavior of polymers and polymeric composites has been approached, in the case of linear viscoelastic behavior, through the development and application of spring-dashpot combination models, while in the case of nonlinear viscoelastic behavior, through more advanced models such as those based on Schapery’s constitutive equation.

1.2

Types of polymers

Depending on the structure and properties, plastics are divided into two main categories: thermoplastics and thermosetting plastics. In addition, both thermoplastics and thermosetting plastics can be reinforced with fibers and thus new materials with unique properties can be developed.

1.2.1 Thermoplastics In a thermoplastic the macromolecules are connected to each other by weak van der Waals forces. An image that would fit in describing a thermoplastic material is that of very fine cotton fibers that are randomly distributed. When the material is heated, the intensity of the intermolecular forces is greatly reduced so that the material becomes soft and flexible and then, at higher temperatures, it becomes a viscous fluid. When it is allowed to cool, it is again converted to a solid. This heating fluidization and cooling solidification cycle can be repeated many times and is an advantage of the molding process of these materials. There is, of course, the disadvantage that with multiple heating-cooling cycles, the properties of the thermoplastic can be degraded. The analogy for understanding these cycles is that of wax that liquefies upon heating and solidifies on cooling. Thermoplastic materials can be further subdivided into crystalline and amorphous materials. The structure of crystalline thermoplastic materials is characterized by the order in the arrangement of their macromolecules, while the structure of amorphous thermoplastics is characterized by the random arrangement of their macromolecules. Of course, it is impossible to make a perfect crystalline thermoplastic because of its complex structure. Some plastics, such as polyethylene and nylon, may have a high degree of crystallinity but are best described by the term semicrystalline or partially crystalline. Other plastics, such as acrylics and polystyrene, are always amorphous. The presence of crystallinity in plastics that can be crystallized depends to a large extent on the heat treatment during their manufacture. Similarly, the mechanical properties of the plastic depend on the percentage of the crystalline phase contained therein. Generally, plastics are characterized by higher density when crystallized, and this is due to the denser accumulation of their macromolecules. Typical characteristics of crystalline materials are (1) High stiffness, especially at elevated temperatures;

Viscoelastic constitutive modeling

5

(2) Low friction coefficient; (3) Hardness; (4) Resistance to breakage; (5) Ability to be reinforced; (6) Ability to distort; and (7) Higher creep resistance. On the other hand, despite these advantages, crystalline plastics are always opaque and have high shrinkage during molding. Many thermoplastics have been accepted as engineering materials and are called “engineering plastics.” This term likely comes from a classification made to distinguish those plastics that can substitute for metals such as aluminum in small constructions from those that do not exhibit such good mechanical properties. This classification, of course, is not absolute, because the properties of thermoplastics are highly sensitive to temperature, with the result that a thermoplastic that can replace a metal at some temperature cannot replace it in another. A more appropriate definition of “engineering plastic” is that a plastic is said to be an engineering material when it can carry mechanical loads for long time periods.

1.2.2 Thermosets A thermosetting plastic is produced by a two-step chemical reaction. In the first step, macromolecular chains are produced, just as with thermoplastics, but they can react further. The second stage occurs when, under the influence of heat and pressure, “crosslink” bonds are created between the macromolecular chains, resulting in the final product after being cooled to be solid and rigid. The structure of thermosetting plastics is characterized as a “network” of molecules where the macromolecules have minimal degrees of freedom of movement. The characteristic of thermosetting plastics is that they are not fluidized by reheating. If heated heavily, they are instead decomposed. This behavior is analogous to that of a boiled egg which, when cooled, is solidified and when reheated it does not return to its original state. Because the crosslinking process is of a chemical nature, thermosetting plastics are stiff and their properties are not strongly influenced by temperature. Some examples of thermosetting plastics are phenol-formaldehyde, melamine-pharmaceutical, ureaformaldehyde, epoxy resins, and some polyesters.

1.2.3 Polymer-matrix composites One of the key factors that make plastics attractive to mechanical applications is their fiber-reinforcing capability. Fibrous composites find applications in the aerospace and automotive industries, along with others. Today, in the United States, these industries absorb 10% of production of composite materials. As mentioned earlier, both thermoplastics and thermosetting plastics can be reinforced with fibers to develop new materials with unique properties. Thermosetting systems, limited by their brittle character due to the crosslinks of the matrix, were initially only reinforced with long fibers, while thermoplastics were only reinforced by short fibers. In the first case, a low viscosity of the matrix was used so that a perfect coating of the long fibers would help to transfer stress from the matrix to the fiber; however, production rates were very low. In the latter case, due to the high resistance to crack propagation of thermoplastic materials and the rapid production of short

6

Creep and Fatigue in Polymer Matrix Composites

thermoplastic composites, these materials were usually used. Today, however, with the discovery of improved techniques, all matrix-fiber combinations are produced. Glass fibers are the most common type of fiber used as a plastic reinforcement because they offer a good combination of strength, stiffness, and cost. Improved strengths and stiffness accompany the “aramid” (Kevlar) fibers, or carbon fibers, which however have much higher costs. In addition, it is possible to manufacture “hybrid composites” that are reinforced with two or more types of fibers or inclusions in order to reduce costs and achieve better mechanical properties. For example, impact behavior of carbon fiber reinforced plastic (CFRP) can be improved by adding glass fibers, while the modulus of elasticity of glass fiber reinforced plastic (GFRP) can be increased by adding carbon fibers.

1.3

Viscoelastic behavior

Plastics, even under normal environmental conditions, exhibit intense viscoelastic behavior, meaning that their behavior presents simultaneously the characteristics of viscous fluids and elastic solids. Thus, when a plastic is mechanically loaded, part of the energy is dissipated in the form of heat or acoustic emission and another part is stored in the form of elastic energy. Viscoelastic behavior is both directly and indirectly dependent (i.e., depends on the rate of loading, deformation, or heating) on time. Static stress-strain diagrams are known for their short-term description of the mechanical behavior of plastics. It should be emphasized that such a characterization is useful only for an initial comparative choice of material suitability and is not a decisive criterion in the final design of long-term behavior of structures. In most cases, plastics exhibit a characteristic stress-strain curve similar to that shown in Fig. 1.1. In this curve, A is the limit of proportionality, B is the mechanical strength, and C is the fracture limit. For small deformations, the curve σ-ε exhibits a linear elastic behavior whereas for large deformations the behavior is nonlinear. For completely unknown plastics design, deformations should be limited to 1%. Smaller deformation values (about 0.5%) are recommended for brittle plastics, such as acrylics and polystyrene, while values of 0.2%–0.3% should be used in designs with thermosetting plastics. s B A

0

C

e

1.1 Typical stress-strain curve of a polymer: (A) Limit of proportionality, (B) mechanical strength, (C) stress at fracture.

Viscoelastic constitutive modeling

7

T=25°C Stress s (MPa)

T=80°C

T=120°C

Strain e(%)

1.2 Effect of temperature on the stress-strain curve of a polymer.

1.3.1 Effect of temperature The influence of temperature on the mechanical behavior of plastics is shown in Fig. 1.2. From this figure we observe that, with increasing temperature, the material becomes more ductile with a lower modulus of elasticity and higher deformation at fracture.

1.3.2 Effect of strain rate Another interesting parameter is the rate of deformation. If a plastic is stretched with a high deformation rate, it exhibits a greater modulus of elasticity than with a low deformation rate. Also, at high rates of deformation the plastics behave in a more brittle way. This phenomenon is shown in Fig. 1.3.

1.3.3 Effect of type of plastic At this point it should be stressed that a plastic may be available in different types and depending on the type the properties may vary considerably. For example, with polypropylene, each change in density of 1 kg/m3 shows a change of 4% in the modulus of elasticity. Fig. 1.4 shows the difference in mechanical behavior of different types of de/dt=1min–1 de/dt=0.8min–1

Stress s (MPa)

de/dt=0.5min–1

Strain e%

1.3 Effect of strain rate on the stress-strain behavior of a plastic.

8

Creep and Fatigue in Polymer Matrix Composites

Type A Type B

Stress s (MPa)

Type C

Strain e(%)

1.4 Effect of polymer type on the stress-strain curve.

acrylonitrile butadiene styrene (ABS). In the same figure, we can see that choosing a particular type of plastic with a specific feature (e.g., high strength) can lead to a corresponding reduction in another property (e.g., tensile strength). Deformations that develop in carriers under the influence of operating loads should be as small as possible, so that the carriers not only do not fail but also still operate unimpeded even when local, limited amounts of yielding are observed. Therefore, high modulus, high strength, and high ductility are the desired combination of construction. However, such an ideal combination is almost impossible, and what is always observed is that one property being improved causes degradation of another.

1.3.4 Creep and creep recovery Creep is the phenomenon in which materials under the influence of a constant mechanical load at constant temperature and humidity show an increase in deformation over time. Similarly, the instant removal of the constant mechanical load is characterized by a time-dependent reduction in deformation. This phenomenon is called recovery. A graphical representation of both phenomena is given in Fig. 1.5.

e (%)

Elastic recovery Viscoelastic deformation Viscoelastic

A 0

Elastic deformation

D Time (t)

1.5 Strain vs. time curve showing the creep and creep-recovery phenomena.

Viscoelastic constitutive modeling

9

1.3.5 Stress relaxation Another important feature of the viscoelastic behavior of plastics is the phenomenon of relaxation. More specifically, as shown in Fig. 1.6, if we apply a constant deformation at a constant temperature and humidity to a plastic, then the stress required to maintain constant deformation decreases with time. Relaxation is very important in the design of shutters, springs, etc.

1.3.6 Creep rupture When a material is loaded at a constant stress, its deformation increases with time to a point where it can break. This phenomenon is called creep rupture, or static fatigue. When designing metal devices, it is important to understand and take this phenomenon seriously into account, because it is usually the classic fault to assume that if a structure withstands static loading, then it will continue to withstand the same loading conditions indefinitely. Therefore, especially in the case of plastics designs, it is necessary to take into account the long-term design data of the materials and not just the shortterm design data (static properties).

1.3.7 Fatigue Plastics, like metals, are sensitive to brittle fracture resulting from cyclic loading. In addition, the rigidity of the plastics increases during their wear due to the heat absorbed by them during testing, especially at high frequencies. The plastics with the best fatigue behavior are polypropylene and ethylene-propylene copolymer.

1.0 0.8 Stress 0.6 s (MPa) 0.4 0.2 0.0 0

100,000

200,000

1.6 Stress relaxation curve of a plastic.

300,000 Time (s)

400,000

500,000

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Creep and Fatigue in Polymer Matrix Composites

1.3.8 Toughness The term toughness refers to resistance to breakage. Some plastics are by nature very ductile and therefore exhibit high resistance to fracture, while others are very brittle. However, things are not as simple as they appear, because materials that are by nature resistant to fracture may show brittle behavior after their production and molding process, or after being attacked by various chemicals, or after prolonged loading, even under constant stress. Thus, if a particular design requires high toughness, it is necessary to carefully examine the conditions and operating environment in relation to all these factors. At room temperature the plastics most resistant to cracking are nylon 66, low density polyethylene (LDPE), linear low density polyethylene (LLDPE), ethylene vinyl acetate (EVA), and polyurethane foam. At temperatures below zero, ABS, polycarbonate, and EVA materials are recommended.

1.4

Properties degradation

1.4.1 Physical and chemical attack One of the most basic research areas of plastics is anticorrosion resistance and, in general, environmental resistance. As in the case of metals, it is difficult to predict the behavior of a plastic in an aggressive environment. Obviously, it is not possible to describe the behavior of each material in any environment. Thus, we will outline some environmental factors and how they affect the degradation of plastic properties. The degradation of a plastic is the result of the destruction of its chemical structure. It should be emphasized that such degradation results not only from attacks on the material by acids and solvents. It can happen from seemingly innocent factors such as water (hydrolysis) or oxygen (oxidation). Degradation is also caused by heat, mechanical load, and radiation. When molding, the plastic is subject to the influence of the first two factors, so it is necessary to add plastic stabilizers and antioxidants to preserve its properties long term. Regarding the general behavior of plastics, it is known that crystalline plastics have a higher resistance to environmental degradation agents than amorphous plastics. This is due to the different structure of these two categories of plastic. Thus, mechanical plastics that are also crystalline, e.g., nylon 66, have the direct advantage of combining relatively high mechanical strength and high resistance to chemical attack. The appearance of new crystalline plastics such as polyetheretherketone (PEEK) and polyphenylene sulfide (PPS) has set new rules on environmental endurance. At ambient temperature there is no PPS solvent, whereas PEEK is only affected by 98% sulfuric acid solution.

1.4.2 Weathering This effect is generally due to the combined effect of water or moisture absorption and exposure of the material to ultraviolet radiation. The absorption of water results in plasticizing the material and thus increasing its flexibility, which, however, after loss

Viscoelastic constitutive modeling

11

of water absorbed, leads to brittleness of the material. Conversely, exposure of the material to ultraviolet radiation results in breakage of macromolecular chain bonds, resulting in the degradation of the physical properties of the materials. Also, the same factors can cause loss of color and clarity of the plastic. Water absorption reduces dimensional stability of molded plastics. Most plastics, and in particular cellulose, polyethylene, PVC, and nylon derivatives, are affected by these factors.

1.4.3 Oxidation Oxidation is caused by contact of the plastic with oxidizing bodies or exposure to the free atmosphere. It leads to the degradation of mechanical properties (brittleness and possibly fracture) and the reduction of clarity. It affects most thermoplastics to some degree, especially polyolefins, PVC, nylons, and cellulose derivatives.

1.4.4 Environmental stress cracking In some plastics, breakage occurs due to contact forces that develop between plastic and other environmental bodies. Loads can be applied externally, and some steps can be taken to avoid them. However, the internal or residual stresses that can develop in the plastic during the molding process are often the main factor in breakage. Most organic fluids promote environmental breakage of plastic, but in some cases the problem may be caused by liquids that are seemingly innocent. A classic example is the brittle fracture of polyethylene glasses due to the remaining stresses that develop at the exit of the injection machine in conjunction with a soap solution used to clean them. Although environmental damage does not directly affect the chemical structure of the plastic, it can be addressed by controlling the structural parameters. For example, the resistance of polyethylene is highly dependent on density, crystallinity, flow index, and molecular weight. Apart from polyethylene, other plastics that are sensitive to environmental breakage are ABS and polystyrene. The mechanism of environmental stress cracking (ESC) relies on the causative agent penetrating through the surface microcracks, resulting in a change in surface energy and ultimately leading to breakage.

1.4.5 Resistance to wear and friction Recently, there has been an increasing use of plastic in load-bearing structures as well as mechanical elements where sliding friction develops, such as gears, pistons, rings, closures, cams, etc. The advantages of plastics are low cost production, small friction coefficients, vibration absorption capacity, silent operation, and low power consumption. Also, when the plastics are fiber-reinforced they exhibit high strength. Typical reinforcements are glass fibers, carbon fibers and kevlar, polytetrafluoroethylene (PTFE) inclusions, and inorganic materials. Friction and wear resistance of plastics are complicated phenomena that are strongly dependent on the nature of the application and the properties of the material. The properties of plastics differ significantly from those of metals. Still, the reinforced plastics have a lower modulus of elasticity than that of the metals.

12

Creep and Fatigue in Polymer Matrix Composites

Therefore, the friction that develops between metals and plastics is characterized by adhesion and deformation that lead to the development of frictional forces that are not proportional to load but speed. The rate of wear of plastics is governed by various mechanisms. The primary one is adhesion wear characterized by the surface loss of plastic particles. This small-scale phenomenon usually occurs in well-functioning bearings. However, the second and most important mechanism occurs when the plastic overheats, leading to the loss of large parts of its mass. Here we have to say that because of the complexity of the wear mechanism, the wear rate values may vary depending on the operating conditions. In linear loading applications, the suitability of a plastic is usually determined by the wear resistance plots P-V, where P is the load divided by the projection of the contact surface and V the linear velocity. Resistance to wear increases when the material is lubricated or interrupted. Instead, resistance to wear decreases if the operating temperature is high. Of course, the wear and friction are not fully quantified and, in many cases, are determined by the manufacturer. Plastics that show the best wear resistance are high molecular weight polyethylene, PTFE, acetals, and PBT.

1.5

Physical properties of polymers

1.5.1 Thermal properties Before considering the classical thermal properties, such as thermal conductivity, it is advisable to briefly refer to the effect of temperature on the mechanical properties of plastics. As mentioned previously, the properties of plastics are highly dependent on temperature. This is a result of their molecular structure. Let’s first assume an amorphous plastic whose macromolecules are randomly distributed and are in constant motion. As the material is heated, the macromolecules absorb thermal energy and convert it into kinetics, making the material more flexible. Conversely, if the material is cooled, the molecular mobility decreases, and the material becomes rigid. In amorphous plastic there is a characteristic temperature called the glass transition temperature, Tg, below which the material behaves like glass, i.e., it is hard and rigid. This helps us to better understand the differences between different plastics. For example, at room temperature, PS and PMMA look like glass because the room temperature is less than their Tg (i.e., they are in the glass phase). In contrast, at room temperature natural rubber (NR) is above its Tg and is therefore flexible. However, if it is cooled down to a temperature below Tg (
75°C), it becomes tough and brittle. Amorphous polymers exhibit more than one transition temperature. The main Tg is called a glass-rubber transition and expresses the transition of the polymer from the glassy state, where the macromolecules have little mobility, to the rubbery state, characterized by high mobility of the macromolecules. At temperatures lower than Tg, other secondary transitions occur due to micromovements of side branches of the main macromolecule chain.

Viscoelastic constitutive modeling

13

At this point it should be stressed that the Tg values given in any table in the international literature do not constitute constants of these materials, but their value depends on the thermal analysis method used to determine them, as well as the heating or cooling rate of the sample. When heating crystalline polymers whose structure is characterized by crystalline regions with macromolecules oriented in one direction as well as by amorphous regions of random orientation of macromolecules, while increasing the mobility of the macromolecules, these materials still remain stiff due to the strong forces that develop between the macromolecules of the crystalline phase. The temperature Tg of the crystalline polymers is mainly due to the thermal transition of the macromolecules of the amorphous phase. For example, at room temperature polypropylene is a solid with high toughness and this is not because of its low Tg but is due to the strong forces that develop between the macromolecules of the crystalline phase. When this material is cooled down to
10°C, it behaves as brittle material because its amorphous phase at these temperatures behaves in a brittle manner. In the past, a serious limitation on engineering polymer designs was the prolonged stay of the design in a high-temperature environment. This limitation results not only from the degradation of the mechanical properties of the polymer at high temperatures and the increase in the probability of creep, but also from the permanent wear that develops in the material under these operating conditions mainly due to its oxidation and thermal fatigue of the polymer. Although the problem can be partially addressed by reinforcing polymers with various inclusions, such as glass and carbon fibers, new polymers need to be developed that can withstand high temperatures. Other thermal properties of polymers related to mechanical design are thermal conductivity and thermal expansion coefficient.

1.5.2 Electrical properties Traditional plastics have been established in applications where electrical insulation is required. PTFE and PE are among the materials with the best electroinsulating properties. The main electrical properties are electrical insulation, dielectric strength, and dielectric resistance. Electrical insulation is destroyed in strong electric fields. The dielectric strength of an insulator is defined as the maximum electric field strength (V/m) that the insulator can withstand. In plastics, dielectric strength can vary between 1 and 1000 MV/m. Even though plastics are good insulators, local electrical failures are often encountered due to their uneven structure and arc development in their surfaces (tracking). An electric arc can develop due to the presence of foreign elements (dust, moisture) on the surface of the polymer. It is interesting to note that, while electrical insulation of plastics is considered to be one of their advantages, relatively recent efforts are being made to increase their conductivity. There are two ways to increase the electrical conductivity of plastics: coating and compounding. In the first method, the surface of the polymer is coated with an electrically conductive material (e.g., carbon or metal), while in the second method the

14

Creep and Fatigue in Polymer Matrix Composites

polymer is reinforced with metal inclusions in the form of fine fibers of long length/ diameter ratio (aspect ratio). In recent years, much effort has been made to develop materials reinforced with metal-coated glass fibers. However, because fibers with a long length/diameter ratio are required, they can be destroyed in the process of making the reinforced polymers. In this case, thermosetting plastics are superior to other types of plastics because they have a simpler construction and therefore the reinforcing fibers are not destroyed. It is now possible to make conductive fibers with special resistances on the order of 7  10
3 Ωm.

1.6

Viscoelastic behavior of polymers

By viscoelastic behavior we mean exactly a behavior in which the response of the material to a specific excitation is time-dependent. Thus, in plastics whose behavior is predominantly viscoelastic, it is observed that stiffness, strength, ductility, and other properties are sensitive to the rate of deformation (indirect dependence on time), the loading rate, the strain or stress history, temperature, and heating or cooling rate. To study the viscoelastic behavior of these materials, different types of experiments are executed, the most basic of which are creep, relaxation, and creep-recovery. Creep is the phenomenon in which the deformation that develops in a body at constant tension and at constant temperature and humidity increases with time. During creep, there is a change in stiffness of the material that comes up against time. Knowing the law governing this change is very important for the proper design of plastic parts or other products. Metals also exhibit viscoelastic behavior, but this is evident only in particularly high temperatures. At ambient temperature, polymers show a strong creep while the creep of metals is negligible. Thus, all materials flow and, depending on environmental conditions and the different loads applied, their creep is more or less intense. Relaxation is the phenomenon in which the stress that develops in a body in constant distortion, temperature, and humidity conditions decreases over time. Creep-recovery is the phenomenon observed in a body immediately after reduction of the fixed imposed creep load and in which deformation decreases exponentially over time. In most design cases with traditional materials, the purpose of the design is to determine the allowable stresses. Conversely, in plastics, the limiting factor is the determination of the deformation that will develop material failure stresses. In a perfectly elastic or Hookean material, the stress σ is directly proportional to the deformation ε and the relation between stress and deformation in a uniaxial stress state can be formulated as follows: σ¼Eε

(1.1)

where the proportionality constant E is the modulus of elasticity of the material.

Viscoelastic constitutive modeling

15

In a perfect or Newtonian fluid, shear stress, τ, is directly proportional to the rate of change of shear deformation, dγ/dt, and the relation that connects these measures is:
 dγ τ¼η ¼ η  γ_ dt

(1.2)

where the constant, η, in this case refers to the viscosity of the fluid. The two preceding cases describe two extreme types of behavior of the materials. In practice, at each point in time the behavior of each material presents both types of behavior simultaneously. Simply, depending on the environmental conditions, each of these types will be intense. That is, the behavior of the materials can be characterized at the same time as both viscous and elastic. Such behavior is called viscoelastic behavior. Polymers, even under normal environmental conditions, exhibit intense viscoelastic behavior. Conversely, in metals viscoelastic behavior occurs only at higher temperatures. In viscoelastic materials, at a given temperature, stress is a function of both deformation and time. That is, a relationship of the following form will apply: σ ¼ f ðε, tÞ

(1.3)

Viscoelastic behavior, just like elastic behavior, is distinguished as both linear and nonlinear. The linear elastic model is used to describe materials that respond as follows: (i) the strains in the material are small (linear) (ii) the stress is proportional to the strain, (linear) (iii) the material returns to its original shape when the loads are removed, and the unloading path is the same as the loading path (elastic) (iv) there is no dependence on the rate of loading or straining (elastic)

This model well represents engineering materials up to their elastic limit. It also models well almost any material, provided the stresses are sufficiently small. At this point we should emphasize the following differences: a) In linear elastic behavior, regardless of the moment of observation, the stress-strain curve is linear and unique. b) In linear viscoelastic behavior, a different rectilinear σ-ε curve corresponds at each time instant. c) In nonlinear viscoelastic behavior, a different nonlinear σ-ε curve corresponds at each time instant.

These three different behaviors are shown in Fig. 1.7.

1.7

Short-term behavior

Simple experimental tensile testing is the most widespread method of metal characterization but is also widely applied to polymers. However, in the case of polymers, we must be particularly careful because in this case the tensile effects can be used only as quality control elements rather than as absolute data to be used in the design of structures. This is because, in the case of plastics, the response strongly depends on the

16

Creep and Fatigue in Polymer Matrix Composites

1.7 Stress-strain curve dependence on time (isochronous curves) for linear elastic, linear viscoelastic and nonlinear viscoelastic behavior.

s

Linear elastic

t 1, t2 Linear viscoelastic

t1 t2 Non-linear viscoelastic

t1

t2

t 2 > t1 0

e

conditions of the experiment, such as load application rate, deformation rate, ambient temperature, humidity, etc. If during stretching of the polymers the load is intermittently interrupted in order to measure the deformation, it should be noted during the temporary cessation of the load whether the material is leaking, in which case if the loading intervals are not equal to each other (which is very likely), then the material will exhibit an apparent nonlinear behavior. In addition, the method and the timing of application of the external load define the so-called “material loading history” that affects the final behavior. Thus, according to this analysis, we should not trust values of polymer properties that result from experiments whose conditions are not clearly described. For this reason, it is always necessary for the experimental testing of materials to follow certain specific standard specifications. The standards followed for experimental testing of the polymers in tensile strength are BS 2782, ASTM D38, and ASTM D790.

1.8

Long-term behavior

Because tensile tests of polymers have all the preceding disadvantages, creep control is the most accurate experimental method for studying the viscoelastic behavior of these materials. As mentioned previously, a constant load is applied at constant temperature and we measure deformation. In the ε-t diagrams we usually use a logarithmic time scale to cover longer time intervals. As shown in Fig. 1.8, the ε-t diagram is characterized by an instantaneous initial elastic deformation followed by a gradual increase in deformation. Most polymers exhibit nonlinear viscoelastic behavior. We will first deal with linear viscoelastic behavior and then with nonlinear viscoelastic behavior. As expected, at any time an increase in stress will cause a corresponding increase in deformation (Fig. 1.9). In the case of creep, if at a given time we double the imposed stress, this does not necessarily mean doubling the deformation. The stress-strain analogy can only be applied over short times. Finally, at certain stress and time, an increase in temperature causes higher strain (Fig. 1.9).

Viscoelastic constitutive modeling

17

1.8 Effect of scaling on creep diagram shape. 1.9 Effect of the applied stress and temperature on creep curves.

1.9

Isochronous and isometric diagrams

Let us assume now that we have a group of creep curves ε-t where each curve corresponds to a different fixed value of externally applied stress. From this group of curves, very useful conclusions can be drawn for the overall viscoelastic behavior of the material we are studying (Fig. 1.10). Thus, if on the plot of the group of curves we draw a line vertical to the deformation axis corresponding to a specific value of the deformation, letting ε ¼ εo, then this vertical will intersect the curves of the group at points having coordinates (σ i, t). Then, based on these points, if we draw the curve σ-t, then the resulting curve is called the isometric curve and describes the relaxation of the material for ε ¼ εo. The same procedure can then be repeated for another deformation value. If we then divide the ordinate values of the points of the isometric diagram (i.e., the stress) by the constant deformation (ε ¼ εo), then we can get the creep modulus variation as a function of time. We notice that the creep modulus decreases with time. Another piece of useful information that may arise from the cluster of creep curves is the isochronous curve. The graph is produced as follows: If on the plot of the ε-t curves we draw a line vertical to the horizontal axis of time corresponding to a certain value of time, letting t ¼ tο, then this vertical line will intersect the curves at points

18

Creep and Fatigue in Polymer Matrix Composites

1.10 Sections through the creep curves at constant time and constant strain give curves of isochronous stress-strain, isometric stress-log (time), and creep modulus-log (time).

each one of them having coordinates (σ i, εi). If we then plot the σ-ε curve based on these points, then the curve obtained is called an isochronous curve. At each time another isochronous curve corresponds. This shows the change in the material response over time. If the isochronous curve is a straight line, then the behavior of the material is linear viscoelastic, whereas if it is not straight, then the behavior of the material is nonlinear viscoelastic. The transition from linear to nonlinear viscoelastic behavior occurs when the applied creep stress exceeds a specific value called the linear-nonlinear viscoelastic threshold. The value of this characteristic stress depends on the nature of the material, the temperature, and the loading history. Here, based on the preceding description, we will make some useful points: a) The creep modulus resulting from the isometric curves, as mentioned here, equals to the slope of the corresponding isochronous curve. b) Isochronous curves better describe the mechanical behavior of polymers than the curves σ-ε resulting from a simple tensile experiment. This is because the isometrics take into account the creep phenomenon. That is why, at any given time, another isochronous curve corresponds. c) Usually the isochronous curves are plotted on a log-log scale because, regardless of whether the behavior is linear or nonlinear, the σ-ε variation in every isochronous curve when plotted on a logarithmic scale is always rectilinear. d) Also, if the behavior is linear viscoelastic, then the slope of the rectilinear isochronous, when plotted on a logarithmic scale, is equal to 45 degrees. Conversely, if the behavior is nonlinear viscoelastic, then this slope is less than 45 degrees. This observation is extremely useful in recognizing the type of viscoelastic behavior the polymer under study exhibits at any specific time instant. e) All the three characteristic curves constitute the characteristic surface of a material when plotted on a 3D diagram (Fig. 1.11).

Viscoelastic constitutive modeling

19

1.11 The characteristic surface.

1.10

Creep-recovery and stress relaxation analysis

The time-dependent behavior of materials may be studied by conducting creeprecovery and stress relaxation experiments.

1.10.1 Creep Creep is a slow, continuous deformation of a material under constant stress. Unlike metals, polymers undergo creep even at room temperature. The creep response to a constant stress applied at time t ¼ 0 is shown in Fig. 1.12a. An instantaneous strain proportional to the applied stress is observed after the application of the stress and this is followed by a progressive increase in strain, as

e

s

Creep

Recovery

s0

er ec

e0 0

(a)

t1

t

0

(b)

1.12 Creep. (a) Application of constant stress; (b) strain response.

t1

t

20

Creep and Fatigue in Polymer Matrix Composites

1.13 Creep stages.

shown in Fig. 1.12b. The total strain at any instant of time is represented as the sum of the instantaneous elastic strain and the creep strain, i.e., εð t Þ ¼ ε0 + εc

(1.4)

The ratio of the total strain to the applied constant stress is called creep compliance and is given by DðtÞ ¼

εðtÞ σ0

(1.5)

The creep compliance at any point of time is the sum DðtÞ ¼ Do + ΔDðtÞ ¼ Dð0Þ + ΔDðtÞ

(1.6)

where Do is the instantaneous creep compliance and ΔD(t) is the transient component of the compliance. In general, creep can be described in three stages: primary, secondary, and tertiary. In the first stage, the material undergoes deformation at a decreasing rate, followed by a region where it proceeds at a nearly constant rate. In the third or tertiary stage, it occurs at an increasing rate and ends with fracture (Fig. 1.13). Understanding the creep behavior of a material is important in design and manufacturing, as this can lead to dimensional instability of the end product, as well as failure at applied constant stresses that are significantly lower than the ultimate tensile strength. Rheological behavior such as creep can be observed in many engineering materials including metals, polymers, ceramics, concrete, soils, rocks, and ice. Creep strains, viscoelastic or viscoplastic deformations, and creep fractures in engineering structures result from time- and temperature- dependent processes that occur when specific materials are subject to stress. Creep of materials often limits their use in practice.

Viscoelastic constitutive modeling

21

Particularly in high-temperature environments, creep has a severe effect on structural or component response. Creep is a very complex phenomenon that depends on many parameters. For example, the creep behavior of fiber-reinforced composites depends on the following factors: the creep behavior of the matrix, elastic and fracture behavior of the fibers, geometry and arrangement of the fibers, and the fiber-matrix interphasial properties. Mechanisms such as load transferring from the matrix to the fiber, increased dislocation density around the fiber, and residual stresses arising from the difference in the coefficients of thermal expansion between the fiber and the matrix should be considered simultaneously. Generally, the creep strain rate of composites can be expressed by the following qualitative equation: ε_ ¼ f ðσ o , T, ε_ m , Vf , λf , θ, Ef , σ uf , Em , Ei , ti Þ

(1.7)

where σ o, external applied stress; T, testing temperature; ε_ m , matrix strain rate, Vf, fiber volume fraction, λf, geometric parameters of the fibers; θ, fiber orientation angle (relative to the applied loading); Ef, fiber modulus; σ uf, fiber ultimate strength; Em, matrix modulus; Ei, fiber-matrix interphase modulus; ti, interphase thickness. Another example showing the complexity and importance of the creep phenomenon is the creep behavior of epoxy-based structural adhesives. Epoxy-based structural adhesives have emerged as a critical component for assembling structural parts due to their high strength-to-weight ratio, excellent adhesion properties, and superior thermal stability. A structural adhesive can be defined as a load-bearing material with high modulus and strength that can transmit stress without loss of structural integrity. Compared with other joining methods, such as welding or bolting, epoxy-based structural adhesives provide exceptional advantages, including distributing stresses equally over large areas while minimizing stress concentrations, joining dissimilar materials, and reducing the overall weight and manufacturing costs. However, epoxy resins, being viscoelastic in nature, exhibit unique time-dependent behavior. This leads to significant concerns in assessing their long-term load-bearing performance.

1.10.2 Recovery If the load is removed, a reverse elastic strain followed by recovery of a portion of the creep strain will occur at a continuously decreasing rate. The amount of the timedependent recoverable strain during recovery is generally a very small part of the time-dependent creep strain for metals, whereas for plastics it may be a large portion of the time-dependent creep strain that occurred (Fig. 1.12b). Some plastics may exhibit full recovery if sufficient time is allowed for recovery. The strain recovery is also called delayed elasticity.

1.10.3 Relaxation Viscoelastic materials subjected to a constant strain will relax under constant strain (Fig. 1.14a) so that the stress gradually decreases, as shown in Fig. 1.14b.

22

Creep and Fatigue in Polymer Matrix Composites

e

s e0

Relaxation

t

0

(a)

t

0

(b)

1.14 (a) Application of constant strain; (b) stress relaxation.

1.11

Linearity

A viscoelastic material is said to be linear if: 1. The stress is proportional to the strain at a given time, that is

ε½cσ ðtÞ ¼ cε½σ ðtÞ

(1.8)

as shown in Fig. 1.15. This also implies that for a linear viscoelastic material, the creep compliance is independent of the stress levels. Thus the compliance-time curves at different stress levels should coincide if the material is linear viscoelastic.

Cs s s

0

t

e

C e (t) e (t)

0

1.15 Linear viscoelastic behavior.

t

Viscoelastic constitutive modeling

23

2. The linear superposition principle holds. This implies that each loading step makes an independent contribution to the final deformation, which can be obtained by the addition of these. This principle is also called the Boltzmann superposition principle.

For the two-step loading case shown in Fig. 1.16, the strain response is given by ε½σ 1 ðtÞ + σ 2 ðt
t1 Þ ¼ ε½σ 1 ðtÞ + ε½σ 2 ðt
t1 Þ

(1.9)

Further, for multistep loading, during which stresses σ 1, σ 2, σ 3… are applied at times τ1, τ2, τ3…, the strain at time t is given by: εðtÞ ¼ σ 1 Dðt
τ1 Þ + σ 2 Dðt
τ2 Þ + σ 3 Dðt
τ3 Þ + …

(1.10)

where D(t) is the creep compliance. Typically in order to determine the linear viscoelastic region, creep and recovery experiments are carried out. A suitable model is developed for the compliance using the creep portion of the experiment and using this model, the recovery strains are predicted. If the predicted and experimental recovery strains match, then the linear superposition principle holds, and the behavior is linear. The linear viscoelastic response to a multiple step loading can be generalized in the integral form (also known as the Boltzmann superposition integral) as: ðt εðtÞ ¼ Do σ +

ΔDðt
τÞ

dσ dτ dτ

(1.11)

0

where Do is the instantaneous creep compliance, ΔD(t
τ) is the transient creep compliance, σ is the applied stress, and τ is a variable introduced into the integral in order to account for the stress history of the material. s

1.16 Boltzmann superposition principle.

s1 + s2 s1 s2

0

t

t1

e e[s1 (t)] + e[s2 (t–t1)]

e[s 1 (t)] 0

t1

e[s 2 (t–t1)] t

24

Creep and Fatigue in Polymer Matrix Composites

The preceding integral is called the hereditary or Volterra integral. The integral basically implies that the strain is dependent on the stress history of the material under consideration. It can be seen from the hereditary integral representation of linear viscoelastic behavior that the creep compliance can be separated into an instantaneous component, D0σ, and a time-dependent component. The transient creep compliance function is often given in the form of a Power law or a Prony series in viscoelastic modeling. The Power law form of this function is as follows: ΔDðtÞ ¼ D1 tn

(1.12)

The benefit of this function is that it is mathematically simple and has been found to provide an adequate prediction of short-term creep behavior. A Prony series expansion would result in a transient creep compliance function and hereditary integral equation of the following forms: ΔDðtÞ ¼

N X

  Di 1
e
t=τi

(1.13)

i¼1

εð t Þ ¼ D o σ +

N X

  Di σ 1
e
t=τi

(1.14)

i¼1

Even though the use of both Power Law and Prony series are common in creep modeling, the use of Prony series is dominant when finite element methods are involved. Similarly, the principle can be used for stress relaxation data, resulting in an analogous relation: ðt σ ðtÞ ¼ Eo ε +

ΔEðt
τÞ

dε dτ dτ

(1.15)

0

where Eo and ΔE(t
τ), are components of the stress-relaxation modulus. The preceding equations are sometimes referred to as linear viscoelastic material functions and are interrelated mathematically. Therefore, if the linear viscoelastic behavior is known under creep loading, the stress relaxation behavior can also be determined without the need to conduct additional experimentation, and vice versa. If any of the conditions for linear viscoelasticity are no longer satisfied, the viscoelastic behavior is nonlinear. The degree of nonlinearity can be influenced by factors such as applied stress level, strain rate, and temperature. For linear elastic behavior, modulus and compliance can be interrelated by: ðt

ðt Dðt
τÞEðτÞdτ ¼ t or

0

Eðt
τÞDðτÞdτ ¼ t 0

(1.16)

Viscoelastic constitutive modeling

25

However, it is to be noted that (instantaneous) analytical integration of the equation is possible only for simple forms of creep compliance. For example, if the compliance can be expressed by power law given by DðtÞ ¼ D1 tn

(1.17)

then it can be shown that the relaxation modulus is given by: Eð t Þ ¼

1 t
n D1 Γð1 + nÞΓð1
nÞ

(1.18)

Ð where Γ(x) ¼ e
ttx
1dt is the gamma function.

1.12

The time-temperature superposition principle

Polymeric materials, because of their viscoelastic nature, exhibit behavior during deformation and flow that is both temperature and time/frequency dependent. For example, if a polymer is subjected to a constant load, the deformation or strain (compliance) exhibited by the material will increase over a period of time. This occurs because the material, when under load, undergoes molecular rearrangement in an attempt to minimize localized stresses. Hence, compliance or modulus measurements performed over a short time span result in lower/higher values respectively than longer-term measurements. This timedependent behavior would seem to imply that the only way to accurately evaluate material performance for a specific application is to test the material under the actual temperature and time conditions the material will meet during its application. This implication, if true, would present real difficulties for the rheologist, because the range of temperatures and/or frequencies covered by a specific instrument might not be adequate, or at best might result in extremely long and tedious experiments. Fortunately, however, there is a treatment of the data, designated as the method of reduced variables or time-temperature superposition principle (TTSP), which overcomes the difficulty of extrapolating limited laboratory tests at shorter times to longer-term, more real-world conditions. According to the TTSP, the viscoelastic response at a higher temperature is identical with the response at the low temperature for a longer time. The underlying bases for time/temperature superpositioning are (Brinson et al., 1978) that the processes involved in molecular relaxation or rearrangements in viscoelastic materials occur at accelerated rates at higher temperatures and (Brueller, 1987) that there is a direct equivalency between time (or frequency of measurement) and temperature. Hence, the time over which these processes occur can be reduced by conducting the measurement at elevated temperatures and transposing (shifting) the resultant data to lower temperatures. To accomplish this step, creep compliance curves, D(T,t), developed at different temperatures are horizontally shifted along the log-time scale to develop temperature dependent shift factors (aT): t0 ¼

t aT

(1.19)

26

Creep and Fatigue in Polymer Matrix Composites

where t’ is the shifted or reduced time; t is the elapsed time of a test; aT is the shift factor specific to a test. The result of this shifting is a “master curve” depicting creep compliance against reduced time (t’), where the material property of interest at a specific end-use temperature can be predicted over a broad time scale. The amount of shifting along the horizontal (x-axis) in a typical TTSP plot required to align the individual experimental data points into the master curve is generally described using one of two common theoretical models. The first of these models is the Williams-Landel-Ferry (WLF) equation: log ðaT Þ ¼


C1 ðT
To Þ C2 + ðT
To Þ

(1.20)

where C1 and C2 are empirical constants, To is the reference temperature (in K), T is the temperature at the accelerated state of interest (in K), and aT is the horizontal shift factor. The WLF equation is typically used to describe the time/temperature behavior of polymers in the glass transition region. The equation is based on the assumption that, above the glass transition temperature, the fractional free volume increases linearly with respect to temperature. The model also assumes that as the free volume of the material increases, its viscosity rapidly decreases. Eq. (1.20) is normally called the WLF equation and was originally developed empirically. It holds extremely well for a wide range of polymers in the vicinity of the glass transition temperature and if T0 is taken as Tg, measured by a static method such as dilatometry, then, log ðaT Þ ¼


Cg1 ðT
To Þ Cg2 + ðT
To Þ

(1.21)

and the new constants Cg1 and Cg2 become “universal” with values of 17.4 and 51.6 K, respectively. In fact, the constants vary somewhat from polymer to polymer, but it is often quite safe to assume the universal values as they usually give shift factors close to measured values. Another model commonly used is the Arrhenius equation: log at ¼

E RðT
To Þ

(1.22)

where E is the activation energy associated with the relaxation, R is the gas constant, T is the temperature at the accelerated state of interest (in K), T0 is the reference temperature (in K), and at is the time-based shift factor. The Arrhenius equation is typically used to describe behavior outside the glass transition region but has also been used to obtain the activation energy associated with the glass transition.

1.13

The time stress superposition principle

In the time stress superposition principle (TSSP), stress is used as an accelerating factor. Short-term, isothermal, creep-recovery tests are carried out at different stress

Viscoelastic constitutive modeling

27

levels. The master curves in the time-stress scale are generated by a numerical procedure using an analytical constitutive equation.

1.14

The time-temperature-stress superposition principle

Increased stress accelerates creep of many viscoelastic materials, similar to the effect from increased temperature. A number of researchers have proposed timetemperature-stress superposition principles (TTSSPs), including Schapery (1969), Yen and Williamson (1990), and Brinson et al. (1978). The fundamental ideas behind TTSSP are: (1) particular environmental conditions such as temperature and stress level can accelerate the viscoelastic deformation process; (2) the creep deformation curves associated with different conditions are of the same shape; (3) an increase in temperature or stress will shift creep deformation curves on a log-time scale; and (4) these curves can be combined to form a smooth continuous curve, known as the master curve. When successful, the master curve formed using TTSSP represents the predicted long-term viscoelastic response at a given reference condition. Time-temperature-stress superposition assumes that creep behavior at one temperature or stress can be related to that at another by simply shifting the data along the logtime scale. This shift implies that, as temperature or stress increases, molecular relaxations accumulate at a constant rate and that the underlying mechanism of creep remains unchanged. Free volume theory is often used to describe this molecular mobility. Free volume is viewed as void space allowing motion of polymer chains. Time-dependent mechanical properties can be directly related to changes in free volume (Knauss and Emri, 1981). Wenbo et al. (2001) proposed a TTSSP that is constructed within the framework of free volume theory. The following discussion summarizes their work. From the free volume theory, the viscosity of a material, η, can be related to the free volume fraction ƒ by:
 1 ln η ¼ ln A + B
1 f

(1.23)

where A and B ¼ material constants. Eq. (1.23), known as the Doolittle equation, is the foundation of time-temperature superposition. Assuming that changes in the free volume fraction are linearly dependent on stress changes, as well as temperature changes, the free volume fraction as a function of temperature and stress can be expressed as: f ¼ f0 + αT ðT
T0 Þ + ασ ðσ
σ 0 Þ

(1.24)

where: αT ¼ coefficient of thermal expansion of the free volume fraction; ασ ¼ stressinduced expansion coefficient of the free volume fraction; and ƒ0 ¼ free volume fraction at a reference temperature and stress.

28

Creep and Fatigue in Polymer Matrix Composites

Presume there exists a shift factor that satisfies ηðT, σ Þ ¼ ηðT0 , σ 0 ÞaTσ

(1.25)

Then Eqs. (1.23), (1.24) can be combined:  log ðaTσ Þ ¼
C1

C3 ðT
T0 Þ + C2 ðσ
σ 0 Þ C2 C3 + C3 ðT
T0 Þ + C2 ðσ
σ 0 Þ

 (1.25a)

where C3 ¼ αf0σ . Eq. [1.25a] reduces to the WLF equation if there is no stress difference. Additionally, the stress shift factor at constant temperature aTσ and the temperature shift factor at constant stress level aσT are defined so that: ηðT, σ Þ ¼ ηðT, σ 0 ÞaΤσ ¼ ηðΤ0 , σ 0 ÞaΤσ aσΤ0 ¼ ηðΤ0 , σ ÞaσΤ ¼ ηðΤ0 , σ 0 ÞaΤσ 0 aσΤ ; therefore, aTσ ¼ aΤσ aσΤ0 ¼ aΤσ 0 aσT Eq. (1.24) shows that time-dependent properties of viscoelastic materials at different temperatures and stress levels can be shifted along the time scale to construct a master curve of a wider time scale at a given temperature, T0, and stress level, σ 0. In a case where the service temperature is chosen as the reference temperature, T0, Eq. (1.25a) reduces to:
   B σ
σo C 1 ðσ
σ 0 Þ log ðaΤσ Þ ¼
¼
2:303fo fo =aσ + σ
σ o C3 + ðσ
σ 0 Þ where aσ ¼ stress shift factor. Now, the nonlinear creep compliance at varied stress levels can be related by the reduced time, t/aσ: Dðσ, tÞ ¼ Dðσ 0 , t=aσ Þ

1.15

(1.26)

Linear viscoelastic models

1.15.1 The linear spring All linear viscoelastic models are made up of linear springs and linear dashpots. Inertia effects are neglected in such models. In the linear spring shown in Fig. 1.17, σ ¼ Eε

(1.27)

Viscoelastic constitutive modeling

29

s s0 s

E

0

t1

t

e s 0/E s

0

t1

t

1.17 Linear spring response to constant stress.

where E can be interpreted as a linear spring constant or a Young’s modulus. The spring element exhibits instantaneous elasticity and instantaneous recovery, as shown in Fig. 1.17.

1.15.2 The linear viscous dashpot A linear viscous dashpot element is shown in Fig. 1.18, where: σ¼η

dε dt

(1.28)

and the constant η is called the coefficient of viscosity. Eq. (1.28) states that the strain rate dε/dt is proportional to the stress or, in other words, the dashpot will be deformed continuously at a constant rate when it is subjected to a step of constant stress, as shown in Fig. 1.18. On the other hand, when a step of constant strain is imposed on the dashpot, the stress will have an infinite value at the instant when the constant strain is imposed and the stress will then rapidly diminish with time to zero at t ¼ 0+ and will remain zero, as shown in Fig. 1.19. This behavior for a step change in strain is indicated mathematically by the Dirac delta function, δ(t). Thus the stress resulting from applying a step change in strain ε0 to Eq. (1.28) is indicated as follows: σ ðtÞ ¼ ηεo δðtÞ

(1.29)

30

Creep and Fatigue in Polymer Matrix Composites

1.18 Linear viscous dashpot response to constant stress.

s s0 s

0

t

t1

h e e(t) = s 0 t/h s

t

0

An infinite stress is impossible in reality. It is therefore impossible to impose instantaneously any finite deformation on the dashpot.

1.15.3 The Maxwell model The Maxwell model is a two-element model consisting of a linear spring element and a linear viscous dashpot element connected in series, as shown in Fig. 1.20. The stress-strain relations of spring and dashpot respectively are: σ ¼ Eε2 σ¼η

(1.30)

dε1 dt

1.19 The linear viscous dashpot stress relaxation response.

(1.31)

e

s e0 s(t) = he 0 d(t)

0

t

+

0 0

t

Viscoelastic constitutive modeling

31

s

e2

E

e e1

h

s s

e s0

s 0/E + s 0 t/h s 0/E s 0 t/h

s 0 /E 0

t1

t

0

t1

t

1.20 The Maxwell model creep and creep-recovery response.

Because both elements are connected in series, the total strain is: ε ¼ ε1 + ε2

(1.32)

By eliminating ε1 and ε2, the following constitutive equation for the Maxwell model is obtained: dε 1 dσ σ ¼ + dt E dt η

(1.33)

The response of the model to various stress or strain conditions can be obtained after applying integration together with appropriate initial conditions.

1.15.3.1 Creep For example, applying a constant stress σ ¼ σ o at t ¼ 0, the following response is obtained: εð t Þ ¼

σo σo + t E η

This result is shown in Fig. 1.20.

(1.34)

32

Creep and Fatigue in Polymer Matrix Composites

1.15.3.2 Recovery If the stress is removed from the Maxwell model at time t1, the elastic strain σEo in the   spring returns to zero at the instant the stress is removed, while σηo t1 represents a permanent strain that does not disappear. This result is also shown in Fig. 1.20.

1.15.3.3 Relaxation If the Maxwell model is subjected to a constant strain εo at time t ¼ 0 (Fig. 1.21), for which the initial value of stress is σ o, the stress response can be obtained by integrating Eq. (1.33) for these initial conditions with the following result: σ ðtÞ ¼ σ o exp ð
Et=ηÞ ¼ Eεo exp ð
Et=ηÞ

(1.35)

where εo is the initial strain at t ¼ 0+, and 0+ refers to the time just after application of the strain. Eq. (1.35) describes the stress relaxation phenomenon for a Maxwell model under constant strain. This phenomenon is shown in Fig. 1.21. The rate of stress change is given by the derivative of Eq. (1.35). dσ ¼
ðσ o E=ηÞ exp ð
Et=ηÞ dt

(1.36)

Thus, the initial rate of change in stress at t ¼ 0+ is dσ dt ¼
σ o E=η. If the stress were to decrease continuously at this initial rate, the relaxation equation would have the following form: σ ¼
ðσ o Et=ηÞ + σ o

(1.37)

and the stress would then reach zero at time tR ¼ η/E, which is called the relaxation time of the Maxwell model. The relaxation time characterizes one of the viscoelastic properties of the material. Actually, most of the relaxation of stress occurred before time tR because the variable factor exp(
t/tR) in Eq. (1.35) converges toward zero very rapidly for t < tR. For example, at t ¼ tR, σ(t) ¼ σ o/e ¼ 0.37σ o. Thus only 37% of the initial stress remains at t ¼ tR.

1.21 The Maxwell model relaxation response.

e

s

e0 s 0 = Ee 0

0

t

0

tR

t

Viscoelastic constitutive modeling

33

1.15.4 The Voigt or kelvin model The Voigt model is shown in Fig. 1.22, where the spring element and dashpot element are connected in parallel. The spring and dashpot have the following stress-strain relations: σ 1 ¼ Eε dε σ2 ¼ η dt Because both elements are connected in parallel, the total stress is:

(1.38) (1.39)

σ ¼ σ1 + σ2

(1.40)

Eliminating σ 1 and σ 2 among these equations yields the following constitutive equation for the Voigt model: dε E σ + ε¼ dt η η

(1.41)

1.15.4.1 Creep The solution of Eq. (1.41) may be shown to have the following form for creep under constant stress applied at σ o applied at t ¼ 0: εð t Þ ¼

σo ½1
exp ð
Et=ηÞ E

(1.42)

s s0

0

t1

s s1

t

e

s2

s 0/E h

E

s

1.22 The Voigt or Kelvin model.

tc

t1

t

34

Creep and Fatigue in Polymer Matrix Composites

As shown in Fig. 1.22, the strain described by Eq. (1.42) increases with a decreasing rate and approaches asymptotically the value of σ 0/E when t tends to infinity. The response of this model to an abruptly applied stress is that the stress is at first carried entirely by the viscous element. Under the stress the viscous element then elongates, thus transferring a greater and greater portion of the load to the elastic spring. Thus finally the entire stress is carried by the elastic element. The behavior just described is appropriately called delayed elasticity. The strain rate dε/dt for the Voigt model in creep under a constant stress σ o is found by differentiating Eq. (1.42): dε σ o ¼ exp ð
Et=ηÞ dt η Thus the initial strain rate at t ¼ 0+ is finite,
 dε σo ¼ dt t¼0 + η

(1.43)

(1.44)

and the strain rate approaches asymptotically to zero when t tends to infinity:
 dε ¼0 dt t!∞

(1.45)

If the strain were to increase at its initial rate σ o/η, it would cross the asymptotic value σ o/E at time tc ¼ η/E, called the retardation time. Actually, most of the total strain σ o/E occurs within the retardation time period, because exp(
Et/η) converges toward the asymptotic value rapidly for t < tc; at  σ 
1 σo o t ¼ tc , εðtc Þ ¼ (1.46) ¼ 0:63 1
e E E Thus, only 37% of the asymptotic strain remains to be accomplished after t ¼ tc.

1.15.4.2 Recovery If the stress is removed at time t1 the strain following stress removal can be determined by the superposition principle. The strain ε0 in the Voigt model resulting from stress σ o applied at t ¼ 0 is: ε0 ¼

σo ½1
exp ð
Et=ηÞ E

(1.47)

The strain ε00 resulting from applying a stress (
σ o) independently at time t ¼ t1 is ε00 ¼


σo ½1
exp ð
Eðt
t1 Þ=ηÞ E

(1.48)

Viscoelastic constitutive modeling

35

If the stress σ o is applied at t ¼ 0 and removed at t ¼ t1 (
σ ois added), the superposition principle yields the strain ε(t) for t > t1, during recovery εðtÞ ¼ ε0 + ε00 ¼

i σ o
Et=η h Et=η e e
1 , t > t1 E

(1.49)

As shown by Eq. (1.49) and illustrated in Fig. 1.22, when t ! ∞ , ε∞ ! 0. Some real materials show full recovery while others show only partial recovery.

1.15.4.3 Relaxation The Voigt model does not show a time-dependent relaxation. Owing to the presence of the viscous element an abrupt change in strain can be accomplished only by an infinite stress. Having achieved the change in strain either by infinite stress (if that were possible) or by slow application of strain, the stress carried by the viscous element drops to zero, but a constant stress remains in the spring. These results are obtained from the Voigt model constitutive equation (see Eq. 1.38): dε E σ + ε¼ dt η η By using the Heaviside H(t) and Dirac δ(t) functions to describe the step change in strain, εðtÞ ¼ εo H ðtÞ,

dε ¼ εo δðtÞ dt

(1.50)

Thus. E σ εo δðtÞ + εo H ðtÞ ¼ η η

(1.51)

where the first term describes the infinite stress pulse on application of the strain and the second the change in stress in the spring. A stress relaxation experiment consists of the application of a known strain to a previously unstrained sample. This strain is maintained for a period during which time the decay in the stress within the polymer is noted. Most experiments have been made by applying the strain at fast strain rates, that is to say, the straining time was very short. Several methods of applying an “instantaneous” strain to a sample have been developed. While these methods are of considerable value for basic investigations, in some relatively long-term applications it is permissible to apply the strain over a longer period. However, there is always an effect of the different strain histories upon the subsequent initial stress relaxation behavior of a polymer.

36

Creep and Fatigue in Polymer Matrix Composites

Fig. 1.23 shows schematically a typical stress/time trace. Several mathematical expressions exist for the description of stress relaxation phenomena in polymers at a given temperature. From the stress-time curves it was evident that the following expression could be used to describe satisfactorily the stress relaxation behavior of all polymers considered, over the time scale of experiments: ðΔσ ðtÞ=σ o Þ ¼ n log t + I

(1.52)

where σ o ¼ the maximum stress developed at the end of the straining phase, σ t ¼ the stress at any time t after the elongation had ceased, t ¼ time after the elongation phase was completed, Δσ(t) ¼ σ o
σ t, n ¼ the slope of the plot of Δσ(t) against log t; this is equivalent to a rate of stress relaxation I ¼ the intercept on the value of Δσ(t)/σ o when log t ¼ 0. This relationship describes the decay in stress achieved after a time t in terms of the maximum stress developed and the time after the strain phase is completed. A plot of Δσ(t)/σ o against logt produces straight lines. The characteristics of these lines, i.e., the values of n and I, depend upon the type of polymer considered at the time. The rate of stress relaxation with logarithmic time is independent of history of straining while the intercepts bear a linear relationship to the logarithm of straining time, logp, the intercept decreasing with increasing straining time. Because I plotted against logp gives a straight line, then. I ¼ m log p + C

(1.53)

where I ¼ the value of the intercept as defined in Eq. (1.52), p ¼ straining time, C ¼ the value of I for logp ¼ 0.

10

Stress

8

Strain applied at a constant rate

Strain held constant Maximum stress

6

Stress decaying due to stress relaxation

4

2

Stress increasing

2 4 Straining time "p"

1.23 Typical stress/time trace.

Stress at time t

6 t=0

8

10 Time

12

14

16

18

Viscoelastic constitutive modeling

37

This expression may now be substituted into Eq. (1.52) to give Δσ ðtÞ ¼ nlog t + m log p + C σo

(1.54)

By taking a mean value of n obtained for a given polymer and obtaining the values of m and C from the appropriate curves, Eq. (1.54) is used to predict the percentage stress relaxation that would take place after certain minutes for different straining times. This equation is significant in that it suggests that for a given polymer the decay in stress as compared to the initial stress developed is purely dependent upon the time in which the initial strain was applied. The actual amount of stress applied initially will of course be dependent upon the amount of strain. The results indicated that, although initial stress decreases with decreasing rate of application of strain, this decrease can be compensated for by increasing the deformation slightly. Neither the Maxwell nor Voigt model described previously accurately represents the behavior of most viscoelastic materials. For example, the Voigt model does not exhibit time-dependent strain on loading or unloading, nor does it describe a permanent strain after unloading. The Maxwell model shows no time-dependent recovery and does not show the decreasing strain rate under constant stress that is a characteristic of primary creep. Both models show a finite initial strain rate, whereas the apparent initial strain rate for many materials is very rapid. Thus, it becomes clear that there is a need for more complex models.

1.15.5 The three-element solid The three-element solid involves a spring and a Voigt model in series, as shown in Fig. 1.24. s

E1

h2 E2

s

1.24 The three-element solid.

38

Creep and Fatigue in Polymer Matrix Composites

To solve for the differential equation, we are making use of the constitutive relation we already know for the Voigt model. The relevant equations are as follows: Equilibrium: σ ¼ σ 1 ¼ σ 2. Kinematic: ε ¼ ε1 + ε2, which becomes, ε_ ¼ ε_ 1 + ε_ 2 . Constitutive: For the spring: σ 1 ¼ E1ε1. For the Voigt model: σ 2 ¼ E2 ε2 + η2 ε_ 2 . which can be written: ε2 ¼ E12 ½σ 2
η2 ε_ 2 . Differentiating both of the constitutive equations, we can substitute them into the kinematic equation and solve for the governing differential equation: σ+

η2 E1 E 2 E1 η2 σ_ ¼ ε+ ε_ E1 + E2 E1 + E2 E1 + E2

(1.55)

1.15.6 The four-element model At constant load the creep data for polymers can be fitted to a four-element model consisting of a Voigt unit and a Maxwell unit in series. Use of this model (Fig. 1.25) assumes linear viscoelastic behavior of the polymer under investigation. The experimental strain behavior of a polymer as a function of time has been represented by conditions in the model corresponding to certain times (Fig. 1.25). The specimens were subjected to a constant stress σ at time t0. During step 1 for a time interval (t1–t0), we observe an immediate elastic deformation of the Maxwell spring (at time t0) corresponding to diagram (a) in Fig. 1.25, followed by a slower extension of the Voigt element, (b); finally the Maxwell dashpot begins to move, corresponding to inelastic deformation, also shown in diagram (b). In step 2, the specimen is quickly returned to zero load at time t1. The Maxwell spring immediately returns to zero extension as shown by diagram (c). Creep recovery occurs during step 3 over the permitted time interval (t2–t1), corresponding to the movement of the Voigt element to its initial position as shown in (d). Only the nonrecoverable deformation of the Maxwell dashpot remains at this time. To determine the parameters and constants for this model, let us consider the following. The total deformation ε is equal to ε1 + ε2 + ε3 where ε1 represents pure elastic deformation of the Maxwell spring only, ε2 is the retarded elastic deformation of the Voigt model only, and ε3 corresponds to the viscous deformation of the Maxwell dashpot. The values of the parameters E1, E2, η2, and η3 can then be calculated from the following relationships: Total deformation: ε ¼ ε1 + ε2 + ε3

(1.56)

Elastic deformation for the Maxwell model: ε1 ¼ σ=E1

(1.57)

Viscoelastic constitutive modeling

39

s

E1

h2 E2

h3 s

Strain

s/E1 s/E2

s/E2 s/E1 t0

(s/h3)(t1–t0) Time

(a)

t1

(b)

t2 (c)

(d)

1.25 Creep-recovery behavior of a four-element model.

The retarded elastic deformation of the Voigt model only: ε2 ¼ ðσ=E2 Þ ½1
exp ð
t=τ2 Þ, where τ2 ¼ η2 =E2

(1.58)

dε2 E2 σ + ε2 ¼ η2 dt η2

(1.59)

40

Creep and Fatigue in Polymer Matrix Composites

The viscous deformation of the Maxwell dashpot: ε3 ¼ ðσ=η3 Þt

(1.60)

Eliminating ε1,ε2, and ε3 among these equations yields the following constitutive equation for the four-element model:
 η η η dσ η2 η3 d 2 σ dε η η d2 ε + σ+ 3 + 3 + 2 ¼ η3 + 2 3 2 (1.61) 2 dt E1 E2 E2 dt E1 E2 dt E2 dt Thus, the creep behavior may be found to be as follows: εð t Þ ¼

σ σ σ + ½1
exp ð
E2 t=η2 Þ + t E1 E 2 η3

Differentiating Eq. (1.62) yields the creep rate as follows:
 dε σ σ E2 t ¼ + exp
dt η3 η2 η2 Thus the creep rate starts at t ¼ 0+ with a finite value. 
 dε 1 1 ¼ + σ ¼ tan α dt t¼0 + η2 η3 and approaches asymptotically to the value.  dε σ ¼ ¼ tan β dt t!∞ η3

(1.62)

(1.63)

(1.64)

(1.65)

It may also be observed that OA ¼ σ/E1, and AA0 ¼ σ/E2. Thus in theory the material constants E1, E2, η2, η3 may be determined from a creep experiment by measuring α, β, OA and AA0 as in Fig. 1.26.

1.15.7 The generalized Maxwell model This model consists of many Maxwell models either in series or in parallel (Fig. 1.27). When several Maxwell models are connected in series, the constitutive equation is given by N N X dε dσ X 1 1 ¼ +σ dt dt i¼1 Ei η i¼1 i

(1.66)

The response of this model is not much different from the earlier mentioned Maxwell model and hence is not significant.

Viscoelastic constitutive modeling

41

e B a

A¢

BC = OA e2 b

C

e3

A

D e1

0

t1

t

1.26 Four-element model parameters determination.

When several Maxwell models are connected in parallel, the resulting model can represent instantaneous elasticity, viscous flow, creep with various retardation times, and relaxation with various relaxation times. However, this model is more convenient when the strain history (stress relaxation) is known. Hence, the response of this model to a constant strain is given by:
 t σ ðtÞ ¼ εo Ei exp
i τr i¼1 N X

(1.67)

1.15.8 The generalized Voigt or kelvin model This model consists of many Voigt or Kelvin models in parallel (Fig. 1.28) or in series (Fig. 1.29). When several Kelvin models are connected in parallel, the constitutive equation is given as. σ¼ε

N X

Ei +

i¼1

N dε X η dt i¼1 i

(1.68)

Again, the response of this model is no different from the earlier mentioned Kelvin model and hence is not significant. When several Kelvin models are connected in series, the resulting constitutive equation is given by ε¼

! 1 σ E + dηi =dt i¼1 i

N X

(1.69)

42 Creep and Fatigue in Polymer Matrix Composites

1.27 The generalized Maxwell model: (a) connection in series; (b) in parallel.

Viscoelastic constitutive modeling

43

1.28 The generalized Voigt or Kelvin model (parallel connection).

This model is more convenient when the stress history is known. The creep response of this model is given by N X








t εð t Þ ¼ σ o Di 1
exp i τc i¼1

 (1.70)

where Di is the creep compliance.

1.16

Nonlinear viscoelastic behavior of polymers

1.16.1 The limits of linearity Regarding the time dependent response, the limits of linearity and the initiation of the nonlinear viscoelastic behavior of a given material depend on the following conditions that must be met in terms of the stress-strain relation. First at any fixed time interval (isochronous) following initiation of a loading history, the strain should be proportional to the stress. This condition (stress-strain linearity) amounts to demanding that the response of the material to a sum of inputs be equal to the sum of the responses. Second, the strain generated by (or recovered from) a load currently applied (or removed) should be independent from any previously applied load. This second condition is the property of time invariance, an assertion that the response to a given input does not change with the chronology of application of input. This implies that the creep compliance is a function of the time lag (t
τ) and is independent of any other function of t and τ. When at least one of these two necessary and sufficient conditions is violated, the general form of Boltzmann’s superposition principle is adequate to describe neither creep nor relaxation of viscoelastic materials and the models described in previous paragraphs are no longer valid and applicable. It follows that nonlinear viscoelasticity is observed when one of the two conditions mentioned previously is denied by the data.

44

Creep and Fatigue in Polymer Matrix Composites

1.29 The generalized Voigt or Kelvin model (connection in series).

For instance, many experimental investigations have shown that the kernel of Boltzmann integral may in the form of creep compliance be found to increase with stress, or alternatively that the rate of recovery under zero load is different than the rate of the preceding creep under constant load.

Viscoelastic constitutive modeling

45

As a result, due to the violation of the previous hypotheses, a different approach must be employed for the description of the nonlinear viscoelastic behavior as the equivalent linear principles no longer exist and will lead to an underestimation of the measured magnitudes of interest. In general, the stress-strain relationship can be alternatively expressed in two forms. The first one is the differential form. This type has been widely used in the linear viscoelastic behavior as the mathematical formulation occurring from linearity is simple in its application. The latter is the integral form which, although beneficial in its general description, is difficult to be applied in the mathematical manipulation. Most of the models describing the nonlinear viscoelastic behavior appearing in the literature are based on the integral form and are reviewed further in the next paragraphs.

1.16.2 Multiple integral representations 1.16.2.1 Green, Rivlin and Spencer model This model was first proposed by Green et al. (1959) and Green and Rivlin (1957, 1960) for the description of the time-dependent mechanical behavior of polymeric systems, but the fundamental approximation is very appealing because it is not limited to a particular material or class of materials. According to the proposed expression, the time-dependent strain response can be given as: ðt εðtÞ ¼ D1 ðt
τ1 Þ

dσ ðτ1 Þ dτ1 + dτ1

0

ðt ðt +

D2 ðt
τ1 , t
τ2 Þ

dσ ðτ1 Þ dσ ðτ2 Þ dτ1 dτ2 + dτ1 dτ2

0 0 ðt ðt ðt

D3 ðt
τ1 , t
τ2 , t
τ3 Þ

+

(1.71)

dσ ðτ1 Þ dσ ðτ2 Þ dσ ðτ3 Þ dτ1 dτ2 dτ3 + … dτ1 dτ2 dτ3

0 0 0

where D1(t), D2(t), … Dn(t), are kernels of time. The first integral of Eq. (1.71) describes the linear viscoelastic behavior, defined by the Boltzmann theory, and the second and higher order integrals are representations of both nonlinearity magnitude and interaction nonlinearity, the latter implying an interaction effect between, e.g., the stress increments at times τ1 and τ2. The Kernel function D1(t) is expressed in terms of a single time parameter t. As a result, D1(t) versus t may be illustrated by a curve in a diagram D1-t as shown in Fig. 1.30. However, the higher order nonlinear kernel functions such as D2(t) and D3(t) require more than one parameter for their descriptions.

46

Creep and Fatigue in Polymer Matrix Composites

D1

D1(t)

t

1.30 A graphic illustration of the linear kernel function D1(t) vs. time.

For example, the second-order Kernel function D2(t, t – ξ1) is described in terms of two time parameters, t and t – ξ1. The variation of D2(t, t – ξ1) with these two time parameters can be illustrated by a surface A, as shown in Fig. 1.31, with coordinates D2, t and t. In this figure the line B in surface A describes the ordinate D2(t,t) vs. time t when the two time parameters are equal. Hence, line B lies in a plane that bisects the angle between the coordinates t and t. Line C in surface A of Fig. 1.31 represents D2(t, t – ξ1) vs. time t, when the two time parameters differ by a particular value ξ1. Thus, line C lies in a plane parallel to the plane of line B. A set of lines like C in parallel planes may be found by using a set of values of ξ1, thus defining surface A. This suggests an experimental method of determining D2 (t, t – ξ1). A set of creep experiments may be performed using two stresses applied at different times, one at t ¼ 0 and the other at t ¼ t – ξ1 where ξ1 is different for each experiment of the set. Thus, the results of such experiments all using the same stresses but suitably chosen values of ξ1 will yield information from which D2(t, t – ξ1) may be obtained. 1.31 A graphic illustration of the second-order kernel functions D2(t) vs. two time parameters.

D2 (t,t)

D2 (t,t–x1)

B

C

D2 (t,t–x1)

t 45 t

x1

Viscoelastic constitutive modeling

47

Because the Kernel functions are considered symmetrical with respect to their time parameters, D2 ðt, t
ξ1 Þ ¼ D2 ðt
ξ1 , tÞ

(1.72)

Thus surface A is symmetrical with respect to the plane defined by line B. A third-order kernel function does not yield to meaningful pictorial representation. The accuracy of the description of the nonlinear viscoelastic behavior increases by taking into account as many terms as possible in the expression of Eq. (1.71). A further investigation of Eq. (1.71) in combination with the appropriate selection of compliance functions has proved that the model can lead to an adequate description of nonlinearity in the viscoelastic behavior of polymers (Smart and Williams, 1972a, 1972b). Ward and Onat (1963) have also shown that the accuracy can be preserved successfully using the first and the third term of the equation. Despite its conceptual generality the previously described model is rather complicated to be applied, mainly due to the large number of tests needed, not only for the determination of the kernels and integrals but also for the numerical instability that arises in the fitting procedure of the experimental data in the integral functions. Moreover, higher-order kernel functions have no physical meaning.

1.16.2.2 Pipkin and Rogers model (Nonlinear superposition theory) An alternative to the Green-Rivlin approach has been formulated by Pipkin and Rogers (1968). This model is often referred to as the nonlinear superposition theory (NLST) and generalizes the formulation of Green and Rivlin. It was applied on the Findley and Lai (1967) experimental results leading to successful agreement. According to this model, the viscoelastic response of a polymer can be given in terms of a series of integrals as follows: ðt εð t Þ ¼

D 1 ðt
τ 1 , σ ðτ 1 ÞÞ

dσ ðτ1 Þ dτ1 + dτ1

0

ðt ðt D2 ðt
τ1 , σ ðτ1 Þ, t
τ2 , σ ðτ2 ÞÞ

+

dσ ðτ1 Þ dσ ðτ2 Þ dτ2 + dτ1 dτ1

0 0

ðt ðt ðt D2 ðt
τ1 , σ ðτ1 Þ, t
τ2 , σ ðτ2 Þ, t
τ3 , σ ðτ3 ÞÞ

+

dσ ðτ1 Þ dσ ðτ2 Þ dτ2 + … dτ1 dτ1

0 0 0

(1.73) It is clear that Eq. (1.73) is nonlinear even in its first approximation, and thus the additivity of incremental stress effects in the Boltzmann superposition sense is preserved.

48

Creep and Fatigue in Polymer Matrix Composites

1.16.3 Single integral representations The concept of mapping the nonlinear viscoelastic response through single integrals is based in principle on the nonlinearization of stress and strain measure through a general form of the following equation as: ð d φ½εðtÞ ¼ Dðt
τÞ ψ ½σ ðtÞdτ dt

(1.74)

All representations used in the literature are alternatives or special cases of this general formulation.

1.16.3.1 Leaderman’s model Leaderman (1943) proposed a generalization of Boltzmann’s linear theory in order to include the nonlinear effects in the time-dependent response, assuming that the strain response of polymers can be given as: ð d εðtÞ ¼ Dðt
τÞ ψ ½σ ðtÞdτ dt

(1.75)

where ψ[σ(t)] is a stress-dependent function.

1.16.3.2 Rabotnov’s model An alternative expression proposed for the description of the stress in time under a strain history was formulated in terms of strain by Rabotnov (1948) as: ð φ½εðtÞ ¼ Dðt
τÞ

dσ ðtÞ dτ dτ

(1.76)

The model allows for a stress or pressure deformable time measure.

1.16.3.3 Brueller’s model In an extensive research effort of the nonlinear viscoelastic behavior of PVC and PMMA as well as glass and carbon fabric reinforced thermoplastics, Brueller (1987, 1993, 1996) developed and applied the following model for the description of the time-dependent behavior under creep loading: ðt dσ εðtÞ ¼ ga D0 σ + gb ΔDðt
τÞ dτ dτ

(1.77)

0

The nonlinearity is controlled by two nonlinearity functions, namely ga, gb, which solely depend on the applied stress level. The function ga attributes the nonlinearity

Viscoelastic constitutive modeling

49

in the instantaneous response while gb attributes the nonlinearity in transient creep. The nonlinearity functions ga and gb can be determined using a mixed iteration procedure based on numerical methods.

1.16.3.4 Schapery’s constitutive equation According to Schapery (1966a, 1966b, 1968, 1969), for the case of uniaxial loading, under given hygrothermal conditions when stress σ is defined as the independent variable, the constitutive equation that describes the time-dependent strain can be formulated as follows: ðt dðg2 σ ðτÞÞ εðtÞ ¼ g0 D0 σ + g1 ΔDðψ
ψ 0 Þ dτ dτ

(1.78)

0

where D0 is the initial, time-independent component of the compliance and ΔD(ψ) is the transient, time-dependent component of compliance, while ψ and ψ 0 are the so-called reduced times defined by: ðt

ðτ 0 dt0 0 dt ψ ¼ ψ ðt Þ ¼ ,ψ ¼ ψ ðτÞ ¼ aσ aσ 0

0

and g0, g1, g2 and aσ are stress-dependent nonlinear material parameters. Each of these parameters defines a nonlinear effect on the compliance of the material. The factor g0 defines stress and temperature effects on the instantaneous elastic compliance and is a measure of state-dependent reduction (or increase) in stiffness. Transient compliance factor g1 has similar meaning, operating on the creep compliance component. The factor g2 accounts for the influence of load rate on creep and depends on stress and temperature. The factor aσ is a time scale shift factor. This factor is in general a stress- and temperature-dependent function and modifies the viscoelastic response as a function of temperature and stress. Mathematically, aσ shifts the creep data parallel to the time axis relative to a master curve for creep strain versus time. Schapery’s constitutive equation also includes the linear case where g0 ¼ g1 ¼ g2 ¼ aσ ¼ 1 and leads to the Boltzmann’s superposition principle. Schapery’s model has been used extensively for the description of polymers and polymer matrix composites (Zhang and Xiang, 1992; Mohan and Adams, 1985). Moreover, according to comparative studies with other nonlinear models, its advantage has been proven in the prediction of the experimental data (Smart and Williams, 1972a, 1972b; Partom and Schanin, 1983). From a numerical point of view, Schapery’s constitutive equation can be modified in order to describe multistep loading. Numerical algorithms have been developed by Tuttle and Brinson (1986), Tuttle et al. (1995), Dillard et al. (1987), and Haj-Ali and Muliana (2008).

50

Creep and Fatigue in Polymer Matrix Composites

1.16.3.5 Determination of the nonlinear parameters Experimentally, the estimation of the nonlinear parameters demands an accurate and precise method, as curve-fitting techniques to the experimental data usually lead to wrong values or values with no physical meaning due to the mutual dependence of the parameters. Howard and Hollaway (1987), taking into account the hyperbolic dependence of the nonlinear parameters of Schapery’s constitutive equation, proposed a model for their dependence on applied stress level in glass-polyester composites. Based on the experimental results on creep-recovery loading on glass-polyester composites, they concluded on the following semiempirical model for the nonlinear parameters of Schapery’s model: for :




σ  σ c , g0 ¼ g1 ¼ aσ ¼ 1, g2 ¼ for : σ  σ c , g0 ¼ g1 ¼ 1 , aσ ¼

"


bsinh ðσ=bÞ σ




#1=n bsinh ðσ=bÞ
1 bsin ðσ c =bÞ σ σc

(1.79)

(1.80)

where σ c is the stress threshold from linear to nonlinear viscoelastic behavior of the material. A method description as well as a generic function for the accurate prediction of the nonlinear parameters was developed by Papanicolaou et al. (1999a, 1999b) and Zaoutsos et al. (1998). The method is capable of analytically evaluating g0 and g1, without having any dependence from the material compliance, by using only limiting values of the creep creep-recovery test, so that any inaccuracies and/or instabilities introduced by multiple numerical treatments are avoided. Then, assuming that the compliance follows a power law, the value of aσ can be easily evaluated with enough accuracy using a oneparameter curve fitting of the recovery data. The nonlinear parameter g2 can also be analytically estimated from the respective creep-recovery tests. Additionally, a prediction of the stress dependence of the nonlinear viscoelastic behavior of continuous fiber reinforced polymeric systems was achieved using the following generic function: 8 1 for σiσ c > > > 1
k <
 + k for σhσ c (1.81) G¼ x
λ > > x
λ > :1 + e 1
x 1
x where G is the parameter of interest, x ¼ σσu , and λ ¼ σσuc , while σ u is the ultimate tensile strength of the material, σ c is the stress threshold from linear to nonlinear behavior, and k is the maximum value of the parameter of interest when σ tends to σ u. This model resulted in the estimation of all four nonlinearity parameters that characterize the respective nonlinear viscoelastic behavior. All variables included have a clear physical meaning and all can be measured through simple experiments.

Viscoelastic constitutive modeling

1.17

51

Applications to different materials

Application of the Papanicolaou et al. (1999a, 1999b) model to cross-ply carbon/ epoxy composites resulted in good agreement between the experimental results and the respective model predictions. Additionally, the method and the model were tested successfully at different loading times (Figs. 1.32 and 1.33).

1.32 Comparison between experimental values and the proposed model predictions of the nonlinear parameters go, and g1 as a function of the applied stress at different loading times.

52

Creep and Fatigue in Polymer Matrix Composites

1.33 Comparison between experimental values and the proposed model predictions of the stress shift factor aσ and the nonlinear parameter g2 as a function of the applied stress at different loading times.

The extension of the model also included the characterization of carbon/epoxy composites at different fiber orientations (Zaoutsos and Papanicolaou, 1999; Papanicolaou et al., 2004). A good agreement between the experimental data and the predicted values of the model was observed, as is shown in Figs. 1.34 and 1.35. The same model was also applied and compared successfully to other semianalytical approximations for polycarbonate (Wing et al., 1995; Papanicolaou et al., 2005), as can be seen in Figs. 1.36 and 1.37.

Viscoelastic constitutive modeling

0 90.0

53

0.80

0 85.0 0.60

0 80.0

go

0 75.0

3.00

0.40

0 70.0

0 65.0

3.20

2.80

0.20

2.60

0 0 60.0 0.0

2.40

3.00

2.20

2.00

2.00 1.80

1.00

1.60 1.40

0.00

0

90.0

0.80

0 85.0

0.60

0 80.0

Fiber o

rienta

0 75.0

0 70.0

tion ( D

ate /Ultim

0.40

egree

0 65.0

s)

0.20 0 0 60.0 0.0

1.20 1.00

Tens

tress

ed S

Appli

ess ile Str

7.50

0

90.0

10.0

1.00

0 85.0

0 80.0

0.80 0

75.0

6.00

0.40

0 70.0

0 65.0

0

g1 8 .00

5.50

0.20

5.00

0 0 60.0 0.0

4.50 4.00

6.00

3.50

4.00

3.00 2.50

2.00

2.00

0.00

0 90.0

7.00 6.50

0.60

0 85.0

1.00

Fiber o

0 80.0

0.80

rienta

tion (

0 75.0

0.60 0.40

0

70.0

Degr

ees)

0.20

0

65.0

0 0 60.0 0.0

imate

ss/Ult

d stre

Applie

1.50 1.00

ess

e Str Tensil

1.34 Characteristic surfaces of the parameter g0 g1 as a function of fiber orientation and the ratio of the applied stress to ultimate tensile stress for carbon/epoxy composite.

Reported results on particulate composites (Papanicolaou et al., 2009) were also very encouraging concerning the analytical model of the description of nonlinear parameters of the Schapery’s constitutive equation, as can be seen in Figs. 1.38 and 1.39.

54

Creep and Fatigue in Polymer Matrix Composites

1.35 Characteristic surfaces of the stress shift factor aσ and the parameter g2 as a function of fiber orientation and the ratio of the applied stress to ultimate tensile stress for carbon/epoxy composite.

Viscoelastic constitutive modeling

55

1.36 Comparison between experimental values, numerical values, and proposed model prediction of the nonlinear parameters g0 and g1 as a function of the applied stress level for the 9034 polycarbonate.

g0 experimental values

1.6

g0 numerical values (Wing et al.) g0 proposed model

g0

1.4 1.2 1.0 0.8 0

10

20

30

40

50

60

Applied stress (MPa) 3.0

g1 experimental values g1 numerical values (Wing et al.) g1 proposed model

2.5

g1

2.0 1.5 1.0 0.5 0

40 10 20 30 Applied stress (MPa)

50

60

1.37 Comparison between experimental values, numerical values, and proposed model prediction of the stress shift factor aσ as a function of the applied stress level for the 9034 polycarbonate.

1.2 1.0

as

0.8 0.6 0.4

as experimental values as numerical values (Wing et al.)

0.2

as proposed model

0.0 0

10

40 20 30 Applied stress (MPa)

50

60

56

Creep and Fatigue in Polymer Matrix Composites

1.15 1.10

g0

1.05 1.00 0.95 0.90 0.85 0

2

4

6

8

10

8

10

Applied stress (MPa) 2.0 1.8 1.6

g1

1.4 1.2 1.0 0.8 0.6 0

2

4

6

Applied stress (MPa)

1.38 Variation of the nonlinear parameters g0 and g1 as a function of the applied stress for aluminum/epoxy particulate composite.

Viscoelastic constitutive modeling

57

1.39 Variation of the stress shift factor aσ and the nonlinear parameter g2 as a function of the applied stress for aluminum/epoxy particulate composite.

1.20 1.15 1.10

as

1.05 1.00 0.95 0.90 0.85 0.80 0

2

0

2

4 6 Applied stress (MPa)

8

10

8

10

5 4 3

g2

2 1 0 –1 –2 4 6 Applied stress (MPa)

References Brinson, H.F., Morris, D.H., Yeow, Y.T., 1978. A new experimental method for the accelerated characterization of composite materials. In: 6th International Conference on Experimental Stress Analysis, Munich, September 18–22. Brueller, O.S., 1987. On the nonlinear characterization of the long term behaviour of polymeric materials. Polym. Eng. Sci. 27, 144–148. Brueller, O.S., 1993. Predicting the behaviour of nonlinear viscoelastic materials under spring loading. Polym. Eng. Sci. 33, 97–99. Brueller, O.S., 1996. creep and failure of fabric reinforced thermoplastics. In: Progress in Durability Analysis of Composite Systems, pp. 39–44.

58

Creep and Fatigue in Polymer Matrix Composites

Dillard, D.A., Straight, M.R., Brinson, H.F., 1987. The nonlinear viscoelastic characterization of graphite/epoxy composites. Polym. Eng. Sci. 27, 116–123. Findley, W.N., Lai, J.S.Y., 1967. A modified superposition principle applied to creep of nonlinear viscoelastic material under abrupt changes in state of combined stress. Trans. Soc. Rheol. 11, 361. Green, A.E., Rivlin, R.S., 1957. The Mechanics of Nonlinear Materials with Memory, Part I, Archives of Rational Mechanics Analysis. Vol. 1. pp. 1–21. Green, A.E., Rivlin, R.S., 1960. The mechanics of nonlinear materials with memory part III. Arch. Ration. Mech. Anal. 4, 387. Green, A.E., Rivlin, R.S., Spencer, A.J.M., 1959. The mechanics of nonlinear materials with memory. Part II Arch. Ration. Mech. Anal. 3, 82. Haj-Ali, R., Muliana, A., 2008. A micro-to-meso sublaminate model for the viscoelastic analysis of thick-section multi-layered FRP composite structures. Mech. Time-Depend. Mater. 12, 69–93. Howard, M., Hollaway, L., 1987. The characterization of the nonlinear viscoelastic properties of a randomly orientated fibre/matrix composite. Composites 18, 317–323. Knauss, W.G., Emri, I.J., 1981. Non-linear viscoelasticity based on free volume consideration. Comput. Struct. 13, 123–128. Leaderman, H., 1943. Elastic and Creep Properties of Filamentous Materials and Other High Polymers. The Textile Foundation, Washighton. Mohan, R., Adams, D.F., 1985. Nonlinear creep-recovery response of a polymer matrix and its composites. Exp. Mech. 25, 262–271. Papanicolaou, G.C., Zaoutsos, S.P., Cardon, A.H., 1999a. Further development of a data reduction method for the nonlinear viscoelastic characterization of FRP’s. Compos. A: Appl. Sci. Manuf. 30, 838–849. Papanicolaou, G.C., Zaoutsos, S.P., Cardon, A.H., 1999b. Prediction of the non-linear viscoelastic behaviour of polymer matrix composites. Compos. Sci. Technol. 59, 1311–1319. Papanicolaou, G.C., Zaoutsos, S.P., Kontou, E., 2004. Fiber Orientation Dependence of Continuous Carbon/Epoxy Composites Nonlinear Viscoelastic Behaviour Composites Science and Technology. Vol. 64, pp. 2535–2545. Papanicolaou, G.C., Zaoutsos, S.P., Kosmidou Th, V., 2005. Describing νonlinearities in the mechanical viscoelastic behaviour of polymers and polymer matrix composites. In: Modern Problems of Deformable Bodies Mechanics.vol. 1, pp. 201–210 Collection of papers, Yerevan. Papanicolaou, G.C., Xepapadaki, A.G., Pavlopoulou, S., Zaoutsos, S.P., 2009. On the investigation of the stress threshold from linear to nonlinear viscoelastic behaviour of polymermatrix particulate composites. Mech. Time-Depend. Mater. 13, 261–274. Partom, Y., Schanin, I., 1983. Modelling nonlinear viscoelastic response. Polym. Eng. Sci. 23, 849–859. Pipkin, A.C., Rogers, T.G., 1968. A nonlinear integral representation for viscoelastic behaviour. J. Mech. Phys. Solids 16, 59–72. Rabotnov, Y.N., 1948. Some Problems of Creep Theory. vol. 10. Vestik Mosk. Univ. Matem., Mekh., pp. 81–91. Schapery, R.A., 1966a. An engineering theory of nonlinear viscoelasticity with applications. Int. J. Solids Struct. 2. Schapery, R.A., 1966b. A theory of nonlinear thermoviscoelasticity based on irreversible thermodynamics. In: Proceedings of the 5th U.S. National Congress in Applied Mechanics, ASME, pp. 511–530.

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59

Schapery, R.A., 1968. On a thermodynamic constitutive theory and its application to various nonlinear materials. In: Proceedings of UITAM Symposium on Thermoinelasticity, East Kilbriole, pp. 259–284. Schapery, R.A., 1969. On the characterization of nonlinear viscoelastic materials. Polym. Eng. Sci. 9, 295–310. Smart, J., Williams, G.C., 1972a. A power law model for the multiple-integral theory of nonlinear viscoelasticity. J. Mech. Phys. Solids 20, 325–335. Smart, J., Williams, J.G., 1972b. A comparison of single integral non-linear viscoelasticity theories. J. Mech. Phys. Solids 20, 313–324. Tuttle, M.E., Brinson, H.F., 1986. Prediction of the long term creep compliance of general composite laminates. Exp. Mech. 26, 89–102. Tuttle, M.E., Pasricha, A., Emery, A.F., 1995. The nonlinear viscoelastic-viscoplastic behaviour of IM7/5260 composites subjected to cyclic loading. J. Compos. Mater. 29, 2025–2046. Ward, I.M., Onat, E.T., 1963. Nonlinear mechanical behaviour of oriented polypropylene. J. Mech. Phys. Solids 11, 217–229. Wenbo, L., Ting-Qing, Y., Qunli, A., 2001. Time-temperature-stress equivalence and its application to nonlinear viscoelastic materials. Acta Mech. Solida Sin. 14, 195–199. Wing, G., Pasricha, A., Tuttle, M.A., Kumar, V., 1995. Time dependent response of polycarbonate and microcellular polycarbonate. Polym. Eng. Sci. 35, 673–679. Yen, S.C., Williamson, F.L., 1990. Accelerated characterization of creep response of an off-axis composite material. Compos. Sci. Technol. 38, 103–118. Zaoutsos, S.P., Papanicolaou, G.C., 1999. The effect of fiber orientation on the nonlinear viscoelastic behaviour of continuous fiber polymer composites. In: Proceedings of the 4rd International Conference on Progress in Durability Analysis of Composite Systems, Vrije Universiteit Brussel, Brussels, Belgium. Zaoutsos, S.P., Papanicolaou, G.C., Cardon, A.H., 1998. On the nonlinear viscoelastic behavior of polymer matrix composites. Compos. Sci. Technol. 58, 883–886. Zhang, S.Y., Xiang, X.Y., 1992. Creep characterization of a fiber reinforced plastic material. J. Reinf. Plast. Compos. 11, 1187–1194.

Further reading Cessna, L.C., 1971. Stress-time superposition of creep data for polypropylene and coupled glass-reinforced polypropylene. Polym. Eng. Sci. 11, 211–219. Findley, W.N., Lai, J.S., Onaran, K., 1976. Creep and Relaxation of Nonlinear Viscoelastic Materials. North-Holland Publishing Company, New York, NY. Flugge, W., 1967. Viscoelasticity. Blaisdell, Waltham, MA. Lai, J., Bakker, A., 1995. Analysis of the non-linear creep of high-density polypropylene. Polymer 36, 93–99. Ma, C.C.M., Tai, N.H., Wu, S.H., Lin, S.H., Wu, J.F., Lin, J.M., 1997. Creep behavior of carbon-fiber-reinforced polyetheretherketone (PEEK) laminated composites. Compos. Part B 28, 407–417. Park, S.W., Kim, Y.R., 2001. Fitting Prony-series viscoelastic models with power-law Presmoothing. J. Mater. Civ. Eng. 26–32. Yannas, I.V., 1974. Nonlinear viscoelasticity of solid polymers in uniaxial tensile loading. J. Polym. Sci. 9, 163–190.

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Time-temperature-age superposition principle for predicting long-term response of linear viscoelastic materials

2

E.J. Barbero West Virginia University, Morgantown, WV, United States

2.1

Correlation of short-term data

Material characterization provides the information needed to support structural analysis and design. The first step in a materials characterization program is to regress experimental data to model equations in order to represent such data. For this purpose, consider a creep test where a constant stress σ 0 is applied at some time te. Denoting by λ the time elapsed since application of the load, the compliance D(λ) may be represented by one of a number of possible equations that fit the strain vs. time data. For example, the standard linear solid (SLS) model is described by: h i DðλÞ ¼ D0 + D1 1  eλ=τ

(2.1)

where the retardation time τ is the time it takes for an exponential e–λ/τ to decay to 100  e1 ¼ 36.8% of its original value. The larger the τ, the longer it takes for the relaxation modulus E(λ) to decay. Since creep tests are easier to perform than relaxation tests, the compliance D(λ) is often measured instead of the relaxation modulus. For a linear, unaging material, they are related by: Eð λ Þ ¼ L

1



1 2 s L½DðλÞ

 (2.2)

where L[], L1[], s, denote the Laplace transform, the inverse Laplace transform, and the Laplace variable, respectively (Barbero, 2007, chapter 7). Momentary data (to be defined shortly) can be transformed as in Eq. (2.2) but long-term data cannot, because aging invalidates Boltzmann’s superposition principle (Barbero, 2007) even if the material is linear (i.e., when the response does not depend on stress). A schematic of the SLS model using spring and dashpot elements is shown in (Barbero, 2007, Figure 7.1.c). Note that, in the context of linear viscoelasticity (Chapter 1), the compliance is not a function of stress. Additionally, a linear viscoelastic material, for which Boltzmann’s superposition applies, must have a constitutive Creep and Fatigue in Polymer Matrix Composites. https://doi.org/10.1016/B978-0-08-102601-4.00002-3 © Woodhead Publishing Limited, 2011.

62

Creep and Fatigue in Polymer Matrix Composites

model that is not a function of the absolute time t but rather is a function of the time λ elapsed since application of the load (Barbero, 2007, Figure 7.3). Such material is said to be unaging (Barbero, 2007; Creus, 1986; Struik, 1978). However, all polymers age at temperatures below their glass transition temperature Tg. Thus all the matrix-dominated properties of polymer-matrix composites are subject to aging for in-service temperature conditions (Sullivan, 1990; Gates and Feldman, 1995). A methodology to deal effectively with the aging problem is presented in Sections 2.3 and 2.4. Furthermore, the constitutive response of all polymers is a function of temperature. Therefore, a methodology to characterize and model temperature effects is presented in Sections 2.2 and 2.6. Eq. (2.1) is a very simple model that may not fit the data well. To obtain a better fit, that is, a better regression between the model equation and the data, more springdashpot elements can be added in series, as follows: D ð λÞ ¼ D 0 +

n X

h i Dj 1  eλ=τ j

(2.3)

j¼1

When the number of elements is very large, one can replace the summation by an integral and the compliance coefficients D0, Dj, by a compliance spectrum Δ(τ) as follows: ∞ ð

D ðλÞ ¼

h i ΔðτÞ 1  eλ=τ dτ

(2.4)

0

While Eqs. (2.3), (2.4), can fit virtually any material compliance provided a large number of terms is used, the generalized Kelvin model is more efficient with only four parameters: h i m DðλÞ ¼ D0 + D01 1  eðλ=τÞ (2.5) In order to reduce the time to complete the material characterization, short-term tests are used. In this case, it may be difficult to regress Eq. (2.5) to the data because shortterm material behavior may be impossible to distinguish from a three-parameter power law (Eq. 2.6). Expanding Eq. (2.5) with a Taylor power series results in: DðλÞ ¼ D0 + D01 ðλ=τÞm ½1  ðλ=τÞm + … DðλÞ  D0 + D1 λm

(2.6)

where D1 ¼ D10 /τ. The power law (2.6) has the advantage that it becomes a straight line with slope m in a log-log plot, as follows: log ðDðλÞ  D0 Þ ¼ log D1 + m log λ

(2.7)

and thus it is very easy to regress the data by performing a linear regression in log-log scale. Finally, the Kohlrausch model (Struik, 1978, (10)): DðλÞ ¼ D0 eðλ=τÞ

m

(2.8)

Time-temperature-age superposition principle

63

has been shown to fit the compliance of a broad variety of materials (Struik, 1978, Figure 34). The parameters D0, τ, shift the curve in the vertical and horizontal directions, respectively, and the parameter m  1 stretches the exponential in time. Since all the data is manipulated in log-log scale, it is best to sample data uniformly in log time, not uniformly in time. Uniform time-sampling yields data points in log scale that are closely packed for long times. Then, regression algorithms used to fit model equations tend to bias the regression towards longer times. However, most automatic data sampling equipment sample uniformly in time. A simple MATLAB algorithm can be used to pick data uniformly spaced in log time from a set of data uniformly spaced in time (Barbero, n.d.), as follows: % log sampling, user picks the initial time and increment time. % xi (:,1) time (equally spaced in time). % xi (:,2) compliance. ndp ¼ length (xi (:, 1)); %# data points read tf ¼ xi (ndp, 1); %final time logti ¼  1; %log of initial time to sample, user choice del_logti ¼ 0.1; %log time interval to sample, user choice logt ¼ [logti: del_logti: %equally spaced in log scale log10(tf )]; tr ¼ 10.^ logt; %back to time scale nr ¼ length (tr); %number of newly sampled data if tr(nr) ¼ tf; tr ¼ [tr, tf]; %add the final time nr ¼ length(tr); %number of newly sampled data end rcount ¼ 1; for i ¼ 1:ndp. if xi (i, 1) > ¼ tr(rcount) %say 10^–0.1 xo (rcount,:) ¼ xi(i,:); %copy data equally spaced in log(time) rcount ¼ rcount + 1; end if rcount > nr, break, end; end

2.2

Time-temperature superposition

In this section, the classical superposition method is described wherein the acceleration factor is temperature. The notion of momentary data is introduced along with a full description of the technique used to obtain the time-temperature momentary master curve and temperature shift factor plot. To illustrate the principle of superposition, let D0 ¼ 1 GPa1, D1 ¼ 9 GPa1, and consider two temperaturesa T > Tr for which the retardation times (in Eq. 2.9) are a

Or two ages te < te,r.

64

Creep and Fatigue in Polymer Matrix Composites 12

10

Compliance

8

6

t t

4

2

0

0

100

200

300

400

500

Time

2.1 Compliance of a material described by Eq. (2.5) with D0 ¼ 1 GPa1, D1 ¼ 9 GPa1, and two values of the retardation time τ ¼ 100 s and τ ¼ 10 s.

τ ¼ 10 s and τr ¼ 100 s, respectively. The compliance vs. time D(λ;T) and D(λ;Tr), are shown in Fig. 2.1. Note that creep strain develops slower for the larger retardation time. The same curves are shown in double logarithmic scale in Fig. 2.2. The creep data at temperatures T and Tr are described by SLS models like Eq. (2.1), as follows:   Dðλ; T Þ ¼ D0 + D1 1  eλ=τ   Dðλ; Tr Þ ¼ D0r + D1r 1  eλ=τr

(2.9)

If one can shift curve T onto curve Tr (Fig. 2.2) and they superpose nicely, it is said that the curves are superposable. To shift the curve T horizontally, one plots the creep values log D(λ;T) vs. log aTλ instead of log λ, where aT(T) is the horizontal shift factor. Because log aTλ ¼ log λ + log aT, then aT > 1 shifts the curve T to the right onto the master curve Tr, the latter having aT ¼ 1 by definition.b To shift the curve T vertically, one divides the values of D(λ; T) by a vertical shift factor b(T). If the curves are superposable, D(λ; T) at time λ is equal to bTD(aTλ; Tr) at time aT λ (see Fig. 2.1). Mathematically, Dðλ; T Þ ¼ bT DðaT λ; Tr Þ

b

(2.10)

Note that the entire formulation could be done by proposing a shift of the form log λ/aT instead of log aT λ. The two formulations can be easily reconciled noting that the shift factor in one is the reciprocal of the same factor in the other.

Time-temperature-age superposition principle

65

10 T, aT > 1 Tr, aT = 1

Log compliance

DT

DTr t = 10 s t = 100 s aT = 10

1 0.01

aT > 1

0.1

t 1

10

aT

100

1000

Log time

2.2 Double logarithmic plot of compliance for the material in Fig. 2.1.

For this simple example, D0 ¼ D0r and D1 ¼ D1r; that is, the only difference between them are the retardation times τ, τr. Then, bT ¼ 1 and, using Eq. (2.9), h i h i D0 + D1 1  eλ=τ ¼ D0 + D1 1  eaT λ=τr

(2.11)

from which τ ¼ τr =aT

(2.12)

Since aT > 1, then τ < τr. Therefore, the well-known fact that creep strain grows faster at temperature T > Tr is described by a shorter retardation time τ < τr. Since creep strain grows faster at temperature T > Tr, one can accelerate a test by running it at a higher temperature, within limits so that the material does not degrade. In this case the retardation times at temperature T are reduced by a factor 1/aT (see Eq. 2.12) and creep is accelerated by a factor aT. This is the basis for the widely used accelerated testing, but in performing accelerated testing, one must be careful that the acceleration factor (temperature in this case) does not affect the physical or chemical characteristics of the material. For aging, the well-known fact that age stiffens polymers is described by a retardation time τ < τr when te < ter, with te being the aging time, or simply age, of the material and ter being another age taken as reference. In this case the aging shift factor is denoted as ae. If the momentary compliance D(λ;te) of a specimen with age te is

66

Creep and Fatigue in Polymer Matrix Composites

plotted vs. time aeλ, it superposes the compliance of a specimen with age ter. Aging time te is the time elapsed since the sample was quenched. In principle, the data obtained at higher temperature can be shifted to lower temperature in order to predict the creep compliance at lower temperature for times that exceed the time available to do the test. However, no other physical or chemical phenomena should interfere with the superpositions being made. If the material ages during the test, the data will not superpose (Matsumoto, 1988; Vleeshouwers et al., 1989). To solve this problem, the individual tests must be of duration short enough that the effects of aging are negligible. This is accomplished by restricting the time of the tests to λ/te < 1/10, where λ is the time of the test started at age te. This is called the snapshot condition and the individual curves thus obtained are called momentary curves (Struik, 1978). The effective time λ is used to describe momentary data in order to distinguish it from the real time t. The total time since the sample was quenched is te + λ. The concept of effective time is formalized in Section 2.4. For now it suffices to say that λ is time elapsed since the application of the load and with no further aging, which is accomplished by testing for short times, within the snapshot condition λ/te < 1/10. To obtain the temperature shift factors aT, bT, a number of experiments are performed at increasingly higher temperatures in such a way that successive momentary curves superpose when shifted vertically and horizontally. This is illustrated in Fig. 2.3 using data from (Sullivan, 1990).

Log compliance (1/GPa)

100

120°C

Data Regression Shifted data Master curve

115°C

100°C

10−0.1 40°C

10−0.2 90°C 80°C 10

60°C

−0.3

10−0.4 100

102

104

106

108

Log time (s) 2.3 Momentary curves D(λ) at various temperatures, all with age te ¼ 166 h; momentary master curve D(λ;te) at Tr ¼ 40°C (solid line under the shifted data at 40°C) and shifted to 100°C (dotted line).

Time-temperature-age superposition principle

67

The objective of superposing data sets is to construct a master curve that spans longer time than the time span of each data set. In Fig. 2.3, all data sets span approximately the same time, from 60 s to about 16 h. The chosen reference temperature is Tr ¼ 40°C. By performing horizontal shifts of magnitude log a(T) on the data sets with T > Tr, the 16-h tail of the curves extends the master curve further and further to the right in log λ scale. Vertical shifts are necessary to obtain the best possible superposition among data sets, but horizontal shifts are solely responsible for extending the time span of the master curve. Both horizontal and vertical shifts are necessary to produce the momentary master curve in Fig. 2.3. If vertical shifts were enough to superpose the curves, the resulting master curve would span the same time interval of the original data sets and the objective of time-temperature superposition would not be achieved. Since the data sets (or the curves representing the data sets) are shifted horizontally to the right, they superpose over a time span shorter than the individual curves. Estimating the time span over which the curves superpose is critical for implementing an accurate algorithm to superpose the curves, that is, to calculate values of aT, bT that yield the best superposition possible. This is illustrated in Fig. 2.4. Assuming horizontal shift only, the solid-line portion of the curve at temperature T superposes on the solid-line portion of the curve at temperature Tr when the T-curve is shifted to the right by plotting the compliance D(aTλ; T) vs. time aTλ. Therefore the overlapping time span starts at λ ¼ λ0 and ends at λ ¼ λf/aT (see Eq. 2.10). In Fig. 2.3, the compliance D(λ) represents the shear compliance S66 of a unidirectional (UD) composite lamina consisting of Derakane 470-36 Vinyl Ester polymer reinforced with 30% by volume of E-glass fibers in a [45°] UD lamina configuration.

Log compliance

T

Tr

l0

lf /a

lf Log time

2.4 Approximate time span over which two momentary curves superpose.

68

Creep and Fatigue in Polymer Matrix Composites

Table 2.1 Regression parameters and shift factors for the TTSP study depicted in Fig. 2.3 T (°C)

D0

D1

m

log aT

log bT

40 60 80 90 115 120 Average COV

0.36 0.396 0.429 0.438 0.518 0.588 0.455 0.184

0.016 0.008 0.004 0.009 0.016 0.02 0.012 0.496

0.166 0.227 0.306 0.264 0.271 0.294 0.255 0.201

1 1.742 4.033 7.758 67.114 229.328 – –

1 1.051 1.103 1.147 1.35 1.496 – –

A typical experimental setup for larger specimens is presented in (Barbero and Ford, 2006). In Fig. 2.3, individual data sets are regressed with the power law model (Eq. 2.7) and the coefficients are given in Table 2.1. If the momentary data can be fitted exactly with a model equation, such as Eq. (2.1)–(2.8), one can fit each curve with a model and then shift the model curves (Bradshaw and Brinson, 1997), instead of shifting actual data. Such an approach is computationally simpler but, if the model does not fit the data exactly, the shift factor for the models might not yield a smooth master curve when used to shift the actual data. Model equations are regressed based on the average error between the model and the data and are prone to yield the largest error at the ends of the data interval, precisely where the curves must be superposed. Therefore, the regression errors may be magnified and accumulated in the shift process. The computer code for determining the temperature shift factors is based on Eq. (2.10). First, the time span where the curves would superpose is approximated (Fig. 2.4) as the interval [λ0,λf/aT], where λ0, λf are the initial and final time of the momentary curve being shifted. Note that while constructing the momentary master curve, specimens are tested at different temperatures but all are aged equally; thus the data for all specimens span approximately the same testing time [λ0,λf], with λf te/10. Then, the values of the shift factors aT, bT are found by minimizing the norm of the error between the two data sets being superposed. For example, a least squares minimization of the error is implemented by writing Eq. (2.10) as: err ¼

n 1X ½Dðλi ; T Þ  bT DðaT λi ; Tr Þ2 n i

(2.13)

where n is the number of data points. Then, the shift factors aT, bT are found by minimizing the error (Barbero, n.d.). For example, in MATLAB: z ¼ fminsearch (@(z)err(@power, ti, tf, beta(k), beta(k–1), z), z0, options); yields the array z containing the horizontal and vertical shift factors that minimize the error computed in the function err. Further, @power is a function fitting the data sets, in this case with Eq. (2.6), with parameters D0, D1, m, passed through the array beta for temperatures k and k–1. Finally, z0 is an initial guess for the array z (Barbero, n.d.).

Time-temperature-age superposition principle

69

The shift process produces a momentary master curve D(λ; Tr, te) for a particular age and temperature te,Tr, such as the one shown in Fig. 2.3, that spans much more time λ than that devoted to individual tests. However, this momentary master curve does not include the effect of further aging, because it is made up of momentary curves, and all of them tested at the same age te, with each of them experiencing negligible aging during testing for a time span shorter than te/10. The corollary is that the momentary master curve obtained cannot be used to predict long-term creep without further treatment. In fact, the shape of this momentary master curve is very different from that of long-term creep, as shown in Fig. 2.5. The shape of the momentary master curve would predict creep to occur much faster than in reality. As long as aging produces changes of stiffness in the material, timetemperature superposition (TTSP) alone cannot predict long-term creep. In fact, TTSP alone can only predict long-term behavior near the glass transition temperature Tg because aging effects become negligible near the glass transition in a relatively short period of time (Struik, 1978; Lee and McKenna, 1990a; Lee and McKenna, 1990b). Note that D(λ; Tr, te) refers to the collection of shifted data in Fig. 2.3. It is not necessary to fit such data with a model equation in order to proceed with the discussion. If a model equation is desired for convenience, the analyst is responsible for assuring that the model equation fits the momentary master curve accurately. Further, the momentary master curve D(λ; Tr, te) can be shifted to any temperature T and age te by using the temperature shift factors aT(T), bT(T) and aging shift factor ae(te), respectively (see Section 2.3); that is,

Log compliance (1/GPa)

10

−0.1

Master curve Prediction Long-term data

90°C m.c. 60°C m.c.

10−0.2 40°C pred

10−0.3

10−0.4 10−2

100

102

104

106

108

Log time (s) 2.5 Momentary master curves D(λ) at temperatures T ¼ 40°C, 60°C, 90°C, all with age te ¼ 1 h, compared to long-term data at those same temperatures and ages. These predictions are based on the discussion in Section 2.4.

70

Creep and Fatigue in Polymer Matrix Composites

Dðλ; T, te Þ ¼ bT DðaT ae λ; Tr , ter Þ

(2.14)

Having performed a series of momentary tests at increasing temperatures, one can plot the shift factor vs. temperature, as shown in Fig. 2.6. A regression using the WilliamsLandel-Ferry (WLF) equation: log aT ¼

C1 ðT  Tr Þ C2 + T  Tr

(2.15)

fits the values of log aT vs. T very well, yielding parameters C1, C2. The same applies to the vertical shift factor bT. Note that C1 < 0, C2 < 0 in Fig. 2.6. Values of C1, C2 obtained from data at or above the glass transition temperature Tg are different and cannot be used below Tg (Sullivan, 1990). Any abrupt change in the shift factor plot provides an indication of a sudden change in physical or chemical properties such as thermal degradation, phase changes, and so on. The shift factor for any temperature can be predicted from Fig. 2.6 using Eq. (2.15), even for temperatures for which no test data is available. That means that the momentary master curve can be shifted to any temperature. For example, the momentary master curve for temperature T ¼ 100°C, for which no experimental data is available, is shown in Fig. 2.3 (dotted line). Usually, an SLS model (Eq. 2.1) does not fit creep data satisfactorily, so more terms need to be used (see Eq. 2.3). Then, a necessary condition for the curves to be 103 Shift factors aT Temperature shift factors log aT, log bT

WLF regression aT Shift factors bT 102

WLF regression bT

101

100

10–1

0

20

40

60 80 Temperature T (⬚C)

100

120

2.6 Temperature shift factor plot for the data in Fig. 2.3. Tr ¼ 40°C, te ¼ 166 h, horizontal C1 ¼  1.22503, C2 ¼ 122.669, and vertical C1v ¼ 0.0762931, C2v ¼ 116.057.

Time-temperature-age superposition principle

71

superposable is that all the retardation times τj shift equally by a single shift factor aT. Phenomenologically, this means that all the physical processes described by those many retardation times must change equally with temperature, or whatever phenomenon is being studied (i.e., age, stress, etc.), for superposition to be feasible. If the data is represented by a compliance spectrum like in Eq. (2.4), then, using the same reasoning described earlier, Eq. (2.10) yields: Δðτ; T Þ ¼ aT ΔðaT τ; Tr Þ

(2.16)

That is, the whole compliance (or retardation) spectrum must be affected equally by a scalar shift factor in order for superposition to apply. The method presented in this section is called the time-temperature superposition principle because it has not been derived from some underlying principle, but rather it is a principle itself, which is valid only as long as the curves are superposable.

2.3

Time-age superposition

In this section, the characterization of aging is described by a technique similar to that of Section 2.2. Since the momentary (unaged) master curve has been obtained already (Fig. 2.3) by time-temperature superposition, constructing an aging-time master curve is only necessary as an intermediate step for obtaining the aging shift factor μe, which is defined in Eq. (2.22) and allows for the calculation of the aging shift factor ae for any age te. In summary, aging can be characterized by just one scalar value, μe, which is the slope of log ae vs. log te. Physical aging begins when the polymer is quenched to a temperature below Tg, whether the material is under load or not. Annealing of the material slightly above Tg erases all the memory in the material; thus, the material is rejuvenated (Struik, 1978). Quenching to a temperature well below Tg starts the age clock for the material. In order to elucidate the effects of aging, a series of creep tests are performed at various ages. At each age te, a creep test is conducted for a time λ no longer than te/10 so that the creep compliance obtained is not tainted by the effect of further aging. This is a momentary curve obtained while satisfying the snapshot condition. To improve the quality of the regression, the ages at which creep testing takes place are chosen to be approximately equidistant on a log scale. This is easily accomplished by testing at 0.1, 0.3162, 1., 3.1623, 10., units of time, and so on (MATLAB: 10([1:0–5:1])). Compliance vs. time curves in double-logarithmic scale are shown in Fig. 2.7 and Table 2.2; the solid lines are obtained regressing the data with the power law model (Eq. 2.6). It can be seen that the regression is excellent. For long aging time, the testing time can be relatively long and thus complex equations, such as Eq. (2.5), may be fitted to the data. However, one is interested in elucidating the effects of aging with as short a time testing as possible. This leads to short aging times and even shorter creep testing times. For short times, there is not enough data to elucidate the many parameters involved in, say, a four-parameter model

72

Creep and Fatigue in Polymer Matrix Composites

Log compliance D (1/GPa)

160 h

10−0.1

879 h

32 h

8h 2h

5000 h

10−0.2 Regression Data Shifted data 101

102

103

104

105

106

Log time (s) 2.7 Compliance vs. time at constant temperature T ¼ 115°C and various ages. Squares represent data. Circles represent shifted data. Solid lines represent power law regression of data. The broken line represents the momentary master curve shifted to 5000 h. Table 2.2 Regression parameters and shift factors for the aging study depicted in Fig. 2.7 Age te

D0

D1

m

log ae

be

879 160 32 8 2 Average COV

0.518 0.512 0.513 0.533 0.53 0.521 0.019

0.012 0.018 0.026 0.033 0.044 0.027 0.468

0.264 0.264 0.264 0.264 0.264 0.264 0

0 0.657 1.201 1.637 2.102 – –

0 0.006 0.005 0.015 0.012 – –

(Eq. 2.5). Therefore, a simpler model is required, such as the power law (Eq. 2.6), rewritten as follows: Dc ðλÞ ¼ DðλÞ  D0 ¼ D1 λm

(2.17)

where Dc is the creep compliance, D0 is the elastic compliance, and D(λ) is the total compliance measured in the experiment. Each data set is initially regressed to the power law model (Eq. 2.6) to determine D0, D1, m. Numerical results are shown in Table 2.3. Assuming that the elastic compliance is independent of age, D0 should be the same for all data sets. This is confirmed

Time-temperature-age superposition principle

73

Table 2.3 Initial regression parameters for the aging study depicted in Fig. 2.7 Age te

D0

D1

m

879 160 32 8 2 Average COV

0.503 0.523 0.511 0.536 0.58 0.518 0.028

0.018 0.014 0.026 0.032 0.018 0.022 0.361

0.239 0.287 0.262 0.269 0.364 0.264 0.075

by the low coefficient of variance (COV) for D0 in Tables 2.2 and 2.3. Similarly, in (Barbero and Julius, 2004; Barbero and Ford, 2004), no correlation was observed between initial compliance D0 and age te. Small variations of D0 are due to small inaccuracies in the test, such as compliance of the equipment, imperfect initial contact between specimens and loading points, etc. For this reason, the data from the first two tests on each specimen were discarded in (Barbero and Julius, 2004; Barbero and Ford, 2004), recognizing the need for mechanical conditioning of the specimens. The resulting power law exponent m might not be identical for all plots, that is, for all ages, but the dispersion usually is very small. For example, for the data in Fig. 2.7, the average is m ¼ 0:228 with a COV ¼ 16.8%. If the first test (at te ¼ 2 h) is neglected, as recommended in (Barbero and Julius, 2004; Barbero and Ford, 2004), the COV reduces to COV ¼ 7%. The remaining dispersion is caused by the variable amount of data (spanning (t0, te/10)) available to perform the correlation for various ages, with tests at long ages having the most data points. Then, the data is regressed again with Eq. (2.17), but this time keeping m ¼ m constant for all curves and adjusting only D0, D1. Numerical results are shown in Table 2.2. As can be seen in Fig. 2.8, the impact of such averaging is minimal (Barbero and Ford, 2006); that is, the model still regresses the data very well. With all the curves having the same power-law exponent, they are parallel lines in a plot of log (D  D0) vs. log t as shown in Fig. 2.8. Then, the log of the shift factor, log ae, can be measured as the horizontal distance between any given curve and the master curve in the same figure. Noting that the creep compliances are represented by parallel lines, they are obviously superposable. Mathematically, to say that creep compliances are superposable means that: Dc ðλ; te Þ ¼ Dc ðae λ; ter Þ

(2.18)

where Dc(λ;ter) is the creep compliance of the reference curve for the reference age ter. Writing the log of the power law model for the two sets of data: log Dc ðλ; ter Þ ¼ log D1r + m log ðae λÞ

(2.19)

log Dc ðλ; te Þ ¼ log D1 + m log ðλÞ

(2.20)

74

Creep and Fatigue in Polymer Matrix Composites

100

Log (D−D 0) (1/GPa)

2h

8h

32 h

160 h

879 h

10−1

Data Regression 10−2 1 10

102

103 104 Log time (s)

105

106

 2.8 Power law model for the creep data in Fig. 2.7 with m ¼ 0:264 Tr ¼ 115° C, ter ¼ 879 h .

and subtracting the second from the first equation yields explicit formulas for the aging shift factors: 

 D1 1=m ae ¼ D1r be ¼ D0r  D0

(2.21)

where D0r , D1r m, are the power law model parameters for the reference curve at age ter and D0 , D1 ,m, are the parameters for the curve at age te. To superpose the data sets in a plot of log (D  D0) vs. log t as shown in Fig. 2.8, each data set is shifted to the left by log ae. To superpose them on a plot of log D vs. log t as shown in Fig. 2.7, one needs to plot be + D(aeλ) vs. aeλ on a double logarithmic scale. Note that the factor be is not used in the same way as the shift factor bT. While be is added to D(aeλ), log bT is added to log D(aTλ). This complication is not relevant because the master curve obtained from the aging study is seldom used; instead the master curve from the temperature study is used. Furthermore, different treatment of vertical shift allowed us to derive a pair of simple formulas, that is, Eq. (2.21), for the horizontal and vertical aging shift factors, whereas the temperature shift factors are computed by a numerical minimization algorithm, that is, in an approximate way. The procedure described in this section, concluding into Eq. (2.21), cannot be used to calculate the temperature shift factors in Section 2.2, because creep compliance curves obtained at different temperatures have markedly different values of D0 (see Table 2.1).

Time-temperature-age superposition principle

75

In this process, it is best to choose the master curve to be the one corresponding to the longest age tested because it is the curve with more data. Also, each specimen is annealed, quenched, and tested for increasingly longer aging time without removing it from the testing equipment, such as a dynamic mechanical analyzer (DMA) (Barbero and Julius, 2004). This means that the data for the longest age is perhaps the best in terms of mechanical conditioning of the sample. Next, Eq. (2.21) is used to obtain the aging shift factors ae for each data set. A linear regression of log ae vs. log te fits the values very well (Fig. 2.9), and the slope is the aging shift factor rate: μe ¼ 

d log ae >0 d log te

(2.22)

which is normally assumed to be constant for a wide range of temperatures, except near the glass transition (Struik, 1978; Lee and McKenna, 1990b). Temperature dependence was reported in (Sullivan, 1990, Figure 9). Once the aging shift factor rate μe has been determined, the aging momentary master curve constructed at a given reference age (say ter ¼ 879 h in Fig. 2.7) can be shifted to any other age by shifting it to log(λ/ae), where the aging shift factor is computed from Fig. 2.9 and Eq. (2.22) as log ae ¼ μe log ðte =ter Þ

(2.23)

103 Shift factors ae Regression me = 0.7885

Log ae

102

101

100 0 10

101

102

103

Log te (h)

2.9 Aging shift factor plot for the creep data in Fig. 2.8. The data point on the lower right corner of the figure is the shift factor ae ¼ 1 at ter ¼ 879 h (Tr ¼ 115°C).

76

Creep and Fatigue in Polymer Matrix Composites

Eq. (2.23) is analogous to the WLF Eq. (2.15). For example, the aging momentary master curve for age te ¼ 5000 h, for which no experimental data is available, is shown in Fig. 2.7 (broken line). Note that shifting to the right of the last data set assumes that the shift factor plot can be extrapolated outside the range of ages for which data is available (2–879 h in this case). Unlike in the case of TTSP (Fig. 2.3), the objective in this section is not to generate a master curve to span longer time than available for experimentation, but to obtain the aging shift factor plot and from it to calculate the aging shift factor rate μe. All the effect of physical aging is characterized by the aging shift factor rate μe. Still, an aging momentary master curve is produced (Fig. 2.7), which is valid only for the time span up to ter/10. Since the power law model fits the data very well, the aging momentary master curve is represented by Eq. (2.17) with D0 , D1 ,m, being material properties determined by the procedure presented in this section. Also, since power law models fit the master curve in Figs. 2.3 and 2.7, an argument can be made to extrapolate the longest momentary curve of an aging study beyond ter/10. Such an argument has no empirical basis because in a real experiment, aging will mar the data if the testing time goes beyond ter/10. Furthermore, lacking a temperature study, one would not know how to apply the momentary curve to any temperature other than that used to conduct the aging study. A comparison between momentary master curves obtained from temperature and aging studies is presented in Fig. 2.10. The original momentary master curves 100 TTSP m.c. @40°C and 166 h TTSP m.c. @40°C and 1 h

Log compliance (GPa)

10–0.1

ETT m.c. @116°C and 879 h ETT m.c. @40°C and 1 h

10–0.2

10–0.3

10–0.4 0 10

102

104 Log time (s)

106

108

2.10 Comparison between momentary master curves obtained separately from the temperature and aging studies. The original momentary master curves have been shifted to a common temperature T ¼ 60°C and age te ¼ 1 h.

Time-temperature-age superposition principle

77

have been shifted to a common temperature T ¼ 60°C and age te ¼ 1 h. To facilitate comparison, a small, additional vertical shift of log 1.04 [1 GPa1] has been applied to the ETT master curve. The curves are close but not identical. The difference may be attributed to experimental errors. These errors need to be minimized in order to predict long-term creep because minor changes in the momentary curve produce large discrepancies in the predicted compliance at long times. For this reason, several replicates should be used to construct the momentary curves, which allows for determination of the mean and variance of the response (Barbero and Julius, 2004). In Fig. 2.10, the momentary curve from the temperature study is to the right of the target age and temperature (i.e., te ¼ 1 h, T ¼ 60°C). Therefore, it is shifted from te ¼ 166 h, Tr ¼ 40°C to te ¼ 1 h, T ¼ 60°C by plotting bT D(λ) vs. [1(aTae)]λ with aT ¼ 1.732, bT ¼ 1.037, and ae ¼ 56.306. The momentary curve from the aging study is to the right of the target age but to the left of the target temperature. Therefore it is shifted from te ¼ 879 h, Tr ¼ 115°C to te ¼ 1 h, T ¼ 60°C by plotting bT D(λ) vs. (aT/ae)λ with aT ¼ 57.057, bT ¼ 1.345, and ae ¼ 209.573.

2.4

Effective time theory

In this section, the relationship between unaged and real time, that is, the concept of effective times, is described and used to correct the momentary master curve (Fig. 2.3) for aging, thus providing a methodology for predicting long-term creep. Effective time theory (ETT) was proposed by Struik in (Struik, 1978). Considering a test started at age te, running for time t, so that the total time since quench is te + t, from Eq. (2.22), we can calculate the aging shift factors at times te and te + t as: d log ae ðte Þ ¼ μe d log te d log ae ðte + tÞ ¼ μe d log ðte + tÞ

(2.24)

Therefore, the shift factor evolves with time as:   ae ðte + tÞ t e μe ¼ ae ð t Þ ¼ 1 because the curves have to be shifted right onto the master curve. At any shorter age, creep accumulates faster with smaller retardation times, the acceleration

78

Creep and Fatigue in Polymer Matrix Composites

being 1/ae. Therefore the same amount of creep accumulates in a shorter real time interval dt than in effective (ageless) time interval dλ, related by: dt ¼ ð1=ae Þdλ

(2.26)

which leads to the definition of the effective time (Struik, 1978, (85)): ðt λ ¼ ae ðξÞdξ

(2.27)

0

Substituting Eq. (2.25) and integrating yields (Struik, 1978, (117–188)): λðtÞ ¼ te ln ½1 + t=te  if μ ¼ 1 i te h ð1 + t=te Þð1μÞ  1 if μ < 1 λðt Þ ¼ 1μ

(2.28) (2.29)

where ln denotes the natural logarithm (base e). If the material did not age, the momentary master curve D(λ;te) in Fig. 2.3 would predict the compliance as a function of time. But since the material does age, the longterm creep compliance D(t) must be smaller, that is: DðtÞ ¼ DðλðtÞ; te Þ

(2.30)

with λ(t) given by Eqs. (2.28), (2.29). The predicted long-term compliance is shown with a solid line in Fig. 2.5. The methodology has been shown to provide good predictions of actual long-term creep data (see Sullivan, 1990, Figs. 14 and 15; Barbero and Rangarajan, 2005). If the material did not age (μ ¼ 0), the time t and effective time λ would be the same. But the material does age, so 0 < μ  1 and the effective time is much shorter than the real time. From Eqs. (2.28), (2.29), the real time can be calculated as: t=te ¼ 1 + exp ðλ=te Þ forμ ¼ 1

(2.31)

t=te ¼ 1 + ðαλ=te + 1Þ1=α forμ < 1

(2.32)

where α ¼ 1  μ. The longest data in the momentary master curve of an aging study, such as Fig. 2.7, is constructed for λ/te ¼ 0.1. According to Eq. (2.31), no prediction can be made for time exceeding t/te ¼ 0.1, that is, t ¼ 879 h in the example at hand. The model equation that fits the data cannot be extended beyond λ/te ¼ 0.1 because it is known that, as the material ages, the creep rate slows down and the model equation will not fit the data well. Shifting the master curve from, say, te ¼ 879 h down to a shorter aging time, say te ¼ 30 min, does not help in predicting long-term creep because approximately the same λ/te time span will be covered up to λ/te ¼ 0.1. Shifting to the right, assuming that the extrapolation of the shift factor plot is valid outside the range for which data is available, does not help to extend the range of the

Time-temperature-age superposition principle

79

predictions to long times because the momentary curves do not shift much to the right. For example, the shift to 5000 h is shown in Fig. 2.7. In order to predict long-term creep, a momentary master curve that extends beyond λ/te ¼ 0.1 is needed. The momentary master curve from the TTSP study (Section 2.2, Fig. 2.3) serves the purpose (Struik, 1978). Another option is to shift the momentary curve in Fig. 2.7 to a lower temperature, but for that one needs the temperature shift factor plot from the TTSP study. In that case, a momentary master curve is available with a long time span, such as in Fig. 2.3. Therefore, we shall use the latter. The time span needed on the momentary master curve (Fig. 2.3) can be calculated easily. For example, if the momentary master curve were constructed with momentary data for te ¼ 166 h, and predictions are sought up to 1 year, then t/te ¼ 8640/166 ¼ 52. Assuming the aging study (Section 2.3) yields μe  1, using Eq. (2.31) yields λ/te ¼ ln(t/te  1) ¼ 3.931, or λ ¼ 652 h. It is not possible to produce experimentally a momentary curve for 652 h with a material that has been aged for only 166 h. However, the momentary master curve for T ¼ 40°C (Fig. 2.3) easily exceeds the 652 h (5.6 106 s) required. Usually, aging studies are conducted at temperatures other than room temperature because the test equipment, such as DMA, can hold temperature more stably above (or below) room temperature. As long as there is heat exchange with the environment, the control system of the instrument can hold the temperature accurately. On the contrary, room temperature control relies on the heating, ventilation, and air conditioning (HVAC) system of the building, which is not nearly as accurate as the control system of a dedicated instrument, such as a DMA (Barbero and Julius, 2004) or environmental chamber (Barbero and Ford, 2006; Barbero and Ford, 2004).

2.5

Summary

The procedure used to predict long-term creep is as follows: 1. Perform a number of creep tests at increasing temperatures for a duration λ not to exceed the snapshot condition λ/te < 1/10. All tests are to be performed with materials aged the same amount, that is, te. 2. Shift the data onto a momentary master curve (Fig. 2.3) by determining the temperature shift factors aT, bT, for each temperature. Construct the shift factor plot (Fig. 2.6) and fit it with Eq. (2.15). The resulting momentary master curve represents the momentary compliance Dte(λ) of the material at age te without the effects of any further aging. Since the temperature shift factors aT, bT, can be calculated at any temperature in terms of C1, C2, C1v, C2v, the master curve predicts the unaging compliance at any temperature. 3. Perform a number of creep tests at increasing ages for a duration λ not to exceed the snapshot condition λ/te < 1/10. All tests are to be performed at the same temperature, usually at room temperature for convenience. 4. Shift the data onto a master curve by determining the aging shift factor ae for each age. Construct the shift factor plot log ae vs. log te. Compute the aging shift factor μe as the slope of the plot. 5. The long-term compliance is given by Eq. (2.30) with λ given by Eqs. (2.28), (2.29) in terms of the real time t.

80

2.6

Creep and Fatigue in Polymer Matrix Composites

Temperature compensation

In this section, effective time-temperature superposition (ETTSP, described in Sections 2.2–2.4) is used to perform temperature compensation of long-term data collected in a fluctuating temperature environment. Field testing of polymer and polymer composite structures, as well as laboratory testing of large structures, are subject to temperature variations due to seasonal and daily temperature fluctuations. Due to temperature fluctuations, the material undergoes changes of creep compliance and those are reflected in the data collected, particularly in strain readings (Barbero and Rangarajan, 2005). Therefore, the data needs to be compensated to report the behavior of the material at a constant temperature TR. Temperature compensation is not possible by using the time-temperature momentary master curve and temperature shift factor of Section 2.2, because those curves represent the material behavior without aging, as it was at the age te used to collect the data in Section 2.2. Obviously, the material undergoes further aging during a long-term test. Furthermore, the field test starts with the material having an age tR that represents the time elapsed between material production and the onset of the field test, and it is unlikely that tR coincides with te. The temperature compensation procedure is as follows. First, the time-temperature momentary master curve D(λ; te) is shifted to the reference temperature TR and age tR by using Eqs. (2.15), (2.23) in terms of the known coefficients μe, C1, C2, C1v, C2v. Next, each time interval Δt ¼ ti  ti1 of a long-term test occurs at a time ti for which the recorded temperature is T(ti). The time ti is shifted to effective time. That is, using Eqs. (2.28), (2.29), compute λi, λi1, and the unaged time interval Δλi ¼ λi  λi1. This interval can be adjusted to the reference temperature as: Δλ0i ¼ Δλi =aT

(2.33)

Then, the accumulated aging time at time tn is computed as: λ0n ¼

n X

Δλ0i

(2.34)

i¼1

which is then transformed to real time using Eqs. (2.31), (2.32).

2.7

Conclusions

Age and temperature affect the creep compliance of polymers in similar yet separate ways. Physical aging changes the material behavior with time, thus invalidating the use of time-temperature superposition for any significant length of time for any temperature except in the vicinity of the glass transition. As a result of aging, the material properties change with time and Boltzmann’s superposition principle no longer applies. Combined use of a time-temperature superposition study, performed for a

Time-temperature-age superposition principle

81

time span short enough to render aging negligible, and an age-time superposition study, enables us to predict the combined effects of temperature and age. The time-temperature-age superposition principle also allows us to recover Boltzmann’s superposition principle and thus a plethora of useful analysis techniques based on it, such as prediction of viscoelastic properties of composites (Barbero and Luciano, 1995; Luciano and Barbero, 1995), and so on. Application of this methodology for the case of stress-induced nonlinearity awaits attention.

References Barbero, E.J., 2007. Finite Element Analysis of Composite Materials. Taylor & Francis. Barbero, E.J., Web Resource. http://www.mae.wvu.edu/barbero/. Barbero, E., Ford, K., 2004. Equivalent time-temperature model for physical aging and temperature effects on polymer creep and relaxation. ASME J.Eng. Mater. Technol. 126 (4), 413–419. Barbero, E., Ford, K., 2006. Determination of aging shift factor rates for field-processed polymers. J. Adv. Mater. 38 (2), 7–13. Barbero, E.J., Julius, M.J., 2004. Time-temperature-age viscoelastic behavior of commercial polymer blends and felt-filled polymers. Mech. Adv. Mater. Struct. 11 (3), 287–300. Barbero, E., Luciano, R., 1995. Micromechanical formulas for the relaxation tensor of linear viscoelastic composites with transversely isotropic fibers. Int. J. Solids Struct. 32 (13), 1859–1872. Barbero, E., Rangarajan, S., 2005. Long-term testing of trenchless pipe liners. J. Test. Eval. 33 (6), 377–384. Bradshaw, R., Brinson, L., 1997. Physical aging in polymers and polymer composites: an analysis and method for time-aging time superposition. Polym. Eng. Sci. 37 (1), 31–44. Creus, J.G., 1986. Viscoelasticity: Basic Theory and Applications to Concrete Structures. Springer-Verlag. Gates, T., Feldman, M., 1995. Time-dependent behavior of a graphite/thermoplastic composite and the effects of stress and physical aging. J. Compos. Technol. Res. 17 (1), 33–42. Lee, A., McKenna, G., 1990a. Viscoelastic response of epoxy glasses subjected to different thermal treatments. Polym. Eng. Sci. 30 (7), 431–435. Lee, A., McKenna, G.B., 1990b. Physical ageing response of an epoxy glass subjected to large stresses. Polymer 31 (3), 423–430. https://doi.org/10.1016/0032–3861(90)90379–D. Luciano, R., Barbero, E.J., 1995. Analytical expressions for the relaxation moduli of linear viscoelastic composites with periodic microstructure. ASME J. Appl. Mech. 62 (3), 786–793. Matsumoto, D., 1988. Time-temperature superposition and physical aging in amorphous polymers. Polym. Eng. Sci. 28 (20), 1313–1317. Struik, L.C.E., 1978. Physical Aging in Amorphous Polymers and Other Materials. Elsevier Scientific Pub. Co, New York. Sullivan, J., 1990. Creep and physical aging of composites. Compos. Sci. Technol. 39 (3), 207–232. https://doi.org/10.1016/0266–3538(90)90042–4. Vleeshouwers, S., Jamieson, A., Simha, R., 1989. Effect of physical aging on tensile stress relaxation and tensile creep of cured epon 828/epoxy adhesives in the linear viscoelastic region. Polym. Eng. Sci. 29 (10), 662–670.

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Effect of moisture on elastic and viscoelastic properties of fiber reinforced plastics: Retrospective and current trends

3

Andrey Aniskevich, Tatjana Glaskova-Kuzmina Institute for Mechanics of Materials, University of Latvia, Riga, Latvia

3.1

Introduction

This chapter is a brief review of some journal papers and books about moisture and composites with polymer matrix (mostly fiber reinforced composites but not exclusively). Most polymer materials and composites absorb moisture. It is well known from many papers (some of them mentioned later) over several decades that absorbed moisture essentially affects mechanical and many other important properties of polymers and polymer-based composites. The key question is: how does absorbed moisture affect mechanical properties of composites? Furthermore, the process of moisture absorption in composite structures may occur over many months and even years. More questions then arise: what is the mechanism of moisture absorption in a polymer composite, how long can this process last, and how much moisture can be absorbed? Looking through recent journal papers, it may be concluded that many of them use a classical theoretical approach that has been known and used for many decades. Some authors apply this classical approach to investigate moisture absorption and its effect on the mechanical properties of new materials such as, for example, nanomodified polymers. It would be useful and important to obtain new knowledge about these new materials but, unfortunately, quite often authors don’t investigate the effects of moisture any deeper than was done 40 years ago. This is why we start our retrospective review from the classic papers of the 1970s (Springer, 1988; Vinson, 1978; Shen and Springer, 1976; Loos and Springer, 1979; Weitsman, 1987; Aniskevich et al., 2012a), and also include modern trends.

3.2

Moisture absorption in polymers

3.2.1 Moisture sorption in stationary humid conditions It is well known for many isotropic polymer materials that diffusion is the dominant sorption mechanism (Springer, 1988). The moisture sorption kinetics in Cartesian coordinates is described by the second Fick’s equation given e.g., in (Crank, 1975) Creep and Fatigue in Polymer Matrix Composites. https://doi.org/10.1016/B978-0-08-102601-4.00003-5 © 2019 Elsevier Ltd. All rights reserved.

84

Creep and Fatigue in Polymer Matrix Composites


2  ∂c ∂ c ∂2 c ∂2 c ¼D + + ∂t ∂x2 ∂y2 ∂z2

(3.1)

where c(x, y, z, t) is a concentration of absorbed moisture depending on coordinates and time, and D is the diffusion coefficient (or diffusivity) independent of concentration. A problem with stationary initial and boundary conditions is a simple one. For example, for a parallelepiped-shaped sample with sides a  b  h, the initial moisture concentration distribution over the sample cross-section is considered as uniform c(t ¼ 0) ¼ c0, and the moisture concentration on the sample boundaries is constant c(x ¼ 0; a) ¼ c(y ¼ 0; b) ¼ c(z ¼ 0; h) ¼ c∞ The solution of Eq. (3.1) is (Springer, 1988): h i k n m ∞ X ∞ X ∞ 1  ð1Þ ½1  ð1Þ  ½1  ð1Þ  cðx, y, z, tÞ  c0 8X ¼1 3 π k¼1 n¼1 m¼1 c∞  c0 knm  2  sin ðλk xÞ sin ðλn yÞsin ðλm zÞ exp λ k,n,m Dt

(3.2)

where λ2k,n,m ¼ λ2k + λ2n + λ2m ¼


2     πk πn 2 πm 2 + + a b h

Relative moisture content in a sample is w ¼ PQ0 , where Q is the absolute moisture conÐ tent Q ¼ V cdv, and P0 is the weight of the dry sample. After integration and some mathematical operations, the equation for describing the moistening processes of a parallelepiped-shaped sample takes the form: h i2 2 2 k ∞ X ∞ X ∞ 1  ð1Þ ½1  ð1Þn  ½1  ð1Þm  X wðtÞ  w0 8 ¼1 6 π k¼1 n¼1 m¼1 w∞  w0 k2 n2 m2  2  exp λ k, n,m Dt

(3.3)

where w0 and w∞ are initial and equilibrium moisture contents, respectively. In the case in which a sample dimension in one direction is significantly smaller than in two others (for example, a ≪ b, a ≪ h), a one-dimensional sorption problem is stated that simplifies the solutions. Typically, sorption methods are used to investigate moisture or water absorption kinetics in polymer materials and composites. Samples of the material are placed in a humid environment and their weight gain is controlled periodically. Sorption curpffi ves are typically plotted in coordinates w versus t since the initial segment is a straight line with a slope proportional to D. The initial segment of the sorption process could be considered as moisture diffusion in a semiinfinite media and the mathematics of this can be found in Crank (1975). Typical sorption curves of epoxy resin Reapox

Effect of moisture on elastic and viscoelastic

85

3.1 Sorption curves for Reapox D526 epoxy resin at different relative humidity (RH): experimental data (dots) and their approximation by Fick’s model Eq. (3.3) (lines).

D526™ at room temperature (RT) and atmosphere with different relative humidity (RH) and their approximation by the Fick’s model Eq. (3.3) are shown in Fig. 3.1 as an example. Several accelerated methods for determination of diffusivity and equilibrium moisture contents were developed previously (Aniskevich, 1985), but one of the most used and effective is simple curve fitting controlled by an aim function. The kinetics of sorption depend on RH and temperature of an environment. In most cases, when a sorption process is mainly associated with the Fick’s diffusion, the equilibrium moisture content or saturation level w∞ depends on RH (Fig. 3.1) and may not have significant changes with varying temperature, whereas the diffusion rate often is independent of RH and increases with temperature growth. For a variety of polymer materials, the temperature dependence of the diffusivity is described by the Arrhenius equation:
 Ud D ¼ D0 exp  Rg T

(3.4)

where D0 is a material constant, Ud is activation energy of diffusion, Rg ¼ 8.31 J/mol K is the universal gas constant, and T is the absolute temperature. The graphs of ln D versus 1/T are straight lines with a slope corresponding to the Ud of a given polymer material. Some experimental results are given in Table 3.1. Equilibrium moisture content w∞ as a function of RH is called a sorption isotherm. Henry’s law states that the amount of dissolved gas is proportional to its partial pressure in the gas phase. The sorption isotherm should be a straight line in this case. Real isotherms for water absorption in polymers and composites deviate from this law and possess nonlinear behavior. Isotherms of some materials are presented in Fig. 3.2. The second power polynomial is typically used for their description: w∞  w0 ¼ a1 ðRHÞ2 + a2 ðRHÞ

(3.5)

86

Table 3.1 The parameters of Fick’s diffusion model for some polymer materials Material

b

Equilibrium moisture contenta

Activation energyb

Swelling coefficient

a1

a2

D, 103 mm2/h

w∞, %

Ud, кJ/mol

α, %/%

0.003 0.0001

0.0163 0.0022

1.4 1.0

5.0 1.44

61 60

0.202 0.346

0.0001

0.0002

0.65

1.42

39

0.227

0.0024 0.0002 0 0.0004 0.0003

0.136 0.0124 0.0038 0.0072 0.0021

2.8 1.4 1.8 – 5.09

0.67 8.8 0.36 – 3.07

– –

–0.31 –0.24 –0.05 0.065 –

Determined at RH ¼ 98% and RT. Determined in water sorption tests under various temperatures.

a

Diffusivitya

Creep and Fatigue in Polymer Matrix Composites

Epoxy binder EDT-10 Polyester binder Norpol440 Vinylester binder Hetron 970/35 Polyoxymethylene Polyamide 6 PEEK Polyimide Upilex Epoxy binder Reapox D526

Parameters of sorption isotherm

Effect of moisture on elastic and viscoelastic

87

where a1 and a2 are approximation parameters. The parameters of diffusion model for some thermosetting and thermoplastic polymers are compared in Table 3.1. Fick’s diffusion model assumes an asymptotical approach to the equilibrium moisture content. For a variety of polymer materials, upon long-term moistening under certain conditions (high temperature, immersion in water or atmosphere with high RH, etc.), after an apparent equilibrium is reached a further weight gain is observed that can be both limited and unlimited. In this case, one says that the sorption behavior is non-Fickian, or anomalous diffusion is observed. Analysis of moisture sorption models as applied to an epoxy resin was given in Glaskova et al. (2007). Models compared are: a model taking into account the two-phase state of absorbed moisture in the material (Langmuir model); a model taking into account the two-phase nature of the material (Jacob’s–Jones model); a model with a time-variable diffusivity; relaxation model; and convection model. Effectiveness of the models is compared in Fig. 3.3, by 3.2 Sorption isotherms for EDT-10 epoxy ( ), Norpol-440 polyester (♦), and Hetron 970/35 vinylester (▲) resins at room temperature (RT).

▪

1.4

6

Absolute deviation, % 1 4 0.8 3 0.6 2 0.4 1

0.2

0

0 Fick’s

Two-phasic water

Variable D

Two-phasic polymer

Relaxation in polymer

Convection in polymer

3.3 Comparison of moisture sorption models by example of epoxy resin at RH ¼ 97% (Glaskova et al., 2007).

Absolute deviation, %

Number of parameters

1.2

Number of parameters

5

88

Creep and Fatigue in Polymer Matrix Composites

example of experimental data on moisture sorption by epoxy resin at RH ¼ 97%. The correspondence between the calculated and experimental sorption curves was evaluated by using the sum of absolute deviations for all experimental points. It can be seen that the models with the larger number of parameters give the better description of experimental data, that is, smaller values of absolute deviation. However, as the choice of parameters is ambiguous, the models with a larger number of parameters cannot be successfully used to predict long-term sorption curves. According to Fig. 3.3, the most suitable sorption models are: (i) the model with variable diffusivity, and (ii) the model taking into account the two-phase of polymer material. The former has only three parameters and agrees well with experimental data. The latter has four independent parameters and its agreement with the experimental data is also fairly good. The choice of appropriate model is dependent on the processes impacting the moisture sorption in polymer materials: for example, physical aging and/or swelling of the polymer material. Given phenomenological consideration does not take into account and does not compare the physical essence of different models.

3.2.2 Effect of a uniaxial load on moisture absorption Uniaxial tension or compression load may essentially change kinetics of moisture sorption (Aniskevich and Aniskevich, 1993). Data from long-term experiments (more than 2500 h) for epoxy resin EDT10 show that the application of tensile loads up to 50 MPa (which exceeds 0.5 of the strength of the dry sample) does not change the mechanism of moisture absorption in the binder. Tensile stress within the range 20–30 MPa had no essential effect on the diffusion coefficient, with the mean value for RT D ¼ (1.4  0.3)106 cm2/h. However, the same loads resulted in a significant increase in equilibrium moisture content—by as much as 20% of the initial value for RH ¼ 98% (Fig. 3.4). This observed effect of loading on moisture absorption goes beyond the limits of the classical Fick’s model. Some explanations based on distribution of the free and occupied volumes in the polymer are given in Aniskevich and Aniskevich (1993). Experimental results for epoxy resin EDT10 show that compression stress of up to 40 MPa also had no effect on the kinetics of moisture absorption.

3.2.3 Moisture sorption in nonstationary humidity Polymer materials during storage and operation are subjected to variable moisture actions. The periodic daily or seasonal changes in the atmospheric RH can be regarded as harmonic changes with a high degree of accuracy. One more specific case is stepwise changes of RH that simulate transfer of polymer samples from one stationary condition to another. The analytical solutions to the problems of moisture sorption for the cases of stepwise and harmonic changes in the atmospheric RH are given in Plushchik and Aniskevich (2002).

Effect of moisture on elastic and viscoelastic

89

w¥ % 5

4

3

2

1 σ MPa 0

10

20

30

40

3.4 Dependence of the equilibrium moisture content of the material w on tensile stress σ for RH ¼ 47%, 77%, 98% (Aniskevich and Aniskevich, 1993).

The change with time of the kinetics of moisture sorption under daily, annual, or other variations in the humidity of the surrounding medium can be described in a general form by a harmonic function of the type: RHðtÞ ¼ Asin ðωt + ΨÞ + B

(3.6)

where A is the oscillation amplitude, B is the stationary component of humidity, ω ¼ 2π/P is the variation frequency, P is the oscillation period, and Ψ is the initial phase shift. The analytical solutions allow calculation of distribution of moisture concentration in a cross-section of a sample and moisture content of a sample with time. Calculations were proved on samples of polyester resin Norpol-440. This data allows one to make sure that the moisture sorption process is periodic and a conditional saturation can be achieved at periodic changes in the atmospheric humidity; they also allow one to determine the stationary component of the moisture content and estimate deepness of moisture oscillation in a sample under nonstationary conditions.

3.3

Moisture absorption in fiber reinforced composites

Moisture sorption by heterogeneous and/or anisotropic composites is generally described by the same models used for isotropic polymer materials. Fick’s model is the most often used with fiber reinforced plastics (FRP). However, the problem of determining the diffusion characteristics of heterogeneous anisotropic materials is not a trivial one. There are two main approaches to evaluation of sorption characteristics in FRP: (i) structural, which is based on determining the sorption

90

Creep and Fatigue in Polymer Matrix Composites

characteristics of the composite by the properties of its constituents (Aniskevich, 1986; Aniskevich and Yanson, 1991; Aniskevich and Jansons, 1998; Starkova and Aniskevich, 2004), and (ii) direct experiment. Let us consider some principles and examples of the structural approach.

3.3.1 Microstructural approach The main advantage of the structural approach is its simplicity and applicability to different types of composites. Taking into account a large variety of existing composites with fillers of different shapes and hygroscopic properties, as well as different filler volume contents, determining the sorption characteristics of composites by using analytical models could essentially reduce a number of required tests. A scheme that illustrates a microstructural approach to calculation of the effective properties of unidirectional (UD) FRP (Fig. 3.5) is widely used in micromechanics. The approach could also be used for calculation of diffusivity. The equilibrium moisture content of a two-phase composite is determined according to the rule of mixtures:

wc∞ ¼ μf w f∞ + 1  μf wm ∞

(3.7)

where the superscripts designate the composite, reinforcement element (filler or fiber), and matrix; μf is the filler mass content. To determine the diffusivity of composites, many structural models assume that the diffusivities of both composite phases and μf (or filler volume content νf) are known. The expressions for calculating diffusivities of two-phase composites are summarized and experimentally validated on epoxy resin based aramid FRP in Aniskevich (1986), Aniskevich and Yanson (1991), and Aniskevich and Jansons (1998). From the additivity of the diffusion fluxes through the filler and matrix, expression for longitudinal diffusivity (along fibers) could be obtained:

D11 ¼ Dm 1  νf + νf D f11

(3.8)

f where Dm, D11 are the matrix (polymer resin) and filler diffusivities, respectively.

3

Unidirectional composite

Matrix

Fiber

2

= 1

3.5 Microscale model of unidirectional (UD) composite.

+

Effect of moisture on elastic and viscoelastic

91

Transversal diffusivity in directions 22 and 33 could be calculated using various equations summarized in Aniskevich (1986), Aniskevich and Yanson (1991), and Aniskevich and Jansons (1998). Many fillers like glass and carbon fibers, ceramic, and other mineral particles are hydrophobic or nearly hydrophobic with Df t0

(3.25)

Taking into account independent action of stress impulses applied at different time moments, Eq. (3.23) takes the form εð t Þ ¼ J 0 σ + σ


 k X t  Ai 1  e τi , 0 < t < t0

(3.26)

i¼1

εrec ðtÞ ¼ σ


tt  k X t  0  Ji e τ i  e τ i , t > t 0

(3.27)

i¼1

where εrec is strain at creep recovery. The creep compliance is defined by the ratio J(t) ¼ ε(t)/σ.

Effect of moisture on elastic and viscoelastic

103

3.6.2 Superposition principles Relaxation properties of polymers are affected by various factors, for example, temperature, absorbed moisture, applied stress, physical aging, etc. This fact is widely used in predicting the long-term creep of the materials by applying various superposition principles: time–temperature, time-moisture, time-stress, time-aging, and other methods (Urzhumtsev, 1972; Brinson and Brinson, 2008). The superposition principles are based on the assumption that time and a factor f, which accelerates the relaxation processes (this can be either the temperature T, the moisture content w, the stress σ, or other factors), are interrelated and interequivalent. In the case of thermorheologically simple material, the action of the factor f leads to a parallel shift of the relaxation spectrum, which is quantitatively characterized by coefficients of the reduction function af, also called shift factors (Fig. 3.18). The shift is carried out by passing to the reduced or effective time, τi0 ¼ τi/af or t0 ¼ taf

(3.28)

where f can be T, w, σ, etc. As seen from Eqs. (3.26) and (3.28), the creep compliances for two different values of a factor f, f0, and f1, differ only by a time scale defined by functions af0 and af1. In other words, they can reach the same values at different time moments t0 and t1. Then, the following relations are valid: t0 af 0 ¼ t1 af 1 ln t0  ln t1 ¼ ln af 1  ln af 0

(3.29)

For simplification, the reference function is usually taken as af 0 ¼ 1. According to Eq. (3.29), the creep compliance curves in logarithmic time axes are parallel and shifted to each other for the value lnaf. For a family of creep curves obtained in short-term tests under the action of accelerated factor f0 < f1 < f2 < … fn, one obtains the master curve (Fig. 3.19). This is the main principle used in prediction of long-term behavior of polymers and polymer-based composites. The effects of a number of environmental factors on viscoelastic material properties can be represented by a time shift, that is, the reduction function af. ln af f0 < f 1

f1

ln t i

f0

ln t

3.18 Schematic shift of the relaxation spectrum due to the variation of an external factor f from f0 to f1.

104

Creep and Fatigue in Polymer Matrix Composites

J Experiment

Prediction

fn

f2 f1 f0 te

lnafn

Master curve

lnaf2 lnt

tp

3.19 Creep curves at the action of factor f and their parallel shift to the master curve.

In a general case, when the accelerating factor changes during the loading history, the shift factor is a function of time and Eq. (3.28) transforms to 0

ðt

t ¼ af ½f ðsÞds

(3.30)

0

Under simultaneous action of a number of factors and assuming their individual (noncoupled) influence on thermorheologically simple materials, the time scale shift is contributed by each factor by a multiplicative function. For example, under simultaneous action of temperature T and moisture w, the effective time Eq. (3.28) is given by t0 ¼ taT aw

(3.31)

The total shift, the time–temperature-moisture reduction function, is then determined by a sum ln aTw ¼ ln aT + ln aw

(3.32)

The time–temperature superposition principle (TTSP) is one of the most widely used methods for prediction of long-term viscoelastic behavior of polymer-based composites, mainly due to technical simplicity and controllability of required testing procedures and tractability of the obtained results. However, other methods, for example, time-moisture (TMSP) and time-stress (TSSP) and other superposition principles are successfully applied, mainly for modeling viscoelastic behavior of polymers under specific operating conditions.

3.6.3 Time-moisture superposition principle: creep of moisturesaturated polymers A close analogy between temperature and moisture effects on viscoelastic behavior of polymers and applicability of the time-moisture superposition principle was first mentioned by Maksimov et al. (1972). Hygrothermal viscoelastic behavior of various

Effect of moisture on elastic and viscoelastic

105

4 3 2 1

3.20 Creep of PE resin samples with moisture contents: w ¼ 0 (1); 0.84% (2); 1.44% (3); 1.64% (4); σ ¼ 10 MPa. Solid lines and symbols are data from short- and long-term tests, respectively; dashed line is prediction for T ¼ 27°C and w ¼ 0.84% (Plushchik and Aniskevich, 2000).

polymers and composites and its description by applying TMSP has also been considered in a number of pioneering works (Urzhumtsev, 1972; Flaggs and Crossman, 1981; Weitsman, 1977). In TMSP, absorbed moisture w is considered as a factor accelerating the relaxation processes in polymers. Accelerated effect of absorbed moisture on viscoelastic response of polymers and applicability of TMSP is demonstrated in Fig. 3.20 by the example of creep of polyester resin (Plushchik and Aniskevich, 2000). The moisture reduction function aw was determined by horizontal shift of the shortterm creep curves to the reference curve for w0 ¼ 0%. The dashed line represents the long-term forecast and is calculated by Eq. (3.26) taking into account Eq. (3.31). It is seen that the calculated curve satisfactorily fit the results of the control long-term test. TMSP has been applied for a number of polymers. The moisture reduction functions of some polymers are shown in Fig. 3.21. In a general case, time-moisture shift factors versus moisture content of tested sample dependences could be described by a relation similar to the well-known Williams–Landell–Ferry (WLF) one for the time– temperature shift factor (Aniskevich et al., 2012a; Urzhumtsev, 1972; Brinson and Brinson, 2008): ln aw ¼ 

d1 ðw  w0 Þ d2 + w  w0

(3.33)

where d1 and d2 are empirical parameters, w is moisture content of tested samples, while w0 is moisture content of the reference sample to which all the curves are shifted. For correct evaluation of the time-moisture shift factors, moisture-saturated samples with w ¼ w∞ are considered. The moisture reduction function could also be described by a polynomial with coefficients b1 and b2 (Aniskevich et al., 1992, 2010, 2012a; Plushchik and Aniskevich, 2000): ln aw ¼ b1 ðw  w0 Þ + b2 ðw  w0 Þ2

(3.34)

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Creep and Fatigue in Polymer Matrix Composites 25

20

Polyester resin Epoxy resin EDT

15 ln aw

Vinylester resin Polyamide 6

10

Poliimide Upilex Epoxy resin DGEBA

5

0 0

2

4 6 w –w0,%

8

10

3.21 Moisture reduction function for various polymers determined in short-term creep tests for samples with equilibrium moisture contents.

The moisture reduction function can be considered a quantitative indicator of a moisture-induced plasticization effect on viscoelastic response of polymers. The higher the ln aw value, the more sensitive the material to moisture absorption. For example, according to the data presented in Fig. 3.21, polyamide is the most sensitive to moisture-induced plasticization compared to other polymers under consideration. Some polymer resins at the same levels of equilibrium moisture contents are characterized by noticeably different time-moisture shift factors. For example, lnaw of epoxy resins at w∞ ¼ 1.5% are close to unity, while polyester and vinylester resins are almost fivefold higher. In other words, despite higher moisture absorption capacity of epoxy resins, the extent of their moisture-induced plasticization is lower compared to other polymer resins under investigation.

3.6.4 Creep of polymers under moisture absorption Polymer composites during their service life are often exposed to nonstationary humid conditions, when moisture content in a material changes during its loading. In this case, viscoelastic deformation is accompanied by swelling (or shrinkage) of the material, and the total creep strain is given by a sum ε½σ, t, w ¼ εel ½σ, wðt ¼ t0 Þ + εve ½σ, t, wðtÞ + εsw ½wðtÞ

(3.35)

For linear viscoelastic material, the viscoelastic term is calculated by Eq. (3.26) taking into account a time-varying moisture reduction function according to Eq. (3.30): ’

ðt

t ¼ aw ½wðsÞds 0

(3.36)

Effect of moisture on elastic and viscoelastic

107

where aw(w) is normally determined in creep tests for samples with equilibrium moisture contents and is described by Eq. (3.33) or Eq. (3.34). The elastic response of a material determined by the instantaneous creep compliance (Eq. 3.26) depends on moisture content of a sample at the start of test, that is, w(t ¼ t0). The swelling strain for most polymers is given by a linear dependence, Eq. (3.19). In the case of nonlinear viscoelastic-viscoplastic material, calculation of the total creep strain under simultaneous moisture absorption is a very complicated task due to (i) appearance of moisture-dependent viscoplastic strains, (ii) the complex form of the reduction function determined by moisture and stress shift factors, and (iii) nonlinearity and stress dependence of the swelling strain. Examples of long-term creep (up to 3 months) of polyester resin and polyamide 6 under stationary and nonstationary moisture conditions are demonstrated in Figs. 3.22 and 3.23, respectively. The tests were performed at RT at a stress level of 10 MPa that corresponded to the range of linear viscoelastic behavior of both materials in the unconditioned state. Samples preconditioned in humid (or dry) atmosphere were subjected to chambers with corresponding humidity levels for ensuring stationary moisture conditions. It was assumed that the moisture content of the samples did not change during these creep tests, that is, w ¼ w∞. Nonstationary conditions were achieved by using predried (at RH ¼ 0%) samples for creep testing in a humid atmosphere with RH ¼ 98%. Moisture content of these samples increased during the tests, that is, w ¼ w(t). As seen from Figs. 3.22 and 3.23, creep of the moisture-absorbing sample is essentially accelerated during the test. The strain levels achieved at the end of the tests are comparable to those for the moisture-saturated samples under the same conditions.

0.003

0.0025

RH = 98%

RH = 98% w = w(t)

J, MPa–1

0.002

RH = 0%

0.0015

0.001

0.0005

0 4

9

14

19

24

ln t, [s]

3.22 Creep of polyester resin in stationary (w ¼ w∞) and nonstationary (w ¼ w(t)) moisture conditions, σ ¼ 10 MPa, T ¼ 20°C. Symbols are experimental data, lines are approximations.

108

Creep and Fatigue in Polymer Matrix Composites 0.0084 RH = 98% w = w(t)

RH = 98%

0.0063

J, MPa–1

RH = 77%

0.0042

Room 0.0021 RH = 0% 0

4

6

8

10

12

14

16

18

ln t, [s]

3.23 Creep of polyamide 6 in stationary and nonstationary (w ¼ w(t)) moisture conditions, σ ¼ 10 MPa, T ¼ 20°C. Symbols are experimental data, lines are approximations.

0.007 0.006

J, MPa–1

0.005 RH = 96% w = w(t)

0.004

RH = 96% w= w

0.003 0.002 RH = 10% w= w

0.001 0

8

10

12

14

16

18

ln t, [s]

3.24 Creep of epoxy resin EDT10 samples in stationary and nonstationary moisture conditions, σ ¼ 20 MPa, T ¼ 20°C. Symbols are experimental data; lines are calculations.

Moisture effect on creep of some polymers in the region of linear viscoelastic deformation can be satisfactorily described by Eq. (3.35) taking into account Eq. (3.26) and (3.36). However, for some polymers under specific loading and humidity conditions, creep of moisture-absorbing samples is much higher than predicted by the TMSP and linear viscoelastic theory. Anomalous creep under moisture absorption is shown in Fig. 3.24 by the example of epoxy resin EDT (Aniskevich et al., 1992). Similarly

Effect of moisture on elastic and viscoelastic

109

to the previously described tests, the applied stress corresponded to the range of linear viscoelastic behavior of the material. As seen from Fig. 3.24, the sample undergoing moisture absorption during the creep test is characterized by substantially greater deformability than the one preconditioned up to saturation. Such anomalous creep cannot be described on the basis of the linear viscoelastic theory and TMSP. A possible reason for this anomalous creep may be internal stresses arising in a polymer sample due to nonuniform swelling during moisture absorption. Analysis of such kinds of behavior is presented in the next section.

3.6.5 Viscoelastic stress–strain analysis during moisture uptake under tensile creep A robust and efficient numerical method for the calculation of the internal stress state that develops within structures subjected to mechanical and steady-state or transient hygroscopic loading conditions has been developed (Aniskevich and Guedes, 2009; Mintzas et al., 2015). The method encompasses a layer-by-layer approach whereby the structure is discretized into plies with different material properties corresponding to the different ply moisture content. The proposed method has been validated against finite element solutions, and results from its application on a fully characterized EDT10 (technical name) polymer binder are presented. The impact of the moisture induced viscoelastic behavior on the structural response of the case studied is highlighted and discussed. The approach could be used for different structures operated both open-air and indoor in automotive, marine, airspace, and other applications, for example, for interior parts in the automobile industry. The objective of the research was to estimate the contribution of viscoelasticity on the stress–strain behavior of a polymer material subjected to tensile loading during moisture uptake. The modeling strategy followed is divided into two main stages. At the first stage, the sample of an originally uniform material is virtually split into p number of plies with different material properties corresponding to the different moisture content of each ply, as shown in Fig. 3.25. Fick’s second law is then used for the determination of the time-dependent ply moisture content. At the second stage, constitutive models are applied at the ply level for the determination of the stress distribution within the layered sample. In order to validate the developed program LAMFLU described previously, linear viscoelastic analyses were performed using the commercial ANSYS finite element code. Due to symmetry, only one-fourth of the sample was modeled using four node layered shell elements that can account for viscoelastic material behavior. Forty layers were used along the sample thickness in order to be consistent with the discretization used within the LAMFLU program. The sorption kinetics model described in Section 3.2.1 was implemented into ANSYS parametric design language via a purpose-built command script. A typical mesh of the model is shown in Fig. 3.26 along with the boundary conditions assigned.

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Creep and Fatigue in Polymer Matrix Composites

3.25 Graphical illustration of a sample divided into p plies with different material properties corresponding to the different ply moisture contents.

3.26 ANSYS finite element model and boundary conditions applied.

Effect of moisture on elastic and viscoelastic

111

3.27 Time variation of normalized (a) transverse σ zz/σ ∞ and (b) longitudinal σ yy/σ ∞ stresses as obtained from elastic (solid lines, black symbols) and viscoelastic (dashed lines, white symbols) analyses. The sample is subjected to a uniform tensile load σ ∞ and constant atmospheric humidity φ ¼ 95.8%.

Fig. 3.27 shows the time variation of the elastic and viscoelastic transverse (σ zz/σ ∞) and longitudinal (σ yy/σ ∞) stresses that develop within the outer and the middle plies, respectively. It should be noted that these two plies represent the two extremes, with the middle ply being under tensile stresses and the outer ply being predominantly under compressive stresses. The stress variations from both elastic and viscoelastic analyses are shown to follow the same trend; however, the time instant at which they attain their maximum or minimum values differs. Moreover, during moisture uptake, viscoelastic stresses are shown to be lower in magnitude than the corresponding elastic, since moisture-induced viscoelastic behavior leads to stress relief. A robust and efficient numerical method for the calculation of the internal stress state that develops within structures subjected to mechanical and steady-state or transient hygroscopic loading conditions has been developed. The method has been validated against finite element analyses and was shown to be mathematically accurate. Results from the application of the proposed method on an epoxy polymer binder indicate that: l

l

Moisture-induced viscoelastic behavior should be taken into account when designing polymer structures, especially if strain-based criteria are to be employed. This is because viscoelastic strains are shown to be up to several times greater (four times in this case study) than those calculated via elastic analysis. A variable stress profile arises within a polymer structure over a significant part of its life span with the stresses at some time instant being several times greater (up to three times in this case study) than the externally applied stress field (i.e., the far field load).

Due to the iterative procedure of the developed method, viscoplastic material behavior can be incorporated in a straightforward manner. It is evident that given research and obtained results do not pretend to general conclusions. One should keep in mind that the epoxy resin under consideration is a typical representative of the epoxies, absorbs a little less than 5% moisture during more than 1 year, and was considered only as a

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model material. Behavior of hydrophilic materials, strongly swelling materials, or anisotropic materials may be essentially different.

3.7

Moisture in nanocomposites

3.7.1 Moisture sorption by polymer nanocomposites The addition of nanoparticles having a high aspect ratio e.g., multiwall carbon nanotubes (MWCNT), nanoclay, etc., to polymers and polymer-based composites may lead to the reduction of negative effects of moisture absorption on their mechanical and physical properties (Glaskova-Kuzmina et al., 2016). These results are mainly explained by hindering action of the nanofiller on the intermolecular movements of the polymers and reduction of the free volume available for moisture absorption (Ray and Rathore, 2014; Starkova et al., 2013). Moreover, according to the tortuous path model and since the moisture permeability is a function of volume fraction and aspect ratio of the nanoparticles, the exfoliated nanocomposites (NC) are preferred over conventional or intercalated composites in terms of the barrier characteristics. The peculiarities of moisture absorption of epoxy-nanoclay composite were estimated in Glaskova and Aniskevich (2009). As shown in Fig. 3.28a, sorption in NC occurred more slowly in comparison with neat epoxy resin. For NC filled with 6 wt % of nanoclay, the diffusivity reduced to about half that of neat epoxy resin. This was explained by the high aspect ratio of the clay platelets, which resulted in an increase of the path length for water molecules upon moisture diffusion through the NC. It was experimentally confirmed that with the increase of nanoclay content in the NC, the equilibrium moisture content increased from 3.2% to 3.4%. The increase of equilibrium moisture content with the increase of clay weight content in NC could be caused by the growth of interphase weight content. The sorption isotherm displays a relationship between equilibrium moisture content in the material and equilibrium RH

3.28 Diffusion coefficient of NC versus nanoclay weight content in φ ¼ 98% (a), and sorption isotherm of epoxy and interphase in NC filled with 1 wt% of nanoclay (b) (Glaskova and Aniskevich, 2009).

Effect of moisture on elastic and viscoelastic

113

3.29 Coefficient of diffusion D (left axis) and equilibrium moisture content w∞ (right axis) of epoxy resin, NC, CFRP, and CFRP-NC (Glaskova-Kuzmina et al., 2016).

of the surrounding atmosphere at a given and constant temperature. In the equilibrium state, it is considered that the total moisture content is distributed homogeneously by the section of the NC samples and according to the sorption isotherm for the moisture concentration in NC. Thus, using the sorption isotherm shown in Fig. 3.28b, it is possible to estimate the effect of the interphase on sorption properties of NC in the whole. Similar results associated with the reduction of the coefficient of diffusion for NC and CFRP filled with 1 wt% of MWCNT were obtained in Glaskova-Kuzmina et al. (2016). As shown in Fig. 3.29, the diffusivity of NC was reduced almost by one-third ( 31%), compared with that of the epoxy resin. The equilibrium moisture content was reduced by approximately 15% in the case of NC, whereas it remained almost identical (reduced by 0.47%) for CFRP-NC. This was explained by the high aspect ratio (30–80) of the MWCNTs, which resulted in increased tortuosity of the path for the water molecules and in a restriction of the molecular dynamics for the polymer chains near the nanoparticles. The improved sorption characteristics also indicated good adhesion between epoxy and MWCNT in NC, and MWCNT and carbon fibers in CFRP-NC systems (Starkova et al., 2013).

3.7.2 Moisture effect on elastic properties of polymer NC and nanomodified FRP The next step after moisture absorption of polymers and polymer-based CM is testing of their mechanical properties and the evaluation of moisture-induced effects. Experimentally measured stress–strain curves of NC filled with 6 wt% of nanoclay moistened at φ ¼ 24%, 77%, and 98% RH are shown in Fig. 3.30 (Glaskova and Aniskevich, 2010). According to these results, the effect of moisture on mechanical behavior is significant, leading to essential plasticization of the NC and change of its fracture character from brittle in a dry atmosphere to plastic in wet atmospheres. The elastic modulus increased by up to 20% and tensile strength of the NC decreased by about the same value with the increase of nanoclay content till 6 wt%. Due to absorbed moisture, the tensile strength of both epoxy resin and NC decreased twice, but the elastic modulus was reduced approximately 1/3 in comparison to a dry state.

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3.30 Typical stress–strain curves for NC with 6 wt% of nanoclay at different RH (indicated on the graph) (Glaskova and Aniskevich, 2010).

3.31 Typical stress–strain curves for epoxy resin (solid lines), nanocomposite (NC; dashed lines), CFRP (solid lines), and CFRP-NC (dashed lines) in dry (1) and moistened (2) states (a), and moisture effect on the relative change of flexural strength and modulus (b) (Glaskova-Kuzmina et al., 2016).

For NC and CFRP filled with 1 wt% of MWCNT, the elastic and viscoelastic properties were studied before (Glaskova-Kuzmina et al., 2014) and after (GlaskovaKuzmina et al., 2016) the moisture absorption in an atmosphere with 98% RH. For this purpose, four cycles of gradually increasing stress, equal to 25%, 50%, 75%, and 90% of the flexural strength, were used for loading during a period of 30 min, followed by 30 min of unloading. The representative stress–strain curves for epoxy, NC, CFRP, and CFRP-NC in dry and moistened states are given in Fig. 3.31a. The relative changes in flexural strength and modulus due to absorbed moisture are shown in Fig. 3.31b for all materials tested. For moistened NC and CFRP-NC, the improved resistance to moisture-induced reduction in both flexural characteristics was revealed. According to Fig. 3.31b, the flexural strengths for moistened EP, NC, CFRP, and CFRP-NC were reduced by 16%, 8%, 12%, and 6%, accordingly. Slightly higher variations were found for the flexural moduli. Thus, the addition of MWCNT to epoxy resin and CFRP resulted in overall reduction of the moisture effect on all these characteristics. The

Effect of moisture on elastic and viscoelastic

115

decrease in the flexural strength of NC and CFRP-NC was twice lower than that of the EP and CFRP systems, whereas the flexural modulus was 1.4 and 3 times lower than that of the EP and CFRP systems.

3.7.3 Moisture effect on viscoelastic properties of polymer NC and nanomodified FRP Generally, the long-term mechanical properties are dependent on moisture content in the polymers and/or NC. To identify such effects, the creep of epoxy-based NC filled with nanoclay at different filler content (2–4 wt%) was examined at a constant tensile load equal to 50% of the breaking one for each NC and at each value of the equilibrium moisture content of the samples moistened previously in Glaskova and Aniskevich (2009). The tests were carried out at RT (20°C  3°C). The creep and creep recovery tests lasted for 7.5 and 17 h, respectively. The results of short-term creep of a dry material showed almost no effect on the creep compliances of the epoxy and NC filled with 4 wt% of nanoclay, while moistening of a material led to a marked increase in the creep compliance. This can likely be explained by the fact that epoxy resin and NC are close to a highly elastic state. The creep and creep recovery curves of NC filled with different filler content at different equilibrium moisture contents were described by using the BoltzmannVolterra linear integral equation and the moisture-time superposition principle (Eqs. 3.26, 3.27, 3.36). Fig. 3.32a presents the approximated compliance curves in creep recovery for NC with 4 wt% of nanoclay. The verification of applicability of the model used showed a satisfactory description of creep experiments (Fig. 3.32b). The functions of moisture-time reduction for NC filled with different filler content of nanoclay

3.32 Experimental compliances in creep recovery (a) and creep (b) of NC filled with nanoclay

▪

(c ¼ 4%) at w ¼ 0 ( ), 2.04% (●), and 3.52% (▲); their approximation by Eq. (3.27) with the account of Eq. (3.36) (a), and calculation by Eq. (3.26) with account of Eq. (3.36) (b) (Aniskevich et al., 2010).

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3.33 Functions of moisture-time reduction of the NC at c ¼ 0 (-◊-), 2% (-□-), 4% (-△-), 6% (--), obtained by approximating the families of creep recovery curves of the material with different moisture content by using Eqs. (3.26) and (3.36) (Aniskevich et al., 2010).

obtained from the approximation are given in Fig. 3.33. In fact, the functions of moisture-time reduction of the NC with different nanoclay content are nonlinear, and their graphs are concave lines describing the growing influence of the absorbed moisture as its content in the material increases. A comparison of the functions of moisture-time reduction of the NC at different values of c showed that the addition of a small amount of nanoclay (c ¼ 2 wt%) weakens the influence of moisture on the viscoelastic properties of the binder in the CM. This was probably caused by the interaction of filler particles with the polymer macromolecules. It is also possible that some part of the absorbed moisture occurs in the interphase (Glaskova and Aniskevich, 2009). Thus the nanomodified materials exhibited improved creep resistance in the moistened state due to the retardation of the relaxation processes on creep in all considered cases. For epoxy resin and NC, the moisture effect normalized to the creep strain of the dry state resulted in an increase at the mean: by 1.33 and 1.32 times for elastic strain, 1.48 and 1.24 times for viscoelastic strain, and 1.53 and 1.28 times for plastic strain, respectively (Glaskova-Kuzmina et al., 2016). For CFRP and CFRP-NC, the effect was insignificant for elastic strain—increase at the mean of 1.09 and 1.03 times, respectively—and more pronounced for viscoelastic—increase at the mean by 1.52 and 1.33 times, respectively—and plastic—increase at the mean by 1.45 and 1.39 times, respectively—strains. The time-stress superposition principle and the Boltzmann-Volterra linear integral Eq. (3.26) was used for the description of a series of creep curves at various stresses for epoxy resin and NC filled with 1 wt% of MWCNT in the moistened state (GlaskovaKuzmina et al., 2016). The master curves of creep compliance for the materials in the moistened state were compared with those obtained for the materials in the dry state (Fig. 3.34a). Obviously, the master curves for moistened epoxy resin and NC were located above the master curves for the same materials in the dry state. Nevertheless, the master curve for moistened NC was positioned under the master curve of moistened

Effect of moisture on elastic and viscoelastic

117

3.34 Master curves for epoxy resin and nanocomposite in dry ( and △) and moistened (● and ▲) states, obtained using the time-stress superposition principle (a) and time-stress shift factor versus difference in applied stress, starting from the stress of the first cycle for epoxy resin and NC filled with 1 wt% of MWCNT in dry (-- and -□-) and moistened (-●- and - -) states (b) (Glaskova-Kuzmina et al., 2016).

▪

epoxy. The results obtained for the time-stress shift factor (Fig. 3.34b) for moistened epoxy and NC completed the preceding analysis of the improved creep resistance due to the addition of MWCNT. Evidently, the time-stress shift factors for the nanomodified epoxy samples, having absorbed moisture, were reduced at all applied stresses. These results provide the possibility to expand the use of such materials to different applications that involve long-term use and require stable mechanical properties under humid conditions. This effect increased with time and provided an improvement in the long-term creep resistance of MWCNT-filled epoxy. Consequently, the operation time of such materials will be longer under both dry and humid conditions.

3.8

Conclusions

Attempts were being made to answer the question of how absorbed moisture affects mechanical properties of composites. The results of complex investigations on sorption and mechanical properties of various polymers and polymer-based composites (including NC) were summarized in this chapter. Moisture diffusion in polymers and composites was described by classical Fick’s model and non-Fickian models depending on experimental conditions and types of materials investigated. For polymer-based FRP, a microstructural approach considering the contribution of each structural component to moisture diffusion was offered. It was shown that moisture can lead to substantial degradation of elastic and viscoelastic properties for both thermosetting and thermoplastic polymers and their composites. To describe the hygrothermal viscoelastic behavior and evaluate long-term

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creep, the time-moisture superposition principle was discussed and applied to various polymers. One of the possibilities for minimizing the negative moisture effect on elastic and viscoelastic properties of polymers and composites is to add moisture impenetrable and stiff nanoparticles (e.g., nanoclay, MWCNT, etc.) to polymers and composites. Thus the nanoparticles act as efficient barriers against moisture transport in the polymers and slightly improve the mechanical properties of the composites, leading to extended durability and lifetime of the polymers and composites, which is crucial for structural applications.

Acknowledgment A. Aniskevich is grateful to ERDF Project No. 1.1.1.1/16/A/141 “Development of nanomodified polyolefin multilayer extrusion products with enhanced operational properties.” T. Glaskova-Kuzmina is grateful to ERDF project No. 1.1.1.2/VIAA/1/16/066 for the support of postdoctoral research “Environmental effects on physical properties of smart composites and fibre-reinforced plastics modified by carbonaceous nanofillers for structural applications.”

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Starkova, O., Aniskevich, A., 2004. Modelling of moisture sorption by CFRP rebars with vinylester matrix. Adv. Compos. Lett. 13 (6), 283–289. Starkova, O., et al., 2013. Water transport in epoxy/MWCNT composites. Eur. Polym. J. 49 (8), 2138–2148. Urzhumtsev, Y.S., 1972. Prediction of the deformation and fracture of polymeric materials. Polym. Mech. 8 (3), 438–450. Vinson, J., 1978. Advanced Composite Materials—Environmental Effects. ASTM International. STP658. Ward, M., 1983. Mechanical Properties of Solid Polymers. Willey. Weitsman, Y., 1977. Hygrothermal viscoelastic analysis of a resin slab under time-varying moisture and temperature. In: AIAA Structures, Dynamics and Materials Conference, San Diego. Weitsman, Y., 1987. Coupled damage and moisture-transport in fiber-reinforced, polymeric composites. Int. J. Solids Struc. 23 (7), 1003–1025.

Micromechanics modeling of hysteretic responses of piezoelectric composites

4

Chien-hong Lin*, Anastasia Muliana† *Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, † Department of Mechanical Engineering, Texas A&M University, College Station, TX, United States

4.1

Introduction

Piezoelectric ceramics, such as lead zirconate titanate (PZT) and barium titanate (BaTiO3), have been widely used as sensors and actuators. Recently, they have also gained popularity for use in energy harvesting devices. Their inherently high electromechanical coupling properties are appealing for actuator applications, where relatively small electric field inputs are sufficient to actuate the ferroelectric ceramics. However, the brittle nature of ceramics limits their applications to only small deformations. In electromechanical devices, several characteristics may be required, such as light weight, high electromechanical coupling constants, low thermal expansion and conductivity, mechanical flexibility and compliance, etc. For this purpose, electroactive composites with several different constituents have been considered. Newnham et al. (1978) discussed the key features in achieving desired properties in developing active composites, which are tailored to the arrangement of the constituents (connectivity). The most common and practical types are composites with active piezoceramic inclusions of particles or long fiber shapes dispersed in continuous soft matrix, for example, polymers. These composites are referred to as 0–3 and 1–3 composites, respectively. Examples of 0–3 and 1–3 piezocomposites are discussed in Tressler et al. (1999) and Babu (2013). Piezoelectric composites are generally fabricated by embedding nonpolarized piezoelectric ceramics into a passive soft matrix, such as polymers and metals. By adjusting spatial concentration and geometry of the piezoelectric inclusions, one can fine tune material properties of the composites to specific device requirements. The composites having piezoelectric particles and polymer matrix are then polarized, which can be quite challenging because the passive polymeric matrix has very low dielectric constants and typically the ceramics inclusions do not form a continuous connection across the composites. Cui et al. (1996) discussed the fact that polarization of the composites with piezoelectric particles can be improved by having a composite with high particle percolations, and/or reducing the ratios of the dielectric constants of the ceramics and polymers. Several experimental studies have shown that Creep and Fatigue in Polymer Matrix Composites. https://doi.org/10.1016/B978-0-08-102601-4.00004-7 © 2019 Elsevier Ltd. All rights reserved.

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piezoelectric composites exhibit more desirable performance than monolithic piezoelectric ceramics: Smith (1986), Safari (1994), and Bent and Hagood (1997). Bent and Hagood (1997) showed that piezocomposites with unidirectional PZT fibers have high electromechanical coupling constants while they are compliant. The overall electromechanical properties and behaviors of piezoelectric composites are strongly influenced by the properties and behaviors of the constituents, that is, inclusions and matrix, microstructural morphologies of the composites such as shape, size, and arrangement of the inclusions, and the volume or weight contents of the constituents. The response of the piezoelectric ceramics also depends on the loading conditions that the piezoelectric ceramics are subjected to. When polarized piezoelectric ceramics are subjected to relatively small electric field inputs, a linear response in stress/strain and electric flux is observed. However, relatively large electric field inputs, even below the coercive electric field limit, lead to nonlinear electromechanical responses; see, for example, Crawley and Anderson (1990) and Park et al. (1998). In such situations, the linear piezoelectricity that was standardized by the IEEE (1987) is no longer applicable to describe the electromechanical behavior of the piezoelectric ceramics. Experimental evidence (Sch€aufele and H€ardtl, 1996; Fett and Thun, 1998; Hall, 2001; Zhou and Kamlah, 2006) shows that the electrical and mechanical responses of the polarized piezoelectric ceramics are time dependent. When subjected to cyclic electric fields with amplitude less than the coercive electric field limit, piezoelectric ceramics show hysteretic strain and electric flux responses, often referred to as minor loop hysteretic responses. The minor loop hysteretic response is frequency dependent and as a result piezoelectric composites also show frequency-dependent hysteretic behaviors; see Khan et al. (2016). When cyclic electric fields with high amplitude, above the coercive electric field limit, are considered, piezoelectric ceramics experience polarization switching, which forms a major loop hysteresis; see, for example, Cao and Evans (1993), Fang and Li (1999), Lente and Eiras (2002), Ren (2004), and Li et al. (2005). The polarization switching response is also time dependent. Furthermore, a polymeric matrix exhibits a viscoelastic response, which eventually affects the hysteretic response of piezoelectric composites (Muliana, 2010). Only limited experimental tests are available on the piezoelectric composites with PZT fibers undergoing polarization switching (Jayendiran and Arockiarajan, 2012, 2013). In order to evaluate the overall properties and responses of piezoelectric composites under various loading histories, several micromechanics models have been formulated. Both linear and nonlinear responses have been considered, including limited studies on the hysteretic responses of piezocomposites. Examples of micromechanical models for piezoelectric composites are Newnham et al. (1980), Banno (1983), Smith and Auld (1991), Hagood and Bent (1993), Nan and Jin (1993), Dunn and Taya (1993), Aboudi (1998), Odegard (2004), Tan and Tong (2001), Li and Dunn (2001), Jiang and Batra (2001), and Lin and Muliana (2013, 2014a). It should be noted that the piezoelectric constitutive models employed in these studies are restricted to piezoelectric phenomena within a polarized state only, in absence of polarization switching. Only limited micromechanical models are available to describe the hysteretic responses of piezoelectric composites due to polarization switching, for example, Aboudi (2005), Muliana (2010), and Jayendiran and Arockiarajan (2012, 2013).

Micromechanics modeling of hysteretic responses of piezoelectric composites

123

Their approaches merely considered rate-independent hysteretic constitutive relations for the ferroelectric constituents. Recently, Lin and Muliana extended their micromechanical models for particle and fiber reinforced piezoelectric composites to include the time-dependent polarization switching behaviors of the piezoelectric inclusions (see Lin and Muliana, 2014b, 2016). In their models, the effect of viscoelastic polymeric matrix on the overall electromechanical response of composites is also studied. In the preceding micromechanics models, a simplified microstructural characteristic was considered in formulating the homogenized (effective) electromechanical response of the piezoelectric composites. For example, Dunn and Taya (1993) considered a microstructural characteristic based on the Mori-Tanaka model, in which a single inclusion is embedded in an effective (homogenized) matrix medium, while Aboudi (2005) considered a unit-cell model with several inclusion and matrix subcells. One of the main advantages of the models with a simplified microstructural characteristic is that it allows incorporating rigorous nonlinear constitutive models for the constituents, while being computationally efficient, with the scope of predicting the overall response of composites. However, this type of micromechanics model is limited in capturing the variations in the field variables (stress, strain, electric flux, electric field, displacement, electric potential), including the effect of localized and/or discontinuity of the field variables on the overall response of the composites. Another type of micromechanics model considers more detailed microstructural morphologies, such as distribution, size, and shape of the inclusions, possible existence of voids, placement of the electrodes, etc. Examples are Nelson et al. (2003), Tajeddini et al. (2014), and Ben-Atitallah et al. (2016). These micromechanics models have an advantage in capturing the variations in the field variables including the localized and/or discontinuity in field variables. In this chapter, we present micromechanical models for fiber and particle reinforced composites (PRCs) in order to obtain the effective hysteretic responses of 1–3 and 0–3 piezoelectric composites, respectively. The 1–3 and 0–3 composites are referred to as fiber-reinforced composite (FRC) and PRC, respectively. The micromechanics models are formulated based on simplified unit-cell models with several subcells (see Fig. 4.1). A phenomenological constitutive model accounting for a rate-dependent polarization switching behavior proposed by Sohrabi and Muliana (2013) is used for the piezoelectric subcells, while the polymeric matrix is considered as a linear viscoelastic material. Detailed micromechanical model formulations are discussed in Lin and Muliana (2014b). Based on these unit-cell models for the fiber and PRCs, another micromechanics model for hybrid composite is formulated. The hybrid composite is composed of a unidirectional fiber reinforcement embedded in a matrix medium reinforced with particles. The intention of adding particulate inclusions to the matrix is to increase the overall properties of the matrix; thus it can minimize the differences in the fiber and matrix properties. For example, increasing the dielectric properties of the matrix can help with polarizing the composites. The micromechanics model for the hybrid composite is formulated by integrating the micromechanics model of the 0–3 piezocomposite to the matrix subcells in the unit-cell model of 1–3 piezocomposite (see Fig. 4.2). This chapter is organized as follows: Section 4.2 briefly discusses the constitutive models for the constituents followed

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Creep and Fatigue in Polymer Matrix Composites

4.1 Fiber- and particle-unit-cell models.

by numerical methods for solving the coupled nonlinear electromechanical constitutive relations. Section 4.3 presents the micromechanical formulation of the fiber-unitcell and particle-unit-cell models. Numerical results of the effective dielectric and strain hysteretic responses of the 1–3 and 0–3 ferroelectric composites are also discussed. Section 4.4 presents the model for hybrid composites and its numerical implementation. Hysteretic responses at different amplitude of electric field inputs are studied. Finally, Section 4.5 is dedicated to conclusions.

Micromechanics modeling of hysteretic responses of piezoelectric composites

125

4.2 Hybrid-unit-cell model.

4.2

Constitutive models

In this section, constitutive material models for PZT inclusions undergoing polarization switching and polymeric matrix with a linear viscoelastic response are briefly discussed.

4.2.1 Polarization switching model For the PZT inclusions, the rate-dependent polarization switching constitutive model proposed by Sohrabi and Muliana (2013) is adopted. Fig. 4.3 illustrates typical hysteretic polarization and strain responses. When at the remanent stage (point B in Fig. 4.3) of a cyclic electric field with relatively small amplitude (below coercive electric field), hysteretic responses form minor loops, shown by dashed lines. The model considered here is capable of capturing both minor and major hysteretic loops. The strain and electric flux are given as: εtij
εij ðtÞ ¼ sijkl σ tkl + 4gtnij κnm gtmkl σ tkl + gtkij Ptk ,

(4.1)

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Creep and Fatigue in Polymer Matrix Composites

4.3 Examples of the (a) polarization and (b) butterfly strain responses in ferroelectric materials due to a cyclic electric field. The operating range of a polarized piezoelectric material is shown by a dashed line.

Dti ¼ 2κim gtmkl σ tkl + Pti ,

(4.2)

where the upper right superscript t denotes the current time variable, and εtij, σ tij, Dti, and Pti are the components of the strain, stress, electric displacement, and polarization tensors at current time t, respectively. The material properties are the elastic compliance tensor sijkl determined at a constant electric field and dielectric coefficient tensor κ ij calibrated at a constant stress. The scalar component of the piezoelectric coefficient gtijk is formulated based on the current polarization Pt3 and stress σ t33 with x3 direction as the poling axis, and is given as: c2 |σ 33 | 3| Pt3 |P e c1 e σ c grijk , c2 ¼ 0, if σ t33 > σ c , Pr t

gtijk ¼

t

(4.3)

where C1 and C2 are the material parameters that need to be calibrated from experiment;σ c is the coercive stress limit; grijk is the scalar component of the piezoelectric coefficient measured at constant (remanent) polarization Pr. It is noted that gr ¼ κ1dr, where dr is the direct piezoelectric constant measured at remanent polarization. Eq. (4.3) indicates that in absence of polarization Pt3 the material does not have electromechanical coupling properties, that is, for piezoelectric materials with randomly oriented dipoles the piezoelectric constants are zero. The scalar components of polarization along the x1 and x2 direction are Pt1 ¼ κ 11 Et1 ,

(4.4)

Pt2 ¼ κ 22 Et2 ,

(4.5)

where Eti is the electric field tensor at current time t. The scalar component of the polarization Pt3 (i.e., along the x3 direction) depends on the applied electric field input, that

Micromechanics modeling of hysteretic responses of piezoelectric composites

127

is, Et3. In order to account for polarization switching, the polarization is decomposed into reversible and irreversible parts:

 Pt3 ¼ R Eτ3 , t + Q Et3 ,

(4.6)

where R(Eτ3, t) is the time-dependent reversible polarization at current time t with R(0, t) ¼ 0 while Q(Et3) is residual (irreversible) polarization. The upper right superscript τ indicates the previous time variable. The reversible polarization is written as:
 Rt ¼ R E03 , t +

ðt

 dEτ3 ∂
τ dτ, τ R E3 , t  τ dτ 0 ∂E3

(4.7)

where   


 t R E03 , t ¼ R0 E03 + R1 E03 1  exp  : τ1

(4.8)

One may consider R(E03, t) as the polarization at current time t due to a constant electric field applied at τ ¼ 0. Both R0(Eτ3) and R1(Eτ3) are functions of Eτ3. The characteristic time τ1measures the rate of the polarization changes with time. The irreversible polarization is given as: Qt ¼

ð Et

3

0

dQτ τ dE : dEτ3 3

(4.9)

The rate of the residual polarization during a polarization switching response is modeled using the following ad hoc expression: 8 0, > > > > > > > >  t n > >   > > > λ  E3  , > E  < t dQ c ¼ t > dE3 > > >   t   > > E3 > > > μ exp ω 1 , > > Ec > > > :

0  Et3 < Em , dEt3 < 0 or  Em < Et3  0, dEt3 > 0, Ec  Et3 < 0, dEt3  0 or 0 < Et3  Ec , dEt3  0,

(4.10)

Em  Et3 < Ec , dEt3  0 or Ec < Et3  Em , dEt3  0,

where λ, μ, ω, n are the material parameters that are calibrated from experiments. The parameters Ec and Em are the coercive and maximum electric field, respectively. A similar function with different material parameters can be used for modeling the initial polarization, as discussed in Muliana (2011).

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Creep and Fatigue in Polymer Matrix Composites

The compressive stresses along the poling axis could significantly affect the hysteretic polarization switching response. In this study, it is assumed that the coercive electric field varies with the compressive stresses along the x3 direction, which has to be calibrated from the experimental data: Ec ¼


 Ec E0c , σ t33 , σ t33 < 0, E0c , σ t33  0,

(4.11)

where E0c is the coercive electric field in the absence of mechanical stresses. The polarization switching model has been compared to experimental data of PZT 51, obtained from Fang and Li (1999). The comparisons between the model and experiment were given in Sohrabi and Muliana (2013). It is seen that the preceding model requires a number of material parameters that need to be calibrated from experiments. It is also noted that the discontinuities in the functions used in Eq. (4.10) for the irreversible part of the polarization may pose some difficulties in the numerical implementation. Alternatively, a new approach based on a multiple natural configuration theory has been formulated for describing the polarization switching response of materials (Song et al., 2018). This approach leads to continuous changes in the microstructures during the hysteretic response and smaller number of material parameters.

4.2.2 Linear viscoelastic model The polymeric matrix is assumed to be an isotropic viscoelastic solid with regards to its mechanical response and the constitutive models are described as: εtij ¼ ð1 + νÞ

ðt

Dðt  τÞ

0

dSτij ð1  2νÞ δij dτ + 3 dτ

Di ¼ κij Ej :

ðt 0

Dðt  τÞ

dσ τkk dτ, dτ

(4.12) (4.13)

For simplicity, we assume that the corresponding linear elastic Poisson’s ratio v is time independent to Eq. (4.12) because the experimental data on the Poisson’s effect of viscoelastic materials are not often available. Here Stij and σ tkk are the components of the deviatoric stress and volumetric stress tensors at current time t, respectively; δij is the Kronecker delta, and D(t) is the extensional (uniaxial) time-dependent compliance, which is expressed as: DðtÞ ¼ D0 +

N X

Dn ð1  exp ½λn tÞ:

(4.14)

n¼1

Here D0 is the instantaneous (elastic) compliance and the transient compliance is expressed in terms of a series of exponential functions, where N is the number of

Micromechanics modeling of hysteretic responses of piezoelectric composites

129

terms, Dn is the nth coefficient of the time-dependent compliance, and λn is the nth reciprocal of retardation time.

4.2.3 Linearized forms for the constitutive models For convenience in analyzing the time-dependent and nonlinear electromechanical behavior, we present a linearized incremental form of the constitutive relations, that is, Eqs. (4.1), (4.2), (4.12), and (4.13). A recursive time-integration algorithm presented in Taylor et al. (1970) is used to numerically evaluate the time integral forms of the constitutive models such as Eqs. (4.7) and (4.12). The linearized incremental formulations are implemented in each subcell within the unit-cell models. The incremental independent field variables at current time t are: Δσt ¼ σt  σtΔt ,

(4.15)

ΔEt ¼ Et  EtΔt ,

(4.16)

where superscript t  Δt denotes the previous time and Δt is the current incremental time. A detailed procedure of the numerical implementation is given in Lin and Muliana (2014b). With the incremental form for the independent field variables, the linearized constitutive relation is expressed as: Ξt ¼ Ot ΔTt + ΞtΔt ,

(4.17)

where Ξt ¼

εt , Dt

# t e0 t e s d O ¼ t t , e e d κ

(4.18)

"

t

ΔTt ¼

Δσt : ΔEt

(4.19)

(4.20)

Ot is a 9  9 matrix and Ξt and ΔTt are 9  1 column vectors; ΞtΔt is the history variables of the dependent field variables Ξt. A factor of two for the shear strains is accounted for in the vector Ξt. This matrix formulation of the linearized constitutive relation will be used in the following micromechanical analysis. After some lengthy algebraic manipulations, the resulting components of Ot and ΞtΔt for PZT and the polymer constitutive model are summarized as follows.

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Creep and Fatigue in Polymer Matrix Composites

For the PZT inclusions with constitutive relations in Eqs. (4.1) and (4.2), the resulting components of Ot and ΞtΔt are: e s tijmn ¼ sijmn + 4gtkij κ kl gtlmn , 0 denij ¼ gtnij Kklt , t

t deimn ¼ 2κij gtjmn , t e κ tin ¼ Kin , 
 ¼ sijkl + 4gtnij κnm gtmkl σ tΔt + gtkij PtΔt + Ftk , εtΔt ij kl k

(4.21)

¼ 2κim gtmkl σ tΔt + PtΔt + Fti , DtΔt i kl i In Eq. (4.21), Ktij is: 8 i ¼ j ¼ 1, κ11 , > > < κ22 , i ¼ j ¼ 2, t Kij ¼ t κ + κ + ΔQ , i ¼ j ¼ 3, > 0 1 > : 0, i 6¼ j:

(4.22)

where κ 0 is referred to as the dielectric constant and κ 1 is the time-dependent part of the dielectric constant, at the initial stage (all dipoles are randomly oriented). Using the rate of residual polarization in Eq. (4.10), the incremental residual polarization at current time t is: ΔQt 

dQt t ΔE : dEt3 3

(4.23)

In addition, in Eq. (4.21), Fti and Pti are: 8 0, i ¼ 1,2, > >    > > Δt > < 1  exp  qtΔt τ1 Fti ¼
 tΔt  >   
t t > > ∂R1 E3 dE3 Δt ∂R1 EtΔt dE3 Δt > 3 > :  , i ¼ 3, + exp  τ1 2 ∂Et3 dt dt ∂EtΔt 3

(4.24)

Pti ¼ PtΔt + ΔPti , i

(4.25)

where the history variable related to the polarization is:
 tΔt      
 Δt tΔt ∂R1 Et3 dEt3 Δt ∂R1 EtΔt dE3 Δt 3 t , q ¼ exp  + exp  + q τ1 τ1 2 ∂Et3 dt dt ∂EtΔt 3 (4.26) and the incremental polarization is determined by: ΔPti ¼ Kijt ΔEtj + Fti :

(4.27)

Micromechanics modeling of hysteretic responses of piezoelectric composites

131

For the polymeric matrix with constitutive relations, on the other hand, in Eqs. (4.12) and (4.13), the resulting components of Ot and ΞtΔt are: e s tijmn

  ^ ^ 1^ 1^ B  J δij δmn + 2 J δik δjl  J δil δjk , ¼ 3 3

0 denij ¼ 0, t

t deimn ¼ 0, κeint ¼ κtin ,  ^  ^ 1 ^ tΔt tΔt tΔt t tΔt t t B σ kk  Vkk , εij ¼ 2 J Sij  dij + δij  J Sij + dij + 3

(4.28)

¼ κij EtΔt : DtΔt i j ^ ^

In Eq. (4.28), B , J , dtij, and Vtkk are: # 1  exp ½λn Δt B ¼ ð1  2νÞ D0 + Dn  Dn , λn Δt n¼1 n¼1 " # N N X X ^t 1  exp ½λn Δt Dn  Dn , J ¼ ð1 + νÞ D0 + λn Δt n¼1 n¼1   N X 1  exp ½λn Δt tΔt t tΔt Sij dij ¼ ð1 + νÞ Jn exp ½λn Δtqij,n  , λn Δt n¼1   N X 1  exp ½λn Δt tΔt t tΔt σ kk , Vkk ¼ ð1  2νÞ Jn exp ½λn Δtqkk,n  λn Δt n¼1 ^t

"

N X

N X

(4.29)

where the history variables related to the deviatoric and volumetric strains are: 

 1  exp ½λn Δt  t , Sij  StΔt ij λn Δt    1  exp ½λn Δt
t t tΔt : qkk,n ¼ exp ½λn Δtqkk,n + σ kk  σ tΔt kk λn Δt

qtij,n ¼ exp ½λn ΔtqtΔt ij,n +

4.3

(4.30)

Fiber- and particle-unit-cell models

This section presents the formulation of the micromechanics models for the 1–3 and 0–3 piezoelectric composites. We also compare predictions of the effective responses of 1–3 and 0–3 piezoelectric composites with existing experimental results. It is noted that limited experimental data are available for the 0–3 composites. We then conduct parametric studies investigating the effects of constituent compositions and loading history on the overall hysteretic performance of both 1–3 and 0–3 ferroelectric composites.

132

Creep and Fatigue in Polymer Matrix Composites

4.3.1 Formulation of the unit-cell models The unit-cell (UC) models for the fiber and PRCs, that is, 1–3 and 0–3 composites, respectively, are discussed in this section. A detailed discussion of the models can be found in Lin and Muliana (2014b). The composite microstructures are idealized with periodically distributed arrays of inclusions in a homogeneous matrix, as shown in Fig. 4.1. For each of the 1–3 and 0–3 models, a unit-cell model comprising several subcells is identified. The inclusions are assumed to be fully surrounded by homogeneous matrix. The first subcell denotes the piezoelectric constituent and the rest of the subcells indicate the polymeric matrix. The interfaces between all subcells are assumed perfectly bonded. The UC models result in rather simple micromechanical relations by satisfying the equilibrium condition and displacement compatibility in both mechanical and electrical domains among all subcells. The time-integration algorithms for the rate-dependent PZT (i.e., Eq. 4.7) and viscoelastic matrix (i.e., Eq. 4.12) are implemented in the subcells within the UC model in order to obtain approximate solutions of the overall nonlinear and time-dependent responses of the 1–3 and 0–3 composites. A volume averaging method is considered in order to determine the overall field variable in a UC, which is: t

Ξ ¼

n X

cðαÞ ΞðαÞ,t ,

(4.31)

α¼1

where the superscript α denotes the subcells’ number and n is the number of subcells (e.g., n ¼ 4 for the fiber UC and n ¼ 8 for the particle UC). The boldface variable indicates the first-order tensor or higher. The variable Ξ(α), t is the average field variable within each subcell with a volume V(α). The UC volume V is given by. n X V ðαÞ : (4.32) V¼ α¼1

A linearized constitutive relation for the overall composites at current time t is written as: t

t

t

tΔt

Ξ ¼ O ΔT + Ξ

,

(4.33)

and also for the subcell (α): ΞðαÞ,t ¼ OðαÞ,t ΔTðαÞ,t + ΞðαÞ,tΔt :

(4.34)

In order to relate the effective incremental independent field variable in the UC to the corresponding field variable in its subcells, a concentration matrix B(α), t needs to be defined. The field variables in the subcells and the ones in the unit cell are related as: ΔTðαÞ,t ¼ BðαÞ,t ΔT + XðαÞ,t , t

(4.35)

Micromechanics modeling of hysteretic responses of piezoelectric composites

133

where X(α), t is the vector of history variables at current time t. Substituting ΔT(α), from Eq. (4.35) into Eq. (4.34) gives. ΞðαÞ,t ¼ OðαÞ,t BðαÞ,t ΔT + OðαÞ,t XðαÞ,t + ΞðαÞ,tΔt : t

t

(4.36)

Substituting Ξ(α), t from Eq. (4.36) into Eq. (4.31) gives. t

Ξ ¼

n X

cðαÞ OðαÞ, t BðαÞ,t ΔT + t

α¼1

n X

 cðαÞ OðαÞ, t XðαÞ,t + ΞðαÞ,tΔt :

(4.37)

α¼1

From Eqs. (4.37) and (4.33), the effective electromechanical properties and history variables of the UC are: t

O ¼

n X

cðαÞ OðαÞ,t BðαÞ,t :

(4.38)

α¼1

Ξ

tΔt

¼

n X

 cðαÞ OðαÞ,t XðαÞ,t + ΞðαÞ,tΔt :

(4.39)

α¼1

In order to evaluate the concentration matrices and history variables B(α), t and X(α), t in the fiber and particle UC models, it is necessary to use the constitutive relations for the piezoelectric and polymer constituents together with the linearized micromechanical relations from the fiber UC and the particle UC. The linearized micromechanical relations can be found in Lin and Muliana (2013). Because of the nonlinear constitutive relations for the constituents, the linearized micromechanical relations generally violate the overall nonlinear responses, and the fixed-point iterative method is used to minimize the residual. A detailed discussion can be found in Lin and Muliana (2014b).

4.3.2 Experimental validation The UC models are first compared to available experimental data in the literature. The UC model for 1–3 piezocomposites is first compared to the experimental data of Chan and Unsworth (1989) for basic mechanical and electrical properties, as summarized in Fig. 4.4. The properties of the constituents, which are reported by Dunn and Taya (1993) and Chan and Unsworth (1989), are listed in Table 4.1. The response from the UC model is also compared to other micromechanics models, for example, Mori-Tanaka (MT) and self-consistent (SC) models. The predictions from the micromechanics models are in good agreement with the experimental data over the range of volume fractions considered. For the coupling parameter kp and compliance s1111 + s1122 shown in Fig. 4.4, the UC and MT models give nearly identical estimations but the predictions from the SC model slightly deviate from those obtained using the UC and MT models. These discrepancies are probably due to the large contrast in the properties of the inclusions and matrix constituents.

134

Creep and Fatigue in Polymer Matrix Composites 200

500

400 (pm/V)

150

UC MT SC Experiment (Chan and Unsworth, 1989)

κ33 /κ0

300

d

333

100

UC MT SC Experiment (Chan and Unsworth, 1989)

50

(a)

0 0

0.2

0.4 0.6 Fiber volume fraction

0.8

200

100

1

(b)

0 0

0.2

0.4 0.6 Fiber volume fraction

0.8

1

140 (x10–12 m2/N)

0.5

1122

0.3

1111

0.2

80 60

0.2

(c)

0.4 0.6 Fiber volume fraction

0.8

40

s

UC MT SC Experiment (Chan and Unsworth, 1989)

0.1 0 0

100

+s

k

p

0.4

UC MT SC Experiment (Chan and Unsworth, 1989)

120

20 0 0

1

(d)

0.2

0.4 0.6 Fiber volume fraction

0.8

1

4.4 Comparison of various micromechanical predictions to the experimental data (Chan and Unsworth, 1989) for the effective (a) piezoelectric strain coefficient d 333 , (b) relative


 0:5 permittivity κ 33 =κ 0 , (c) coupling parameter kp ¼ 1  κ ε33 cD , and 3333 =κ 33 c3333 (d) compliance s1111 + s1122 for the PZT-7A/Araldite D polarized 1–3 piezocomposite as a function of polarized PZT-7A fiber VF. cD ijkl is the elastic stiffness at a constant reference electric displacement.

Table 4.1 Electromechanical material properties for the PZT-7A and Araldite D

PZT-7Ab Araldite Dc

c1111 (GPa)

c1122 (GPa)

c1133 (GPa)

c3333 (GPa)

c2323 (GPa)

d311 (pm/V)

d333 (pm/V)

d113 (pm/V)

κ11 a κ0

κ33 κ0

148 8

76.2 4.4

74.2 4.4

131 8

25.4 1.8

60 0

150 0

362 0

840 4

460 4

κ 0 ¼ 8.85  1012 (F/m) denotes vacuum permittivity. Dunn and Taya (1993). Transversely isotropic PZT-7A with longitudinal axis and poling direction along the x3-axis. c Chan and Unsworth (1989). Elastic (instantaneous) isotropic properties. a

b

Micromechanics modeling of hysteretic responses of piezoelectric composites

135

Jayendiran and Arockiarajan (2013) conducted experiments on studying effective dielectric hysteresis and butterfly strain responses for a stress-free PZT-5A1/epoxy 1–3 ferroelectric composite with various PZT-5A1 fiber volume fractions (VFs) due to a cyclic triangular bipolar electric field loading with amplitude of E3 ¼ 2 MV/m along the poling direction (x3 direction). The material properties of the PZT-5A1 are calibrated according to experimental data from the dielectric hysteresis and the butterfly strain curves. The calibrated material properties listed in Tables 4.2–4.5 are able to capture the experimental results quite well, even though there are several discrepancies between the calibration and the experiments of strain responses at the coercive electric field, as discussed in Lin and Muliana (2014b). Fig. 4.5 depicts the comparisons between micromechanical predictions and experimental data. Again, the agreement of the effective dielectric hysteresis between the micromechanical predictions and experimental data is good. Table 4.2 Material parameters for the time-dependent polarization of PZT-5A1 (calibrated from experimental data of Jayendiran and Arockiarajan, 2013) and PZT-51 (Sohrabi and Muliana, 2013)

PZT-51 PZT-5A1 _

_

E0c (MV/m)

κ0 × 1029 (F/m)

κ1 × 1029 (F/m)

0.67 1.19

70 50

225 0

_

τ 1 (s)

λ × 1026 (F/m)

1 

0.35 0.50

n

μ × 1026 (F/m)

ω

3 4

1.6 2.8

4 6

_

Note: λ ¼ 0:5λ; n ¼ n; μ ¼ 0:5μ; ω ¼ ω.

Table 4.3 Electromechanical coupling parameters for PZT-5A1 (calibrated from experimental data of Jayendiran and Arockiarajan, 2013) and PZT-51 (Sohrabi and Muliana, 2013)

PZT-51 PZT-5A1

dr333 × 10212 m/V

dr311 × 10212 m/V

κr11 × 1029 F/m

κr33 × 1029 F/m

Pr (C/m2)

C1

1520 440

570 185

38.0 16.7

42.0 16.4

0.194 0.200

0.19 0.30

Table 4.4 Elastic constants for PZT-5A1 (calibrated from experimental data of Jayendiran and Arockiarajan, 2013) and PZT-51 (Sohrabi and Muliana, 2013)

PZT-51 PZT-5A1

E11 5 E22 (GPa)

E33 (GPa)

G12 (GPa)

G31 5 G32 (GPa)

v12

v31 5 v32

34.48 59.62

33.00 48.97

13.19 21.85

12.37 21.00

0.307 0.364

0.334 0.474

136

Creep and Fatigue in Polymer Matrix Composites

Table 4.5 Material parameters above the coercive stress limit for PZT-5A1 (calibrated from experimental data of Jayendiran and Arockiarajan, 2013) and PZT-51 (Sohrabi and Muliana, 2013)

PZT-51 PZT-5A1

σ c (MPa)

C2

λ × 1026 F/m

n

μ × 1026 F/m

ω

25 28

0.3 0.5

0.40 0.51

3 4

1.1 2.4

4 6

0.5

7000

σ33 = 0 MPa, FRC

6000

σ33 = 0 MPa, FRC

ε 33 (microstrain)

D (C/m2)

5000

3

0

4000 3000 2000 1000 0 –1000

–0.5 –2

(a)

–1

0 E (MV/m) 3

1

2

–2000 –2

(b)

–1

0

1

2

E (MV/m) 3

4.5 Comparison of micromechanical predictions (solid lines) to experimental data (circles) of Jayendiran and Arockiarajan (2013) for the dielectric hysteresis and butterfly strain responses for the PZT-5A1/epoxy fibrous ferroelectric composite with fiber VF ¼ 0.8 subjected to both a cyclic triangular bipolar electric loading with amplitude up to E3 ¼ 2 MV=m.

Fig. 4.6 shows the UC predictions of the polarized 0–3 piezocomposites. Experimental data from Furukawa et al. (1976) and Zeng et al. (2002) are considered. In Furukawa et al. (1976) the polarized 0–3 piezocomposite consists of spherical polarized PZT-5 particles embedded in an epoxy I medium. The properties of the constituents used for the numerical prediction were reported by Dunn and Taya (1993) and are listed in Table 4.6. In the data obtained by Zeng et al. (2002), the polarized 0–3 piezocomposite comprises spherical polarized PZTs (Navy type II) particles embedded in a P(VDF-TrFE) matrix. The material parameters used in the UC model are listed in Table 4.7. Overall good predictions are observed. Another prediction of a micromechanics model for 0–3 piezocomposite was carried out on PZT particles dispersed in PA matrix, which was experimentally conducted by Babu (2013). The material properties for the PZT and PA matrix are listed in Table 4.8. Fig. 4.7 gives the comparisons between the micromechanics models and experimental data, showing large discrepancies. We believe that the source is due to large mismatches in the properties of the inclusion and the matrix. As the volume content of particles increases, the conductive inclusion accelerates the conduction process in the composites, termed a conductive chain mechanism. This behavior, which was observed in heat and electrical conduction processes, has been previously discussed by Maxwell (1954), Agri and Uno (1985), Zhou et al. (2007), Khan and Muliana

Micromechanics modeling of hysteretic responses of piezoelectric composites

UC MT SC Experiment (Furukawa et al., 1976)

–d311 (x10–14 m/V)

15,000

10,000

5000

0 0

0.2

0.4

0.6

0.8

1

Particle volume fraction

137

4.6 Comparison of various micromechanical predictions to the experimental data. (Top) The effective piezoelectric strain coefficient d 311 for the PZT-5/epoxy I polarized 0–3 piezocomposite as a function of polarized PZT-5 particle VF. (Bottom) The effective relative permittivity for the PZT (Navy type II)/P(VDFTrFE) polarized 0–3 piezocomposite as a function of polarized PZT (Navy type II) particle VF.

2000 UC MT SC Experiment (Zeng et al., 2002)

k33/k0

1500

1000

500

0

0

0.2

0.4 0.6 0.8 Particle volume fraction

1

Table 4.6 Electromechanical material properties for PZT-5 and epoxy I

PZT-5a Epoxy Ib

c1111 (GPa)

c1122 (GPa)

c1133 (GPa)

c3333 (GPa)

c2323 (GPa)

d311 (pm/V)

d333 (pm/V)

d113 (pm/V)

κ11 κ0

κ33 κ0

121 8

75.4 4.4

75.2 4.4

111 8

21.1 1.8

171 0

374 0

584 0

1700 4.2

1730 4.2

a

Dunn and Taya (1993). Transversely isotropic PZT-5 with poling direction along the x3-axis. Dunn and Taya (1993). Elastic (instantaneous) isotropic properties.

b

(2010), and Xing et al. (2018). The micromechanical models that rely mainly on the constituent properties and geometrical parameters are unable to capture the chain conductive mechanism. In the previous examples (Fig. 4.6), the particle content was relatively low or the mismatches in the properties of the constituents were smaller than the data shown in Fig. 4.7.

138

Creep and Fatigue in Polymer Matrix Composites

Table 4.7 Electromechanical material properties for PZT (Navy Type II) and P(VDF-TrFE)

PZT (Navy type II)a P(VDFTrFE)b

d333 (pm/V)

d113 (pm/V)

κ11 κ0

κ33 κ0

175

400

580

1800

1800

0

0

0

9.9

9.9

c1111 c1122 c1133 (GPa) (GPa) (GPa)

c3333 c2323 d311 (GPa) (GPa) (pm/V)

98.2

44.1

44.1

98.2

27

4.8

3.2

3.2

4.8

0.8

a

Manufacturer’s data sheet for powder PKI502 supplied by piezo kinetics. Transversely isotropic PZT (Navy type II) with poling direction along the x3-axis. Zeng et al. (2002). Elastic isotropic properties.

b

Table 4.8 Electromechanical material properties for the PZT and PA polymer

PZTa PAb

c1111 (GPa)

c1122 (GPa)

c1133 (GPa)

c3333 (GPa)

c2323 (GPa)

d311 (pm/V)

d333 (pm/V)

d113 (pm/V)

κ11 κ0

κ33 κ0

120 1.12

75 0.48

75 0.48

140 1.12

21 0.32

240b 0.25b

480 0.5

580 0.4

1800 9.9

1800 9.9

a

Babu (2013) with an assumption of Poisson ratio of 0.45. It is assumed to be half of d333.

b

4.3.3 Parametric studies In order to examine the effect of loading history (i.e., various loading frequencies) we consider a rate-dependent PZT-51 as an inclusion, that is, fiber or particle, and a timedependent 934 viscoelastic epoxy as a matrix to 1–3 and 0–3 ferroelectric composites. The material property of the 934 viscoelastic epoxy whose dielectric constant is 0.064  109 F/m is listed in Table 4.9. The effective hysteretic responses of PZT51/934 composites with inclusion volume content of 0.5 subjected to cyclic electric loadings E3 ¼ 1:2 sin ð2πftÞ along the poling axis, with different frequencies f ¼ 0.5, 1, and 10 Hz are shown in Fig. 4.8 for the 1–3 piezoelectric composites and in Fig. 4.9 for the 0–3 piezoelectric composites. From Figs. 4.8 and 4.9, it is seen that the matrix dominates the overall responses of the 0–3 ferroelectric composites, while the fibers dominate the overall longitudinal responses of the 1–3 ferroelectric composites, as expected. For the 1–3 piezocomposites (Fig. 4.8), lower frequency loading results in larger hysteretic response because slower loading allows for the materials to experience more pronounced time-dependent response. In this analysis, PZT-51 fibers experience a creep-like polarization response, while the matrix exhibits viscoelastic deformation. For the higher frequency loading, smaller hysteretic responses are seen and a saturated (steady-state) condition is reached after the first cycle, indicating negligible time-dependent response. Under high-frequency loading (10 Hz) the hysteretic response is mainly due to the irreversible polarization during dipolar switching. This is because the loading rate is significantly higher than the characteristics time of the materials. In Fig. 4.9, the 0–3 ferroelectric composites are

Micromechanics modeling of hysteretic responses of piezoelectric composites

139

4.7 Comparison of various micromechanical predictions to the experimental data of 0–3 composites of PZT particle and PA matrix, obtained from Babu (2013). (a) Relative permittivity and (b) piezoelectric constant.

Table 4.9 Time-dependent compliance, instantaneous time-dependent (elastic) compliance and Poisson’s ratio for the viscoelastic 934 epoxy at 22°C n

λn (min21)

Dn (GPa21)

1 2 3

1 101 102

0.0150 0.0050 0.0120

D0 ¼ 0.2217(GPa1 ). v ¼ 0.311. The coefficients of Prony series are determined from Yancey, R. N., and M.-J. Pindera. 1990. Micromechanical analysis of the creep response of unidirectional composites. J. Eng. Mater. Trans. ASME, 112 (2), 157–163.

140

Creep and Fatigue in Polymer Matrix Composites 0.2

3500 f = 0.5 Hz, FRC Time elapsed = 6 s

0.15

ε 33 (microstrain)

0

3

D (C/m2)

0.1 0.05

–0.05 –0.1

–0.2 –1.5

–1

–0.5

(a)

0

0.5

1

1000

0 –1.5

3

–1

–0.5

0

0.5

1

1.5

0.5

1

1.5

0.5

1

1.5

E (MV/m) 3

3500 f = 1 Hz, FRC Time elapsed = 3 s

f = 1 Hz, FRC Time elapsed = 3 s

3000

ε 33 (microstrain)

0.1 0.05 0

3

D (C/m2)

1500

(b)

0.2

–0.05 –0.1

2500 2000 1500 1000 500

–0.15 –0.2 –1.5

–1

–0.5

(c)

0

0.5

1

0 –1.5

1.5

–1

–0.5

(d)

E (MV/m) 3

0.2

0 E (MV/m) 3

3500 f = 10 Hz, FRC

f = 10 Hz, FRC Time elapsed = 0.3 s

3000

ε 33 (microstrain)

0.1 0.05 0

3

D (C/m2)

2000

1.5

E (MV/m)

0.15

–0.05 –0.1

–0.2 –1.5

Time elapsed = 0.3 s

2500 2000 1500 1000 500

–0.15

(e)

2500

500

–0.15

0.15

f = 0.5 Hz, FRC Time elapsed = 6 s

3000

–1

–0.5

0 E3 (MV/m)

0.5

1

0 –1.5

1.5

(f)

–1

–0.5

0 E (MV/m) 3

4.8 Frequency effect (f ¼ 0.5 Hz for a, b; 1 Hz for c, d; and 10 Hz for e, f ) on the effective dielectric hysteresis and butterfly strain responses for the stress-free PZT-51/934 viscoelastic epoxy fibrous piezocomposite with fiber VF ¼ 0.5 undergoing a cyclic sinusoidal electric loading E3 ðtÞ ¼ 1:2 sin ð2πftÞ MV/m. Only the first three cycles are plotted.

insensitive to the change of electric loading frequency, because only a small fraction of the applied electric field reaches the ferroelectric particles and consequently the ferroelectric behavior of the ferroelectric particles does not significant impact the overall responses of the 0–3 ferroelectric composites. It is also noted that, due to

Micromechanics modeling of hysteretic responses of piezoelectric composites

x 10

–4

0.025 f = 0.5 Hz, PRC

Time elapsed = 6 s Span = 5.259 x 10 –4 C/m2

0.02

ε 33 (microstrain)

2

D (C/m2) 3

1 0 –1

–1.5

–1

–0.5

0

0.5

1

1.5

E (MV/m) 3

x 10

0.01

(b)

0 –1.5

–1

–0.5

0

0.5

1

1.5

E (MV/m) 3

–4

0.025 Time elapsed = 3 s Span = 5.255 x 10 –4 C/m2

0.02

ε 33 (microstrain)

2 1 0

3

D (C/m2)

Span = 2.021 x 10 –2 με

0.015

f = 1 Hz, PRC

–1

f = 1 Hz, PRC Time elapsed = 3 s

Span = 2.016 x 10 –2 με

0.015

0.01

0.005

–2

(c)

f = 0.5 Hz, PRC Time elapsed = 6 s

0.005

–2

(a)

141

–1.5

–1

–0.5

0

0.5

1

1.5

E (MV/m) 3

(d)

0 –1.5

–1

–0.5

0

0.5

1

1.5

0.5

1

1.5

E (MV/m) 3

–4

x 10

0.025 f = 10 Hz, PRC

f = 10 Hz, PRC

Time elapsed = 0.3 s Span = 5.251 x 10 –4 C/m2

0.02

ε 33 (microstrain)

2

0

3

D (C/m2)

1

–1

(e)

Span = 2.008 x 10 –2 με

0.015

0.01

0.005

–2

–1.5

Time elapsed = 0.3 s

–1

–0.5

0 E (MV/m) 3

0.5

1

0 –1.5

1.5

(f)

–1

–0.5

0 E (MV/m) 3

4.9 Frequency effect (f ¼ 0.5 Hz for a, b; 1 Hz for c, d; and 10 Hz for e, f ) on the effective dielectric hysteresis and butterfly strain responses for the stress-free PZT-51/934 viscoelastic epoxy particulate piezocomposite with particle VF ¼ 0.5 undergoing a cyclic sinusoidal electric loading E3 ðtÞ ¼ 1:2 sin ð2πftÞ MV/m. Only the first three cycles are plotted.

a low dielectric property of the matrix, the overall polarization and strain obtained from applying an external electric field is significantly small. By considering a 0–3 piezocomposite with conductive matrix, that is, silver, much more pronounced electromechanical hysteretic response can be obtained (Fig. 4.10). As expected, the conductive matrix can efficiently transfer the electric field to the piezoelectric particles.

142

Creep and Fatigue in Polymer Matrix Composites 0.3

–200 Particle VF = 0.35 σ33 = –45 MPa

0.2

–400

ε 33 (microstrain)

–600 D (C/m2)

0.1

3

0 –0.1

Particle VF = 0.35 σ33 = –45 MPa using silver matrix whose dielectric constant is 88.54 x 10 –9 F/m

–800 –1000 –1200 –1400

–0.2

using silver matrix whose dielectric constant is 88.54 x 10 –9 F/m

–2

–1

(a)

0

1

–1600 –1800 –2

2

–1

0

(b)

E (MV/m) 3

1

2

E (MV/m) 3

0.3

–2000

Particle VF = 0.10 σ = –45 MPa

0.2

ε 33 (microstrain)

0

3

D (C/m2)

0.1

–0.1 –0.2

using silver matrix whose dielectric constant is 88.54 x 10 –9 F/m

–2500

–3000

using silver matrix whose dielectric constant is 88.54 x 10 –9 F/m

–2

(c)

Particle VF = 0.10 σ = –45 MPa 33

33

–1

0 E (MV/m) 3

1

2

–3500 –2

(d)

–1

0

1

2

E (MV/m) 3

4.10 Effective hysteretic polarization and butterfly strain responses for the PZT-5A1/silver 0–3 active composite with PZT-5A1 particle undergoing both a cyclic electric field and a constant mechanical stress σ 33 ¼ 45 MPa.

Piezocomposites with higher particle contents result in larger hysteretic area, associated with the more pronounced polarization effect. However, adding more brittle particles increases the stiffness of the composites, thus reducing the flexibility of the composites.

4.4

Hybrid piezocomposite model

Currently available studies have shown great promise in the development of piezoelectric composites comprising piezoelectric ceramic inclusions and a polymeric matrix. This type of piezoelectric composite can achieve a balance between high electromechanical coupling properties while being relatively compliant. One of the main drawbacks with using a polymeric matrix is the inherently low electromechanical properties, which can lead to poor performance, particularly for the 0–3 piezoelectric composites. One approach for improving the properties of the matrix is adding inclusions to the matrix. For example, conductive particles can be added to the polymeric matrix in the 1–3 and 0–3 piezocomposites, in order to increase the dielectric properties of the matrix. Hagood and Bent (1993) and Bent and Hagood (1997) showed

Micromechanics modeling of hysteretic responses of piezoelectric composites

143

that improvement of the overall dielectric of a 1–3 piezocomposite can be achieved by adding PZT powder into the epoxy matrix. Bent and Hagood (1997) also claimed that a matrix with both dielectric and conductive fillers will reduce the needed voltage for poling and operation of the 1–3 piezocomposites. Bent et al. (1995) mentioned that 1–3 piezocomposites that have a relatively low transverse stiffness are unable to sustain large transverse loads and provide structural stiffness. Because the transverse behavior of 1–3 composite is governed by the matrix properties, improving the matrix properties by adding fillers can overcome this issue. There are currently limited micromechanical models available to predict the overall response of hybrid piezocomposites. Aldraihem et al. (2007) used the Mori-Tanaka model to evaluate the effective loss factor for a hybrid piezocomposite having shunted piezoelectric particles embedded in a conductive particle–reinforced matrix. Their model was extended by Aldraihem (2011) to derive the effective loss factor for a hybrid piezocomposite with orientation-dependent piezoelectric inclusions embedded in conductive inclusions dispersed in a viscoelastic polymer. Recently, Lin and Muliana (2016) presented a unit-cell model for a hybrid piezocomposite comprising piezoelectric fiber and particle inclusions embedded in a homogeneous matrix. They showed that adding particles to the matrix improves the stiffness in the lateral direction, and as a consequence the composite becomes less compliant.

4.4.1 Formulation of the unit-cell model The hybrid-unit-cell model is formulated by integrating the unit-cell model of 0–3 composite to the matrix subcells in the unit-cell model of 1–3 composite. Using a volume-averaging scheme, the effective field variables over the 1–3 unit-cell model at current time t are written as: t

T ¼

IV X

cðαÞ TðαÞ,t ,

(4.40)

cðαÞ ΞðαÞ,t :

(4.41)

α¼I

t

Ξ ¼

IV X α¼I

The superscript (α) denotes the subcell’s number of the fibrous unit cell. The fiber volume fraction is defined as c(I) ¼ V(I)/V (i.e., volume fraction of the fibers P to the (α) hybrid piezocomposite) and the fibrous unit-cell volume is given by V ¼ α¼IV α¼I V . In an average sense, a linearized constitutive relation for the 1–3 piezocomposite at current time t is written as: t

t

t

Ξ ¼O T ,

(4.42)

144

Creep and Fatigue in Polymer Matrix Composites

and for the subcell (α) it is: ΞðαÞ,t ¼ OðαÞ,t TðαÞ,t :

(4.43)

In order to relate the field variables in the fibrous unit cell to those in its subcells, a concentration matrix B(α), t at current time t is defined as: TðαÞ,t ¼ BðαÞ,t T : t

(4.44)

Substituting T(α), t from Eq. (4.44) into Eq. (4.43) gives. ΞðαÞ,t ¼ OðαÞ,t BðαÞ,t T : t

(4.45)

Substituting Ξ(α), t from Eq. (4.45) into Eq. (4.41) gives. t

Ξ ¼

IV X

cðαÞ OðαÞ,t BðαÞ,t T : t

(4.46)

α¼I

Comparing Eq. (4.46) to Eq. (4.42) gives the effective generalized compliance of the fibrous unit cell. t

O ¼

n X

cðαÞ OðαÞ,t BðαÞ,t :

(4.47)

α¼1

Next, the hybrid-unit-cell model is derived from the fiber-unit-cell model by equating of the field variables of the fibrous subcells II, III, and IV with the volume-averaging field variables of the particulate unit cell II, III, and IV, respectively; that is, TðαÞ,t ¼

8 X

cðα, βÞ Tðα, βÞ,t , α ¼ II,III,IV,

(4.48)

cðα, βÞ Ξðα, βÞ,t , α ¼ II, III, IV:

(4.49)

β¼1

ΞðαÞ,t ¼

8 X β¼1

The superscript (α, β) indicates the subcells’ number of each particulate unit cell, similarly. The particle volume fraction is defined as c(II,1) ¼ V(II,1)/V(II) (i.e., volume fraction of the particles to the host medium) which should be the same as bothP c(III,1) and (IV,1) (α) (α,β) c . The corresponding particulate unit-cell volumes are given by V ¼ β¼8 β¼1V with α ¼ II, III, IV. The linearized constitutive relation for the particulate subcell (α, β) at current time t is written as: Ξðα, βÞ,t ¼ Oðα, βÞ,t Tðα, βÞ,t , α ¼ II, III,IV, β ¼ 1 ,2, …, 8:

(4.50)

Micromechanics modeling of hysteretic responses of piezoelectric composites

145

The concentration matrix B(α,β), t at current time tis defined as: Tðα, βÞ,t ¼ Bðα, βÞ,t TðαÞ,t , α ¼ II,III, IV, β ¼ 1, 2, …,8,

(4.51)

which relates the independent field variables of the fibrous subcell II, III, and IV (i.e., the particulate unit cell II, III, and IV), to the corresponding particulate subcells, respectively. Substituting Eq. (4.51) into Eq. (4.50) we replace T(α,β), t and then eliminate Ξ(α,β), t by using Eq. (4.49). We arrive at. ΞðαÞ,t ¼

8 X

cðα, βÞ Oðα, βÞ,t Bðα, βÞ,t TðαÞ,t , α ¼ II, III,IV:

(4.52)

β¼1

Comparing Eq. (4.52) to Eq. (4.43) gives the effective generalized compliance of the particulate unit cells. OðαÞ,t ¼

8 X

cðα, βÞ Oðα, βÞ,t Bðα, βÞ,t , α ¼ II, III,IV:

(4.53)

β¼1

Finally, in order to evaluate the concentration matrices B(α), t and B(α,β), t, the hybridunit-cell model uses the subcells’ constitutive relations together with linearized micromechanical relations from the fibrous unit cell and the particulate unit cell II, III, and IV. The linearized micromechanical relations for the fibrous and particulate unit cells can be found in Lin and Muliana (2013). Because of the nonlinear constitutive relations for the piezoelectric constituent, the linearized micromechanical relations would generally violate the overall nonlinear responses, which results in error predictions on the effective responses, defined by the following residual vector: 8 ðI Þ, t 9 T > > > > > > > > ⋮ > > > > > ðIV Þ, t > > > T > > > > > ðII , 1Þ, t > > > T > > > > > > > > ⋮ > > > = < ðII, 8Þ, t > n o T t  ½ Q  T , f R t g ¼ ½ Pt  ðIII , 1Þ, t T > > 2521 252252 > 2529 > > > 91 > > ⋮ > > > > > > > > > TðIII, 8Þ, t > > > > ðIV, 1Þ, t > > > > > T > > > > > > > > ⋮ > ; : ðIV, 8Þ, t >

(4.54)

T 2521

where the Pt matrix is a function of the electric fields, material parameters, and the volume fraction of each subcell at current time tand the Q matrix is a constant matrix from the micromechanical relations. The dimension of each matrix is denoted at its bottom. A set of trial solutions for the unknown T(α), t with α ¼ I, II, III, IV and

146

Creep and Fatigue in Polymer Matrix Composites

T(α,β), t with α ¼ II, III, IV; β ¼ 1, 2, … , 8 at current time t together with the Newton– Raphson iterative method is used to minimize the residual vector at each time step. Once the residual vector has been minimized, the concentration matrices B(α), t and B(α,β), t at current time t can be evaluated. Consequently, the effective generalized t t compliance O and the effective field variables Ξ for the hybrid piezocomposite at current time t can be determined from Eqs. (4.47), (4.53) and (4.41), (4.45), (4.53), respectively.

4.4.2 Numerical implementation First, we consider a hybrid piezocomposite with PZT-G1195 inclusions and viscoelastic matrix subjected to large electric driving fields. The matrix properties follow those of the FM73 polymer whose material parameters are taken from Muliana and Khan (2008) and are listed in Table 4.10. The relative dielectric constants of the FM73 matrix are assumed κ11/κ0 ¼ κ22/κ0 ¼ κ33/κ0 ¼ 4.43. Several boundary conditions are considered in the following discussion. First, a hybrid piezocomposite, PZT-G1195 fibers embedded in the FM73 viscoelastic matrix reinforced by PZT-G1195 filler, subjected to a constant electric field E3 ¼ 1 MV/m with a fully constrained displacement (Fig. 4.11) and a stress-free boundary condition (Fig. 4.12) are both examined. Fig. 4.11 depicts the effective stress relaxations for the PZT particle VF ¼ 0, 0.1, 0.3, and 0.5, respectively. As the PZT particles increase, higher transverse blocked stresses are expected (Fig. 4.11a) and this therefore results in more pronounced stress relaxations, even though the corresponding VF of the viscoelastic matrix decreases. Stress relaxations in the longitudinal direction are also observed (Fig. 4.11b). Creep responses for a stress-free boundary condition are shown in Fig. 4.12. No significant differences in the creep behaviors due to various volume contents of the PZT particles are observed. However, increasing the PZT particles stiffens the matrix and as a result decreases both instantaneous compressive (Fig. 4.12a) and tensile (Fig. 4.12b) free strains. We also study the effect of loading frequency on the overall hysteretic electromechanical responses of the hybrid piezocomposite. The predictions of the cyclic electric loadings, E3(t) ¼  0.5cos(wt) + 0.5 MV/m with the frequencies w ¼ 0.5, 1, and 10 Hz along the poling direction, on a fully constrained PZT-G1195/FM73 hybrid piezocomposite with PZT-G1195 fiber VF ¼ 0.2 along with PZT-G1195 particle Table 4.10 Time-dependent compliance, instantaneous time-dependent (elastic) compliance and Poisson’s ratio for the FM73 polymer at 30–60°C n

λn (min21)

Dn (GPa21)

1 2 3

1 101 102

0.0210 0.0216 0.0118

D0 ¼ 0.369(GPa1 ). v ¼ 0.35.

Micromechanics modeling of hysteretic responses of piezoelectric composites

0.8

Particle VF = 0.5

s 11 (MPa)

0.7 0.6 Particle VF = 0.3

0.5 0.4 0.3 0

Particle VF = 0.1 Particle VF = 0.0 100

(a)

200 Time (s)

300

147

Fig. 4.11 Effective stress relaxations on (a) transverse σ 11 and (b) longitudinal σ 33 responses for the fully constrained displacement of the PZT-G1195/FM73 piezoelectric hybrid composite with a fixed PZTG1195 fiber volume fraction 0.2 and various PZT-G1195 particle volume fractions, 0.0, 0.1, 0.3, and 0.5, due to a constant electric field E3 ¼ 1 MV/m along the poling direction.

–5.2 Particle VF = 0.5

s 33 (MPa)

–5.25

–5.3 Particle VF = 0.3

–5.35 Particle VF = 0.1 Particle VF = 0.0 –5.4 0

(b)

100

200 Time (s)

300

content VF ¼ 0.5 are shown in Fig. 4.13. For the frequency w ¼ 0.5 Hz, the transverse stress σ 11 (Fig. 4.13b) has more significant hysteresis than the longitudinal stress σ 33 (Fig. 4.13a) does, because the transverse response is dominated by the viscoelastic matrix (which was reinforced by the PZT-G1195 particles). The same trends are also observed for other frequencies w ¼ 1 and 10 Hz. Due to the viscoelastic matrix, the lower frequency (slower loading) shows the broader hysteretic loop, because it allows for more stress relaxation or creep deformation, and thus higher energy dissipation in one cycle. The maximum tensile transverse stress σ 11 happens when the periodic loading E3 ¼  0.5cos(wt) + 0.5 MV/m reaches 1 MV/m from zero during the first period. Fig. 4.14 depicts the maximum σ 11 in the first cycle vs. frequency. Lower frequency loading leads to lower maximum σ 11. This is due to the time-dependent effect of the viscoelastic matrix. Thus, a relatively slow electric loading (i.e., lower frequency) gives sufficient time for the stress to experience relaxation behavior.

148

Particle VF = 0.5

–140 –160 e 11 (microstrain)

4.12 Effective strain creeps on (a) transverse ε11 and (b) longitudinal ε33 responses for the stress-free PZTG1195/FM73 piezoelectric hybrid composite with a fixed PZT-G1195 fiber volume fraction 0.2 and various PZTG1195 particle volume fractions, 0.0, 0.1, 0.3, and 0.5, due to a constant electric field E3 ¼ 1 MV/m along the poling direction.

Creep and Fatigue in Polymer Matrix Composites

Particle VF = 0.3

–180 –200 –220

Particle VF = 0.1

–240 –260 –280

Particle VF = 0.0

–300 0

100

(a)

200 Time (s)

300

750 Particle VF = 0.0 700

e 33 (microstrain)

650 Particle VF = 0.1 600 550 500 Particle VF = 0.3 450 400 Particle VF = 0.5 350 0

(b)

100

200 Time (s)

300

Next, we present simulation results on a piezoelectric hybrid composite under cyclic electric field inputs at different amplitude of electric field. We consider PZT-51/[PZT-51/FM73 polymer] hybrid piezocomposite with 40% PZT-51 fiber volume content and 20% PZT-51 particle content. Figs. 4.15–4.17 show the results under different electric field amplitudes, that is, 0.12, 0.6, and 1.2 MV/m, respectively. The response after 6 cycles of loading is presented, showing closed hysteretic loops. It is seen that at low loading amplitude the polarization shows nearly linear hysteretic response with a symmetric butterfly strain response. A linear hysteretic response is expected when the loading amplitude is small. It is also seen that a low remanent strain is associated with a nearly complete depolarization when the electric field is zero. Fig. 4.16 shows the hysteretic responses when the amplitude is close to the coercive electric field of the composites, at which the dipolar motions occur easily. It is interesting to notice the nonsymmetric shape of the butterfly strain at this amplitude, compared to the symmetric butterfly strain responses at low amplitude (Fig. 4.15)

Micromechanics modeling of hysteretic responses of piezoelectric composites

149

0 s 11 (MPa)

0.8 –1 0.6 s 11 (MPa)

s 33 (MPa)

–2 –3

0.81 0.8 0.79 0.78 0.97

0.4

0.98 0.99 E3 (MV/m)

1

0.2

–4

w = 0.5 Hz

w = 0.5 Hz

0

–5 –6

(a)

0

0.2

0.4 0.6 E3 (MV/m)

0.8

1

–0.2 0

0.2

(b)

0.4 0.6 E3 (MV/m)

0.8

1

0

0.6 s 11 (MPa)

s 33 (MPa)

–2 –3

s 11 (MPa)

0.8 –1

0.81 0.8 0.79 0.78 0.97

0.4

0.98 0.99 E3 (MV/m)

1

0.2

–4

w = 1 Hz

w = 1 Hz

0

–5

(c)

–6 0

0.2

0.4 0.6 E3 (MV/m)

0.8

1

–0.2 0

0.2

(d)

0.4 0.6 E3 (MV/m)

0.8

1

0

0.6 s 11 (MPa)

s 33 (MPa)

–2 –3

0.4

0.78 0.97

0.98 0.99 E3 (MV/m)

1

0

–5

(e)

0.8 0.79

w = 10 Hz

w = 10 Hz

0.2

0.81

0.2

–4

–6 0

s 11 (MPa)

0.8 –1

0.4 0.6 E3 (MV/m)

0.8

1

–0.2 0

(f)

0.2

0.4 0.6 E3 (MV/m)

0.8

1

4.13 Effective stress responses for the fully constrained displacement of the PZT-G1195/FM73 piezoelectric hybrid composite with PZT-G1195 fiber volume fraction 0.2 and PZT-G1195 particle volume fractions 0.5 due to a set of cyclic electric field loadings E3(t) ¼  0.5cos(wt) + 0.5 MV/m with the frequencies w ¼ 0.5, 1, and 10 Hz along the poling direction. Total time is 2 s.

and saturated polarization (Fig. 4.17). The trends in the changes of the hysteretic polarization and butterfly strain responses at different amplitudes of electric field inputs are similar to the experimental observation of PVDF piezoelectric polymers (Gookin et al., 1984). However, no further explanation is available on the asymmetric butterfly shapes at moderate electric field inputs.

150

0.85

Maximum s 11 (MPa)

4.14 Effective maximum tensile transverse stress σ 11 in the first cycle vs frequency for a fully constrained displacement of the PZTG1195/FM73 piezoelectric hybrid composite with PZTG1195 fiber volume fraction 0.2 and PZT-G1195 particle volume fraction 0.5 undergoing various frequencies of cyclic electric loadings E3(t) ¼  0.5cos(wt) + 0.5 MV/m along the poling direction.

Creep and Fatigue in Polymer Matrix Composites

0.8

0.75

2

4 6 Frequency (Hz)

8

10

4.15 Effective electric displacement D3 and longitudinal strain ε33 responses for the PZT-51/ [PZT-51/FM73 polymer] hybrid piezocomposite with PZT-51 fiber VF ¼ 0.4 and PZT-51 particle VF ¼ 0.2 subjected to both a cyclic electric loading E3 ðtÞ ¼ 0:12 sin ð2πftÞ MV/m with frequency f ¼ 1 Hz along the poling direction.

Micromechanics modeling of hysteretic responses of piezoelectric composites

151

4.16 Effective electric displacement D3 and longitudinal strain ε33 responses for the PZT-51/ [PZT-51/FM73 polymer] hybrid piezocomposite with PZT-51 fiber VF ¼ 0.4 and PZT-51 particle VF ¼ 0.2 subjected to both a cyclic electric loading E3 ðtÞ ¼ 0:6 sin ð2πftÞ MV/m with frequency f ¼ 1 Hz along the poling direction.

4.5

Conclusions

We have presented micromechanical models for piezoelectric composites having fiber and particle inclusions dispersed in a homogeneous matrix. The micromechanics models are based on simplified unit-cell models of idealized composite microstructures. The unit-cell models incorporate nonlinear and time-dependent response of the constituents and are capable of capturing the overall hysteretic response of piezocomposites at different frequencies and loading amplitudes. It is noted that the unit-cell models give crude approximations of the overall behaviors of the composites, which can be useful for preliminary design of piezocomposites. The unit-cell models are limited in capturing detailed variations in the field variables in the piezocomposites. Several parametric studies have been performed to investigate the overall response of the composites and the behaviors of the constituents when the piezocomposites are subjected to various loading histories. We also highlighted the influence of the viscoelastic matrix on the performance of piezocomposites. It is concluded that higher electromechanical coupling performance in the piezoelectric

152

Creep and Fatigue in Polymer Matrix Composites

4.17 Effective electric displacement D3 and longitudinal strain ε33 responses for the PZT-51/ [PZT-51/FM73 polymer] hybrid piezocomposite with PZT-51 fiber VF ¼ 0.4 and PZT-51 particle VF ¼ 0.2 subjected to a cyclic electric loading E3 ðtÞ ¼ 1:2 sin ð2πftÞ MV/m with frequency f ¼ 1 Hz along the poling direction.

composites can be achieved by increasing the connectivity between the conductive inclusions and/or improving the dielectric properties of the matrix. Experimental investigations suggest that piezocomposites with high contrasts in the dielectric properties of the particles and matrix can lead to accelerated conductive behaviors for moderate to high volume contents of the particles. The presented unit-cell models that rely mainly on the properties and compositions of the constituents are not capable of capturing the accelerated conductive behaviors.

Acknowledgment This research is sponsored by the National Science Foundation under grant CMMI-1437086.

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Tajeddini, V., Lin, C.-H., Muliana, A., Levesque, M., 2014. Average electro-mechanical properties and responses of active composites. Comput. Mater. Sci. 82, 405–414. Tan, P., Tong, L., 2001. Micro-electromechanics models for piezoelectric-fiber-reinforced composite materials. Compos. Sci. Technol. 61 (5), 759–769. Taylor, R.L., Pister, K.S., Goudreau, G.L., 1970. Thermomechanical analysis of viscoelastic solids. Int. J. Numer. Meth. Eng. 2 (1), 45–59. The Institute of Electrical and Electronics Engineers, 1987. IEEE Standard on Piezoelectricity. IEEE, New York. Tressler, J.F., Alkoy, S., Dogan, A., Newnham, R.E., 1999. Functional composites for sensors, actuators and transducers. Compos. Part A: Appl. Sci. 30 (4), 477–482. Xing, J., Radovic, M., Muliana, A., 2018. Elastic and dielectric properties of active Ag/BaTiO3 composites. Exp. Mech. 58 (4), 645–660. Zeng, R., Kwok, K.W., Chan, H.L.W., Choy, C.L., 2002. Longitudinal and transverse piezoelectric coefficients of lead zirconate titanate/vinylidene fluoride-trifluoroethylene composites with different polarization states. J. Appl. Phys. 92 (5), 2674–2679. Zhou, D., Kamlah, M., 2006. Room-temperature creep of soft PZT under static electrical and compressive stress loading. Acta Mater. 54, 1389–1396. Zhou, H., Zhang, S., Yang, M., 2007. The effect of heat-transfer passages on the effective thermal conductivity of high filler loading composite materials. Compos. Sci. Technol. 67 (6), 1035–1040.

Further reading Yancey, R.N., Pindera, M.-J., 1990. Micromechanical analysis of the creep response of unidirectional composites. J. Eng. Mater. Trans. ASME 112 (2), 157–163.

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Predicting the viscoelastic behavior of polymer composites and nanocomposites

5

A. Beyle University of Texas at Arlington, Arlington, TX, United States

5.1

Preface

This chapter is based on Chapter 6, “Viscoelasticity characterization of nanocomposites,” of the previous edition of this book (Beyle and Ibeh, 2011), written jointly with the late Dr. C.C. Ibeh. Comparison of the titles shows that the chapter has not only been updated but also that the content has shifted more toward viscoelasticity of conventional composites, having increased practical applications.

5.2

Specific features of constituents

The mechanical behavior of heterogeneous materials depends on mechanical and physical properties of constituents, geometrical factors (shapes and orientations of constituents, architecture of the material), and the quality of the bonds between constituents. Conventional fibers are made mostly with circular cross-sections. Attempts to make square or hexagonal cross-sectional fibers (for increased maximal possible volume concentrations of fibers) as well as gear-shaped, bean-shaped, and ellipticalshaped cross-sections (for increased transversal shear resistance) have failed, due to poor mechanical properties of the peripheral parts of the fiber cross-sections. Fibers provide ideal reinforcement only if the local stresses acting on them are either axial tension or axial compression. Misalignment and waviness of fibers sharply decrease strength and stiffness of composites as well as result in elevated creep and decrease of the fatigue resistance. The majority of fibers are linearly elastic. However, polymeric fibers (ultra high molecular weight polyethylene (UHMWPE), polyparaphenylenebenzobisoxazole (PBO), etc.) have some creep, especially at elevated temperatures. Some conventional fibers are isotropic (glass fibers, for example), but some fibers are anisotropic (carbon fibers, many polymeric fibers). Anisotropic fibers are either cylindrically monotropic or cylindrically orthotropic. This means that these fibers must have either five or nine independent elastic constants. Unfortunately, for the majority of anisotropic fibers only longitudinal modulus of elasticity, and sometimes longitudinal-transversal Poisson ratio, are known. Creep and Fatigue in Polymer Matrix Composites. https://doi.org/10.1016/B978-0-08-102601-4.00005-9 © 2019 Elsevier Ltd. All rights reserved.

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A wide range of mechanical properties of conventional fibers is available on the market now (and a wide range of prices as well). The longitudinal modulus of fibers can range from a few tens of gigapascals up to 1000 GPa. The tensile strength of fibers can vary from decimal parts of gigapascals up to 6.4 GPa. Because the strength of fibers increases with the decrease of their diameters (thanks to the classical work of Griffith in 1920–24), investigations into nanofibers have become popular during the last 30 years. The data on strength of nanofibers are not very consistent, because many important factors are ignored in recalculations of measured data to acting stresses. However, it is clear that the strength of the best nanofibers can be several times higher than the strength of conventional fibers. On the other hand, nanocomposites are much weaker than conventional composites. There are several reasons for this; the major ones being that nanofibers are not continuous yet and they have low flexible stiffness. Because only short nanofibers have been produced to this point, the reinforcing effect is similar to that using chopped conventional fibers. Continuous straight fiber works in concert with the matrix in a purely parallel scheme, in which the properties of the fibers are dominant and they determine the longitudinal properties of the composite. In a consecutive scheme, the matrix properties are dominant. Reinforcement by short fibers corresponds to a mixed parallel and consecutive scheme, in which matrix properties contribute significantly. Additionally, the reinforcing effect of the end zones (with length equal to many diameters) of the short fiber is negligible. Due to the small diameter of nanofibers their bending stiffness is low despite the high longitudinal modulus (diameter participates in higher power than modulus). As a result, nanofibers are often bent by matrix flow during technological processes. Fibers with waviness have a small reinforcing effect, as mentioned previously (see also Fig. 5.1). A revolution in materials science and technology can occur when continuous nanofibers are produced and the technology of their pretension (without damage to their surfaces) is elaborated. Among all types of matrices, polymeric matrices are the most popular due to their technological convenience. Both thermoplastic matrices, such as PEEK, polysulfones, etc., and thermoset matrices, such as epoxies and polyesters, have low Young’s modulus, low strength, and are characterized by some creep; in addition it is typical for these matrices to have some material aging features (Sullivan et al., 1993; Honorio, 2017) (properties deterioration with time due to chemical and physical interaction with the environment).

5.1 Demonstrational experiment with rope having initial waviness. A small force is needed to move the ends apart by distance δ until an ideally straight configuration is reached. Further elongation providing the same displacement δ requires significantly greater force. Conclusion for composites: fibers with waviness can carry nonsignificant part of the load and they do not contribute to stiffness and resistance of composite.

Predicting the viscoelastic behavior

159

Despite the fact that polymeric matrices have much better adhesion in comparison with other types of matrices (metallic, cement, ceramic, amorphous carbon, etc.), each time new fibers have appeared on the market, the problem of bonding had to be solved. Carbon fibers and carbon nanofibers have low adhesion and it is necessary to functionalize their surfaces using chemically very active substances, having other groups able to react with matrices also (Thakur and Thakur, 2018). In many cases, it is reasonable to replace a two-phase model for composites by a three-phase model including a thin intermediate layer having significantly different properties from the matrix. Insufficient adhesion can be realized in additional creep mechanisms, including some sliding of fibers with respect to the matrix. Fillers and reinforcing fibers affect the matrix morphology and properties. As a result, properties of the matrix determined from macroscale samples made from pure matrix materials can differ significantly from the local properties of the matrix in composite materials. This is especially sharply seen in the case of nanocomposites, where the ratio of interphase area to the volume of inclusions is very high. Mechanical properties of heterogeneous systems depend on such factors as properties of constituents, concentration of constituents, shape of fillers, orientation of fillers, etc. Existing theories used for calculation of the effective elastic properties are not sensitive to the absolute sizes of inclusions, and only their volumetric concentration is important. In the case of nanoparticles, some modifications are possible due to the influence of the interfacial interaction between phases on the bulk matrix properties; this role increases in nanostructured systems due to the high surface-to-volume ratio of nanoinclusions. However, the classical analysis still describes the main effects in the case of elastic properties of nanocomposites. Deviations in experimental data from theoretical ones are mostly related to the effect of aggregation of nanoparticles. The theoretical prediction of viscoelastic properties is less developed, despite the principle of an elastic-viscoelastic analogy. Some nanoparticles have irregular shape. However, many types of nanoparticles have quasiclassical shapes, and are spherical, cylindrical-long fibers, or cylindrical-plane disks. Nanoparticles such as nanoclay and nanographene are platelets, carbon nanofibers have cylindrical shapes, and silicon carbide nanoparticles can be considered as quasispherical ones. Carbon nanotubes are considered as hollow cylinders, whereas bucky-balls are hollow spheres. These classical shapes will be analyzed using elastic-viscoelastic analogy and Rabotnov’s algebra of resolvent integral operators of viscoelasticity. Nanoparticles have excellent mechanical and physical properties and very high total surface to total volume ratio. However, the improvement of properties in comparison to the matrix properties is not as high as expected by many researchers. The causes of such contradictions are discussed and illustrated by the results and data of a modeling approach. Starting with Griffith’s classical works from the 1920s, it becomes clear that the huge difference between theoretical strength of materials and their real strength (two to three decimal orders) can be reduced if the sizes of the solid body are decreased. Following Griffith’s ideas, the first high-strength glass fibers were produced industrially in the 1940s. The idea of building strong bulk materials using thin strong fibers and a binding matrix (initially polymeric, later metallic, ceramic, cement,

160

Creep and Fatigue in Polymer Matrix Composites

etc.)—that is, the idea of composites—was taken from the architecture of natural materials such as wood, bones, and so forth, which are natural composites. Similarly, the reinforced concrete appeared few decades before the composites. Progress in technology resulted in the ability to make nanosized fillers, the individual strength of which approaches the theoretical value. Data on many unique properties of nanoparticles are collected in (Dasari and Njuguna, 2016). However, the expectation that the mechanical properties of nanocomposites would be much higher than the properties of conventional composites was not realized. Some progress has been achieved in dynamic applications and in other areas, but it is not very pronounced. Information on nanocomposite properties, technology, and applications can be found in multiple sources, for example in (Dasari and Njuguna, 2016; Yiu-Wing and ZhongZhen, 2006; Liu et al., 2006; Bhattacharya et al., 2007; Njuguna, 2014; Shaker, 2016; Jlassi et al., 2017; Tripathy and Sahoo, 2017; Koo, 2016). In this situation a critical review of the existing methods for prediction of the mechanical behavior of heterogeneous materials could be useful. All mechanical properties of heterogeneous systems depend on properties of constituents, their concentrations, the shape of fillers, the fillers’ orientations, the type of spatial lattice, etc. Existing theories used for calculations of the effective elastic properties are insensitive to the absolute sizes of the inclusions, and only their volumetric concentration is important. In the case of nanoparticles, some modifications are possible due to the influence of the interfacial interaction between phases on the bulk matrix properties; this effect increases in nanostructured systems due to their high surface-to-volume ratio. In reality, the volume of the individual inclusion VI is related to the total volume V of the composite as c¼

NI VI V

(5.1)

where c is the volume concentration of inclusions and NI is the total number of the identical inclusions in the volume V; the ratio of the total surface S of all inclusions to the total volume of composite can be written as S NI SI N I SI SI 1 ¼ ¼c ¼ c ¼ cξI V V N I VI VI ðVI Þ1=3

(5.2)

where SI is the outer surface area of the individual inclusion, and ξI is the dimensionless inclusion’s form factor ξI ¼

SI ðVI Þ2=3

(5.3)

The values of the form factor for different types of inclusions are presented in Table 5.1. According to Eq. (5.2) both form factor and especially the decreasing volume of the individual inclusion play significant roles in elevated surface effects in

Predicting the viscoelastic behavior

161

Table 5.1 Form factors ξI for different types of inclusions Shape type

Sizes ratio

Form factor ξI

Spherical Cubical Cylindrical Cylindrical Cylindrical Cylindrical Circular platelet Circular platelet Circular platelet Circular platelet Square platelet Square platelet Square platelet Square platelet

Any Any L/R ¼ 10 L/R ¼ 100 L/R ¼ 1000 L/R ! ∞ D/h ¼ 10 D/h ¼ 100 D/h ¼ 1000 D/h ! ∞ L/h ¼ 10 L/h ¼ 100 L/h ¼ 1000 L/h ! ∞

4.836 6 6.942 13.732 29.321 ∞ 10.278 40.550 184.896 ∞ 11.140 43.950 200.400 ∞

nanocomposites. The simplest model, which takes into account surface effects, includes three phases: inclusions, matrix, and a thin layer of modified matrix, separated matrix and inclusions (see e.g., Beil’ et al., 1975). In the case of a thermoplastic matrix, this layer is formed due to lower mobility of the macromolecular segments near the solid surface; the thickness of the layer is a few segments. In the case of thermosets, the kinetics of chemical reactions near the surfaces of inclusions are different from the kinetics of reactions in bulk, and as a result the macromolecular structure and properties are different. In some cases, when inclusions are acting as nucleates of physical or morphological transitions, the thickness of the modified layer of matrix can be comparable with the size of the inclusions. A stepwise change of the matrix properties is an idealized model replacing continuous monotonic change of the matrix properties from the surface of the inclusion to bulk properties. Unfortunately, there is not enough data on the effective thickness of modified matrix layers or on the properties of such modified layers. The classical models of composites, taking into account the properties of the matrix and inclusions, shapes of inclusions, their orientations, type of spatial lattice, etc., still describe the main effects in the case of the elastic properties of nanocomposites. Theoretical modeling was done for the classical shapes of inclusions: long cylindrical, hollow cylindrical, spherical, hollow spherical, and cylindrical platelets. Differences between experimental data and theoretical data are primarily related to the effect of the aggregation of nanoparticles, the shape deviations from the ideal (mostly occurring as waviness), the nonuniformity of the nanoparticle distributions over the volume of the nanocomposite, and the imperfections in orientation, lattice, etc. The aggregation of nanoparticles is the biggest obstacle in nanocomposite technology. During transportation and storage, nanoparticles can accumulate electrical

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Creep and Fatigue in Polymer Matrix Composites

charges and nanoparticles with opposite charges can form the aggregate much more easily than the conventional fillers, simply due to smaller masses. The number of nanoparticles Nn providing the same volume concentration c as a number Nc of the conventional particles is inversely proportional to the cube of their size ratios: Nn ¼ Nc

R3c R3n

(5.4)

Because conventional fillers have sizes mostly in the range of 1 μm–1 mm but nanoparticles are mostly in the range of 10–100 nm, then the number of nanoparticles replacing the same mass of the conventional fillers must be from 103 to 1015 times larger. Taking into account that the distances between neighboring particles are proportional to their sizes (if volume concentration is the same) and that hydrodynamic resistance is proportional to the size of the particle, the probability of nanoparticle aggregation during the technological process of mixing with the matrix in liquid form (before solidification of the nanocomposite) is very high. The presence of aggregates is worse than the presence of voids. They are not only stress concentrators for the matrix, but they can also carry very low tensile and shear stresses and in this aspect are practically no better than voids; but the main negative effect is that they work as levers opening cracks in the matrix. Sonication technology allows destruction of the aggregates, but this technology also changes the polymeric matrix (Sha et al., 2008). Atoms on the surface are less active than atoms on the edges and especially atoms on the corners. With decrease of the sizes of the particles, the number of such active atoms grows exponentially. This explains both the modifying effect on the matrix properties and the tendency to aggregation of nanoparticles.

5.3

Distinctive characteristics of behavior of heterogeneous materials with polymeric matrix

Mechanical behavior of matrices significantly depends on temperature: at high temperatures a matrix can have significant creep, whereas at low temperatures a matrix can be brittle. The meaning of what is considered “high temperature” and “low temperature” is determined by melting temperature (thermoplastic matrix) or by temperature of the beginning of decomposition of a polymer (thermoset matrix) from one side and glassy temperature from the other side. A temperature below the glassy limit is considered to be “low” and a temperature a few tens of degrees below the upper limit (melting or decomposition) is considered “high.” The upper limit is determined by the chemical nature of the polymer but the brittleness is also dependent on the technology. In many cases, technologists are trying to cure thermoset matrices at the highest allowable temperature to run processes faster and increase productivity. This has resulted in a brittle matrix. It is interesting to note that the creep of the matrix is typical for high

Predicting the viscoelastic behavior

163

temperatures and for low temperatures, but the mechanisms of creep are different. At high temperatures, creep is related to flexibility of the macromolecular chains. At very low temperatures, creep is related to damage accumulation and it is somehow similar to the creep of concrete. Creep is intensified not only when the temperature rises but also when stress increases, especially when it approaches the strength. Absorbed moisture content intensifies creep also. A typical set of creep curves for epoxy resin is shown in Fig. 5.2. Most creep curves have a horizontal asymptote (restricted creep) but at stresses approaching the strength, the asymptote can have some slope (unrestricted creep). Usually the creep curve ε ¼ ε(t); σ ¼ const is characterized by a few parameters: (a) (b) (c) (d)

Instantaneous modulus E0 ¼ σ/ε(0). Long-time modulus (in the case of restricted creep) E∞ ¼ σ/ε(∞).
 Long-time viscosity (in the case of unrestricted creep) η∞ ¼ σ= dε dt ð∞Þ .   1 Time t1/2 when half of the creep is achieved ε t1=2 ¼ 2 ½εð0Þ + εð∞Þ (in the case of restricted creep). (e) Effective retardation time te defined from     εðte Þ ¼ εð0Þ + ½εð∞Þ  εð0Þ 1  1e ¼ εðe0Þ + εð∞Þ 1  1e (in the case of restricted creep).

The main characteristic of short-time behavior is instantaneous modulus (such as E0) and long-time behavior is described with the help of a protracted modulus (such as E∞). These concepts are illustrated in Fig. 5.3. In many practical applications, it is enough to know only two of these characteristics (E0 and E∞). If both of them predict stresses, strains, displacements, and rotations in the allowable limits of the particular design, the details of the transient period can be ignored. If some of the mentioned values are outside the allowable range, knowledge of the transient period is very important, because it gives an estimation of the longevity of the structure.

e,%

4 3

2.4 2 1 1.6

0.8

0

400

800

1200

t, h

5.2 Creep of epoxy-maleic resin (Van Fo Fy, 1971a); curve 1: 40 MPa; 2: 50 MPa; 3: 55 MPa; 4: 60 MPa.

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Creep and Fatigue in Polymer Matrix Composites

e

e¥ s E¥ = e ¥

s E0 = e 0

e0

t

5.3 Determination of instantaneous and long-time moduli from the creep curve.

In the case of unrestricted creep, the modified creep diagrams can be plotted by subtracting final viscous flow ε°(t) ¼ ε(t)  η∞ t. Dependence of ε°(t) on time is similar to restricted creep and all the previously mentioned characteristics E∞, t1/2, te can be found. Qualitatively mechanical behavior of a polymeric matrix is described with the help of a four-element mechanical model consisting of two springs and two dashpots (Fig. 5.4). The equation of this model is    d2 σ 1 1 E1 dσ E0 E1 d2 ε E0 E1 dε + + E + σ ¼ E0 2 + + 0 2 η∞ η1 η1 dt η∞ η1 η1 dt dt dt

(5.5)

s

E0

E1

h1

h¥ s

5.4 Model of viscoelastic solid body (with unrestricted creep) or viscoelastic fluid. The consecutive dashpot has to be removed from the model in the case of restricted creep.

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165

The creep curve corresponds to conditions σ ¼

0, t < 0 . The solution of σ 0 ¼ const, t  0

Eq. (5.5) is

    σ0 E0 E0 E1 1+ t+ 1  exp  t ε¼ E0 η ∞ E1 η1

(5.6)

In the case of cyclic loading σ ¼ σ 0 exp(iωt) the complex compliance J∗ is used for expressing cyclic strains produced by cycling stresses ε ¼ J∗(ω)σ 0 exp(iωt), where J ∗ ¼ J 0 ðωÞ  iJ 00 ðωÞ;J 0 ðωÞ ¼

1 E1 1 η1 ω + + ;J 00 ðωÞ ¼ E0 E21 + η21 ω2 η∞ ω E21 + η21 ω2

(5.7)

Here the prime denotes the real part and the double prime denotes the imaginary part of the complex variable. The complex modulus E∗ ¼ E0 + iE00 ¼ J1∗ is calculated via the complex compliance: 8 > 0 > > < storage modulus E ¼ > > > : loss modulus E00 ¼

J0 ðJ 0 Þ2 + ðJ 00 Þ2

;

J 00

(5.8)

ðJ 0 Þ2 + ðJ 00 Þ2

In the case of restricted creep, the dashpot consecutively attached to the standard three-element model is removed or η∞ ! ∞. For such a model, t1/2 ¼ te ln 2  0.693te. From the data in Fig. 5.2 it is seen that the dependence of creep on stresses is nonlinear. This means that the moduli calculated from these curves are the secant ones. The creep of a matrix is realized in the directions and planes of the composite where the contribution of matrix compliance in the total compliance of the composite is significant (shear, transversal loading). Longitudinal creep is related either to defects in the composite structure (misalignments, length disparity, and waviness), which load the matrix with additional shear, or with damage accumulation (local ruptures of some fibers, or local cut of adhesion bonds between fiber and matrix, for example), which is related to local stress redistributions in the matrix. The decrease of the cross-section during creep under tension is observed. The Poisson ratio increases by approximately 10% (see Fig. 5.5).

5.4

Viscoelasticity of matrix

The model shown in Fig. 5.4 describes the mechanical behavior of a polymeric matrix qualitatively well but quantitatively is insufficiently precise. One of the ways to better approximate experimental data is to increase the number of elements in the model (Fig. 5.6).

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Creep and Fatigue in Polymer Matrix Composites

0.42

n

0.41 0.40 0.39 0.38

0

400

800

1200

t, h

5.5 Change of the Poisson ratio with time during creep of the maleic-epoxy matrix (Van Fo Fy, 1971a).

EN

hN

E3

h3

E2

h2

h0

E2

E3

EN

E1 E1 h2

h3

hN h0

(a)

(b)

5.6 Generalized Maxwell (a) and Kelvin-Voigt (b) models of mechanical behavior of polymeric matrix. For ideally solidified polymer, usually the viscosity η0 ! ∞ (i.e., this damping element can be removed from the model). However, for unrestricted creep at high stresses, this element is needed.

Complex compliance for a generalized Kelvin-Voigt model is described as J∗ ¼

N X Ei  iωηi 1 1 + i 2 2η 2 E ωη E + ω 1 0 i i¼2 i

(5.9)

In practical applications, models with more than six elements are rarely used due to insufficient reliability of the results from finding too many constants describing the

Predicting the viscoelastic behavior

167

same process. The order of the differential equation increases with the rise of the number of elements. As the limit of this process, two models with an infinite number of elements can be created, that play a big role in the development of the theory of viscoelasticity. These models are described with the help of differential equations of infinite order and contain an infinite number of coefficients (constants to be found experimentally). The intuitive idea is to replace this set of constants by a continuous distribution described by a formula containing few constants to be determined experimentally. Also, it is necessary to replace a differential equation of infinite order by some other equivalent but more convenient form, such as an integral equation. These transitions can be made in different ways, some of which are described in the following text.

5.5

Other factors contributing to viscoelastic behavior

Temperature dependence of the viscosity of the elements is described with the help of the Arrhenius law: 

E ηðT Þ ¼ η0 exp RT

 (5.10)

There are other empirical formulae expressing temperature dependence of viscosity. Temperature dependencies of moduli in a narrow range can be adequately described as linear ones. The same can be done with respect to the dependency of moduli on moisture content. Qualitatively, such an approach models the thermal-viscoelastic behavior of polymeric matrices. A more popular approach is based on the idea of the time–temperature analogy and time-moisture analogy, which are used for accelerated tests on creep at elevated temperatures and humidity and are applied for prediction of long-time creep of materials at regular service temperatures.

5.5.1 Mathematical Methods of Viscoelasticity The constitutive law for a viscoelastic body can be written in the most general form (see Rabotnov, 1977 for an example) as εij ¼ ε0ij + αij ΔT + Fijkl t∞ σ kl

(5.11)

Here εij is strain tensor, ε0ij is tensor of free strains (chemical shrinkage, for example) except thermal expansion, αij is tensor of coefficients of linear thermal expansion, Fijklkt∞ is tensor-functional, and σ kl is stress tensor. Functional Fkt∞ connects the current value of some function f(t) not to the current value of the function g(t) as it is used in the parametric form of functions, but connects its value to the whole history of the function g(τ) from infinitely far in the past until the current moment of time:

f ðtÞ ¼ F t∞ gðτÞ;  ∞ < τ  t

(5.12)

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Creep and Fatigue in Polymer Matrix Composites

In the theory of viscoelasticity, the functional is usually written in the form of the Volterra integral operator: 1 εð t Þ ¼ σ ð t Þ + E0

ðt Ψð t  τ Þ ∞

σ ðτ Þ dτ E0

(5.13)

where the kernel Ψ(t  τ) describes the fading memory of previous actions on the material. It has the dimension of inverse time. In the particular case of the kernel K ðt  τ Þ ¼

X

Am exp ½βm ðt  τÞ

(5.14)

m

the Volterra integral operator describes the mechanical behavior of the models shown in Fig. 5.6. Expression (5.13) can be rewritten in the alternative symbolic form εðtÞE0 ¼ ð1 + Π Þσ ðtÞ

(5.15)

where Π is an operator. Expression (5.13) can be also rewritten as 1 εð t Þ ¼ σ ð t Þ + E0

ðt Λ ðt  τ Þ ∞

1 dσ ðτÞ dτ E0 dτ

(5.16)

where Λ is a dimensionless function. The Boltzmann superposition principle leads to the same Eq. (5.16) with more detailed expression for Λ: Λ¼

E0 ðt  τ Þ + Λ 1 ðt  τ Þ η0

(5.17)

To find the expression for σ(t) from Eq. (5.13), it is necessary to solve the Volterra integral equation of the second kind. The solution has the following general form: ðt σ ðtÞ ¼ E0 εðtÞ  E0

Φðt  τÞεðτÞdτ ¼ E0 ½1  Υ εðtÞ

(5.18)

∞

Here Φ(t  τ) is the resolvent of the Volterra integral equation of the second kind and Υ is the operator of the solution. Formally, the relationship between two operators can be written as 1 1  ¼ 1 + Π  ¼1Υ ; 1+Π 1  Υ

(5.19)

Predicting the viscoelastic behavior

169

In the more general case 1 1  ¼ 1 + λΠ ðλÞ  ¼ 1  λΥ ðλÞ; 1 + λΠ 1  λΥ

(5.20)

where operators Π(λ) and Υ(λ) are mutually resolvent operators, depending on the parameter λ, because the kernels are dependent on λ now: ðt σ ðtÞ ¼ E0 εðtÞ  E0

Φðλ, t  τÞεðτÞdτ ¼ E0 ½1  Υ ðλÞεðtÞ

(5.21)

∞

The rules of Rabotnov algebra of resolvent operators (Rabotnov, 1948, 1969, 1977) follow: 1 ¼ 1 + μΥ ðλ + μÞ 1  μΥ ðλÞ 1 Υ ðλÞΥ ðμÞ ¼ ½Υ ðλÞ  Υ ðμÞ λμ ∂Υ ðλÞ ½Υ ðλÞ2 ¼ ∂λ Remembering that the product of Volterra operators W  ¼ L M 

(5.22) (5.23) (5.24)

(5.25)

with corresponding kernels L(t  τ) and M(t  τ) is the operator with kernel ðt

ðt

W ðt  τÞ ¼ Lðt  sÞMðs  τÞds ¼ Mðt  sÞLðs  τÞds τ

(5.26)

τ

This means that the square of the resolvent operators on the left side of Eq. (5.24) containing the iterated kernel in Eq. (5.26) can be replaced by the operator containing a single kernel, as is shown on the right side of Eq. (5.24). Resolvent operators represent most operators used in the theory of viscoelasticity. There are some additional Rabotnov’s algebra rules following from the preceding discussion: ½Υ ðλÞn ¼

1 ∂n1 Υ ðλÞ ðn  1Þ! ∂λn1

1 + μ 1 Υ  ðλ1 Þ μ ðλ1  λ2 Þ  ¼1+ 1 Υ ðλ1 Þ  1 + μ 2 Υ ðλ2 Þ λ1  λ2 + μ2 μ ðλ1  λ2 + μ2  μ1 Þ   2 Υ ðλ2  μ2 Þ λ1  λ2 + μ2

(5.27)

(5.28)

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Creep and Fatigue in Polymer Matrix Composites

In the particular case λ1 ¼ λ2 ¼ λ: 1 + μ 1 Υ  ðλÞ ¼ 1 + ðμ1  μ2 ÞΥ ðλ  μ2 Þ 1 + μ 2 Υ  ðλÞ

(5.29)

½1 + μ1 Υ ðλÞ2 μ21  ðμ1  μ2 Þ2  ¼ 1 + Υ ð λ Þ + Υ ðλ  μ 2 Þ 1 + μ 2 Υ  ðλÞ μ2 μ2

(5.30)

½1 + μ1 Υ ðλÞ½1 + μ2 Υ ðλÞ μ μ ðμ  μ1 Þðμ3  μ2 Þ  ¼ 1 + 1 2 Υ ðλÞ + 3 Υ ðλ  μ 3 Þ  1 + μ3 Υ ðλÞ μ3 μ3 (5.31) The more complicated case of mutual resolvents of a sum of operators is not written here due to the lack of space, but it is used in calculations of some numerical examples shown in the following sections of the chapter. Let’s demonstrate an application of the preceding written formulae for the case of a three-element model (Fig. 5.6b). The version of the Kelvin-Voigt model containing only elements E1, E2, η2 can be described with a particular form of Eq. (5.13): εð t Þ ¼

 ðt 1 E1 E2 ð t  τ Þ σ ð τ Þ σ ðt Þ + exp  dτ E1 η2 E1 η2

(5.32)

0

Using Eq. (5.22) (from the right side to the left side) and taking into account that in this case μ ! Eη 1 ; λ + μ !  Eη 2 ; λ !  E1 η+ E2 . 2 2 2 it is possible to easily find the inverse form  ðt E1 ðE1 + E2 Þðt  τÞ σ ðtÞ ¼ E1 εðtÞ  E1 εðτÞexp  dτ η2 η2

(5.33)

0

the derivation of which is much longer if traditional methods are used. Let’s denote module-operators as: In shear:
 G ¼ G 0 1  μG Υ  G ðλG Þ

(5.34)

In tension-compression:
 E ¼ E0 1  μE Υ E ðλ E Þ

(5.35)

Then the corresponding compliance-operators will have the form: In shear: JG ¼

 1
1 + μG Υ  G ðλG + μG Þ G0

(5.36)

Predicting the viscoelastic behavior

171

In tension-compression: JE ¼

 1
1 + μE Υ E ðλ E + μ E Þ E0

(5.37)

Here μi  0; λi < 0, where i has to be replaced by G or E, correspondingly. There is a set of formulae for calculation of the effective elastic properties of different types of composites via properties of constituents and their volume concentrations. The change of viscoelastic properties of a composite or nanocomposite with a concentration of fillers can be calculated from the change of corresponding elastic properties. If elastic moduli are replaced in the formulae by instantaneous viscoelastic moduli, the calculated results give the instantaneous effective moduli of the composite. If elastic moduli are replaced in the formulae by long-time viscoelastic moduli, the calculations results give the long-time effective moduli of the composite. The remaining problem is to find a way to predict transient properties. Volterra proved that the solution of the viscoelastic problem can be obtained from the solution of the corresponding elastic problem by replacing the elastic moduli by corresponding viscoelastic operators. Despite the fact that nowadays complicated expressions containing several operators can be numerically estimated using appropriate software, it is reasonable to use Rabotnov’s algebra of resolvent operators (Rabotnov, 1977) to convert some expressions with operators into one operator with shifted parameters, or at least to decrease the number of operators participating in the expressions for the effective characteristics. Some examples of derivations can be found in references (Van Fo Fy, 1971a,b; Vanin, 1985; Van Fo Fy, 1970; Shermergor, 1977). Another way to arrive at viscoelastic characteristic dependencies on volumetric concentrations of fillers is to replace elastic characteristics such as moduli by complex viscoelastic characteristics (Christensen, 1979, 2003) and to convert the corresponding algebraic expression containing many complex numbers to the standard form of complex numbers using trivial operations such as 1 a  ib a  ib ¼ ¼ a + ib ða  ibÞða + ibÞ a2 + b2

(5.38)

The attempt to replace elastic moduli by complex viscoelastic moduli discussed in reference (Gerasimov, 1948) appears unattractive, because the hypothesis of constant Poisson ratio in the viscoelastic process was used, which is a very rough one. The choice of the particular form of operator of viscoelasticity is subjective: it depends on the experience of the researcher, the precision of experimental determination of characteristics, the required precision of prediction of viscoelastic behavior, etc. The most popular types of operators are (a) Operator (5.14) containing sum of exponents (b) Abel operator of fractional derivative with kernel

172

Creep and Fatigue in Polymer Matrix Composites

Iα ðxÞ ¼

xα Γð1 + αÞ

(5.39)

where Γ(x) is the gamma function, which is a generalization of the factorial of noninteger values of the argument: ∞ ð

ΓðzÞ ¼ ðz  1ÞΓðz  1Þ ¼

tz1 et dt; Γð1Þ ¼ 1

(5.40)

0

(a) Rabotnov’s operator of fractional exponent with kernel

Эα ðβ, xÞ ¼ xα

∞ X

βn xnð1αÞ Γ½ðn + 1Þð1  αÞ n¼0

(5.41)

(b) Rzhanitsin’s operator with kernel

ΘðxÞ ¼ exp ðβxÞIα ðxÞ

(5.42)

(c) Koltunov’s operator with kernel

ΞðxÞ ¼ exp ðγxÞЭα ðβ, xÞ

(5.43)

(d) Sum of Rabotnov’s operators

The most popular operators are sum of exponents and Rabotnov’s operator. The sum of exponents may contain a different number of viscoelastic characteristics necessary to determine experimentally (3,5,7, etc., depending on the number of elements in the model in Fig. 5.1). A longer time range requires more and more terms to be taken into account. The precision and statistical determination of the viscoelastic characteristics fall with a rise in the number of exponents. Rabotnov’s operator contains only four constants and successfully approximates experimental data over a wide time range, but its application requires more complicated calculations. The key to understand the majority of operators is to learn how Abel’s operator works. Let D be the operator of differentiation D ¼ dtd . The square of this operator is the operator of the second derivative; the nth power of this operator is the derivative of nth order, where n is an integer positive number. It is obvious that D0 is equal to a constant, which can be set equal to unity. Operator D1 is obviously the operator of integration. If D[F(x)] ¼ f(x), then D1[f(x)] ¼ J[f(x)] ¼ F(x)  F(0). If additionally Ðx F(0) ¼ 0, then D1 ½f ðxÞ ¼ FðxÞ ¼ f ðtÞdt. 0

But what about other powers of this operator, including fractional ones? This question was asked by Leibniz in the 18th century and was answered starting with

Predicting the viscoelastic behavior

173

Liouville in the 19th century. First, let’s consider operator D2, which is repeated integration 2 3 ðx ðt ðx ðx ðx 2 4 5 J ½f ðxÞ ¼ f ðuÞdu dt ¼ x f ðtÞdt  tf ðtÞdt ¼ ðx  tÞf ðtÞdt 0

0

0

0

(5.44)

0

Generalization of the last expression is the famous Cauchy formula for repeated integrals ðx

ðx J ½ f ðx Þ ¼ n

f ðxÞdx…dx ¼ |fflfflffl{zfflfflffl}

, 0

ðx

|{z}

n

0

0

ðx  tÞn1 f ðtÞdt ðn  1Þ!

(5.45)

n

Here zero conditions are used for the lower limit in all steps of integration. The gamma function, one of the most popular special functions of mathematical physics, is a generalization of the concept of the factorial for a nonobligatory integer argument. It reduces to the factorial if the argument is integer Γ(n) ¼ (n  1)!. Replacement of the integer argument n by the real argument λ in Eq. (5.45) gives J λ ½ f ðxÞ ¼ Dλ ½ f ðxÞ ¼

ðx 1 f ðtÞ dt ΓðλÞ ðx  tÞ1λ

(5.46)

0

The formula for differentiation never be obtained from the formula for integration by simple replacement of λ by ( λ), which is easy to ascertain using the example of the power function f(x) ¼ xk. Taking the derivative from Eq. (5.46),   Ðx f ðtÞ d D Dβ ½ f ðxÞ ¼ D1β ½ f ðxÞ ¼ Γð1βÞ dx dt; and replacing β ¼ 1  λ, the formula ðxtÞ1β 0

for fractional derivative can be obtained. λ

D ½ f ðx Þ ¼ D

2λ



ðx 1 d f ðtÞ D ½ f ðxÞ ¼ dt Γð1  λÞ dx ðx  tÞλ λ



(5.47)

0

Let (Gerasimov, 1948). σ ðtÞ ¼ κDλ εðtÞ

(5.48)

When λ ¼ 0 and κ ¼ E the relationship in Eq. (5.48) represents Hook’s law. When λ ¼ 1 and κ ¼ 3η the relationship shown in Eq. (5.48) represents Newton’s law of viscous flow (in tension, not in shear, which is why the multiplier 3 appears). It is attractive

174

Creep and Fatigue in Polymer Matrix Composites

to use an intermediate value of λ for description of some aspects of viscoelastic behavior. In the case of creep 1 εðtÞ ¼ Dλ σ ðtÞ κ

(5.49)

all three modes of behavior are compared in Fig. 5.7. Let’s now add one new element to viscoelastic models (Fig. 5.8), the corresponding fractional derivative. Direct application of the law shown in Eq. (5.49) with 0 < λ < 1 was successful for some materials’ creep description, but not for all materials. The curve in Fig. 5.7 has an obvious deficiency: the tangential viscosity at the time t ¼ 0 is infinitely high. It is easy to correct this by adding the consecutive dashpot of (Slonimskiy, 1961) corresponding to a Newtonian liquid. The whole diversity of viscoelastic behavior can be well represented with the help of the model shown in Fig. 5.9. The power of the operator can be represented as an infinite power series of operators with integer powers. There is no power series for xλ but the series exists for xm/n. The real number λ can be approximated with the precision required for engineering calculations via a ratio of two integer numbers. The most often used power series is for λ ¼ 0.5: 1 11 113 1135 ðD  1Þ2 + ð D  1Þ 3  ðD  1Þ4 + …; D1=2 ¼ 1 + ðD  1Þ  2 24 246 2468 1 13 135 1357 ðD  1Þ2  ð D  1Þ 3 + ðD  1Þ4  … D1=2 ¼ 1  ðD  1Þ + 2 24 246 2468 (5.50) Thus the use of fractional derivatives is equivalent to the use of a differential equation of infinite order, that is, it is equivalent to use the models in Fig. 5.6 with infinite numbers of elements. However, among an infinite number of constants, only two are independent: all constants are calculated using κ and λ as well as the coefficients participating in the series, similar to Eq. (5.50). A model containing few exponents 5.7 Different creep curves depending on parameter λ: elastic (λ ¼ 0), viscous (λ ¼ 1), and viscoelastic (0 < λ < 1).

e

l > 2 > > > > 9 1 + 2ð1  2νM0 Þ + + 17ð1  2νM0 Þ > > = < EM0 EM0 EM0 5  1+c    2 00 > > 2 E0M 2 > > 2 EM > > > > 3 + 5 1  2ν ð Þ + 25 ð 1  2ν Þ M0 M0 ; : EM0 EM0 (5.73) Numerical examples of calculating dependencies of the components of the complex modulus on frequency and on concentrations of inclusions are shown in Figs. 5.20 and 5.21. From two known effective elastic characteristics of an isotropic body, the other ones can be easily calculated using well-known formulae of Eqs. (5.51a), (5.55). For the Young’s modulus, the final formula following from Eqs. (5.60), (5.67), (5.55) can be written as

Predicting the viscoelastic behavior

189

áG¢ñ/GM0 1.2 c= 0 0.02 0.04 0.06 0.08 0.10

1

0.8

wtEcr 0.6

0

2

4

6

8

10

0

5.20 Dependencies of the effective shear storage modulus hG i (real part of the complex shear modulus) of polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here EM∞/EM0 ¼ 0.6 and GM0 is instantaneous shear modulus of the matrix. The shear storage modulus is approaching to the instantaneous one at infinite frequency.

áG²ñ/GM0 0.2 c= 0 0.02 0.04 0.06 0.08 0.10

0.1

wtEcr 0

0

2

4

6

8 00

10

5.21 Dependencies of the effective shear loss modulus G (imaginary part of the complex shear modulus) of polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardant time in the creep τEcr at tension). Here EM∞/EM0 ¼ 0.6 and GM0 is instantaneous shear modulus of the matrix.

190

Creep and Fatigue in Polymer Matrix Composites





2

GI 15ð1  νM Þ 1 GM

 3

7 7 GI 5 7  5νM + 2ð4  5νM Þ   GM9 h Ei ¼ E M 8 G I > > > > 5 1 = < 1 1  2νM GM 1 + 2cð1  νM Þ + GI > > ð1  cÞ 2 > ; : 7  5νM + 2ð4  5νM Þ > GM 1 + 3c

ð1  ν M Þ ð1  c Þð1 + ν M Þ

6 61 + c 4



(5.74)

For rigid inclusions this formula degenerates into  h Ei ¼ E M

 1  νM 15ð1  νM Þ 1 + 3c 1+c 2ð4  5νM Þ ð1  cÞð1 + νM Þ 1  νM ð1  νM Þð1  2νM Þ 1 + 2c + 5c 2ð4  5νM Þ ð1  cÞ

(5.75)

A more complicated solution, which is valid for slightly larger volume concentration of inclusions, is given by Vanin (1985). He derived a direct formula for the effective Young’s modulus instead of using the solution for the shear modulus  3 9 8 2 GI KI > > > > ð 1 + ν Þ ð 7  5ν Þ  1  1 M M > > > 6 7 > 1 G > > K M M > > 6 7 + 1 + 2c 2 > > > > 4 5 G > > K 2 ð 1 + ν Þ 3 I I M > > > > ð7  5νM Þ + 2ð4  5νM Þ +4 = < GM KM ð1  2νM Þ   3 h Ei ¼ E M 2 > > GI KI > > > > 2ð4  5νM Þ 1 1 > > > > 6 7 2 ð 1 + ν Þ G > > K M M M > > 6 7 1c + > > > 4 KI 2ð1 + νM Þ 5> G > > 3 I > > ; : + 4 ð7  5νM Þ + 2ð4  5νM Þ GM KM ð1  2νM Þ (5.76) For absolutely rigid inclusions this formula degenerates into

  EM ð1  2νM Þ ð1 + νM Þð7  5νM Þ 1+c + h Ei ¼ 3ð4  5νM Þ 1c ð1 + νM Þ

(5.77)

The results of the calculations are shown in Fig. 5.22. In a similar manner used for shear, it is possible to derive formulae for the effective viscoelastic module-operator of the filled material for tension-compression and also the inverse viscoelastic compliance-operator. Unfortunately, the final formulae are too bulky to be printed here. An example of the calculations for the complianceoperator is shown in Fig. 5.23.

5 áEñ EM

GI = 25 GM

4

GI = 50 GM GI ®¥ GM

3

2

c 1

0

0.2

0.4

0.6

5.22 Relative increase of Young’s modulus of composite made from polymeric matrix (epoxy)   filled by glass microspheres GGMI ¼ 25 , by micro- or nanospheres made from material with ¼ 50 (MgO, low grades of carbon, silicon carbide, etc.), or by absolutely rigid micro- or   nanospheres GGMI ! ∞ as a function of the volume concentration c of inclusions. GI GM

according to the formulae (Christensen, 1979) and Eqs. (5.63), (5.64) derived for small concentrations; according to the formulae (Vanin, 1985) and Eqs. (5.65), (5.66) derived for medium concentrations. 1.8 áJEñEM0 1.6

0% 1% 2% 3% 4% 5% 10%

1.4

1.2

1

0.8 t/tEcr 0.6

0

0.5

1

1.5

2

5.23 Tension/compression creep function JE (creep curve divided by the stress) of composite made from matrix filled by different volume percentages of absolutely rigid spherical particles; τEcr is the strain retardant time in the creep at unidirectional tension/compression; here EM∞/ EM0 ¼ 0.6 and EM0 is the instantaneous Young’s modulus of the matrix.

192

Creep and Fatigue in Polymer Matrix Composites

1.4 áE′ñ/EM0 c=

1.2

0 0.02 0.04 0.06 0.08 0.10

1

0.8

wtEcr 0.6

0

2

4

6

8

10

0

5.24 Dependencies of the effective storage modulus hE i (real part of the complex modulus) of composite made from polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardant time in the creep τEcr at tension). Here EM∞/EM0 ¼ 0.6 and EM0 is the instantaneous shear modulus of the matrix.

The complex Young’s modulus can be calculated using the previously found complex bulk modulus from Eq. (5.65) and the complex shear modulus in Eqs. (5.71)–(5.73):     h    i 3G0 K 0 2 + K 00 2 + K 0 G0 2 + G00 2 + i 3G00 K 0 2 + K 00 2 + K 00 G0 2 + G00 2 E∗ ¼ 9 ð3K 0 + G0 Þ2 + ð3K 00 + G00 Þ2 (5.78) Numerical examples of the calculations are shown in Figs. 5.24 and 5.25. It is necessary to mention that viscoelastic behavior for shear and tension/compression are similar, which is natural because the bulk creep of the matrix was ignored in the model; however, this similarity is not absolute because the stress concentrations in the matrix near inclusions are different for shear and tension/compression. Other effective elastic properties can be easily calculated from two known effective elastic characteristics of an isotropic body using the well-known formulae from Eq. (5.51b) and G¼

E 3

E 3K

(5.79)

Results of calculations of Poisson ratio change with concentration of inclusions are shown in Fig. 5.26. This is the only characteristic that may change

Predicting the viscoelastic behavior

193

áE²ñ/EM0 0.2 c= 0 0.02 0.04 0.06 0.08 0.10

0.1

wtEcr 0

0

2

4

6

8

10

00

5.25 Dependencies of the effective loss modulus hE i (imaginary part of the complex modulus) of polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardant time in the creep τEcr at tension). Here EM∞/EM0 ¼ 0.6 and EM0 is the instantaneous shear modulus of the matrix. 0.45 ánñ

GI = 25 GM

0.4

GI = 50 GM GI ®¥ GM

0.35

c 0.3 0

0.2

0.4

0.6

5.26 Change of Poisson ratio of composite made from polymeric matrix (epoxy) filled by glass   solid microspheres GGMI ¼ 25 , by micro- or nanospheres made from material with GGMI ¼ 50 (MgO, low   grades of carbon, silicon carbide, etc.) or by absolutely rigid micro- or nanospheres GI ! ∞ as a function of the volume concentration c of inclusions. —according to the GM formula (5.51b) and formulae (Christensen, 1979) and Eqs. (5.74), (5.75) derived for small concentrations; —according to the formula (5.51b) and formulae (Vanin, 1985) and Eqs. (5.76), (5.77) derived for medium concentrations; in both cases Eqs. (5.73), (5.74), (5.51a), (5.55) are used also.

194

Creep and Fatigue in Polymer Matrix Composites

án¢ñ 0.42

c= 0.4

0 0.02 0.04 0.06 0.08 0.10

0.38

wtEcr 0.36

0

2

4

6

8

10

0

5.27 Dependencies of the real part hν i of the effective complex Poisson ratio of polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardant time in the creep τEcr at tension). Here EM∞/EM0 ¼ 0.6. 0 án²ñ

c= 0 0.02 0.04 0.06 0.08 0.10

–0.01

–0.02 wtEcr 0

2

4

6

8

10

00

5.28 Dependencies of the effective imaginary part hν i of the complex Poisson ratio of composite made from polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardant time in the creep τEcr at tension). Here EM∞/EM0 ¼ 0.6.

nonmonotonically with filler concentration. Results of calculations of the viscoelastic Poisson effect are shown in Figs. 5.27 and 5.28. It is interesting that the frequency dependence of the imaginary part of the Poisson ratio is practically independent of filler concentration.

Predicting the viscoelastic behavior

5.8

195

Viscoelasticity of particulate composites: elongated inclusions

Chopped fibers and nanofibers (which are short due to existing technology) are used for filling polymeric materials, as the layers between continuous fiber-reinforced layers, and as matrices for composites. Whiskers (monocrystalline needle-like short fibers) are used for reinforcing polymeric, metal, and ceramic matrices. There are few models proposed for analysis of such materials. It is necessary to mention that not only the properties of constituents (fibers and matrix) and the volume concentration of the fibers define effective properties of the material, but also the geometrical architecture of the material (fiber orientation distribution, lattice parameters distribution). A model (Russel, 1973; Christensen, 1979) of material reinforced by a set of parallel prolate spheroidal inclusions (in small volume concentration) is the most rigorous mathematically, but it does not adequately describe the geometry of real inclusions. A model of a finite length cylindrical inclusion is possible but an infinitely large stress concentration is expected near plane cuts of the ends of the cylinder. This can be overcome in selection of semispherical or semispheroidal bottoms, but the result will be sensitive to the selected configuration of the ends. Approximate engineering models were also proposed that separated the middle part of the inclusion interacting with the matrix exactly as in the case of continuous fiber and analyzed the end zones with the help of a set of simplifying hypotheses. The most primitive models of material reinforced by prismatic parallel inclusions are used for rough estimations of possible effects. The condition of small concentrations of inclusions allows use of the solution of single inclusion in an infinitely large volume of matrix. Obtained effective anisotropic properties of microvolumes can be used in the procedures of angle averaging for the cases of random orientation of inclusions in planes or in space. The type of lattice of the parallel set of fibers is important also. In the plane perpendicular to the fibers, if the packing is random or ideal hexagonal, the microvolume will be effectively monotropic (characterized by five independent elastic constants). If packing is quadratic, the microvolume will be quadratically orthotropic (characterized by six independent elastic constants). If packing is rectangular, the microvolume will be orthotropic (characterized by nine independent elastic constants). Packing in the plane parallel to the fibers is even more important for creep (Fig. 5.29). The final formulae for effective elastic properties are bulky (the formula for longitudinal Young’s modulus can be found in (Christensen, 1979)). For the prediction of viscoelastic behavior, the simplified formula for the case of much stiffer inclusions than the matrix is used here: 8 > >
> =

c  E11 ¼ EM 1 + > 2 5  4νM > > > ; : 2ð1 + νM Þκ2 ln  κ 4ð1  νM Þ

(5.80)

196

Creep and Fatigue in Polymer Matrix Composites

5.29 Different possible mutual positions of short reinforcing fibers: upper case provides minimal creep of polymeric matrix, the middle case provides maximal creep, and the bottom case gives a more realistic prediction (Beyle and Cocke, 2003).

Here κ is the ratio of the diameter of short fiber to its length. Inverse ratio (length to diameter) is called aspect ratio. Nevertheless, the more general formula was used in numerical calculations of the effect of the presence of aligned short fibers. The results of calculations are presented in Figs. 5.30 and 5.31. The effective longitudinal modulus depends on the relative stiffness nonlinearly and it approaches the asymptote corresponding to absolutely rigid short inclusions. This picture dramatically changes only for the case of straight continuous fibers. Dependence of the effective longitudinal modulus on the aspect ratio is also nonlinear (Fig. 5.31).

5.9

Viscoelasticity of particulate composites: oblate and platelet inclusions

Let’s consider two cases of platelet orientation in matrix: parallel and random. Bulk modulus is calculated according to the same formula, in Eq. (5.60), as for spherical inclusions and it is independent on inclusion orientation. Along the plane of inclusions, the Young’s modulus can be roughly calculated on the basis of primitive model “rectangular parallelepiped inclusion in rectangular parallelepiped cell,” neglecting the Poisson effect. Before applying it to platelet inclusion, let’s verify the model of cubical inclusions arranged in a cubical lattice. For simplicity of the verification, let’s consider inclusions as absolutely rigid. The upper estimation of Young’s modulus for this particular case gives

Predicting the viscoelastic behavior

197

12 áE1ñ/EM

10 1/k = 8 5 15 25 50 100 ¥

6

4

2 EI /EM

0

(a)

50

0

100

150

200

25 áE1ñ/EM

20 1/k = 5 15 25 50 100 ¥

15

10

5 EI /EM

0

0

50

100

150

200

(b) 5.30 Dependence of the effective Young’s modulus hE1i of the material made from matrix with Young’s modulus EM and parallel fibrous inclusions having Young’s modulus EI on the relative stiffness for different aspect ratio (length to diameter). Calculations are based on the Russel formulae (Christensen, 1979; Russel, 1973). These results and the following discussion were presented in (Beyle and Cocke, 2003; Beyle et al., 2004). Volume concentration of inclusions is 5% (A) and 10% (B).

198

Creep and Fatigue in Polymer Matrix Composites

áE1ñ/EM EI /EM = 15

80 140 200 ¥

10

200 1/k

5

¥

1/k 0

0

20

40

60

80

100

5.31 Dependence of the effective Young’s modulus hE1i of the material made from matrix with Young’s modulus EM and parallel fibrous inclusions having Young’s modulus EI on the aspect ratio (length to diameter) for inclusions having different relative stiffness. Calculations are based on the Russel formulae (Christensen, 1979; Russel, 1973) and were presented in (Beyle and Cocke, 2003; Beyle et al., 2004). Volume concentration of inclusions 5%. The asymptotic line for the case EEMI ¼ 200 is shown for comparison.

 hEiupp ¼ EM 1 +

p ffiffiffi  3 c ffiffiffi p 1 3 c

(5.81)

The lowest estimation of Young’s modulus for this case gives  hEilow ¼ EM 1 +

c pffiffiffi 1 3 c

 (5.82)

The results of the calculations according to Eqs. (5.81), (5.82) are compared in Fig. 5.32 with previous calculations according to Vanin’s formula in Eq. (5.77) for spherical inclusions. The results for spherical inclusions are very close to the results of the lowest estimation using the model described here. Calculations for the platelet inclusions were carried out according to the lowest estimate for the case of the cubic lattice of platelet inclusions and for the case of a lattice in which the individual cell has a shape similar to the inclusion. For the cubical lattice of platelet inclusions oriented parallel to each other, the formula for the effective Young’s modulus is 2 6 h Ei ¼ E M 6 41 + c

3 7 EI  EM 7 rffiffiffi 5 3 c EI  ð EI  EM Þ κ

(5.83)

Predicting the viscoelastic behavior

199

áEñ EM

6

4

2

c 0

0

0.2

0.4

0.6

5.32 Dependencies of the effective Young’s modulus of composite on concentration of absolutely rigid inclusions; cubic inclusions, upper estimation according to Eq. (5.81), cubic inclusions, lowest estimation according to Eq. (5.82), spherical inclusions according to Eq. (5.77).

Here κ is the ratio of thickness of the plate to its size in the plane. Inverse ratio is called aspect ratio. Dependence of the effective modulus on the ratio of Young’s moduli of inclusion and the matrix for platelets with different aspect ratio is shown in Fig. 5.33. It follows from these data that the use of majority of platelet fillers for polymer matrices (ratio of moduli 25–50) is practically equivalent to the use of absolutely rigid platelets. For this case, Eq. (5.83) degenerates into 2 6 h Ei ¼ E M 6 41 + c

3 1 7 rffiffiffi7 c5 1 3 κ

(5.84)

In the case of a lattice having cells with a shape similar to the shape of the inclusions, the effective Young’s modulus is calculated according to the same formula (Eq. 5.82) as for cubical inclusions: that is, the result is independent of the aspect ratio and the effectiveness of the reinforcement is much lower than in the case of the cubic lattice. In reality, the random package of the inclusions in the matrix can be considered as an intermediate case between these two types of lattices. It is necessary to mention that the cubic lattice for platelets with high aspect ratio can be realized only at small volumetric concentrations c  κ. Increase of the aspect ratio leads to higher effectiveness of the reinforcement, but in very high aspect ratios the problem of keeping the plane

200

Creep and Fatigue in Polymer Matrix Composites

1.6 áE1ñ/EM

1/k = 1 5 10 15

1.4

1.2

EI /EM

1

0

50

100

150

200

5.33 Dependence of the effective Young’s modulus hE1i in the direction parallel to the plane of the platelet inclusions on the ratio of the Young’s moduli of inclusion to matrix and on the aspect ratio of inclusions.

shape of the inclusions appears. Due to insufficient bending and torsion stiffness of very thin platelet inclusions, their shape can be distorted by matrix flow during the technological process, by shrinkage of the matrix, etc. Effectiveness of the reinforcement of distorted inclusions is much lower than the effectiveness of ideally plane platelets. For randomly oriented platelet inclusions, it is possible to derive the formula for the effective Young’s modulus as follows: 2 3 6 3 h Ei ¼ E M 6 41 + 5 c

7 EI  EM 7 rffiffiffi 5 3 c EI  ð EI  EM Þ κ

(5.85)

which for the large ratios of Young’s moduli degenerates into 2 3 6 3 h Ei ¼ EM 6 41 + 5 c

1 7 rffiffiffi7 c5 1 3 κ

(5.86)

This formula is valid only for small concentrations of inclusions, especially if the aspect ratio is large. The high theoretical effectiveness of reinforcement by platelet inclusions mentioned in many works is obtained when extrapolating the aspect ratio to infinity. For a finite aspect ratio, the effectiveness is not so high.

Predicting the viscoelastic behavior

201

The viscoelasticity of platelet inclusion reinforced material is mostly determined by the viscoelasticity of the polymer matrix, because it follows from the formulae derived for absolutely rigid inclusions.

5.10

Viscoelasticity of fibrous composites

The set of recommended formulae for prediction of the effective properties of unidirectional-fiber reinforced composite is: 1. Effective longitudinal Young’s modulus EL ¼ cEIL + ð1  cÞEM +

 2 2cð1  cÞEWM νM  νILT WM =EWI  ½ð1 + c + νWM  νWM cÞ + ð1  cÞð1  νWI TT ÞE T

(5.87)

here the plane strain constants are denoted by index W; indices M (matrix) and I (inclusion, in this case fiber) are moving up for convenience; index L denotes longitudinal direction; and index T denotes transversal direction 1 1  νITL νILT νWI νI + νI νI ¼ ; TT ¼ TT ITL LT ; WI I WI ET ET ET ET 1 EWM

2

¼

2

1  ðνM Þ νWM νM + ðνM Þ ; WM ¼ ; EM E EM EWM EM ¼ GM ¼ WM 2ð1 + ν Þ 2ð1 + νM Þ

(5.88)

In most cases, the last term in Eq. (5.87) can be neglected and the “mixture rule” can be used for the effective longitudinal Young’s modulus: hEL i ¼ cEIL + ð1  cÞEM

(5.89)

2. Effective longitudinal-transversal shear modulus:   GM  GI 1c M G + GI  hGLT i ¼ GM  GM  GI 1+c M G + GI

(5.90)

In the case when fibers are much stiffer than the matrix: hGLT i ¼ GM

1+c 1c

(5.91)

3. Longitudinal-transversal Poisson ratio: hνLT i ¼ νM + 2c

νI  νM  LT  EWM  1 + c + ð1  cÞ νWM + WI 1  νWI TT ET

(5.92)

202

Creep and Fatigue in Polymer Matrix Composites

4. Bulk modulus in the plane strain state: 8 9 > > > > > > < = 1 2 4c WM  W  ¼ WM  1 + c  ν ð1  c Þ   > E ð1  c Þ > EWM  KTT > > > > 1 + c + ð1  cÞ νWM + WI 1  νWI : ; TT ET (5.93) 5. Transversal shear modulus: 8 2 39 GM > > > > > 1  = 6 7> 1 1 < GITT 6 7   ¼ M 1  c6 7 M 4 hGTT i G > G ð1  c Þ 5 > > > > > 1 1 I : ; GTT 4ð1  νM Þ

(5.94)

In the case of infinitely rigid fibers, this formula degenerates to  1 1 ð1  νM Þ 1  4c ¼ 3  4νM + c hGTT i GM

(5.95)

Some constants, which are more often used in engineering practice, can be derived from the previously written five independent elastic constants. Transversal Poisson ratio at plane strain state:



 νW TT ¼

hGTT i 1 W  KTT hGTT i 1+  W  KTT

(5.96)

Transversal Young’s modulus for plane strain state: 

  W   W  EW T ¼ 2 KTT 1  νTT

(5.97)

Transversal Young’s modulus:  W ET  W hE T i ¼ 2 ET 1 + hνLT i hEL i

(5.98)

Transversal Poisson ratio:   hET i hE i  W   hνLT i2 T hνTT i ¼ νW TT hEL i ET

(5.99)

The derivations for some of these equations can be found in Christensen (1979) and some in Vanin (1985). Replacing elastic characteristics by corresponding viscoelastic operators and using the algebra of resolvent operators, it is possible to arrive at a prediction of the viscoelastic

Predicting the viscoelastic behavior

203

behavior of unidirectional fibrous composites. It is reasonable to consider plane strain bulk modulus as pure elastic. Some research using slightly different methods can be found in the literature (see, e.g., (Christensen, 1979; Beyle et al., 2004) to (Pobedrya, 1979)). The main conclusions are: 1. If fibers are elastic, not viscoelastic, the longitudinal creep of the matrix is for practical purposes blocked. Experimentally observed longitudinal creep is related to technological defects such as misalignment and waviness of the fibers, length disparity, etc. 2. The main viscoelastic effects are in interlaminar shear, which result in additional deflections of bent structures and in decreasing critical buckling loads. 3. Some viscoelasticity is realized in transversal loading, which leads to redistribution of stresses in thick-walled pressure vessels, flywheels, and other structures.

5.11

Hollow fillers: buckyballs, microballoons

Hollow fillers are used widely in composite and nanocomposite technology, providing light weight, good thermal insulation properties, better resistance to dynamic loads, and so forth. Syntactic foams use hollow microspheres or microballoons; hollow nanospheres called buckyballs have been used during the last two decades. Other hollow nanosized objects, such as fullerenes, are also used. Syntactic foams are used widely in many industries, such as electrical machinery and shipbuilding. The behavior of solid glass spheres in polymeric matrices, and especially that of carbon spherical particles under hydrostatic pressure, is very similar to the behavior of absolutely rigid inclusions. As a result, hollow glass spheres, having one-half or less of the mass of solid spheres, provide almost the same reinforcing effect (Fig. 5.34). Even glass hollow microspheres having 10% of the mass of solid microsphere still have a reinforcing effect. For carbon hollow nanospheres, this “quasi-neutral” limit is much lower. Only microspheres with very thin walls act as voids. Similar conclusions can be drawn with respect to the effective Young’s modulus (Fig. 5.35).

5.12

Hollow fibers and nanotubes

Hollow glass fibers and some other fibers have been produced for decades to make materials lighter, increase ratio of compression strength to density, decrease thermal conductivity, and to use in filtration systems, mass transfer, etc. Nanotube-based materials modeling can have similarities to hollow fiber reinforced materials modeling. The majority of natural fibers are hollow. Hollow fibers have been used for reinforcing clay-based and lime-based materials since ancient times. Most research in nanofillers deals with nanotubes. Continuous straight fiber replacement of solid fibers by hollow fibers does not change the longitudinal effective Young’s modulus if the mass concentration of fibers is the same. However, for short hollow fibers the result of such replacement is different (see Fig. 5.36).

204

Creep and Fatigue in Polymer Matrix Composites 4 m = 0 pore

áK ñ/KM

m = 0.02 3 m = 0.1 m = 0.2 m = 0.4

2

m = 0.6 m = 1 solid 1

0

Rigid

0

0.1

0.2

0.3

0.4

0.6

0.5

c

5.34 Dependence of the effective bulk modulus hKi of material made from matrix with bulk modulus KM and hollow spherical inclusions on the volume concentration of inclusions. Here  3 r0 is the volume concentration of the solid phase in single inclusion. Calculations m¼ 1 r1 are made for vinyl-ester matrix and glass microballoons. 3.5 áE ñ/EM

3 2.5

m = 0.02 m = 0.1

2

m = 0.2 m = 0.4

1.5

m = 0.6 1 0.5 c

0

0

0.1

0.2

0.3

0.4

0.5

0.6

5.35 Dependence of the effective Young’s modulus hEi of material made from matrix with Young’s modulus EM and hollow spherical inclusions on the volume concentration of  3 r0 inclusions. Here m ¼ 1  is volume concentration of the solid phase in single inclusion. r1 Calculations are made for vinyl-ester matrix and glass microballoons.

Predicting the viscoelastic behavior

205

áE 1ñ/EM

c=0.05 1/k =100

15

c=0.05 1/k =25

10

c=0.025 1/k =100

5

m 0 0.4

0.2

0.6

0.8

5.36 Dependence of the effective longitudinal Young’s modulus hE1i of short hollow fibers with the same mass concentration as solid fibers on the concentration of solid phase in the cross-section  2 r0 of the fiber m ¼ 1  . Initial volume concentration of solid fibers and aspect ratio are shown r1 in the figure. Other values used in calculations: EI/EM ¼ 300; νI ¼ 0.25; νM ¼ 0.35.

This phenomenon can be understood if we return to the results of the calculations shown in Fig. 5.32. If a solid fiber is replaced by two hollow fibers with the same total mass, the effective stiffness drops by two times but is still sufficiently high and the effect of the reinforcement of one short hollow fiber is comparable to the effect of the reinforcement of one solid fiber. This happens due to the significant nonlinearity of the curves in Fig. 5.32. However, because the number of fibers and their volume concentration in the nanocomposite is doubled, the total reinforcing effect is almost doubled. This is the reason why nanotubes are more promising than solid nanofibers. When the fiber wall thickness is decreased more, the effect is decreased. Moreover, there is an optimal wall thickness of nanotubes providing the maximal reinforcing effect. The true optimum is shifted toward thicker walls than is shown in Fig. 5.36, because too thin nanotubes behave not as hollow rods but as shells; they can thus be deformed more easily and in different ways (local wall bending or buckling, etc.).

5.13

Viscoelasticity of foams and nanoporous materials

In the case of spherical voids, Eq. (5.60) degenerates into 0

1 4 GM  B 1  νM 3 KM C C ¼ KM 1  3c 1  c hK i ¼ KM B @ 4 GM A 2ð1  2νM Þ + cð1 + νM Þ c+ 3 KM 1+

(5.100)

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Creep and Fatigue in Polymer Matrix Composites

The corresponding viscoelastic operator can be found by replacing the Poisson ratio by the operator in Eq. (5.57) and by using Eq. (5.29); the result has the following form: 3 ð1  νM0 Þ  1  3c 7 6 2ð1  2νM0 Þ + cð1 + νM0 Þ 68 97  7 6> ð1  2νM0 Þ ð4  cÞð1  2νM0 Þ μE > 7 6> > 1 >  hK  i ¼ KM 6 > 7 = < 7 6 1  νM0 2ð1  2νM0 Þ + cð1 + νM0 Þ 2 7 6   5 4> > ð4  cÞð1  2νM0 Þ μE > >  > > ; : Υ E λE + 2ð1  2νM0 Þ + cð1 + νM0 Þ 2 2

(5.101)

In the case of spherical voids, Eq. (5.67) degenerates into  hG i ¼ G M

15ð1  νM Þ 1c 7  5νM

(5.102)

The Vanin formula in Eq. (5.78) degenerates into 9 5 > > = 1 c 3 h Ei ¼ EM ð1 + νM Þð13  15νM Þ> > > > ; :1 + c 2ð7  5νM Þ 8 > >
dfC for domain 3,

(14.22a)

dfb  dfC  0 for domain 2,

(14.22b)

Damage (D)

DdfBb DfA

A DDAC

DfB

B b

DC

C 0

0.5 Location factor (df)

DDBC 1

14.5 Damage as a function of general location factor (df) on S–N plan for three points b, B, and C in Fig. 14.4: the upper curve represents damage (D) for point B corresponding to df of point C; and the lower curve represents damage (D) for points b and C for an unknown S–N curve.

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Creep and Fatigue in Polymer Matrix Composites

and dfb  dfC  1 for domain 1:

(14.22c)

The necessary and sufficient conditions for a valid dfb cannot possibly be found without a known S–N curve and a damage function (D).

14.6

Damage function and validity

According to the monotonicity of damage in the axioms, a damage function (D) is required to monotonically vary between df ¼ 0 and 1 on the scale of log N at a given peak stress level. The following candidate damage function (D) may be considered to see if it satisfies the damage validity conditions: D ¼ Df dfn

(14.23)

where n is an unknown constant. It is seen that Eq. (14.23) satisfies the boundary conditions in domains 1 and 2 (i.e., D ¼ 0 at df ¼ 0 or log N ¼ log 0.5 ¼  0.3, and D ¼ Df at df ¼ 1 or log N ¼ log Nf). However, it is not known yet whether it satisfies Eq. (14.22a), which is a necessary condition, for the compatibility in domain 3, given neither n value nor an S–N curve is known. The value of n depends on the damage characteristics determining the shape of the S–N curve. For example, if an S–N curve is concave up (or the upper curve in Fig. 14.5 is concave down), it should be 1 < n < ∞. If a single critical value of n exists for a given S–N plane satisfying Eq. (14.22a), it would be sufficiently valid because it represents a unique characteristic value for a given S–N curve. The n value can be determined numerically by incrementing n starting from 1 until it reaches the critical value at which Eq. (14.22a) is satisfied for a given S–N curve. In other words, all the values of (dfb – dfC) at various stress levels should be positive at the critical value of n for Eq. (14.23). If multiple critical values of n due to different stress levels exist in an acceptable small range, a highest critical value of n may be taken as an approximately valid one.

14.7

Isodamage point b

The isodamage points can practically be used for predicting the remaining fatigue life at a different stress level (σ max). Let us consider damage on two horizontal lines DA and D0 B on the S–N plane (Fig. 14.4) to find an isodamage point b whose damage is equal to damage at point B. A damage function (DA) for the points on line DA is given by n DA ¼ DfA dfA

(14.24a)

S–N curve and fatigue damage for practicality

455

and another damage function (DB) for the points on line D0 B by n DB ¼ DfB dfB

(14.24b)

according to Eq. (14.23). The general location factor for point b (dfb) at a stress level of point A (Fig. 14.4) for an isodamage point, then, is found by equating these two equations:  1 DfB n log Nb + 0:3 dfb ¼ ¼ log NfA + 0:3 DfA

(14.25)

where DfA and DfB are known damage values at points A and B, respectively. Eq. (14.25) can be used for finding a position of an isodamage line on the S–N plane for predicting the remaining fatigue life. For example, fatigue cycling is conducted for two sequential blocks of loading—the first one is cycled at a peak stress (σ maxA) and then changed at point b to a lower peak stress σ maxH (Fig. 14.4). In this example, another isodamage point at σ maxH can be obtained by replacing DfA with DfH in Eq. (14.25), 

DfB dfb ¼ DfH

1 n

¼

log Nb + 0:3 log NfH + 0:3

(14.26)

where dfb is known, DfB and DfH are known damage values, and NfH is the number of cycles at failure for point H. Accordingly, the number of cycles Nb (or a new position, dfb at σ maxH) can be found such that the remaining fatigue life at σ maxH is predicted to be NfH  Nb.

14.8

Numerical determination of n value

To determine numerically the n value in Eq. (14.23), two different stress levels of σ max should be chosen within the highest and lowest σ max bounds of an S–N curve, and then dfb and dfC are calculated until a positive value of (dfb – dfC) is obtained. The two σ max bounds should be within a range between tensile (or compressive) strength and assumed fatigue limit (or substantially low σ max). Two different numerical schemes may be possible for calculating (dfb – dfC): (1) the highest σ max (σ Hmax) is fixed and then the lowest σ max (σ Lmax) is incremented, and this process is repeated with the next highest (σ Hmax(i )) until σ Hmax(i ) ¼ σ Lmax ; and (2) both two adjacent σ Hmax and σ Lmax (¼ σ Lmax < σ Hmax) are decremented, or both two adjacent σ Hmax and σ Lmax are incremented between the two σ max bounds. The first scheme may be comprehensive but requires many iterations, whereas the second scheme is relatively simple and efficient for execution. Thus, the second scheme may be sufficient for most of the cases because the angle θb [see Eq. (14.20)] tends to be

456

Creep and Fatigue in Polymer Matrix Composites

low for a small interval (¼ σ Hmax – σ Lmax). A sequence of the second scheme for calculating (dfb – dfC) is given in Fig. 14.6, which is readily executable using Microsoft Excel.

14.9

Examples for predicting the remaining fatigue life

Experimental results obtained by Broutman and Sahu (1972) were employed to demonstrate predicting the remaining fatigue life. Fig. 14.7 shows an S–N curve represented by Eqs. (14.8a) and (14.8b) fitted to the experimental data for cross-ply GFRP laminate with an ultimate strength of 448 MPa and a stress ratio (R) of 0.05 (which is close to zero). Fitting parameters for Eqs. (14.8a) and (14.8b) were found to be α ¼ 2.8  1041 and β ¼ 14.2. The dashed lines on both sides of the solid line represent experimental failure probabilities 10% and 90%. The experiment for predicting the remaining fatigue life was conducted with two sequential blocks of loading. Each specimen was cycled at a peak stress level 386, 338, 290, or 241 MPa for the first blocks of loading and then changed to another peak stress level for the second blocks of loading until failure occurred as detailed in Table 14.1.

Determine

sHmax and sLmax with a small interval (=sHmax –sLmax) for a chosen n value Calculate DfB and DfA for s Hmax and s Lmax, respectively, using

CalculateLog Nf for point B at sHmax and point C at sLmax using

Calculate dfC at sLmax using Calculate dfb using

with

obtained DfB and DfA where NfB and NfC are N f at s Hmax and s Lmax, respectively

Calculate dfb – dfC

14.6 A sequence for calculating (dfb – dfC).

S–N curve and fatigue damage for practicality

457

n=1

450

n=2 400

n=3 n = 3.4

350 300

90%

smax

250 200 150

10% 100 50 0 –0.3

0.7

1.7

2.7

3.7

4.7

5.7

6.7

Log N 14.7 An S–N curve fitted to data of Broutman and Sahu (1972) for cross-ply glass fiber reinforced plastic laminate with an ultimate strength of 448 MPa and a stress ratio (R) of 0.05. Fitting parameters for Eqs. (14.8a) and (14.8b) are α ¼ 2.8  1041 and β ¼ 14.2. The dotted lines on both sides of the solid line represent failure probabilities 10% and 90%. Other lines are isodamage loci affected by n value of Eq. (14.23).

14.9.1 Determination of n value To find a valid n in Eq. (14.23), a value starting from n ¼ 1 is incremented with an interval of 0.1 until all the values of (dfb – dfC) become positive, according to the sequence given in Fig. 14.6. A pair of two stress levels with an interval (¼ σ Hmax – σ Lmax) of 1 MPa was used to calculate dfb and dfC for a stress starting from σ Hmax ¼ 201 MPa (paired with σ Lmax ¼ 200 MPa) incrementally up to 446 MPa (paired with 445 MPa < σ uT ¼ 448 MPa). As a result, n was found to be 3.4. The effect of the n value on isodamage lines is shown in Fig. 14.7. As expected, no effect of n value on domains 1 and 2 is seen. However, the effect becomes sensitive in the middle of domain 3. Also, it is seen that once the n value is higher than 3, no violation of the validity (Eq. 14.20) is visually detected, and not much difference between n ¼ 3 and 3.4 in damage variation is expected.

Table 14.1 Two stress level tests in sequence for comparison with theoretical predictions (Broutman and Sahu, 1972) Log10 Nf1 (cycles)

σ 2max (MPa)

Log10 Nf2

N1 Nf 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

386 386 386 386 386 386 338 338 338 338 290 290 241 241 241 241 241 241 290 290 290 290 338 338

2.693 2.693 2.693 2.693 2.693 2.693 3.393 3.393 3.393 3.393 4.168 4.168 5.236 5.236 5.236 5.236 5.236 5.236 4.168 4.168 4.168 4.168 3.393 3.393

241 241 290 290 338 338 241 241 290 290 241 241 290 290 338 338 386 386 338 338 386 386 386 386

5.236 5.236 4.168 4.168 3.393 3.393 5.236 5.236 4.168 4.168 5.236 5.236 4.168 4.168 3.393 3.393 2.693 2.693 3.393 3.393 2.693 2.693 2.693 2.693

0.507 0.203 0.507 0.203 0.507 0.203 0.405 0.101 0.405 0.101 0.680 0.136 0.290 0.116 0.290 0.116 0.290 0.116 0.68 0.136 0.680 0.136 0.405 1.020

σ 1max is the peak stress level first applied in the sequence. Nf1 is cycles to failure at σ 1max. σ 2max is the second peak stress level applied after σ 1max. Nf2 is cycles to failure at σ 2max. N1 is the number of cycles the specimen was subjected to at the first stress level σ 1max. ΔN2 is the number of cycles to failure at the second stress level σ 2max. df1 [¼(log n1 + 0.3)/(log Nf1 + 0.3)] is a general relative location factor at the end of cycling at σ 1max. df2 is a general relative location factor at σ 2max equivalent to df1 in damage.

Log10 ΔN2 (cycles) experiment

Log10 ΔN2 (cycles) predicted

df1

df2

5.283 5.286 3.766 4.078 3.097 3.214 4.934 5.211 3.938 3.903 5.236 5.236 3.572 3.977 2.592 2.905 N/A 2.093 2.467 3.111 N/A 2.550 2.473 2.702

5.287 5.287 3.813 4.086 3.194 3.243 4.971 5.214 4.045 3.930 5.171 5.076 4.279 4.212 3.749 3.524 N/A 3.152 3.685 3.374 N/A 2.975 2.928 2.817

0.897 0.759 0.897 0.759 0.897 0.759 0.894 0.730 0.894 0.730 0.963 0.811 0.905 0.834 0.905 0.834 0.905 0.834 0.963 0.811 0.963 0.811 0.894 0.730

0.630 0.533 0.681 0.576 0.758 0.641 0.743 0.607 0.803 0.656 0.891 0.750 0.978 0.902 1.088 1.033 – 1.188 1.072 0.902 – 1.069 1.058 0.865

Creep and Fatigue in Polymer Matrix Composites

σ 1max (MPa)

458

No.

S–N curve and fatigue damage for practicality

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500 450 400 350

smax

300 250 200 150 100 50 0 –0.3

0.7

1.7

2.7

3.7

4.7

5.7

6.7

7.7

Log N

(a) 500 450 400 350

smax

300 250 200 150 100 50 0 –0.3

0.7

1.7

2.7

(b)

3.7

4.7

5.7

6.7

7.7

Log N

500 High–Low 450 Low–High 400 350

smax

300 250 200 150 100 50 0 –0.3

(c)

0.7

1.7

2.7

3.7

Log N

4.7

5.7

6.7

7.7

14.8 Predictions of the remaining fatigue life and experimental results: (a) high-low sequence loading paths; (b) low-high sequence loading paths—two of the data points for premature failure before the second stress level cycling are denoted by the symbol “◊”; and (c) combined failure points without loading sequence paths after being subjected to second stress level. The dotted lines on both sides of the S–N curve (solid line) represent failure probabilities, 10% and 90%.

460

Creep and Fatigue in Polymer Matrix Composites

14.9.2 Prediction of the remaining fatigue life Fig. 14.8a and b shows the loading paths indicated by horizontal straight lines between data points for high-low and low-high sequences, respectively, with predicted remaining number of cycles (Nf2) at the second stress level (σ 2max). Each isodamage point at the second stress level following the first stress level (σ 1max) is shown with a diagonal line. Fig. 14.8c shows experimental data points obtained from both high-low and low-high sequence loadings without loading paths in comparison with the S–N curve. The load sequence paths between the first and second stress levels closely follow, as expected, the predicted isodamage lines represented by the dashed lines (Eqs. 14.25 and 14.26). In general, a majority of experimental data points (Fig. 14.8c) for Nf2 appear to follow the S–N curve, indicating a close agreement between prediction (S–N curve) and experiment, particularly for the high-low loading (Fig. 14.8a). All the experimental data points at failure predicted are within, or sufficiently close to, the scatter band represented by dotted lines of 10% and 90% probability failures. On the other hand, data from low-high sequence loading (Fig. 14.8c) appear scattered a little to one side of the S–N curve compared to those of the high–low sequence loading. It may be interesting to understand the difference in statistical behavior between the two types of loading sequences. An isodamage line between two stress levels for high–low sequence loading never intersects the S–N curve before failure. In the low– high sequence loading, however, it tends to have a higher probability of intersecting the S–N curve before failure at a higher number of cycles, resulting in a lopsided distribution of failure points (Nf2) with some deviation from the S–N curve. The scatter of Nf2 (Fig. 14.8b) to one side may be expected when individual data points are examined as follows: (a) many of the data points for the number of cycles (N1) at the first stress level are already close to the failure points or within the scatter band (see bold face numbers in Table 14.1); (b) some specimens indicated by a large diamond symbol (◊) for two data points (Fig. 14.8b) were already broken even before cycling at the second level stress; and (c) there are three other data points that are already within the scatter band at the first stress level but survived for the second stress level cycling. Of course, one might consider, at the same time, the possibility of load sequence effect for the distribution of Nf2 points apart from the scatter, as reported in the literature (Subramanyan, 1976; Gamstedt and Sjogren, 2002). However, such an effect has not been quantitatively understood yet because the differences found due to the load sequences in the past were described in terms of cyclic ratio (i.e., N1/Nf1 ¼ N2/Nf2), which is not based on the valid damage analysis. Nonetheless, if there would still be an experimental difference in well-chosen sampling between two different loading sequences, some effect of load sequence on the prediction of Nf2 may be possible.

14.10

Concluding remarks

The Kim and Zhang S–N curve model for practical use is discussed within the historical context of development of other S–N curve models. Subsequently, the validity conditions for fatigue damage function for predicting the remaining fatigue life at zero

S–N curve and fatigue damage for practicality

461

stress ratio are also discussed. A damage function approximately satisfying the validity conditions is introduced in conjunction with the Kim and Zhang S–N curve model. A practical procedure for predicting the remaining fatigue life is demonstrated using experimental data. Further work for various other stress ratios than a zero stress ratio may be required before progressing to deal with complex loading cases.

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Gathercole, N., Reiter, H., Adam, T., Harris, B., 1994. Life prediction for fatigue of T800/5245 carbon–fibre composites: I. Constant-amplitude loading. Fatigue 16, 523–532. Gerber, W.Z., 1874. Bestimmung der zulassigen Spannungen in Eisen-Constructionen. [calculation of the allowable stresses in iron structures]. Z. Bayer. Arch. Ing. Ver. 6, 101–110. Goodman, J., 1899. Mechanics Applied to Engineering, first ed. Longmans, Green and Co, London. Haery, H.A., Kim, H.S., 2013. Damage in hybrid composite laminates. JMC 1 (2), 127–137. Hahn, H.T., Kim, R.Y., 1976. Fatigue behavior of composite laminates. J. Compos. Mater. 10 (2), 156–180. Hertzberg, R. W. and Manson, J. A., Fatigue of Engineering Plastics, Academic Press, 1980. P147. Hwang, W., Han, K.S., 1986a. Cumulative damage models and multi-stress fatigue life prediction. J. Compos. Mater. 20 (2), 125–153. Hwang, W., Han, K.S., 1986b. Fatigue of composite—fatigue modulus concept and life prediction. J. Compos. Mater. 20, 154–165. Kassapoglou, C., 2007. Fatigue life prediction of composite structures under constant amplitude loading. J. Compos. Mater. 41, 2737–2754. Kawai, M., Itoh, N., 2014. A failure-mode based anisomorphic constant life diagram for a unidirectional carbon/epoxy laminate under off-axis fatigue loading at room temperature. J. Compos. Mater. 48, 571–592. Kawai, M., Koizumi, M., 2007. Nonlinear constant fatigue life diagrams for carbon/epoxy laminates at room temperature. Compos. Part A Appl. Sci. Manuf 38, 2342–2353. Kensche, C.W., 1995. Influence of composite fatigue properties on lifetime predictions of sailplanes. In: Presented at the XXIV OSTIV Congress, Vol. XIX, no. 3. Omarama, New Zealand, pp. 69–76. Kim, H.S., 2018. Mechanics of Solids and Fracture, third ed. Ventus Publishing ApS. https:// bookboon.com/en/mechanics-of-solids-and-fracture-ebook#download. Kim, H.S., Zhang, J., 2001. Fatigue damage and life prediction of glass/vinyl ester composites. J. Reinf. Plast. Compos. 20, 834–848. Kitagawa, H., Takashima, S., 1976. Application of fracture mechanics to very small fatigue cracks or the cracks in the early stage. In: Proceedings of the Second International Conference on Mechanical Behavior of Materials, American Society for Metals, Materials Park, OH, pp. 627–631. Kohout, J., Vechet, S., 2001. A new function for fatigue curves characterization and its multiple merits. Int. J. Fatigue 23, 175–183. Mandell, J.F., Meier, U., 1983. Effects of stress ratio, frequency, and loading time on the tensile fatigue of glass-reinforced epoxy. In: O’Brien, T.K. (Ed.), Long-Term Behavior of Composites. ASTM STP 813. American Society for Testing and Materials, Philadelphia, p. 55. Mao, W., Ringsberg, J.W., Rychlik, I., 2011. Theoretical development and validation of a fatigue model for ship routing. Ships Offshore Struct. 7 (4), 399–415. Marco, S.M., Starkey, W.L., 1954. A concept of fatigue damage. Transs ASME 76, 627–632. Marin, J., 1962. Mechanical Behavior of Engineering Materials. Prentice-Hall, Englewood Cliffs, NJ. Mesmacque, G., Garcia, S., Amrouche, A., Rubio-Gonzalez, C., 2005. Sequential law in multiaxial fatigue, a new damage indicator. Int. J. Fatigue 27 (4), 461–467. Miner, M.A., 1945. Cumulative damage in fatigue. J. Appl. Mech. 12 (3), 159–164. Owen, M., Howe, R., 1972. The accumulation of damage in a glass-reinforced plastic under tensile and fatigue loading. J. Phys. D. Appl. Phys. 5, 1637–1649.

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Palmgren, A.G., 1924. “Die Lebensdauer von Kugellagern” [Life Length of Roller Bearings]. Zeitschrift des Vereines Deutscher Ingenieure (VDI Zeitschrift) 68 (14), 339–341. Philippidis, T.P., Passipoularidis, V.A., 2007. Residual strength after fatigue in composites: theory vs. experiment. Int. J. Fatigue 29 (12), 2104–2116. Post, N.L., Case, S.W., Lesko, J.J., 2008. Modeling the variable amplitude fatigue of composite materials: a review and evaluation of the state of the art for Spectrum loading. Int. J. Fatigue 30 (12), 2064–2086. Poursartip, A., Beaumont, P.W.R., 1986. The fatigue damage mechanics of carbon fibre composite laminate: II. Life prediction. Compos. Sci. Technol. 25, 283–299. Poursartip, A., Ashby, M.F., Beaumont, P.W.R., 1986. The fatigue damage mechanics of carbon fibre composite laminate: I. Development of the model. Compos. Sci. Technol. 25, 193–218. Rotem, A., 1988. Residual strength after fatigue loading. Int. J. Fatigue 10, 27–31. Sch€utz, W., 1996. A history of fatigue. Eng. Fract. Mech. 54, 263–300. Sendeckyj, G.P., 1981. Fitting models to composite materials fatigue data. In: Chamis, C.C. (Ed.), Test Methods and Design Allowables for Fibrous Composites. In: vol. 734. ASTM STP, p. 245. Shokrieh, M.M., Lessard, L.B., 1997. Multiaxial fatigue behavior of unidirectional plies based on uniaxial fatigue experiments: I. modelling. Int. J. Fatigue 19 (3), 201–207. Soderberg, C.R., 1930. Factor of safety and working stress. Trans. Am. Soc. Mech. Eng. 52 (Part AP), 13–28. Stromeyer, C.E., 1914. The determination of fatigue limits under alternating stress conditions. Proc. R. Soc. A Math. Phys. Eng. Sci. 90 (620), 411–425. Subramanyan, S.A., 1976. Cumulative damage rule based on the knee point of the S-N curve. J. Eng. Mater. T. ASME 98 (4), 316–321. Sutherland, H.J., Mandell, J.F., 2004. The effect of mean stress on damage predictions for spectral loading of fiberglass composite coupons. Wind Energy 8 (1), 93–108. Tamuz, V., Dzelzitis, K., Reifsnider, K., 2004. Fatigue of woven composite laminates in offaxis loading II. Prediction of the cyclic durability. Appl. Compos. Mater. 11, 281–292. Weibull, W., September 1951. A statistical distribution function of wide applicability. J. Appl. Mech. 103 (730), 293–297. Weibull, W., 1952. The statistical aspect of fatigue failures and its consequences. In: Presented at the Fatigue and Fracture of Metals. Massachusetts Institute of Technology. John Wiley & Sons, New York, pp. 182–196. Weyrauch, P. J. J., "Strength and Determination of the Dimensions of Structures of Iron and Steel with Reference to the Latest Investigations [Translated by Du Bois, A. J.]”, third ed. New York: John Wiley and Sons. https://archive.org/details/ strengthanddete01thurgoog (1891). W€ ohler, A., 1870. Ueber die Festigkeits-Versuche mit Eisen und Stahl. Zeitschrift fur Bauwes XX, 74–106. Yang, J.N., 1978. Fatigue and residual strength degradation for graphite/epoxy composites under tension-compression cyclic loadings. J. Compos. Mater. 12, 19–39. Yang, J.N., Liu, M.D., 1977. Residual strength degradation model and theory of periodic proof tests for graphite/epoxy laminates. J. Compos. Mater. 11, 176–203.

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Modeling, analysis, and testing of viscoelastic properties of shape memory polymer composites and a brief review of their space engineering applications

15

Wessam Al Azzawia,b, Madhubhashitha Heratha,b, Jayantha Epaarachchia,b a School of Mechanical and Electrical Engineering, University of Southern Queensland, Toowoomba, QLD, Australia, bCentre for Future Materials, University of Southern Queensland, Toowoomba, QLD, Australia

15.1

Introduction

Shape memory polymers (SMPs) have the ability to retain a temporary shape when deformed above the glass transition temperature (Tg) and subsequently cooled below Tg. When reheated above Tg, SMPs can recover up to 100% of the original shape (Fejo˝s and Karger-Kocsis, 2013). The key advantages of SMPs over shape memory alloys (SMAs) are their good manufacturability, high shape deformability, and high shape recoverability (Lendlein and Kelch, 2002; Leng et al., 2011; Ohki et al., 2004). However, the relatively poor mechanical properties and recovery stress are the intrinsic drawbacks of SMPs, which in the past has caused significant attention from material researchers to be devoted instead to SMAs (Abrahamson et al., 2003). Interestingly, when SMPs have been reinforced with a suitable material, remarkable improvements in their properties have been achieved (Ohki et al., 2004). Particulate composites have been used to improve the thermal and electrical properties, which are critical for biomedical applications. Fiber reinforcement has been used to improve the mechanical properties. Regardless of the reinforcing material, a significant alteration of shape memory behavior has been reported due to reinforcement of SMPs. Fejo˝s et al. (2012) investigated the effect of glass-fiber reinforcement on the recovery behavior of shape memory polymer composites (SMPCs), and a comparison was made between the shape memory characteristics of the glycerol-based aliphatic SMP and its glass fiber SMPC. A substantial increase of recovery stress of SMPC (from 0.4 to 42 MPa) was reported compared to the neat SMP. However, no effect of the reinforcement on the shape recovery ratio has been reported. Similarly, the effect of carbon-fiber reinforcement on shape memory behavior has been investigated Creep and Fatigue in Polymer Matrix Composites. https://doi.org/10.1016/B978-0-08-102601-4.00015-1 © 2019 Elsevier Ltd. All rights reserved.

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by Roh et al. (2014). The inclusion of woven carbon-fiber fabric reinforcement in an SMPC has been shown to significantly increase the shape recovery rate and recovery ratio relative to neat SMPs. Further, Sun et al. (2014) investigated the improvement in mechanical properties achieved by reinforcing the styrene-based SMP with elastic fibers and they have reported improved SMPC toughness at room temperature and increased strength and modulus at high temperatures. Moreover, on another study to investigate the feasibility of using the SMPC in a deployable structure hinge, Lan et al. (2009) have shown increased shape recovery behavior of the styrene-based SMP reinforced by plain-weave carbon fiber. In their study an improved SMPC storage modulus and a 90% shape recovery ratio of the SMPC was achieved. Various modeling techniques have been introduced in recent years to model the thermomechanical behavior of SMPs (Al Azzawi et al., 2017b). One of the modeling approaches assumes an amorphous crosslinked shape memory polymer as a two-phase material with a glassy phase and a rubbery phase (Baghani and Arghavani, 2012; Chen and Lagoudas, 2008; Liu et al., 2004; Wang et al., 2009). However, this approach has been criticized by Gilormini and Diani (2012) due to its debility in prediction of the shape memory response of a material under varying heating rates and heating profiles. On the other hand, many researchers, including Diani et al. (2006), Srivastava et al. (2010) and Tobushi et al. (1996), have proposed another modeling approach using the viscoelastic properties of the polymers. Some of those models require parameter fitting to the experimental data of shape memory behavior to determine either mechanical behavior or the volume fraction, whereas other researchers, such as Srivastava et al. (2010), rely on a series of constant stress or strain tests. Recent research work of Roh et al. (2015), Tan et al. (2014) and Al Azzawi et al. (2017a) has stressed the need for immediate action on a robust model for fiber-reinforced SMPCs to analyze the fiber effect on the shape memory behavior. The finite element method (FEM) has been used for many years to simulate SMP component behavior under applied loads and at different temperatures. The FEM uses SMP’s time-dependent thermomechanical behavior through user-defined subroutines, which involves significantly time-consuming FORTRAN programming. In addition, due to mismatches of implicit key codes, many errors are generated that are extremely hard to debug. The recent work by Tao et al. (2016) has developed a three-dimensional FE procedure to simulate the shape memory effect based on a constitutive model. Furthermore, Baghani et al. (2013) have proposed an interesting finite element model to simulate and solve boundary value problems of tension and compression in SMP bars, and a three-dimensional SMP beam. Yu et al. (2013) have proposed an FE model for a particle-reinforced SMPC to simulate the coupling between heat transfer from spherically shaped particles and heat-induced shape recovery of SMPCs. Further, Taherzadeh et al. (2016) have used a UMAT subroutine to input a 3D constitutive model into the nonlinear FE software ABAQUS/Standard to characterize the effects of aspect ratio and volume fraction of nanofillers on the SMP properties. The ABAQUS commercial finite element analysis (FEA) software has useful builtin facilities within the materials module, such as the Williams-Landel-Ferry (WLF) equation and the finite strain extension of the generalized Maxwell model specified by Simo (1987). This facility has provided extended opportunities for users to input the relaxation time and modulus pairs of the Prony series into the materials module of

Modeling, analysis, and testing of viscoelastic properties

467

the ABAQUS software. The research work detailed in this chapter utilizes the method and the procedure presented in Diani et al. (2012), Shrotriya and Sottos (1998), and Yu et al. (2014) to develop a finite element simulation using finite element code ABAQUS to simulate and predict the shape memory behavior of a glass fiberreinforced SMPC. For many years, researchers have been investigating the use of SMPCs for various space engineering applications, due to its shape-changing ability and its relatively low density. However, the most challenging concerns in selecting SMPC materials for space applications are that the material endures uniquely and is stable in the presence of micrometeoroids, space debris, extreme radiation and temperatures, and vacuum, while maintaining structural permanence. Therefore SMPCs have also been investigated under the effects of UV and γ radiation, impact failure, and long-term behavior. It has been noted that the shape-memory effect of the SMPCs does not degrade considerably due to such circumstances. Accordingly, the use of SMPCs for space engineering applications is being discussed.

15.2

Modeling the viscoelastic behavior of SMP and SMPC

The generalized Maxwell model shown in Fig. 15.1 has been used to designate the viscoelastic parameters of the SMP material. It represents the most general model for viscoelasticity, because it takes into account that the relaxation process does not occur at a single time, but at a distribution of times. Stress-relaxation tests, usually carried out using a dynamic mechanical analyzer (DMA) at constant strain and different temperatures, have been used to find the relaxation master curves and determine the time temperature dependency of the material as described by Hu et al. (2006). The stress relaxation modulus of the material has been obtained for a 10-min relaxation test at 0.25% strain for different temperatures between 50°C and 85°C with a 5°C increment. Fig. 15.2 presents the results of the relaxation behavior for each sample. It has been observed that the relaxation modulus shows a significant dependence on time at the glass transition temperature region. 15.1 Schematic of generalized Maxwell model (Al Azzawi et al., 2017b).

s

k1

k2

kj

ke t1

t2

tj

Relaxation modulus (MPa)

800 700 600 500 400 300

4000

50°C 55°C 60°C 65°C 70°C 75°C 80°C 85°C

3500 3000 2500

468

50°C 55°C 60°C 65°C 70°C 75°C 80°C 85°C

900

Relaxation modulus (MPa)

1000

2000 1500 1000

200 500 100 0

0

(a)

2

4

6

8

10

Time (min)

4000

3000 2500 2000 1500 1000

0

2

4

(b)

6

8

10

12

Time (min)

5000

50°C 55°C 60°C 65°C 70°C 75°C 80°C 85°C

4500 4000 3500 3000 2500 2000 1500 1000

500

500

0

(c)

0 0

2

4

6 Time (min)

8

10

12

(d)

0

2

4

6

8

10

12

Time (min)

15.2 Relaxation modulus vs time under varying isothermal conditions measured by DMA tests: (a) neat SMP, (b) 20% SMPC, (c) 25% SMPC, and (d) 30% SMPC (Al Azzawi et al., 2017b).

Creep and Fatigue in Polymer Matrix Composites

50°C 55°C 60°C 65°C 70°C 75°C 80°C 85°C

3500 Relaxation modulus (MPa)

12

Relaxation modulus (MPa)

0

Modeling, analysis, and testing of viscoelastic properties

469

The dependency on time is less pronounced at temperature levels higher and lower than the transition region. The time–temperature superposition principle (TTS) was utilized to characterize the viscoelastic behavior of the material over an extended time. This technique relies on the experimental observation that polymer viscoelastic behavior at a reference temperature (Tref) can be related to the behavior at another arbitrary temperature (T) by changing the experimental time, as described in following equation (Ferry, 1980):
 E t, Tref ¼ EðaT t, T Þ

(15.1)

where E is the relaxation modulus, t is the time, and T and Tref are the arbitrary temperature and reference temperatures, respectively. The underlying concept of Eq. (15.1) is the relaxation modulus at any temperature (T) and time (aTt) is equivalent to the relaxation modulus at reference temperature (Tref) and time (t). It is widely accepted that for polymer, the glass transition temperature (Tref) and the shift factor (log aT) follows the WLF equation (Williams et al., 1955).
 C1 T  Tref log 10 ðaT Þ ¼  C2 + T  Tref

(15.2)

Here, C1 and C2 are constants, obtained by fitting the WLF equation to the shift factors-temperature curve derived experimentally. Furthermore, the TTS technique has been used to generate the relaxation master curve for all sample types based on the experimental stress relaxation data given in Fig. 15.2. In this process, the relaxation curve at the Tref was taken as the reference, and the curves at lower temperatures were shifted to the left while those at higher temperatures were shifted right to make them fall on the intermediate master curve. The master curve of the 30% SMPC sample is presented in Fig. 15.3. Master curves for other samples are not shown here for clarity. The shifting factors log aT of the preceding curve shifting are temperature dependent. Fig. 15.4 presents those factors against temperature for the 30% SMPC sample at its reference temperature. Then, the WLF Eq. (15.2) was fitted to the shifting factorstemperature curve to determine the fitting parameters (C1 and C2). The same procedure was repeated for all sample types, and the fitting parameters are given in Table 15.1. Relaxation master curves were then fitted with a Prony series Eq. (15.3) to find the series parameters Eð t Þ ¼ E∞ +

N X

Ei e

t τ

i

(15.3)

i¼1

Here, E∞ is the equilibrium modulus of the sample when it has fully relaxed; Ei and τi are Prony parameter and relaxation time, respectively. Prony series fitting of the relaxation master curve of the 30% SMPC sample is shown in Fig. 15.3; a similar procedure has been used to determine the parameters for the other sample types. Tables 15.2–15.5 list Prony series coefficient-relaxation time pairs for all samples.

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Creep and Fatigue in Polymer Matrix Composites 5.0E+03

Relaxation master curve Prony series fit

Relaxation modulus (MPa)

4.5E+03 4.0E+03 3.5E+03 3.0E+03 2.5E+03 2.0E+03 1.5E+03 1.0E+03 5.0E+02 0.0E+00 1.E–04

1.E–02

1.E+00

1.E+02

1.E+04

1.E+06

Time (s)

15.3 Relaxation master and Prony series fitting curves for the 30% SMPC sample obtained for 0.25% strain at reference temperature 68°C (Al Azzawi et al., 2017b). 5 Experimental WLF equation fiting

4 3

Log aT

2 1 0 45

50

55

60

65

70

75

80

85

90

–1 –2 –3 –4

Temperature (°C)

15.4 Shifting factors (log aT) vs temperature of the 30% SMPC sample at reference temperature 68°C (Al Azzawi et al., 2017b). Table 15.1 WLF equation fitting parameters and reference temperature for neat SMP and three different SMPC samples (Al Azzawi et al., 2017b) Sample

Tref (°C)

C1

C2

Neat SMP 20% SMPC 25% SMPC 30% SMPC

60 64 66 68

11.83 11.51 45.83 29.32

56.99 69.76 231.5 170.5

Modeling, analysis, and testing of viscoelastic properties

471

Table 15.2 Relaxation times and Prony series coefficients for neat SMP sample for 0.25% strain master curve (Al Azzawi et al., 2017b) i

τi

Ei (MPa)

1 2 3 4 5

1.0E 03 1.0E 01 1.0E +00 1.0E +03 1.0E +05

7.1E +01 5.3E +02 8.2E +02 3.0E +01 2.0E +01 E∞ ¼ 4.0E +01

Table 15.3 Relaxation times and Prony series coefficients for 20% SMPC sample at 0.25% strain master curve (Al Azzawi et al., 2017b) i

τi

Ei (MPa)

1 2 3 4 5

1.0E 02 1.0E 01 1.0E +00 1.0E +02 1.0E +04

7.0E +02 1.1E +03 1.6E +03 2.5E +02 1.7E +02 E∞ ¼ 2.8E +02

Table 15.4 Relaxation times and Prony series coefficients for 25% SMPC sample at 0.25% strain master curve (Al Azzawi et al., 2017b) i

τi

Ei (MPa)

1 2 3 4 5

1.0E 04 1.0E 02 1.0E +00 1.0E +02 1.0E +04

6.2E +02 0.89E + 02 1.1E +03 2.9E +02 2.6E +02 E∞ ¼ 3.0E +02

Table 15.5 Relaxation times and Prony series coefficients for 30% SMPC sample at 0.25% strain master curve (Al Azzawi et al., 2017b) i

τi

Ei (MPa)

1 2 3 4 5

1.0E 03 1.0E 01 1.0E +00 1.0E +03 1.0E +05

8.0E +02 1.5E +03 2.2E +03 3.1E +02 2.0E +02 E∞ ¼ 5.0E +02

472

15.3

Creep and Fatigue in Polymer Matrix Composites

Finite element simulation procedure for modeling of viscoelastic properties of SMPC

Finite element simulation of the thermomechanical cycle of both SMP and SMPC cantilever beam models was implemented in the ABAQUS standard finite element commercial software. A rectangular beam model of 200  15  3 mm3 was built and meshed with 9000 solid 8-node quadrilateral brick elements (C3D8R). The simulation of the shape programming stage started with elevating the model temperature to 120°C, and then bending by applying a specific displacement at the free end of the beam corresponding to a 45-degree bending angle. Then, the model temperature was reduced below Tg while the bending angle was kept constant by maintaining the constraint deformation unchanged. Later, the constraint was released to simulate the unloading process and to obtain the temporary shape. Subsequently, during the simulation of the shape recovery stage, the temperature of the beam model was increased above Tg. A job in the solver module of ABAQUS was created with five steps: the Initial step and four Visco steps. In the Initial step, a high temperature (120°C) was assigned to the model as a Predefined Field-1 and it was considered as constant throughout the region. Additional model boundary conditions, such as all degrees of freedom being constrained on the fixed end of the cantilever beam, were also created in this step. These boundary conditions were propagated through to all subsequent steps of the simulation job. In contrast, the predefined field-1 (model temperature) was propagated from the Initial step to the first Visco step only, and then it was modified in the subsequent Visco steps whenever cooling or heating was required. Next, in the first Visco step, the beam-bending process was simulated by applying instantaneous vertical displacement at the free end of the beam (as shown in Fig. 15.5) while the boundary conditions and temperature were propagated unchanged for the previous (Initial) step. As the effect of deformation rate is not considered here, the deformation was set to take place promptly (in 1 s) for reducing overall job running time. Furthermore, in order to facilitate the nonlinear effect due to the large deformation, the Nlgeom option in ABAQUS was turned on at this step to allow the use of large displacement formulation and to include geometric nonlinearity effects. Then, in the second Visco step, the Predefined Field-1 was modified to simulate the cooling process. In this modification, cooling amplitude was introduced according to the time-temperature history of the cooling process, which was obtained experimentally. However, both deformation and boundary conditions were propagated constantly from the first Visco step. Thereafter, in the third Visco step, the unloading process of the SMP thermomechanical cycle was simulated by deactivation of the applied deformation while the same temperature and boundary conditions were allowed to propagate from the second Visco step. For the same reason explained in the first Visco step, a short time span was considered here for the unloading process. By the end of the third Visco step, the simulation of the shape programming part of the thermomechanical cycle was finished and the temporary shape was achieved. Finally, the fourth Visco step was introduced to simulate the shape recovery part of the thermomechanical cycle. In this step, the FEA model has no deformation constraint and has the same initial boundary condition. However, it was modified in the Predefined Field-1 to simulate a heating process.

Modeling, analysis, and testing of viscoelastic properties

473

15.5 Configuration of the cantilever beam FEA model illustrates the boundary condition of the fixed end and the applied vertical deformation of the free end (Al Azzawi et al., 2017b).

This was done by introducing a new heating amplitude defined as tabulated timetemperature data of the heating process obtained experimentally. Table 15.6 summarizes the aforementioned steps (Initial step and four Visco steps) and describes the actions done in each of them. Results obtained by the simulation were then used to calculate the bending angle fa fixity and recovery ratios using Rf ¼  100%, where Rf is the bending angle fixity ba ratio, fa is the fixity angle achieved after removing the bending constraint, and ba is the applied bending angle, which was 45 degrees; (ba) and (fa) (shown in the schematic 3δmax PL2 diagram in Fig. 15.6) were determined using the formula ba and fa ¼ θ ¼ ¼ 2l 2EI (Hibbeler, 2015), where E is the modulus of SMPC.

15.4

Finite element simulation results

The finite element simulation was performed using ABAQUS/Standard software. The initial configuration of the simulated beams is shown in Fig. 15.7. A neat SMP beam is shown in Fig. 15.7a and a composite beam with an intermediate reinforcement layer is shown in Fig. 15.7b. The beam model has a flat shape with a length of 200 mm, width of 15 mm, and thickness of 3 mm. The fiber volume fraction of the SMPC beam was varied by changing the intermediate fiber layer thickness while the total thickness of the beam was kept constant. Figs. 15.8–15.10 depict the simulation results of cantilever beam bending during the four steps of the thermomechanical cycle. Fig. 15.8 presents the shape

474

Creep and Fatigue in Polymer Matrix Composites

Table 15.6 Finite element model steps and the action incorporated in each step (Al Azzawi et al., 2017b)

Boundary condition Predefined field-1

Deformation constraint

Initial step

First Visco step

Second Visco step

Third Visco step

Fourth Visco step

Created

Propagated

Propagated

Propagated

Propagated

Created (Assign a temperature above Tg to the model)

Propagated (Keep same model temperature while applying deformation)

Modified (Reduce model temperature below Tg)

Modified (Model temperature increased above Tg according to defined amplitude)

—

Created (Deformation applied on the model)

Propagated (Model deformation kept same as in previous step)

Propagated (Model temperature remains as in previous step while deformation constraint inactivated) Inactive (Model deformation removed)

P q y

Inactive (Model has no applied deformation)

x dmax

l

15.6 Schematic diagram showing the configuration adopted for the calculation the bending angle of a cantilever beam bended by applying deformation at the free end (Al Azzawi et al., 2017b).

(a)

(b)

15.7 Beam configuration, (a) neat SMP and (b) SMPC beam with intermediate fiber layer (Al Azzawi et al., 2017b).

Modeling, analysis, and testing of viscoelastic properties

475

U, U2 +1.300e+02 +1.192e+02 +1.083e+02 +9.750e+01 +8.666e+01 +7.583e+01 +6.500e+01 +5.416e+01 +4.333e+01 +3.249e+01 +2.166e+01 +1.083e+01 –7.036e–03

1

Y

Z

X

U, U2 +1.300e+02 +1.192e+02 +1.083e+02 +9.750e+01 +8.666e+01 +7.583e+01 +6.500e+01 +5.416e+01 +4.333e+01 +3.249e+01 +2.166e+01 +1.083e+01 –7.899e–03

2

Y

Z

X

3

U, U2 +1.220e+02 +1.119e+02 +1.017e+02 +9.153e+01 +8.136e+01 +7.119e+01 +6.102e+01 +5.084e+01 +4.067e+01 +3.050e+01 +2.033e+01 +1.016e+01 –9.114e–03

Y

Z

X

15.8 Simulation results for the shape programming part of the thermomechanical cycle: (1) deformation above Tg, (2) cooling below Tg under constraint, and (3) spring back due to constraint removal (Al Azzawi et al., 2017b).

U, U2

+1.177e+02 +1.079e+02 +9.806e+01 +8.826e+01 +7.845e+01 +6.864e+01 +5.883e+01 +4.903e+01 +3.922e+01 +2.922e+01 +1.961e+01 +9.798e+00 –9.727e–03

Y

Z

X

U, U2 +1.322e–03 –5.499e–03 –1.232e–02 –1.914e–02 –2.596e–02 –3.278e–02 –3.960e–02 –4.642e–02 –5.324e–02 –6.006e–02 –6.689e–02 –7.371e–02 –8.053e–02

Y

Z

X

15.9 Simulation results of the shape recovery step of the thermomechanical (Al Azzawi et al., 2017b).

50 45

Bending angle (degree)

40

Step 2

Step

35 30 Step 4

25

Step 1

20 15 10 5 0 0.85

0.95

1.05

1.15

1.25

T/Tg

15.10 Finite element simulation results of the thermomechanical cycle of SMP cantilever beam bending angle (Al Azzawi et al., 2017b).

Modeling, analysis, and testing of viscoelastic properties

477

programming part of the cycle where the model was initially deformed at a temperature above Tg, cooled below Tg with the existence of the constraints, and then released the constraints and let the specimen elastically relax. Further, Fig. 15.9 shows the recovery part of the thermomechanical cycle, that is, the bent beam commenced the release of stored strain and eventually recovered the original flat shape. A schematic diagram of the numerical simulation results of the preceding steps is presented in Fig. 15.10. The bending angle is plotted against the model temperature. In step 1, the beam temperature was 1.3Tg and a 45-degree bending angle was applied to it in a short time period of 1 s. Then, in step 2, the model temperature was reduced from 1.3Tg to 0.87Tg while the bending angle was kept constant at 45 degrees. In step 3 the model was kept isothermally at 0.87Tg and the bending constraint was released. It is obvious that the FEA simulation has apprehended the spring-back action occurring in the material due to the higher stiffness at the lower temperature. This is an important parameter to evaluate the capacity of the material to retain the temporary shape. Subsequently, in step 4, the model temperature was increased to 1.3Tg again and the bending angle recovery was started closer to the temperature 0.95Tg, and the beam was returned to its original flat shape at the end of the step.

15.5

Validation of the FEA technique

15.5.1 Bending angle fixity Fig. 15.11 shows the proposed FEA prediction of the thermomechanical cycle for four sample types (SMP and three fiber-reinforced SMPCs). As anticipated, the behavior in steps 1 and 2 are identical for all samples. However, the behavior in steps 3 and 4 was dissimilar because of the stiffness added by the fibers. Step 3 is the last step of the shape programming part of the thermomechanical cycle, where deformation is normally stored inside the model to achieve the temporary shape. Fig. 15.11 shows that the developed FEA technique was capable of predicting the effect of the reinforcement on the SMPC’s ability to store the applied bending deformation. Simulation results have shown a bending angle fixity ratio of 98% for the neat SMP, and 85% for the 30% SMPC model, and the fixity ratios lies between those two ratios for the other two SMPCs. Table 15.7 presents a comparison between the experimental data and FE results of the bending angle and fixity ratio for all simulated material types.

15.5.2 Experimentation Styrene-based SMP supplied by Harbin Institute of Technology China was used to prepare four rectangular plates (250 mm  200 mm): one neat SMP plate and three SMPCs plates with different fiber volume fraction produced using woven glass fiber AR177100 W/C 450 g/m2 0/90 (supplied by COLAN Australia). A rectangular glass mold coated with a thin film of Teflon was used to cast the plates. Then, the plates

478

Creep and Fatigue in Polymer Matrix Composites 50

SMP 20% SMPC 25% SMPC 30% SMPC

Bending angle (degree)

40

30

20

10

0 0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

T/Tg

15.11 Finite element simulation of the thermomechanical cycles of SMP and three SMPC beams, bending angle fixity ratio diminishes with the increase of the fiber fraction (Al Azzawi et al., 2017b).

Table 15.7 Fixity angle and fixity ratio of bending angle of neat SMP and three different SMPC cantilever beams (Al Azzawi et al., 2017b) Sample

Neat SMP 20% SMPC 25% SMPC 30% SMPC

Experimental

Simulation

Fixity angle (degree)

Fixity ratio (%)

Fixity angle (degree)

Fixity ratio (%)

44

97.7

44.2

98

42

93

42.7

95

41

91

40.5

90

38

84

38.2

85

were cured in a temperature-controlled oven, at 75°C for 24 h. The cured plates were cut into two groups of specimens: small specimens of (35  14  1.5 mm3), according to the single cantilever beam standard of the TA Q800 dynamic mechanical analysis equipment, and bigger specimens (200  15  3 mm3), which were used to investigate the bending shape memory behavior. Fig. 15.12 shows the DMA machine used to conduct the dynamic mechanical analysis to identify the specimens’ Tg, characterize their storage modulus, and determine the viscoelastic parameters

Modeling, analysis, and testing of viscoelastic properties

479

15.12 Dynamic mechanical analyser (DMA Q800) used to characterize the specimens storage modulus, transition temperature and viscoelastic parameters (Al Azzawi et al., 2017b).

using stress relaxation tests. A multifrequency mode on DMA equipment was used for the determination of Tg and storage modulus. In the test, the frequency was set to 1.0 Hz, and temperature ramped at 10°C/min over temperature span (30–115°C) using the single cantilever bending clamp arrangement. The same specimen types were then used to determine the material viscoelastic parameters by conducting a set of relaxation tests with applied 0.25% strain on the samples at different temperatures between 50°C and 80°C with 5°C increment over a 10 min stress relaxation time. Then the TTS principle was applied for the relaxation modulus curves and the relaxation data were exported to the Rheology module of the Advantage (v5.5.22) software package to generate the material’s master curves by using a relaxation curve shifting technique. Fig. 15.13 shows a special tool designed to test the shape memory behavior and measure the beam’s bending angle fixity and recovery. The Instron temperaturecontrolled environment chamber, shown in Fig. 15.14, was used to provide the environment at the required temperatures during the test, and a thermocouple was placed inside the thermal chamber as close as possible to the specimen to ensure accurate monitoring of the specimen temperature. The bending angle fixity and recovery test was done in four steps. First the specimen was heated inside the thermal chamber above the Tg and kept for 5 min to ensure a uniform temperature distribution inside the specimen. Then the specimen was bent to 45 degrees around a spindle of 40 mm diameter, shown in Fig. 15.13. Then the specimen and the bending tool were removed from the thermal chamber to cool down. The 45-degree bending angle was kept constant by a firm constraint imposed on the sample, and the sample was allowed to naturally cool down to room temperature before

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Creep and Fatigue in Polymer Matrix Composites

15.13 Shape fixity and recovery ratio testing tool (Al Azzawi et al., 2017b). 15.14 Instron temperature controlled environment chamber (Al Azzawi et al., 2017b).

releasing the constraints. Subsequently, in the shape recovery part of the test, the specimen reentered into the thermal chamber, where it was heated above Tg and a video camera was used to record the shape recovery process. Table 15.7 presents a comparison between the experimental data and FEA results of the bending angle and fixity ratio of different samples, showing excellent agreement between the two results.

Modeling, analysis, and testing of viscoelastic properties

481

50

SMP

Recovery angle (degree)

45

20% SMPC

40

25% SMPC

35

30% SMPC

30 25 20 15 10 5 0 0

50

100 Time (s)

150

200

15.15 Experimental result of the bending angle recovery vs time of cantilever beam samples made of neat SMP and three different fiber fraction SMPCs (Al Azzawi et al., 2017b).

15.5.3 Experimental results for SMPC bending angle recovery Fig. 15.15 depicts the experimental results of the bending angle recovery of neat SMP and three SMPC cantilever beam samples vs time. It can be seen that the increase of fiber fractions in the SMPC of 20%, 25%, and 30% reduced the recovery time of the neat SMP by 70%, 75%, and 80%, respectively. This is attributed to the increased stiffness of the SMPCs, which caused faster shape change during shape recovery (Roh et al., 2014). Fig. 15.15 also shows that the angle recovery was initiated gradually as the temperature approached Tg, which is caused by the partial glass transition; then it became more significant in the region from Tg to 1.15Tg, which is due to the fast glass transition and strain energy dissipation (Lan et al., 2009). This phenomenon was also justified by Liu et al. (2006) as the shape recovery being dominated by thermal expansion at temperatures below Tg and due to both thermal strain and release of the strain stored inside the polymer molecules at temperatures above Tg.

15.5.4 Bending angle recovery Fig. 15.16 presents a comparison between FEA results and the experimental results of the bending angle recovery of neat SMP and three SMPCs. A good agreement between the two sets of results was found. This indicates that the proposed simulation technique was accurate and robust enough for predicting the bending angle recovery behavior of both SMP and SMPC specimen types. However, a slight deviation between the two sets of results was noticed, which can be attributed to unavoidable error in monitoring the time rate of the heating process of the shape recovery test. Furthermore, Fig. 15.16 shows that all models recovered their original shape during step 4, which indicates the effectiveness of the technique in simulating the shape recovery process.

482

Creep and Fatigue in Polymer Matrix Composites 50

Recovery angle (degree)

SMP experimental 45

SMP simulation

40

20% SMPC experimental 20% SMPC simulation

35

25% SMPC experimental 30

25% SMPC simulation

25

30% SMPC experimental 30% SMPC simulation

20 15 10 5 0 0

20

40

60

80

100

120

140

160

180

200

Time (s)

15.16 Comparison between the experimental and simulation results for the bending angle recovery of cantilever beam made of neat SMP and different fiber fraction SMPCs (Al Azzawi et al., 2017b).

15.6

Competence of SMPCs for space engineering applications

15.6.1 Space environment Nearly three decades of continuous research on shape memory polymers has inspired engineers and scientists to seek numerous opportunities in many branches of engineering and has given hope to actualizing their dreams and imaginings for space exploration (Liu et al., 2014). In recent years, growing breakthroughs in shape memory materials have revealed the immense potential of lightweight materials for space exploration. Having a relatively low density comparable to other matrix materials, SMPs have shown their potential to enter the race for future materials in the space age. However, these smart materials have counteractive limitations on their survival in the harsh environments prevailing in space. The most challenging concerns for a lightweight material selected for space applications include the need to endure uniquely in a harsh environment and maintain stability in the presence of micrometeoroids traveling as fast as 60 km s1, space debris traveling at an average velocity of 10 km s1, a vacuum (1.33322E4 to 1.3332E7 Pa), ultraviolet (UV) radiation, particulate radiation, plasma, atomic oxygen, temperature extremes, and thermal cycling (from 120°C to +120°C), all while maintaining structural permanence (Finckenor and Groh, 2017). Importantly, UV and particulate radiation can damage SMPs by either cross-linking (hardening) or chain scission (weakening), which may affect their shape memory behavior (Al Azzawi et al., 2018). Moving micrometeoroids and debris can cause impact damage (Finckenor and Groh, 2017). Also, the low-Earth orbit (LEO) environment (200–1000 km above Earth’s surface) is a particularly harsh environment for most nonmetallic materials, as single-oxygen atoms (atomic oxygen

Modeling, analysis, and testing of viscoelastic properties

483

[AO]) are present along with all other environmental components (Yang and de Groh, 2010). AO reacts strongly with any material containing carbon, nitrogen, sulfur and hydrogen bonds, meaning that many polymers react and erode. Determining how long-term exposure to space conditions impacts various materials and, thus, which materials are best suited for spacecraft construction can most effectively be accomplished through actual testing in space and ground laboratory facilities. Prominently, the Materials and Processes Technical Information System (MAPTIS) of the National Aeronautics and Space Administration (NASA) maintains 50 years of space material data that is open for all registered users. The NASA’s database contains the test results from ASTM E1559, Standard Test Method for Contamination Outgassing Characteristics of Spacecraft Materials, and the older ASTM E595, Standard Test Method for Total Mass Loss and Collected Volatile Condensable Materials from Outgassing in a Vacuum Environment. In addition to the metal’s data, nonmetals data of MAPTIS consist of test results, including flammability, fluid compatibility, odor, outgassing (thermal vacuum stability), toxicity (offgassing), and vacuum condensable material compatibility with optics (VCMO) (Finckenor and Groh, 2017). At present much research is devoted to SMPC applications in aerospace and the space sector. However, until now the material characteristics of SMPCs as a spacecraft material are not available in the literature.

15.6.2 Durability of SPMCs Researchers have investigated the effects of UV and γ radiation, impact mitigation techniques, and cycle life for different SMPCs (Liu et al., 2014). Leng et al. have investigated epoxy-based shape memory polymers, exposed to simulated γ-radiation environments up to 140 days for an accelerated irradiation (Leng et al., 2014). The influence of γ radiation on thermal and mechanical properties was evaluated by differential scanning calorimetry, dynamic mechanical analysis, and tensile tests. Four different samples, named SMP1, SMP2, SMP3, and SMP4, were used, which contained 0, 5, 10, and 15 wt.% of an active linear epoxy monomer composed of a long linear chain of C-O bonds respectively. Fig. 15.17 shows the changes to the storage modulus and tan δ before and after exposure to the γ-radiation of 1  105 Gy and 1  106 Gy (Leng et al., 2014). The loss modulus is commonly used to express the polymer deformation energy lost or converted to heat, which corresponds to the damping. The storage modulus describes the elastic recovery ability of the polymer material after deformation. It represents the energy stored in the elastic deformation of a viscoelastic material. The storage modulus is essentially the Young’s modulus. The ratio of the loss modulus to the storage modulus is defined as the loss factor, tan δ. The change of tan δ reflects the viscoelastic characteristics of the molecule (Meyers and Chawla, 2008). The results revealed that the glass transition temperatures (Tg) have decreased no less than 10%. A sharp decrease in the storage modulus around the glass transition could be observed for both before and after γ radiation, which is essential for a good shape memory material. Also, a large elastic modulus ratio is beneficial for good shape memory capability, since it gives larger shape fixity upon cooling and realizes a larger strain with a small stress at high temperature. The tensile

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Creep and Fatigue in Polymer Matrix Composites 2

104

2

104

SMP2

SMP1

150.3°C

0 Gy

1

1 ´ 105 Gy 1 ´ 106 Gy

101

100 –50

0

0

50

100

150

200

103

102

1

1 ´ 105 Gy 1 ´ 10 Gy 6

101

100 –50

250

147.8°C 150.7°C

0 Gy

Tan d

102

Storage modulus (MPa)

160.0°C 162.8°C

103

Tan d

Storage modulus (MPa)

142.1°C

0

0

50

Temperature (°C)

100

150

200

250

Temperature (°C) 2

104

2

104

SMP3

SMP4

100.0°C

137.9°C

1

1 ´ 105 Gy 1 ´ 106 Gy

101

100 –50

0

0

50

100

150

Temperature (°C)

200

250

103

105.3°C 110.3°C

0 Gy 102

1

1 ´ 105 Gy

Tan d

102

Storage modulus (MPa)

133.5°C

0 Gy

Tan d

Storage modulus (MPa)

131.5°C

103

1 ´ 10 Gy 6

101

100 –50

0

0

50

100

150

200

250

Temperature (°C)

15.17 DMA curves of the epoxy-based SMPs before and after γ-radiation (Leng et al., 2014).

strength and elastic modulus could, respectively, maintain 26 and 1.36 GPa after being irradiated by 1  106 Gy radiation, showing great potential for aerospace structural materials. Furthermore, according to Leng et al., the shape recovery speed improved after γ radiation of 1  105 Gy (Leng et al., 2014). Xu et al. investigated the durability of an SMP-based close-celled syntactic foam, under environmental attacks with accelerated aging of up to 90-day exposure to UV radiation (Xu et al., 2011). Combined UV radiation and water immersion (rain water and saturated salt water) were also investigated. Accordingly, UV exposure resulted in discoloration on the exposed surface of the foam in air, and yellowing increased with duration of exposure. UV exposure resulted in cracks on the foam surface along the compression direction. Also, the UV radiation caused a decrease in mechanical strength and ductility, and an increase in modulus. Cracks in the foam had less effect on tensile strength than on compressive strength, but had a greater influence on ductility. Combined effects of UV and water led to a larger decrease in strength and ductility than that of UV alone, and the decrease for foams immersed in salt water was less than that in rainwater. Water immersion also caused a reduction in Tg of the foam (Xu et al., 2011). Xie et al., performed vacuum outgassing and UV radiation exposure tests on a cyanate-based SMP with a glass transition temperature of 206°C (Xie et al., 2017). Vacuum UV radiation deepened the color of the surface, but had little effect on the thermal stability of the SMP sample. The irradiation induced some instability of the molecular structure within the material, and this effect was gradually strengthened with the increase of exposure time.

Modeling, analysis, and testing of viscoelastic properties

485

15.18 Effects of UV radiation on cyanate-based SMP. (a) Mechanical properties before and after UV radiation. (b) Storage modulus and loss factor (tan δ) before and after UV radiation. (c) Shape memory characteristic before exposed to UV radiation. (d) Effect of UV radiation on the shape memory effect after exposed to 3000ESH (Xie et al., 2017).

However, UV radiation did not detectably change the mechanical properties of the cyanate-based SMP. Fig. 15.18a compares the mechanical properties between the cyanate-based SMP samples before and after UV radiation (Xie et al., 2017). The tensile strength remained at 66 within 2 MPa standard deviation after UV radiation. As the irradiation dose increased, strain at break of the four samples was 7  1%, 6  1%, 7  1%, and 6  1%, respectively. These values lie between 5% and 8%. The tensile modulus of the material has not decreased even after 3000ESH UV radiation. The elastic modulus remained essentially constant at 1940  80 MPa. According to the DMA curves shown in Fig. 15.18b, the peak of the tan δ curves has not significantly changed with 1000ESH and 2000ESH irradiation compared to the sample before UV irradiation (Xie et al., 2017). However, the peak of the tan δ curves has moved to a lower temperature, from 206°C to 197°C, when the irradiation dose reached 3000ESH. SMCR-3000ESH also exhibited a lower storage modulus at high temperature (>150°C) than that of the other samples. Fig. 15.18c and d shows the shape memory behavior for before (0ESH) and after exposure to UV radiation of 3000ESH. Calculations indicated that the shape fixity rates of three shape memory cycles were 97.8%, 97.9%, and 97.1% for sample

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Creep and Fatigue in Polymer Matrix Composites

SMCR-0ESH with an average shape fixity rate of 97.6%. The shape recovery rates were 96.9%, 97.0%, and 99.0% with an average shape recovery rate of 97.9%. The shape fixity rates of three shape memory cycles were 97.8%, 97.9%, and 98.0% for sample SMCR-3000ESH with an average shape fixity rate of 97.9%. The shape recovery rates were 98.2%, 98.8%, and 98.9% with an average shape recovery rate of 98.6%. The average shape fixity rate and average shape recovery rate before and after UV radiation were all above 97.6%, and the repeatability was satisfactory (Xie et al., 2017). Wong et al. investigated the decoloration, UVR absorbability, surface hardness, dynamic mechanical properties, and shape memory effects under UVA degradation of styrene-based and epoxy-based SMPs (SSMPs and EPSMPs) filled with different contents of nano-ZnO particles (Wong et al., 2015). The samples were kept under accelerated UVA aging at intensity 11 times higher and a temperature 3 times higher than those of the ambient environment, which is equivalent to a situation where the samples were exposed to 22 days under the sunlight in terms of UVA intensity. Results of the surface hardness tests are shown in Fig. 15.19 (Wong et al., 2015). After accelerated UVA aging, the surfaces of all SSMP samples were hardened while

16

Hardness index

15 14 13

Before UVA degradation

12

After UVA degradation

11 10 Neat SSMP

2 wt. % ZnO/ SSMP

4 wt. % ZnO/ SSMP

5 wt. % ZnO/ SSMP

7 wt. % ZnO/ SSMP

(a) 16

Hardness index

15 14 13

Before UVA degradation

12

After UVA degradation

11 10 Neat

(b)

2 wt. % ZnO/ EPSMP

4 wt. % ZnO/ EPSMP

5 wt. % ZnO/ EPSMP

7 wt. % ZnO/ EPSMP

15.19 Hardness of SMPCs before and after UVA degradation: (a) Neat SSMP and all ZnO/ SSMPs. (b) Neat EPSMP and all ZnO/EPSMPs (Wong et al., 2015).

Modeling, analysis, and testing of viscoelastic properties

487

those of all EPSMP samples were softened. This implied that all SSMP samples behaved more like amorphous thermoplastics which formed crosslinks and became a thermoset plastic, while all EPSMP samples behaved like thermoset plastics in which their crosslinks were broken and which became more flexible and tough but less strong and stiff after UVA degradation (Wong et al., 2015). Tandon et al. conducted a baseline assessment of the durability of styrene-based (VF62) and epoxy-based (VFE2–100) thermosetting shape memory polymer resin materials being considered for morphing applications when exposed to service environment (Tandon et al., 2009). The approach for the experimental evaluation was a measurement of the shape memory properties and elastomeric response before and after separate environmental exposure to (i) water at 49°C for 4 days, (ii) in lube oil at room temperature and at 49°C for 24 h, and (iii) after exposure to xenon arc (63°C, 18 min water and light/102 min light only) and spectral intensity of 0.3–0.4 W/m2 for 125 cycles (250 h exposure time). Two shape memory cycles were performed on each specimen in order to get a preliminary estimate of repeated cycling on the shape-memory behavior. Table 15.8 shows the influence of service environment on the maximum stress, recovery stress, and unconstrained linear shape recovery of VF62 and VFE2–100 SMP resins. For VF62 resin under a service environment, it is noted that there is a decrease in the stress recovery ratio for UV-exposed samples, while water immersion results in a small increase. Moreover, the stress recovery ratio seems to improve slightly during the second SM cycle, except for UV-conditioned specimens. Furthermore, it is seen that almost complete unconstrained recovery (97%–100%) was possible under all conditions studied for VF62. For VFE2–100 there is some decrease in the recovery stress ratio for samples immersed in water at room temperature, while the recovery ratio remains relatively unchanged with heated water and UV exposure. Lastly, oil conditioning at 25°C and 49°C results in a small knockdown in the stress recovery ratio compared with the unconditioned samples. Moreover, with the exception of heated oil, the stress recovery ratio seems to reduce slightly during the second SM cycle. Also, for VFE2–100, it is seen that almost complete recovery (97%–100%) was possible under all conditions studied. Kong and Xiao have investigated the behavior of a shape memory polyimide over 1000 shape memory cycles (Kong and Xiao, 2016). The SMP exhibited shape fixity of 98%–100% and recovery of 100% during the >1000 bending deformation cycles at 250°C. Fig. 15.20a and b illustrates the storage modulus and UV-Vis transmittance before and after different shape memory cycles, respectively. The storage modulus is almost the same for all different shape memory cycles, which may cause 100% recovery. However, the percentage of UV–Vis transmittance is reduced with increasing number of cycles (Kong and Xiao, 2016). Nji et al. have proposed a 3D woven fabric-reinforced SMPC for impact mitigation (Nji and Li, 2010). The shape memory functionality of SMPs has been utilized to selfrepair impact damage in 3D woven fabric-reinforced SMP based syntactic foam composites under repeated impact loadings. It is found that the impact energy has a significant effect on the healing efficiency. Control specimens impacted with 32 J of impact energy were perforated at the 9th impact while healed specimens lasted until the 15th impact; control specimens impacted with 42 J of impact energy were

Table 15.8 The influence of service environment on the maximum stress, recovery stress and unconstrained linear shape recovery of VF62 and VFE2-100 SMP resins (Tandon et al., 2009) Material

VF62 resin

VFE2100 resin

Specimen condition

Unconditioned UV conditioning (average of 25 cycles and 75 cycles) Water conditioning at 49°C for 4 days Unconditioned Water conditioning at 25°C for 4 days Water conditioning at 49°C for 4 days UV conditioning 125 cycles Oil conditioning at 25°C for 24 h Oil conditioning at 49°C for 24 h

SM cycle 01

SM cycle 02

σ max (MPa)

σ recovery (MPa)

σ recovery/ σ max

Unconstrained linear shape recovery

σ max (MPa)

σ recovery (MPa)

σ recovery/ σ max

Unconstrained linear shape recovery

0.259 0.221

0.176 0.141

0.677 0.637

97.211 100.82

0.249 0.275

0.177 0.148

0.711 0.538

99.970 99.38

0.254

0.184

0.724

100.78

0.170

0.133

0.781

99.34

0.726 0.782

0.559 0.560

0.770 0.717

98.26 97.84

0.785 0.800

0.578 0.552

0.736 0.690

98.54 98.66

0.735

0.564

0.767

100.25

0.765

0.553

0.722

97.98

0.645

0.503

0.780

97.33

0.729

0.558

0.765

99.21

0.813

0.591

0.727

100

0.834

0.571

0.685

98.82

0.848

0.575

0.678

98.09

0.800

0.577

0.721

96.25

Modeling, analysis, and testing of viscoelastic properties

489 90

100

Initial sample After 86 cycles After 320 cycles After 780 cycles After 937 cycles After 1084 cycles

Transmittance (%)

Storage modulus (MPa)

1000

60

Initial sample After 86 cycles After 320 cycles After 780 cycles After 937 cycles After 1084 cycles

30

10

0

(a)

100

150 200 Temperature (°C)

250

(b)

300

500 600 400 Wavelength (nm)

700

800

15.20 Proprieties of SMP after different shape memory cycles. (a) Storage modulus. (b) UV-Vis transmittance (Kong and Xiao, 2016).

perforated at the 5th impact while healed specimens lasted until the 7th impact. This is because some unrecoverable damage such as microballoon crushing and fiber fracture increase as impact energy increases and the nonhealable damage accumulates under repeated impacts. The appearance of nonhealable damage is represented by a considerable reduction in either the peak impact load or the initiation energy, or an increase in propagation energy. Perforation is signified by a sharp drop in peak load, total energy, initiation energy, and propagation energy (Nji and Li, 2010).

15.6.3 SMPC space engineering applications Shape memory polymers have the unique ability to be programmed into a compressed shape and recover back to the expanded original shape. Hence SMPCs offer potential benefits in deployable space applications. In addition to the strength and durability requisite for space materials, the weight and occupied volume are significant for space travel. Lightweight and compressed structures help in transporting space materials from Earth to outer space. The storage modulus of the SMP materials indicates the ability to store deformation energy in an elastic manner. Higher storage modulus is helpful in achieving a better shape memory performance. However, it will inversely affect the damping performance of the material. Both constrained strain stress recovery and stress-free strain recovery characteristics are important factors to be considered for space engineering applications. When space travelers stay for long durations in outer space, a shelter or habitat must be used to protect them from extreme space environment conditions. Such habitats can be constructed on Earth and transported into space. Thus an important factor in the development of such deployable structures is to reduce the room occupied in the spacecraft and make the best use of the deployed volume in space. SMPCs are one of the lightweight materials that can satisfy this purpose, as they can be packaged and stowed in a smaller volume and deployed using a particular stimuli in space. Fig. 15.21 shows a model deployable structure for a space habitat, which has been recovered at 130°C in a 80 kPa vacuum. The model was made out of carbon

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Creep and Fatigue in Polymer Matrix Composites

15.21 Deployment steps of a model deployable structure made out of carbon fiber reinforced shape memory polymer composites.

15.22 A space habitat developed by ILC Dover and Folded Structures Company (Hinkle et al., 2011).

fiber-reinforced SMPC. The deployable structure was compressed into an almost three times smaller volume during the shape programming phase. The importance of using SMPC materials for space habitat is that the deployable structure can be fabricated as a single unit. There are no moving joints and assembly components. Space travelers would not need to engage with any assembling or construction work in space, saving their time and energy. Furthermore, travelers would not need to take any tools or equipment for construction work. Due to the temperature and radiation in space, the SMPC material will be activated by its own once it is exposed to the surrounding. Fig. 15.22 shows a space habitat developed by ILC Dover and Folded Structures Company, under a NASA contract (Hinkle et al., 2011). Antennas are used to communicate between Earth and space. Satellites, Mars Rovers, and other space devices are equipped with antennas that transmit important information. The reflector aperture and precision are two main parameters in measuring the working properties of deployable antennas. Therefore, a precise recovery into the designed shape is crucial for such SMP applications. Fig. 15.23 shows the deployment steps of an antenna developed by Composite Technology Development, Inc. (Keller et al., 2006).

Modeling, analysis, and testing of viscoelastic properties

491

15.23 Deployment steps of an SMPC antenna developed by Composite Technology Development, Inc. company (Keller et al., 2006).

15.24 Model of a deployable solar panel array for a satellite and the shape recovery behaviors due to near infrared radiation (Herath et al., 2018).

Herath et al., have demonstrated a model of a deployable solar panel array to be activated by near infrared radiation (Herath et al., 2018). Fig. 15.24a shows the recovery steps of the proposed solar panel array. Light is one of the most suitable activation methods for SMPCs to be used in the space environment, as the light can travel a long distance without any media. Furthermore, localized activation can be achieved since the light can be focused on a particular area. Fig. 15.24b–f present the recovery steps from the initial shape to the final shape. Advances in structural performance and durability of SMPCs will improve future space applications (Herath et al., 2018). However, at present SMPC space engineering applications are limited to experimental demonstrations at ground level. Unfortunately, the transformation of breakthrough research on SMPCs into space applications suffers from many problems, including lack of analytical tools for initial design studies. Over the past few years, many finite element models have been presented for the analysis of SMPCs, but most of them have required complicated user-defined material

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Creep and Fatigue in Polymer Matrix Composites

subroutines to be integrated with standard finite element software packages. The subroutines are problem specific and extremely difficult to use for generic cases. Moreover, they require skilled programmers and need extra running and debugging time. Therefore, the viscoelastic properties-based model developed in Section 15.3 will be useful in implementing realistic space applications.

15.7

Conclusion

In this chapter, a novel finite element simulation technique was proposed to simulate thermomechanical behavior of shape memory polymers and the effects of glass fiber reinforcement on their composites’ shape fixity and shape recovery behavior. The proposed simulation technique was based solely on the viscoelastic characteristics of the material, which were acquired by application of the time-temperature superposition (TTS) technique on the material’s stress relaxation behavior. The commercially available finite element software ABAQUS was used to build cantilever beam models. The material’s viscoelastic parameters, which were obtained experimentally, have been incorporated into the property module using the proposed procedure. The bending angle of the cantilever beam configuration was simulated for the verification of the proposed technique. FEA predictions were compared with the experimental results of the bending tests for both neat SMP and fiber-reinforced SMPC samples. A good correlation has been found between the two sets of results for both sample types. As a consequence, it can be concluded that the proposed procedure, that is, inclusion of the viscoelastic properties and the time temperature dependence of the SMP materials into the ABAQUS software, has proven its accuracy and the robustness of simulating the shape memory behavior of SMPs and their fiber composites in a significantly reduced time frame. Further, experimental results of the SMPC samples have revealed that the fiber reinforcement has significantly improved the bending angle recovery. Increasing the fiber volume fraction by 20%, 25%, and 30% has reduced the recovery time by 70%, 75%, and 80%, respectively, comparing to the neat SMP. Furthermore, it has been found that the fiber reinforcement has marginally reduced the bending angle fixity ratio, from 98% for neat SMP to 85% for 30% SMPC. In contrast, reinforcement did not affect the shape recoverability of the SMPC samples, as all samples fully recovered their original (flat) shape at the end. Because of its enhanced mechanical properties and light activation ability, SMPC will be a potential candidate for space engineering applications. Though the structural performance of SMPCs has shown relatively notable improvements over its parent SMP material, further improvements are necessary in order to function adequately in the harsh space environment. Interestingly, SMPCs can be used in deployable space applications which can fabricate on earth, compress, and pack in a spacecraft, transport to an outer space location, and ultimately deploy into the original shape.

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References Abrahamson, E.R., et al., 2003. Shape memory mechanics of an elastic memory composite resin. J. Intell. Mater. Syst. Struct. 14 (10), 623–632. Al Azzawi, W., et al., 2017a. Quantitative and qualitative analyses of mechanical behavior and dimensional stability of styrene-based shape memory composites. J. Intell. Mater. Syst. Struct. 28 (20), 3115–3126. Al Azzawi, W., et al., 2017b. Implementation of a finite element analysis procedure for structural analysis of shape memory behaviour of fibre reinforced shape memory polymer composites. Smart Mater. Struct.. 26(12), 125002. Al Azzawi, W., et al., 2018. Investigation of ultraviolet radiation effects on thermomechanical properties and shape memory behaviour of styrene-based shape memory polymers and its composite. Compos. Sci. Technol. 165, 266–273. Baghani, R.N., Arghavani, J., 2012. A constitutive model for shape memory polymers with application to torsion of prismatic bars. J. Intell. Mater. Syst. Struct. 23 (2), 107–116. Baghani, M., Naghdabadi, R., Arghavani, J., 2013. A large deformation framework for shape memory polymers: constitutive modeling and finite element implementation. J. Intell. Mater. Syst. Struct. 24 (1), 21–32. Chen, Y.-C., Lagoudas, D.C., 2008. A constitutive theory for shape memory polymers. Part I: large deformations. J. Mech. Phys. Solids 56 (5), 1752–1765. Diani, J., Liu, Y., Gall, K., 2006. Finite strain 3D thermoviscoelastic constitutive model for shape memory polymers. Polym. Eng. Sci. 46 (4), 486–492. Diani, J., et al., 2012. Predicting thermal shape memory of crosslinked polymer networks from linear viscoelasticity. Int. J. Solids Struct. 49 (5), 793–799. Fejo˝s, M., Karger-Kocsis, J., 2013. Shape memory performance of asymmetrically reinforced epoxy/carbon fibre fabric composites in flexure. Express Polym. Lett. 7, 528–534. Fejo˝s, M., Romhany, G., Karger-Kocsis, J., 2012. Shape memory characteristics of woven glass fibre fabric reinforced epoxy composite in flexure. J. Reinf. Plast. Compos. 31 (22), 1532–1537. Ferry, J.D., 1980. Viscoelastic Properties of Polymers. John Wiley & Sons. Finckenor, M.M., Groh, K.K.d., 2017. Steele, D. (Ed.), A Researcher’s Guide to: Space Environmental Effects. In: ISS Researcher’s Guide Series, NASA ISS Program Science Office. Gilormini, P., Diani, J., 2012. On modeling shape memory polymers as thermoelastic two-phase composite materials. Comptes Rendus Mecanique 340 (4–5), 338–348. Herath, H.M.C.M., et al., 2018. Structural performance and photothermal recovery of carbon fibre reinforced shape memory polymer. Compos. Sci. Technol. 167, 206–214. Hibbeler, R.C., 2015. Structural analysis. Vol. 9. Pearson Prentice Hall, Upper Saddle River, NJ. Hinkle, J., Lin, J.H., Kling, D., 2011. Design and materials study of secondary structures in deployable planetary and space habitats. In: 52nd AIAA/ASME/ASCE/AHS/ASC Structures. In: Structural Dynamics and Materials Conference, Structures, Structural Dynamics, and Materials and Co-located Conferences. American Institute of Aeronautics and Astronautics. Hu, G., et al., 2006. Characterization of viscoelastic behaviour of a molding compound with application to delamination analysis in IC packages. In: 2006 8th Electronics Packaging Technology Conference. IEEE. Keller, P., et al., 2006. Development of elastic memory composite stiffeners for a flexible precision reflector. In: 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. American Institute of Aeronautics and Astronautics.

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Kong, D., Xiao, X., 2016. High cycle-life shape memory polymer at high temperature. Sci. Rep. 6, 33610. Lan, X., et al., 2009. Fiber reinforced shape-memory polymer composite and its application in a deployable hinge. Smart Mater. Struct.. 18(2), 024002. Lendlein, A., Kelch, S., 2002. Shape-memory polymers. Angew. Chem. Int. Ed. 41 (12), 2034–2057. Leng, J., et al., 2011. Shape-memory polymers and their composites: stimulus methods and applications. Prog. Mater. Sci. 56 (7), 1077–1135. Leng, J., et al., 2014. Effect of the γ-radiation on the properties of epoxy-based shape memory polymers. J. Intell. Mater. Syst. Struct. 25 (10), 1256–1263. Liu, Y., et al., 2004. Thermomechanics of shape memory polymer nanocomposites. Mech. Mater. 36 (10), 929–940. Liu, Y., et al., 2006. Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling. Int. J. Plast. 22 (2), 279–313. Liu, Y., et al., 2014. Shape memory polymers and their composites in aerospace applications: a review. Smart Mater. Struct. 23 (2), 023001. Meyers, M.A., Chawla, K.K., 2008. Mechanical Behavior of Materials, second ed. Cambridge University Press. Nji, J., Li, G., 2010. A self-healing 3D woven fabric reinforced shape memory polymer composite for impact mitigation. Smart Mater. Struct. 19(3), 035007. Ohki, T., et al., 2004. Mechanical and shape memory behavior of composites with shape memory polymer. Compos. A Appl. Sci. Manuf. 35 (9), 1065–1073. Roh, J.-H., Kim, H.-J., Bae, J.-S., 2014. Shape memory polymer composites with woven fabric reinforcement for self-deployable booms. J. Intell. Mater. Syst. Struct. 25 (18), 2256–2266. Roh, J.-H., Kim, H.-I., Lee, S.-Y., 2015. Viscoelastic effect on unfolding behaviors of shape memory composite booms. Compos. Struct. 133, 235–245. Shrotriya, P., Sottos, N., 1998. Creep and relaxation behavior of woven glass/epoxy substrates for multilayer circuit board applications. Polym. Compos. 19 (5), 567–578. Simo, J., 1987. On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput. Methods Appl. Mech. Eng. 60 (2), 153–173. Srivastava, V., Chester, S.A., Anand, L., 2010. Thermally actuated shape-memory polymers: experiments, theory, and numerical simulations. J. Mech. Phys. Solids 58 (8), 1100–1124. Sun, J., Liu, Y., Leng, J., 2014. Mechanical properties of shape memory polymer composites enhanced by elastic fibers and their application in variable stiffness morphing skins. J. Intell. Mater. Syst. Struct. 26 (15), 2020–2027. Taherzadeh, M., et al., 2016. Modeling and homogenization of shape memory polymer nanocomposites. Compos. B Eng. 91, 36–43. Tan, Q., et al., 2014. Thermal mechanical constitutive model of fiber reinforced shape memory polymer composite: based on bridging model. Compos. A Appl. Sci. Manuf. 64 (0), 132–138. Tandon, G.P., et al., 2009. Durability assessment of styrene- and epoxy-based shape-memory polymer resins. J. Intell. Mater. Syst. Struct. 20 (17), 2127–2143. Tao, R., et al., 2016. Parametric analysis and temperature effect of deployable hinged shells using shape memory polymers. Smart Mater. Struct. 25(11), 115034. Tobushi, H., et al., 1996. Thermomechanical properties in a thin film of shape memory polymer of polyurethane series. Smart Mater. Struct. 5 (4), 483–491. Wang, Z., et al., 2009. Modeling thermomechanical behaviors of shape memory polymer. J. Appl. Polym. Sci. 113 (1), 651–656.

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Characterization of viscoelasticity, viscoplasticity, and damage in composites

16

Janis Varna, Liva Pupure Lulea˚ University of Technology, Lulea˚, Sweden

16.1

Introduction

Due to polymer resin behavior and microdamage development, polymeric composites with carbon and glass fibers are, in general, inelastic materials. For unidirectional (UD) long fiber composites the behavior in the longitudinal (fiber) direction is governed by elastic fibers: the stress-strain relationship is mostly elastic with a very limited amount of viscoelasticity, which due to low strains is in the linear region, and almost zero viscoplasticity. One of the exceptions is man-made cellulosic fibers; see an example in Fig. 16.1a, exhibiting all the features of time-dependent inelastic behavior (hysteresis loops, residual strains after load removal). In the transverse direction the behavior of UD composites is mostly related to the matrix performance and therefore is expected to be time dependent and nonlinear. However, since the strain to UD ply transverse failure is low, the specimen usually breaks before large viscoelastic (VE) or viscoplastic (VP) strains are observed. The shear stress-strain response (usually studied in tensile loading of [45/
45]s specimens) of UD composites is very nonlinear and hence the time-dependent phenomena are observed best. The behavior of UD composites has been extensively studied by Guedes et al. (1998), Megnis and Varna (2003), Giannadakis et al. (2011b), Giannadakis and Varna (2014), and Pupure et al. (2018a). Linear viscoelastic and nonlinear viscoplastic material model for UD composite with experimentally confirmed linear VE behavior was developed by Megnis and Varna (2003). The assumption that the creep compliance does not depend on the stress level in creep tests significantly simplifies the necessary testing. However, in most of the referred papers the viscoelasticity, especially in shear, was highly nonlinear. In short-fiber composites the inelastic behavior is very common, especially for composites with biobased fibers and biobased resins. They show very nonlinear response to mechanical loading: stress-strain curves are nonlinear, in loading-unloading cycles hysteresis loops are typical (see Fig. 16.1b where several loading-unloading cycles for flax/starch composite are shown). The stress-strain curve depends on the loading rate and elastic properties degrade after loading to high stress. The main possible sources for this behavior are: (a) microdamage development with increasing load leading to elastic properties degradation; (b) viscoelastic effects that would lead to time dependence and loading rate dependence; (c) viscoplastic Creep and Fatigue in Polymer Matrix Composites. https://doi.org/10.1016/B978-0-08-102601-4.00016-3 © 2019 Elsevier Ltd. All rights reserved.

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(a)

(b)

16.1 Mechanical behavior of (a) regenerated cellulosic fiber (RCF) bundles; (b) flax/starch composite with 40 wt% fiber content at room temperature and relative humidity of 34%.

strains that develop with time and depend on the whole loading history but there is no fading memory. The viscoelasticity (and most certainly viscoplasticity) may be nonlinear with respect to stress. At very high stresses, interaction between these phenomena is expected: for example, microdamage is expected to influence macroscopic VP-strains and viscoplasticity can influence damage evolution. Lou and Schapery (1971) and Schapery (1997) developed a general thermodynamically consistent material model for nonlinear viscoelastic materials and this is the model used in this chapter. Since the presented data mostly correspond to uniaxial loading, the model is presented for this case. Nonlinear viscoplasticity was also included in Schapery’s model (Schapery, 1997), considered as an extreme viscoelastic response case with an infinite retardation time. However, our experience is that this model has difficulty in describing experimental trends in VP-strain, εVP(t, σ). The irreversible (viscoplastic) strain development in a general stress ramp was described by a nonlinear function by Zapas and Crissman (1984) and used by Tuttle et al. (1993). This representation is used in the following discussion. According to this model, in creep tests the VP-strain is a power function with respect to time and with respect to stress. Hence, exponents in these functions can be determined in creep tests and the identified functions used to simulate VP-strains in an arbitrary stress ramp. In Marklund et al. (2008) the constitutive equation including VP-strains was further modified to account for microdamage. The material model will be analyzed and an efficient test methodology will be suggested for complete material characterization to identify the stress dependent parameters in the material model. In this chapter, we demonstrate that the necessary experimental data to identify the material model accounting for these phenomena can be obtained in the following tests: (a) creep tests at several load levels with a following strain recovery; (b) elastic properties reduction measurement tests after loading to a certain stress level.

16.2

Material model

In building the material model, we assume that the viscoelastic and viscoplastic responses may be decoupled. The test results used in this chapter are from uniaxial loading cases (in most of the cases even Poisson’s effect-related strains were not

Characterization of viscoelasticity, viscoplasticity, and damage in composites

499

recorded). In addition to VE and VP terms, damage function d(σ max) with value determined by the maximum (most damaging) stress state experienced during the previous service life is incorporated to account for microdamage. As demonstrated later, the physical meaning of the damage function is the elastic compliance increase in the composite, which is degrading due to the experienced high stress/strain. The basic assumption of the material model is that strain decomposition is possible: the microdamage influenced viscoelastic strain response can be separated from the viscoplastic response, which is also affected by damage: εðσ, tÞ ¼ d ðσ max ÞεVE ðσ, tÞ + dðσ max ÞεVP ðσ, tÞ

(16.1)

Certainly, the product form of the interaction between d(σ max) and the VP-strain εVP used in Eq. (16.1) is rather questionable. It was concluded from a limited number of empirical observations that this form fits the test results better than a stand-alone term of εVP. Actually, it says that larger VP-strains develop in damaged composites than in undamaged composites. Since damage in the form of microcracks introduces local stress concentrations, this assumption is reasonable. The nonlinear viscoelastic material behavior is described in this chapter by the thermodynamically consistent theory developed by Lou and Schapery (1971) and Schapery (1997). In the one-dimensional case, the viscoelastic model contains three stress-dependent functions characterizing the nonlinearity. The material model is   ðt d ð g2 σ Þ ε ¼ dðσ max Þ  ε0 + g1 ΔSðψ
ψ 0 Þ dτ + εVP ðσ, tÞ dτ 0

(16.2)

In Eq. (16.2) “reduced time” is introduced as ðt

dt0 and ψ 0 ¼ ψ¼ 0 aσ

ðτ

dt0 0 aσ

(16.3)

Here ε0 represents the elastic strain which, generally speaking, may be nonlinear with respect to stress. ΔS(ψ) is the transient component of the linear viscoelastic creep compliance; g1 and g2 are stress-dependent material functions; aσ is the time-stress shift factor, which in fixed environmental conditions is a function of stress only. Actually, these functions also depend on temperature and humidity. For sufficiently small stresses a linear region may exist where g1 ¼ g2 ¼ aσ ¼ 1, and thus Eq. (16.1) turns into a strain-stress relationship for linear viscoelastic nonlinear viscoplastic materials. Schapery (1997) showed that the viscoelastic creep compliance can be written in the form of a Prony series: ΔSðψ Þ ¼

X i

   ψ Ci 1
exp
τi

Ci are constants and τi are the retardation times.

(16.4)

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Creep and Fatigue in Polymer Matrix Composites

The physical meaning of the damage related function d(σ max) is revealed by comparing at given stress σ the elastic strain ε0 in the undamaged and strain εel in the damaged composite: ε0 ¼ σ=E0 ; εel ¼ σ=Eðσ max Þ

(16.5)

From here εel ¼

E0 ε0 Eðεmax Þ

(16.6)

Since the elastic response is given by the first term in Eq. (16.2), we conclude that d ðσ max Þ ¼

E0 Eðσ max Þ

(16.7)

Obviously, d(σ max) has the physical meaning of composite normalized elastic compliance—it is inverse to the normalized elastic tangent modulus dependence on the “worst” experienced stress. The optimal set of experiments needed to determine the stress-dependent functions in the material model Eq. (16.2) and develop reliable methodology for data reduction is still a debatable issue. Most researchers agree that, due to large scatter between specimens, as much as possible data should be obtained from each individual “representative” specimen (see Pupure et al., 2015). At the end, an averaging must be performed, but it is better to average the found functional relationships than to use all available data pooled at once to define the model “on average”; some obvious trends observed for each specimen may “disappear” in the data pool. The development of viscoplastic strains is described in this chapter by a function presented in Zapas and Crissman (1984): (ð εVP ðσ, tÞ ¼ CVP

)m  σ ðτ Þ M dτ σ∗

t=t∗ 

0

(16.8)

where CVP, M, and m are constants to be determined, t/t∗ is normalized time, where t∗ is a characteristic time constant: for example, 1 h, if this is the time interval of interest. Constant σ ∗ could be, for example, 1 MPa. Actually t∗ and σ ∗ are introduced to have dimensionless strain expression. This model was successfully used in Marklund et al. (2006), Nordin and Varna (2006), Marklund et al. (2008), Giannadakis et al. (2011a), Giannadakis and Varna (2014), Pupure et al. (2016), and Pupure et al. (2018a,b). If the data fitting by Eq. (16.8) is not good, generalization is possible by replacing σ(τ)M by a more general function f(σ(τ)), to be identified (see Schapery, 1997): εVP ðtÞ ¼

ð t 0

α

ðf ðσ ÞÞ dτ

β (16.9)

Characterization of viscoelasticity, viscoplasticity, and damage in composites

16.3

501

Microdamage effect on stiffness

16.3.1 Elastic modulus The elastic modulus of the composite may degrade as the result of microdamage developing at high stresses. Therefore, elastic properties must be determined in a relatively low strain region ε 2 [ε1, ε2] where we expect that damage and irreversible phenomena will not develop. The most commonly used is a linear fit to relevant stress-strain data in a loading-unloading cycle. For the composites analyzed in this chapter, the maximum strain value in the modulus determination interval was 0.2% < ε2 < 0.3%, whereas ε1 was 0.05%. As demonstrated in Fig. 16.1b, the loading and unloading curves for flax/starch composite even in the low strain region have very different slopes. Hence, three different values of the modulus may be defined (in loading, in unloading, and the average of both). A detailed description of this material and the material properties is given in Sparnins et al. (2011, 2012). If ε2 is too high for the given material, the unloading slope in the elastic modulus measurements may be slightly lower than the loading slope due to damage accumulated in the loading part. However, in Fig. 16.1b the unloading slope is higher. The higher elastic modulus in unloading is certainly an artifact caused by viscoelastic behavior and the specific loading ramp used in Fig. 16.1b. Simple simulation of this type of loading ramp for linear viscoelastic materials shows that at the upper “turning point” the unloading slope may be very high and definitely higher than the elastic modulus given by input data in simulation. On the other hand, the simulated unloading slope further away from the “turning point” due to viscoelasticity is usually lower than the input elastic modulus. One can see it also in experimental data in Fig 16.1b. A practical suggestion for modulus determination based on these observations is to have the strain at the turning point at least 0.05% above ε2. Since neither the loading nor the unloading slope represents the elastic response, the strain rate in the tests has to be increased or at least kept constant during all tests using the same fitting interval. Sometimes the average of loading and unloading modulus is used as an “apparent modulus,” which makes sense only if the damage state during the test does not change.

16.3.2 Stiffness reduction measurements Loading to high strain/stress causes microdamage in the composite that accumulates during the loading history. Microdamage affects the elastic modulus and the elastic modulus dependence on the microdamage state has to be characterized. In cases when the damage state is very complex, as it is in short fiber composites, a more pragmatic approach is to find the modulus reduction as a function of the applied strain (or stress) in tensile tests. The tensile loading ramp in this test is shown in Fig. 16.2. The test consists of blocks, each containing a sequence of four steps: (a) new damage introduction by loading to a certain (high) strain level εi and unloading to “almost zero” stress;

502

Creep and Fatigue in Polymer Matrix Composites

16.2 Example of the applied strain steps during one block of the modulus determination test. (b) waiting at “almost zero” stress for a time at least five times longer than the length of the previous step for decay of all viscoelastic effects (some irreversible VP-strains called εresid i in Fig. 16.2 can be present at the end of the waiting time); (c) the elastic modulus determination applying high strain rate and low level loadingunloading; (d) VE-strain recovery after modulus measurement for a time that is at least five times longer than the time interval for modulus measurement.

Then the same steps are repeated in the next block using a higher level of applied strain εi+1 in step (a). After unloading to “almost zero” stress and a relatively large recovery period during step (b), rather large irreversible strains εresid may still be present. Therefore, it i is not suitable to keep the strain region ε 2 [ε1, ε2], defined in Section 16.3.1, in the following modulus determination step. Instead the strain region used should be shifted with respect to the irreversible residual strain after each loading cycle, ε 2 [ε1 + εresid , ε2 + εresid ]. As an alternative, a fixed stress interval can be used for elasi i tic properties determination defining the stress interval to use in the first elastic modulus test. This is the stress interval corresponding to ε 2 [ε1, ε2] for an undamaged composite.

16.3.3 Microdamage and stiffness degradation in fiber composites The main reason for elastic properties degradation in composites with increasing load is microcracking. Cracks open and crack surfaces slide, reducing the average stress in the composite at a given applied strain. Since the stress averaged over the volume is equal to the macroscopically applied stress, the macroscopic stress to reach the given

Characterization of viscoelasticity, viscoplasticity, and damage in composites

503

strain in the damaged composite is lower, which means that the elastic modulus is reduced due to damage (Lundmark and Varna, 2005; Varna, 2013). In order to use this reasoning in simulations of stiffness reduction, the relevant damage entities have to be identified and characterized by size and orientation. This is relatively easy to do in laminated composites where microcracks are well defined (intralaminar cracks, interlayer delaminations, debonds). They usually follow the fiber or the interface orientation and their number per unit length of the specimen can be characterized by crack density. Typical methods for quantification are based on optical microscopy, acoustic emission during failure events, edge replicas taken from loaded specimens, X-rays, or CT-microtomography. In so-called micromechanics modeling (see review in Nairn and Hu, 1994), stress perturbation due to the presence of cracks is analyzed. Most of the models focus on approximate analytical determination of the local stress state in the repeating element between two cracks. The simplest calculation schemes used are based on shear lag assumptions or on variational principles (Hashin, 1985; Smith and Wood, 1990; Varna and Berglund, 1991, 1994). To deal with complex laminate lay-ups with damage in several layers, damage in one layer was considered explicitly replacing the rest of the damaged and undamaged layers with one effective layer, which would have the same constraint effect on the damaged layer as the original lay-up (see Zhang et al., 1992). Most of the analytical solutions and available experimental data are for cross-ply types of laminates with cracks in 90-layers only, which is the case when stiff and elastic 0-layers suppress viscoelastic and viscoplastic effects, making the laminate response rather elastic. In this chapter the main focus is on viscoelasticity and viscoplasticity that may be accompanied by damage. Therefore, the only case of laminates analyzed here is the [45/
45]s laminate for which the applied axial stress is easily transformed to shear stress and the shear strain is calculated from measured axial and lateral strains, thus presenting an excellent case for analyzing the shear stress-strain nonlinearity of the composite. This loading may also introduce some cracks in layers that are not so easy to see and to count but their presence changes the laminate stiffness. Fig. 16.3 shows axial modulus and Poisson’s ratio change with increasing applied strain in GF/VE noncrimp fabric (NCF) [45/
45]s composite (see Giannadakis et al., 2011b for details). As expected, the axial modulus decreases ( t1 the loading can be written as

Characterization of viscoelasticity, viscoplasticity, and damage in composites

σ ðtÞ ¼ σ 0 ½HðtÞ
H ðt
t1 Þ

519

(16.23)

where H(t) is the Heaviside step function. The material model of Eq. (16.2) may therefore be applied separately to the creep interval t 2 [0, t1] and to the strain recovery interval t > t1. From Eq. (16.2) we obtain the following expressions for creep strain εcreep and recovery strain εrec respectively:     t εcreep ¼ dðσ max Þ ε0 + g1 g2 △S σ + εVP ðσ, tÞ 0 < t < t1 aσ

(16.24)

εrec ¼ d ðσ max Þfg2 σ ½△Sðψ Þ
△Sðψ
ψ 1 Þ + εVP ðσ, t1 Þg t > t1

(16.25)

where ψ1 ¼

t1 t1 and ψ ¼ + t
t1 aσ aσ

(16.26)

Using the Prony series (Eq. 16.4) for creep compliance in Eq. (16.24), we obtain the following expression to describe the strain development during the creep test: " εcreep ¼ dðσ max Þ ε0 + g1 g2 σ

X i

#    t Ci 1
exp
+ εVP ðσ, tÞ aσ τ i

(16.27)

Substituting the compliance of Eq. (16.4) in Eq. (16.25), we obtain a strain expression for the recovery interval: ( εrec ¼ d ðσ max Þ g2 σ

X i

)      t1 t
t1 Ci 1
exp
exp
+ εVP ðσ, t1 Þ aσ τ i τi (16.28)

Expressions (16.27) and (16.28) are used to fit the experimental creep and strain recovery data. Stress independent constants Ci, i ¼ 1, 2,… , I and stress dependent ε0, aσ , g1 and g2 are found in the result of the fitting. The retardation times τi in the Prony series are chosen arbitrarily, but the largest τi should be at least a decade larger than the length of the conducted creep test. A good approximation to experimental data may be achieved if the retardation times are spread more or less uniformly over the logarithmic time scale, typically with a factor of about 10 between them. The τi selection can be optimized comparing the accuracy of the obtained fit for different selections using the least-square method as described later. Even if it is not a necessary condition, the additional requirement Ci > 0 usually leads to improved fitting of test results. If stiffness reduction can be ignored, then Eqs. (16.27) and (16.28) are reduced to εcreep ¼ ε0 + g1 g2 σ

X i

   t Ci 1
exp
+ εVP ðσ, tÞ aσ τ i

(16.29)

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Creep and Fatigue in Polymer Matrix Composites

εrec ¼ g2 σ

X i

     t1 t
t1 Ci 1
exp
exp
+ εVP ðσ, t1 Þ aσ τ i τi

(16.30)

The stress and time-dependent VP-strains enter Eq. (16.27). The term εVP(σ, t1) in Eq. (16.27) is the VP-strain developed during the creep test and it comes directly from the test as the last data point at the end of strain recovery. However, in the creep strain expression (Eq. 16.27) the VP-strain at each time instant during this test is required. If known, it could be subtracted from the measured strain during the creep test, with the rest being a pure nonlinear viscoelastic strain. There are three alternatives to account for VP-strains when viscoelasticity is analyzed using Eqs. (16.27) and (16.28): (a) Using low stress levels in creep tests so that viscoplasticity is almost not present. However, then the viscoelasticity is usually linear and this alternative is not applicable in the nonlinear viscoelasticity region; (b) Performing specimen “conditioning” by subjecting to high stress creep before the viscoelastic characterization. If the viscoplasticity development rate decays with time as it does for many materials (Giannadakis et al., 2011a; Marklund et al., 2006, 2008; Nordin and Varna, 2006; Pupure et al., 2018a) and if damage does not develop during this test, the specimen after conditioning can be used for nonlinear viscoelasticity analysis. If performing creep tests at any level below the conditioning stress level, the VP-strains in these specimens will be small and may be negligible. In other words, the conditioning means that VP-strains in the specimen have already developed and if the creep test is now performed at lower stress, the new (additional) VP-strains will be small. This must be checked from the curves after the strain recovery: the irreversible strain should be small. (c) If the viscoplastic behavior is already described by equations and the previous loading history for the given specimen is known, the time dependence of the VP-strain during the current creep test can be calculated and subtracted from the measured strains to have pure viscoelastic response to analyze. The methodology is described in the Appendix to this chapter. If the loading history effect is correctly described, the same specimen can be used again at different stress levels.

The steps for using creep and strain recovery data to determine coefficients Ci, τi,aσ , g1, g2 and also the elastic term ε0 are summarized as follows: 1. The damage-related function d(σ max) is found as described in Section 16.3. 2. Viscoplastic strains (if present) are subtracted from creep strain using expression (A6) or (A7), obtaining pure viscoelastic response data. 3. For creep and strain recovery data in a low stress region where linear response could be expected, (g1 ¼ g2 ¼ aσ ¼ 1) are used to determine Ci by fitting simultaneously the viscoelastic creep and strain recovery data (method of least squares (LSQ)) using Eqs. (16.29) and (16.30). Retardation times τi are selected rather arbitrarily with about a one-decade step and adjusted to insure that all Ci are positive. 4. Data at higher stress (possible nonlinear viscoelastic region) are fitted by Eqs. (16.27) and (16.28) using the previously obtained Cm, τm. In the procedure used, the initial value of aσ is selected and then increased with a selected step. For each value the method of LSQ is used to find the best g1, g2 and ε0. For each set of aσ , g1, g2 and ε0 the misfit function (sum of squares of deviations with test data) is calculated. The set of aσ , g1, g2 and ε0 that gives the minimum

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of the misfit function is considered to be the correct set. Certainly, other available numerical nonlinear minimization routines may be more efficient. 5. The procedure described in step 4 is repeated for all available stress levels to obtain the stress dependence of aσ , g1, g2 and ε0.

16.5.2 Examples of nonlinear viscoelastic behavior Creep and strain recovery tests are performed at several stress levels and time intervals, which is very time consuming. It is therefore crucial for the characterization to use representative specimens for the analyzed material. The specimen may be singled out based on its elastic properties being intermediate in this group of specimens. It can also be suggested that each specimen in the nonlinear viscoelastic analysis be analyzed separately, using the methodology described in Section 16.5.1, which is a preferable strategy since the data reduction procedure otherwise easily becomes both tedious and impractical and may contain some artificial trends when averages are used. In all figures discussed in this section, the VP-strains are already subtracted. The viscoelastic compliance of two composites, defined as viscoelastic strain divided by stress level in the creep test, is shown for several stress levels in Fig. 16.19. In Fig. 16.19a the shear compliance of a GF/EP UD composite is shown, derived by subtracting VP-strains in Giannadakis and Varna (2014) from axial loading data of [45/
45]s laminates. The creep compliance of the flax/PLA composite in Fig. 16.19b is from Varna et al. (2012). For clarity, the elastic response is subtracted presenting the time-dependent VE part only. The VE creep compliance for both composites increases with increasing stresses. There seems to be a nearly linear VE region for the GF/EP composite in the stress range 10–20 MPa, where the VE

16.19 Nonlinear viscoelastic creep compliance: (a) shear compliance of UD layer in GF/EP [45/
45]s laminate after subtracting the elastic part; (b) flax/PLA at RH ¼ 34%. Panel (a): Data from Giannadakis, K., Varna, J., 2014. Analysis of nonlinear shear stress-strain response of unidirectional GF/EP composite. Compos. Part A 62, 67–76. Panel (b): Data from Varna, J., Rozite, L., Joffe, R., Pupurs, A., 2012. Nonlinear behavior of PLA based flax composites. Plast. Rubber Compos. 41 (2), 49–60.

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16.20 Viscoelastic shear compliance of UD layer in GF/EP [45/
45]s laminate after subtracting the elastic part at two different stress levels for creep and recovery. Data from Giannadakis, K., Varna, J., 2014. Analysis of nonlinear shear stress-strain response of unidirectional GF/EP composite. Compos. Part A 62, 67–76.

creep compliance is almost stress independent. Linearity for the flax/PLA composites can be assumed in the range between 6 and 10 MPa. The VE-strain development during creep loading and during the following VE-strain recovery interval for a GF/EP composite is shown in Fig. 16.20. Some initial experimental points and some points after stress removal have been deleted because in test the load application and removal are not step functions, as in theory. It took approximately 5 s to upload and, therefore, data in 20-s intervals were considered as not usable for fitting with expressions for step loading. In this work, constants Ci, τi were found using VE creep and recovery data for the shear stress level of 20 MPa. These values were used at all stress levels, finding parameters aσ , g1, g2 and ε0 from the fitting procedure described in Section 16.5.1. The two sets of shift factors in Fig. 16.21a (open symbols and filled) and also the data in Fig. 16.22 were obtained using the same procedure but with different reference specimens at 20 MPa for finding Ci, τi. The sensitivity of results with respect to this choice is rather low. Two specimens were used at each stress. The shift factor slightly decreases at high stress, which means that at high stress slightly less time is required to reach the same VE response. Similarly, as with the time-temperature effect, the decrease of the stress-related shift parameter can be interpreted as a deformation of the time axis. However, the stress dependence is much less than the temperature dependence, which usually is over several decades.

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16.21 High stress-related time shift factors for (a) shear response of UD GF/EP; (b) flax/PLA at RH ¼ 34% Panel (a): Data from Giannadakis, K., Varna, J., 2014. Analysis of nonlinear shear stress-strain response of unidirectional GF/EP composite. Compos. Part A 62, 67–76. Panel (b): Data from Varna, J., Rozite, L., Joffe, R., Pupurs, A., 2012. Nonlinear behavior of PLA based flax composites. Plast. Rubber Compos. 41 (2), 49–60.

16.22 Dependence on stress of nonlinear shear viscoelasticity functions: (a) g1(σ LT); (b) g2(σ LT). Data from Giannadakis, K., Varna, J., 2014. Analysis of nonlinear shear stress-strain response of unidirectional GF/EP composite. Compos. Part A 62, 67–76.

In Fig. 16.21b the shift factor dependence on stress for flax/PLA composite is shown. The variation is small and there is no clear trend in the data. The small increase with stress is most likely a mathematical curiosity occurring when the method of least squares with four parameters is applied to a small amount of rather uncertain accuracy test data (the VE strains in the strain recovery region are very small). It would probably be reasonable to state that for these composites the shift factor does not depend on stress and this parameter can be included in τi. The two parameters g1 and g2 in Fig. 16.22 are clearly higher at higher stress, which means that the time-dependent integral in Eq. (16.2) and also the second term in Eq. (16.27) grow faster than the applied stress. The elastic response of the GF/EP composite shown in Fig. 16.23 is linear with respect to stress, but it also could be nonlinear.

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16.23 Elastic shear strain vs shear stress in UD layer of GF/EP. Data from Giannadakis, K., Varna, J., 2014. Analysis of nonlinear shear stress-strain response of unidirectional GF/EP composite. Compos. Part A 62, 67–76.

16.24 Quasistatic shear stress-shear strain curves for UD layer in [45/
45]s GF/EP laminate at two strain rates. Ddata from Giannadakis, K., Varna, J., 2014. Analysis of nonlinear shear stress-strain response of unidirectional GF/EP composite. Compos. Part A 62, 67–76.

The material model with VP and VE nonlinearity parameters defined following the methodology described in this chapter was used to predict the shear stress-strain response of the GF/EP UD composite in tensile loading until failure. Experimental data and simulations for two strain rates are presented in Fig. 16.24 showing good agreement, even in the highly inelastic region. In this study, the damage effect was estimated as negligible. The damage-caused stiffness reduction in the constitutive model was neglected also in similar studies on paper/phenol-formaldehyde composite (Nordin and Varna, 2005, 2006). An example of a study where the damage effect was included is presented in Marklund et al. (2008) for a hemp/lignin composite. The elastic modulus reduction is shown in Fig. 16.9 and VP-strains are presented in Fig. 16.15. The values of the

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16.25 Stress-dependent functions in Schapery’s model for hemp/lignin composite. Data from Marklund, E., Eitzenberger, J., Varna, J., 2008. Nonlinear viscoelastic viscoplastic material model including stiffness degradation for hemp/lignin composites, Compos. Sci. Technol. 68, 2156–2162.

stress-dependent functions in the nonlinear viscolelastic model are presented in Fig. 16.25. One may observe that for this composite g1, g2 slightly increase with stress, whereas the aσ (time shift) increase is not large and may be an artifact. Linearity of the viscoelasticity, if it exists, holds until 3 MPa only. For validation, the developed material model was used to simulate the composite strain response in a stress-controlled test with low stress rate. The stress was first increasing, then reduced, and finally the strain increased to failure with a constant stress rate. Simulation results together with experimental curves for two specimens are shown in Fig. 16.26. The good agreement confirms that the developed material model is adequate for this material.

16.26 Simulated and experimental loading-unloading curves for hemp/lignin composite. Data from Marklund, E., Eitzenberger, J., Varna, J., 2008. Nonlinear viscoelastic viscoplastic material model including stiffness degradation for hemp/lignin composites, Compos. Sci. Technol. 68, 2156–2162.

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16.6

Conclusions

Sources of inelastic behavior of short-fiber and long-aligned fiber composites were analyzed in this chapter. The main identified phenomena included in the constitutive model are: (a) microdamage related reduction of thermoelastic properties; (b) nonlinear viscoplastic strains developing with time at high stresses; (c) nonlinear viscoelasticity. A methodology for experimental determination of parameters and stress-dependent functions in the material model is suggested and validated. The necessary tests are tensile quasistatic loading-unloading tests and creep tests at different stress levels with following strain recovery after load removal. The methodology is demonstrated on (a) GF/EP prepreg tape and NCF GF/VE [45/
45]s laminates; (b) SMC with glass fiber bundles; (c) flax and hemp fiber/lignin matrix composites; (d) paper/phenol-formaldehyde composites. In these materials, significant microdamage evolves in the final stage just before the specimen failure. Therefore, damage does not affect the strain response in creep tests used for viscoelasticity and viscoplasticity characterization, performed at lower stress levels to avoid creep rupture. For the composites used, viscoplasticity and nonlinear viscoelasticity affect the macroscopic inelastic behavior more than the microdamage accumulation.

Appendix

Time dependence of VP-strain in one creep test

Viscoplastic strains develop on a similar time scale and at similar stresses as the nonlinear viscoelastic strains. Therefore, the VP-strain, being a function of time, has to be subtracted from the creep test data to leave a pure viscoelastic response to analyze. Since the accumulation of VP-strain in one creep test depends on the previous loading history, the approach is as described in the following paragraphs. We consider the current (k + 1) creep test of length tk+1 at stress σ k+1 where t 2 [tP(k), tP(k+1)]. Since the intervals when the specimen is unloaded (strain recovery after creep) can be ignored in the VP-strain analysis, the time t in the analysis corresponds to the time in the loaded condition. The VP-strains can be correctly subtracted from data in the (k + 1)th step only if the previous loading history in terms of accumulated VP-strains for this particular specimen is known and accounted for. The previous loading in our case consists of k creep tests with different length and stress levels. We assume that the “history” is all that happened over the loading P time tP(k). The viscoplasticity developed during the “history” we denote as εVP(k) (in a particular case, it can be the sum of VP-strains developed in all earlier creep tests). According to Eqs. (16.8) and (16.12) further development of the total viscoplastic strain during the current creep test is described by

εVP ðtÞ ¼ CVP

8 > >

> :

0

 ð t=t∗ =t∗  σ ðτ Þ M ðk Þ dτ + σ∗ tX

9m > > =

σ M k+1 dτ > σ∗ =t∗ > ; ðk Þ

(A1)

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The “new” viscoplastic strain developing in the current step is X εkVP+ 1 ðtÞ ¼ εVP ðtÞ
εVP

ðkÞ

(A2)

The upper index in the first term indicates that this is new VP-strain developing during the k + 1 creep test. 0 P 11=m ðkÞ ε According to Eq. (16.8), the first term in Eq. (A1) is equal to @ VP A and CVP therefore Eq. (A1) can be rewritten as 9m 80 X 11=m X > > ð k Þ t
t > > < ε σ M ðk Þ = B VP C k+1 εVP ðtÞ ¼ CVP @ A + > > CVP σ∗ t∗ > > ; : 8 X ! 9m X 1=m t
t < =   M ð k Þ ðk Þ 1=m σ k + 1 ¼ εVP + CVP t > tX ðkÞ : ; σ∗ t∗

(A3)

Using Eq. (A3) in Eq. (A2), εkVP+ 1 ðtÞ ¼

8 < :

X !1=m εVP

ðk Þ

1=m + CVP

σ

k+1

X 9m M t
t ðkÞ =

σ∗

t∗

;

X
εVP

ðkÞ

(A4)

Expression (A4) can be used directly for VP-strain developing in the k + 1 creep test, assuming that CVP and M have been previously found. Unfortunately, the variation between specimens is very large and using the average characteristics is not the best way to subtract VP-strain for a particular specimen in a particular creep test. It is much more accurate to use the experimental value of the final VP-strain for the particular specimen in this particular creep test and to use the model to describe its development in time. Expression (A4) can be modified for this purpose as described later. From strain recovery we know the “new” viscoplastic strain at the end of the current creep P step, εcVP ¼ εk+1 VP (t (k+1)) and we can use it in Eq. (A4) to express 1=m

CVP

σ

k+1

σ∗

M

2 ¼

t∗ tk + 1

X !1=m

4 εc + ε VP VP

ðk Þ

X !1=m 3 ðk Þ 5
εVP

(A5)

Substituting Eq. (A5) back into Eq. (A4), we obtain the time dependence of viscoplastic strain in the current creep step to be used to subtract from the creep strain before performing viscoelastic analysis:

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εkVP+ 1 ðtÞ ¼

8 < :

t
tX

X !1=m εVP

ðkÞ

X


εVP

+

tk + 1

2 ðkÞ

X !1=m

4 εc + ε VP VP

ðk Þ

X !1=m 39m = ðkÞ 5
εVP ;

ðk Þ

(A6) This form is more complex than the previous one (Eq. A4). However, it has a very significant advantage: it does not contain CVP obtained from a different specimen or from some averaging procedure over several specimens. Certainly, m comes from independent experiments. The most important required information comes from tests on this particular specimen: (a) the sum of viscoplastic strain during P previous loading steps, εVP(k); (b) the “new” viscoplastic strain during the current test εcVP. P (0) During the first step εVP ¼ 0, tP(0) ¼ 0 and Eq. (A6) simplifies: ε1VP ðtÞ ¼ εcVP

 m t t1

(A7)

References Giannadakis, K., Varna, J., 2014. Analysis of nonlinear shear stress-strain response of unidirectional GF/EP composite. Compos. Part A 62, 67–76. Giannadakis, K., Szpieg, M., Varna, J., 2011a. Mechanical performance of recycled carbon fibre/PP. Exp. Mech. 51, 767–777. Giannadakis, K., Mannberg, P., Joffe, R., Varna, J., 2011b. The sources of inelastic behavior of Glass Fibre/Vinylester non-crimp fabric [45/
45]s laminates. J. Reinf. Plast. Compos. 30 (12), 1015–1028. Guedes, R.M., Marques, A.T., Cardon, A., 1998. Analytical and experimental evaluation of nonlinear viscoelastic-viscoplastic composite laminates under creep, creep-recovery, relaxation and ramp loading. Mech. Time-Depend. Mater. 2 (2), 113–128. Hashin, Z., 1985. Analysis of cracked laminates: a variational approach. Mech Mater. 4 (1985), 121–136. Hull, D., 1981. An Introduction to Composite Materials. Cambridge Solid State Science. Kabelka, J., Hoffman, L., Ehrenstein, G.W., 1996. Damage process modeling in SMC. J. Appl. Polym. Sci. 62, 181–198. Lou, Y.C., Schapery, R.A., 1971. Viscoelastic characterization of a nonlinear fiber-reinforced plastic. J. Compos. Mater. 5, 208–234. Lundmark, P., Varna, J., 2005. Constitutive relationships for laminates with ply cracks in in-plane loading. Int. J. Damage Mech. 14 (3), 235–261. Marklund, E., Varna, J., Wallstr€om, L., 2006. Nonlinear viscoelasticity and viscoplasticity of flax/polypropylene composites. JEMT 128, 527–536.

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Marklund, E., Eitzenberger, J., Varna, J., 2008. Nonlinear viscoelastic viscoplastic material model including stiffness degradation for hemp/lignin composites. Compos. Sci. Technol. 68, 2156–2162. Megnis, M., Varna, J., 2003. Nonlinear viscoelastic, viscoplastic characterization of unidirectional GF/EP composite. Mech. Time-Depend. Mater. 7, 269–290. Nairn, J., Hu, S., 1994. Matrix microcracking. In: Pipes, R.B., Talreja, R. (Eds.), Damage Mechanics of Composite Materials. In: Composite Materials Series, vol. 9. Elsevier, Amsterdam, pp. 187–243. Nordin, L.-O., Varna, J., 2005. Nonlinear viscoelastic behavior of paper fiber composites. Compos. Sci. Technol. 65, 1609–1625. Nordin, L.-O., Varna, J., 2006. Nonlinear viscoplastic and nonlinear viscoelastic material model for paper fiber composites in compression. Compos. Part A 37, 344–355. Oldenbo, M., Varna, J., 2005. A constitutive model for non-linear behavior of SMC accounting for linear viscoelasticity and micro-damage. Polym. Compos. 26 (1), 84–97. Pupure, L., Varna, J., Joffe, R., Pupurs, A., 2013. An analysis of the nonlinear behavior of ligninbased flax composites. Mech. Compos. Mater. 49 (2), 139–154. Pupure, L., Varna, J., Joffe, R., 2015. On viscoplasticity characterization of natural fibers with high variability. Adv. Compos. Lett. 24 (6), 125–129. Pupure, L., Varna, J., Joffe, R., 2016. Natural fiber composites: challenges simulating inelastic response in strain controlled tensile tests. J. Compos. Mater. 50 (5), 575–587. Pupure, L., Saseendran, S., Varna, J., Basso, M., 2018a. Effect of degree of cure on viscoplastic shear strain development in layers of [45/
45]s glass fibre/epoxy resin composites. J. Compos. Mater. 52 (24), 3277–3288. Pupure, L., Varna, J., Joffe, R., 2018b. Methodology for macro-modeling of bio-based composites with inelastic constituents. Compos. Sci. Technol. 163, 41–48. Rozite, L., Varna, J., Joffe, R., Pupurs, A., 2013. Nonlinear behavior of PLA and lignin based flax composites subjected to tensile loading. J. Thermoplast. Compos. Mater. 26 (4), 476–496. Schapery, R.A., 1997. Nonlinear viscoelastic and viscoplastic constitutive equations based on thermodynamics. Mech. Time-Depend. Mater. 1, 209–240. Smith, P.A., Wood, J.R., 1990. Poisson’s ratio as a damage parameter in the static tensile loading of simple cross-ply laminates. Compos. Sci. Technol. 38, 85–93. Sparnins, E., Pupurs, A., Varna, J., Joffe, R., Nattinen, K., Lampinen, J., 2011. The moisture and temperature effect on mechanical performance of flax/starch composites in quasi-static tension. Polym. Compos. (12), 2051–2061. Sparnins, E., Varna, J., Joffe, R., Nattinen, K., Lampinen, J., 2012. Time dependent behavior of flax/starch composites. Mech. Time-Depend. Mater. 16 (1), 47–70. Tuttle, M.E., Pasricha, A., Emery, A.F., 1993. Time-temperature behavior of IM7/5260 composites subjected to cyclic loads and temperatures. In: Mechanics of Composite Materials: Nonlinear Effects, AMD-159, ASMEpp. 343–357. Varna, J., 2013. Modeling mechanical performance of damaged laminates. J. Compos. Mater. 47 (20
21), 2443–2474. Varna, J., Berglund, L.A., 1991. Multiple transverse cracking and stiffness reduction in crossply laminates. J. Compos. Technol. Res. 13 (2), 97–106. Varna, J., Berglund, L.A., 1994. Thermo-elastic properties of composite laminates with transverse cracks. J. Compos. Technol. Res. 16 (1), 77–87.

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Varna, J., Rozite, L., Joffe, R., Pupurs, A., 2012. Nonlinear behavior of PLA based flax composites. Plast. Rubber Compos. 41 (2), 49–60. Zapas, L.J., Crissman, J.M., 1984. Creep and recovery behavior of ultra-high molecular weight polyethylene in the region of small uniaxial deformations. Polymer 25 (1), 57–62. Zhang, J., Fan, J., Soutis, C., 1992. Analysis of multiple matrix cracking in [θm/90n]s composite laminates. Part 1. In-plane stiffness properties. Composites 23 (5), 291–304.

Further reading Lundmark, P., Varna, J., 2006. Crack face sliding effect on stiffness of laminates with ply cracks. Compos. Sci. Technol. 66, 1444–1454.

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Sreenivas Alampalli New York State Department of Transportation, Albany, NY, United States

17.1

Introduction

The transportation infrastructure industry is facing severe challenges with rapidly deteriorating infrastructure due to increased use, change in customer expectations, and constrained resources. Due to an emphasis on uninterrupted mobility and high reliability of the transportation infrastructure, reduction in travel time delays and service interruptions due to reconstruction and maintenance are emphasized by bridge owners to meet customer expectations. Hence, to meet customer expectations with constrained resources, advanced materials with improved durability and maintainability, innovative and cost-effective construction methodologies, and new design procedures are under consideration by bridge owners. Composite materials, such as fiber-reinforced polymer (FRP), are becoming increasingly popular due to the advantages they offer, which include their light weight, better corrosion resistance, shop fabrication capabilities, high strength, and perceived low life-cycle costs. During the last 20 years, several bridge superstructures and decks have been built around the world using such materials (Alampalli et al., 2002; Triandafilou and O’Connor, 2009; NCHRP, 2015). There has also been an increased use of external wrapping to protect bridge components for strengthening and protection from adverse environment (Hag-Elsafi et al., 2002; Sen, 2003; NCHRP, 2015; Michels et al., 2016; Peiris and Harik, 2015), as well as limited use of these materials for internal reinforcement, form materials, posttensioning of existing structures, and repair of damaged bridge elements (NCHRP, 2015). Considering these materials are still relatively new, as compared to steel and concrete, to the industry and to the harsh in-service environment bridge structures are subjected to, more data on their in-service structural behavior, durability, maintainability, and serviceability are required to use them appropriately and cost-effectively. This chapter explores the use of structural health monitoring (SHM) of composite materials in bridge structural applications. The next two sections of this chapter discuss the use of FRP structures in bridge applications and SHM issues. The fourth section discusses how SHM can benefit in understanding FRP structures, improving durability, and assisting in implementation issues. This section is followed by case studies to further illustrate the use of SHM in FRP applications. Finally, the last section presents some recommendations for future research and practice. Creep and Fatigue in Polymer Matrix Composites. https://doi.org/10.1016/B978-0-08-102601-4.00017-5 © 2019 Elsevier Ltd. All rights reserved.

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17.2

Creep and Fatigue in Polymer Matrix Composites

FRP structures in bridge industry

FRP materials have been widely used for several decades in the aerospace industry due to their light weight and high strength. They have also become popular in the automotive and aerospace industries due to their light weight and noncorrosive properties. Compared to these industries, the application of composites to infrastructure applications is relatively new. Most of the bridge applications utilizing FRP materials were started in the early 1990s, on an experimental basis, primarily to increase the service life of existing structures by taking advantage of their lightweight characteristics. Thus, early applications included bridge decks for old deteriorated truss bridges to replace the heavy concrete decks with asphalt overlays to cost-effectively extend the service life and avoid complete replacement. They were also used for wrapping deteriorated bridge piers and beams to protect them from salt water ingress and to arrest/decrease further deterioration. With these experimental applications, engineers also started paying attention to other potential advantages these materials could offer besides their light weight. These included higher strengths, noncorrosive properties, engineerable characteristics, water resistance, easy transportation, shop fabrication, ease of erection, and perceived long-term durability. The capability to shop fabricate these components, coupled with short erection times compared to conventional materials, are attractive features to prevent long traffic interruptions and costly work-zone control required during the construction. In general, the use of FRP materials in bridge applications can be broadly divided into four areas: superstructures/decks (see Figs. 17.1–17.5), external reinforcement/ wrapping for strengthening (bond-critical) applications (see Fig. 17.6), maintenance/

17.1 An FRP slab bridge during construction: Bennetts Creek Bridge in United States.

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17.2 Bennetts Creek Bridge after construction during the proof load testing.

17.3 An FRP slab bridge after construction: Troups Creek Bridge in United States.

temporary (nonbond-critical) applications (see Figs. 17.7 and 17.8), and internal reinforcement (see Fig. 17.9). Each of these applications differ significantly in both structural and nonstructural requirements and thus require very different structural and material characteristics, workmanship, quality control and quality assurance methods, and inspection requirements. For example, fatigue properties are extremely important

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17.4 An FRP bridge deck during construction: Bentley Creek Bridge in United States.

17.5 An FRP bridge deck after construction: Schroon River Bridge in United States.

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17.6 External FRP reinforcement for strengthening application.

17.7 External reinforcement for maintenance application: during construction.

for a composite superstructure or deck, whereas creep is more important for internal reinforcement in prestressed applications or strengthening applications for concrete piers. More than 100 superstructures and bridge decks have been built around the world, several on an experimental basis or using research funding (Alampalli et al., 2002;

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Creep and Fatigue in Polymer Matrix Composites

17.8 Columns and pier-cap wrapped with CFRP: Everett Road, New York in United States.

17.9 Bridge deck with FRP rebars. Courtesy: Dr. GangaRao, West Virginia University.

Triandafilou and O’Connor, 2009; NCHRP, 2015). There have also been hundreds of column/beam wrappings, mostly for maintenance applications but with a few for strengthening (Hag-Elsafi et al., 2002; Sen, 2003). In some instances these materials have been used as internal reinforcement, but this application is still not widely found

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(e.g., Chen et al., 2008). Several standard specifications have emerged in recent years (AASHTO, 2008, 2009, 2012a, b; ACI, 2004, 2007, 2008a, b, 2010, 2012, 2014, 2015a, 2015b), but are still in their infancy due to the limited long-term data available on their behavior in the harsh in-service conditions in which these bridge structures operate.

17.3

Structural health monitoring

All structures are built for a purpose and it is the responsibility of owners to make sure the intended purpose is served at minimal or optimal cost while ensuring safety. This requires knowing the condition (or health) of the structure and taking appropriate actions, in a cost-effective manner, just-in-time, to make sure that the condition or actions proposed have minimal adverse impact on the structure fulfilling its intended purpose. In the case of a building, it could be the time of occupation, occupant comfort, and ability to do the work for which the building was intended. In the case of a bridge, it is making sure that the bridge can carry the loads it is intended to carry with uninterrupted mobility during operational hours. To ensure required safety, security, and mobility, bridge owners use several tools to assess structural condition and capacity. This is loosely defined in the literature as structural health monitoring (SHM). SHM can be accomplished in several ways, depending on the decision(s) to be made—periodically or continuously, visually or using sensors and instrumentation, and manually or remotely. Appropriate decision making depends on two factors: structural capacity, which accounts for the condition of the structure at any given time, and corresponding loading. In most cases, SHM examples presented in the literature deal with the structural capacity and seldom with loading. In most cases, loading is defined by the codes and specifications effective during the original construction or reconstruction. Ensuring safety is the predominant reason for SHM in infrastructure applications. But in the last decade, besides safety, there has been more emphasis on uninterrupted service and reliability of the service. While security was taken for granted before, this has emerged as another challenging item to consider in design and maintenance of structures. All these reasons have had several implications in the infrastructure arena and thus, to meet stakeholder expectations in the face of multiple hazard environments, new designs, innovative construction and maintenance procedures (such as design-build and design-build-maintain), and new materials are being explored and increasingly used. These are making the field of SHM more popular and thus have increased its use in recent years. This trend is expected to continue in the coming years. It is argued by Ettouney and Alampalli (2011a, b) that SHM contains three distinct phases: measurements, structural identification, and damage detection. They introduced the term structural health in civil engineering by adding another phase of “decision making” with an argument that any SHM project that does not integrate decision-making (or cost-benefit) ideas in all tasks cannot be a successful project (see Fig. 17.10). Full treatment of the subject with detailed applications can be found in Ettouney and Alampalli (2011a, b).

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Damage detection: Global vs. local, location, type, and extent; effect on the structure

Second base

First base

Third base Decision making: Do nothing, rehabilitation, replacement, or maintenance: type, when, etc.

Structural identification: Mathematical modeling and analysis based on measurements

Home plate

Measurements: Type, number, and location; Sensors and layout

17.10 Structural health in civil engineering concept.

17.4

FRP structures and SHM

As noted earlier, even though there have been numerous applications of FRP materials in infrastructure and bridge applications, most of these applications were funded by special grants or research funding on an experimental basis and are still considered by some as relatively new to civil engineering. There has been considerable research on their behavior through analysis, load testing, and monitoring (Alampalli, 2005; Reising et al., 2004; Farhey, 2005; Reay and Pantelides, 2006; Cai et al., 2008; Mufti and Neale, 2008; Czaderski and Meier, 2017; Holden et al., 2014; Charalambidi et al., 2016; Manalo et al., 2016). Most monitoring has been for relatively short periods when compared to the expected service life, due to budgetary considerations and change in personnel involved with these projects. Owners are still hesitant to widely adopt these materials for civil applications due to the following factors: l

l

l

l

l

l

l

l

l

l

l

Limited knowledge and understanding of long-term behavior and durability of FRP materials High initial costs compared to conventional materials Highly restrained resource environment Inadequate understanding and unavailability of maintenance and inspection procedures Unavailability of documented repair procedures Lack of specifications Lack of design software Lack of adequate training materials and sources Lack of quick analysis to determine capacity in case of accidental damage while in-service for unforeseen conditions such as fire, impact, or snow-plow damage Lack of data on the system performance under extreme events such as fire Lack of adequate industry support during the structure’s life compared to that provided by steel and concrete industries

The behavior of FRP materials for mechanical and environmental demands has been extensively researched in the aerospace and automotive industries and is relatively

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well understood. But the knowledge gained cannot be used directly in civil applications due to the following differences. l

l

l

l

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In-service conditions for civil structures are quite different due to geographical location. For example, within the United States, a structure built in California needs high seismic resistance, whereas in the northeastern states they face corrosive road salts due to their use in winter for traction. The service lives are long compared to aerospace and automotive applications. At present, the expected design life of a new bridge is 75 years in the United States and there are efforts to extend this to 100 or more years. Inspection efforts are quite different for bridge applications. In the United States, by federal mandate, all bridges require an inspection at least once in 2 years. Most inspections are still visual based on nondestructive test methods used on a limited basis, as needed, based on visual inspection findings (Alampalli and Jalinoos, 2009). Most FRP applications do not lend themselves well to visual inspections and thus need more advanced methodologies. Corrective and preventive maintenance plans are also very different. In civil applications, most of these plans are reactive, not proactive. Deterioration and failure mechanisms of structural components made using conventional materials are well understood by civil engineers and thus details are designed to offer time for reactive maintenance. But deterioration/ failure mechanisms associated with FRP components tend to accelerate faster than in conventional materials and thus require proactive maintenance. Civil FRP structures such as bridge decks are sandwich structures with several laminates, joints, connections, etc. Often these are integrated (or connected) with conventional materials such as steel and concrete. Even though there is considerable data available on individual components, there is not much data available on the performance and durability of the entire system. There have been very few or no long-term studies reported in the literature that monitored the behavior of the entire system rather than individual components.

Due to these complexities and differences from other applications that have been well studied, more data is required for improving the existing knowledge of FRP materials and to better maintain and manage them once they are built. SHM has great potential to bridge this gap and is an essential ingredient to promote the use of FRP materials in civil engineering applications and enhance their cost-effective management. Along with conventional sensors and instrumentation, fiber optic sensors and other instrumentation methodologies are under investigation, as they can be better integrated into FRP structures during their construction (e.g., Amano et al., 2007). The next section gives three case studies, one in each of the areas of bridge superstructure, bridge decks, column wrapping for strengthening, and column wrapping for temporary repairs. These studies were supported by the New York State Department of Transportation, where SHM was used to better understand the behavior and durability of FRP bridge applications.

17.5

Case studies

The New York State Department of Transportation has used FRP materials for several applications in the past. Realizing that FRP structures and the structures retrofitted with these materials should be monitored to ensure their adequate in-service

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Creep and Fatigue in Polymer Matrix Composites

performance and to gain more knowledge on their behavior and durability, they have monitored several of these applications and evaluated further with advanced analyses (Halstead et al., 2000; Hag-Elsafi et al., 2000; Hag-Elsafi et al., 2002; Alampalli et al., 2002; Alampalli and Kunin, 2002; Chiewanichakorn et al., 2003; Aref et al., 2005a, b; Alampalli, 2005; Alampalli and Ettouney, 2006; Alnahhal et al., 2007; Alampalli and Hag-Elsafi, 2013; Hag-Elsafi et al., 2002). This section briefly describes three bridge applications the author was directly involved with using FRP materials in New York and discusses the SHM used to evaluate their in-service performance.

17.5.1 Bridge deck One of the primary applications of FRP decks has been their use as an alternative to replace old, deteriorated, heavy concrete bridge decks to increase the live load capacity of old steel superstructures with minimal repairs. The behavior of a few of the bridges fitted with FRP decks has been studied under static loads in the literature. Even though FRP decks have been tested for fatigue by several researchers (Duatta et al., 2007; Kitane et al., 2004; Brown and Berman, 2010), there have been limited studies available on how the entire bridge system performs due to the lighter deck and thus careful evaluation is required. This case study gives one such example where SHM, including field testing followed by experimentally validated finite element (FE) models, was used to make appropriate recommendations. More details on this case study can be found in Alampalli and Kunin (2002, 2003), and Chiewanichakorn et al. (2006).

17.5.1.1 Reason for SHM The heavy concrete deck of an old deteriorated truss bridge was replaced with a lighter FRP deck. Verification of design assumptions, such as no composite action between the deck and the floor beams it is attached to, effectiveness of field joints in transferring the loads between FRP panels, and an evaluation of the effects of the rehabilitation process on the remaining fatigue life of the structure was needed.

17.5.1.2 Structure Bentley Creek Bridge, 42.7 m long and 7.3 m wide, is a highway bridge located on State Route 367 in Chemung County, New York, in the United States. The floor system was made up of steel transverse floor-beams at 4.27 m center-to-center spacing with longitudinal steel stringers. It was originally built as a single simple-span, steel truss bridge with a reinforced concrete slab.

17.5.1.3 Repairs In 1997, based on a capacity analysis, due to additional dead load from asphalt overlays and the deterioration of the steel trusses and floor system due to corrosion, the New York State Department of Transportation rehabilitated this bridge by replacing

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17.11 FRP bridge deck on a truss bridge that replaced an old concrete deck with asphalt overlays (elevation view).

17.12 FRP bridge deck on a truss bridge that replaced an old concrete deck with asphalt overlays (plan view).

the reinforced concrete slab with an FRP deck to prolong the structure’s service life as well as satisfy new load rating requirements (see Figs. 17.11 and 17.12). The FRP deck consists of top and bottom face skins and a web core. The face skins are composed of two plies of QM6408 and six plies of Q9100 E-glass stitched fiber

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Creep and Fatigue in Polymer Matrix Composites

fabric for a total thickness of 15 mm. The web core structure is made of two plies (3.7 mm) of QM6408 E-glass stitched fiber fabric wrapped around 150 mm  300 mm  350 mm isocycrinate foam blocks used as stay-in-place forms. The deck was designed using FE analysis. Orthotropic in-plane properties were used in the analysis. Stresses in the composite materials were limited to 20% of their ultimate strength and deflection was limited to span/800. The deck panels were designed to span between the floor beams. The steel stringers were left in place to provide bracing to the structure, although they no longer function in carrying live load. A total of six FRP panels were used to replace the roadway. Bearing pads made of 6-mm thick neoprene pads were placed across the full length of the floor beams to provide uniform bearing between the structural steel and the FRP deck. A polymer concrete haunch was placed on top of the bearing pads to provide a cross slope to the bridge deck. Three-inch diameter holes were drilled in through the top face skin and foam core of the deck panels. A one-inch diameter hole was then drilled through the bottom face of the composite deck, haunch material, and the top floorbeam flange. A structural bolt secured with a locking nut attached the deck to the superstructure. The drilled holes were then filled with a nonshrink grout. The panels were connected to each other using epoxy and splice plates. The joints consist of a longitudinal joint that runs the entire length of the bridge and four transverse joints that each span one lane. Vertical surface joints between panel sections were glued together with epoxy. Top and bottom splice plates were bonded using an acrylic adhesive. A 10-mm thick epoxy thin polymer overlay was used as the wearing surface of both the deck and sidewalk. Most of the wearing surface was applied to the panels during fabrication. Portions of the wearing surface covering panel joints and bolt lines were applied on-site after the FRP surface was lightly sandblasted and cleaned.

17.5.1.4 SHM instrumentation Sensors and instrumentation were designed appropriately to suit the objectives of the SHM. Conventional, general-purpose, uniaxial 350-ohm, self-temperature compensating, constantan foil strain gages were used to measure strains during the testing (see Fig. 17.13). The strain gages were bonded to steel and the FRP deck with adhesive and then waterproofed. A total of 18 strain gages was used, 6 placed on a steel floorbeam and 12 placed on the FRP deck. The data was collected using a computerized data acquisition system.

17.5.1.5 Testing and analysis Two fully loaded trucks of required configuration were used to load the bridge. The loads were positioned on the deck in such a way that the SHM objectives could be accomplished and enough data could be collected for the calibration of the FE models that would be developed for further analysis (see Fig. 17.14). Truck configuration and

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3.67 m

#6 0.46 m Fourth Floorbeam

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#9

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CL

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#12

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17.13 Strain gage locations on FRP deck during load testing for measuring joints effectiveness and data for calibration of finite element models.

weights used in the testing can be found in Alampalli and Kunin (2002, 2003). Loaded trucks were also driven across the bridge at crawl speeds to create influence lines for calibration of a detailed FE model. An FE model of the entire bridge was developed, according to the construction drawings, using a commercial FE modeling software and required analysis was performed using a general-purpose FE analysis package. This model with the FRP deck system was validated against load test results obtained from the field testing. The FRP

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17.14 Load testing of the FRP deck.

deck was also replaced by a generic reinforced concrete deck in a model to simulate a prerehabilitated deck system. Implicit dynamic time-history analyses were conducted with appropriate loading configuration for a moving design fatigue truck. Fatigue life of all truss members, floor beams, and stringers were determined based on a fatigue-resistance formula in the appropriate specifications used for bridge design. The modeling method used in this study is described in detail in Chiewanichakorn et al. (2006).

17.5.1.6 Conclusions The field test results indicated that the FRP deck was designed and fabricated conservatively. As assumed in the design, no composite action between the deck and the superstructure was verified. But the study showed, contrary to assumptions made, that the joints are only partially effective in load transfer between different panels. Thus, it was recommended that a future load test should be considered to determine if the combination of in-service loads and environmental exposure weakens the joints. Based on the FE analyses, it was found that this bridge would expect to have 354 years or, presumably, infinite fatigue life based on anticipated average daily truck traffic and new construction assumptions. The results indicated that the fatigue life of the FRP deck system almost doubles when compared with the prerehabilitated reinforced concrete deck system. Based on the estimated truck traffic that the bridge carries, stress ranges of the FRP deck system lie in an infinite fatigue life regime and this implies that fatigue failure of the trusses and floor system would not be expected during its service life (Chiewanichakorn et al., 2006). Fatigue life of critical members

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1500

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ADTT(2002) = 275, Fatigue Life = 1485 ADTT(2020) = 385, Fatigue Life = 1063

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17.15 Estimated fatigue life comparison (in years) of critical truss members after deck replacement.

in one of the trusses of the FRP deck was found to be more than 1000 years, as illustrated in Fig. 17.15.

17.5.2 Bridge wrapping (nonbond-critical) FRP materials have been widely used for increasing structural durability against environmental (mostly corrosion) damage through wrapping substructure components, such as columns and pier caps, cost effectively when compared to conventional concrete repairs. Lightweight characteristics, resistance to expansive tendencies of the corrosion products, relatively easy assessment with visual and simple nondestructive evaluation (NDE) methods, and relatively minor changes to the original structural geometry and dimensions, in addition to their cost-effectiveness, make these

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Creep and Fatigue in Polymer Matrix Composites

wrappings attractive. Sen (2003) provides a good overview of the application of FRP materials for repairing corrosion-damaged structures by external wrapping. The primary conclusions from this study included that the wrapping does not stop corrosion but reduces the corrosion rate; better performance is achieved when the component is fully wrapped than when partially wrapped; and effectiveness is increased when used with epoxies that offer a better barrier to chloride penetration. The New York State Department of Transportation has used FRP wrapping for maintenance applications since 1998 in numerous applications. Both glass and carbon FRP materials were used with a combination of surface preparation methods and labor skills (Halstead et al., 2000; Alampalli, 2005). Application of FRP materials to sound concrete may be considered a long-term repair, as these materials may provide protection against the environment—especially to chloride penetration and moisture ingress. Thus, design life can be extended to match the remaining life of the bridge elements. The short-term performance of these materials has been generally satisfactory. Long-term monitoring is in progress. This case study gives a general overview of one such application where SHM has been very useful in evaluating the surface preparation options available to bridge owners on the durability of FRP repairs for shortterm applications. More details on this case study can be found in Alampalli (2005).

17.5.2.1 Reason for SHM Different surface preparations (or concrete removal methods) and the number of FRP layers for short-term repairs can have significant cost differences. Hence, effectiveness of one layer of FRP wrapping along with various concrete repair strategies in reducing the corrosion rate in the reinforcing bars were investigated. The three concrete repair strategies considered were: (1) Removal of unsound concrete to a depth of no less than 25 mm from the rear-most point of reinforcement to sound concrete at an estimated cost of about $750/m2, (2) removal of unsound concrete to rebar depth at an estimated cost of about $270/m2, and (3) no removal of concrete except for minor patching of the depressions and uneven areas at a minimal cost. Note that all these costs do not include costs for FRP wrapping (material or labor), pressure washing the concrete, and sandblasting the concrete surface for a good bond between the concrete and FRP materials. The cost for FRP repairs was estimated, in 2002, at $125/m2 per layer of E-glass and $175/m2 per layer of carbon.

17.5.2.2 Structure The 430-m long and 23-m wide bridge carrying Route 2 over the Hudson River in Troy, NY in the United States, built in 1969 with eight spans of steel stringers and a concrete deck, was chosen for this experimental project (see Fig. 17.16). The columns (three in total) in one of the spans were deteriorated, partly due to leaking deck joints above. These columns were rectangular and tapered from bottom to top. The deterioration was nonstructural and was repaired using concrete patch work in 1991–92. These repairs failed quickly and hence further nonstructural (cosmetic) repairs were again needed in 1999 to slow or avoid future deterioration.

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17.16 The bridge intended for FRP wrapping for investigation of surface preparation effects on durability.

Hence, FRP materials were used as a cost-effective way to repair the concrete. At the same time, it was also decided to evaluate the three possible concrete repair strategies described in the previous section with one layer of FRP materials to see their longterm influence on the durability of the repair in terms of rebar corrosion rate and bond between the FRP materials and concrete surface.

17.5.2.3 Repairs The North column was repaired using repair strategy 1, the South column was repaired using strategy 2, and the center column was repaired using repair strategy 3, except for minor patching. Once the repairs were done, concrete surfaces were pressure washed and sandblasted to obtain a good bond between the concrete and FRP wrapping. One layer of Sika Wrap Hex 106G, which is a bidirectional E-glass fabric, was used to wrap all the columns. Sikadur 330, a high modulus, high strength, impregnating resin was used. It was covered with Sikadur 670W, which is a water-dispersed acrylic, protective, anticarbonation coating. The repair work was conducted in August and September of 1999.

17.5.2.4 SHM instrumentation The durability was evaluated using the rate of corrosion and bond between the concrete and FRP materials. The corrosion rates of the longitudinal rebar in the column were measured using corrosion probes from Concorr, Inc., which were installed inside the column during the concrete repair (see Fig. 17.17). PR500 data acquisition

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17.17 Corrosion sensor installation during the concrete repairs.

17.18 Humidity sensor after installation.

equipment was used to measure the corrosion rate from the probes. A total of nine corrosion probes (three for each column) were installed based on the measured half-cell potentials, which are an indication of the probability of corrosion activity. Probes were embedded at locations that showed the maximum corrosion activity. Vaisala HMP44 humidity/temperature probes, three per column, next to the corrosion probes, with an HM141 indicator were used to measure humidity and temperature inside the columns (see Fig. 17.18).

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5/00

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17.19 Typical time history of corrosion rates in column.

17.5.2.5 Monitoring Corrosion rates, humidity, and temperature were collected periodically from August 1999 through 2008. Humidity levels inside the columns were found to be around 90% indicating constant moisture levels, and this was attributed to water ingress from the unsealed top of the columns. The data also indicated no correlation between the concrete temperature inside the column and the rebar corrosion rates. A typical time history plot of rebar corrosion rates for the center column is shown in Fig. 17.19.

17.5.2.6 Conclusions The corrosion rates initially went up, then gradually slowed down and decreased with time. After about 2 years, they converged to values of about (or less than) 2 mils/year and stayed constant after that, indicating that the FRP wrapping is effective in controlling the corrosion rates regardless of the concrete repair strategy used. Visual inspections and thermographic inspections indicate that, in general, the bond quality did not significantly deteriorate compared to the time of construction in 1999 (see Fig. 17.20). Thus, results indicate that FRP wrapping was effective, for temporary (5–7 years) repairs, in confining the repaired/delaminated concrete columns and that concrete removal strategies did not influence the durability during the 4-year monitoring duration.

17.5.3 External reinforcement (bond-critical) FRP laminates were used to strengthen a T-beam bridge in Rensselaer County, New York in 1999 to demonstrate the application of FRP materials for cost-effective rehabilitation of deteriorated reinforced concrete bridges to improve capacity and extend service life. This case study briefly describes this project and the use of SHM to evaluate the durability of the FRP strengthening system after 2 years in service. More details on this case study can be found in Hag-Elsafi et al. (2001, 2004).

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17.20 The columns after 7 years in-service.

17.5.3.1 Reason for SHM In strengthening applications, the bond between the FRP laminates and the concrete surface is very crucial. Hence, appropriate surface preparation and proper application are very important for the long-term durability of the retrofit strengthening system. Hence, NDE-based SHM techniques are often required to ensure desired quality installation and assess bond effectiveness. Coin tapping and thermographic imaging are generally used, respectively, for local and global assessment of bond quality and effectiveness of the FRP retrofit system. But SHM using load testing gives a better picture of the durability of the strengthening application. This case study briefly describes such an application.

17.5.3.2 Structure The 12.19-m long and 36.58-m wide reinforced-concrete structure with 26 simply supported T-beams was built in 1932 and carries Route 378 over the Wynantskill Creek in the City of South Troy, New York (see Fig. 17.21). The bridge carries five lanes of traffic with annual daily traffic of about 30,000 vehicles. Concerns over section loss of the reinforcing steel to corrosion and the overall safety of the structure prompted the bridge strengthening using FRP laminates to improve flexural and shear capacities.

17.5.3.3 Repairs The Replark® laminate system consisting of Replark 30® unidirectional carbon fibers and three types of Epotherm materials (primer, putty, and resin), all manufactured exclusively by Mitsubishi Chemical Corporation of Japan, was used. The ultimate

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17.21 T-beam bridge before strengthening. 17.22 T-beam bridge strengthened with FRP laminates for additional flexure and shear strength.

strength of the laminate system is 3400 MPa corresponding to a guaranteed ultimate strain of 1.5%. Details of the laminate system are described in Hag-Elsafi et al. (2001). Laminates were located at the bottom of the webs and between beams oriented parallel to the beams. Those at the flange soffits, spanning between the beams, are oriented at a right angle to the beams. The U-jacket laminates, applied on the bottom and sides of the beams, are oriented parallel to the legs of the U-jackets (see Fig. 17.22).

17.5.3.4 SHM instrumentation The initial instrumentation and loading were to collect data to ascertain the effectiveness of the FRP retrofit system in reducing the steel rebar stresses, ensuring the bond between the laminate and the concrete, and the effect of the retrofit system on transverse

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To city of Troy 36.58 m Beam Numbers 2

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17.23 Typical strain gage scheme for load testing and data for calibration of finite element models.

load distribution, effective flange width, and neutral axis location. Nine beams were instrumented to provide information on transverse load distribution on the bridge. Foil strain gages, mounted directly on the reinforcing steel and FRP laminates, and concrete strain gages with large measuring grids were bonded using an epoxy resin (see Fig. 17.23). All gages were made watertight and protected from the environment for long-term monitoring purposes. System 6000, a general-purpose data acquisition system, manufactured by the Measurements Group®, was used for data collection.

17.5.3.5 Monitoring/testing The bridge was instrumented and load tested before and after installation of the FRP laminates to evaluate effectiveness of the strengthening system. Four trucks with known loads were used for the load testing (see Fig. 17.24). The load tests were repeated after 2 years to monitor in-service performance of the installed system. The analysis includes general flexural behavior of the most heavily stressed beam during the testing, bond between the FRP laminates and concrete, effective flange width, and neutral axis location.

17.5.3.6 Conclusions The load tests generally indicated lower strains than those measured during the test immediately after the construction, good quality of the bond between the FRP laminates and concrete, and no change in the effectiveness of the retrofit system after 2 years in-service (see Fig. 17.25). Based on recent visual inspections, the FRP

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17.24 Load testing of the T-beam bridge. 17.25 Effectiveness of repairs, using rebar strain, after 2 years in-service.

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laminates appear to remain in a fair condition after 18 years in service with some localized delamination and minor deterioration in the underlying concrete.

17.6

Summary

FRP materials are relatively new to bridge applications. Hence, to overcome the knowledge gap and to widely use these materials in infrastructure and bridge applications, more in-service long-term performance data is required. Thus FRP structures

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and structures retrofitted with FRP materials should be monitored to ensure their adequate in-service performance and to collect in-service performance data. SHM is not only useful in evaluating the performance of these materials but is also helpful in improving our knowledge to develop rational, cost-effective design and construction procedures. This chapter briefly described some bridge applications using FRP materials in the state of New York and discussed the test methods used to evaluate their in-service performance. Alampalli and Ettouney (2006) reviewed the long-term issues related to structural health of bridge decks. The following summarizes the potential use of SHM in various stages of the FRP materials use. Design and Analysis: FRP decks are still designed mostly by manufacturers using finite element analysis as these are yet to be standardized. Many properties used in the analysis are not openly revealed due to their proprietary nature. At the same time, as noted earlier, not much performance data is available and hence the designs are overconservative. Integrating SHM into these applications can help verify the design assumptions made on various performance characteristics. This data can also lead to development of simplified approaches to verify the designs by owners for quality assurance purposes, new design development, and to ascertain capacity once they are applied in the field (e.g., Aref et al., 2001; Alnahhal and Aref, 2008). The data can then be used for calibration of the standards and specifications that can lead to design procedures that can be easily adopted by owners’ engineers. Planning: At present, very limited data is available on the long-term performance of these bridges (e.g., Alampalli, 2006). Integration and incorporation of SHM can fill this gap by understanding the deterioration rate and cycle of the FRP applications such that maintenance, rehabilitation, and replacement activities can be planned appropriately. Construction: Considering most FRP components are shop fabricated and transported to site, quality control and assurance are required to make sure that they are not damaged enroute or during construction. Since visual methods are not very suitable for inspection of these components, integrated SHM offers utilizing sensors such as fiber optic sensors that can accommodate quality assurance during the transportation and construction process. In-Service Issues: As noted earlier, durability of FRP materials is not yet well documented and these materials are also not as forgiving as conventional materials. Thus, they require proactive maintenance and SHM (either passive or active depending on the decisions required) can assist immensely. One of the big drawbacks faced by owners is the lack of available standardized procedures to make quick decisions when these structures suffer in-service damage due to conditions such as truck impact, snow-plow, vandalism, fire, etc. In such situations, owners have to make a quick decision on what actions, such as closing the lane or entire bridge or not to close, should be taken. Effectiveness of wearing surfaces has been a big issue. There has been little study done in this area (Kalny et al., 2004; Wattanadechachan et al., 2006; Alnahhal et al., 2006; NCHRP, 2015) and integrating SHM with these applications could help develop such procedures.

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Durability: Durability issues have been investigated by several researchers in the literature (Singhvi and Mirmiran, 2002; Harries, 2005; Aref and Alampalli, 2001; ElRagby et al., 2007). There is limited data on FRP civil system performance, especially on fatigue, creep, moisture, temperature, UV, etc. Thus, integrated SHM considering all these issues can help advance the knowledge that can further lead to better and more durable systems.

References AASHTO (Ed.), 2008. Guide Specifications for Design of FRP Pedestrian Bridges. first ed. AASHTO, Washington, DC. AASHTO (Ed.), 2009. LRFD Bridge Design Guide Specifications for GFRP-Reinforced Concrete Bridge Decks and Traffic Railings. first ed. AASHTO, Washington, DC. AASHTO (Ed.), 2012a. Guide Specifications for Design of Bonded FRP Systems for Repair and Strengthening of Concrete Bridge Elements. first ed. AASHTO, Washington, DC. AASHTO (Ed.), 2012b. LRFD Guide Specifications for Design of Concrete-Filled FRP Tubes. first ed. AASHTO, Washington, DC. ACI, 2004. Prestressing Concrete Structures with FRP Tendons (Reapproved 2011). American Concrete Institute, Farmington Hills, MI. ACI, 2007. Report on Fiber-Reinforced Polymer (FRP) Reinforcement for Concrete Structures. American Concrete Institute, Farmington Hills. ACI, 2008a. Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures. American Concrete Institute, Farmington Hills, MI. ACI, 2008b. Specification for Carbon and Glass Fiber-Reinforced Polymer Bar Materials for Concrete Reinforcement. American Concrete Institute, Farmington Hills, MI. ACI, 2010. Guide for Design & Construction of Externally Bonded FRP Systems for Strengthening Unreinforced Masonry Structures. American Concrete Institute, Farmington Hills, MI. ACI, 2012. Guide Test Methods for Fiber-Reinforced Polymers (FRPs) for Reinforcing or Strengthening Concrete Structures. American Concrete Institute, Farmington Hills, MI. ACI, 2014. Specification for Carbon and Glass Fiber-Reinforced Polymer Materials Made by Wet Layup for External Strengthening of Concrete and Masonry Structures. American Concrete Institute, Farmington Hills, MI. ACI, 2015a. Guide for the Design and Construction of Structural Concrete Reinforced with Fiber-Reinforced Polymer Bars. American Concrete Institute, Farmington Hills, MI. ACI, 2015b. Guide to Accelerated Conditioning Protocols for Durability Assessment of Internal and External Fiber-Reinforced Polymer (FRP) Reinforcement. American Concrete Institute, Farmington Hills, MI. Alampalli, S., 2005. Effectiveness of FRP Materials with Alternative Concrete Removal Strategies for Reinforced Concrete Bridge Column Wrapping. Int. J. Mater. Product Technol., Intersci. Publ. 23 (3/4), 338–347. Alampalli, S., 2006. Field Performance of an FRP Slab Bridge. J. Compos. Struct. 72 (4), 494–502. Alampalli, S., Ettouney, M.M., 2006. Long-Term Issues Related to Structural Health of FRP Bridge Decks. J. Bridge Struct. Assess. Des. Construct. 2 (1), 1–11. Alampalli, S., Hag-Elsafi, O., 2013. Load testing of bridge FRP applications. In: Zoghi, M. (Ed.), International Handbook of FRP Composites in Civil Engineering. CRC Press, Boca Raton, FL, pp. 607–622.

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Creep and Fatigue in Polymer Matrix Composites

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Index Note: Page numbers followed by f indicate figures and t indicate tables. A ABAQUS, 466–467, 472–473, 492 Accelerated testing methodology (ATM), 326–331 applicability of, 336–338, 338–339t creep strength, master curve of, 328–329, 329f CSR strength, master curve of, 327, 328f experimental verification for, 331–336 flexural creep strength, 333–335, 334f flexural CSR strength, 333, 333–334f flexural fatigue strength, for arbitrary stress ratio, 336, 337f flexural fatigue strength, for zero stress ratio, 335–336, 335–336f specimen and testing method, 331–332 fatigue strength for zero stress ratio, master curve of, 329–331, 330f fatigue strength prediction, for arbitrary frequency, stress ratio, and temperature, 331, 332f future trends and research, 344–345 procedure of, 326–327, 327f statistical prediction of creep life, 341–344 CFRP strand, creep failure tests of, 343–344, 345t, 346f CFRP strand, CSR strength of, 343, 344–345f, 345t creep compliance of matrix resin, 342–343, 343f creep failure time, formulation of, 341 experiments, 341–342, 342t, 342f theoretical verification of, 339–340, 340t Acoustography, for visualization of fatigue damage, 426 AFRL. See Air Force Research Lab (AFRL) Aging, 65–66, 71 shift factor, 75–76 Airbus A350, polymer matrix composites in, 303 Air Force Research Lab (AFRL), 350

Anisotropic composite lamina, diffusion in, 91–92, 91–92f, 93t Anisotropic composite laminate, diffusion in, 93–94, 93–94f Arrhenius equation, 26, 85 ATM. See Accelerated testing methodology (ATM)

B Basquin model, 442–444 Bending angle fixity, of shape memory polymer composites, 477, 478t, 478f Bending angle recovery, of shape memory polymer composites, 481, 482f experimental results for, 481, 481f Bending fatigue, 412–413 Bennetts Creek Bridge FRP slab bridge during construction, 532f proof load testing, 533f Bentley Creek Bridge bridge deck, structure of, 540 FRP bridge deck during construction, 534f Biaxially loaded specimens, topology optimization in, 417–418, 419–420f Boeing 727, polymer matrix composites in, 303 Boeing 787, polymer matrix composites in, 303 Boltzmann superposition principle, 23, 23f, 168 Bond-breaking process, 354 Bond breaking rate, 354–355 Bragg grating, 410 Bridge deck, 540–545 Bentley Creek Bridge, 534f fatigue life of critical truss members after deck replacement, 545f with FRP rebars, 536f repairs, 540–542, 541f SHM instrumentation for, 542, 543f

560

Bridge deck (Continued) structural health monitoring, reasons for, 540 structure of, 540 testing and analysis of, 542–544, 544f Bridge industry, FRP structures in, 532–537, 532–536f Bridge wrapping (nonbond-critical), 545–549 columns after 7 years in-service, 550f monitoring of, 549, 549f repairs, 547 SHM instrumentation for, 547–548, 548f structural health monitoring, reasons for, 546 structure of, 546–547, 547f Brueller’s model, 48–49 Buckyballs, 203 Bulk modulus, 184, 202 loss, 185f storage, 185f Butterfly strain responses, 142f

C Camera vision techniques, for visualization of fatigue damage, 425–426, 426f Cantilever beam FEA mode, 473f Carbon fiber-reinforced plastics (CFRPs), 253 failure morphology, 253–254 FEA and critical-point stress, 254–256 interface modeling, 262–263 interfacial debonding, 264–265 matrix crack, 264–265 matrix modeling, 261–262 periodic boundary condition, 259–261 RVE approach, 258–259 strand creep failure tests of, 343–344, 345t, 346f CSR strength of, 343, 344–345f, 345t stress–strain curves, 265–266 tensile directional normal stress, 264–265 time and temperature dependence, interface strength, 256–258 Cauchy formula, 173 CDM. See Continuum damage mechanics (CDM) CFRPs. See Carbon fiber-reinforced plastics (CFRPs)

Index

Chemical attack, on polymers, 10 CML. See Curtin-McLean model (CML) Complex Poisson ratio, 177 Complex shape pultruded composite profiles, durability evaluation procedures for, 94–95, 95f Complex Young’s modulus, 192 Composite fatigue life, 354–355 Composite structures for durability, structural health monitoring of bridge deck, 540–545, 541f, 543–545f bridge wrapping (nonbond-critical), 545–549, 547–549f external reinforcement (bond-critical), 549–553, 551–553f FRP structures, in bridge industry, 532–537, 532–536f Compression-compression fatigue, 411–412 Constant Poisson ratio, 171 Constant strain rate (CSR), 326 and creep lifetime curves, relationship between, 294–296, 295f loading condition, 294 rupture curves and creep rupture, relationship between, 283–286 strength of CFRP strand, 343, 344–345f, 345t flexural, 333, 333–334f master curve of, 327, 328f Constitutive models linearized forms for, 129–131 PZT, 125–131 Continuum damage mechanics (CDM), 276–278 Conventional fibers, 157–158 Creep, 8, 8f, 19–21, 19–20f, 162–163, 165, 176–179 deformation, 351 of linear viscoelastic materials, 102 Maxwell model for, 31, 31f of moisture-saturated polymers, 104–106, 105–106f of polymers under moisture absorption, 106–109, 107–108f stages of, 20f strain, 65, 361–363, 398 tests, 71 Voigt or Kelvin model for, 33–34 Creep analysis, of polymer matrix composites

Index

creep-failure time prediction, 225–230, 228–232f future trends of, 243–244 viscoplastic creep modeling, 218–242 finite strain viscoplasticity, 231–242, 234f small strain–framework constitutive analysis, 219–225, 222–225f, 226–227t, 227–228f Creep compliance, of matrix resin, 342–343, 343f Creep failure tests, of CFRP strand, 343–344, 345t, 346f Creep failure time formulation of statistical prediction of, 341 prediction, of polymer composites, 225–230, 228–232f Creep lifetime curves and constant strain rate, relationship between, 294–296, 295f Creep-recovery, 8, 8f, 21 behavior, of four-element model, 39f Maxwell model for, 31f, 32 Voigt or Kelvin model for, 34–35 Creep rupture, 9 and constant strain/stress rate rupture curves, relationship between, 283–286 models, 351, 361, 398 Creep strength flexural, 333–335, 334f master curve of, 328–329, 329f Creep stress loading condition, 293–294 Creep test nonlinear viscoelasticity in, 518–521 viscoelasticity melting in, 508–511 experimental procedure, 512–513, 512–513f Crochet time-dependent yielding model, 273–275 C-scans, for visualization of fatigue damage, 424 CSR. See Constant strain rate (CSR) Curtin-McLean model (CML), 293–297, 295–296f, 296t D DABF. See Damage accumulated before failure (DABF) Damage accumulated before failure (DABF), 450–451

561

Damage function, S–N curve model, 454 Damage validity, in S–N curve model, 454 conditions at R=0, 450–454 boundary conditions and damage in domain 2, 450–451 boundary conditions and domain 1, 450 compatibility conditions between N and σmax in domain 3, 451–454, 453f Darwin, Charles, 418 Degree of freedom (DOF) method, 259 Delamination failure, 352 DIC. See Digital image correlation (DIC) Digital image correlation (DIC) for visualization of fatigue damage, 422 Direct piezoelectric constant, 125–127 DMA. See Dynamic mechanical analyzer (DMA) DOF. See Degree of freedom (DOF) method Doolittle equation, 27 Durability failure, 352 of a material, 269 of SMPCs, 483–489, 484–486f, 489f Dynamic mechanical analyzer (DMA), 75, 467–469, 468f, 477–479, 479f, 483–485, 484f

E Edge replication, of composite specimens, 423 EDT-10 epoxy, sorption isotherms for, 87f Effective bulk modulus, 182f Effective hysteretic polarization, 142f Effective shear creep function, 187f Effective shear storage modulus, 189f Effective stress, 366 Effective time theory (ETT), 77–79 Elastic moduli/modulus, 181–183, 501 Elastic properties of polymers and composites, moisture effect on, 97–102, 98–101f Electrical properties of polymers, 13–14 Energy-based failure criteria for multidirectional polymer matrix composites, 279–282, 280–281f for viscoelastic homogenous materials, 271–273, 273f Engineering plastics, 5

562

Environmental stress cracking (ESC), 11 Epoxy-based shape memory polymers (EPSMPs), 485–487, 486f Epoxy resins, moisture sorption of, 87f EPSMPs. See Epoxy-based shape memory polymers (EPSMPs) Equivalent static strength, 443 ESC. See Environmental stress cracking (ESC) ETT. See Effective time theory (ETT) Everett Road, New York columns and pier-cap wrapped with CFRP, 536f External reinforcement (bond-critical), 549–553 monitoring/testing of, 552, 553f repairs, 550–551, 551f, 553f SHM instrumentation, 551–552, 552f structural health monitoring, reasons for, 550 structure of, 550, 551f Extrinsic damages, in polymer composite laminates multidirectional laminates, 305–306, 306f prediction of TDD initiation and evolution, criterion for, 316–317 time-dependent evolution of modes of, 310–313, 312f

F Failure caused by damage (FCD), 450–451 Failure modes, 349–350 Fatigue, 9 data characterisation, 442–445 of FRP composites, 349–350 life, 363 life models, 350 load cycles, 379–380 methodology, 366–385 prediction models, 350 stress, 365–366 Fatigue damage, 441–442, 442f mechanisms, quantification of, 428 in structural composites, 418–422, 421f visualization, inspection techniques for, 422–426, 423f, 425–427f Fatigue strength flexural. (see Flexural fatigue strength)

Index

prediction, for accelerated frequency, stress ratio, and temperature, 331, 332f for zero stress ratio, master curve of, 329–331, 330f Fatigue testing, of polymer matrix composites advanced instrumentation methods, 418–426 fatigue damage in structural composites, 418–422, 421f fatigue damage visualization, inspection techniques for, 422–426, 423f, 425–427f boundary conditions and specimen geometry, effect of, 415–418 stress state near tabbed regions in uniaxial fatigue loading, 415–417, 416f topology optimization in biaxially loaded specimens, 417–418, 419–420f future trends and challenges, 427–428 fatigue damage mechanisms, quantification of, 428 fatigue performance of new composite materials, faster assessment of, 428 methods, 403–415, 405f bending fatigue, 412–413 compression-compression fatigue, 411–412 multiaxial fatigue, 414–415 shear dominated fatigue, 413–414, 415f tension-compression fatigue, 411–412 tension-tension fatigue, 407–410, 408–409f, 411f sources of further information and advice, 428 FCD. See Failure caused by damage (FCD) FEA. See Finite element analysis (FEA) FE code Abaqus, 376–378 progressive creep strain and rupture, 388–390 FE combined, 390–394 FE creep strain predictions, 390 progressive fatigue implementation, 376–378 FEM. See Finite element models (FEM) Fiber- and particle-unit-cell models, 131–142 experimental validation, 133–137 formulation of, 132–133 parametric studies, 138–142

Index

Fiber composites microdamage and stiffness degradation in, 502–508, 504–507f viscoelasticity of, 513–518, 514–518f Fiber-reinforced composites, moisture absorption in, 89–96 microstructural approach, 90–91, 90f Fiber-reinforced polymers (FRP), 269 structures, and structural health monitoring, 538–539 structures, in bridge industry, 532–537, 532–536f Fick’s diffusion model, for polymer materials, 85, 86t Fillers, 159 Finite element analysis (FEA), 258–259 validation of, 477–481 bending angle fixity, 477, 478t, 478f Finite element implementation, 366–385 of creep rupture and strain methodology, 385–394 fatigue prediction, 378–385 layup [0/45/90/-45]2S, 379–380 layup [30/60/90/-60/-30]2S, 383–385 layup [60/0/-60]3S, 380–383 material characterization, 367–372, 386–387 activation energy and activation volume, delamination layer matrix, 369–372 activation energy and activation volume, in-situ matrix, 368–369 constituent property extraction, 386–387 elastic properties calibration, 367–368 static testing, 386 modeling strategy, 373–376 in-plane mesh convergence, 373–375 open hole tension coupon, 373 through-thickness mesh convergence, 375–376 progressive creep strain and rupture, 388–390 progressive fatigue implementation, 376–378 Finite element models (FEM) stress state near tabbed regions in uniaxial fatigue loading, 416–417 Finite element simulation for modeling of SMPC viscoelastic properties, 472–473, 473–474f, 474t results, 473–477, 474–476f

563

Finite linear viscoelasticity, 249 Finite strain viscoplasticity, 231–242, 234f First ply failure (FPF), 303–304, 310–311 Flax fibers in lignin matrix composites, 506, 507f Flax/PLA composites, 506 high stress-related time shift factors for, 522, 523f Flax/starch composites elastic modulus reduction in, due to presumed microdamage development with increasing stress, 506–507, 507f mechanical behavior of, 498f VP-strains in, 518, 518f Flexural creep strength, 333–335, 334f Flexural CSR length, 333, 333–334f Flexural fatigue strength for arbitrary stress ratio, 336, 337f for zero stress ratio, 335–336, 335–336f Form factor, 160–161 Four-element model, 38–40 creep-recovery behavior of, 39f parameters determination, 41f FPF. See First ply failure (FPF) Fracture mechanics, of viscoelastic materials, 275–276 Free volume theory, 27 Frequency effect, 141f FRP. See Fiber-reinforced polymers (FRP) G Gamma function, 173 Generalized Kelvin model, 62 Generalized Kelvin-Voigt model, 41–43, 43–44f, 166 Generalized Maxwell model, 40–41, 42f, 467, 467f GF/EP composites creep loading and viscoelastic recovery for, subject to axial tensile loading, 506, 513f GF/EP laminate damaged layer of shear modulus reduction versus shear stress, 503, 504f UD layer of elastic shear strain vs shear stress in, 523, 524f

564

GF/EP laminate (Continued) quasistatic shear stress-shear strain curves, 524, 524f shear response of, high stress-related time shift factors for, 522, 523f viscoelastic shear compliance of, after subtracting elastic part, 522, 522f viscoelastic shear strain development in layers of, subject to shear stress levels, 514, 515f GF/VE NCF composite elastic properties degradation with applied stress in, 503, 504f GLOB-LOC model, 503 Gough-Joule effect, 249–250 Green, Rivlin and Spencer model, 45–47, 46f

H Hemp/lignin composite normalized elastic modulus versus strain for, 508, 508f simulated and experimental loadingunloading curves for, 525, 525f stress-dependent functions in Schapery’s model for, 524–525, 525f Henry’s law, 85–87 Heterogeneous materials characteristics of behavior, 162–165 mechanical behavior of, 157 Heterogeneous systems, mechanical properties of, 159–160 Hetron 970/35 vinylester, sorption isotherms for, 87f HFGMCs. See High-fidelity generalized method of cells (HFGMCs) High-cycle fatigue, 406 High-fidelity generalized method of cells (HFGMCs), 249–250 High-resolution 3D X-ray micro-tomography, for visualization of fatigue damage, 424 Hollow fibers, 203–205 Hollow fillers, 203 Hook’s law, 173–174 Hybrid piezocomposite model, 142–150 numerical implementation, 146–150 unit-cell model formulation, 143–146

Index

Hysteresis heating, 354–355 Hysteretic responses, 122–123 I ILC Dover and Folded Structures Company, 489–490, 490f Incremental polarization, 130–131 In-plane mesh convergence, 373–375 In-situ micro-CT, for visualization of fatigue damage, 422 Interface matrix, 366 Intrinsic damages, in polymer composite laminates in [0] lamina, 304–305, 305f prediction of TDD initiation and evolution, criterion for, 313–316, 315f time-dependent evolution of modes of, 307–310, 308–309f Inverse ratio, 199 Irreversible polarization, 127 Isochronous diagrams, 17–18, 18–19f Isodamage points, 454–455 Isometric diagrams, 17–18, 18–19f K Kim and Zhang’s S–N curve model, 445–450, 447f, 449f Kinetic rate theory for time-dependent failure, 270–271 Kohout model, 444 KTF-based durability modeling of FRP composites, 394–397 KTF-based methodology, 352–366, 382–383 application, composite laminae, 364–366 creep rupture of polymers, 360–361 fatigue failure of polymers, 352–360 L LAMFLU, 279–281 LCD. See Linear cumulative damage (LCD) Leaderman’s model, 48 Lead zirconate titanate (PZT), 121–122 constitutive models, 125–131 electromechanical material properties for, 138t Leaku lamb wave technique, for visualization of fatigue damage, 426

Index

Lifetime prediction, of polymers and polymer composite structures micromechanical model, 292–297 constant strain rate loading condition, 294 creep lifetime curves and constant strain rate, relationship between, 294–296, 295f creep stress loading condition, 293–294 two-step creep loading, 296–297, 296t, 296f multiple step creep loading, 287–291, 288–292f, 290t, 295t polymer-based matrix composites, timedependent failure criteria for, 278–282 long-term failure, accelerated experimental methodologies for, 282–287 multidirectional polymer matrix composites, energy-based failure criterion for, 279–282, 280–281f orthotropic state failure theories, of timedependent creep rupture, 278–279 viscoelastic homogeneous materials, timedependent failure criteria for, 270–278 continuum damage mechanics, 276–278 Crochet time-dependent yielding model, 273–275 energy-based failure criteria, 271–273, 273f fracture mechanics, 275–276 kinetic rate theory for time-dependent failure, 270–271 Linear cumulative damage (LCD), 326, 328–329, 329f, 335–336, 340 Linearity, 22–25, 22f limits of, 43–45 Linear-nonlinear viscoelastic threshold, 17–18 Linear spring, 28–29, 29f Linear viscoelastic materials, creep of, 102 Linear viscoelastic models, 28–43, 128–129 four-element model, 38–40, 39f, 41f generalized Maxwell model, 40–41, 42f generalized Voigt or Kelvin model, 41–43, 43–44f linear spring, 28–29, 29f linear viscous dashpot, 29–30, 30f

565

Maxwell model, 30–32 creep, 31, 31f creep-recovery, 31f, 32 strain relaxation, 32, 32f three-element solid, 37–38, 37f Voigt or Kelvin model, 33–37, 33f creep, 33–34 creep-recovery, 34–35 stress relaxation, 35–37, 36f Linear viscous dashpot, 29–30, 30f Load history, 357 Lockheed Corporation, 279 Longitudinal creep, 165 Longitudinal-transversal Poisson ratio, 201 Long-term behavior, 16, 17f Long-term creep, 79–80 Long-term failure of polymer-based matrix composites, accelerated experimental methodologies for, 282–287 creep rupture and constant strain/stress rate rupture curves, relationship between, 283–286 semiempirical extrapolation, 286–287 stepped isostress method, 286 time-temperature superposition principle, 283 Low-cycle fatigue, 406

M MAPTIS. See Materials and Processes Technical Information System (MAPTIS) Material loading history, 16 Material model, 498–500 Materials and Processes Technical Information System (MAPTIS), 482–483 Maximum strain (MS) criterion, 272 Maximum work stress (MWS) criterion, 272 Maxwell model, 30–32 for creep, 31, 31f for creep-recovery, 31f, 32 for strain relaxation, 32, 32f MDF. See Multidirectional fabric (MDF) laminates MDT. See Multidirectional tape (MDT) laminates Microballoons, 203

566

Microcrack accumulation, 352–354 Microdamage effect, on stiffness, 501–508 elastic modulus, 501 stiffness degradation in fiber composites, 502–508, 504–507f stiffness reduction measurements, 501–502, 502f Micromechanical (fiber/matrix level) failure, of FRP composites, 349–350 Micromechanical model, for lifetime prediction of polymers and polymer composite structures, 292–297 constant strain rate loading condition, 294 creep lifetime curves and constant strain rate, relationship between, 294–296, 295f creep stress loading condition, 293–294 two-step creep loading, 296–297, 296t, 296f Mitsubishi Chemical Corporation, 550–551 Modified Reiner–Weissenberg (MR–W) criterion, 272 Moire interferometry, for visualization of fatigue damage, 426 Moisture absorption creep of polymers under, 106–109, 107–108f in fiber reinforced composites, 89–96 complex shape pultruded composite profiles, durability evaluation procedures for, 94–95, 95f diffusion in anisotropic composite lamina, 91–92, 91–92f, 93t diffusion in anisotropic composite laminate, 93–94, 93–94f microstructural approach, 90–91, 90f multiphase multilayer system, 95–96, 96f in polymers, 83–89 nonstationary humidity, moisture sorption in, 88–89 stationary humid conditions, moisture sorption in, 83–88, 87f uniaxial load, effect of, 88, 89f Moisture effect, on elastic and viscoelastic properties of reinforced plastics elastic properties and strength, 97–102, 98–101f

Index

fiber reinforced composites, moisture absorption in, 89–96 complex shape pultruded composite profiles, durability evaluation procedures for, 94–95, 95f diffusion in anisotropic composite lamina, 91–92, 91–92f, 93t diffusion in anisotropic composite laminate, 93–94, 93–94f microstructural approach, 90–91, 90f multiphase multilayer system, 95–96, 96f moisture absorption in polymers, 83–89 nonstationary humidity, moisture sorption in, 88–89 stationary humid conditions, moisture sorption in, 83–88, 87f uniaxial load, effect of, 88, 89f nanocomposites, 112–117 elastic properties of polymer NC and nanomodified FRP, 113–115, 114f polymer nanocomposites, moisture sorption by, 112–113, 112–113f viscoelastic properties of NC and nanomodified FRP, 115–117, 115–117f swelling, 96–97, 97f viscoelastic behavior, 102–112 creep of linear viscoelastic materials, 102 creep of polymers under moisture absorption, 106–109, 107–108f superposition principles, 103–104, 103–104f time-moisture superposition principle, 104–106, 105–106f viscoelastic stress–strain analysis during moisture uptake under tensile creep, 109–112, 110–111f Moisture sorption in nonstationary humidity, 88–89 by polymer nanocomposites, 112–113, 112–113f in stationary humid conditions, 83–88, 87f Momentary master curves, 66, 69, 76–79 MS. See Maximum strain (MS) criterion Multiaxial fatigue, 414–415 Multidirectional fabric (MDF) laminates, 303

Index

Multidirectional polymer composite laminates, time-dependent damage evolution in characterization methods, 306–307 modes of, 304–306, 305–306f extrinsic damage modes, 310–313, 312f intrinsic damage modes, 307–310, 308–309f prediction of TDD initiation and evolution, criterion for, 313–317 extrinsic damage modes, 316–317 intrinsic damage modes, 313–316, 315f Multidirectional polymer matrix composites, energy-based failure criterion for, 279–282, 280–281f Multidirectional tape (MDT) laminates, 303 Multiphase multilayer system, 95–96, 96f Multiple integral representations, of polymer’s nonlinear viscoelastic behavior, 45–47 Green, Rivlin and Spencer model, 45–47, 46f Pipkin and Rogers model (nonlinear superposition theory), 47 Multiple step creep loading, 287–291, 288–292f, 290t, 295t MWS. See Maximum work stress (MWS) criterion

N Nanocomposites, moisture effect on, 112–117 elastic properties of polymer NC and nanomodified FRP, 113–115, 114f moisture sorption, 112–113, 112–113f viscoelastic properties, 115–117, 115–117f Nanomodified FRP, moisture effect on elastic properties, 113–115, 114f viscoelastic properties, 115–117, 115–117f Nanoparticles, 159–162, 209–210 Nanotubes, hollow fibers and, 203–205 NASA. See National Aeronautics and Space Administration (NASA) National Aeronautics and Space Administration (NASA) Materials and Processes Technical Information System, 482–483

567

NCF GF/VE laminate axial VP strain in, subject to axial tensile loading, 513, 514f stress dependence of axial VP-strain in, 514, 515f NDE. See Nondestructive evaluation (NDE) New York State Department of Transportation, 540–541, 546 Nondestructive evaluation (NDE), 545–546 Nonlinear superposition theory, 47 Nonlinear viscoelastic behavior of polymers, 43–50 limits of linearity, 43–45 multiple integral representations, 45–47 Green, Rivlin and Spencer model, 45–47, 46f Pipkin and Rogers model (nonlinear superposition theory), 47 single integral representations, 48–50 Brueller’s model, 48–49 Leaderman’s model, 48 parameters determination, 50 Rabotnov’s model, 48 Schapery’s constitutive equation, 49 Nonlinear viscoelasticity, 518–525 behavior, examples of, 521–525, 521–525f in creep and strain recovery tests, 518–521 Nonstationary humidity, moisture sorption in, 88–89 Normalized rupture free energy, 272 Norpol-440 polyester, sorption isotherms for, 87f Number of cycles to failure, 359–360 O One creep test, time dependence of VP-strain in, 526–528 Open hole tension coupon modeling, 373 Optical fiber sensing for visualization of fatigue damage, 422 Optical microscopy, for visualization of fatigue damage, 423, 423f Oxidation, 11 P Paper/phenol-formaldehyde composite, viscoplastic strains in, 516–517, 516f development of, 517–518, 517f

568

Particle reinforced composites (PRCs), 123–124 Phenomenological models, 350 Physical aging, 71 Physical attack, on polymers, 10 Physical properties of polymers, 12–14 electrical properties, 13–14 thermal properties, 12–13 Physics-based multiscale models, 350 Piezoelectric composites, 121–122 electromechanical properties, 122 micromechanical models for, 151–152 Pipkin and Rogers model (Nonlinear superposition theory), 47 Plain weave glass/epoxy composites loaded in bending fatigue, 423, 423f Plain woven glass/epoxy laminate, fatigue damage in, 421–422, 421f, 424, 425f Plastic type, effect on polymers, 7–8, 8f Poisson Ratio, 176–179 Polarization switching model, 125–128 Polarization switching response, 122, 127 Polarized piezoelectric ceramics, 122 Polymer-based matrix composites, timedependent failure criteria for, 278–282 long-term failure, accelerated experimental methodologies for, 282–287, 282f creep rupture and constant strain/stress rate rupture curves, relationship between, 283–286 semiempirical extrapolation, 286–287 stepped isostress method, 286 time-temperature superposition principle, 283 multidirectional polymer matrix composites, energy-based failure criterion for, 279–282, 280–281f Polymer elastic modulus degradation, 363 Polymeric materials, 249 Polymer-matrix composites, 5–6 Polymer matrix composites, fatigue testing of advanced instrumentation methods, 418–426 fatigue damage in structural composites, 418–422, 421f fatigue damage visualization, inspection techniques for, 422–426, 423f, 425–427f

Index

boundary conditions and specimen geometry, effect of, 415–418 stress state near tabbed regions in uniaxial fatigue loading, 415–417, 416f topology optimization in biaxially loaded specimens, 417–418, 419–420f future trends and challenges, 427–428 fatigue damage mechanisms, quantification of, 428 fatigue performance of new composite materials, faster assessment of, 428 methods, 403–415, 405f bending fatigue, 412–413 compression-compression fatigue, 411–412 multiaxial fatigue, 414–415 shear dominated fatigue, 413–414, 415f tension-compression fatigue, 411–412 tension-tension fatigue, 407–410, 408–409f, 411f sources of further information and advice, 428 Polymer nanocomposites, moisture effect on elastic properties, 113–115, 114f moisture sorption, 112–113, 112–113f Polymers, types of, 4–6 Power law, 24 Power law model, 72–74 PRCs. See Particle reinforced composites (PRCs) Progressive fatigue implementation, 376–378 Prony series, 24 Proof load testing, in Bennetts Creek Bridge, 533f Properties degradation environmental stress cracking, 11 oxidation, 11 physical and chemical attack, 10 resistance to wear and friction, 11–12 weathering, 10–11 PZT. See Lead zirconate titanate (PZT) R Rabotnov algebra, 169 Rabotnov’s algebra, 171 Rabotnov’s model, 48

Index

Radiography, for visualization of fatigue damage, 424 Rate theory of fracture, 270–271 RCF. See Regenerated cellulosic fiber (RCF) Recursive time-integration algorithm, 129 Regenerated cellulosic fiber (RCF) bundles, mechanical behavior of, 498f Reiner–Weissenberg (R–W) Criterion, 272 Reinforcing fibers, 159 Remaining fatigue life, 456–460, 457f, 458t n value, determination of, 459f, 460 prediction of, 459f, 460 Replark® laminate system, 550–551 Representative volume element (RVE), 254–256, 372 Residual polarization, 127 Resistance to wear and friction, 11–12 RVE. See Representative volume element (RVE) S Scaling effect, on creep diagram shape, 17f Scanning electron microscopy (SEM) of fatigue damage in structural composites, 422 for visualization of fatigue damage, 423 Schapery’s constitutive equation, 49 Schroon River Bridge FRP bridge deck after construction, 534f SDM. See Synergistic damage mechanics (SDM) model SEM. See Scanning electron microscopy (SEM) Shape memory alloys (SMAs), 465–466 Shape memory polymer composites (SMPCs), 465–467 finite element analysis technique, validation of, 477–481 bending angle fixity, 477, 478t, 478f bending angle recovery, 481, 482f bending angle recovery, experimental results for, 481, 481f experimentation, 477–480, 479–480f finite element simulation procedure results, 473–477, 474–476f for space engineering applications, competence of, 482–492, 490–491f durability, 483–489, 484–486f, 489f

569

space environment, 482–483 viscoelastic behavior of, modeling, 467–471, 467–468f, 470–471t, 470f finite element simulation procedure, 472–473, 473–474f, 474t Shape memory polymers (SMPs), 466 -based close-celled synthetic foam, durability of, 483–485 beam configuration, 473, 474f bending angle recovery of, 481, 481–482f cyanate-based vacuum outgassing and UV radiation exposure tests on, 483–485, 485f epoxy-based, 485–487, 486f glycerol-based aliphatic, 465–466 materials, storage modulus of, 489 proprieties of, after shape memory cycles, 487, 489f space engineering applications, 490 styrene-based, 465–466, 477–479, 485–487, 486f thermomechanical cycle, finite element simulation of, 472–473 beam bending angle, 476f, 477 bending angle fixity, 477, 478t, 478f fixity ratio of bending angle, 477, 478t 3D woven fabric-reinforced, 487–489 VF62 and VFE2-100 resins, 487 viscoelastic behavior of, modeling, 467–471, 467–468f, 471t Shape parameter, 363 Shear dominated fatigue, 413–414, 415f SHM. See Structural health monitoring (SHM) Short-term behavior, 15–16 Short-term data correlation, 61–63 Simple MATLAB algorithm, 63 Single integral representations, of polymer’s nonlinear viscoelastic behavior, 48–50 Brueller’s model, 48–49 Leaderman’s model, 48 Rabotnov’s model, 48 Schapery’s constitutive equation, 49 SLERA. See Strength-life equal rank assumption (SLERA) SLS. See Standard linear solid (SLS) model Small strain–framework constitutive analysis, 219–225, 222–225f, 226–227t, 227–228f

570

SMAs. See Shape memory alloys (SMAs) SMC composite damaged specimen edge, micrograph of, 505, 505f elastic modulus reduction in, after loading to high strain, 505, 506f SMPCs. See Shape memory polymer composites (SMPCs) SMPs. See Shape memory polymers (SMPs) Snapshot condition, 66 S–N curve model, 325, 330–331, 335–336, 335f, 442–445 damage function and validity, 454 damage validity conditions at R=0, 450–454 boundary conditions and damage in domain 2, 450–451 boundary conditions and domain 1, 450 compatibility conditions between N and σmax in domain 3, 451–454, 453f fatigue data characterisation, 442–445 isodamage points (B), 454–455 n value, numerical determination of, 455–456, 456f proposed by Kim and Zhang, 445–450, 447f, 449f remaining fatigue life, 456–460, 457f, 458t n value, determination of, 457–459 prediction of, 459f, 460 stress ratio effect, 445 Sonication technology, 162 Space engineering applications, competence of SMPCs for, 482–492, 490–491f durability, 483–489, 484–486f, 489f space environment, 482–483 SPATE (Stress Patterns Analysis by the measurement of Thermal Emissions) for visualization of fatigue loss, 426 SSMPs. See Styrene-based shape memory polymers (SSMPs) Standard linear solid (SLS) model, 61 schematic of, 61–62 Stationary humid conditions, moisture sorption in, 83–88, 87f Statistical prediction of creep life, 341–344 CFRP strand, creep failure tests of, 343–344, 345t, 346f CFRP strand, CSR strength of, 343, 344–345f, 345t

Index

creep compliance of matrix resin, 342–343, 343f creep failure time, formulation of, 341 Stepped isostress method, 286 Stiffness degradation curve, 380–382 Stiffness, microdamage effect on, 501–508 elastic modulus, 501 stiffness degradation in fiber composites, 502–508, 504–507f stiffness reduction measurements, 501–502, 502f Strain rate, effect on polymers, 7, 7f Strain recovery test, nonlinear viscoelasticity in, 518–521 Strength-life equal rank assumption (SLERA), 365–366 Strength of polymers and composites, moisture effect on, 97–102, 98–101f Stress ratio effect, 445 Stress relaxation, 9, 9f, 21, 22f linear viscous dashpot, 30f Maxwell model for, 32, 32f Voigt or Kelvin model for, 35–37, 36f Structural composites, fatigue damage in, 418–422, 421f Structural health monitoring (SHM), 537, 538f of composite structures for durability bridge deck, 540–545, 541f, 543–545f bridge wrapping (nonbond-critical), 545–549, 547–549f external reinforcement (bond-critical), 549–553, 551–553f FRP structures, in bridge industry, 532–537, 532–536f FRP structures and, 538–539 Styrene-based shape memory polymers (SSMPs), 465–466, 477–479, 485–487, 486f Superposition principles Boltzmann, 23, 23f, 168 time-moisture, 104–106, 105–106f time-stress, 26–27, 269–270 time-temperature-stress, 27–28 time-temperature, 25–26, 63–71, 269–270, 283, 325, 327, 467–469, 477–479, 492 viscoelastic behavior of polymers and composites, 103–104, 103–104f

Index

Swelling, 96–97, 97f Synergistic damage mechanics (SDM) model, 316

T Temperature compensation, 80 Temperature effect on polymers, 7, 7f Temperature shift factors, 68, 70f Tension/compression creep function, 191f Tension-compression fatigue, 411–412 Tension-tension fatigue, 407–410, 408–409f, 411f Thermal properties of polymers, 12–13 Thermoelastic inversion effect, 249–250 Thermography, for visualization of fatigue damage, 425 Thermoplastics, 4–5 Thermosets, 5 Thermoviscoelastic polymer matrix composites, 249–250 Three-element solid model, 37–38, 37f Through-thickness mesh convergence, 375–376 Time-age superposition, 71–77 Time-dependent damage (TDD) evolution, in unidirectional and multidirectional polymer composite laminates characterization methods, 306–307 modes of, 304–306, 305–306f extrinsic damage modes, 310–313, 312f intrinsic damage modes, 307–310, 308f prediction of TDD initiation and evolution, criterion for, 313–317 extrinsic damage modes, 316–317 intrinsic damage modes, 313–316, 315f Time-dependent failure criteria for polymer-based matrix composites, 278–282 long-term failure, accelerated experimental methodologies for, 282–287 multidirectional polymer matrix composites, energy-based failure criterion for, 279–282, 280–281f orthotropic state failure theories, of timedependent creep rupture, 278–279 for viscoelastic homogeneous materials, 270–278

571

continuum damage mechanics, 276–278 Crochet time-dependent yielding model, 273–275 energy-based failure criteria, 271–273, 273f failure mechanics, 275–276 kinetic rate theory, 270–271 Time-dependent reversible polarization, 125–127 Time-moisture superposition principle (TMSP) creep of moisture-saturated polymers, 104–106, 105–106f Time-stress superposition principle (TSSP), 26–27, 269–270 Time-temperature shift factors (TTSFs), 326–327 Time-temperature-stress superposition principle (TTSSP), 27–28 Time-temperature superposition principle (TTSP), 25–26, 63–71, 269–270, 325, 327, 467–469, 477–479, 492 long-term failure of polymer-based matrix composites, 283 Topology optimization, in biaxially loaded specimens, 417–418, 419–420f Toughness, 10 Transversal Poisson ratio, 202 Transversal shear modulus, 202 Transversal Young’s modulus, 202 Transverse crack, 253 Troups Creek Bridge FRP slab bridge after construction, 533f Tsai–Hill criterion, 278, 280 TSSP. See Time-stress superposition principle (TSSP) TTSFs. See Time-temperature shift factors (TTSFs) TTSP. See Time-temperature superposition principle (TTSP) TTSSP. See Time-temperature-stress superposition principle (TTSSP) Two-step creep loading, 296–297, 296t, 296f

U ULF. See Ultimate laminate failure (ULF) Ultimate laminate failure (ULF), 303–304

572

Ultrasonics, for visualization of fatigue damage, 423–424 Uniaxial fatigue loading, stress state near tabbed regions in, 415–417, 416f Uniaxial load on moisture absorption, effect of, 88, 89f Unidirectional composites failure mechanisms of, subject to various loading types, 340t Unidirectional polymer composite laminates, time-dependent damage evolution in characterization methods, 306–307 modes of, 304–306, 305–306f extrinsic damage modes, 310–313, 312f intrinsic damage modes, 307–310, 308–309f prediction of TDD initiation and evolution, criterion for, 313–317 extrinsic damage modes, 316–317 intrinsic damage modes, 313–316, 315f

V VF62 shape memory polymer resins, 487 VFE2-100 shape memory polymer resins, 487 Vibrothermography, for visualization of fatigue damage, 426 Viscoelastic behavior of polymers and composites, 6–10, 6f, 14–15 creep and creep recovery, 8, 8f creep of polymers under moisture absorption, 106–109, 107–108f creep rupture, 9 fatigue, 9 moisture effect on, 102–112 creep of linear viscoelastic materials, 102 plastic type, effect of, 7–8, 8f strain rate, effect of, 7, 7f stress relaxation, 9, 9f superposition principles, 103–104, 103–104f temperature effect, 7, 7f time-moisture superposition principle, 104–106, 105–106f toughness, 10 viscoelastic stress–strain analysis during moisture uptake under tensile creep, 109–112, 110–111f

Index

Viscoelastic constitutive modeling of creep and stress relaxation applications to different materials, 51–56, 51–57f creep, 19–21, 19–20f creep-recovery, 21 isochronous and isometric diagrams, 17–18, 18–19f linearity, 22–25, 22f linear viscoelastic models. (see Linear viscoelastic models) long-term behavior, 16, 17f nonlinear viscoelastic behavior of polymers. (see Nonlinear viscoelastic behavior of polymers) physical properties, 12–14 electrical properties, 13–14 thermal properties, 12–13 polymers, types of, 4–6 polymer-matrix composites, 5–6 thermoplastics, 4–5 thermosets, 5 properties degradation environmental stress cracking, 11 oxidation, 11 physical and chemical attack, 10 resistance to wear and friction, 11–12 weathering, 10–11 short-term behavior, 15–16 stress relaxation, 21, 22f time-stress superposition principle, 26–27 time-temperature-stress superposition principle, 27–28 time-temperature superposition principle, 25–26 viscoelastic behavior of polymers. (see Viscoelastic behavior of polymers) Viscoelastic homogeneous materials, timedependent failure criteria for, 270–278 continuum damage mechanics, 276–278 Crochet time-dependent yielding model, 273–275 energy-based failure criteria, 271–273, 273f fracture mechanics, 275–276 kinetic rate theory for time-dependent failure, 270–271

Index

Viscoelasticity, 508–518 elongated inclusions, 195–196 factors contributing to, 167–176 fiber composites, experimental results for, 513–518, 514–518f of fibrous composites, 201–203 of foams and nanoporous materials, 205–208 mathematical methods of, 167–176 of matrix, 165–167 melting, in creep test, 508–511 experimental procedure, 512–513, 512–513f nonlinear, 518–525 behavior, examples of, 521–525, 521–525f in creep and strain recovery tests, 518–521 oblate and platelet inclusions, 196–201 of particulate composites, 179–201 spherical inclusions, 179–194 Viscoelastic properties of shear, 178 of polymers and composites, moisture effect on, 115–117, 115–117f of tension/compression, 178 Viscoelastic stress–strain analysis during moisture uptake under tensile creep, 109–112, 110–111f

573

Viscoplastic creep modeling, for polymer composites, 218–242 finite strain viscoplasticity, 231–242, 234f small strain–framework constitutive analysis, 219–225, 222–225f, 226–227t, 227–228f Visual inspection, of fatigue damage, 422–423 Voigt or Kelvin model, 33–37, 33f for creep, 33–34 for creep-recovery, 34–35 generalized, 40–43, 42–44f for stress relaxation, 35–37, 36f Volterra integral operator, 168–169 W Weathering, polymer-matrix composites, 10–11 Weibull model, 443–444 Williams-Landel-Ferry (WLF) equation, 466–467, 469, 470t Wynantskill Creek, City of South Troy, New York, 550 Y Young’s modulus, 190

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