Creep and Fatigue in Polymer Matrix Composites (Woodhead Publishing in Materials)

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Creep and fatigue in polymer matrix composites

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Related titles: Fatigue life prediction of composites and composite structures (ISBN 978-1-84569-525-5) This important book will address the highly topical subject of fatigue life prediction of composites and composite structures. Fatigue is the progressive and localised structural damage that occurs when a material is subjected to cyclic loading. The use of composites is growing in structural applications and they are replacing traditional materials, primarily metals. Many of the composites being used have only recently been developed and there are uncertainties about the long-term performance of these composites and how they will perform under cyclic fatigue loadings. The book will provide a comprehensive review of fatigue damage and fatigue life modelling. Failure analysis and fractography of composites (ISBN 978-1-84569-217-9) The growing use of polymer composites is leading to an increasing demand for fractographic expertise. Fractography is the study of fracture surface morphology and is an essential tool for advancing the understanding in composite structural behaviour. Failure analysis and fractography of polymer composites gives an insight into damage mechanisms and underpins the development of physically based failure criteria. It provides a vital link between predictive models and experimental observations. Finally, it is essential for post-mortem analysis of failed or crashed structures, the findings of which are used to optimise future designs. This authoritative work provides both an overview of the field and a reference text for engineers. Delamination behaviour of composites (ISBN 978-1-84569-244-5) Delamination is a phenomenon that is of critical importance to the composite industry. It involves a breakdown in the bond between the reinforcement and the matrix material of the composite. With growing use of composites in aerospace and other sectors, understanding delamination is essential for preventing catastrophic failures. This important book reviews the phenomenon of delamination in composites. Part I focuses on delamination as a mode of failure. Part II covers testing of delamination resistance, while Part III analyses detection and characterisation. Further parts cover analysis of delamination behaviour from tests, modelling delamination, analysis of structural performance under delamination and preventing and mitigation of delamination. Details of these and other Woodhead Publishing materials books can be obtained by: • visiting our web site at www.woodheadpublishing.com • contacting Customer Services (e-mail: [email protected]; fax: +44 (0) 1223 893694; tel.: +44 (0) 1223 891358 ext. 130; address: Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington, Cambridge CB21 6AH, UK) If you would like to receive information on forthcoming titles, please send your address details to: Francis Dodds (address, tel. and fax as above; e-mail: francis.dodds@ woodheadpublishing.com). Please confirm which subject areas you are interested in.

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Creep and fatigue in polymer matrix composites

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Edited by Rui Miranda Guedes

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Published by Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington, Cambridge CB21 6AH, UK www.woodheadpublishing.com Woodhead Publishing, 525 South 4th Street #241, Philadelphia, PA 19147, USA Woodhead Publishing India Private Limited, G-2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi 110002, India www.woodheadpublishingindia.com First published 2011, Woodhead Publishing Limited © Woodhead Publishing Limited, 2011 The authors have asserted their moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials. Neither the authors nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. ISBN 978-1-84569-656-6 (print) ISBN 978-0-85709-043-0 (online) The publisher’s policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elemental chlorine-free practices. Furthermore, the publisher ensures that the text paper and cover board used have met acceptable environmental accreditation standards. Typeset by RefineCatch Limited, Bungay, Suffolk, UK Printed by TJI Digital, Padstow, Cornwall, UK

iv © Woodhead Publishing Limited, 2011

Contents

Contributor contact details Part I Viscoelastic and viscoplastic modeling 1

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

xiii 1

3

G. C. Papanicolaou, University of Patras, Greece and S. P. Zaoutsos, Technological Educational Institute of Larissa, Greece

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Introduction Creep Linearity The time–temperature superposition principle (TTSP) The time–stress superposition principle (TSSP) The time–temperature–stress superposition principle (TTSSP) Linear viscoelastic models Nonlinear viscoelastic behavior References Time–temperature–age superposition principle for predicting long-term response of linear viscoelastic materials

3 4 7 11 13 13 14 29 45

48

E. J. Barbero, West Virginia University, USA

2.1 2.2 2.3 2.4 2.5 2.6

Correlation of short-term data Time–temperature superposition Time–age superposition Effective time theory Summary Temperature compensation

48 50 58 64 66 67 v

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vi

Contents

2.7 2.8

Conclusions References

68 68

3

Time-dependent behavior of active/intelligent polymer matrix composites incorporating piezoceramic fibers

70

K.-A. Li and A. Muliana, Texas A&M University, USA

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

Introduction Linearized time-dependent model for materials with electromechanical coupling Simplified micromechanical model of homogenized active polymer matrix composites (PMCs) FE models of representative volume elements (RVEs) of the active PMCs Effective electromechanical and piezoelectric properties Applications of active PMCs as actuators Conclusions Acknowledgement References

70

83 87 104 109 110 110

Predicting the elastic-viscoplastic and creep behaviour of polymer matrix composites using the homogenization theory

113

73 75

T. Matsuda, University of Tsukuba, Japan and N. Ohno, Nagoya University, Japan

4.1 4.2 4.3 4.4 4.5 4.6 4.7 5

Introduction Homogenization theory for non-linear time-dependent composites Elastic-viscoplastic analysis of CFRP laminates and experimental verification Elastic-viscoplastic analysis of plain-woven GFRP laminates and experimental verification Creep analysis of unidirectional CFRP laminates at elevated temperature Summary References Measuring fiber strain and creep behavior in polymer matrix composites using Raman spectroscopy

113 115 118 132 142 144 146

149

T. Miyake, Nagoya Municipal Industrial Research Institute, Japan

5.1

Introduction: creep mechanism of composites reinforced unidirectionally with long fibers

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5.2 5.3 5.4 5.5 5.6 5.7 6

Contents

Stress or strain measurement by Raman spectroscopy Experiments on stress relaxation in broken fibers Time-dependent variation in fiber stress during pull-out tests Discussion Summary References Predicting the viscoelastic behavior of polymer nanocomposites

vii

152 157 164 175 181 182 184

A. Beyle and C. C. Ibeh, Pittsburg State University, USA

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 7

Specific features of nanoparticles and nanocomposites Viscoelasticity of polymer matrix Viscoelasticity of polymers filled by quasi-spherical nanoparticles Viscoelasticity of polymers filled by platelet-shape nanoparticles Viscoelasticity of polymers filled by nanofibers Viscoelasticity of polymers filled by buckyballs and nanotubes Viscoelasticity of nanoporous polymers Viscoelasticity of fibrous composites with nano-filled matrices Concluding remarks Notation Acknowledgement References Constitutive modeling of viscoplastic deformation of polymer matrix composites

184 188 200 217 219 220 224 227 229 230 231 231 234

M. Kawai, University of Tsukuba, Japan

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Introduction Framework for constitutive modeling of the viscoplastic deformation of anisotropic materials Modeling of tension–compression asymmetry in initial anisotropy Modeling of transient creep softening due to stress variation Conclusions Future trends Acknowledgements References

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Contents

8

Creep analysis of polymer matrix composites using viscoplastic models

273

E. Kontou, National Technical University of Athens, Greece

8.1 8.2 8.3 8.4 8.5

Introduction Viscoplastic creep modeling for polymer composites Concluding remarks Future trends References

273 276 296 296 297

9

Micromechanical modeling of viscoelastic behavior of polymer matrix composites undergoing large deformations

302

J. Aboudi, Tel Aviv University, Israel

9.1 9.2 9.3 9.4 9.5 9.6 9.7

Introduction Finite strain viscoelasticity coupled with damage model of monolithic materials Finite strain micromechanical analysis Computational procedure Applications Conclusions References

Part II Creep rupture 10

Fibre bundle models for creep rupture analysis of polymer matrix composites

302 304 309 311 312 322 323 325 327

F. Kun, University of Debrecen, Hungary

10.1 10.2 10.3 10.4 10.5 10.6

Introduction Fibre bundle model Fibre bundle models for creep rupture Summary and outlook Acknowledgement References

327 329 333 345 347 347

11

Micromechanical modeling of time-dependent failure in off-axis polymer matrix composites

350

J. Koyanagi, Institute of Space and Astronautical Science, Japan

11.1 11.2 11.3 11.4 11.5

Introduction Experiments Finite element analysis Discussion Conclusion

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Contents

ix

11.6 11.7

Future trends References

363 363

12

Time-dependent failure criteria for lifetime prediction of polymer matrix composite structures

366

R. M. Guedes, Faculdade de Engenharia da Universidade do Porto, Portugal

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11

Introduction Energy-based failure criteria Creep rupture based on simple micromechanical models Experimental cases The Crochet model (time-dependent yielding model) Kinetic rate theory Fracture mechanics extended to viscoelastic materials Continuum damage mechanics Damage accumulation models for static (creep) and dynamic fatigue Conclusions References

Part III Fatigue modeling, characterization and monitoring 13

Testing the fatigue strength of fibers used in fiber-reinforced composites using fiber bundle tests

366 370 373 381 390 397 398 398 399 401 401

407 409

P. K. Mallick, University of Michigan-Dearborn, USA

13.1 13.2 13.3 13.4 13.5 13.6

Introduction Determination of fiber strength distribution parameters Fiber bundle model for fatigue Stress-life diagram of fiber bundles Conclusion References

409 411 416 419 421 423

14

Continuum damage mechanical modeling of creep damage and fatigue in polymer matrix composites

424

D. Perreux and F. Thiebaud, MaHyTec Ltd, France

14.1 14.2 14.3 14.4 14.5

Introduction Mesomodel: viscoelastic strain, damage and viscoplastic strain of a layer From meso- to macroscopic behavior Conclusion References

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Contents

15

Accelerated testing methodology for predicting long-term creep and fatigue in polymer matrix composites

439

M. Nakada, Kanazawa Institute of Technology, Japan

15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8

Introduction Accelerated testing methodology Experimental verification for ATM Applicability of ATM Theoretical verification of ATM Future trends and research Conclusions References

439 440 447 450 452 457 458 458

16

Fatigue testing methods for polymer matrix composites

461

W. Van Paepegem, Ghent University, Belgium

16.1 16.2 16.3 16.4 16.5 16.6 16.7

Introduction Fatigue testing methods Effect of boundary conditions and specimen geometry Typical fatigue damage in structural composites Future trends Sources of further information and advice References

461 461 474 477 482 483 483

17

The effect of viscoelasticity on fatigue behaviour of polymer matrix composites

492

J. A. Epaarachchi, University of Southern Queensland, Australia

17.1 17.2 17.3 17.4 17.5 17.6 18

Introduction Linear viscoelastic analysis of the characteristics of viscoelastic materials under static and dynamic loading Fatigue behaviour of composite materials Concluding remarks Acknowledgements References Characterization of vicoelasticity, viscoplasticity and damage in composites

492 494 500 508 511 511 514

J. Varna, Lulea University of Technology, Sweden

18.1 18.2 18.3 18.4 18.5 18.6

Introduction Material model Microdamage effect on stiffness Viscoplasticity Nonlinear viscoelasticity Conclusions

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Contents

xi

18.7 18.8

References Appendix: time-dependence of VP-strain in a creep test

539 540

19

Structural health monitoring of composite structures for durability

543

S. Alampalli, New York State Department of Transportation, USA

19.1 19.2 19.3 19.4 19.5 19.6 19.7

Introduction FRP structures in the bridge industry Structural health monitoring FRP structures and SHM Case studies Summary References

Index

543 544 547 551 552 566 569 572

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Contributor contact details

(* = main contact)

Editor and Chapter 12

Chapter 2

R. M. Guedes Departamento de Engenharia Mecânica e Gestão Industrial Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias s/n 4200-465 Porto Portugal

E. J. Barbero Department of Mechanical and Aerospace Engineering West Virginia University Morgantown, WV 26506 USA

E-mail: [email protected]

Chapter 3

Chapter 1 Professor George Papanicolaou* Department of Mechanical and Aeronautical Engineering University of Patras Patras 26500 Greece E-mail: [email protected]

Professor Stephanos P. Zaoutsos Department of Mechanical Engineering Technological Educational Institute of Larissa Larissa 41110 Greece E-mail: [email protected]

E-mail: [email protected]

Kuo-An Li and Anastasia Muliana* Department of Mechanical Engineering Texas A&M University College Station TX 77843-3123 USA E-mail: [email protected]

Chapter 4 Dr T. Matsuda Department of Engineering Mechanics and Energy University of Tsukuba 1-1-1 Tennodai Tsukuba 305-8573 Japan E-mail: [email protected]

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Contributor contact details

Professor N. Ohno Department of Mechanical Science and Engineering Nagoya University Furo-cho, Chikusa-ku Nagoya 464-8603 Japan E-mail: [email protected]

Chapter 5 Dr T. Miyake Department of Electronics and Information Technology Nagoya Municipal Industrial Research Institute 3-4-41, Atsuta-ku Nagoya 456-0058 Japan E-mail: [email protected]. nagoya.jp

Chapter 6 A. Beyle* W224b Kansas Technology Center Pittsburg State University 1701 South Broadway Pittsburg, KS 66762 USA E-mail: [email protected]

C. C. Ibeh Center for Nanocomposite and Multifunctional Materials W123 Kansas Technology Center Pittsburg State University 1701 South Broadway Pittsburg, KS 66762 USA E-mail: [email protected]

Chapter 7 Professor M. Kawai Department of Engineering Mechanics and Energy University of Tsukuba Tsukuba 305-8573 Japan E-mail: [email protected]

Chapter 8 Professor E. Kontou Department of Mechanics School of Applied Mathematical and Physical Sciences National Technical University of Athens 5 Heroes of Polytechnion Av. Athens 15773 Greece E-mail: [email protected]

Chapter 9 Jacob Aboudi Faculty of Engineering Tel Aviv University Ramat Aviv 69978 Israel E-mail: [email protected]

Chapter 10 Ferenc Kun Department of Theoretical Physics University of Debrecen H-4010 Debrecen PO Box 5 Hungary E-mail: [email protected]

© Woodhead Publishing Limited, 2011



Contributor contact details

Chapter 11

Chapter 16

Jun Koyanagi Japan Aerospace Exploration Agency Institute of Space and Astronautical Science 3-1-1 Yoshinodai Sagamihara Kanagawa 229-8510 Japan

Professor Wim Van Paepegem Department of Materials Science and Engineering Ghent University Sint-Pietersnieuwstraat 41 9000 Ghent Belgium

E-mail: [email protected]

Chapter 13 Professor P. K. Mallick Department of Mechanical Engineering University of Michigan-Dearborn 4901 Evergreen Road Dearborn, MI 48128 USA

E-mail: [email protected]

Chapter 17 J. A. Epaarachchi Faculty of Engineering and Surveying University of Southern Queensland Australia E-mail: [email protected]

Chapter 18

E-mail: [email protected]

Janis Varna Lulea University of Technology Sweden

Chapter 14

E-mail: [email protected]

Dominique Perreux* and Frédéric Thiebaud MaHyTec Ltd 210 Avenue de Verdun 39100 Dole France

Chapter 19

E-mail: dominique.perreux@ mahytec.com

xv

S. Alampalli Bridge Evaluation Services Bureau New York State Department of Transportation Albany, NY USA E-mail: [email protected]

Chapter 15 Masayuki Nakada Materials System Research Laboratory Kanazawa Institute of Technology 3-1 Yatsukaho Hakusan Ishikawa 924-0838 Japan E-mail: [email protected]. ac.jp

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1 Viscoelastic constitutive modeling of creep and stress relaxation in polymers and polymer matrix composites G. C. Papanicolaou, University of Patras, Greece and S. P. Zaoutsos , Technological Educational Institute of Larissa, Greece Abstract: This chapter discusses the basic concepts of viscoelastic behavior of polymers and polymeric composites as well as respective modeling for both the linear and nonlinear behavior. The material is discussed in four main sections. The first discusses creep-recovery and stress relaxation experiments. The second section discusses the concept of linearity in viscoelasticity and a short presentation of the time–temperature, time–stress and time–temperature–stress superposition principles is made. The third section discusses the different linear viscoelastic models, from the basic viscoelastic elements up to the generalized models. Finally, the fourth section discusses the nonlinear viscoelastic behavior of polymers and polymeric composites and different predicting models and methods along with specific applications are presented. Key words: creep, stress relaxation, viscoelastic modeling, linear viscoelastic behavior, nonlinear viscoelastic behavior modeling.

1.1

Introduction

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. A viscous material exhibits timedependent behavior when a stress is applied while under constant stress and deforms at a constant rate, and when the load is removed, the material has ‘forgotten’ its original configuration, remaining in the deformed state. On the other hand, an elastic material deforms instantaneously when stretched and ‘remembers’ its original configuration, returning instantaneously to its original state once the stress is removed. Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain showing a ‘fading memory’. Such a behavior may be linear (stress and strain are proportional) or nonlinear. Whereas elasticity is usually the result of bond stretching along crystallographic planes in an ordered solid, viscoelasticity is the result of the diffusion of atoms or molecules inside an amorphous material. Polymers are characterized by the fact that their behavior under load or deformation is, to a large extent, time dependent even at room temperature. Moreover, their response to a load or deformation will depend, in some cases, upon any previous load, deformation or temperature history. This time dependence 3 © Woodhead Publishing Limited, 2011

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Creep and fatigue in polymer matrix composites

manifests itself in several forms: two of these are creep, that is to say a progressive increase in deformation under a constant load; and stress relaxation, a gradual decrease in stress under a constant deformation. Both these phenomena influence and, in many cases, limit the application of plastics for structural and load-bearing applications. In a plastics pipe assembly, a pipe with its coupling may have to withstand a continuous internal pressure. Under these conditions both the pipe coupling and the pipe will slowly creep; unless the coupling creeps radially less than the pipe, a leak will occur. It is therefore a matter of design to ensure that the relative movement is kept to a minimum. Sometimes a plastics component or part is deliberately deformed so that it should exert a mechanical force due to its elasticity. A spring will grip or support an article as long as the external forces exerted on the article are less than the force exerted on it by the grip or spring. If the grip or spring is made of a plastics material, stress relaxation will take place and the force exerted by it, when it is deformed, will progressively decrease with time until it is no longer able to oppose the external forces. A similar problem can be encountered with bottle closures; these remain in position by virtue of the fact that they are strained: if stress relaxation is present these closures may eventually fail. The property of stress relaxation does not preclude the use of plastics for this type of application but knowledge of the degree to which it occurs under various conditions is essential for design purposes. In general, given suitable laboratory information on the behavior of polymers under long-term stress or strain, it is possible for design engineers to overcome some of the obstacles to the use of thermoplastics under load-bearing conditions. The study of creep and stress relaxation phenomena is also of considerable importance in the examination of fundamental viscoelastic properties of polymers and can lead to the determination of such basic viscoelastic constants as retardation and relaxation times.

1.2

Creep

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

1.2.1 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 = 0 is shown in Fig. 1.1a. An instantaneous strain (ε0) proportional to the applied stress is observed after the application of the stress and this is followed by a progressive increase in strain as shown in Fig.1.1b. The total strain at any instant of time is represented as the sum of the instantaneous elastic strain and the creep strain, i.e.,

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Viscoelastic constitutive modeling of creep and stress relaxation

5

1.1  Creep: (a) application of constant stress; (b) strain response.

e(t) = e0 + ec

[1.1]

The ratio of the total strain ε(t) to the applied constant stress σ0 is called creep compliance and is given by:

e(t) D(t) = σ   0

[1.2]

The creep compliance at any point of time is the sum: D(t) = D0 + DD(t) = D(0) + DD(t)

[1.3]

Where D0 is the instantaneous creep compliance and ∆D(t) 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. 1.2). 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 such as metals, polymers, ceramics, concrete, soils, rocks, ice, etc. Creep strains, viscoelastic or viscoplastic deformations and creep fractures in engineering structures result from time- and temperature-dependent processes which occur when specific materials are subject to stress. Creep of materials often limits their use in practice. Particularly in high-temperature environments, creep has a severe effect on structural or component response. Creep is a very complex phenomenon which depends on many parameters. For example, the creep behavior of fiber-reinforced composites depends on factors

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Creep and fatigue in polymer matrix composites

1.2  Creep stages.

such as 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: . . e = f(σ0, T, em, Vf, lf, θ, Ef, σuf, Em, Ei, ti) [1.4] . where σ is external applied stress; T, testing temperature; ε , matrix strain rate; 0

m

Vf, fiber volume fraction; λf, geometric parameters of the fibers; θ, fiber orientation angle (relative to the applied loading); Ef, fiber modulus; σuf, 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. Epoxybased 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 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 a large area while minimizing stress concentrations, joining dissimilar materials, and reducing the overall weight and manufacturing costs. However, epoxy resins, being viscoelastic

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Viscoelastic constitutive modeling of creep and stress relaxation

7

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

in nature, exhibit unique time-dependent behavior. This leads to a great concern in assessing their long-term load-bearing performance.

1.2.2 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 time-dependent 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 which occurred (Fig. 1.1b). Some plastics may exhibit full recovery if sufficient time is allowed for recovery. The strain recovery is also called delayed elasticity.

1.2.3 Relaxation Viscoelastic materials subjected to a constant strain will relax under constant strain (Fig. 1.3a) so that the stress gradually decreases as shown in Fig. 1.3b.

1.3

Linearity

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

ε[cσ (t)] = cε[σ (t)]

[1.5]

This is shown in Fig. 1.4. 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. 2. The linear superposition principle holds. This implies that each loading step makes an independent contribution to the final deformation, which can be

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Creep and fatigue in polymer matrix composites

1.4  Linear viscoelastic behavior.

obtained by the addition of these. This principle is also called the Boltzmann superposition principle. For a two-step loading case shown in Fig. 1.5, the strain response is given by:

ε[σ1(t) + σ2(t – t1)] = ε[σ1(t)] + ε[σ2(t – t1)]

[1.6]

Further, for multi-step loading, during which stresses σ1, σ2, σ3 . . . applied at times τ1, τ2, τ3 . . . the strain at time ‘t’ is given by:

ε(t) = σ1D(t – τ1) + σ2D(t – τ2) + σ3D(t – τ3) + . . . where D(t – τ) is the creep compliance.

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[1.7]

Viscoelastic constitutive modeling of creep and stress relaxation

0

0

9

t

t1

t1

t

1.5  Boltzmann superposition principle.

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 Boltzmann superposition integral) as:

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Creep and fatigue in polymer matrix composites t

ε(t) = D0σ + ∫∆D(t – τ)dσ dτ [1.8] dτ        0 where D0 is the instantaneous creep compliance, ∆D(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. The above 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, D0, and a time-dependent component, ∆D(t). The transient creep compliance function ∆D(t) 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: ∆D(t) = D1tn

[1.9]

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: N

∆D(t) = ∑  Di(1 – e–t/τi) i=1

[1.10]

N

ε(t) = D0σ + ∑  Diσ (1 – e–t/τi) i=1

[1.11]

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: t

σ (t) = E0ε + ∫∆E(t – τ) dε dτ dτ           0

[1.12]

where E0 and ∆E(t – τ) are components of the stress-relaxation modulus. The above 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 considered to be 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 related by:

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Viscoelastic constitutive modeling of creep and stress relaxation

11 [1.13]

However, it is to be noted that D(0)E(0) = 1 (instantaneous). Analytical integration of equation is possible only for simple forms of creep compliance. For example, if the compliance can be expressed by power law given by: D(t) = D1tn

[1.14]

then it can be shown that the relaxation modulus is given by: E(t) =

1 t–n D1Γ (1 + n)Γ (1 – n)

[1.15]

where Γ(x) = ∫ e–t tx–1dt, is the gamma function.

1.4

The time–temperature superposition principle (TTSP)

Polymeric materials, because of their viscoelastic nature, exhibit behavior during deformation and flow which 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 under a 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 time-dependent 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 see in the 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 (1) that the processes involved in molecular relaxation or rearrangements in viscoelastic materials occur at accelerated rates at higher temperatures and (2) 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

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Creep and fatigue in polymer matrix composites

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 (a T): t t′ = a  T

[1.16]

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(a T) =

–C1(T – T0)  C2 + (T – T0)

[1.17]

where C1 and C2 are empirical constants, T0 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. The above equation 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, and if T0 is taken as Tg, measured by a static method such as dilatometry, then: g

–C (T – T0) log(aT) = g 1 C2 + (T – T0) g

[1.18] g

and the new constants C1 and C2 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 which are close to measured values. The other model commonly used is the Arrhenius equation: log at =

E R(T – T0)

[1.19]

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

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Viscoelastic constitutive modeling of creep and stress relaxation

13

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.

1.5

The time–stress superposition principle (TSSP)

Here stress is used as the accelerating factor. Short-term, isothermal, creeprecovery tests are carried out at different stress levels. The generation of the master curves in the time–stress scale is performed by a numerical procedure using an analytical constitutive equation.

1.6

The time–temperature–stress superposition principle (TTSSP)

Increased stress accelerates creep of many viscoelastic materials, similar to the effect from increased temperature. A number of researchers have proposed time– temperature–stress superposition principles (TTSSP) (Schapery, 1969; Yen and Williamson, 1990; 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 log-time 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 construed within the framework of free volume theory. The following discussion summarizes their work. From free volume theory, the viscosity of a material, η, can be related to the free volume fraction, ƒ by:

( )

[1.20] ln η = ln A + B 1f –1 where A and B = material constants. Equation 1.20, known as the Doolittle equation, is the foundation of time–temperature superposition.

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Creep and fatigue in polymer matrix composites

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 = f0 + α T (T – T0) + ασ (σ – σ0)

[1.21]

where αT = coefficient of thermal expansion of the free volume fraction; ασ = stress-induced expansion coefficient of the free volume fraction; and ƒ0 = free volume fraction at a reference temperature and stress. Presume there exists a shift factor (aTσ) that satisfies

η(T, σ) = η(T0, σ0)a Tσ

[1.22]

then Eq. 1.20 and 1.21 can be combined:

[

log(a Tσ) = –C1

]

C3(T – T0) + C2 (σ – σ0) C2C3 + C3 (T – T0) + C2 (σ – σ0)

[1.22a]

f where C3 = α 0 . Equation 1.22a reduces to the WLF equation if there is no stress σ difference. Additionally, the stress shift factor at constant temperature aTσ and the temperature shift factor at constant stress level aσT are defined so that: 0 aσ η(T, σ) = η(T, σ0)aTσ = η(T0, σ0)aσT aσT0 = η(T0, σ)a σT = η(T0, σ0)aTσ   T

therefore, a Tσ = aσT aσT0 = aTσ0 aσT Equation 1.21 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, σ0. In a case where service temperature is chosen as the reference temperature, T0, Eq. 1.22a reduces to: log(a Tσ) = –

(

) [

]

B σ – σ0 C (σ – σ0) =– 1 2.303f0 f0 /aσ + σ – σ0   C3 + (σ – σ0)

where aσ = stress shift factor. Now nonlinear creep compliance at varied stress levels can be related by the reduced time, t/aσ: D(σ, t) = D(σ0, t/aσ )

1.7

[1.23]

Linear viscoelastic models

1.7.1 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. 1.6 © Woodhead Publishing Limited, 2011

Viscoelastic constitutive modeling of creep and stress relaxation

σ = Eε

15 [1.24]

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. 1.6.

1.6  Linear spring response to constant stress.

1.7.2 The linear viscous dashpot A linear viscous dashpot element is shown in Fig.1.7 where: σ = η dε dt and the constant η is called the coefficient of viscosity. © Woodhead Publishing Limited, 2011

[1.25]

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1.7  Linear viscous dashpot response to constant stress.

Equation 1.25 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.1.7. 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 = 0+ and will remain zero, as shown in Fig. 1.8. This behavior for a step change in strain is indicated mathematically by the Dirac delta function δ(t) where δ(t) = 0, for t ≠ 0, δ(t) = ∞ for t = 0. Thus the stress resulting for applying a step change in strain ε0 to Eq. 1.25 is indicated as follows:

σ (t) = ηε0 δ (t)

[1.26]

An infinite stress is impossible in reality. It is therefore impossible to impose instantaneously any finite deformation on the dashpot. © Woodhead Publishing Limited, 2011

Viscoelastic constitutive modeling of creep and stress relaxation

17

1.8  The linear viscous dashpot response to constant strain.

1.7.3 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. 1.9. The stress–strain relations of spring and dashpot respectively are:

σ = Eε2 σ=η

[1.27]

dε1 dt

[1.28]

Since both elements are connected in series, the total strain is:

ε = ε1 + ε2

[1.29]

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

[1.30]

The response of the model to various stress or strain conditions can be obtained after applying integration together with appropriate initial conditions. Creep For example, applying a constant stress σ = σ0 at t = 0, the following response is obtained:

ε(t) =

σ0 σ0 + t E  η

[1.31]

This result is shown in Fig. 1.9.

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1.9  The Maxwell model.

Recovery If the stress is removed from the Maxwell model at time t1, the elastic strain σ0/E in the spring returns to zero at the instant the stress is removed, while (σ0/η)t1 represents a permanent strain which does not disappear. This result is also shown in Fig. 1.9. Relaxation If the Maxwell model is subjected to a constant strain ε0 at time t = 0 (Fig. 1.10), for which the initial value of stress is σ0, the stress response can be obtained by integrating Eq. 1.30 for these initial conditions with the following result

σ (t) = σ0 exp(– Et/η) = Eε0 exp(– Et/η)

[1.32]

where ε0 is the initial strain at t = 0+ and 0+ refers to the time just after application of the strain. Equation 1.32 describes the stress relaxation phenomenon for a

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Viscoelastic constitutive modeling of creep and stress relaxation

19

1.10  Maxwell model relaxation response.

Maxwell model under constant strain. This phenomenon is shown in Fig. 1.10. The rate of stress change is given by the derivative of Eq. 1.32. dσ = –(σ E/η)exp(– Et/η) 0 dt

[1.33]

dσ = –σ0E/η. If the stress were Thus the initial rate of change in stress at t = 0+ is dt to decrease continuously at this initial rate, the relaxation equation would have the following form:

σ = –(σ0 Et/η) + σ0

[1.34]

and the stress would then reach zero at time tR = 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 since the variable factor exp(–t/tR) converges toward zero very rapidly for t < tR. For example at t = tR, σ (t) = σ0 /e = 0.37σ0. Thus only 37% of the initial stress remains at t = tR.

1.7.4 The Voigt or Kelvin model The Voigt model is shown in Fig. 1.11 where the spring element and dashpot element are connected in parallel. The spring and dashpot have the following stress–strain relations:

σ1 = Eε

[1.35]

σ2 = η dε dt Since both elements are connected in parallel, the total stress is: σ = σ1 + σ2

[1.36]

[1.37]

Eliminating σ1 and σ2 among these equations yields the following constitutive equation for the Voigt model:

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Creep and fatigue in polymer matrix composites dε Eε σ + = dt  η  η

[1.38]

Creep The solution of Eq. 1.38 may be shown to have the following form for creep under constant stress σ0 applied at t = 0,

ε(t) =

σ0 [1 – exp(– Et/η)] E

[1.39]

1.11  The Voigt or Kelvin model.

As shown in Fig. 1.11, the strain described by Eq. 1.39 increases with a decreasing rate and approaches asymptotically the value of σ0/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 σ0 is found by differentiating Eq. 1.39:

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Viscoelastic constitutive modeling of creep and stress relaxation dε σ0 = exp(– Et/η) dt  η

21

[1.40]

Thus, the initial strain rate at t = 0+ is finite:

()

dε σ = 0 dt  t = 0  + η

[1.41]

and the strain rate approaches asymptotically to zero when t tends to infinity:

(ddtε )

t→∞

=0

[1.42]

If the strain were to increase at its initial rate σ0/η, it would cross the asymptotic value σ0/E at time tc = η/E, called the retardation time. Actually, most of the total strain σ0/E occurs within the retardation time period since exp(–Et/η) converges toward the asymptotic value rapidly for t < tc.

( )( )

σ 1 σ At t = tc,      ε (tc) = 0 1 – = 0.63 0 [1.43]         E  e  E Thus, only 37% of the asymptotic strain remains to be accomplished after t = tc. Recovery If the stress is removed at time t1 the strain following stress removal can be determined by the superposition principle. The strain ε′ in the Voigt model resulting from stress σ0 applied at t = 0 is:  σ ε ′ = 0 [1 – exp(– Et/η)] [1.44]  E The strain ε ″ resulting from applying a stress (–σ0) independently at time t = t1 is: σ ε ″ = – 0 [1 – exp(–E(t – t1)/η)] [1.45]  E If the stress σ0 is applied at t = 0 and removed at t = t1 (– σ0 is added) the superposition principle yields the strain ε(t) for t > t1 during recovery:

ε(t) = ε′ + ε″ =

         

σ0 –Et/η Et/η e [e – 1]        t > t1 E

[1.46]

As shown by Eq. 1.46 and illustrated in Fig.1.10, when      t → ∞,    ε∞ → 0 Some real materials show full recovery while others show only partial recovery. Relaxation The Voigt model does not show a time-dependent relaxation. Owing to the presence of the viscous element an abrupt change in strain ε0 can be accomplished

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Creep and fatigue in polymer matrix composites

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 ε = σ dt  η  η By using the Heaviside H(t) and Dirac δ(t) functions to describe the step change in strain:

ε (t) = ε0 H(t),    dε = ε0δ (t) dt

[1.47]

E ε H(t) = σ ε0 δ (t) + η 0 η   

[1.48]

Thus,

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. The majority of 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. Whilst 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. Figure 1.12 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: (∆σ (t)/σ0) = n log t + I

[1.49]

where σ0 = the maximum stress developed at the end of the straining phase, σt = the stress at any time t after the elongation had ceased, t = time after the elongation phase was completed, ∆σ (t) = σ0 – σt, n = the slope of the plot of ∆σ (t) against log t; this is equivalent to a rate of stress relaxation, I = the intercept on the value of ∆σ (t)/σ0 when log t = 0. This relationship describes the decay in stress which has been achieved after a time t in terms of the maximum stress developed and the time after the strain phase is completed.

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Viscoelastic constitutive modeling of creep and stress relaxation

23

1.12  Typical stress/time trace.

A plot of ∆σ (t)/σ0 against log t 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 bore a linear relationship to the logarithm of straining time, log p, the intercept decreasing with increasing straining time. Since I plotted against log p gives a straight line relationship it can be expressed in the form: I = m log p + C

[1.50]

where I = the value of the intercept as defined in Eq. 1.49, p = straining time and C = the value of I for log p = 0. This expression may now be substituted into Eq. 1.49 to give:

[1.51]

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. 1.51 is used to predict the percentage stress relaxation which 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 which is applied initially will of course be dependent upon the amount of strain. The results indicate that, although initial stress decreases with decreasing rate of application of strain, this decrease can be compensated for by increasing the deformation slightly.

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Creep and fatigue in polymer matrix composites

Neither the Maxwell nor Voigt model described above 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 which 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.

1.7.5 The three-element solid The three-element solid involves a spring and a Voigt model in series as shown in Fig. 1.13.

1.13  The three-element solid.

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Viscoelastic constitutive modeling of creep and stress relaxation

25

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 = σ1 = σ2

Kinematic: ε = ε1 + ε2, which becomes, ε· = ε·1 + ε·2 Constitutive: For the spring: σ1 = E1ε1

    For the Voigt model: σ2 = E2ε2 + η2ε·2

  which can be written: ε = 1 [σ – η ε· ] 2 2 2              2 E2 Differentiating both of the constitutive equations, we can substitute them into the kinematic equation and solve for the governing differential equation:

σ

σ⋅ =  

ε·

[1.52]

1.7.6 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. 1.14) 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. 1.14). 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. 1.14, 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 non-recoverable 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 = ε1 + ε2 + ε3

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[1.53]

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

Elastic deformation for the Maxwell model:

ε1 = σ/E1

[1.54]

The retarded elastic deformation of the Voigt model only:

ε2 = (σ/E2) [1 – exp(–t/τ2)],  where        τ2 = η2/E2 The viscous deformation of the Maxwell dashpot:

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[1.55] [1.56]

Viscoelastic constitutive modeling of creep and stress relaxation

ε3 = (σ/η3) t

27 [1.57]

Eliminating ε1, ε2 and ε3 among these equations yields the following constitutive equation for the four-element model:

[1.58]

Thus, the creep behavior may be found to be as follows:

[1.59]

Differentiating Eq. 1.59 yields the creep rate dε/dt as follows:

[1.60]

Thus the creep rate starts at t = 0+ with a finite value:

[1.61]

and approaches asymptotically to the value:

[1.62]

It may also be observed that OA = σ/E1 and AA′ = 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 AA′ as in Fig. 1.15.

1.15  Four-element model parameters determination.

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1.7.7 The generalized Maxwell model This model consists of many Maxwell models either in series or in parallel (Fig. 1.16). When several Maxwell models are connected in series, the constitutive equation is given by:

[1.63]

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

The response of this model is not much different from the Maxwell model mentioned earlier and hence is not significant. When several Maxwell models are connected in parallel, the resulting model is capable of representing 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:

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1.7.8 The generalized Voigt or Kelvin model This model consists of many Voigt or Kelvin models in parallel (Fig. 1.17) or in series (Fig. 1.18). When several Kelvin models are connected in parallel, the constitutive equation is given as

[1.65]

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:

[1.66]

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

[1.67]

where Di is the creep compliance.

1.8

Nonlinear viscoelastic behavior

1.8.1 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.

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

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

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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 chapters are no longer valid and applicable. It follows that the nonlinear viscoelasticity is observed when one of the two conditions mentioned above is denied by the data. For instance, a lot of experimental investigations have shown that the kernel of the Boltzmann integral may, in the form of creep compliance, be found to increase with stress or alternately that the rate of recovery under zero load is different than the rate of the preceding creep under constant load. As a result, due to the violation of the above 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 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 apply 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 in the following paragraphs.

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

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[1.68] where D1(t), D2(t), . . . Dn(t), are kernels of time. The first integral of Eq. 1.68 describes the linear viscoelastic behavior defined by the Boltzmann theory and the second and higher order integrals are representations of both magnitude nonlinearity 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. 1.19. However, the higher-order nonlinear kernel functions such as D2(t) and D3(t) require more than one parameter for their descriptions. For example, the second-order kernel function D2(t, t – ξ1) is descri bed in terms of two time parameters t and t – ξ1. The variation of D2(t, t – ξ1) with these two time parameters may be illustrated by a surface A as shown in Fig. 1.20 with coordinates D2, t and t. In Fig. 1.20 the line B in surface A describes the ordinate D2(t,t) versus time t when the two time parameters are equal. Hence, line B lies in a plane which bisects the angle between the coordinates t and t. Line C in surface A of Fig. 1.20 represents

1.19  A graphic illustration of the linear kernel function D1(t) versus time.

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1.20  A graphic illustration of the second-order kernel functions D2(t) versus two time parameters.

D2(t, t – ξ1) versus 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 = 0 and the other at t = 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. Since the kernel functions are considered symmetrical with respect to their time parameters D2 (t, t – ξ1) = D2 (t – ξ1, t)

[1.69]

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. 1.68. A further investigation of Eq. 1.68 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 et al., 1972). Ward

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and Onat (1963) have also shown that the accuracy can be preserved successfully using the first and the third term of equation. Despite its conceptual generality, the model described above is rather complicated to be applied, mainly due to the large number of the tests needed not only for the determination of the kernels and the 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. Model of Pipkin and Rogers: nonlinear superposition theory (NLST) An alternative to the Green and Rivlin approach has been formulated by Pipkin and Rogers (1968). This model is often referred to as nonlinear superposition theory and generalizes the formulation of Green and Rivlin. It was applied in the experimental results of Findley et al. (1976), 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:

[1.70] It is clear that Eq. 1.70 is nonlinear even in its first approximation, thus the additivity of incremental stress effects in the Boltzmann superposition sense is preserved.

1.8.3 Single integral representations The concept of mapping the nonlinear viscoelastic response through single integrals is based in principle in the nonlinearization of stress and strain measure through a general form of the equation:

[1.71]

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

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

ϕ



[1.73]

The model allows for a stress or pressure deformable time measure. Brueller’s model In an extensive research effort into 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: [1.74]



The nonlinearity is controlled by two nonlinearity functions, namely ga, gb, that solely depend on the applied stress level. The function ga attributes the nonlinearity in the instantaneous response while gb attributes the nonlinearity in transient creep. The determination of the nonlinearity functions ga and gb can be achieved using a mixed iteration procedure on the basis of numerical methods. Schapery’s constitutive equation According to Schapery (1966, 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:

[1.75]

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

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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 = g1 = g2 = aσ = 1 and leads to the Boltzmann Superposition Principle. Schapery’s model has been used extensively for the description of polymers and polymer matrix composites (Zhang et al., 1992; Mohan and Adams, 1985). Moreover, according to comparative studies with other nonlinear models it has proved its advantage in the prediction of the experimental data (Smart et al., 1972; Partom et al., 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 et al. (1986; 1995), Dillard et al. (1987) and Rami et al. (2008).

1.8.4 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 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 the following semiempirical model for the nonlinear parameters of Schapery’s model: for:





for:

[1.76]

[1.77]

where σc is the stress threshold from linear to nonlinear viscoelasatic behavior of the material.

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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-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 one-parameter curve fitting 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: for  σ > σc

where G is the parameter of interest,

for  σ < σc

and

[1.78]

, while σu is the ultimate

tensile strength of the material, σc is the stress threshold from linear to nonlinear behaviour and k is the maximum value of the parameter of interest when σ tends to σu. 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.

1.8.5 Application to different materials Application of the Papanicolaou et al. model to cross-ply carbon/epoxy composites resulted in a good agreement between the experimental results and the respective model predictions. Additionally, the method and the model were tested successfully in different loading times (Fig. 1.21, 1.22). 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 Fig. 1.23 and 1.24. The same model was also applied and compared successfully to other semi-analytical approximations for polycarbonate (Wing et al., 1995; Papanicolaou et al., 2005), as can be seen in Fig. 1.25 and 1.26. Reported results on particulate composites were also very encouraging concerning the analytical model of the description of nonlinear parameters of

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Schapery’s constitutive equation as can be seen in Fig. 1.27 and 1.28 (Papanicolaou et al., 2009).

1.21  Comparison between experimental values and the proposed model predictions of the nonlinear parameters g0 (a) and g1 (b) as a function of the applied stress at different loading times.

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

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1.23  Characteristic surfaces of the parameters g0 (a) and g1 (b) as a function of fiber orientation and the ratio of the applied stress to ultimate tensile stress for carbon/epoxy composite.

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1.24  Characteristic surfaces of the stress shift factor aσ (b) and the parameter g2 (a) as a function of fiber orientation and the ratio of the applied stress to ultimate tensile stress for carbon/epoxy composite. © Woodhead Publishing Limited, 2011

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

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1.26  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.

1.27  Variation of the nonlinear parameters g0 (a) and g1 (b), as a function of the applied stress for aluminium/epoxy particulate composite. (Continued)

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1.27  Continued.

1.28  Variation of the stress shift factor aσ (a) and the nonlinear parameter g2 (b), as a function of the applied stress for aluminium/ epoxy particulate composite.

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1.28  Continued.

1.9

References

Brinson H F, Morris D H, Yeow, Y T (1978), A New Experimental Method for the Accelerated Characterization of Composite Materials. 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, Polymer Engineering and Science, 27, 144–148. Brueller O S (1993), Predicting the Behaviour of Nonlinear Viscoelastic Materials Under Spring Loading, Polymer Engineering and Science, 33, 97–99. Brueller O S (1996), Creep and Failure of Fabric-Reinforced Thermoplastics, Progress in Durability Analysis of Composite Systems, 39–44. Cessna L C (1971), Stress–Time Superposition of Creep Data for Polypropylene and Coupled Glass-Reinforced Polypropylene, Polymer Engineering and Science, 11, 211–219. Dillard D A, Straight M R, Brinson H F (1987), The Nonlinear Viscoelastic Characterization of Graphite/Epoxy Composites, Polymer Engineering and Science, 27, 116–123. Findley W N, Lai J S, Onaran K (1976), Creep and Relaxation of Nonlinear Viscoelastic Materials, North-Holland Publishing Company, New York, NY. Findley W N, Lai J S Y (1967), A Modified Superposition Principle Applied to Creep of Nolinear Viscoelastic Material Under Abrupt Changes in State of Combined Stress, Transactions of The Society of Rheology, 11, 361. Flugge W (1967), Viscoelasticity, Blaisdell, Waltham, MA.

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Green A E, Rivlin R S (1957), The Mechanics of Nonlinear Materials with Memory, Part I, Archives of Rational Mechanics Analysis, 1, 1–21. Green A E, Rivlin R S, Spencer A J M, (1959), The Mechanics of Nonlinear Materials with Memory, Part II, Archives of Rational Mechanics Analysis, 3, 82. Green A E., Rivlin R S (1960), The Mechanics of Nonlinear Materials with Memory, Part III, Archives of Rational Mechanics Analysis, 4, 387. Haj-Ali Rami, Mulian A (2008), A micro-to-meso sublaminate model for the viscoelastic analysis of thick-section multi-layered FRP composite structures, Mechanics of TimeDependent Materials, 12, 69–9312. 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), Nonlinear Viscoelasticity Based on Free Volume Consideration, Computers and Structures, 13, 123–128. Lai J, Bakker A (1995), Analysis of the Nonlinear Creep of High-Density Polypropylene, Polymer, 36, 93–99. Leaderman H (1943), Elastic and Creep Properties of Filamentous Materials and Other High Polymers, The Textile Foundation, Washington, DC. Ma C C M, Tai N H, Wu S H, Lin S H, Wu J F, Lin, J M (1997), Creep Behavior of CarbonFiber-Reinforced Polyetheretherketone (PEEK) Laminated Composites, Composites Part B, 28, 407–417. Mohan R, Adams D F (1985), Nonlinear creep-recovery response of a polymer matrix and its composites, Experimental Mechanics, 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 Composites Part A, Applied Science and Manufacturing, 30, 838–849. Papanicolaou G C, Zaoutsos S P, Cardon A H (1999b), Prediction of The Non-Linear Viscoelastic Behaviour of Polymer Matrix Composites, Composites Science and Technology, 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, 64, 2535–2545. Papanicolaou G C, Zaoutsos S P, Kosmidou Th V (2005), Describing Nonlinearities in the Mechanical Viscoelastic Behaviour of Polymers and Polymer Matrix Composites, Modern Problems of Deformable Bodies Mechanics Volume I: Collection of Papers, Yerevan, 201–210. 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 polymer-matrix particulate composites, Mechanics of Time-Dependent Materials, 13, 261–274. Park S W, Kim Y R (2001), Fitting Prony-Series Viscoelastic Models with Power-Law Presmoothing, Journal of Materials in Civil Engineering, 13, 26–32. Partom Y, Schanin I (1983), Modelling Nonlinear Viscoelastic Response, Polymer Engineering and Science, 23, 849–859. Pipkin A C, Rogers T G (1968), A Nonlinear Integral Representation for Viscoelastic Behaviour, Journal of the Mechanics and Physics of Solids, 16, 59–72. Rabotnov, Yu. N (1948), Some problems of creep theory, Vestik Mosk. Univ. Matem., Mekh., 10, 81–91. Schapery R A (1966), An Engineering Theory of Nonlinear Viscoelasticity with Applications, International Journal of Solids and Structures, 2.

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Schapery R A (1966), A Theory of Nonlinear Thermoviscoelasticity Based on Irreversible Thermodynamics, Proceedings of the 5th U.S. National Congress in Applied Mechanics, ASME, 511–530. Schapery R A (1968), On a Thermodynamic Constitutive Theory and Its Application to Various Nonlinear Materials, Proceedings of UITAM Symposium On Thermoinelasticity, East Kilbride, 259–284. Schapery R A (1969), On the Characterization of Nonlinear Viscoelastic Materials, Polymer Engineering and Science, 9, 295–310. Smart J, Williams G C (1972), A Power Law Model for the Multiple-Integral Theory of Nonlinear Viscoelasticity, Journal of the Mechanics and Physics of Solids, 20, 325–335. Smart J, Williams J G (1972), A Comparison of Single Integral Nonlinear Viscoelasticity Theories, Journal of the Mechanics and Physics of Solids, 20, 313–324. Tuttle M E, Brinson H F (1986), Prediction of the Long Term Creep Compliance of General Composite Laminates, Experimental Mechanics, 26, 89–102. Tuttle M E, Pasricha A, Emery AF (1995), The Nonlinear Viscoelastic-Viscoplastic Behaviour of IM7/5260 Composites Subjected to Cyclic Loading, Journal of Composite Materials, 29, 2025–2046. Ward I M, Onat E T (1963), Nonlinear Mechanical Behaviour of Oriented Polypropylene, Journal of the Mechanics and Physics of Solids, 11, 217–229. Wenbo L, Ting-Qing Y, Qunli A (2001), Time–Temperature–Stress Equivalence and its Application to Nonlinear Viscoelastic Materials, Acta Mechanica Solida Sinica, 14, 195–199. Wing G, Pasricha A, Tuttle M A, Kumar V (1995), Time-Dependent Response of Polycarbonate and Microcellular Polycarbonate, Polymer Engineering and Science, 35, 673–679. Yannas I V (1974), Nonlinear Viscoelasticity of Solid Polymers in Uniaxial Tensile Loading, Journal of Polymer Science, 9, 163–190. Yen S C, Williamson F L (1990), Accelerated Characterization of Creep Response of an Off-Axis Composite Material, Composite Science and Technology, 38, 103–118. Zaoutsos S P, Papanicolaou G C (1999), The Effect of Fiber Orientation on the Nonlinear Viscoelastic Behavior of Continuous Fiber Polymer Composites, 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, Composites Science and Technology, 58, 883–886. Zhang S Y, Xiang X Y (1992). Creep Characterization of a Fiber-Reinforced Plastic Material, Journal of Reinforced Plastics and Composites, 11, 1187–1194.

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2 Time–temperature–age superposition principle for predicting long-term response of linear viscoelastic materials E. J. B arbero , West Virginia University, USA Abstract: Viscoelastic responses such as creep and relaxation are strongly affected by temperature and age for all materials in the range of time when they exhibit viscoelastic effects. Effective time–temperature superposition (ETTSP) is introduced in this chapter to predict long-term viscoelastic behavior from short-term experimental data. Since the material responds to both temperature and age in the time span of interest, both phenomena are studied, isolated, and described. First, the traditional time–temperature superposition (TTSP) is described and the need to use momentary curves to construct the momentary master curve is addressed. Next, the time–age superposition is described and modeled. Then, the concept of effective time brings everything together into a useful predicting tool. Finally, the methodology is applied to the problem of temperature compensation during long-term testing. Key words: aging, creep, effective time, master curve, momentary data, shift factor, superposition, temperature compensation, effective time theory, TTSP.

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 σ0 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: D(λ) = D0 + D1 [1 – e–λ/τ ]

[2.1] e–λ/τ

where the retardation time τ is the time it takes for an exponential to decay to 100 × e–1 = 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: E(λ) = L–1

[

]

1 s2 L [D(λ)]

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[2.2]



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49

where L[], L–1[], s, denote the Laplace transform, the inverse Laplace transform, and the Laplace variable, respectively [1, 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 [1] 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 [1, 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 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 [1, Figure 7.3]. Such material is said to be unaging [1, 2, 3]. 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 [4, 5]. 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. Equation 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 spring-dashpot elements can be added in series, as follows: n

D(λ) = D0 + ∑ Dj [1 – e–λ/τ j ] j =1

[2.3]

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 ∆(τ) as follows: ∞

D(λ) = ∫ ∆(τ) [1 – e–λ/τ dτ] 0

[2.4]

While Eq. 2.3 and 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: D(λ) = D0 + D1′ [1 – e(– λ/τ ) ] m

[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 short-term material behavior may be impossible to distinguish from a threeparameter power law (Eq. 2.6). Expanding Eq. 2.5 with a Taylor power series results in: D(λ) = D0 + D1′ (λ/τ)m [1 – (λ/τ)m + . . .] D(λ) ≈ D0 + D1 λm

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[2.6]

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where D1 = D1′/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 (D(λ) – D0) = log D1 + m log λ

[2.7]

and thus it is very easy to regress to data by performing a linear regression in log-log scale. Finally, the Kohlrausch model [3, (10)]: m

D(λ) = D0 e(λ/τ)

[2.8]

has been shown to fit the compliance of a broad variety of materials [3, 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 [6], as follows: % log sampling, user picks the initial time and increment time % xi (:,1) time (equally spaced in time) % xi (:,2) compliance ndp = length (xi (:, 1)); %# data points read tf = xi (ndp, 1); %final time logti = –1; %log of initial time to sample, user choice del_logti = 0.1; %log time interval to sample, user choice logt = [logti: del_logti: %equally spaced in log scale log10(tf)]; tr = 10.ˆlogt; %back to time scale nr = length (tr); %number of newly sampled data if tr(nr)~ = tf;   tr = [tr, tf]; %add the final time   nr = length(tr); %number of newly sampled data end rcount = 1; for i = 1:ndp   if xi (i, 1) > = tr(rcount) %say 10ˆ –0.1    xo (rcount,:) = xi(i, :); %copy data equally spaced in log(time)    rcount = 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 © Woodhead Publishing Limited, 2011



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2.1  Compliance of a material described by Eq. 2.5 with D0 = 1 GPa-1, D1 = 9 GPa-1, and two values of the retardation time τ = 100 s and τ = 10 s.

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 = 1 GPa–1, D1 = 9 GPa–1, and consider two temperatures* T > Tr for which the retardation times (in Eq. 2.9) are τ = 10 s and τr = 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: D(λ;T) = D0 + D1 [1 – e–λ/τ ] D(λ;Tr) = D0r + D1r [1 – e–λ/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. Since log aT λ = log λ + log aT, then aT > 1 shifts the curve T to the right onto the master curve Tr, the latter having aT = 1 by definition.† To shift the curve T vertically, one divides the * Or two ages te < te,r. † 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.

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

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, D(λ; T ) = bT D(aT λ; Tr)

[2.10]

For this simple example, D0 = D0r and D1 = D1r; that is, the only difference between them are the retardation times τ, τr. Then, bT = 1 and using Eq. 2.9, D0 + D1 [1 – e–λ/τ ] = D0 + D1 [1 – e–aTλ/τr]

[2.11]

from which

τ = 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 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 © Woodhead Publishing Limited, 2011



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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 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 [7, 8]. 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 [3]. 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 2.3. 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 [4]. 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 hr. The chosen reference temperature is Tr = 40°C. By performing horizontal shifts of magnitude log a(T ) on the data sets with T > Tr, the 16 hr-tail of the curves extend 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, i.e., 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

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2.3  Momentary curves D(λ) at various temperatures, all with age te = 166 hr ; momentary master curve D(λ;te) at Tr = 40ºC (solid line under the shifted data at 40ºC ) and shifted to 100ºC (dotted line).

2.4  Approximate time span over which two momentary curves superpose.

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right by plotting the compliance D(aT λ; T) vs. time aTλ. Therefore the overlapping time span starts at λ = λ0 and ends at λ = 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. A typical experimental setup for larger specimens is presented in [9]. 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, then shift the model curves [10], 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 value 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: n

err = 1n ∑[D(λi; T ) – bT D(a T λi; Tr)]2

[2.13]

i

Table 2.1  Regression parameters and shift factors for the TTSP study depicted in Fig. 2.3 T [ºC]

D0

D1

m

log aT

log bT

40 60 80 90 115 120

0.36 0.396 0.429 0.438 0.518 0.588

0.016 0.008 0.004 0.009 0.016 0.02

0.166 0.227 0.306 0.264 0.271 0.294

1 1.742 4.033 7.758 67.114 229.328

1 1.051 1.103 1.147 1.35 1.496

Average COV

0.455 0.184

0.012 0.496

0.255 0.201

– –

– –

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where n is the number of data points. Then, the shift factors aT, bT are found by minimizing the error [6]. For example, in MATLAB: z = 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 [6]. 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 to that of long-term creep, as shown in Fig. 2.5.

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

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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, time-temperature 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 [3, 11, 12]. 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 ageing shift factor ae(te), respectively (see section 2.3), i.e., D(λ; T, te) = bT D(aT ae λ; Tr, ter)

[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 Williams-Landel-Ferry (WLF) equation:  C1 (T – Tr) log aT =  C2 + T – Tr

2.6  Temperature shift factor plot for the data in Fig. 2.3. Tr = 40ºC, te = 166 hr, horizontal C1 = -1.22503, C2 = 122.669, and vertical C1v = 0.0762931, C2v = -116.057.

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[2.15]

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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 [4]. 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 = 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 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 above, Eq. 2.10 yields:

∆(τ ; T) = aT ∆(aT τ ; Tr)

[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 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, wether 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 [3]. 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

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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 2.3; 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 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 (Eq. 2.5). Therefore, a simpler model is required, such as the power law (Eq. 2.6), re-written as follows: Dc(λ) = D(λ) – D0 = D1λm

[2.17]

where Dc is the creep compliance, D0 is the elastic compliance, and D(λ) is the total compliance measured in the experiment.

2.7  Compliance vs. time at constant temperature T = 115ºC 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 5,000 hr.

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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 2.2. Assuming that the elastic compliance is independent of age, D0 should be the same for all data sets. This is confirmed by the low coefficient of variance (COV) for D0 in Tables 2.2 and 2.3. Similarly, in [13, 14], 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 [13, 14], recognizing the need for mechanical conditioning of the specimens. The resulting power law exponent m might not be identical for all plots, i.e., for all ages, but the dispersion usually is very small. For example, for the data in Fig. 2.7, the average is m = 0.228 with a COV = 16.8%. If the first test (at te = 2 hr) is neglected, as recommended in [13, 14], the COV reduces to COV = 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 = m constant for all curves and adjusting only D0, D1. Numerical results are shown in Table 2.2  Initial regression parameters for the aging study depicted in Fig. 2.7 Age te

D0

D1

m

879 160 32 8 2

0.503 0.523 0.511 0.536 0.58

0.018 0.014 0.026 0.032 0.018

0.239 0.287 0.262 0.269 0.364

Average COV

0.518 0.028

0.022 0.361

0.264 0.075

Table 2.3  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

0.518 0.512 0.513 0.533 0.53

0.012 0.018 0.026 0.033 0.044

0.264 0.264 0.264 0.264 0.264

0 0.657 1.201 1.637 2.102

0 0.006 0.005 -0.015 -0.012

Average COV

0.521 0.019

0.027 0.468

0.264 0

– –

– –

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Table 2.3. As it can be seen in Fig. 2.8, the impact of such averaging is minimal [9]; i.e., 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(λ;te) = Dc(aeλ;ter)

[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(λ;ter) = log D1r + m log(aeλ)

[2.19]

log Dc(λ;te) = log D1 + m log(λ)

[2.20]

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

( )

D   1/ m ae = 1 D1r 

[2.21]

be = D0r – D0

– = 0.264 2.8  Power law model for the creep data in Fig. 2.7 with m (Tr = 115ºC, ter = 879 hr).

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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, i.e., 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). 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) [13]. 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:  d log ae > 0 µe = –  d log te

[2.22]

which is normally assumed to be constant for a wide range of temperatures, except near the glass transition [3, 12]. Temperature dependence was reported in [4, 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 = 879 hr 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 = – μe log(te /ter)

[2.23]

Equation 2.23 is analogous to the WLF equation (2.15). For example, the aging momentary master curve for age te = 5,000 hr, 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 to 879 hr in this case).

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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 = 1 at ter = 879 hr (Tr = 115ºC ).

Unlike 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 materials properties determined by the procedure presented in this section. Also, since power law models fit the master curve in Fig. 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 on 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 have been shifted to a common temperature T = 60°C and age te = 1 hr. To facilitate comparison, a small, additional vertical shift of log 1.04 [1/GPa] 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

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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 = 60ºC and age te = 1 hr.

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 [13]. In Fig. 2.10, the momentary curve from the temperature study is to the right of the target age and temperature (i.e., te = 1 hr, T = 60°C). Therefore, it is shifted from te = 166 hr, Tr = 40°C to te = 1 hr, T = 60°C by plotting bT D(λ) vs. [1/(aT ae)]λ with aT = 1.732, bT = 1.037, and ae = 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 = 879 hr, Tr = 115°C to te = 1 hr, T = 60°C by plotting bT D(λ) vs. (aT /ae)λ with aT = 57.057, bT = 1.345, and ae = 209.573.

2.4

Effective time theory

In this section, the relationship between unaged and real time, i.e., 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 [3]. 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:

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–d log ae(te) = μe d log te –d log ae(te + t) = μe d log(te + t) Therefore, the shift factor evolves with time as: a (t + t) te µe ae(t) = e e = <1  ae (te) te + t 

( ) 

[2.24]

[2.25]

On the momentary master curve of the time–age superposition study reported in Fig. 2.7, one may assume that aging has stopped at te = 879 hr because the testing time is so short (less than te /10) that no further aging is allowed to manifest itself during the test time. Superposition of momentary compliance curves obtained for increasing ages te indicates that the retardation times τ, which represent material behavior, are longer as the sample ages. For t → ∞, aging stops, and choosing that long age as the reference state to construct the master curve, one has ae(∞) = 1. Any shorter age has a shift factor ae > 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 being 1/ae. Therefore, the same amount of creep accumulates in a shorter real time interval dt than in effective (ageless) time interval dλ, which are related by: dt = (1/ae)dλ

[2.26]

which leads to the definition of the effective time [3, (85)]: t

λ = ∫ ae(ξ)dξ

[2.27]

0

Substituting Eq. 2.25 and integrating, yields [3, (117–188)]:

λ(t) = te ln [1 + t/te]                                  if μ = 1 [2.28] te λ(t) = [(1 + t/te)(1 – μ) – 1]                if μ < 1 [2.29] 1–μ where ln denotes the natural logarithm (base e). If the material did not age, the momentary master curve D(λ;te) on Fig. 2.3 would predict the compliance as a function of time. But since the material does age, the long-term creep compliance D(t) must be smaller, that is: D(t) = D(λ(t); te)

[2.30]

with λ(t) given by Eq. 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 references [4, Fig. 14–15] and [15]. If the material did not age (μ = 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 Eq. 2.28, 2.29, the real time can be calculated as: t/te = –1 + exp(λ/te)                       for μ = 1

[2.31]

t/te = –1 + (αλ/te + 1)1/α                for μ < 1

[2.32]

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where α = 1 – μ. The longest data in the momentary master curve of an aging study, such as Fig. 2.7, is constructed for λ/te = 0.1. According to Eq. 2.31, no prediction can be made for time exceeding t/te = 0.1, that is t = 879 hr in the example at hand. The model equation that fits the data cannot be extended beyond λ/te = 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 = 879 hr down to a shorter aging time, say te = 30 min, does not help in predicting longterm creep because approximately the same λ/te time span will be covered up to λ/te = 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 predictions to long times because the momentary curves do not shift much to the right. For example, the shift to 5,000 hr is shown in Fig. 2.7. In order to predict long-term creep, a momentary master curve that extends beyond λ/te = 0.1 is needed. The momentary master curve from the TTSP study (section 2.2, Fig. 2.3) serves the purpose [3]. 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 = 166 hr, and predictions are sought up to one year, then t/te = 8640/166 = 52. Assuming the aging study (section 3) yields μe ≈ 1, using Eq. 2.31 yields λ/te = ln(t/te – 1) = 3.931, or λ = 652 hr. It is not possible to produce experimentally a momentary curve for 652 hours with a material that has been aged for only 166 hours. However, the momentary master curve for T = 40°C (Fig. 2.3) easily exceeds the 652 hr (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 [13] or environmental chamber [9, 14].

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, i.e., te.

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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 Eq. 2.28 and 2.29 in terms of the real time t.

2.6

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 [15]. 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 Eq. 2.15 and 2.23 in terms of the known coefficients μe, C1, C2, C1v, C2v. Next, each time interval ∆t = ti – ti–1 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 Eq. 2.28 and 2.29, compute λi, λi–1, and the unaged time interval ∆λi = λi – λi–1. This interval can be adjusted to the reference temperature as:

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∆λ′i = ∆λi /aT

[2.33]

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

λ′n = ∑ ∆λ′i

[2.34]

i=1

which is then transformed to real time using Eq. 2.31 and 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 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 [16, 17], and so on. Application of this methodology for the case of stress induced non-linearity awaits attention.

2.8

References

[1]  E. J. Barbero, Finite Element Analysis of Composite Materials, Taylor & Francis, 2007. [2] J. G. Creus, Viscoelasticity: Basic Theory and Applications to Concrete Structures, Springer-Verlag, 1986. [3] L. C. E. Struik, Physical Aging in Amorphous Polymers and Other Materials, Elsevier Scientific Pub. Co.; New York, 1978. [4] J. Sullivan, Creep and physical aging of composites, Composites Science and Technology 39 (3) (1990) 207–32. URL http://dx.doi.org/10.1016/0266–3538(90)90042–4 [5] T. Gates, M. Feldman, Time-dependent behavior of a graphite/thermoplastic composite and the effects of stress and physical aging, Journal of Composites Technology and Research 17 (1) (1995/01/) 33–42. [6] E. J. Barbero, Web resource. URL http://www.mae.wvu.edu/barbero/ [7] D. Matsumoto, Time-temperature superposition and physical aging in amorphous polymers, Polymer Engineering and Science 28 (20) (1988) 1313–1317. [8] S. Vleeshouwers, A. Jamieson, R. Simha, Effect of physical aging on tensile stress relaxation and tensile creep of cured epon 828/epoxy adhesives in the linear viscoelastic region, Polymer Engineering and Science 29 (10) (1989) 662–670. [9] E. Barbero, K. Ford, Determination of aging shift factor rates for field-processed polymers, Journal of Advanced Materials 38 (2) (2006) 7–13.

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[10] R. Bradshaw, L. Brinson, Physical aging in polymers and polymer composites: An analysis and method for time-aging time superposition, Polymer Engineering and Science 37 (1) (1997) 31–44. [11] A. Lee, G. McKenna, Viscoelastic response of epoxy glasses subjected to different thermal treatments, Polymer Engineering and Science 30 (7) (1990) 431–435. [12] A. Lee, G. B. McKenna, Physical ageing response of an epoxy glass subjected to large stresses, Polymer 31 (3) (1990) 423–430. URL http://dx.doi.org/10.1016/0032–3861(90)90379–D [13] E. J. Barbero, M. J. Julius, Time-temperature-age viscoelastic behavior of commercial polymer blends and felt-filled polymers, Mechanics of Advanced Materials and Structures 11 (3) (2004) 287–300. [14] E. Barbero, K. Ford, Equivalent time-temperature model for physical aging and temperature effects on polymer creep and relaxation, ASME Journal of Engineering Materials and Technology 126 (4) (2004) 413–419. [15] E. Barbero, S. Rangarajan, Long-term testing of trenchless pipe liners, Journal of Testing and Evaluation 33 (6) (2005) 377–384. [16] E. Barbero, R. Luciano, Micromechanical formulas for the relaxation tensor of linear viscoelastic composites with transversely isotropic fibers, International Journal of Solids and Structures 32 (13) (1995) 1859–1872. [17] R. Luciano, E. J. Barbero, Analytical expressions for the relaxation moduli of linear viscoelastic composites with periodic microstructure, ASME J Appl Mech 62 (3) (1995) 786–793.

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3 Time-dependent behavior of active/intelligent polymer matrix composites incorporating piezoceramic fibers K.-A. Li and A. Muliana, Texas A&M University, USA Abstract: Piezoceramic fibers, such as lead zirconate titanate (PZT) fibers, dispersed in viscoelastic polymer matrix form an active composite which can be used as sensor, actuator, or energy harvester. We discuss electromechanical coupling and time-dependent effects in active materials and micromechanical models for predicting effective time-dependent properties of active composites. Finally, we present an application of using active composites to control deformation in load-bearing structural components. Key words: active fiber composite, piezoelectric ceramics, viscoelastic material, micromechanical model, electromechanical coupling.

3.1

Introduction

Polymer matrix composites (PMCs) are widely used in many engineering applications, such as aircraft structural components, military vehicles, and bridges. Modern military vehicles and commercial aircraft require the ability to sustain complex loading, such as fatigue and impact, and various environmental conditions, such as extreme changes in temperature and moisture. For this reason, the current generation of advanced composite materials are expected to incorporate multiple functions, in that they not only serve as load-bearing components, but also are capable of monitoring and controlling the performance of the structures while adapting to various external stimuli. One approach in developing intelligent composite structures is to integrate active materials, such as piezoelectric materials, into laminated composite systems or other host structures. Various types of piezoelectric materials, such as barium titanate (BaTiO3), lead zirconate titanate (PZT), and polyvinylidene fluoride (PVDF), have been used in intelligent composites. Piezoceramics generally have higher electromechanical and piezoelectric properties compared to those of piezoelectric crystals and polymers, which make them capable of sustaining a wider range of loadings. Brittle characteristics of piezoceramics may lead to early failure that limits their applications. PVDF can overcome a drawback of brittle piezoceramics; however, it has low piezoelectric constants (1–2 orders of magnitude less than the PZT) and time-dependent behaviors. PZT fibers dispersed in ductile polymers provide more flexible and compliant transducers compared to the monolithic PZT wavers (Bent and Hagood, 1997) and higher electromechanical and piezoelectric properties compared to those of 70 © Woodhead Publishing Limited, 2011

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PVDF. The existence of polymer matrix in the active fiber composites prevents catastrophic failure due to fiber breaking. Active PMCs have the following prominent features: they have high stiffness and large bandwidth, making it possible to use a wider range of signals in actuator applications; and they have better strength and conformability, creating more flexible and pliable structures and improving resistance to brittle damage. Polymer matrix is known for its viscoelastic characteristics especially at elevated temperatures. In addition, PZTceramics exhibit time-dependent behaviors under both mechanical and electrical loadings (Fett and Thun 1998; Cao and Evans, 1993; Schäufele and Hardtl, 1996). Experimental studies have shown that the dielectric and piezoelectric properties of polymer and PZT ceramics (Strobl, 1997; Hall, 2001) can strongly depend on time and rate of loading. The different time-dependent behaviors of the fiber and matrix constituents result in complex electromechanical responses of the active PMCs. Thus, understanding overall time-dependent performance of active PMCs becomes important to improve reliability in using intelligent composites. Two kinds of active PMCs have been commercialized, which are active fiber composite (AFC) and macro fiber composite (MFC). AFC was developed by Bent and Hagood (1997) at Massachusetts Institute of Technology and MFC was developed at NASA Langley Research Center. These two active composites do not have significant differences in their overall structures, but only in the types and shapes of the fibers. AFC uses a circular fiber cross-section while MFC has a rectangular fiber cross-section. The maximum fiber volume content of AFC is less than 0.785 because of the restriction of the fiber geometry. The fiber volume content of MFC could reach up to 0.824 (Williams et al., 2004). Increasing fiber volume contents enhances the performance and improves the stiffness and strength of the active composites. An understanding of the linear and nonlinear electromechanical performance of AFC and MFC has recently been gained, which can be found in Wickramasinghe and Hagood (2004), Williams (2004), Williams et al. (2006), among others. Lin and Sodano (2008) suggested a new piezoelectric composite called an active structure fiber (ASF). The structure of an ASF has a circular carbon fiber as the core covered by layers of piezoelectric material and electrode. The ASF, which can be dispersed in polymer matrix, is good for controlling vibration, power harvesting, and structural health monitoring. Odegard (2004) has examined linear electromechanical responses of four active PMCs, i.e., graphite/PVDF fiber composite, SiC/PVDF particulate composite, PZT-7A/LaRC-Si fiber composite, and PZT-7A/LaRC-Si particulate composite. The use of LaRC-Si polymer matrix is appealing for high-temperature applications. Micromechanical modeling approaches have been widely used to determine effective electromechanical properties of active composites. Pedersen (1989) discussed a generalization of a volume averaging scheme to obtain effective properties of composites with coupled mechanical and non-mechanical problems, such as thermo-mechanics, piezoelectric, and electromagnetic. Dunn (1993), Dunn and Taya (1993), Aboudi (2001), Odegard (2004), Lee et al. (2005),

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and Muliana (2010a), among others, have developed micromechanical models for predicting linear field-coupling responses of active composites. Limited micromechanical models that incorporate time-dependent effects on the overall electromechanical behaviors of active composites have been developed. Li and Dunn (2001) presented complex electromechanical properties of linear viscoelastic piezocomposites. They combined the correspondence principle and homogenized micromechanical models to determine the complex properties. The finite element (FE) method has been used to generate detailed microstructures of active PMCs. Tan and Tong (2001) established a rectangle model (r model) and a rectangle-cylinder model (RCR model) for piezoelectric-fiber-reinforced composites to predict mechanical and piezoelectric properties of the composites at several fiber volume contents. They compared the predictions of the effective properties with the ones obtained from FE models of detailed composite microstructures and existing experimental data. Beckert and Kreher (2003) established two FE micromechanical models for (a) bulk film Interdigitated Electrode (IDE) actuator and (b) composite IDE actuator. For the bulk film, the FE model consists of 2D IDE-containing electrode, isolator coating and piezoelectric material. For the composite IDE model, they incorporated isolator coating, electrode layer, dielectric interlayer, polymer matrix, homogenized composite layer and piezoceramics fiber. The results suggested that for the piezoelectric layer with thickness of 200 µm the electrode width should be 300 µm and the spacing of electrodes should be 1000 µm. The results also showed that reducing the thickness of the dielectric interlayer or enhancing the dielectric constant of the layer could improve the overall performance of active PMCs. Nelson et al. (2003) used a representative volume element (RVE) model generated using FE to study responses of an IDE on a bulk PZT substrate. They observed that the optimal actuation occurred at electrode widths equal to half of the substrate thickness. For thin substrates, the spacing between the electrode fingers can be reduced to enable lower driving voltages. This chapter examines time-dependent behaviors of active PMCs consisting of active PZT fibers and polymer matrix. Two micromechanical modeling approaches are considered. The first approach uses a simplified micromechanical model to obtain effective electromechanical responses of the active PMCs. This approach treats the composites as fictitiously homogeneous materials and limited information on the local field variables can be incorporated in predicting overall responses of the active PMCs. The second approach generates RVEs of composite microstructures using the FE method, which allow quantifying detailed variations in the field variables of the constituents in the active composites. However, the RVE models with detailed FE meshes generally result in high computational cost. This chapter is organized as follows. Section 3.2 discusses time-dependent constitutive material models for the constituents in the active PMCs. A general time-integral function is used for the electromechanical and piezoelectric components. This integral representation can be easily reduced to model linear

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piezo-elastic behaviors. Section 3.3 deals with a simplified micromechanical model for fiber-reinforced active composites. RVE models for both AFC and MFC generated using FE are discussed in Section 3.4. Prediction of the effective electromechanical responses obtained from the simplified micromechanical model and FE analyses of the AFC and MFC microstructures is given in Section 3.5. Section 3.6 presents a practical application of using active PMCs to control deflection in the structures. Section 3.7 is dedicated to concluding remarks.

3.2

Linearized time-dependent model for materials with electromechanical coupling

The linearized strain-electric flux coupled equations with a time-dependent effect are expressed as:

[3.1]



[3.2]

where are the scalar components of the compliance tensor, and are the scalar components of the piezoelectric and dielectric constants, respectively. Variable Di is the scalar component of the electric displacement (electric flux), and Ei is the scalar component of the electric field strength (negative of the electric potential gradient). Variables σij and εij represent the scalar components of the stress and strain tensors, respectively. The material properties and are defined at zero electric field (short-circuit) and zero stress, respectively. It is noted that the piezoelectric constants defined at zero stress and at zero electric field are thermodynamically identical; thus, their superscript will be dropped. Other expressions for the linearized piezoelectric relations are obtained by arbitrarily choosing independent field variables, as discussed in Damjanovic (1998). An alternative expression for time-dependent electromechanical coupling is:

[3.3] [3.4]

where are the scalar components of the stiffness at zero electric field, are the scalar components of the piezoelectric constant, and are the scalar components of the dielectric constants measured at zero strain. The above electromechanical properties of piezoelectric materials can be characterized at constant stresses (strains) or constant electric fields (electric displacements), which might result in substantially different values due to the field-dependent (nonlinear)

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behaviors. It is also possible to characterize the elastic stiffness (or compliance) under an open-circuit condition. The material properties in the linear electromechanical relations (Eq. 3.1–3.4) are related, which for the time-dependent behaviors are expressed in terms of the convolution integral form as:

[3.5]



[3.6]



[3.7]

where δij is the Kronecker delta and the convolution relation is defined as:

[3.8]

The coupling effects in the equations governing the responses of the materials require solving the mechanical and non-mechanical constitutive equations simultaneously. The above integral forms of the constitutive equations incorporate the effects of loading histories such as those of viscoelastic materials. When histories of loading have insignificant effects on the electromechanical response of the materials, the above constitutive equations can be reduced to linearized piezo-elastic relations. In this study, Eq. 3.1–3.2 or Eq. 3.3–3.4 are considered for predicting response of the active PZT fibers. When non-piezoelectric polymer matrix is considered, only the first term of Eq. 3.1 or 3.3 and the second term of Eq. 3.2 or 3.4 are used to model responses of the polymers. A recursive method is used to solve the convolution integrals in Eq. 3.1–3.4. Discussion of the recursive method for integrating viscoelastic constitutive material models can be found in Taylor et al. (1970), Haj-Ali and Muliana (2004), and Sawant and Muliana (2008). Experimental studies on PZT ceramics and PVDF polymers show timedependent behaviors of the electromechanical and piezoelectric properties (Hall, 2001; Vinogradov et al., 2004). The moduli and dielectric constants of PZT and PVDF are shown to relax with time, while piezoelectric properties dijk under various constant stresses increase as time progresses. Here, we examine the effects of time-dependent properties on the overall performance of piezoelectric materials, i.e., PZT-5, subject to various loading histories. Time-dependent properties of the PZT-5 along its polarization axis (x3) are given as:

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75 [3.9] [3.10] [3.11]

where κo = 8.854 × 10–12 F/m is the dielectric permittivity at vacuum and the unit of t is in seconds. It should be noted that the above time-dependent properties resemble qualitative measures of the experimental data of PZT specimens at room temperature. Available time-dependent responses of mechanical, electrical, and piezoelectric properties are reported separately for different PZTs. For the purpose of our analyses, we calibrate time-dependent material constants in Eq. 3.9–3.11 to simulate available experimental data in Fett and Thun (1998) and Hall (2001). Due to limited experimental data, we assume all other electromechanical properties of the PZT-5 to be independent of time. To examine the significance of the time-dependent behaviors we perform the following case studies. The first case study subjects the PZT sample to a constant stress of 20 MPa along its polarization axis (x3). The mechanical strain and electrical displacement due to a constant stress are illustrated in Fig. 3.1. Significant time-dependent responses are exhibited, which is expected. In the second case study, we apply a ramp electric field E3 = 1t kV/m. The axial strain and electrical displacement responses are given in Fig. 3.2. The responses are also compared to those generated using constant (time-independent) piezoelectric and dielectric properties. It is seen that a significant increase in strain is observed when the time-dependent piezoelectric constant is considered due to the increasing value of d333 with time and accumulated strain response from history of the applied electric field. On the contrary, the time-dependent dielectric properties result in smaller electrical displacement due to the relaxation behavior of κ33. In this case, the differences in the electrical displacements for the time-dependent and constant material properties are negligible. Finally, we perform the third case study by prescribing a PZT sample with a constant stress of σ33=20 MPa and a ramp electric field of E3 = 1t kV/m. Figure 3.3 illustrates the corresponding strain and electric displacement. It is seen that time-dependent properties can significantly influence overall electromechanical behavior of active materials.

3.3

Simplified micromechanical model of homogenized active polymer matrix composites (PMCs)

A simplified micromechanical model is used to provide effective electromechanical and piezoelectric properties of active fiber-reinforced PMCs. The studied active PMC consists of unidirectional ferroelectric fibers dispersed in polymer matrix. The microstructures of the active PMC are idealized by a periodically distributed arrays of fibers embedded in a homogeneous polymeric

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3.1  Time-dependent strain and electrical displacement under a constant stress 20 MPa: (a) axial strain; (b) electrical displacement.

matrix. Interphases between the fibers and matrix are assumed perfect. A unit-cell model consisting of four fiber and matrix subcells is generated as illustrated in Fig. 3.4. Haj-Ali and Muliana (2004) and Muliana and Sawant (2009) have previously introduced this unit-cell model for analyzing nonlinear viscoelastic behaviors of fiber-reinforced PMCs. The first subcell is the fiber constituent and subcells 2, 3, and 4 are the matrix constituents. For convenience in presenting basic formulations of the homogenized properties of active PMCs, the following linearized constitutive relation is adopted:

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3.2  Strain and electrical displacement subject to a linear ramp electric field: (a) axial strain; (b) electrical displacement.

Z = ΩΣ

[3.12]

ΣT = {σ11, σ22, σ33, σ12, σ13, σ23, E1, E2, E3} ZT = {ε11, ε22, ε33, ε12, ε13, ε23, D1, D2, D3}

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[3.13]

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3.3  Strain and electrical displacement subject to σ33 = 20 MPa and a ramp electric field of E3 = 1t kV/m: (a) axial strain; (b) electrical displacement.

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3.4  Simplified microstructures of active fiber PMC.

The effective responses of the composites are formulated based on volume averaging of the field variables in the constituents in a representative unit-cell model. Each unit-cell model is divided into a number of subcells and the spatial variation of the field variables in each subcell is assumed uniform (Haj-Ali and Muliana, 2004). The effective field variables, indicated by an over-bar, are defined as:

 k = 1,2,3 and i = 1,2,…, 9

[3.14]

[3.15]

Superscript (β) denotes the subcell’s number and N is the number of subcells. Variables and are the components of the average field quantities in each subcell. The unit-cell volume V is defined by

.

Following Hill’s (1964) micromechanical formulation for linear elastic materials, local average field quantities can be expressed in terms of the effective (homogenized) field variables which are written as:

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[3.16]

where and are the concentration tensors for the subcell β. Since the two field quantities in Eq. 3.14–3.16 are related by material constants, it is not necessary to define two concentration tensors in formulating the micromechanical model. In case the concentration tensors are used, substituting Eq. 3.16 into 3.14 gives the following effective field quantities:

[3.17]

which implies that:



[3.18]

Imposing the linear piezoelectric relations as in Eq. 3.12 for each constituent and using relations between the local average field quantities and the effective field quantities as in Eq. 3.16, the average (homogenized) strains and electric displacements are now written as:

[3.19]

It is seen from Eq. 3.19 that the linearized material constants for the homogenized composites are given by:

[3.20]

Next, we form micromechanical relations between the fiber and matrix subcells in the unit-cell model (Fig. 3.4) by imposing traction continuity and displacement compatibility at the interphases of all subcells. The linearized micromechanical relations for active PMCs, having unidirectional fibers aligned in the x3 direction, are summarized as follows. The relations along the fiber axial direction (x3) are:

[3.21]



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[3.22]

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The micromechanical relations in the x2 direction are:



[3.23]



[3.24]



[3.25]

The micromechanical relations in the x1 direction are:



[3.26]



[3.27]

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[3.28]

The transverse shear relations are given as:

[3.29]

The above linearized micromechanical relations (Eq. 3.14–3.29) give exact effective properties1 of the homogenized composites only when linear piezoelectric constitutive equations are used for all constituents (subcells) in the unit-cell model. Here, we deal with different time-dependent constitutive models for the fiber and matrix constituents that require incorporating different histories of loadings from all constituents in predicting effective performance of the active composites. Thus, using time-dependent constitutive models in Section 3.2 could lead to violation in the linearized micromechanical relations (Eq. 3.21–3.29), which is expressed by the following residual:



[3.30]

instead of the above It is also possible to define the residual in terms of equation. In order to obtain effective time-dependent response of active PMCs, we need to minimize the residual in Eq. 3.30. The linearized micromechanical relations are first used to perform trial solutions at every instant of time and a corrector scheme is added to minimize the residual. This study uses the NewtonRaphson iterative method as a corrector scheme. Once the residual has been minimized, the components of the concentration matrices are determined as:

1 The exact effective properties in this context are attributed to satisfying the volume average schemes in Eq. 3.14–3.15 and linear electromechanical relations for all subcells. It should be noted from Eq. 3.14–3.15 that average field variables (a mean field approach) of each subcell are used in the volume averaging scheme, while the exact field variables generally vary with the locations in each subcell. The use of the mean field approach results in approximate solutions of the effective field variables.

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Time-dependent behavior of active polymer matrix composites



83

[3.31]

Upon determining the concentration matrices at each instant of time, the field variables in each subcell due to the prescribed stress and electric field at the macro (homogenized composite) level can be determined using Eq. 3.16 and the constitutive equations in Section 3.2 are used to calculate the field variables in each subcell. Finally, the effective field variables at every instant of time are evaluated using the volume average schemes in Eq. 3.14 and 3.15.

3.4

FE models of representative volume elements (RVEs) of the active PMCs

The effective electromechanical and piezoelectric responses of the active PMCs are also analyzed using FE. Microstructures of AFC and MFC are generated using ABAQUS FE. To reduce complexity in generating the microstructures of the AFC and MFC, the existence of interdigitated electrode fingers is ignored and the electric field is assumed to be uniformly distributed along the fiber longitudinal axis. The microstructures of the AFC and MFC are similar except in the shape of the fibers. MFC has fibers with a rectangular cross-section, while AFC uses fibers with a circular cross-section. Various types of PZT fibers have been utilized in the AFC and MFC. MFC configuration could reach a volume fraction up to 0.824, while the maximum fiber volume content of AFC is 0.785. Figure 3.5 illustrates the cross-section of the AFC consisting of soft PZT fibers and epoxy matrix placed in between two electrode layers.

3.5  Microstructures of AFC having PZT-5A and epoxy matrix placed between two electrode layers (source: Advanced Cerametrics, www.advancedcerametrics.com).

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Two RVE-FE meshes are generated for both AFC and MFC. The first RVE model consists of a single fiber of a square or circular cross-section surrounded by epoxy matrix (Fig. 3.6). The second RVE model consists of five fibers embedded in epoxy as illustrated in Fig. 3.7. AFC and MFC, having 20%, 40%, and 60% fiber contents, are generated by varying the fiber dimensions. Table 3.1 reports dimensions of fibers used for the AFC and MFC RVE models at three different fiber contents. The AFC has a circular fiber of a radius R, while the MFC has a square fiber with a side length L. We generate five-fiber RVE models only for AFC and MFC with 40%

3.6  Single-fiber RVE models for AFC and MFC.

3.7  Five-fiber RVE models for AFC and MFC. Table 3.1  The dimensions of fiber for AFC and MFC RVE models per unit area Active PMC

Fiber volume content



20%

40%

60%

AFC MFC

R=0.252 L=0.447

R=0.357 L=0.632

R=0.437 L=0.775

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fiber volume fraction. We examine response of the AFC and MFC obtained using the single and five-fiber RVE models. A three-dimensional continuum piezoelectric element, C3D8E, in ABAQUS software is used for the RVE models. The single fiber model has 10 000 elements, while the five-fiber model has 50 000 elements. To determine effective electromechanical and piezoelectric properties, different boundary conditions are prescribed to the RVE models. Figures 3.6 and 3.7 illustrate the RVE models with a coordinate system given in Fig. 3.6. Surfaces 1, 2, and 3 have unit-outward normal vectors in the positive x1, x2, and x3 directions, respectively, and surfaces 4, 5, and 6 have unit-outward normal vectors in the negative x1, x2, and x3 directions, respectively. Table 3.2 summarizes the prescribed boundary conditions used to evaluate the effective properties of the RVE models, where Ui represents the components of the displacement applied in xi direction, σij denotes the components of the applied stress with a unit-outward normal vector in xi direction and a traction component in xj direction, and C is the applied surface charge. The effective linear elastic modulus and Poisson’s ratio , are determined as follows:



[3.32]

Table 3.2  Prescribed boundary conditions on RVE models Property/surface

1

2

3

4

5

6

U1=0 U1=0

U2=0 U2=0

U3=0 U3=0

σ31=σo U2=0 U2=0 U3=0 U3=0

U1=0 U2=0 U3=0

U2=0 U3=0

G12

σ12=σo U2=0 U2=0 U2=0 U2=0 U2=0 U3=0 U3=0 U3=0 U3=0 U3=0

U1=0 U2=0 U3=0

e113 = e131 = e223 = e232

C=Q U2=0 U2=0 U3=0 U3=0

C=–Q U2=0 U2=0 U3=0 U3=0

U1=0 U2=0 U3=0

e311 = e322 σ11=σo C=Q U1=0 U2=0

C=–Q U3=0

e333

C=–Q U3=0

E11=E22 σ11=σo E33 σ33=σo G23=G31 U2=0 U2=0 U3=0 U3=0

κ11/κ o

σ31=σo U2=0 U3=0

σ33=σo C=Q U1=0 U2=0

C=Q U2=0 U2=0 U2=0 U3=0 U3=0 U3=0

C=–Q U2=0 U2=0 U3=0 U3=0

U1=0 U2=0 U3=0

κ33/κ o C=Q U1=0 U2=0

C=–Q U3=0

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In a similar way, the effective linear properties for

,

, and

are:



[3.33]

Once the effective mechanical properties at zero (constant) electric field have been determined from Eq. 3.32 and 3.33, the linear elastic compliance SE and stiffness CE matrices can be obtained. Next, the effective linear piezoelectric constants are calculated as:



[3.34]

The effective dielectric properties along and transverse to the fiber direction are obtained as:



[3.35]

The above formulations are used to determine effective linear properties of the AFC and MFC by prescribing constant average field variables at the surfaces of the RVE models. When the nonlinear and time-dependent responses are incorporated to the fiber and matrix constituents, the relations in Eqs. 3.32–3.35 are imposed to provide linearized trial solutions and corrector schemes are added to minimize errors from the linearized relations. The predictor-corrector scheme for the RVE models is performed within the ABAQUS FE code (ABAQUS 2008). At every incremental time step in the piezoelectric analyses within a quasistatic loading, the following conditions are defined:

[3.36]

where and are the residual vectors, M and N denote the nodal number, uN, N M M ϕ , P , and Q are the displacement, electric potential, mechanical force, and

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electric charge vectors, respectively. The displacement stiffness matrix, dielectric matrix, and piezoelectric coupling matrix are expressed as:



[3.37]

and are the spatial derivative of a shape function NN that interpolates where the nodal displacement and electric potential, respectively. The material constants in the above equations are defined in Eq. 3.3–3.4.

3.5

Effective electromechanical and piezoelectric properties

The simplified micromechanical model provides effective anisotropic electromechanical and piezoelectric properties of active PMCs in that the composite systems are treated as homogeneous media. As discussed in Section 3.3, the effective properties obtained using the simplified micromechanical model are derived based on selected unit-cell models of unidirectional fiber composites. Although the present homogenization scheme recognizes the different time-dependent responses of the fiber and matrix constituents, field variables obtained using the homogenization scheme represent approximate (average) field variables with limited spatial variations. The homogenization scheme does not quantify detailed variations in local field variables of the constituents as in the heterogeneous materials (only four subcells are considered). These variations are often pronounced near the fiber-matrix interphase regions, such as discontinuities in the stress field. These discontinuities potentially lead to fiber-matrix debonding. In this study, we examine average (effective) properties, determined using the four-cell micromechanical model, and compare them to the properties of heterogeneous composite systems. The heterogeneous composites are simulated for AFC and MFC microstructures2 in ABAQUS FE software. The first study discusses the effective linear properties and the second study presents the effective time-dependent properties.

3.5.1 Linear electromechanical and piezoelectric properties of active PMCs We verify the two micromechanical modeling approaches by comparing their effective material properties to the experimental data. Experimental data on 2 We

use microstructures of AFC and MFC, which are represented by RVE models with single- and five-fiber arrangements, to examine detailed variations in field variables of the different constituents. By selecting these RVEs we made a tacit assumption that fibers are uniformly distributed in the entire matrix medium, which contradicts with real microstructures of composites.

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PZT-5A/epoxy active fiber PMCs reported by Nelson et al. (2003) are used for comparisons. The linear electromechanical and piezoelectric properties of the PZT-5A are given in Table 3.3. Fibers are aligned in the x3 direction. The properties of the isotropic epoxy matrix are given in Table 3.4. Prediction of the simplified micromechanical model and the RVE models of the AFC and MFC having single and five fibers are illustrated in Fig. 3.8. The coupling coefficient k33 and compliances measured under short-circuit and open-circuit are defined as:

[3.38]

[3.39]

Table 3.3  Electromechanical and piezoelectric properties of PZT-5A (Nelson et al., 2003) SD3333 ν12 ν13 = ν23  SE3333 (10–3 1/GPa) (10–3 1/GPa)

d333 d311 (10–12 C/N) (10–12 C/N)

κσ33 κo

15.3

263

1350

9.7

0.352 0.438

–102

Table 3.4  Electromechanical properties of epoxy (Nelson et al., 2003) E (GPa)

ν

κσ κo

2.8

0.381

5

E 3.8  Effective properties of PZT-5A/epoxy composites: (a)S3333 ; (b)S D3333; (c)d333; (d)d311; (e)k33; (f)κ33/κo.

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Time-dependent behavior of active polymer matrix composites

3.8  Continued.

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(Continued )

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3.8  Continued.

Overall, the simplified micromechanical model and the RVE models of AFC and MFC are in good agreement with the experimental data. We further investigate the overall responses obtained using the two modeling approaches. For this purpose, we characterize effective linear properties of PZT-7A/LaRC-Si active PMCs. The properties of PZT-7A and LaRC-Si are given in Table 3.5. The responses are also compared to the ones obtained using the Mori-Tanaka and self-consistent micromechanical models. Figures 3.9 to 3.11 illustrate the effective elastic properties, piezoelectric constants, and dielectric constants of the PZT-7A/LaRC-Si piezocomposites. It is seen that for the electromechanical and piezoelectric properties along the longitudinal fiber direction, all micromechanical models are in good agreement. This indicates that the effective © Woodhead Publishing Limited, 2011

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Table 3.5  Electromechanical and piezoelectric properties of PZT-7A (Odegard, 2004) Properties

PZT-7A

LaRC-Si

C1111 (GPa) C1122 C1133 C2222 C2233 C3333 C1212 C1313 C2323 e113 = e113 (C/m2) e311 = e322 e333 κ ε11/κo = κ ε22/κo κ ε33/κo

148.0 76.2 74.2 148.0 74.2 131.0 35.9 25.4 25.4 9.2 –2.1 9.5 460 235

8.1 5.4 5.4 8.1 5.4 8.1 1.4 1.4 1.4 –

2.8

(κo=8.854187816*10–12 F/m)

3.9  Effective elastic properties of PZT-7A/LaRC-Si composites: (a)E11 = E22; (b)E33; (c)G13 = G23; (d)G12. (Continued )

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3.9  Continued.

properties for long unidirectional fiber composites along the longitudinal fiber axis are independent of the shape, size, and arrangement of the fibers in the matrix medium, but they mainly depend on the compositions and properties of the constituents. All properties in the transverse fiber direction predicted by the different micromechanical models are in good agreement for composites with lower fiber volume contents, i.e., less than 40% fiber contents. At higher fiber volume contents, some mismatches are observed. This might be due to an existence of localized field variables, i.e., stress, electric field, in the matrix constituents between the fiber spacing when the composites are loaded in the transverse fiber directions. As fiber contents increase, the spacing between fibers decreases, resulting in higher localized field variables, whose magnitudes and distributions

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3.10  Effective piezoelectric constants of PZT-7A/LaRC-Si composites: (a)e311 = e322; (b) e333; (c)e113 = e223.

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3.11  Effective dielectric properties of PZT-7A/LaRc-Si composites: (a) κ11 = κ22; (b)κ33.

strongly depend on the shape, size, and arrangements of the fibers. Observing the shear properties, all micromechanical models show some mismatches even at low fiber contents (less than 20%). This condition is expected, since performance of heterogeneous materials subject to shear boundary conditions is strongly influenced by microstructural arrangements of the constituents.

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3.5.2 Time-dependent properties of active PMCs Piezoelectric ceramics have high stiffness and show mild creep at room temperature, while at elevated temperatures ceramics could experience significant time-dependent mechanical responses such as creep. Fett and Thun (1998) conducted creep tests on poled and unpoled PZT ceramics at room temperature under various constant stresses. They showed that non-negligible creep strains and the creep responses are more pronounced at higher stresses. Heiling and Härdtl (1998) examined time-dependent responses of piezoelectric and dielectric properties of soft PZT ceramics. The surface charges of the PZT samples increases with time when the PZTs are subjected to constant stresses. Active PMCs are often utilized for applications under high mechanical and electrical stimuli, where the driving voltage can take up to 1500 V. Under such conditions, a significant amount of heat could be generated, increasing the temperatures in the active PMCs. At elevated temperatures, most materials, especially polymers, show significant time-dependent behaviors. In order to understand responses of active PMCs at high-applied mechanical and electrical stimuli and at elevated temperatures, we investigate and discuss time-dependent behaviors of active composites comprising unidirectional long PZT fibers embedded in epoxy matrix. We examine the time-dependent performance of PZT-7A/LaRC-Si active PMCs. Viscoelastic data for LaRC-Si matrix are obtained from Nicholson et al. (2002), which show significant viscoelastic responses at elevated temperatures: 213–223°C. Here, we examine the responses at temperature 218°C. Nicholson et al. (2002) used the Kolrausch-William-Watts (KWW) model to express timedependent compliance of LaRC-Si at different temperatures. The KWW model of the isotropic matrix is given as:

[3.40]

where So = 0.371(GPa–1) is the initial compliance, t is the current time, τ = 3.21×10–5 sec is the retardation time, β  = 0.411 is the parameter that determines the shape of the creep compliance. Figure 3.12 presents the creep compliance for LaRC-Si using the KWW model. For our numerical simulation, we use the Prony series representation (Eq. 3.41) for the time-dependent compliance due to the advantage of using the exponential function in solving the time-integral function in a recursive manner. The material constants for the Prony series model are reported in Table 3.6.

[3.41]

where So = 0.371(GPa–1) is the initial compliance, N is the number of terms in the Prony series, Sn and τn are the nth coefficient and retardation time in the Prony series.

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3.12  Creep compliance for LaRC-Si matrix at 218°C. Table 3.6  Prony parameters for LaRC-Si matrix at 218°C n

Sn (1/GPa)

τn (sec)

1 2 3 4 5 6

0.03 0.04 0.04 0.05 0.2 0.25

100 1000 5000 10000 15000 20000

Limited data of the PZT properties at elevated temperatures are available, which for the PZT-7A at 218°C are reported only for d311 = –130.5 × 10–12C/N and κ σ33 /κo = 1463 (Jaffe and Berlincourt, 1965). To study the effect of temperatures on the electromechanical and piezoelectric properties, we first convert the piezoelectric and dielectric properties of the PZT-7A in Table 3.5 to the ones presented in Eq. 3.1–3.2, which are expressed as:

[3.42]

The piezoelectric and electrical properties of the PZT-7A at room temperature are given in Table 3.7. To simulate overall responses of active PMCs at temperature 218°C, we assume that the rest of the piezoelectric and dielectric properties vary linearly with temperatures similar to the two values obtained from Jaffe and

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Berlincourt (1965). The piezoelectric and dielectric properties at 218°C are reported in Table 3.7. Creep responses of the active PMCs having PZT-7A fibers and LaRC-Si matrix are now examined. Figures 3.13 and 3.14 illustrate the time-dependent behavior of the elastic compliances and piezoelectric constants for the active PMCs with 20, 40, and 60% fiber contents obtained from the simplified micromechanical model. As expected, active composites having low fiber contents exhibit more pronounced time-dependent responses as compared to the ones with higher fiber contents. Responses are monitored for 25 000 seconds. All properties along the longitudinal fiber direction (x3) show mild time-dependent effects since most of the loads are carried by the fibers, whose properties are assumed Table 3.7  Piezoelectric and dielectric properties of PZT-7A Properties

PZT-7A

PZT-7A at 218°C

d113 = d113 (10–12 C/N) d311 = d322 d333

362 –53.4 133

439 –130.5* 210

e113 = e113 (C/m2) e311 = e322 e333

9.2 –2.1 9.5

20 –5.1 23.4

σ /κ = κ σ /κ κ 11 o 22 o κ σ33/κo

836 403

1896 1463*

* Properties obtained from Jaffe and Berlincourt, 1965.

3.13  Time-dependent elastic constants for PZT-7A/LaRC-Si composites: (a)S3333; (b)S1133; (c)S1111; (d)S1313. (Continued ) © Woodhead Publishing Limited, 2011

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3.13  Continued.

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3.14  Time-dependent piezoelectric constants for PZT-7A/LaRC-Si composites: (a)d333; (b)d311.

time-independent. The transverse and shear properties experience significant time-dependent response. This is due to the significant amount of loads carried by the matrix constituent under the transverse loading and the added shear responses. Next, we examine time-dependent response of two active PMCs, i.e., AFC and MFC, generated using the RVE-FE models. Only RVE models with a single fiber are considered in the analyses. We run the analyses for a longer period (up to 6000 minutes) to predict long-term responses of the active PMCs. Figure 3.15 shows long-term elastic constants of the AFC and MFC at various fiber contents. As expected, active composites with lower fiber contents experience significant

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time-dependent behaviors. This is due to higher stresses in the epoxy, increasing overall time-dependent response. It is also seen that AFC models exhibit higher creep behaviors than the MFC models for the same fiber contents, which are more significant for the transverse compliances (S1111). This might be due to the round shape of fibers that provide smoother variations in the field variables near the fiber-matrix interphases and allow higher shear deformation in the matrix. The

3.15  Long-term elastic responses of AFC and MFC: (a) S1111; (b) S3333; (c) S1212; (d) S2323. (Continued)

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3.15  Continued.

square fiber in the MFC results in higher stress discontinuities near the interphase regions, providing higher resistance to transverse loading. Figure 3.16 (Plate I in color section between pages 288 and 289) presents the von Mises stress contours of the AFC and MFC, at fiber contents 40%, subject to the transverse normal stress (σ–11). The stress contours are reported at time 5000 minutes. Figure 3.17 illustrates long-term effective piezoelectric constants for the AFC and MFC for various fiber contents. For the d311 and d333, significant time-dependent response are observed for the composites with lower fiber contents. Slight mismatches between the AFC and MFC response are also seen for the d311, which might be

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3.16  Von Mises stress contour for the AFC and MFC with 40% fiber contents due to transverse loading σ11.

due to the same behaviors as in the transverse elastic compliances. The piezoelectric constant d113 seems to have mild time-dependent response. For the AFC and MFC at different fiber contents, the rate of change of the d113 values with time seems to be the same. Significant mismatches between the AFC and MFC d113 properties are observed at 60% fiber contents, which might be due to different distributions of the electric fields in the two RVE-FE models and the added shear deformation from prescribing σ–13 on the boundaries of the RVEs (Fig. 3.18 and Plate II in color section).

3.17  Long-term piezoelectric responses of AFC and MFC: (a) d311; (b) d333; (c)d113. (Continued )

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3.17  Continued.

Now we consider both fiber and matrix experience time-dependent response. As discussed in Section 3.2, electromechanical and piezoelectric constants of PZT are also time-dependent. It is seen from Fig. 3.15b and 3.17b that the time-dependent responses of S3333 and d333 properties with 40 and 60% fiber contents, when only the viscoelastic matrix are considered, are less significant. When loadings are applied along the longitudinal fiber axis, most of the loads are carried by the fibers, whose properties are time-independent. We now examine the longitudinal

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3.18  Electric fields in the AFC and MFC models with 60% fiber contents from d311 characterization.

compliance S3333 and piezoelectric constant d333 of the AFC and MFC, in which the properties of the fiber and matrix constituents are time-dependent. We perform the analyses for composites with 40% fiber content at temperature 218°C. The relaxation modulus and piezoelectric constant for the PZT fibers are expressed as:

[3.43]

Figure 3.19 illustrates the long-term response of S3333 and d333 for the AFC and MFC with 40% fiber contents. As expected, more pronounced time-dependent behavior is observed when both constituents exhibit time-dependent responses.

3.6

Applications of active PMCs as actuators

We present an application of using ACFs to control deformation in a cantilever beam subject to a transverse load at the free end. Figure 3.20 illustrates the steel cantilever beam with an AFC bonded to its top surface, termed as a smart beam. The AFC has 40% fiber volume fraction with fiber radius of 178.4 mm and is polarized along the fiber longitudinal axis. The beam is made of linear elastic stainless steel SUS304 having the following properties: density 8 gr/cm3, modulus of elasticity of 193 GPa, and Poisson’s ratio of 0.3. The AFC consists of PZT-7A fiber and LaRC-Si matrix (Table 3.5). FE model is used to analyze the deformation in the smart beam. Continuum elements C3D8R and C3D8E are used for the beam and AFC actuator, respectively. The cantilever beam has 40 000 elements and AFC actuator has 15 000

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3.19  Long-term compliance and piezoelectric constant for AFC and MFC with time-dependent fiber and matrix constituents (vf = 40%): (a) S3333; (b)d333.

elements (Fig. 3.21 and Plate III, in color section between pages 288 and 289). Fixed boundary conditions are imposed to the end of the smart beam where the AFC is placed and the transverse load is applied at the free end. The lateral deflection in the smart beam is monitored at the free end (point C in Fig. 3.20). Figure 3.22 illustrates the load-displacement curve in the smart cantilever beam subject to the transverse load. The maximum deflection is about 27 µm when the transverse load is 1 N. The stresses and strains in the smart beam are also recorded

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3.20  A cantilever beam with AFC actuator (units are in mm).

at several locations. Figure 3.23 shows normal and shear stresses at the beam and AFC regions. Point A and B are located at the mid-section and at the end, respectively, of the interphase line between the AFC and steel beam (Fig. 3.20). High stresses at the top and bottom faces of the beam and localized stresses at the interphase regions due to bending of the smart beam could lead to failure and/or

3.21  FE mesh for the smart beam.

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Time-dependent behavior of active polymer matrix composites

3.22  Lateral deflections in the smart cantilever beam.

3.23  Normal and shear stresses in the deformed smart beam.

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debonding in the AFC and beam. Since the AFC possesses electromechanical coupling properties, the existence of the stress/strain in the AFC due to the applied lateral force in the smart beam generates electric charge/potential. The accumulated electric field in the AFC during the mechanical loading is given in Fig. 3.24. This underlines another potential application of AFC to harvest energy. In order to minimize the lateral deflection in the smart beam, an electric charge or electric potential can be applied to the AFC. Figure 3.22 shows that applying either charge or electric potential continuously to the AFC during loading in the smart beam can compensate the lateral deflection. In the above analyses, the AFC with linear elastic properties is considered and a quasi-static loading is applied. Next, we simulate a sinusoidal load applied to the elastic beam (F=sin π t N) and a sinusoidal charge (Q=43 sin π t nC) is simultaneously applied to minimize deformation at each instant of time. Figure 3.25 illustrates tip deflection in the smart cantilever beam during the sinusoidal loading. It is seen that applying charge could compensate the tip-deflection to less than 1 µm. As discussed in Section 3.5, AFCs with 40% fiber content at room temperature show insignificant time-dependent effect. Thus, applying charge with the same frequency as the applied load could properly compensate the deformation. At elevated temperatures, PZT fiber and polymer matrix exhibit significant time-dependent electromechanical properties. This would cause phase lagging in the overall deflection when the charge is applied with the same frequency as the applied load.

3.24  Accumulated electric field (magnitude only) in the AFC during the deformation of the smart beam (without applied electric boundary condition to control deformation).

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3.25  Tip-deflection in the smart cantilever beam due to a sinusoidal point load at the free end.

3.7

Conclusions

We have studied the effect of time-dependent material properties on the overall performance of active PMCs having unidirectional piezoceramic fibers, such as PZT fibers, dispersed in viscoelastic polymer matrix. Two micromechanical modeling approaches have been considered: a simplified micromechanical model with four subcells and RVE models of the AFC and MFC generated using FE. AFC has a circular fiber cross-section, while MFC has a square fiber cross-section. The simplified micromechanical model is used to obtain the effective electromechanical properties of the homogenized active PMCs. The use of RVE models allows quantifying detailed variations in the field variables, such as stress discontinuities at the fiber-matrix interphases, in the active PMCs. The effective properties determined from the simplified micromechanical RVE models have been compared to available experimental data, and good agreements are observed. Responses obtained from the two micromechanical modeling approaches are also compared to the ones of the Mori-Tanaka and self-consistent models. It can be concluded that effective properties along the longitudinal fiber direction mainly depend on the compositions and properties of the constituents and seem to be unaffected by the shape, size, and arrangement of the fibers. The transverse and

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shear properties of the active PMCs depend strongly on the shape, size, and arrangement of the fibers, especially for composites with higher fiber contents. This might be due to the effect of the localized field variables in the matrix regions between the fibers. The effects of time-dependent and field coupling behaviors of the fiber and matrix constituents on the overall performance of active PMCs have been examined. Two case studies have been performed. The first study assumes only the elastic properties of the polymer matrix vary with time, while in the second study, both fiber and matrix properties are time-dependent. The elastic and piezoelectric properties in the fiber direction from the first case study show mild time-dependent behaviors especially at higher fiber contents, while the transverse and shear responses are strongly timedependent. As expected, when time-dependent elastic and piezoelectric properties of the fibers and matrix constituents are considered, significant time-dependent responses even in the fiber direction are observed. It is also seen that AFC shows higher time-dependent transverse compliance and piezoelectric constants than those of the MFC. This might be due to the round shape of fibers that provides smoother variations in the field variables near the fiber-matrix interphase regions, allowing more shear deformation. The square fibers result in higher stress discontinuities near the interphase regions, providing higher resistance to transverse loading. We have shown the capability of active PMCs, i.e., AFC to control time-dependent deformation in host structures by continuously applying electric charge or potential.

3.8

Acknowledgement

This research is sponsored by the Air Force Office of Scientific Research (AFOSR) under grants FA9550-09-1-0145 and FA 9550-10-1-0002 (Program manager: Dr David Stargel).

3.9

References

ABAQUS Theory Manual v6.8 (2008), DS Simulia Corp. Aboudi, J. (2001), ‘Micromechanical Analysis of Fully Coupled Electro-magnetothermo-electro-elastic Multiphase Composites,’ Smart Mater. Struct., 10, 867–877. Beckert, W. and Kreher, W.S. (2003) ‘Modelling piezoelectric modules with interdigitated electrode structures,’ Computational Materials Science, 26 (2003), 36–45. Bent, A.A. and Hagood, N.W. (1997), ‘Piezoelectric Fiber Composites,’ J. Intell. Mater. Syst. Struct., 8, 903–919. Cao, H. and Evans, A.G. (1993), ‘Nonlinear Deformation of Ferroelectric Ceramics,’ J. Amer. Ceramic Soc., 76, 890–896. Damjanovic, D. (1998), ‘Ferroelectric, Dielectric and Piezoelectric Properties of Ferroelectric Thin Films and Ceramics,’ Rep. Prog. Phys., 61, 1267–1324. Dunn, M.L. (1993), ‘Micromechanics of coupled electroelastic composites: effective thermal expansion and pyroelectric coefficients,’ J. Appl. Phys., 73, 5131–5140. Dunn, M.L. and Taya, M. (1993), ‘Micromechanics Predictions of the Effective Electroelastic Moduli of Piezoelectric Composites,’ Int. J. Solids and Structures, 30, 161–175.

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Fett, T. and Thun, G. (1998), ‘Determination of Room-temperature Tensile Creep of PZT,’ J. Materials Science Letter, 17, 1929–1931. Haj-Ali, R.M. and Muliana, A. H. (2004a), ‘Numerical Finite Element Formulation of the Schapery Nonlinear Viscoelastic Material Model,’ Int. J. of Numer. Meth. in Engrg, 59, 25–45. Haj-Ali, R. and Muliana, A. (2004b), ‘A Multi-scale Constitutive Formulation for the Nonlinear Viscoelastic Analysis of Laminated Composite Materials and Structures,’ Int. Journal of Solids and Structures, 41, 3461–3490. Hall, D.A. (2001), ‘Review Nonlinearity in Piezoelectric Ceramics,’ J. Materials Sci., 36, 4575–4601. Heiling, C. and Härdtl, KH. (1998), ‘Time Dependence of Mechanical Depolarization in Ferroelectric Ceramics’, Proc. Eleventh IEEE International Symposium of Applications on Ferroelectrics (ISAF), pp. 503–508. Hill, R. (1964), ‘Theory of Mechanical Properties of Fiber-strengthened Materials–I. Elastic Behavior,’ J. Mech. Phys. Solids, vol. 12, 199–212. Jaffe, H. and Berlincourt, D.A. (1965), ‘Piezoelectric Transducer Materials,’ IEEE Proceedings, 53, 1327–1386. Lee, J., Boyd, J.G., and Lagoudas, D. (2005), ‘Effective Properties of Three-phase Electro-magneto-elastic Composites,’ Int. J. Engrg Sci., 43, 790–825. Li, J.Y. and Dunn, M.L. (2001), ‘Viscoelastic Behavior of Heterogeneous Piezoelectric Solids,’ J. Applied Physics, 89, 2893–2903. Lin, Y. and Sodano, H.A. (2008), ‘Concept and model of a piezoelectric structural fiber for multifunctional composites,’ Composites Science and Technology 68, 1911–1918. Muliana, A. (2010a), ‘A Multi-scale Formulation for Smart Composites with Field Coupling Effects,’ part of Advances in Mathematical Modeling and Experimental Methods for Materials and Structures. The Jacob Aboudi Volume, Vol. 168, pp. 73–87. Muliana, A. (2010b) ‘A Micromechanical Formulation for Piezoelectric Fiber Composites with Nonlinear and Viscoelastic Constituents,’ Acta Materialia, 58, 3332–3344. Muliana, A.H. and Sawant, S. (2009), ‘Viscoelastic Responses of Polymer Composites with Temperature and Time Dependent Constituents,’ Acta Mechanica, 204, 155–173. Nelson, L.J., Bowen, C.R., Stevens, R., Cain, M., and Stewart, M. (2003), ‘Modelling and Measurement of Piezoelectric Fibers and Interdigitated Electrodes for the Optimisation of Piezofiber Composites,’ Proc. of SPIE’s Symposium on Smart Structures and Materials, 5053, pp. 556–567. Nicholson, L.M., Whitley, K.S. and Gates, T.S. (2002) ‘The Role of Molecular Weight and Temperature on the Elastic and Viscoelastic of a Glassy Thermoplastic Polyimide,’ International Journal of Fatigue, 24, 185–195. Odegard, G.M. (2004), ‘Constitutive Modeling of Piezoelectric Polymer Composites,’ Acta Materialia, 52, 5315–5330. Pedersen, O.B. (1989), ‘Thermomechanical Hysteresis and Analogous Behavior of Composites,’ in Micromechanics and Inhomogeneity, The Toshio Mura Anniversary Volume, Edited by G.J. Weng, M. Taya, and H. Abe, Springer-Verlag. Sawant, S. and Muliana, A. (2008), ‘A Thermo-mechanical Viscoelastic Analysis of Orthotropic Media,’ Composite Structures, 83, 61–72. Schaäufele, A. and Härdtl, K.H. (1996), ‘Ferroelastic Properties of Lead Zirconate Titanate Ceramics,’ J. Amer. Ceramic Soc., 79, 2637–2640. Strobl, G. (1997), The Physics of Polymers, 2nd Ed., Springer-Verlag. Tan, P. and Tong, L. (2001) ‘Micro-electromechanics models for piezoelectric-fiberreinforced composite materials,’ Composites Science and Technology 61, 759–769.

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Taylor, R.L., Pister, K.S., and Goudreau, G.L. (1970), ‘Thermomechanical Analysis of Viscoelastic Solids,’ Int. Journal for Numerical Methods in Engineering, Vol. 2, 45–59. Vinogradov, A.M., Schmidt, V.H., Tithill, G.F., and Bohannan, G.W. (2004), ‘Damping and Electromechanical Energy Losses in the Piezoelectric Polymer PVDF,’ Mechanics of Materials, 36, 1007–1016. Wickramasinghe, V. and Hagood, N. (2004), ‘Material Characterization of Active Fiber Composites for Integral Twist-actuated Rotor Blade Application,’ Smart Mater. Struct., 13, 1155–1165. Williams, B., Inman, D., Schultz, M., and Hyer, M. (2004) ‘Tensile and Shear Behavior of Macro Fiber Composite Actuators,’ Journal of Composite Materials, 38, 855–870. Williams, B., Inman, D., and Wilkie, W.K. (2006), ‘Nonlinear Responses of the Macro Fiber Composite Actuator to Monotonically Increasing Excitation Voltages,’ J. Intel Mater. System Struct., 17, 601–608. Williams, R.B. (2004), ‘Nonlinear Mechanical and Actuation Characterization of Piezoceramic Fiber Composites,’ PhD Thesis, Virginia Polytechnic Institute and State University.

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Plate I Von Mises stress contour for the AFC and MFC with 40% fiber contents due to transverse loading σ11.

Plate II Electric fields in the AFC and MFC models with 60% fiber contents from d311 characterization.

Plate III FE mesh for the smart beam.

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4 Predicting the elastic-viscoplastic and creep behaviour of polymer matrix composites using the homogenization theory T. MATSUDA, University of Tsukuba, Japan and N. OHNO, Nagoya University, Japan Abstract: This chapter presents the elastic-viscoplastic and creep analyses of polymer matrix composites using the homogenization theory for non-linear time-dependent composites, and their experimental verification. First, the homogenization theory is introduced as a basis for all of the analyses in this chapter. Subsequently, the elastic-viscoplastic analyses of carbon fibrereinforced plastic (CFRP) laminates and plain-woven glass fibre-reinforced plastic (GFRP) laminates are performed. Uniaxial tensile tests of the laminates are also conducted to verify the results of the analyses. Finally, the creep behaviour of a unidirectional CFRP laminate at elevated temperature is predicted using the homogenization theory. Key words: homogenization, viscoplasticity, creep, carbon fibre-reinforced plastic (CFRP) laminate, plain-woven glass fibre-reinforced plastic (GFRP) laminate.

4.1

Introduction

The recent development of high-end industrial products in the aerospace, auto and energy-related industries requires lighter, stiffer and stronger materials. Some of the most promising materials which satisfy these requirements are polymer matrix composites (PMCs). Their high specific stiffness and high specific strength are highly desirable features for industrial products, resulting in their abundant use. For example, PMCs make up around half of the weight of next generation aircraft such as the Airbus A350 and the Boeing B787. More recently, auto companies have begun to employ PMCs as structural components in vehicles, and their usage in the auto industry is expected to increase rapidly over the next decade. In the above situations, PMCs can be subjected to severe environments, including high stress and high temperature, which cause time-dependent behaviour such as viscoplastic and creep behaviour. Thus, analysis of the time-dependent behaviour of PMCs has become an increasingly important issue. Such analysis, however, generally involves difficulties because PMCs are extremely heterogeneous materials comprised of reinforcements, which are mainly fibres, and polymeric materials. Moreover, these constituents of PMCs have completely 113 © Woodhead Publishing Limited, 2011

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different time-dependent properties; in general, polymeric materials exhibit clear time-dependent behaviour, whereas reinforcements do not. Accurate analysis of the time-dependent behaviour of PMCs requires a theory which describes their macroscopically homogenized time-dependent behaviour by explicitly considering their microscopic heterogeneity. The homogenization theory for non-linear time-dependent composites developed by the present authors (Wu and Ohno, 1999; Ohno et al., 2000) is one of the most advantageous theories for such analysis. This theory is based on the mathematical homogenization theory (Babuska, 1976; Bensoussan et al., 1978; Sanchez-Palencia, 1980; Bakhvalov and Panasenko, 1984), and provides a mathematically rigorous correlation between the macroscopic and microscopic time-dependent behaviour of heterogeneous materials, which will be described in the next section. This gives small calculation errors, and avoids accumulated inaccuracy due to incremental computation, which results in its excellent applicability to the time-dependent analysis of composites. The authors, therefore, have applied the above-mentioned theory to the elastic-viscoplastic analysis of two kinds of PMCs, i.e. carbon fibre-reinforced plastic (CFRP) laminates (Matsuda et al., 2002, 2003), and plain-woven glass fibre-reinforced plastic (GFRP) laminates (Matsuda et al., 2007). In these studies, tensile tests of the laminates were also performed to provide the experimental verification of the theory. Comparison between the experimental and predicted results proved the ability of the theory to analyze the elastic-viscoplastic behaviour of PMCs. In more recent years, the authors (Fukuta et al., 2008) have further applied the theory to predict the creep behaviour of CFRP laminates at elevated temperature reported in the literature (Kawai and Masuko, 2004), verifying the usefulness of this theory in the creep analysis of PMCs. Incidentally, the mathematical homogenization theory has been successfully employed for not only time-dependent analysis but also for many other non-linear analyses of composites, including cellular solids and porous materials (e.g., Suquet, 1987; Agah-tehrani, 1990; Jansson, 1991, Terada 1995; Ghosh and Moorthy, 1995; Ohno et al., 2002; Takano et al., 2002; Okumura et al., 2004; Asada and Ohno, 2007; Tsuda et al., 2009). This chapter presents the elastic-viscoplastic and creep analyses of PMCs using the homogenization theory and their experimental verification, demonstrating the high applicability of the theory to the time-dependent analysis of PMCs. First, in the next section, the homogenization theory is introduced as a basis for all of the following analyses in this chapter. The subsequent two sections deal with the elastic-viscoplastic analyses of CFRP laminates and plain-woven GFRP laminates, respectively. In these sections, uniaxial tensile tests of the laminates at a constant strain rate are also conducted at room temperature to verify the results of the analyses. Finally, the creep behaviour of a unidirectional CFRP laminate at elevated temperature is predicted using the homogenization theory.

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4.2

Predicting behaviour using the homogenization theory

115

Homogenization theory for non-linear time-dependent composites

In this section, the homogenization theory for non-linear time-dependent composites (Wu and Ohno, 1999; Ohno et al., 2000) is introduced to prepare the analyses performed in the following sections, 4.3, 4.4 and 4.5. For this, consider a body Ω with a periodic internal structure comprised of more than two kinds of constituents, and then define its smallest representative volume element which is referred to as the unit cell Y (Fig. 4.1). The Cartesian coordinates yi (i = 1, 2, 3) are employed for Y, and time is denoted by t. It is assumed that Ω is subjected to macroscopically uniform stress and strain, and exhibits infinitesimal strain both macroscopically and microscopically.

4.1  Body Ω with a periodic internal structure and its unit cell Y.

4.2.1 Basic equations Describing the microscopic distributions of stress and strain in Y as σij( y, t) and εij ( y, t), respectively, the equilibrium of σij can be expressed in a rate form as: ,

[4.1]

where (.) and (    ), j denote the differentiation with respect to t and yj, respectively. The constituents of Ω are assumed to exhibit linear elasticity and non-linear viscoplasticity as characterized by: ,

[4.2]

where cijkl and βkl indicate the elastic stiffness and viscoplastic function of the constituents, respectively, satisfying: , .

[4.3] [4.4]

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. The microscopic velocity field ui(y, t) in Y has the following expression when Ω has macroscopically uniform deformation: ,

[4.5]

. where Fij(t) and ui# denote the gradient of macroscopic displacement and the . perturbed displacement from the macroscopic displacement Fij(t)yj, respectively. . Then the microscopic strain rate ε ij in Eq. 4.2 is expressed as: , [4.6] . .# where Eij and εij stand for the macroscopic strain rate and the perturbed strain rate, respectively: ,

[4.7]

.

[4.8]

. It is noted that u#i satisfies the periodicity with respect to Y, which results from the periodic internal structure of Ω. Such periodicity is called Y-periodicity.

4.2.2 Solution for a perturbed velocity field . Now, the problem for finding the current field of perturbed velocity in Y, u #i(y, t), is considered, on the assumption that the current distribution of the microscopic stress in Y, σij(y, t), is known. Let υi(y, t) be an arbitrary Y-periodic velocity field defined in Y at t. Then, the integration by parts and the divergence theorem allow Eq. 4.1 to be transformed to:

[4.9]

where Γ denotes the boundary of Y, and ni indicates the unit vector outward normal to Γ. The second term in the above equation vanishes because σij and υi are Y-periodic, while ni takes opposite directions on the opposite boundary surfaces of Y. Thus, Eq. 4.9 becomes:

[4.10]

Substitution of Eq. 4.2 and 4.6 into Eq. 4.10 results in:

[4.11]

  Now, let χ ikl and ϕi be functions which are determined by solving the following boundary value problems for Y, respectively: with Y -periodicity of χ ikl,



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[4.12]



Predicting behaviour using the homogenization theory c ϕ υ dY Y ijpq p,q i,j

c β υ  dY  Y ijkl kl i,j

=

with Y-periodicity of ϕi.

117 [4.13]

In general, these problems are solved numerically based on the .finite element . method (FEM). Then Eq. 4.11, in which u #i depends linearly on Eij through the first term on the right-hand side, has the following solution for the perturbed velocity field:

[4.14]

4.2.3 Microscopic stress evolution equation and macroscopic constitutive relation An evolution equation of microscopic stress in Y is obtained by substituting Eq. 4.6 and 4.14 into Eq. 4.2:

[4.15]

where

[4.16] [4.17]

and δij indicates Kronecker’s delta. Let    be a volume average operator with respect to Y expressed as:

[4.18]

where |Y| signifies the volume of Y. Applying the above operator to Eq. 4.15, the following rate-type macroscopic constitutive relation of Ω is derived: . where Σij indicates the macroscopic stress rate defined as:

[4.19]

[4.20]

Moreover, applying the volume average operator to Eq. 4.6, and then using the . divergence theorem and the Y-periodicity of u#i, the relationship between the macroscopic and microscopic strains is obtained as:

[4.21]

4.2.4 Computational procedure Using the homogenization theory described above, the time-dependent behaviour of Ω is able to be computed providing that the macroscopic boundary condition,

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i.e. the time history of Σij or Eij, or a combination of them both, is prescribed, and σij(y, t) at the current time t is known. The computational procedure from the current time t to the subsequent time t + ∆t is summarized as follows: (1) The functions χ ikl and ϕi are determined by solving the boundary value problems 4.12 and 4.13 based on the FEM. (2) Calculating aijkl and rij from Eq. 4.16 and 4.17, respectively, their volume averages aijkl and rij are computed. (3) The macroscopic relation 4.19 is used to find unknown . constitutive . components of Σij and Eij. . (4) The microscopic stress rate σij is calculated using Eq. 4.15. (5) Adding the increments to the current values, the next time step of the computation is started.

4.3

Elastic-viscoplastic analysis of CFRP laminates and experimental verification

Carbon fibre-reinforced plastic (CFRP) laminates are now regarded as the most popular polymer matrix composites. They particularly draw the attention of transport equipment industries such as the aerospace, auto and marine industries, because their high specific stiffness and strength may allow dramatic improvement in the energy efficiency of transportation. For example, in recent years, almost all of the aviation manufacturing companies have used large amounts of CFRP laminates for not only secondary but also primary structural components of aircraft including main wings and main bodies. Such usage of CFRP laminates raises the probability of their exposure to high stress and high temperature, which can cause the viscoplastic deformation of the laminates. In this section, therefore, the analysis of elastic-viscoplastic behaviour of CFRP laminates (Matsuda et al., 2002, 2003) is performed using the homogenization theory described in the previous section. For this, an inplane elastic-viscoplastic constitutive relation for long fibre-reinforced laminates is derived, based on the homogenization theory in conjunction with the standard lamination theory. Subsequently, to verify the present method, uniaxial tensile tests of CFRP laminates at a constant strain rate are conducted, using unidirectional, cross-ply and quasi-isotropic laminates. The experimental results are then compared with the predictions obtained from the present theory. Finally, the effects of randomness of fibre distribution in the laminae on the macroscopic homogenized behaviour of CFRP laminates are discussed by considering two types of fibre arrays, i.e. hexagonal and random arrays.

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4.3.1 Elastic-viscoplastic constitutive equation for laminates Modelling of laminate and basic assumptions Consider a laminate in which unidirectional long fibre-reinforced laminae are stacked symmetrically as shown in Fig. 4.2, where N and f (α) indicate the number of laminae and the volume fraction of the α th lamina, respectively. In each lamina, the fibres are assumed to be arranged periodically; for example, a hexagonal fibre array as illustrated in Fig. 4.2(c). For the periodic microstructure of the α th lamina, a unit cell Y(α) is defined. Three kinds of Cartesian coordinates, i.e. Xi, x (iα) and y(iα) (i = 1, 2, 3), are defined for the laminate, the α th lamina, and Y(α), respectively. The X2-axis is taken in the stacking direction, and the x (2α) -axis is parallel to the X2-axis. On the other hand, the x(3α) -axis is taken in the fibre direction of the α th lamina, and makes an angle θ (α) with the X3-axis. The y(iα) -axis is parallel to the x (iα) -axis, but is employed solely for Y(α). The laminate is assumed to be infinitely large in the X1- and X3-directions. When the laminate is subject to an in-plane load, the macroscopic stress in the laminate, Σij, and the overall stresses in the laminae, Σ (ijα), have the following relation: , in-plane components, ,

[4.22] [4.23]

 ,

[4.24]

4.2  Structure of a long fibre-reinforced laminate with three kinds of Cartesian coordinates; (a) laminate, (b) lamina, and (c) unit cell.

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where X(  ) stands for the components with respect to the Xi coordinate system. The macroscopic strain in the laminate, Eij, and the overall strains in the laminae, E(ijα), are expressed as: ,

,

,

,

,

[4.25] [4.26]

,

[4.27] ,

[4.28]

Equations 4.24 and 4.28 also hold with respect to the x(iα) coordinate system: ,



,

.

[4.29] [4.30]

It is noted that no bending occurs on the laminate because of the symmetrical laminate configuration. Homogenization of laminae The microscopic distributions of stress and strain in Y(α), σ (ijα)(y, t) and ε (ijα) (y, t), respectively, are assumed to have the same type of constitutive relation as Eq. 4.2: ,

[4.31]

where and indicate the elastic stiffness and viscoplastic function of ( α ) constituents in Y , respectively. Then, according to the homogenization theory described in Section 4.2, the following relations are derived:

[4.32] [4.33]

where

[4.34]



[4.35]

.

[4.36]

and

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Here, |Y(α)| denotes the volume of Y(α), and and are the functions determined by solving the following boundary value problems: ,

[4.37]

,

[4.38]

where υ(iα) signifies any Y-periodic velocity field defined in Y(α) at t. In-plane elastic-viscoplastic constitutive equation of laminates Equation 4.33 is solved for

, and then transformed to a matrix form:

[4.39]

The above equation is reduced using Eq. 4.29 and 4.30 to:

[4.40]

[4.41]

_ where ( ) indicates that the term consists of only in-plane components, i.e.,

[4.42]

[4.43]

and so on. Here (  )T denotes the transpose. Now, introduce the in-plane vectors of

and

furthermore, i.e.,



[4.44]



[4.45]

Because the x (1α) - and x3(α) -axes are at angle θ(α) to the X1- and X3-axes (see Fig. 4.2), respectively, the following relations are obtained:

[4.46]



[4.47]

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where

[4.48]

[4.49]

  Substitution of Eq. 4.46 and 4.47 into 4.40 yields:

[4.50]

Finally, by substituting Eqs. 4.50 and 4.25 into Eq. 4.22, a macroscopic in-plane constitutive equation of the laminate is derived as:

[4.51]

where

[4.52]



[4.53]

4.3.2 Experimental procedure To verify the present theory, uniaxial tensile tests of CFRP laminates at a constant strain rate were performed at room temperature using coupon specimens cut out from TR30/#340 carbon fibre/epoxy laminates of 300 × 300 mm, manufactured by Mitsubishi Rayon Co. Ltd. Three kinds of laminates, i.e. unidirectional, crossply and quasi-isotropic laminates were used in the tests. Their laminate configurations were [0]12, [0/90]4s and [0/±60]3s, and their thicknesses were 1.5, 2.0 and 2.25 mm, respectively. The fibre volume fraction Vf in each ply was 56%. Figure 4.3 shows a schematic diagram of a coupon specimen with rectangular GFRP tabs used in the tests. A 5 mm long strain gauge was adhered on each side of the specimens. In the figure, ψ indicates the angle between the longitudinal direction (loading direction) of the specimens and the fibre direction of 0°-plies, i.e. the off-axis angle. The values of ψ selected in the tests are listed in Table 4.1. A closed-loop servohydraulic testing machine with a load/strain computer . controller was used for the tests. The longitudinal strain rate of each specimen, Eψ, was controlled to be 10–5 s–1.

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4.3  Schematic diagram of a coupon specimen with dimensions in mm. Table 4.1  Off-axis angle ψ for unidirectional, cross-ply and quasi-isotropic laminates Laminate

ψ (degree)

Unidirectional Cross-ply Quasi-isotropic

0, 0, 0,

10, 10, 10,

20, 20, 20,

30, 30, 30

45, 45

60,

90

4.3.3 Analysis conditions The array of carbon fibres in each lamina was modelled as a perfectly hexagonal array on the x (1α) – x (2α) plane, as illustrated in Fig. 4.2(c) and 4.4(a). This fibre array provides a transverse isotropy for both elastic and viscoplastic behaviour (Ohno et al., 2000), accurately simulating random fibre arrays in actual laminae. Moreover, a hexagonal fibre array is convenient to use when generating the

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4.4  Microstructure of laminae; (a) hexagonal fibre array, and (b) unit cell Y (α) and finite element mesh.

geometry of a unit cell and its finite element mesh, which is a major advantage for numerical analysis. Thus, a hexagonal fibre array was adopted as the microstructure of each lamina in the present analysis. The effects of random fibre arrays in laminae on the homogenized elastic-viscoplastic behaviour of CFRP laminates will be discussed in Section 4.3.5. For the hexagonal fibre array mentioned above, a hexagonal unit cell Y(α) was defined, and divided into four-node isoparametric finite elements as shown in

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Fig. 4.4(b). This unit cell was adopted for all of the laminae in the laminates. The fibre volume fraction Vf was taken to be 56% in accordance with that of the TR30/#340 carbon fibre/epoxy laminates used in the experiment. It should be noted that Y(α) was taken to be two-dimensional rather than three-dimensional, because each lamina was assumed to have no microscopic variation in the longitudinal direction of fibres. Also, it is sufficient to consider only half of the unit cell as the domain of analysis for solving the boundary value problems 4.37 and 4.38, because the hexagonal unit cell has the point symmetry with respect to its centre (Ohno et al., 2001). The carbon fibres were regarded as transversely isotropic elastic materials, possessing five independent elastic constants, i.e. two Young’s moduli Ef 1 and Ef 3, two Poisson’s ratios νf 12 and νf 31, and one shear rigidity Gf 31, where the subscripts 1, 2 and 3 denote the y (1α) , y (2α) and y (3α) directions, respectively. The five elastic constants used in the present analysis are listed in Table 4.2; Ef 3 was provided by the manufacturer, Mitsubishi Rayon Co. Ltd., while the other constants were obtained by referring to the literature (Kriz and Stinchcomb, 1979). The epoxy matrix, on the other hand, was regarded as an elastic-viscoplastic material characterized as: [4.54] where Em, νm and n are material constants, g(ε− p) is a material function depending . on an accumulated viscoplastic strain ε− p = ∫[(2/3) ]1/2 dt, ε 0p is a reference strain rate, sij indicates the deviatoric part of σij, and σeq = [(3/2)sijsij]1/2. The material constants and material function in Eq. 4.54 were determined by simulating . 45° off-axis tensile tests of the unidirectional and cross-ply laminates at E45° = 10–3, 10–5 and 10–7 s–1 (Fig. 4.5(a) and (b)). This was because the effect of matrix viscoplasticity was expected to be significant in such off-axis tests. Table 4.2 lists the material constants determined. It is noted that Eq. 4.54, which is based on the assumption of isotropic hardening, is valid as far as a monotonic loading is concerned.

Table 4.2  Material constants of the carbon fibres and epoxy matrix for CFRP laminates Carbon fibre Epoxy

Ef1 = 1.55 × 104 Ef 3 = 2.40 × 105 Gf 31 = 2.47 × 104 νf12 = 0.49 νf 31 = 0.28 . Em = 3.5 × 103 νm = 0.35 ε p0 = 10–5 n = 35 − − g(ε p) = 141.8(ε p)0.165 + 10

MPa (stress), mm/mm (strain), s (time).

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4.5  Macroscopic tensile curves of CFRP laminates with the 45° off-axis angle at constant strain rates; (a) unidirectional laminate, and (b) cross-ply laminate.

4.3.4 Comparison between experiments and predictions In this section, the relations between the macroscopic tensile stress Σψ and strain Eψ of the CFRP laminates, obtained from the uniaxial tensile tests described in Section 4.3.2, are compared with predictions obtained from the present theory. © Woodhead Publishing Limited, 2011



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First, the results of the unidirectional laminate are shown in Fig. 4.6(a). The markers indicate the experimental results, while the solid lines denote the predictions. As seen from the experimental data, the macroscopic behaviour of the unidirectional laminate had significant dependence on ψ ; when ψ = 0°, the tensile behaviour was almost linear, suggesting that it was governed by the carbon

4.6  Macroscopic stress versus strain relations of laminates at . Eψ = 10–5 s–1; (a) unidirectional laminate, (b) cross-ply laminate, and (c) quasi-isotropic laminate. (Continued )

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4.6  Continued.

fibres. However, even a small deviation of ψ from 0°, such as 10° and 20°, caused noticeable nonlinearity due to the influence of the viscoplasticity of the epoxy matrix. The increase in ψ brought about the drop in the viscoplastic flow stress, and had a negligible influence on the macroscopic behaviour beyond ψ = 45°. This marked dependence on ψ was accurately predicted using the present theory, as depicted in Fig. 4.6(a), i.e. the predictions agreed well with the experiments for all ψ. Therefore, the present theory is successful in describing the in-plane elastic-viscoplastic behaviour of unidirectional laminates. Next, Fig. 4.6(b) compares the experimental results and the predictions obtained for the cross-ply laminate. It is seen from the experimental results in the figure that almost the same anisotropic behaviour was observed as in the unidirectional laminate. This implies that the interaction between the 0°- and 90°-plies in the crossply laminate was considerably weak, resulting in similar macroscopic behaviour to the unidirectional laminate. Such anisotropic behaviour of the cross-play laminate was predicted well by the present theory, demonstrating that the present theory is also applicable to cross-ply laminates as successfully as unidirectional laminates. Incidentally, the minor difference between the predicted and experimental values observed when ψ = 10° seems to be attributable to microscopic damages, such as matrix cracking and interfacial debonding (Tohgo et al., 2001). Finally, the experimental and predicted results for the quasi-isotropic laminate are shown in Fig. 4.6(c). The quasi-isotropic laminate exhibited almost linear behaviour at all ψ until a macroscopic fracture occurred around Σψ = 600 MPa, which was completely different from the macroscopic behaviour of the other two laminates

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already discussed. The fracture stresses of the quasi-isotropic laminate were much higher than the viscoplastic flow stresses of the unidirectional and cross-ply laminates of ψ = 45°, which were about 100 MPa. These results imply that the viscoplasticity of the matrix did not exert any noticeable influence on the macroscopic behaviour of the quasi-isotropic laminate. It is also seen from the figure that the quasi-isotropic laminate showed negligible dependence on ψ, i.e. it exhibited inplane isotropy. Comparing these experimental results with the predictions, which are almost identical at all ψ, a good agreement is observed as shown in Fig. 4.6(c), even though the macroscopic behaviour of the quasi-isotropic laminate was totally different from that of the unidirectional and cross-ply laminates. This shows the applicability of the preset theory to quasi-isotropic laminates. In summary, the theory described in Section 4.3.1 has turned out to be useful for analyzing the homogenized elastic-viscoplastic behaviour of CFRP laminates.

4.3.5 Effects of fibre distribution randomness in laminae In the preceding analysis, the hexagonal fibre array on the x (1α) – x (2α) plane was employed as the fibre array in each lamina. However, in real laminates, the distribution of fibres on the x (1α) – x (2α) plane in the laminae exhibits some randomness. In this section, therefore, taking an example of random fibre distribution as a fibre array in the laminae, the same elastic-viscoplastic analysis of CFRP laminates as in the preceding analysis is performed. This allows the influence of the fibre distribution on the homogenized elastic-viscoplastic behaviour of CFRP laminates to be investigated. To consider random distribution of constituents in the framework of the homogenization theory, in general, a unit cell Y is defined in which constituents are randomly distributed, and is then arranged Y-periodically as depicted in Fig. 4.7(a). In this section, however, another arrangement of Y, the point-symmetric cell arrangement (Matsuda et al., 2003), is employed to increase the randomness of the distribution of the constituents. Figure 4.7(b) illustrates such an arrangement; Y is arranged not Y-periodically but point-symmetrically with respect to the cell edge centres indicated by the small solid circles. In the Y-periodic cell arrangement, Y is repeated simply, as shown in Fig. 4.7(a). On the other hand, in the point-symmetric cell arrangement, Y ~ and its conjugate cell Y alternate as illustrated in Fig. 4.7(b). Here, the conjugate cell ~ Y is the transformation of Y with respect to one of the cell edge centres. Using the Y-periodic and point-symmetric cell arrangements mentioned above, two kinds of fibre distributions are generated from an example of Y with nine randomly distributed fibres, as depicted in Figs. 4.8(a) and (b), respectively. Their fibre volume fractions are taken to be 56% as in the preceding experiment and analysis. Comparison between Fig. 4.8(a) and (b) suggests that the pointsymmetric cell arrangement has increased the randomness of fibre distribution compared with the Y-periodic arrangement; specifically, the fibre distribution in Fig. 4.8(b) has more matrix-rich areas and clusters of fibres than that in Fig. 4.8(a).

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4.7  Two types of arrangements of unit cell Y; (a) Y-periodic arrangement, and (b) point-symmetric arrangement.

4.8  Two kinds of transverse fibre distributions in laminae based on (a) Y-periodic cell arrangement and (b) point-symmetric cell arrangement.

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This enhancement of randomness of fibre distribution has been quantitatively demonstrated in terms of the standard deviations of the fibre distribution and the radial distribution function (Ziman, 1979) by Matsuda et al. (2003). In the point-symmetric cell arrangement, the internal structure has pointsymmetry with respect to all of the cell edge centres. This results in a pointsymmetric distribution of the perturbed velocity with respect to these points. Utilizing this point-symmetry as a boundary condition for the boundary value problems 4.12 and 4.13 (Ohno et al., 2001), the unit cell Y is able to be used as the domain of analysis. This means the computational costs are the same for both the point-symmetric and Y-periodic cell arrangements, even though the pointsymmetric cell arrangement has a more random fibre distribution than the Y-periodic one. In the present analysis, therefore, the elastic-viscoplastic analysis of CFRP laminates was performed adopting the fibre distribution shown in Fig. 4.8(b) as the fibre array in the laminae. The unit cell Y in the figure was employed as Y(α), and then Y(α) was divided into four-node isoparametric finite elements as illustrated in Fig. 4.9. The other analysis conditions were the same as in the preceding analysis, which were described in Sections 4.3.2 and 4.3.3.

4.9  Unit cell Y (α) and finite element mesh.

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The relations between macroscopic tensile stress Σψ and strain Eψ obtained in the present analysis are depicted by the dashed lines in Figs. 4.6(a)–(c) for the unidirectional, cross-ply and quasi-isotropic laminates, respectively. In these figures, the analysis results assuming the hexagonal fibre array, which were already discussed in the preceding analysis, are also shown by the solid lines. Comparing these two sets of analysis results, they agree well with each other for all of the laminates and off-axis angles examined. Remember that the fibre distribution based on the point-symmetric cell arrangement was quite random in contrast to the hexagonal fibre array. Nevertheless, they provided almost the same results for the macroscopic tensile behaviour of the laminates. Additionally, the Y-periodic cell arrangement shown in Fig. 4.8(a) also gave almost the same results in all cases (not shown). These results indicate that the transverse randomness of fibre distribution in laminae has little influence on the homogenized elasticviscoplastic behaviour of CFRP laminates. Therefore, it is not necessary to take into account the randomness of fibre distribution as far as the macroscopic behaviour is concerned. On the other hand, the microscopic distributions of stress and strain in a unit cell can be affected by the fibre distribution. Discussion about the influence of fibre distribution at the microscopic level can be found in the literature (Matsuda et al., 2003). It is therefore important to consider the randomness of fibre distribution when studying microscopic problems such as interfacial damage (e.g., Lene and Leguillon, 1982), microscopic fracture (e.g., Takano et al., 1999; Carvelli and Poggi, 2001) and microscopic instability (e.g., Ohno et al., 2002; Okumura et al., 2004).

4.4

Elastic-viscoplastic analysis of plain-woven GFRP laminates and experimental verification

Plain-woven laminates made of plain fabrics and polymeric materials are greatly important PMCs in many industrial sectors today. They have not only high specific stiffness and strength but also excellent formability, making them extremely useful in high-end industrial products such as components of aircraft and vehicles, and the blades of wind power generators. These situations can cause the viscoplastic deformation of plain-woven laminates, which makes their viscoplastic analysis highly important. However, such analysis is not straightforward because plainwoven laminates have noticeably complex-shaped reinforcements, i.e. plain fabrics (see Fig. 4.10). Thus, for the analysis of the viscoplastic behaviour of plain-woven laminates, a theory which is able to deal with such complex-shaped reinforcements is required. One of the most suitable theories is the homogenization theory described in Section 4.2, because, in this theory, the shapes of reinforcements are not restricted to specific cases such as spheres and ellipsoids, but can be defined arbitrarily. In this section, therefore, the homogenization theory in Section 4.2 is applied to the elastic-viscoplastic analysis of plain-woven GFRP laminates (Matsuda et al.,

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4.10  Plain-woven laminates with two types of laminate configurations of plain fabrics; (a) in-phase, and (b) out-of-phase.

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2007). First, it is shown that plain-woven laminates with in-phase or out-of-phase laminate configurations have point-symmetric internal structures, utilizing this point-symmetry as a boundary condition for the boundary value problems. As a result, the domains of analysis for the two laminate configurations are reduced to 1/4 and 1/8, respectively. The homogenization theory described in Section 4.2 is then applied to these reduced domains of analysis. Furthermore, uniaxial tensile tests of a plain-woven GFRP laminate at a constant strain rate are conducted at room temperature. Finally, the results of these experiments are compared with those predicted by the present theory.

4.4.1 Laminate configurations and domains of analysis In the homogenization analysis of plain-woven laminates, two types of laminate configurations of plain fabrics, i.e. in-phase and out-of-phase, are generally employed as the internal structures of plain-woven laminates as illustrated in Fig. 4.10 (Guedes and Kikuchi, 1990; Takano et al., 1995, 1999; Carvelli and Poggi, 2001). The in-phase laminate configuration has no offset of the fabric layers in the y1- and y2-directions (Fig. 4.10(a)). The out-of-phase laminate configuration, on the other hand, has a phase shift of the fabric layers by π in the y1- and y2-directions (Fig. 4.10(b)). According to previous reports (Guedes and Kikuchi, 1990; Takano et al., 1995, 1999; Carvelli and Poggi, 2001), the assumption of these laminate configurations provides fairly reliable results, although the actual microstructures of plain-woven laminates are not perfectly periodic but random to a greater or less extent. In this section, therefore, we consider these two laminate configurations as the internal structures of plain-woven laminates. For the above-mentioned laminate configurations, point-symmetry is found in their internal structures. Such point-symmetry can be utilized as a boundary condition for the boundary value problems, allowing the possibility that the domain of analysis is reduced. This issue is discussed in the following paragraphs. First, a unit cell Y of a laminate with an in-phase laminate configuration is defined as depicted by the dashed lines in Fig. 4.11(a). Now, focus attention on a part of Y, which is indicated by the solid lines in the figure, and is referred to as a basic cell A, hereafter. A careful look at the figure reveals that the internal structure of the laminate has point-symmetry with respect to the centres of the lateral facets of A, which are denoted by the open circles in the figure. The perturbed velocity in the laminate, therefore, distributes point-symmetrically with respect to these points. In contrast, the perturbed velocity at the top and bottom facets of A satisfies Y-periodicity because the internal structure is periodic in the y3-direction (stacking direction) in terms of A. The use of the point-symmetry and Y-periodicity of the perturbed velocity as a boundary condition, along with the results from a previous study (Ohno et al., 2001), enables the boundary value problems (4.12 and 4.13) to employ A as the domain of analysis. In consequence, Eq. 4.12 and 4.13 result in Eq. 4.58 and 4.59, respectively, which are shown in the next section.

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Next, a unit cell Y with an out-of-phase laminate configuration is defined as indicated by the dashed lines in Fig. 4.11(b). This unit cell has twice the volume of that of the in-phase laminate configuration. However, the same basic cell A depicted by the solid lines in the figure is taken again. In this case, the internal structure of the laminate has point-symmetry with respect to the centres of not only the lateral facets but also the top and bottom facets of A, which are denoted by the open circles in Fig. 4.11(b). This means that the perturbed velocity in the laminate distributes point-symmetrically with respect to all the cell facet centres of A. Employing this point-symmetric distribution of the perturbed velocity as a boundary condition, the same boundary value problems for A, Eq. 4.58 and 4.59, are derived as for the in-phase laminate configuration.

4.11  Unit cells Y and basic cells A of plain-woven laminates; (a) in-phase, and (b) out-of-phase. (Continued )

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4.11  Continued.

As mentioned above, the basic cell A is able to be used as the domain of analysis for both the in-phase and out-of-phase laminate configurations. This reduces the domains of analysis of the two laminate configurations to 1/4 and 1/8, respectively, of the conventional unit cells. These reductions in the domains of analysis lead to

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significantly less computational costs being required to solve the boundary value problems, which is of great use for incremental analysis as dealt with in the present chapter.

4.4.2 Homogenization theory for plain-woven laminates Based on the above discussion, the homogenization theory described in Section 4.2 is restated for A as follows. Consider that a plain-woven laminate is subjected to macroscopically uniform load and exhibits infinitesimal deformation both macroscopically and microscopically, and that the constituents of the laminate are assumed to have the elastic-viscoplastic properties characterized by Eq. 4.2. Then, the evolution equation . of microscopic. stress σij, and the relation between macroscopic stress rate Σij and strain rate Eij, are derived in the same form as Eq. 4.15 and 4.19, i.e., [4.55]



[4.56]

Here, it is noted that   represents the volume average in A, not in Y, as: [4.57] where | A| stands for the volume of A. Moreover, χ kli and ϕi are the functions determined by solving the following boundary value problems for A:

[4.58]

[4.59] where υi denotes an arbitrary velocity field satisfying the point-symmetry with respect to the cell facet centres of A or the Y-periodicity. Note that the above problems are solved using FEM with the following boundary conditions. For the in-phase laminate configuration, the point-symmetric condition with respect to the centres of the lateral facets of A, and the Y-periodic condition with respect to the top and bottom facets of A, are imposed on χ kli and ϕi. For the out-of-phase laminate configuration, on the other hand, the point-symmetric condition with respect to all of the cell facet centres of A is imposed on χ kli and ϕi.

4.4.3 Experimental procedure For the verification of the present theory, uniaxial tensile tests of a plain-woven GFRP laminate at a constant strain rate were carried out at room temperature. Coupon specimens as illustrated in Fig. 4.12 were cut out from a plain-woven glass

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4.12  Schematic diagram of a coupon specimen of a plain-woven GFRP laminate with dimensions in mm.

fibre/epoxy laminate (1000 mm × 1000 mm, 10 plain fabrics stacked) manufactured by Nitto Shinko Corporation. Strain gauges and rectangular GFRP tabs were attached on the both sides of each specimen. Defining the off-axis angle ψ as the angle between the longitudinal direction of specimens (loading direction) and the warp direction of the plain fabrics, four values of ψ, i.e. ψ = 0°, 15°, 30° and 45°, were considered. The tensile tests were done by the same closed-loop servohydraulic testing machine as in Section 4.3.2. The strain rate was detected by the crosshead of the testing machine, and the machine . was controlled so that the specimens were elongated at a constant strain rate of Eψ = 10–5 s–1. It was confirmed by the strain gauges that the strain rate was kept precisely at 10–5 s–1 during the tests.

4.4.4 Analysis conditions Plain-woven GFRP laminates were assumed to have in-phase or out-of-phase laminate configurations as described in Section 4.4.1. For the laminates, a basic © Woodhead Publishing Limited, 2011



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cell was defined as illustrated in Fig. 4.13. This geometry of the basic cell was determined based on the average values of measurements obtained from the microscope observation of eight arbitrary aspects of the laminate from which the specimens had been cut out. The basic cell was then divided into eight-node isoparametric elements (1624 elements, 1995 nodes) as shown in Fig. 4.13. This finite element mesh corresponds to the meshes of unit cells with 6496 and 12 992 elements for the in-phase and out-of-phase laminate configurations, respectively. Fibre bundles, i.e. the warp and weft in Fig. 4.13(b), were regarded as glass fibre/epoxy unidirectional composites that exhibit linear elastic behaviour. The material properties of the fibre bundles were calculated using the mathematical homogenization theory, on the assumptions that the fibre volume fraction was 75 % in accordance with the microscope observation, and that the bundles had a hexagonal fibre array. The elastic properties of the glass fibres and epoxy used in the calculation are listed in Table 4.3.

4.13  Basic cell and finite element mesh; (a) full view with dimensions in mm, and (b) fibre bundles in the basic cell.

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Creep and fatigue in polymer matrix composites Table 4.3  Material constants of the glass fibres and epoxy matrix for plain-woven GFRP laminates Glass fibre Epoxy

Ef = 8.0 × 104 νf = 0.30

. Em = 5.0 × 103 νm = 0.35 ε p0 = 10–5 −p − − p p 0.50 n = 20 g(ε ) = (ε ) + 24.5 / 2.5ε

MPa (stress), mm/mm (strain), s (time).

4.14  Macroscopic tensile curves of plain-woven GFRP laminate with the 45° off-axis angle at constant strain rates.

In contrast, the epoxy matrix was regarded as an isotropic elastic-viscoplastic material obeying the same type of constitutive equation as Eq. 4.54 in Section 4.3.3. The identification of the material constants in Eq. 4.54 was achieved by simulating 45° off-axis tensile tests of the plain-woven GFRP laminate at constant . strain rates of E45° = 10–3, 10–5 and 10–7 s–1 (Fig. 4.14), which was the same methodology as in Section 4.3.3. The material constants determined are shown in Table 4.3.

4.4.5 Comparison between experiments and predictions Macroscopic stress–strain relations obtained from the tensile tests and predictions using the present theory are shown in Fig. 4.15. First, as seen from the experimental

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4.15  Macroscopic stress versus strain relations of plain-woven GFRP . laminates at Eψ = 10–5 s–1.

results indicated by the open circles in the figure, almost linear behaviour of the plain-woven GFRP laminate was observed at ψ = 0° (on-axis loading). In contrast, with off-axis loading, i.e. ψ = 15°, 30° and 45°, the laminate exhibited marked nonlinearity caused by the viscoplasticity of the matrix material. It is further seen from the figure that the increase in ψ gave rise to a sudden decrease in the viscoplastic flow stress, showing the significant in-plane anisotropy of the laminate. Comparing these experimental results with the predictions indicated by the lines in the figure, they agreed well for all ψ. Therefore, the present theory can be successfully applied to the prediction of the macroscopic elastic-viscoplastic behaviour of plain-woven GFRP laminates. Next, the results of the in-phase and out-of-phase laminate configurations, which are indicated by the solid and dashed lines, respectively, are compared with each other. As seen from Fig. 4.15, almost no difference was observed between the results in the elastic range. This is in contrast to the observation in the viscoplastic range; the differences of the viscoplastic flow stresses between the two laminate configurations reached 5–15% at Eψ = 0.01. A closer look at the figure reveals that, for the on-axis loading (ψ = 0°), the flow stress of the out-of-phase laminate configuration is higher than that of the in-phase one, whereas for the off-axis loading (ψ = 15°, 30° and 45°), the relation is reversed. These results suggest that the laminate configurations of plain fabrics affect the viscoplastic behaviour of plain-woven GFRP laminates, the explanation of which from a microscopic point of view can be found in the literature (Matsuda et al., 2007).

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4.5

Creep analysis of unidirectional CFRP laminates at elevated temperature

The final demonstration of the present theory in this chapter is the creep analysis of a CFRP laminate (Fukuta et al., 2008). As already stated in Section 4.3, CFRP laminates incorporated into modern technology have a possibility of encountering severe conditions such as high stress and high temperature, which can induce creep deformation of the laminates. Thus, attention also has to be paid to the creep behaviour of CFRP laminates. In this section, therefore, homogenized creep behaviour of a unidirectional carbon fibre/epoxy laminate at elevated temperature is analyzed using the theory described in Section 4.3. The results of this analysis are then compared with experimental data from the literature (Kawai and Masuko, 2004), verifying the applicability of the theory to the creep analysis of CFRP laminates. Because the theory used in the present analysis is the same as in Section 4.3, only the conditions and results of the analysis are described below.

4.5.1 Analysis conditions The off-axis creep behaviour of a unidirectional carbon fibre/epoxy laminate T800H/#3631 manufactured by Toray Industries, Inc. at elevated temperature (100 °C) was analyzed in accordance with an experiment in the literature (Kawai and Masuko, 2004). Two kinds of off-axis angles, i.e. ψ = 30° and 45°, were considered. For each ψ, three kinds of creep stress levels were selected, i.e. 44, 63 and 84 MPa for ψ = 30°, and 35, 51 and 68 MPa for ψ = 45°. These stress levels were 40, 60 and 80%, respectively, of the static tensile strength of the laminate at 100 . °C for each ψ. The laminate was first elongated at a constant strain rate of Eψ = 1.67 × 10–4 s–1 (1.0%/min) until the prescribed creep stress was achieved. The stress was then held constant for five hours. The laminate was assumed to have a hexagonal fibre array similarly to Section 4.3, in which the fibre volume fraction Vf was taken to be 51.7%. Then, a unit cell Y was defined, and its upper half was divided into four-node isoparametric elements as illustrated in Fig. 4.16. The carbon fibres were regarded as transversely isotropic elastic materials, whereas the epoxy matrix as an elastic-creep material characterized by Eq. 4.54. The procedure for determining the material constants of the fibres and epoxy was the same as in Section 4.3; for the carbon fibres, Ef 3 was provided by the manufacturer, Toray Industries, Inc., while the other constants were obtained by referring to the literature (Kriz and Stinchcomb, 1979). Regarding the epoxy, the material constants were determined by simulating 45° . off-axis tensile tests of the uni­ directional CFRP laminate T800H/#3631 at E45° = 1.67 × 10–4 and 1.67 × 10–6 s–1 (Fig. 4.17). The material constants determined are listed in Table 4.4.

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4.16  Unit cell Y (α) and finite element mesh.

4.17  Macroscopic tensile curves of a unidirectional CFRP laminate with the 45° off-axis angle at constant strain rates (100 °C).

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Creep and fatigue in polymer matrix composites Table 4.4  Material constants of the carbon fibres and epoxy matrix for the creep analysis of a unidirectional CFRP laminate Carbon fibre

Ef1 = 1.58 × 104 Ef 3 = 2.94 × 105  Gf 31 = 1.97 × 104



νf 12 = 0.49

Epoxy

Em = 3.6 × 103 νm = 0.35 g(ε− p) = 140.4(ε− p)0.254 + 10



νf 31 = 0.28

. ε p0 = 1.67 × 10–4

n = 35

MPa (stress), mm/mm (strain), s (time).

4.5.2 Results of analysis Figures 4.18(a) and (b) show the creep curves of the unidirectional CFRP laminate at 100 °C for ψ = 30° and 45°, respectively. In the figures, the markers represent experimental data from the literature (Kawai and Masuko, 2004), whereas the solid lines indicate the results of the present analysis. First, as seen from the experimental data, the laminate clearly exhibited creep behaviour in all cases, in which the creep rates were considerably high initially, but rapidly decreased during the early stage of the creep curves. Moreover, a marked dependency of the creep behaviour on the creep stress levels was observed regardless of the off-axis angle; the creep strain became larger as the creep stress increased. Such creep behaviour of the laminate is successfully predicted by the present theory in all the cases. This suggests that the theory in Section 4.3 is also applicable to the creep analysis of CFRP laminates.

4.6

Summary

In this chapter, the elastic-viscoplastic behaviour of CFRP laminates and plainwoven GFRP laminates was analyzed based on the homogenization theory for nonlinear time-dependent composites. Moreover, uniaxial tensile tests of the laminates at a constant strain rate were performed to verify the results of the analyses. Finally, the homogenization theory was applied to the creep analysis of a unidirectional CFRP laminate at elevated temperature. This chapter is summarized as follows: First, in Section 4.2, the homogenization theory for nonlinear time-dependent composites was introduced as a basis for all of the analyses in this chapter. In Section 4.3, an in-plane elastic-viscoplastic constitutive relation of long fibre-reinforced laminates was derived, using the homogenization theory in conjunction with the standard lamination theory. To verify the resulting theory, uniaxial tensile tests of carbon fibre/epoxy laminates at a constant strain rate were performed at room temperature using three kinds of laminates, i.e. unidirectional, cross-ply, and quasi-isotropic laminates. The experimental results for all of the

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4.18  Macroscopic creep curves of a unidirectional CFRP laminate at three kinds of creep stress levels (100 °C); (a) ψ = 30°, and (b) ψ = 45°.

laminates were accurately predicted by the theory, indicating the excellent applicability of the theory to the elastic-viscoplastic analysis of CFRP laminates. Moreover, it was demonstrated that the randomness of the fibre distribution in the laminae had little influence on the homogenized behaviour of CFRP laminates.

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Section 4.4 was devoted to the elastic-viscoplastic analysis of plain-woven GFRP laminates using the homogenization theory, and its experimental verification. First, the point-symmetry of the internal structures of plain-woven laminates with in-phase or out-of-phase laminate configurations was utilized as a boundary condition for the boundary value problems, reducing the domains of analysis and the computational costs. Then, uniaxial tensile tests of a plain-woven glass fibre/epoxy laminate at a constant strain rate were carried out at room temperature. Comparison of the experimental and predicted results revealed that the homogenization theory was able to successfully predict the homogenized elastic-viscoplastic behaviour of plain-woven GFRP laminates. Moreover, it was also shown that the laminate configurations of the plain fabrics within a laminate affected its viscoplastic behaviour. Finally, in Section 4.5, the theory described in Section 4.3 was applied to the creep analysis of a unidirectional carbon fibre/epoxy laminate at elevated temperature. Two kinds of off-axis angles, with three kinds of creep stress levels for each off-axis angle, were considered in the analysis. The results of the analysis agreed well with previously published experimental data, showing that the theory was also applicable to the creep analysis of CFRP laminates. In conclusion, the present authors can say that the homogenization theory for non linear time-dependent composites has high applicability to the viscoplastic and creep analyses of polymer matrix composites.

4.7

References

Agah-tehrani A (1990), ‘On finite deformation of composites with periodic microstructure’, Mech Mater, 8, 255–268. Asada T and Ohno N (2007), ‘Fully implicit formulation of elastoplastic homogenization problem for two-scale analysis’, Int J Solids Struct, 44, 7261–7275. Babuska I (1976), ‘Homogenization approach in engineering’, in Glowinski R and Lions J L, Computing methods in applied sciences and engineering, lecture notes in economics and mathematical systems 134, Berlin, Springer-Verlag. Bakhvalov N and Panasenko G (1984), Homogenisation: Averaging processes in periodic media, Dordrecht, Kluwer Academic Publishers. Bensoussan A, Lions J L and Papanicolau G (1978), Asymptotic analysis for periodic structures, Amsterdam, North-Holland. Carvelli V and Poggi C (2001), ‘A homogenization procedure for the numerical analysis of woven fabric composites’, Compos Part A, 32, 1425–1432. Fukuta Y, Matsuda T and Kawai M (2008), ‘Homogenized creep behaviour of CFRP laminates at high temperature’, Int J Mod Phys B, 22, 6161–6166. Ghosh S and Moorthy S (1995), ‘Elastic–plastic analysis of arbitrary heterogeneous materials with the Voronoi cell finite element method’, Comput Meth Appl Mech Eng, 121, 373–409. Guedes J M and Kikuchi N (1990), ‘Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods’, Comput Meth Appl Mech Eng, 83, 143–198.

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Jansson S (1991), ‘Mechanical characterization and modeling of non-linear deformation and fracture of a fiber reinforced metal matrix composite’, Mech Mater, 12, 47–62. Kawai M and Masuko Y (2004), ‘Creep behavior of unidirectional and angle-ply T800H/3631 laminates at high temperature and simulations using a phenomenological viscoplasticity model’, Compos Sci Technol, 64, 2373–2384. Kriz R D and Stinchcomb W W (1979), ‘Elastic moduli of transversely isotropic graphite fibers and their composites’, Exp Mech, 19, 41–49. Lene F and Leguillon D (1982), ‘Homogenized constitutive law for a partially cohesive composite material’, Int J Solids Struct, 18, 443–458. Matsuda T, Nimiya Y, Ohno N and Tokuda M (2007), ‘Elastic-viscoplastic behavior of plain-woven GFRP laminates: Homogenization using a reduced domain of analysis’, Compos Struct, 79, 493–500. Matsuda T, Ohno N, Tanaka H and Shimizu T (2002), ‘Homogenized in-plane elastic-viscoplastic behavior of long fiber-reinforced laminates’, JSME Int J, Ser A, 45, 538–544. Matsuda T, Ohno N, Tanaka H and Shimizu T (2003), ‘Effects of fiber distribution on elastic-viscoplastic behavior of long fiber-reinforced laminates’, Int J Mech Sci, 45, 1583–1598. Ohno N, Matsuda T and Wu X (2001), ‘A homogenization theory for elastic-viscoplastic composites with point symmetry of internal distribution’, Int J Solids Struct, 38, 2867–2878. Ohno N, Okumura D and Noguchi H (2002), ‘Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation’, J Mech Phys Solids, 50, 1125–1153. Ohno N, Wu X and Matsuda T (2000), ‘Homogenized properties of elastic-viscoplastic composites with periodic internal structures’, Int J Mech Sci, 42, 1519–1536. Okumura D, Ohno N and Noguchi H (2004), ‘Elastoplastic microscopic bifurcation and post-bifurcation behavior of periodic cellular solids’, J Mech Phys Solids, 52, 641–666. Sanchez-Palencia E (1980), Non-homogeneous media and vibration theory, lecture notes in physics 127, Berlin, Springer-Verlag. Suquet P M (1987), ‘Elements of homogenization for inelastic solid mechanics’, in Sanchez-Palencia E and Zaoui A, Homogenization techniques for composite media, lecture note in physics 272, Berlin, Springer-Verlag. Takano N, Uetsuji Y, Kashiwagi Y and Zako M (1999), ‘Hierarchical modelling of textile composite materials and structures by the homogenization method’, Model Simul Mater Sci Eng, 7, 207–231. Takano N, Zako M, Kubo F and Kimura K (2003), ‘Microstructure-based stress analysis and evaluation for porous ceramics by homogenization method with digital imagebased modeling’, Int J Solids Struct, 40, 1225–1242. Takano N, Zako M and Sakata S (1995), ‘Three-dimensional microstructural design of woven fabric composite materials by the homogenization method: 1st report, effect of mismatched lay-up of woven fabrics on the strength’, Trans Jpn Soc Mech Eng, Ser A, 61, 1038–1043 (in Japanese). Terada K, Yuge K and Kikuchi N (1995), ‘Elasto-plastic analysis of composite materials using the homogenization method: 1st report, formulation’, Trans Jpn Soc Mech Eng, Ser A, 61, 2199–2205 (in Japanese). Tohgo K, Kawahara K and Sugiyama Y (2001), ‘Off-axis tensile properties of CFRP laminates and non-linear lamination theory based on micromechanics approach’, Trans Jpn Soc Mech Eng, Ser A, 67, 1493–1500 (in Japanese).

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Tsuda M, Takemura E, Asada T, Ohno N and Igari T (2010), ‘Homogenized elasticviscoplastic behavior of plate-fin structures at high temperatures: Numerical analysis and macroscopic constitutive modeling’, Int J Mech Sci, 52, 648–656. Wu X and Ohno N (1999), ‘A homogenization theory for time-dependent nonlinear composites with periodic internal structures’, Int J Solids Struct, 36, 4991–5012. Ziman J M (1979), Models of disorder, Cambridge, Cambridge University Press.

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5 Measuring fiber strain and creep behavior in polymer matrix composites using Raman spectroscopy T. Miyake , Nagoya Municipal Industrial Research Institute, Japan Abstract: This chapter discusses experimental approaches for measuring temporal stress change in a reinforcing fiber, which plays an important role in the creep of composites reinforced unidirectionally by continuous long fibers. The chapter first describes the application of micro-Raman spectroscopy to long-term measurement of temporal change in fiber stress during stress relaxation tests and constant-load pull-out tests for model composites consisting of a long fiber monofilament and transparent resin matrix. The features of the experimental results are then discussed in comparison with theoretical models. Key words: creep of unidirectional composites, local stress measurement by micro-Raman spectroscopy, temporal stress change in reinforcing fibers, stress relaxation in broken fibers, interfacial slippage.

5.1

Introduction: creep mechanism of composites reinforced unidirectionally with long fibers

There has been little attention paid to the creep strength of unidirectional composites reinforced with long continuous fibers because the reinforcing fibers do not creep at all, and therefore the composites are supposed not to creep in normal working environments. However, some experimental results show that such composites can be subjected to rupture in longitudinal creep, even if the fibers do not creep at all (Lifshitz and Rotem, 1970; Phoenix et al., 1988; Otani et al., 1991; Weber et al., 1993; Ohno et al., 1994; Ohno et al., 1996, Weber et al., 1996). Figure 5.1 shows an example of the experiments conducted by Ohno et al. (1996) using a metal matrix composite consisting of continuous SiC fibers and a beta titanium alloy. The following features can be seen from the figure: rupture occurs without tertiary creep, and strain increases little before rupture. Such creep characteristics make it hard to forecast when the creep rupture of the composite will happen. In the case of such unidirectional composites, it is difficult to make precise predictions for creep life based on the creep mechanism. It has been reported that matrix viscosity and stochastic fiber fracture play important roles in the creep mechanism of unidirectional composites (Ohno, 1998) and the following two mechanisms, which are peculiar to unidirectionally reinforced composites, have been suggested so far. Firstly, when matrix tensile stress relaxation occurs, the matrix stress is transferred to fiber stress, inducing 149 © Woodhead Publishing Limited, 2011

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5.1  Creep curves of SCS-6/Beta21S at 1200 MPa.

fiber breaks, which result in the overall rupture of composites. This mechanism is dominant in short- or medium-term creep, at relatively high applied stress. Secondly, matrix shear creep around fiber breaks causes the broken fibers to relax, so that stress in intact fibers is increased, leading to further fiber breaks. This mechanism is dominant in long-term creep at relatively low applied stress. Unidirectional polymer matrix composites reinforced with long brittle fibers, whose elongation is smaller than those of matrix resins, may suffer from fiber breaks under initial loading and subsequent creep stress holding. In a broken fiber, the axial stress diminishes at the break and builds in intact fibers a certain distance away (Fig. 5.2). The distance between the break and where the axial stress builds up is referred to as the stress recovery length (or an ineffective length), and plays a substantial role in the strength characteristics of those composites, especially longterm failure behaviors, such as creep. In the stress recovery length, a large shear stress builds at the interface between fiber and matrix. As the matrix undergoes shear creep around the stress recovery region, the interfacial shear stress decreases and the stress recovery length evolves along the fiber with time. This results in a decrease in the load-bearing ability of the broken fibers (as shown in Fig. 5.2) and the neighboring intact fibers becoming overloaded and inducing further fiber breaks, Consequently, creep failure can occur even under constant loading applied in the fiber direction. This has motivated investigations from several theoretical viewpoints regarding the time-dependent change in the stress profiles of broken fibers (Lifshitz and

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5.2  Stress profile and stress relaxation in broken fibers embedded in matrix.

Rotem, 1970; Phoenix et al., 1988; Otani et al., 1991; Lagoudas et al., 1989; Mason et al., 1992; Kelly and Barbero, 1993; Du and McMeeking, 1995; Iyengar and Curtin, 1997; Ohno and Miyake, 1999). These studies have revealed that stress recovery length grows with time as the matrix creeps, but different assumptions employed by different authors have yielded divergent predictions. For instance, on the basis of perfect bonding at the fiber/matrix interface, the stress recovery length is found to increase rapidly after fiber breakage (Lifshitz and Rotem, 1970; Lagoudas et al., 1989; Mason et al., 1992). On the other hand, the evolution of this length turns out to be quite slow when incipient slippage at the fiber/matrix interface is accounted for (Du and McMeeking, 1995; Ohno and Miyake, 1999). Raman spectroscopy can be used to measure the stress distributions in fibers in a transparent matrix, based on the principle that the Raman bands shift with applied stress or strain. Raman spectroscopy has been used in a considerable number of experimental studies on the fiber stress in composites, but only the

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present authors (Miyake et al., 1998; Miyake et al., 2001) have used it to conduct investigations into the time-dependent change in stress profiles of broken fibers.

5.2

Stress or strain measurement by Raman spectroscopy

Raman spectroscopy can be used to measure the stress or strain distributions in fibers embedded in transparent matrices. In this method, initiated by Galiotis et al. (1984), a Raman spectrometer equipped with an optical microscope enables us to determine local stress with a high spatial resolution of µ m. This method has been employed in several works to measure stress distributions in the fibers of model composite specimens (Galiotis, 1991; Fan et al., 1991; Melanitis et al., 1993; Huang and Young, 1994; Andrews and Young, 1995; Grubb et al., 1995; Wagner et al., 1996; Chohan and Galiotis, 1997). This section first describes the fundamentals of stress or strain measurement using Raman spectroscopy and presents a result for the carbon fiber employed in the current study as an actual case. It then goes on to describe the microscopic system of the Raman spectrometer and to specify micro-Raman system measurements of fibers in single-fiber model composite specimens, which consist of a carbon fiber monofilament and epoxy resin.

5.2.1 Fundamentals of stress measurement by Raman spectroscopy The use of Raman spectroscopy is based on the finding that the position of the Raman bands in most highly oriented materials (e.g., high-performance fibers and single crystals) are sensitive to applied stress or strain. Raman scattering occurs due to the inherent atomic vibrations in materials under the irradiation of a laser beam. The deformation of such high-oriented fibers or crystals causes molecular deformation and the resulting changes in the interatomic distances and angles alter the frequencies of the atomic vibrations (Anastassakis et al., 1970; Batchelder and Bloor, 1979). This appears as the shift in the vibrational frequency of a particular Raman band, as shown in Fig. 5.3 for the case of a carbon fiber. in general, the peak frequency of the Raman band shifts linearly with the applied stress or strain. Therefore, using a calibration line obtained from the relationship between the shift in the Raman peaks and the axial stress or strain applied to a fiber, it is possible to estimate the axial stress or strain generated in the embedded fiber in the composite from the measured Raman frequencies. Indeed, for the carbon fibers tested in the experiments later described, an almost linear relation between peak frequency ν and applied tensile stress σf was obtained for the Raman band at 2700 cm–1, as shown in Fig. 5.4. The relation was determined by measuring the Raman spectra of a free standing single carbon fiber, stressed

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5.3  Raman spectra of free standing carbon fiber with and without tensile stress.

5.4  Dependence of 2700 cm–1 band peak position on applied tensile stress.

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incrementally with dead weights. The linear least squares fit gives the peak frequency ν as:

ν = 27005.0 ± 0.2 – (7.5 ± 0.1)σf ,

[5.1]

where ν and σf are given in cm–1 and GPa, respectively. The peak frequency ν is also converted to fiber strain using Hook’s law, which is indicated on the upper abscissa axis of Fig. 5.4. The above relation was found to be applicable to the embedded fiber in composite specimens: the peak frequency ν  in the 2700 cm–1 band of the embedded fiber was measured by pulling incrementally a single-fiber composite specimen. At the same time, axial strain in the region very close to the fiber was measured with a resistance strain gauge adhered adjacent to the fiber, on the surface of the specimen. The fiber strain, converted from the peak frequency ν, and the axial strain measured with strain gauges, agreed well with each other, as shown in Fig. 5.5. Thus, Eq. 5.1 was also found to be valid in the case of the embedded fibers in the composite specimens.

5.2.2 Raman spectrometer and measurement The Raman measuring system used in the experimental works is a micro-Raman spectrometer (Model 750-1, Instruments S. A. Inc.) equipped with a single

5.5  Comparison between fiber strain and specimen overall strain in a single-fiber model composite specimen, measured by Raman spectroscopy and a strain gauge, respectively.

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polychromator coupled to a modified optical microscope. Figure 5.6 shows the block diagram of the Raman spectrometer. An Ar+ laser, operating at 514.5 nm, is employed as the light source. The laser beam is focused to a spot of approximately 1 µ m on the surface of the object to be measured (i.e., a reinforcing fiber) using a microscope with a ×80 objective lens. The 180° backscattered light from the fiber is collected using the microscope objective and filtered through a confocal spatial filter with a 200 µ  m pin hole to remove the scattering from out-of-focus surroundings. The scattering light is dispersed by the polychromator. Raman spectra are recorded using a highly sensitive and low-noise charge-coupled device (CCD) cooled with liquid nitrogen. The peak frequencies of Raman spectra are determined by fitting the raw data with a Lorentzian function, and rectified using a neon light source as a reference to remove the fluctuation with time. Thus the peak positions (i.e., Raman frequencies) of the Raman bands are obtained with an accuracy of better than 0.5 cm–1. Because the reinforcing fibers were embedded in transparent matrix and the Raman spectroscopy measurement was taken through matrix resin, the power of

5.6  Micro-Raman spectroscopy system.

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the incident light on the fiber surface was kept below 0.8 mW to avoid a shift of the Raman band due to local overheating. The Raman microprobe was used to map carbon fibers embedded in epoxy resins along the length of the fiber to determine stress or strain profiles. An x–y stage was used to translate the specimens. In the case of the carbon fiber/epoxy resin composite specimens employed in these experiments, the Raman band at 2700 cm–1 was adopted to monitor the stress of the embedded fiber for the following reasons: Fig. 5.7 shows the Raman spectra obtained from a free standing carbon fiber and from the epoxy resin, respectively. The fiber exhibited two major peaks, at 1580 and 2700 cm–1, whereas the epoxy resin had many peaks in the zone smaller than 1600 cm–1. The firstorder peak of the carbon fiber, around 1580 cm–1, was superimposed almost completely by a peak of the scattered light from the epoxy resin. On the other hand, the second-order peak of the carbon fiber, around 2700 cm–1, was not superimposed by the Raman bands of the epoxy resin. Figure 5.8 shows the Raman spectrum near 2700 cm–1 obtained from the carbon fiber embedded in an epoxy resin. It can be seen from the figure that although the spectrum was slightly affected by the epoxy resin, fitting the Lorentzian curve allowed us to determine the peak position in the 2700 cm–1 band of the carbon fiber by removing the two weak peaks in the spectra of the epoxy resin.

5.7  Raman spectra of free standing carbon fiber and epoxy resin.

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5.8  Raman spectrum for 2700 cm–1 band of carbon fiber embedded in epoxy resin together with fitted Lorenzian curves.

5.3

Experiments on stress relaxation in broken fibers

As described in the introduction, broken fibers play an important role in creep in composites reinforced unidirectionally with continuous long fibers. The stress relaxation in broken fibers has been studied both analytically and numerically in several works. Despite these extensive studies, further works, especially experiments to observe the stress relaxation, seem to be necessary. This is because different tendencies have been predicted on the basis of different assumptions. For example, according to the linear viscoelastic solutions of Lifshitz and Rotem (1970) and Lagoudas et al. (1989), stress redistributes significantly in broken fibers just after fiber breaks. In the numerical analysis of Du and McMeeking (1995), on the other hand, stress in broken fibers in a power-low creeping matrix relaxes very slowly in comparison with the matrix normal stress in the fiber direction, as was later expressed analytically in a simple form by Ohno and Miyake (1999). In terms of experimental studies, Raman spectroscopic measurements of the stress distribution of broken fibers in model composite specimens have been carried out in several works (Galiotis et al., 1984; Galiotis, 1991; Fan et al., 1991; Melanitis et al., 1993; Huang and Young, 1994; Andrews and Young, 1995; Grubb et al., 1995; Wagner et al., 1996; Chohan and Galiotis, 1997). The reported works, however, dealt with the stress distributions just after fiber breaks in fiber fragmentation tests. It is therefore worthwhile applying Raman spectroscopy to the study of time-dependent change in stress profiles in broken fibers. This work features long-term monitoring of the stress profiles in broken fibers using Raman

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spectroscopy. Measurements were taken over 1000 hours for stress in a broken fiber relaxed under a constant strain.

5.3.1 Materials and specimens The fibers employed in these experiments were commercially available highmodulus PAN-type carbon fibers (SR-40K, Mitubishi Rayon Co. Ltd). The fiber had been subjected to a surface treatment of epoxy compatible oxidation. The matrix was a room-temperature curing bisphenol A-based epoxy resin AER260 (AsahiCiba Ltd) mixed with a polyamine hardener HY847 (Asahi-Ciba Ltd) in the 10:4 weight proportion. Some mechanical properties of the fibers and the matrix resin are given in Table 5.1. It will be noticed that the fiber properties in the table, which were measured using polymer impregnated strands, can be applied only approximately to a monofilament fiber. A scanning electron microscope determined the fiber diameter to be 5.2 µm from a cross-section of a fiber embedded in the epoxy resin. The single-fiber composite specimens used in this study were prepared as follows: the epoxy resin, which was mixed thoroughly with the hardener and then degassed in a vacuum, was cast into rectangular molds until the molds were half-filled. The epoxy resin in the molds was then cured at room temperature for 24 hours. Next, a fiber monofilament was placed on the epoxy resin in each mold, and the molds were topped up with the epoxy resin. After curing at room temperature for 7 days, the samples were machined into dog-bone shaped specimens with the fiber aligned in the axial direction at a depth of 200 µm from the top surfaces (Fig. 5.9). The specimens were polished with 0.01 µm alumina slurry, so they became smooth and transparent enough to prevent attenuation of the scattered light from the fibers.

5.3.2 Stress relaxation tests and Raman spectroscopy measurement Each specimen was elongated in the fiber direction by means of a straining rig, as shown in Fig. 5.10. The rig was mounted directly on the microscope stage in the Raman system. The axial strain, measured with strain gauges adhered to the top Table 5.1  Mechanical properties of carbon fiber and epoxy resin

Carbon fiber (SR-40K)

Epoxy resin (AER260/HY847)

Young’s modulus (GPa) Tensile strength (GPa) Elongation (%) Diameter (m)

484* 4.6* 0.95* 5.2

3.7 0.057 2.0 –

* Measured using impregnated strand.

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5.9  Shape of single-fiber model composite specimen.

5.10  Loading device for stress relaxation tests.

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surface of the specimen, was increased stepwise by 0.1% increments. After every increment the fiber was inspected within about 2.5 mm distances from the specimen center to ascertain whether the fiber was broken or not. The inspection was carried out by viewing the matrix birefringence around the fiber break with a polarizedlight microscope. As soon as the fiber was found fractured, the axial strain was kept constant for 1000 hours in air, which was conditioned to be 25 °C (298 K) and dehumidified to have a relative humidity below 50 %. While holding the axial strain, the fiber was scanned in the axial direction at appropriate times using the Raman microprobe; i.e., the Raman frequency in the 2700 cm–1 band was measured at points spaced at intervals of 50 µm within a 1000 µm distance from a break in the fiber, as shown in Fig. 5.11. Then, using Eq. 5.1, the Raman frequency ν was converted to fiber stress σf to obtain the stress profile along the fiber near the break. The axial force induced in the specimen was monitored continuously by a load cell connected in series to the specimen in the straining rig (Fig. 5.10). The force was divided by the cross-sectional area in the gauge section to evaluate the matrix tensile stress in the fiber direction. The matrix stress, which was an averaged one in the gauge section, is simply referred to as matrix normal stress and indicated as σm from now on.

5.3.3 Experimental results Three specimens were subjected to constant overall strain for 1000 hours, as was described in Section 5.3.2. The fibers in them fractured when the axial strain measured with strain gauges, ε, was increased to 0.7, 1.0 and 1.4% in the case of each fiber, respectively. The experiments of ε = 0.7 and 1.0% had similar results. Therefore, the results in the experiment of ε  = 1.0% are discussed in the reference

5.11  Location of Raman spectra measurement in broken fiber.

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(Miyake, 2003) and only the experiments of ε  = 0.7% and 1.4%, which had very different results, are reported here. Stress relaxation at 0.7% constant strain Figure 5.12 shows the axial stress profiles in the broken fiber at the elapsed time of t = 1, 10, 100 and 1000 hours after the fiber break in the experiment of ε  = 0.7%. The following features can be seen from the figure: the stress recovery length (the ineffective length) where fiber stress σf built up from zero to that in intact fibers, was estimated as 300 µm just after the fiber break. The stress profiles at t = 1, 10 and 100 hours were almost the same, but from t = 100 to 1000 hours, stress relaxation occurred a little in the fiber and caused the stress recovery length to increase from approximately 300 to 400 µm. The profiles of interfacial shear stress τ around the fiber break were computed from those of σf shown in Fig. 5.12 using an equilibrium equation:

[5.2]

where df denotes the diameter of fibers. The differential σf /z was found to be easily affected by the errors in measuring σf . Savitzky-Golay’s coefficient for a quadratic polynomial was therefore employed for smoothing the differential, though it became impossible to evaluate τ at the two points closest to the break. The profiles of τ obtained with such smoothing are shown in Fig. 5.13. As seen from the figure, τ varied little and slowly with time, and consequently the maximum value of τ decreased by about 10% in 1000 hours after the fiber break.

5.12  Profile of axial stress in broken fiber at ε = 0.7%.

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5.13  Profile of interfacial shear stress in broken fiber at ε = 0.7%.

5.14  Relaxation of normal stress in matrix, σm, at ε = 0.7%.

Matrix normal stress σm, on the other hand, relaxed very significantly, as shown in Fig. 5.14. It took only 200 hours for σm to relax to about one third of the initial value. This relaxation of σm in the broken fiber was in marked contrast to that of σf .

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Therefore, we can say that in the experiment of ε  = 0.7% the relaxation of σf in the broken fiber was little and very slow in comparison with that of σm. In other words, matrix shear stress around the fiber break relaxed much less significantly than the matrix normal stress. Stress relaxation at 1.4% constant strain Another experiment at the higher strain of 1.4% was reported here. As the embedded fiber was greatly overstrained compared to the mean 0.9% breaking elongation of the carbon fiber employed (Table 5.1), the increase of ε from 1.3 to 1.4% induced three fiber breaks in the observable section within about ±2.5 mm from the specimen center. A spontaneous increase of stress recovery length took place during the stress relaxation, as will later be described in detail. The profiles of fiber stress near one of the breaks at t = 1, 10, 50, 100 and 1000 hours are shown in Fig. 5.15. As seen from the figure, the experiment had a spontaneous increase of stress recovery length in the time t = 10 to 50 hours, which was not observed in the experiment of ε  = 0.7%. Such an increase did not take place, and the fiber axial stress profiles changed little after t = 50 hours until the experiment was terminated at t = 1000 hours. The corresponding profiles of interfacial shear stress τ, calculated using Eq. 5.2 from the data of fiber stress, are shown in Fig. 5.16. The profiles in the figure are characterized by peaks, which were not seen in the experiment of ε  = 0.7%. In each profile, τ increased from a relatively low value to a peak, then decreased to zero, with an increase of z from the break. When a polarizer was inserted into the

5.15  Profile of axial stress in broken fiber at ε = 1.4%.

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5.16  Profile of interfacial shear stress in broken fiber at ε = 1.4%.

microscope of the Raman system, the break-to-peak sections were found to be suffering from interfacial debonding, and birefringence patterns were observed on the specimen. Moreover, the peak values of t at t = 1 to 1000 hours were almost the same, around 20 MPa (Fig. 5.16), and so it is appropriate to regard such a value of τ as the shear strength of the carbon/epoxy interface in the present composite. We can thus conclude that the increase of stress recovery length mentioned above was due to the propagation of interfacial debonding in the axial direction of the fiber, and the time-dependent change in fiber stress caused by matrix shear creep was also only little in the case of 1.4% strain. Let us compare the relaxation behaviors of fiber stress σf and matrix normal stress σm. As seen from Fig. 5.16, the peak value of τ decreased incrementally with time after t = 50 hours. This means that σf had a very weak decrease with time, with respect to the maximum gradient in the stress recovery section. On the other hand, σm relaxed to about a quarter of the initial value in only 200 hours after the fiber break (Fig. 5.17). Thus the experiment of ε  = 1.4%, in which the propagation of interfacial debonding occurred, also had the feature that σm relaxed much more quickly than σf in the broken fiber and τ around the break.

5.4

Time-dependent variation in fiber stress during pull-out tests

The experiment for stress relaxation in broken fibers in Section 5.3 revealed that the temporal change in stress of broken fibers is not significant, despite the fact

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5.17  Relaxation of normal stress in matrix, σm, at ε = 1.4%.

that the matrix itself exhibits notable stress relaxation. On the other hand, change in the stress of a broken fiber with time is expected to depend on characteristics of matrix shear creep. It will be necessary to carry out further experimental studies of different material systems or under a range of environmental conditions in order to clarify the influence of matrix creep on the stress relaxation in broken fibers. Time-dependent change in fiber stress profiles similar to those in broken fibers can occur in fiber pull-out tests where the pull-out loads are kept constant. In the pull-out tests, due to the shear stress at the fiber/matrix interface, the surrounding matrix undergoes shear creep. This causes the axial distribution of fiber stress to change with time t, as depicted in Fig. 5.18. Accordingly, the stress transfer length (a length scale corresponding to the stress recovery length and represented by the distance over which σf decays from the pull-out stress σ0 to zero) increases with time. In fact, the time variations in fiber stress profiles in both tests can be described in the same manner as shown in Fig. 5.19 if the shear-lag assumption is employed (Mason et al., 1992). Moreover, constant-load pull-out tests have certain advantages over the stress relaxation tests for the composites with fiber breakage reported in 5.3, since in the pull-out tests merely dead loads need to be applied to fiber ends. In this study, three types of resins with different creep properties are employed as the matrix to elucidate the influence of creeping behavior on the temporal change in fiber stress. Time-dependent variation in stress profiles of fibers embedded in creeping matrices is examined in detail under constant-load pull-out for 500 to 1000 hours.

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5.18  Time-dependent change in fiber stress profiles in constant-load pull-out tests.

5.19  Schematic illustration of fiber stress profile: (a) fiber breakage model and (b) fiber pull-out model.

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5.4.1 Materials The carbon fibers utilized in the present experiment were the same as those used in the experiments of stress relaxation described in Section 5.3. For the matrix, three different kinds of resins were employed, namely, (I) a common roomtemperature curing bisphenol-A based epoxy resin (Epotek 301-2, Epoxy Technology, Inc.), (II) a 1:1 weight-ratio mixture of the above epoxy resin (I) and a more flexible one (Epotek 310, Epoxy Technology, Inc.) having a sub-ambient glass transition temperature, and (III) a UV-curable acrylic (Light-Weld 425, Dymax Corp.). Hereafter, these three resins are referred to as epoxy-1 (normal epoxy), epoxy-2 (flexible epoxy) and acrylic (UV-curable acrylic), respectively. Some of the mechanical properties of these resins are summarized in Table 5.2.

5.4.2 Preparation of single-fiber pull-out specimens Figure 5.20 illustrates schematically the preparation of single-fiber pull-out specimens. A single fiber was first mounted onto a plastic frame with a window 50 mm long and of an elliptical shape. Transparent cover glasses were attached to both faces of the frame to make a cavity to mold a resin. Each of the resins was drawn into a syringe in a vacuum-degassed state and introduced into the cavity, where the resin was subsequently subjected to curing. After curing, the fiber was cut at a certain distance from the resin and fixed to a tab. The frame was finally split down the middle to complete a pull-out specimen. Table 5.2  Mechanical properties of resins used for model pull-out composites

Epoxy-1

Epoxy-2

Acrylic

Young’s modulus (GPa) Tensile strength (MPa) Poisson’s ratio Elongation (%) Durometer hardness D

3.6 60 0.29 5 82

1.2 20 0.30 9 78

1.5 26 0.32 11 80

5.20  Preparation of pull-out test specimens.

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The pull-out specimens made of epoxy-1 and epoxy-2 were cured at room temperature for 7 and 30 days, respectively. The specimen made of acrylic was exposed to a UV lump for 1 hour by means of a parabolic reflector to achieve a uniform light intensity over the resin surface. In order to avoid local shrinkage and the resulting residual stress, the UV intensity was kept to a relatively low level. Durometer hardness D was measured for each resin during the cure to confirm sufficient curing. The saturated values of hardness are also listed in Table 5.2. The geometry and dimensions of the specimens, prepared as outlined above, are illustrated in Fig. 5.21. Raman scattering measurements of each embedded fiber were made through the upper cover glass and the surrounding resin. To prevent severe attenuation of the scattered light from the fiber, it was embedded at a depth of 150 µm, which, though quite small, was fairly large compared to the fiber diameter. Moreover, the portion of the fiber embedded in the matrix was no less than 10 mm, so that the system well approximated an infinite matrix containing a single fiber.

5.4.3 Long-term pull-out tests at constant loads Each of the single fiber specimens was fixed onto a loading device, which was then placed on the stage of the optical microscope in the Raman spectrometer. As shown in Fig. 5.22, the fiber was pulled out by a thin nylon line connected to the fiber end

5.21  Shape of pull-out test specimens.

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5.22  Schematic illustration of constant-load pull-out tests. Table 5.3  Fiber pull-out test conditions for three kinds of model composites

Carbon fiber/ epoxy-1

Carbon fiber/ epoxy-2

Carbon fiber/ acrylic

Pull-out load (mN) Fiber diameter (m) Fiber stress, σ0 (GPa) Pull-out duration (h)

52.4 5.2 2.5 500

52.4 5.3 2.4 1000

62.2 5.4 2.7 500

tab, to which a dead weight was hung via a roller. Pull-out tests were carried out on the specimens with different resins for 500 to 1000 hours in an atmospheric environment at 25 °C and at a relative humidity below 50%. Table 5.3 summarizes the experimental conditions for the three specimens. The fiber diameters in Table 5.3 were determined from scanning electron microscope photographs of cross-sections taken after the pull-out tests. The listed fiber stresses are those applied to the fibers in the axial direction at exposed parts outside the resins, and were about 2.5 GPa for all tests corresponding to the axial strain of 0.5%. Raman measurements were carried out, translating each specimen in the axial direction on the x–y stage in order to obtain a stress profile along the fiber. The measurements were made not only for the portion of the fiber embedded in the resin, but also for the part extending outside. They were made prior to and just after the loading, and when the elapsed time reached 1, 10, 100 and 500 hours (300 hours for some specimens).

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5.4.4 Stress relaxation tests for matrix resins In the results of the stress relaxation tests described in Section 5.3, matrix normal stress relaxation was very significant in comparison with that in broken fibers. In the pull-out tests, the time-dependent deformation behavior of the resins can also be compared with the time variations in the fiber stress. To characterize the timedependent deformation behavior of the three resins employed in the study, stress relaxation tests were carried out at an overall strain of 0.5 %. Small dog-bone specimens of the same shape as in the experiments in Section 5.3 were used in the stress relaxation tests. Durometer hardness D was used to confirm whether the specimens were sufficiently cured. The relaxation tests were made at a strain of 0.5 %, corresponding to the stress applied to the fiber in the pull-out tests. The strain was controlled by a linear actuator, based on the mean output from the two strain gages, and the axial stress was monitored continuously by a load cell, connected serially to the specimen.

5.4.5 Results Pull-out tests at constant loads Figure 5.23 shows the profiles of fiber axial stress in the three pull-out tests at t = 0 hour and 500 hours. The spatial coordinate in the axial direction is denoted by z, with the origin (0) corresponding to the point where the fiber intersects the resin-free surface, and positive values of z corresponding to the inside of the resin

5.23  Comparison of the profiles of fiber axial stress in pull-out tests at t = 0 hour and 500 hours for three matrix resins.

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(see Fig. 5.18). It can be seen from the figure that the time-dependent variation in fiber axial stress was very different with matrix resins, i.e., the variation was little in epoxy-1, whereas it was noticeable in epoxy-2, and significant in acrylic. Details of profiles of the fiber axial stress are shown in Fig. 5.24(a)–(c) for the pull-out specimens with the three resins, epoxy-1, epoxy-2, and acrylic, respectively.

5.24  Profiles of fiber axial stress in pull-out test specimens for (a) epoxy-1 matrix, (b) epoxy-2 matrix, and (c) acrylic matrix. (Continued )

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5.24  Continued.

For all stress profiles in Fig. 5.24, the axial stress σf of fibers embedded inside resins decreases to zero with increasing distance from the origin. However, timedependent variations in these stress profiles show some distinct features, depending on the particular matrix resin. In the case of epoxy-1 shown in Fig. 5.24(a), the stress profile appears to exhibit some variation until one hour after the moment of loading. Later, however, it ceases to show any notable change, and the stress recovery length retains a value of about 300 µm. For epoxy-2 in Fig. 5.24(b), the profile also changes for up to 10 hours after loading, but with little subsequent variation. In contrast, when acrylic is employed as the matrix, as shown in Fig. 5.24(c) the stress profile continuously changes as time elapses, even in a later stage between 300 and 500 hours. In particular, the stress recovery length is doubled from approximately 1000 to 2000 µ m after 500 hours of pull-out loading, which is a trend not seen in epoxy-1 or epoxy-2. It is noted here that for Fig. 5.24(a) and 5.24(b), the fiber stress outside the resin (z < 0) differs slightly from that at z ≈ 0. This is considered to be due mainly to the residual stress in fibers generated during the specimen fabrication, as delineated by the ‘+’ plots in Fig. 5.24. From each fiber stress distribution shown above, the interfacial shear stress τ is calculated using the previously described Eq. 5.1 and the smoothing procedure. The results are shown in Fig. 5.25(a)–(c), respectively. It may be observed in Figs. 5.25(a)–(c) that the time variation in profiles of τ depends on the resin type, in a manner similar to that in axial stresses σf discussed previously. Namely, the degree of temporal change in shear stress profiles is more significant in the order of acrylic, epoxy-2, and epoxy-1. Particular attention is drawn here to the

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maximum shear stress τmax at each elapsed time. This maximum value tends to decrease with time for the three specimens. Particularly in the case of an acrylic matrix, τmax continues to decrease and results in roughly half the initial value after 500 hours. Moreover, the stress transfer length extends and the shear stress distribution tends to be uniform over the stress transfer length with time.

5.25  Profiles of interfacial shear stress in pull-out test specimens for (a) epoxy-1 matrix, (b) epoxy-2 matrix, and (c) acrylic matrix. (Continued ) © Woodhead Publishing Limited, 2011

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5.25  Continued.

It is also noted here that in the results of epoxy-2 and acrylic, just after loading the shear stress profiles attain certain extrema not at z = 0 but at finite distances into the resins, as shown by the ‘’ plots in Figs 5.25(b) and 5.25(c). As further discussed in Section 5.5, this is indicative of the occurrence of interfacial debonding at initial loading. Stress relaxation tests for matrix resins The fiber stress in a pull-out test under a constant load can be described in the same manner as that in broken fibers under a constant strain, as shown in Fig. 5.19, if the shear-lag model is employed. Therefore temporal changes in fiber stress in Fig. 5.24(a)–(c) can be compared with the stress relaxation behavior of the matrix resins at constant strain of 0.5%, as in Section 5.3. Figure 5.26 shows the results of the stress relaxation tests for the three resins at 0.5% fixed strain, corresponding to pull-out fiber strain. For epoxy-1, the matrix normal stress was found to relax to a quarter of the original after 100 hours elapsed. On the other hand, the fiber stress or the interfacial shear stress changed very little, even after 500 hours elapsed, as shown in Fig. 24(a). Epoxy-2 showed relaxation such that the normal stress decreased to about one third of the initial value in 3 hours. The maximum value of the interfacial shear stress, τmax, in epoxy-2 decreased by a few percentages even after 1000 hours elapsed. Acrylic also shows significant relaxation, as the normal stress fell to one fifth of the initial value in 50 hours. In contrast, the maximum shear stress decreased to half of the

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5.26  Stress relaxation curves of matrix resins at tensile strain of 0.5%.

original value, even after 500 hours. The temporal change in the fiber stress was not so significant as the stress relaxation which the matrix resins exhibited. These behaviors were in accord with the results of the stress relaxation experiments described in Section 5.3.

5.5

Discussion

5.5.1 Influence of matrix creep on time-dependent change in fiber stress The time-dependent change in fiber stress profiles during stress relaxation tests and pull-out tests is brought about by matrix creep, driven by shear stress at the fiber/matrix interface. Therefore, the creep behavior of the matrix resins was characterized by creep tests. The influence of matrix resins creep on the timedependent change in fiber stress was discussed for the pull-out tests in Section 5.4, where three different kinds of resin were employed as matrix. Creep tests for matrix resins Figure 5.9 shows small dog-bone-shaped specimens, machined from the three kinds of resin, i.e., the epoxy-1, epoxy-2 and acrylic described in Section 5.4.1, which were used in the creep tests. Durometer hardness D was used to confirm that these specimens attained sufficient curing, as in the model composites used in

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the pull-out tests. The creep strain was measured by two strain gauges, attached on both sides of the gauge section. The creep tests were performed with dead-weight loading. The loading levels were chosen to simulate the matrix shear creep in the pull-out tests as follows: in order to correlate the creep test results for the three resins with the matrix creep behavior in the pull-out tests, the stress level in the creep test for each resin was set equal to the maximum of shear stress, τmax, observed in the corresponding pull-out test in Fig. 5.25. Kitagawa et al. (1992) showed experimentally that there was a certain equivalent relation, such as Tresca or Mises, between tension and shear for the inelastic deformation behavior of some thermoplastic resins. Their study suggests that a similar equivalence also holds between the tensile and shear creep for the resins discussed here. If a simple criterion of the Tresca type is invoked, the tensile creep tests at a stress of σ can be regarded as response under a shear stress of τ  = σ/2. Here σ was chosen to be equal to τmax, so the present tensile creep tests were considered to be equivalent to those under shear stress of τmax/2. Accordingly, the tensile creep tests at σ = τmax reflect the averaged shear creep behavior in the matrix in the neighborhood of the fiber/matrix interface with shear stress distributed from τmax to zero. As a result, the extent of the matrix creep in the proximity of the stress transfer length of the fiber can be evaluated by tensile creep tests with stress levels as described above. Influence of matrix creep The results of the creep tests for the three resins are shown in Fig. 5.27. It is seen that epoxy-1 shows no profound creep throughout the test period. Epoxy-2 exhibits a certain amount of transient creep in an early stage, which subsequently decelerates. Acrylic undergoes significant creep, which progresses in a steady manner through the entire test period. The creep test results enable a qualitative interpretation of the pull-out test results in terms of matrix creep behavior. First, in the case of epoxy-1 matrix, the shear stress at the fiber/matrix interface does not generate profound creep in the matrix, and the stress profile along the fiber varies little with time. In the case of epoxy-2, the matrix creeps somewhat noticeably in an early stage, resulting in a certain time variation in the fiber stress profile. However, this matrix subsequently exhibits creep hardening, which slows the creep strain rate and decelerates the change in fiber stress profiles. Finally, where acrylic is used as the matrix, immediate creep deformation occurs, with less creep hardening. As a consequence, the fiber stress profile continues to change throughout the entire loading process. The above reasoning explains fairly well the major time-dependent characteristics observed in the pull-out tests, except for the change in the epoxy-1 specimen during the initial one-hour period. For this time interval, according to the creep results in Fig. 5.27 the epoxy-1 matrix appears to show no significant creep. The short-term

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5.27  Tensile creep curves of three matrix resins.

change observed in this specimen is considered to be the result of an interface slip due to high-shear stressing. On the whole, the present set of experiments verifies that the time-dependent change in fiber stress profiles is highly influenced by the matrix creep characteristics.

5.5.2 Effect of interfacial debonding and slippage On the basis of these experimental observations, the effect of interfacial debonding and matrix shear yielding on the stress relaxation in fibers is discussed in comparison with linear viscoelastic analysis based on perfect bonding at the interface, for both the stress relaxation tests and the pull-out tests. Stress relaxation tests The experiments in Section 5.3 produced results which indicated that the fiber stress and interfacial shear stress around fiber breaks relaxed a little and very slowly in comparison with matrix normal stress. This result is now compared qualitatively with linear viscoelastic solutions. Lifshitz and Rotem (1970) pointed out that stress in broken fibers can relax as a result of matrix shear creep around fiber breaks in unidirectional composites. They considered a long elastic fiber, broken in a linear viscoelasitc matrix in a cylindrical cell. They obtained an approximate solution from the corresponding elastic solution, based on a shear-lag model, in which perfect bonding at the fiber/ matrix interface was assumed to be prevailing, even near the fiber break. They

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thus showed that the time-dependent extension of stress recovery length, δ (t), under constant overall strain satisfies:

δ (t) ∝ √J (t),

[5.3]

where J(t) denotes the shear creep compliance of the matrix. This relation was almost ascertained in detailed numerical analysis (Lagoudas et al., 1989). The above equation suggests the following: the time-dependent extension of stress recovery length, i.e., the stress relaxation in broken fibers, takes place as significantly as the relaxation of the matrix normal stress σm, since the greater the creep compliance J(t) is, the more significant the relaxation of σm is. The prediction mentioned above, however, does not agree with the tendency in the experimental results in Section 5.3, in which fiber stress relaxed a little and very slowly in comparison with σm. This inconsistency is attributable to the assumptions in the analysis, i.e., perfect bonding at the fiber/matrix interface and linear viscoelasticity of the matrix. According to the assumptions, the profile of interfacial shear stress τ just after a fiber break in the cylindrical cell is expressed in the following form, in which the elastic solution of Roseen is modified by taking into account the radial gradient of matrix shear stress in the cell (Clyne and Withers, 1993):

τmax = Ef ε

[

Gm 2Ef ln(re /rf )

]

1/2



,

[5.4]

where Ef and Gm denote the elastic moduli of the fiber tension and matrix shear, respectively, re/rf indicates the ratio of fiber radius to cell radius. It is appropriate to take re/rf ≈ 4 for single-fiber composites, such as in the present work (Li and Grubb, 1994). Then, with the material constants in Table 5.1 and Gm ≈ 1.5 GPa estimated by Young’s modulus and Poisson’s ratio of the matrix, Eq. 5.4 gives τmax in the experiment of ε  = 0.7 % as:

τmax ≈ 110 MPa.

[5.5]

We notice that the value above is about five times larger than the experimental one obtained just after the break (Fig. 5.13). The linear viscoelastic solution, based on the assumption of perfect bonding at the fiber/matrix interface, did not quantitatively reflect the experimental results. This wide difference can be attributed to matrix shear yielding and interfacial debonding. Pull-out tests Although Lifshitz and Rotem’s (1970) approximate solution is for the time-dependent change in fiber stress profiles for a broken fiber in a linearly viscoelastic matrix, it is applicable to the fiber stress in pull-out tests under constant load in the same manner. When their solution is applied to the case of a constant-load pull-out test, it implies that the stress transfer length grows on a scale of the square root of J(t), the same as described in Eq. 5.3. According to this finding, and with reference to © Woodhead Publishing Limited, 2011



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the creep results in Fig. 5.27, the time variation in fiber stress profiles should be much greater than that observed experimentally, as shown in Fig. 5.24. For example, in Fig. 5.27, after a 100-hour creep, epoxy-2 exhibits 13 times the strain as in the instantaneous value. Thus according to the theory, the stress transfer length in the corresponding pull-out test should be about 3.6 times as long after the same period, which is far beyond the actual observation. A major source of the above discrepancy lies in the assumption of interface bonding. According to the elastic shear-lag model, which assumes perfect bonding (Li and Grubb, 1994), the maximum interfacial shear stress in the pull-out tests is given in the same form as Eq. 5.4,

[

τmax = σ0

Gm 2Ef ln(R/rf)

]

1/2



.

[5.6]

When R/rf is taken to be 4, as in the above case of stress relaxation, substitution of the material constants of Table 5.2 into Eq. 5.6 yields τmax for the three specimens, as shown in Table 5.4, in comparison to those measured experimentally at the moment of loading. Table 5.4 shows that the theoretical values produced by Eq. 5.6 are substantially higher than those found experimentally, which is considered to be due to instantaneous interfacial slipping or shear yielding in the matrix. The experimental profiles of interfacial shear stress τ, just after the moment of loading, for epoxy-2 and acrylic, exhibit certain extrema as already described. As a result of Raman spectroscopy and polarization measurements (Melanitis et al., 1993; Gu et al., 1995), shear stress profile with such extremum proved to be due to interfacial debonding, as shown in Fig. 5.28. Furthermore, there is also the likelihood of an interfacial slip in the epoxy-1 specimen, accounting for the interfacial shear stress for slipping in similar epoxy resins, which was found to be about 20 MPa in the study described in Section 5.3. Since the occurrence of interface slippage suppresses the growth of interface shear stress, the creep rate of the matrix is reduced. Thus, the time variation in fiber stress profiles takes place much slower than that predicted by the approximate solution of Lifshitz and Rotem (1970). On the contrary, a theoretical examination, taking into account possible interface slippage, has been made and the results obtained for the stress profile variation in pull-out in the present acrylic matrix specimen showed good agreement with the experimental results, as shown in Fig. 5.29 (Ohno et al., 2002). Table 5.4  Comparison of theoretical (based on perfect interfacial bonding) and experimental results on maximum interfacial shear stress, τmax, immediately after loading

Carbon fiber/ epoxy-1

Carbon fiber/ epoxy-2

Carbon fiber/ acrylic

Theoretical (MPa) Experimental (MPa)

80 15

44 2.4

55 4.5

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5.28  A schematic representation of the interfacial shear stress (ISS) distribution at high values of applied strain with debonding (Melantis et al., 1993, adapted from J. Mater. Sci. 28 (1993), p.1654 (Chapman & Hall)).

5.29  Comparison of experimental and analytical solution with and without interfacial debonding (Ohno et al., 2002, adapted from Int. J. Solids Struct. 39 (2002), p.171 (Elsevier)).

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After all, a theory based on perfect interface bonding tends to overestimate the time-dependent change in the stress transfer length in the fiber of unidirectional composites.

5.6

Summary

Raman spectroscopy was used to study the temporal stress change in fibers in unidirectional model composites, which plays an important role in the creep rupture of such unidirectional composites. Single-fiber model composites, consisting of a carbon fiber monofilament and transparent resins, were employed as specimens. (1) stress relaxation tests of a broken fiber and (2) constant-load pull-out tests using three different kinds of matrix resin were performed. They were carried out for 500 to 1000 hours, during which time fiber stress profiles were monitored by Raman spectroscopy. From these long-term Raman spectroscopic measurements, the factual time-dependent changes in stress of broken fibers were first revealed as follows. The main results obtained in (1) stress relaxation tests of a broken fiber were: 1. In the experiment of 0.7 % overall strain, the stress profile in the broken fiber changed only a little, even after 1000 hours. In the experiment of 1.4% overall strain, spontaneous increase of stress recovery length occurred as a result of the propagation of interfacial debonding, but the broken fiber demonstrated only a little change of stress distribution after the increase. 2. Matrix normal stress relaxed to about one third or quarter of the initial value within 200 hours. Therefore, the above-mentioned stress relaxation in the broken fibers was much less significant in comparison with that in matrix normal stress. The main results obtained in (2) constant-load pull-out tests were: 3. The change in the temporal stress in the fiber during pull-out tests using three different matrix resins varied depending on the matrix resins. That is, the fiber stress in normal epoxy resin matrix exhibited only a negligible change with time, whereas the flexible and UV-curable acrylic matrices allowed, respectively, considerable and significant changes in fiber stress. Then, creep tests for the matrix resins themselves were performed and the results were correlated with the matrix creep behavior in the pull-out tests. The main result obtained was: 4. The temporal changes in fiber stress described above are in fair accord with the creep characteristics of matrix resins, which implies that the time-dependent evolution of the stress transfer length can be well correlated with the shear creep behavior of the matrix. Finally, the effects of matrix yielding and interfacial debonding on the temporal changes in fiber stress were discussed by comparing them with linear viscoelastic solution based on perfect bonding at the interface. The main result was:

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5. A certain indication of the occurrence of interfacial debonding was found in some experiments. The experimental interfacial shear stress was much lower than an elastic prediction based on perfect bonding at interface. Actual timedependent change in the stress transfer length in the fiber of unidirectional composites was not as significant as the prediction by a linear viscoelastic theory based on perfect interface bonding.

5.7

References

Anastassakis E, Pinczuk A, Burstein E, Pollak F H and Crodona M (1970), ‘Effect of static uniaxial stress on the Raman spectrum of silicon’, Solid State Commun, 8, 133–138. Andrews M C and Young R J (1995), ‘Fragmentation of aramid fibers in single-fiber model composites’, J Mater Sci, 30, 5607–5616. Batchelder D N and Bloor D (1979), ‘Strain dependence of the vibrational modes of a diacetylene crystal’, J Polymer Sci: Polym Phys Ed, 17, 569–581. Chohan V and Galiotis C (1997), ‘Effects of interface, volume fraction and geometry on stress redistribution in polymer composites under tension’, Compos Sci Technol, 57, 1089–1101. Clyne T W and Withers P J (1993), An introduction to metal matrix composites, London, Cambridge University Press. Du Z-Z and McMeeking R M (1995), ‘Creep models for metal matrix composites with long brittle fibers’, J Mech Phys Solids, 43, 701–726. Fan C F, Waldman D A and Hsu S L (1991), ‘Interfacial effects on stress distribution in model composites’, J Polym Sci: Polym Phys, 29, 235–246. Galiotis C, Young R J, Yeung P H J and Bachelder D N (1984), ‘The study of model polydiacetylenr/epoxy composites Part I: The axial strain in the fiber’, J Mater Sci, 19, 3640–3648. Galiotis C (1991), ‘Interfacial studies on model composites by laser Raman spectroscopy’, Compos Sci Technol, 42, 125–150. Grubb D T, Li Z-L and Phoenix S L (1995), ‘Measurement of stress concentration in a fiber adjacent to a fiber break in a model composite’, Compos Sci Technol, 54, 237–249. Gu X H, Young R J and Day R J (1995), ‘Deformation micromechanics in model carbon fiber-reinforced composites Part I: The single-fiber pull-out test’, J Mater Sci, 30, 1409–1419. Huang Y H and Young R J (1994), ‘Analysis of the fragmentation studies test for carbonfiber/epoxy model composites by means of Raman spectroscopy’, Compos Sci Technol, 52, 505–517. Iyengar N and Curtin W A (1997), ‘Time-dependent failure in fiber-reinforced composites by matrix and interface shear creep’, Acta Mater, 45, 3419–3429. Kelly K W and Barbero E (1993), ‘The effect of fiber damage on the longitudinal creep of a CFMMC’, Int J Solids Struct, 30, 3417–3429. Kitagawa M, Onoda T and Mizutani K (1992), ‘Stress-strain behaviour at finite strains for various strain paths in polyethylene’, J Mater Sci, 27, 13–23. Lagoudas D C, Hui C-Y and Phoenix S L (1989), ‘Time evolution of overstress profiles near broken fibers in a composite with a viscoelastic matrix’, Int J Solids Struct, 25, 45–66. Li Z-F and Grubb D T (1994), ‘Single-fiber polymer composites Part I: Interfacial shear strength and stress distribution in the pull-out test’, J Mater Sci, 29, 189–202.

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Lifshitz J M and Rotem A (1970), ‘Time-dependent longitudinal strength of unidirectional fibrous composites’, Fiber Sci Technol, 3, 1–20. Mason D D, Hui C-Y and Phoenix S L (1992), ‘Stress profiles around a fiber break in a composite with nonlinear, power law creeping matrix’, Int J Solids Struct, 29, 2829–2854. Melantis N, Galiotis C, Tetlow P L and Davies C K L (1993), ‘Monitoring the micromechanics of reinforcement in carbon fiber/epoxy resin systems’, J Mater Sci, 28, 1648–1654. Miyake T, Yamakawa T and Ohno N (1998), ‘Measurement of stress relaxation in broken fibers embedded in epoxy using Raman spectroscopy’, J Mater Sci, 33, 5177–5183. Miyake T, Kokawa S, Ohno N and Biwa S (2001), ‘Evaluation of time-dependent change in fiber stress profiles during long-term pull-out tests at constant loads using Raman spectroscopy’, J Mater Sci, 36, 5169–5175. Miyake (2003), ‘Study on creep mechanism of unidirectional composites based on experimental investigation using micro-Raman spectroscopy’ (in Japanese), PhD Thesis, Nagoya, Nagoya University. Ohno N, Toyoda K, Okamoto N, Miyake T and Nishide S (1994), ‘Creep behavior of a unidirectional SCS-6/Ti-15-3 metal matrix composite at 450°C’, ASME J Engng Mater Tech, 116, 208–214. Ohno N, Fujita T, Miyake T, Nakatani H and Imuta M (1996), ‘Creep rupture of a unidirectional SCS-6/BETA21S metal matrix composite at 450, 500 and 550 °C’, Mater Sci Res Int, 2, 199–205. Ohno N (1998), ‘Recent progress in creep rupture analysis of unidirectional composites reinforced with long brittle fibers’, JSME Int J Series A, 41, 167–177. Ohno N and Miyake T (1999), ‘Stress relaxation in broken fibers in unidirectional composite: modeling and application to creep rupture analysis’, Int J Plasticity, 15, 167–189. Ohno N, Ando T, Miyake T and Biwa S (2002), ‘A variational method for unidirectional fiber-reinforced composites with matrix creep’, Int J Solids Struct, 39, 159–174. Otani H, Phoenix S L and Petrina P (1991), ‘Matrix effects on lifetime statistics for carbon fiber-epoxy microcomposites in creep rupture’, J Mater Sci, 26, 1955–1970. Phoenix S L, Schwartz P and Robinson IV H H (1988), ‘Statistics for the strength and lifetime in creep-rupture of model carbon/epoxy composites’, Compos Sci Technol, 32, 81–120. Wagner H D, Amer M S and Schadler L S (1996), ‘Fiber intractions in two-dimensional composites by micro-Raman spectroscopy’, J Mater Sci, 31, 1165–1173. Weber C H, Yang J Y, Löfvander P A, Levi C G and Evans A G (1993), ‘The creep and fracture resistance of γ-TiAl reinforced with Al2O3 fibers’, Acta Metall Mater, 41, 2681–2690. Weber C H, Du Z-Z and Zok F W (1996), ‘High temperature deformation and fracture of a fiber reinforced titanium matrix composite’, Acta Mater, 44, 683–695.

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6 Predicting the viscoelastic behavior of polymer nanocomposites A. Beyle and C. C. Ibeh , Pittsburg State University, USA Abstract: Prediction of elastic moduli and some physical properties of heterogeneous media on the basis of known characteristics of constituents, their volumetric concentrations, shapes and orientation of inclusions and the type of lattice formed by the set of inclusions, is a developed area of the theory. This chapter is devoted to expansion of the theory to viscoelastic systems mostly made from viscoelastic polymeric matrices and elastic fillers. The method is based on elasto-viscoelastic analogy, in which the elastic moduli of constituents participating in the formulae for the effective moduli are replaced by corresponding integral operators of the theory of viscoelasticity. The next step is the decoding of the functions of operators, i.e. in replacing the expressions containing several operators by one equivalent operator with effective parameters. The difference between application of the theory to composites and nanocomposites is discussed. The prediction of elastic and viscoelastic behavior of composites and nanocomposites of geometrically similar structures is proposed. Prediction of the failure for composites and nanocomposites has to be done differently. Prediction of viscoelastic behavior is described for composites made from polymeric matrices and spherical, short cylindrical inclusions, platelets, hollow spherical and cylindrical inclusions, and spherical voids. Influence of stiffness ratio of matrix and inclusion on the effective properties is analyzed. Key words: composites, nanocomposites, effective properties, viscoelasticity, creep, loss modulus, storage modulus, integral operators of viscoelasticity, syntactic foams, rigid fillers.

6.1

Specific features of nanoparticles and nanocomposites

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 calculations 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 nano-structured systems due to the high surface-to-volume ratio of nano-inclusions. 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 184 © Woodhead Publishing Limited, 2011

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viscoelastic properties is less developed despite the principle of elastic-viscoelastic analogy. In this chapter, it is planned to demonstrate the results of theoretical prediction of viscoelastic behavior and to illustrate them by our experimental data as well as via published literature data. Some nanoparticles have irregular shape. However many types of nanoparticles have quasi-classical shapes, and are spherical, cylindrical-long fibers or cylindricalplane disks. Such nanoparticles as nanoclay and nanographene are platelets; carbon nanofibers have cylindrical shape; silicon carbide nanoparticles can be considered as quasi-spherical ones. Carbon nanotubes are considered as hollow cylinders whereas buckyballs 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 ratio of the total surface of matrix-inclusions boundaries to the total volume of nanocomposite. However, the improvement of properties in comparison to the matrix properties is not very 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 from Griffith’s classical works of 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 binding matrix (initially polymeric, later metallic, ceramic, cement, etc.), i.e. the idea of composites, was taken from the architecture occurring in natural materials: wood, bones, etc. are the natural composites. Progress in technology resulted in the ability to make nano-sized fillers, the individual strength of which approaches to the theoretical value. However, the expectations that the properties of nanocomposites would be much higher than the properties of conventional composites are not realized. Some progress has been achieved in dynamic applications and in other areas but it is not very pronounced. Information about nanocomposites’ properties, technology, and applications can be found in multiple sources, for example in references 1 to 3. In this situation the critical review of the existing methods for prediction of the mechanical behavior of heterogeneous materials could be useful. This chapter is devoted to applicability of the methods of prediction of viscoelastic behavior of heterogeneous materials for the particular case of nanocomposites. All mechanical properties of heterogeneous systems depend on properties of constituents, on their concentrations, on shape of fillers, on fillers’ orientation, on type of spatial lattice, etc. Existing theories used for calculations of the effective elastic properties are non-sensitive 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

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in nano-structured systems due to their high surface-to-volume ratio. Really, volume of the individual inclusion, VI is related to the total volume V of composite as

[6.1]

where c is the volume concentration of inclusions and NI 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:

[6.2]

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

[6.3]

The values of the form factor for different types of inclusions are presented in Table 6.1. According to Eq. 6.2 both form factor and especially the decreasing volume of the individual inclusion play significant roles in elevated surface effects in nanocomposites. The simplest model, which takes into account surface effects, includes three phases: inclusions, matrix, and thin layer of modified matrix separated matrix and inclusions (see, for example, reference 4). In the case of thermoplastic matrix this layer is formed due to lower mobility of the macro­ molecular segments near the solid surface; thickness of the layer is a few segments. Table 6.1  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 = 10 L / R = 100 L / R = 1000 L / R → ∞ D / h = 10 D / h = 100 D / h = 1000 D / h → ∞ L / h = 10 L / h = 100 L / h = 1000 L / h → ∞

4.836 6 6.942 13.732 29.321

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∞

10.28 40.55 184.9

∞

11.14 43.95 200.4

∞

Predicting the viscoelastic behavior of polymer nanocomposites

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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 nucleators of physical or morphological transitions, the thickness of the modified layer of matrix can be comparable with the size of inclusions. Stepwise change of the matrix properties is an idealized model replacing continuous monotonic change of the matrix properties from the surface of inclusion to bulk properties. Unfortunately there is not enough data on the effective thickness of modified matrix layers as well as on properties of such modified layers. The classical models of composites taking into account properties of matrix and inclusions, shapes of inclusions, their orientations, type of spatial lattice, etc., still describe the main effects in the case of elastic properties of nanocomposites. Differences between experimental data and theoretical ones are mostly related to the effect of aggregation of nanoparticles, to the shape deviation from the ideal one mostly as waviness, to the nonuniformity of the nanoparticles’ distribution over the volume of nanocomposites, to imperfections in orientation, lattice, etc. The aggregation of nanoparticles is the biggest obstacle in nanocomposite technology. During transportation and storage nanoparticles can accumulate electrical 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 inversed proportional to the cube of their sizes ratio:

[6.4]

Because conventional fillers have sizes mostly in the range 1 µm to 1 mm but nanoparticles mostly in the range of 10 to 100 nm, then the number of nanoparticles replacing the same mass of the conventional fillers has to be from 103 to 1015 bigger. Taking into account that the distances between neighbor 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 technologically mixing them with matrix in liquid form (before solidification of nanocomposite) is very high. The presence of aggregates is worse than the presence of voids. They are not only stress concentrators for the matrix; they can carry very low tensile and shear stresses and practically in this aspect are no better than voids; but the main negative effect is that they are working as levers opening cracks in the matrix. Sonication technology allows the destruction of the aggregates but it also changes the polymeric matrix.5

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6.2

Viscoelasticity of polymer matrix

Viscoelasticity can be described with the help of creep curve, relaxation curve, spectrum of retardance times, spectrum of relaxation times, and dependencies of complex modulus or compliance on frequency. Each of the above-mentioned characteristics can be calculated if one of them is known (in the case of known time spectra it is necessary to know also instantaneous and long time moduli). Usually behavior of polymeric matrix is described with the help of the generalized Maxwell (Fig. 6.1a) or generalized Kelvin-Voigt (Fig. 6.1b) model. These models are equivalent, i.e. the set of characteristics (elastic moduli Ei and viscosities ηi) of one model can be recalculated from the whole set of characteristics of another model. It is necessary to mention that Ek in one model is not the same as Ek in another model; similar can be noted with respect to ηk! In addition to the strains related to the stresses, the strains related to the thermal expansion, to the swelling due to humidity change, and to the chemical or physical shrinkage due to chemical reactions or physical transitions have to be added, as

6.1  Generalized Maxwell (a) and Kelvin-Voigt (b) models of mechanical behavior of polymeric matrix. For ideally solidified polymer the viscosity η0 → ∞ (i.e. this damping element can be removed from the model).

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well as in some cases strains related to the influence of electrical and magnetic fields which must also be taken into account. If free strains (not related to the stresses) are not satisfied in the equations of strains compatibility, the stresses appear to compensate this mismatch and the whole system of equations of mechanics of the particular continuum must be solved for the determination of stresses and strains. All parameters of the models described above are dependent on temperature, on depth of chemical reaction, on humidity, etc. Especially sensitive is the viscosity element η0. In a technological process of mixing matrix with fillers, the matrix material is in the liquid form and η0 is finite. After the depth of the chemical reaction of solidification (initiated by adding some curing agent or by rising the temperature) is rising, the viscosity η0 is rising exponentially and in some critical value of the depth of chemical reaction, the gel point is achieved and the viscosity η0 becomes infinitely high. The reaction is continued after the gel point but it is connected with further changes of the properties of other elements participating in the models shown in Fig. 6.1. In general, all viscous elements in the models shown in Fig. 6.1 are more sensitive to the temperature change and to the depth of chemical reaction than the elastic elements. Change of viscoelastic properties with temperature is often studied with the help of Dynamic Mechanical Thermal Analysis (DMTA). Polymeric matrices are the most technologically convenient ones but their own mechanical characteristics are not impressive: the Young’s modulus of the majority of polymeric matrices at room temperature is in the range 0.1–10 GPa (for comparison: steel has Young’s modulus 210 GPa; the best carbon fibers, carbon nanofibers, and carbon nanotubes have Young’s modulus 1000 GPa). The tensile strength of polymeric matrices is in the range 1–200 MPa (for comparison: good-quality steel has tensile strength over 1000 MPa, the strength of the best carbon fibers is approaching 6000 MPa, the strength of carbon nanofibers is in the range 10 000–50 000 MPa but the theoretical strength of carbon materials is about 150 GPa). Polymeric matrices are characterized by pronounced viscoelastic behavior. The constitutive law for viscoelastic bodies can be written in the most general form (see Rabotnov6 for example) as:

[6.5]

Here εij is strain tensor, is tensor of free strains except thermal expansion, αij is tensor of coefficients of linear thermal expansion, is tensor-functional, and σkl is stress tensor. Functional connects the current value of some function f(t), not to the current value of the function g(t) as it is used in parametric form of functions, but connects its value to the whole history of the function g(τ):

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[6.6]

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In the theory of viscoelasticity the functional is written usually in the form of the Volterra integral operator:

[6.7]

where the kernel Ψ(t – τ) describes the fading memory about previous actions on the material. It has a dimension of the inversed time. In the particular case of the kernel:

[6.8]

the Volterra integral operator describes the mechanical behavior of the models shown in Fig. 6.1. Expression 6.7 can be rewritten in the alternative symbolic form:

[6.9]

where Π × is an operator. Expression 6.7 can be also rewritten as:

[6.10]

where Λ is a dimensionless function. To find the expression for σ (t) from Eq. 6.7, it is necessary to solve a Volterra integral equation of the second kind. The solution has the following general form:

[6.11]

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:

[6.12]

In the more general case:

[6.13]

where operators Π × (λ) and ϒ  × (λ) are mutually resolvent operators depending on the parameter λ because the kernels are dependent on λ now:

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[6.14]

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The rules of Rabotnov's algebra of resolvent operators6–8 are as follows:

[6.15]

[6.16]



[6.17]

Remembering that the product of Volterra operators:

[6.18]

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

[6.19]

It means that the square of the resolvent operators in the left side of Eq. 6.17 containing iterated kernel (Eq. 6.19) can be replaced by an operator containing a single kernel as shown on the right side of Eq. 6.17. Resolvent operators represent the majority of operators used in the theory of viscoelasticity. There are some additional Rabotnov’s algebra rules following from those above:

[6.20]

[6.21] In the particular case:

[6.22]

[6.23]





[6.24]

[6.25]

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The more complicated case of mutual resolvents of the 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 above formulae for the case of a threeelement model (Fig. 6.1b). The version of the Kelvin-Voigt model containing only elements E1, E2,η2 can be described with a particular form of Eq. 6.7: [6.26] Using Eq. 6.15 (from right side to the left side) and taking into account that in this case:

it is possible to find easily the inverse form:

[6.27]

derivation of which is much longer if traditional methods are used. Let’s denote module-operators as: In shear:

[6.28]



[6.29]

In tension-compression:

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

[6.30]

In tension-compression:

[6.31]

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 composite or nanocomposite with 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

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effective moduli of composite. If elastic moduli are replaced in the formulae by long-time viscoelastic moduli, the calculations results give the long-time effective moduli of composite. The remaining problem is to find the way for predicting transient properties. Volterra proved that the solution of the viscoelastic problem can be obtained from the solution of the corresponding elastic problem by replacing elastic moduli by corresponding viscoelastic operators. The complicated expressions containing several operators can nowadays be numerically estimated using appropriate software. However, for analytical approach and for simplification of the numerical calculations it is reasonable to use Rabotnov’s algebra of resolvent operators6 to convert some expressions with operators into one operator with shifted parameters, or at least to decrease a number of operators participating in the expressions for the effective characteristics. Some examples of derivation can be found in the literature.9–13 Another way to come to viscoelastic characteristics’ dependencies on volumetric concentrations of fillers is to replace elastic characteristics such as moduli by complex viscoelastic characteristics14, 15 and to convert the corresponding algebraic expression containing many complex numbers to the standard form of complex number using trivial operations such as:

[6.32]

The attempt to replace elastic moduli by complex viscoelastic moduli done in the work16 does not look very attractive because the hypothesis about constant Poisson’s ratio in viscoelastic process was used, which is a very rough one. Choice of the particular form of operator of viscoelasticity is subjective: it depends on the experience of the researcher, on the precision of experimental determination of characteristics and the required precision of the prediction of viscoelastic behavior, etc. The most popular types of operators are: (a) Operator (Eq. 6.8) containing sum of exponents (b) Abel operator of fractional derivative with kernel: [6.33]



 here Γ (x) is gamma function, which is a generalization of the factorial on w non-integer values of argument:

[6.34]

(c) Rabotnov’s operator of fractional exponent with kernel:

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[6.35]

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(d) Rzhanitsin’s operator with kernel:

[6.36]

(e) Koltunov’s operator with kernel:

[6.37]

(f ) Sum of Rabotnov’s operators. The most popular operators are sum of exponents and Rabotnov’s operator. Sum of exponents may contain different numbers of viscoelastic characteristics, which are necessary to determine experimentally (three, five, seven, etc. depending on the number of elements in the model in Fig. 6.1). A bigger time range is asking to take into account more and more terms. Precision and statistical determination of the viscoelastic characteristics drop down with the rise of the number of exponents. Rabotnov’s operator contains four constants only and successfully approximates experimental data in a wide time range but its application requires more complicated calculations. Before application of elastic-viscoelastic analogy for calculation of the effective viscoelasticity of nanocomposites, let’s demonstrate it for the calculation of the viscoelastic Poisson’s ratio of polymer matrix. For an isotropic body the Poisson’s ratio can be calculated via any pair from three elastic moduli: Young’s modulus E, shear modulus G, and bulk modulus K according to the formulae:



[6.38]

Let us use, for example, the third formula. If the compliances operators in tension and in shear are: [6.39] Then using Eq. 6.21 it can be found that:

[6.40]

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If complex moduli on tension and shear are known, the complex Poisson’s ratio can be calculated:

[6.41]

For isotropic polymers, the commonly used simplifying hypothesis is that the volume change is happening as a result of pure elastic deformation, but the shape change is a result of viscoelastic deformation. This hypothesis is valid only if no orientation occurs during deformation and the polymer remains isotropic after it. For thermosets this hypothesis works and it is validated by multiple experiments, but for thermoplastics this hypothesis is too rough because of the pronounced orientation effects. If this hypothesis is used, then the first two formulae (6.38) are more convenient than the third one. It is enough to know viscoelastic properties on tension or on shear only and two elastic characteristics in the case of using the hypothesis of pure elastic bulk strain. Operators describing shear and tensile/ compression properties are interrelated in this case:

[6.42]

The operator form is the following: In the case of known viscoelastic properties on shear:



[6.43]

In the case of known viscoelastic properties on tension/compression:



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[6.44]

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The complex form is the following: In the case of known viscoelastic properties on shear:



[6.45]

In the case of known viscoelastic properties on tension/compression:

[6.46]

Experimental data on creep of typical polymeric matrix (epoxy) obtained at different levels of stresses show that up to 70–80% of static strength, the viscoelastic behavior of polymeric matrix is linear, i.e. strains are increasing with stresses linearly at any time but at higher stresses strain dependence on stress becomes nonlinear (Fig. 6.2). Viscoelasticity of new polymeric fibers (polyaramid, polyparaphenylenebenzobisoxazole (PBO), ultra high molecular weight polyethylene (UHMWPE), etc.) is more nonlinear with stresses. These fibers are characterized also by very high anisotropy. Different versions of viscoelastic operators describe transient process of creep differently but the short-time behavior and long-time behavior are described similarly. The main characteristic of short-time behavior is instantaneous modulus (such as E0) and long-time behavior is described with help of protracted modulus (such as E∞). These concepts are illustrated by Fig. 6.3. Creep curves allow finding the main parameters of viscoelasticity (see Fig. 6.3). During creep, the deformation in the perpendicular direction is also increasing; moreover, the Poisson’s ratio is increasing with time (Fig. 6.4). Such increase (about 10%) means that the shear modulus during the creep is changed by about 50–60% if change of the volume is purely elastic. Ignoring change of the Poisson’s ratio means in this case that the shear creep is ignored also. Or it means that the creep on tension has happened due to the volumetric creep only, which is absurd. It is the reason why the hypothesis about constant Poisson’s ratio used in some simplified methods of the solution of viscoelasticity problems (see for example16) is a very rough one. Using data from Fig. 6.2 and Eq. 6.44 it is possible to predict the change of the Poisson’s ratio with time during the creep. The results of the calculations are shown in Fig. 6.5. They are close to experimental data shown in Fig. 6.4. Using

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6.2  Creep curves of malein-epoxy matrix at different levels of stresses (shown in MPa under each curve).11

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

6.4  Change of the Poisson's ratio with time during the creep of the malein-epoxy matrix.11

the same data and Eq. 6.46 the complex Poisson’s ratio dependencies on the frequency were calculated and plotted in Fig. 6.6 and 6.7. Systematic study of Poisson’s effect in viscoelasticity was undertaken by Lakes and colleagues (see, for example, references 17 and 18).

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6.5  Calculated change of the Poisson's ratio with time during the creep on tension. E0 / E∞ = 1.5. Retardance time in the tensile creep τEcr = 200 hours.

6.6  Dependence of the real part of the complex Poisson's ratio on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here E0 / E∞ = 1.5.

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6.7  Dependence of the imaginary part of the complex Poisson's ratio on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here E0 / E∞ = 1.5.

6.3

Viscoelasticity of polymers filled by quasi-spherical nanoparticles

Many nanoparticles made from substances such as oxides and carbides of different chemical elements have irregular shapes, which could be statistically equated to the spherical shape. If conventional fillers content can reach several tens of percents of the whole volume of composites, the maximal volume content of nanofillers is usually about several percents. The solution of the problem of calculation of effective elastic properties of a matrix filled by small volume fraction of spherical inclusions is well known (see, for example Christensen’s book14):



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[6.47]

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where K is bulk/hydrostatic modulus, G is shear modulus; indices M and I denote matrix and inclusion, correspondingly; c is volume concentration of inclusions. Effective properties are denoted by putting the corresponding property in angular brackets. Poisson’s ratio ν can be used in this formula instead of shear modulus taking into account the well-known formula:

[6.48]

and:



[6.49]

Formula 6.47 is valid in a wide range of volume concentrations. In the case when inclusions are much stiffer than matrix under hydrostatic pressure, Expression 6.47 degenerates into:



[6.50]

Examples of the calculation of the effective bulk modulus for the epoxy-solid glass microspheres and epoxy-silicon carbide nanofillers according to Eq. 6.44 as well as for the degenerated case (Eq. 6.50) are shown in Fig. 6.8. The majority of modern fillers can be considered with respect to polymeric matrices as practically approaching absolutely rigid inclusions (Fig. 6.8a). It is possible to note that the Poisson’s ratio of matrix material provides a bigger effect than the relative stiffness of inclusions (Fig. 6.8b). Replacing elastic moduli by viscoelastic operators in Eq. 6.47 and using Rabotnov algebra of resolvent operators, it is possible to derive the final formula. Because it is too bulky, here the simplified formula is shown, which is derived using two assumptions: the volumetric strain of the matrix is ideally elastic and the filler is much stiffer than matrix:



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[6.51]

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6.8  Relative increase of the effective bulk modulus of the material filled by spherical inclusions as a function of the volume concentration c of inclusions: (a) composite made from polymeric matrix (epoxy) (νM = 0.382) filled by glass solid microspheres (solid line), by magnesium oxide or low grade silicon carbide or carbon micro or nanospheres (dashed line) or by absolutely rigid micro or nanospheres (dotted line); (b) composites made from absolutely rigid spherical inclusions in the matrices having different Poisson's ratio. © Woodhead Publishing Limited, 2011

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Here νM0 is the instantaneous Poisson’s ratio of matrix, ϒ ×EM is the viscoelastic operator describing behavior of matrix under unidirectional tension/compression, and λEM and µEM are the parameters participating in this operator (see Eq. 6.29 and 6.31). The results of calculations of the volume creep are presented in Fig. 6.9. Creep of materials containing small volume fractions of inclusions is shown in Fig. 6.9a, and containing big volume fractions of inclusions in Fig. 6.9c. Filling materials with solid particles increases their stiffness and as a result the instantaneous strains are decreasing with fillers concentrations. However, the creep is increasing more clearly if creep curves are made dimensionless by dividing full strain by instantaneous strain at the same level of filling (Fig. 6.9b, 6.9d). The effect is small but nevertheless it is remarkable: with increase of concentration of rigid particles the bulk creep is increasing! It is explained by the increase of the total part of the zones in the matrix that is subjected to the stress concentration and because the shear stresses are pronounced. In the framework of the used hypotheses the creep is provided by shear only; these zones contribute more and more to the total volume creep. At small concentration of the fillers the effect is practically proportional to the filler concentrations (Fig. 6.9a, 6.9b) but at higher concentrations the dependence of the effect on concentration becomes nonlinear (Fig. 6.9c, 6.9d). Complex effective bulk modulus can be calculated as:



[6.52]

where:

[6.53]

and E"M, E"M, and EM0 are storage, loss, and instantaneous moduli of matrix on tension/compression, respectively; νM0 is instantaneous Poisson’s ratio of matrix. Numerical examples of calculations are shown in Fig. 6.10 and 6.11. Data in Fig. 6.10 support the conclusion made from the analysis of Fig. 6.9. At high frequency the behavior of viscoelastic material always corresponds to instantaneous reaction on load in creep but behavior at low frequency is similar to long-time creep behavior.

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6.9  Effective bulk creep function 〈JK〉 (volume creep curve divided by the hydrostatic stress) of material containing different volume percentages of absolutely rigid spherical particles; τEcr is the strain retardance time in the creep at unidirectional tension/compression; here EM ∞ / EM 0 = 0.6; 〈K0〉 is the instantaneous bulk modulus of matrix.

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6.9  Continued.

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6.10  Dependencies of the bulk storage modulus 〈K’〉 (real part of the complex bulk 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 ∞ / EM 0 = 0.6 and KM is the instantaneous bulk modulus of matrix.

6.11  Dependencies of the bulk loss modulus 〈K”〉 (imaginary part of the complex bulk 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 ∞ / EM 0 = 0.6 and KM is the instantaneous bulk modulus of matrix.

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Effective shear modulus for the case of small volume concentration of inclusions can be found using Eq. 6.14:



[6.54]

For the case of significantly stiffer inclusions the formula (Eq. 6.54) degenerates into:

[6.55]

Examples of calculations using Eq. 6.54 and 6.55 are shown in Fig. 6.12. In Christensen’s book,14 the method of calculation of the effective shear modulus at medium and high concentration of inclusions is shown. It is based on a three-phase concentric spherical cell model. Unfortunately, the final formulae are bulky. Here we use another approach based on Vanin’s book,9 where he

6.12  Relative increase of the shear modulus of composite made from polymeric matrix (epoxy) filled by glass microspheres (GI /GM = 25), magnesium oxide or low grade silicon carbide or carbon micro or nanospheres (GI /GM = 50) or by absolutely rigid micro or nanospheres (GI /GM → ∞) as function of the volume concentration c of inclusions.  according to Eq. 6.14 and 6.54–6.55 derived for small concentrations;  according to Eq. 6.9 and 6.65–6.66 derived for medium concentrations; in the last case it is necessary to use also Eq. 6.68 and 6.47–6.50.

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derived a more compact formula for the effective Young’s modulus of concentrated solid suspension. From Vanin’s formula and known effective bulk modulus (Eq. 6.47), the effective shear modulus can be found easily. The viscoelastic operator can be found on the basis of Eq. 6.54, 6.44, and 6.23:

[6.56]

By a similar method it is possible to derive the formula for the inversed operator of shear compliance:



[6.57]

 Operator 6.57 gives shear compliance function and creep curves, which are dependent on filler concentration (see Fig. 6.13). Complex shear modulus:

[6.58]

can be found by replacing the shear modulus of matrix and Poisson’s ratio of matrix in Eq. 6.55 by corresponding complex values and by using Eq. 6.46, 6.32,

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6.13  Effective shear creep function 〈JG〉 (shear strain creep curve divided by the shear stress) of matrix filled by different volume percentages of absolutely rigid spherical particles; τEcr is the strain retardance time in the creep at unidirectional tension/compression. Here EM∞ / EM0 = 0.6 and GM0 is the instantaneous shear modulus of the matrix.

6.38, 6.42, 6.48, and 6.49, taking into account that the bulk modulus of matrix can be expressed with the help of instantaneous Young’s modulus and Poisson’s ratio. The final formula for the shear storage modulus is:

[6.59]

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The final formula for the loss modulus is:

[6.60]

The numerical examples of calculations of the dependencies of the components of complex modulus on frequency and on concentration of inclusions are shown in Fig. 6.14 and 6.15.

6.14  Dependencies of the effective shear storage modulus 〈G´〉 (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 = 0.6 and GM0 is instantaneous shear modulus of matrix. Shear storage modulus approaches the instantaneous one at infinite frequency.

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6.15  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 retardance time in the creep τEcr at tension). Here EM∞ / EM0 = 0.6 and GM0 is instantaneous shear modulus of matrix.

From two known effective elastic characteristics of an isotropic body the other ones can be easily calculated using well-known formulae:

[6.61] [6.62]



For Young’s modulus the final formula following from Eq. 6.47, 6.54, and 6.62 can be written as:

[6.63]

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For the rigid inclusions this formula degenerates into:

[6.64] A more complicated solution, which is valid for slightly bigger volume concentrations of inclusions, is given by Vanin.9 He derived a direct formula for the effective Young’s modulus instead of using solution for the shear modulus:

[6.65]

For the absolutely rigid inclusions this formula degenerates into: [6.66] The results of calculations are shown in Fig. 6.16. In a similar way to that done for the shear, it is also possible to derive formulae for the effective viscoelastic module-operator of the filled material for tension/ compression and the inversed viscoelastic compliance-operator. Unfortunately, the final formulae are too bulky to be printed here. An example of calculations using the compliance-operator is shown in Fig. 6.17. Complex Young’s modulus can be calculated via previously found complex bulk modulus (Eq. 6.52) and complex shear modulus (Eq. 6.58–6.60):



[6.67]

Numerical examples of calculations are shown in Fig. 6.18 and 6.19. It is necessary to mention that viscoelastic behavior on shear and on tension/ compression are similar, which is natural because the bulk creep of the matrix was

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6.16  Relative increase of the Young's modulus of composite made from polymeric matrix (epoxy) filled by glass microspheres (GI /GM = 25), by micro or nanospheres made from material with (GI /GM = 50) MgO, low grades of carbon, silicon carbide, etc.) or by absolutely rigid micro or nanospheres (GI /GM → ∞) as function of the volume concentration c of According to Eq. 6.14 and 6.63–6.64 derived for small inclusions. according to Eq. 6.9 and 6.65–6.66 derived for concentrations; medium concentrations.

ignored in the model, but this similarity is not absolute because the stress concentration in matrix near inclusions on shear and on tension/compression are different. Other effective elastic properties can be easy calculated from two known effective elastic characteristics of an isotropic body using well-known formulae:



[6.68]

The results of calculations of Poisson’s ratio change with concentration of inclusions are shown in Fig. 6.20. This is the only characteristic which may

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6.17  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 retardance time in the creep at unidirectional tension/compression. Here EM∞ / EM0 = 0.6 and EM0 is the instantaneous Young's modulus of the matrix.

6.18  Dependencies of the effective storage modulus 〈E´〉 (real 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 retardance time in the creep τEcr at tension). Here EM∞ / EM0 = 0.6 and EM0 is the instantaneous shear modulus of the matrix. © Woodhead Publishing Limited, 2011

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6.19  Dependencies of the effective loss modulus 〈E”〉 (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 retardance time in the creep τEcr at tension). Here EM∞ / EM0 = 0.6 and EM0 is the instantaneous shear modulus of the matrix.

6.20  Change of Poisson's ratio of composite made from polymeric matrix (epoxy) filled by glass solid microspheres (GI /GM = 25), by micro or nanospheres made from material withGI /GM = 50 (MgO, low grades of carbon, silicon carbide, etc.) or by absolutely rigid micro or nanospheres (GI /GM → ∞) as function of the volume concentration c of According to Eq. 6.68, and Eq. 6.14 and 6.63–6.64 inclusions. according to Eq. 6.68 and derived for small concentrations; Eq. 6.9 and 6.65–6.66 derived for medium concentrations; in both cases Eq. 6.47–6.50 are also used.

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change non-monotonically with fillers concentration. Results of calculations of viscoelastic Poisson’s effect are shown in Fig. 6.21 and 6.22. It is interesting that the frequency dependence of the imaginary part of the Poisson’s ratio is practically independent of fillers concentration.

6.21  Dependencies of the real part 〈ν´〉 of the effective complex Poisson's ratio 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 = 0.6.

6.22  Dependencies of the effective imaginary part 〈ν”〉 of the complex Poisson's ratio of composite made from 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 = 0.6. © Woodhead Publishing Limited, 2011

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6.4

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Viscoelasticity of polymers filled by platelet-shape nanoparticles

Let’s consider two cases of platelets orientation in the matrix: parallel and random. Bulk modulus is calculated according to the same formula (Eq. 6.47) as for spherical inclusions and it is independent of inclusions orientation. Along the plane of inclusions, the Young’s modulus can be roughly calculated on the basis of the primitive model ‘rectangular parallelepiped inclusion in rectangular parallelepiped cell’, neglecting Poisson’s effect. Before applying it to platelet inclusion, let’s verify the model on cubical inclusions arranged in 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:

[6.69]

The lowest estimation of Young’s modulus for this particular case gives

[6.70]

The results of calculations according to Eq. 6.69 and 6.70 are compared in Fig. 6.23 with previous calculations according to Vanin’s formula (Eq. 6.66) for

6.23  Dependencies of the effective Young's modulus of composite on concentration of absolutely rigid inclusions; cubic inclusions, upper estimation according to Eq. 6.69, cubic inclusions, lowest estimation according to Eq. 6.70, spherical inclusions according to Eq. 6.66. © Woodhead Publishing Limited, 2011

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spherical inclusions. The results for spherical inclusions are very close to the results of the lowest estimation using the model described above. Calculations for the platelet inclusions were done according to the lowest estimate for the case of cubic lattice of platelet inclusions and for the case of the lattice in which the individual cell has the shape similar to inclusion. For the cubical lattice of platelet inclusions oriented parallel to each other, the formula for the effective Young’s modulus is:



[6.71]

Here κ is the ratio of thickness of the plate to its size in plane. Inversed ratio is called aspect ratio. Dependence of the effective modulus on the ratio of Young’s moduli of inclusion and matrix for platelets with different aspect ratio is shown in Fig. 6.24. It is following from these data that the use of majority platelet fillers for polymer matrices (ratio of moduli 25–50) is practically equivalent to the use of absolutely rigid platelets. For this case the formula (Eq. 6.71) degenerates into:



[6.72]

6.24  Dependence of the effective Young's modulus 〈E1〉 in the direction parallel to the plane of the platelet inclusions on the ratio of the Young's moduli of inclusion to the matrix and on the aspect ratio of inclusions.

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In the case of the lattice having cells with a shape similar to the shape of inclusions, the effective Young’s modulus is calculated according the same formula (Eq. 6.70) as for cubical inclusions, i.e. the result is independent of the aspect ratio and 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 some intermediate case between these two types of lattices. It is necessary to mention that the cubic lattice for the platelets with a 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 the very high aspect ratios the problem of keeping plane shape of the inclusions is encountered. 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 matrix, etc. Effectiveness of the reinforcement by 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:



[6.73]

which for the big ratios of Young’s moduli degenerates into:



[6.74]

This formula is valid only for the small concentration of inclusions, especially if the aspect ratio is big. The high theoretical effectiveness of reinforcement by platelet inclusions mentioned in many works is obtained for the extrapolation of the aspect ratio to infinity. For the finite aspect ratio, the effectiveness is not so high. Viscoelasticity of platelet inclusions reinforced material is mostly determined by viscoelasticity of polymer matrix as it follows from the formulae derived for absolutely rigid inclusions.

6.5

Viscoelasticity of polymers filled by nanofibers

There are several models for describing finite length fiber-reinforced materials. None of these models is rigorous and they are based on the use of some simplified hypotheses. One of the best models made by Russel19 is based on three

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hypotheses: that the concentration of short fibers is small; the shape of the fibers is approximated by prolate ellipsoid; and all fibers are oriented parallel with each other. The final formulae are bulky (the formula for longitudinal Young’s modulus can be found in Eq. 6.14). For the prediction of viscoelastic behavior, the simplified formula for the case of much stiffer inclusion than the matrix is used here:



[6.75]

Here κ is the ratio of the diameter of short fiber to its length. Inversed 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 Fig. 6.25 and 6.26. The qualitative effect of the relative stiffness of inclusions for the case of short fibers (Fig. 6.25) is the same as for platelet inclusions (Fig. 6.24). The effective longitudinal modulus depends on the relative stiffness nonlinearly and it approaches the asymptote corresponding to absolutely rigid short inclusions. The picture is dramatically changed only for the case of straight continuous fibers. Dependence of the effective longitudinal modulus on the aspect ratio is also nonlinear (Fig. 6.26).

6.6

Viscoelasticity of polymers filled by buckyballs and nanotubes

Hollow fillers are used widely in composite and nanocomposite technology providing light weight, good thermal insulation properties, better resistance to dynamic loads, etc. Syntactic foams use hollow microspheres/microballoons; hollow fibers have been used for reinforcing from ancient time; hollow nanospheres, called buckyballs, have been used for the last couple of decades. Other hollow nano-size objects called fullerenes are used also. The biggest number of investigations in nanofillers is done in nanotubes. 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 carbon spherical particles under hydrostatic pressure is close to the behavior of absolutely rigid inclusions. As a result, hollow glass spheres having a half or less of the mass of solid spheres give almost the same reinforcing effect (Fig. 6.27). Even glass hollow microspheres with 10% of the mass of a 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 done with respect to the effective Young’s modulus (Fig. 6.28).

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6.25  Dependence of the effective Young's modulus 〈E1〉 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 Russel formulae.14, 19 These results and the following discussion were presented in references 20 and 21. Volume concentration of inclusions is 5% (a) and 10% (b).

For continuous straight fibers, 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. For short hollow fibers the result of such replacement is different (see Fig. 6.29).

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6.26  Dependence of the effective Young's modulus 〈E1〉 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 Russel formulae14, 19 and were presented in references 20 and 21. Volume concentration of inclusions is 5%. The asymptotic line for the case EI /EM = 200 is shown for comparison.

This phenomenon can be understood if we return to the results of calculations shown in Fig. 6.25. If the solid fiber is replaced by two hollow fibers with the same total mass, their effective stiffness drops twice but it is still sufficiently high and the effect of reinforcement by one short hollow fiber is comparable to the effect of reinforcement by one solid fiber. This is due to the significant nonlinearity of the curves in Fig. 6.25. However, because the number of fibers and their volume concentration in a nanocomposite is doubled, the total reinforcing effect is almost doubled. It is the reason why nanotubes are more promising than solid nanofibers. When the thickness of the walls of fibers is decreased more, the effect is decreased. Moreover, there is the optimal thickness of the walls of nanotubes providing the maximal reinforcing effect. The real optimum will be shifted toward more thick walls than is shown in Fig. 6.29 because too thin nanotubes behave not as hollow rods but as shells and can be deformed more easily and by different ways (local wall bending or buckling, etc.).

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6.27  Dependence of the effective bulk modulus 〈K〉 of material made from matrix with bulk modulus KM and hollow spherical inclusions on the volume concentration of inclusions. Here m = 1 – (r0 /r1)3 is the volume concentration of the solid phase in single inclusion. Calculations are made for vinyl-ester matrix and glass microballoons.

6.28  Dependence of the effective Young's modulus 〈E〉 of material made from matrix with Young's modulus EM and hollow spherical inclusions on the volume concentration of inclusions. Here m = 1 – (r0 /r1)3 is the volume concentration of the solid phase in single inclusion. Calculations are made for vinyl-ester matrix and glass microballoons.

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6.29  Dependence of the effective longitudinal Young's modulus 〈E1〉 of short hollow fibers with the same mass concentration as solid fibers on the concentration of solid phase in the cross-section of the fiber m = 1 – (r0 /r1)2. Initial volume concentration of solid fibers and aspect ratio are shown in the figure. Other values used in calculations: EI / EM = 300; νI = 0.25; νM = 0.35.

6.7

Viscoelasticity of nanoporous polymers

In the case of spherical voids Eq. 6.47 degenerates into: [6.76]

The corresponding viscoelastic operator can be found by replacing Poisson’s ratio by Eq. 6.44 and by using Eq. 6.20); the result has the following form:



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[6.77]

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In the case of spherical voids Eq. 6.54 degenerates into: [6.78] Vanin’s formula (Eq. 6.65) degenerates into:

[6.79] Eq. 6.63 degenerates into:

[6.80] The results of calculations of the effective elastic properties are shown in Fig. 6.30–6.33. It is necessary to mention that at concentrations of voids of 40–50% their shape begins to change and after 60% converts to a polyhedron,22 for which all these formulae are incorrect. Moreover, due to some simplifications

6.30  Relative drop of the effective bulk modulus 〈K〉 of porous material as a function of the volume concentration c of spherical voids. Calculations are made for different values of matrix Poisson's ratio νM.

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6.31  Relative drop of the shear modulus of porous material as a function of the volume concentration c of spherical voids: according to Eq. 6.14 and 6.78 derived for small concentrations; according to Eq. 6.9 and Eq. 6.79, 6.68 and 6.76 derived for medium concentrations.

6.32  Relative drop of the Young's modulus of porous material as a function of the volume concentration c of spherical voids: according to Eq. 6.14 and Eq. 6.78, 6.62 and 6.76 derived for small concentrations; according to Eq. 6.9 and Eq. 6.79, derived for medium concentrations; according to Eq. 6.80.

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6.33  Change of Poisson's ratio of porous material as a function of the volume concentration c of spherical voids: according to Eq. 6.14 and Eq. 6.68, 6.78 and 6.76 derived for small concentrations; according to Eq. 6.9 and Eq. 6.68, 6.79 and 6.76 derived for medium concentrations; according to Eq. 6.68, 6.80 and 6.76.

used in the derivation of these formulae, they become incorrect, giving zero or negative shear and Young’s moduli values for volume concentrations of pores over 50–55% instead of asymptotic approaching zero at c = 1. The results of the bulk creep calculation for porous media are shown in Fig. 6.34. It is interesting to compare it with similar calculations for the bulk creep of the medium with absolutely rigid inclusions (Fig. 6.9). With increase of void concentration the instantaneous bulk modulus is decreasing and the instantaneous strain is increasing (the opposite to the rigid inclusion case). However, the creep is increasing in both cases. It is happening due to the same reason – shear stress concentration near inclusions.

6.8

Viscoelasticity of fibrous composites with nano-filled matrices

A matrix filled by nanoparticles is not only stiffer and stronger but is also working slightly differently in its mechanism of load transmission from fiber to fiber in the composites compared with conventional reinforcing fibers. This effect is especially pronounced in the dynamic loading of composites. The example of an experimental study23, 24 of 3D fabric reinforced plastic filled by SiC nanoparticles is shown in Fig. 6.35. It shows that the dissipation of energy by material is significantly increasing due to the presence of nanoparticles in the

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6.34  Effective bulk creep function 〈JK〉 (volume creep curve divided by the hydrostatic stress) of material containing different volume percentages of spherical voids; τEcr is the strain retardance time in the creep at unidirectional tension/compression. Here EM∞ / EM0 = 0.6, 〈K0〉 is the instantaneous bulk modulus of composite and KM0 is the instantaneous bulk modulus of matrix.

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6.35  Dependencies of tangent of the angle of mechanical losses at glass transition temperature on weight concentrations of SiC nanoparticles in 3D fabric reinforced vinyl ester system in different fiber directions: (a) 90°, (b) 45°, (c) 0°.

polymer matrix. In Sun’s works (see for example reference 25) it was shown that the adding of nanoclay particles to polymeric matrix increases longitudinal compression strength by several tens of percent, probably due to additional support of fibers in matrix in buckling. Adding nanoparticles to the matrix of the syntactic foams also improves their properties,26 probably due to a different mechanism of the load transfer from one microballoon to another one. There are more investigation results described in the literature.27–40 However, due to complexity of the system it is necessary to accumulate more experimental data before making definite qualitative conclusions.

6.9

Concluding remarks

Despite the enthusiasm of scientists working on synthesis of new nanoparticles based on their success in achieving much higher stiffness and especially strength of nanoparticles, composites industry experts remain skeptical about the possibility of the realization of such improvements in composites. Particles and short fibers do not improve properties in comparison with matrix properties even by one decimal order despite the difference in properties of matrix and fillers which achieved two or three decimal orders. The behavior of the majority of conventional fillers, as it is shown in multiple examples in this chapter, is very close to the behavior of absolutely rigid particles. Further improvement of fillers’ mechanical properties going from micro- to nano-scale gives in many cases small gains in composites’ properties only. Nevertheless, it is not related to the physical and chemical properties. Significant effect is possible if continuous nanofibers can be produced on an industrial scale. In this case the increase in longitudinal mechanical properties of nanocomposites in comparison with conventional composites can be expected in the order of several tens of percent.

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Viscoelasticity of nanocomposites is determined first of all by the interaction of the nanoparticles and the viscoelasticity of the polymeric matrix. It can never be completely suppressed by using even ideally elastic or absolutely rigid inclusions if all sizes of these inclusions are finite ones.

6.10 Notation c

volumetric concentration of inclusions m volumetric concentration of solid phase in single hollow inclusion t time E Young’s modulus F functional G shear modulus H thickness I Abel operator J compliance K bulk modulus L, M, W viscoelastic operators N number of inclusions R radius S surface area V volume α, β, γ, λ, µ parameters of viscoelastic operators αij tensor of coefficients of linear thermal expansion εij strain tensor σij stress tensor ν Poisson’s ratio η viscosity φ diameter ξ form factor τ time before the moment t χ defined by Equation 6.53 κ inversed aspect ratio ω angular frequency Π,Υ,Λ viscoelastic operators Φ(t – τ), kernels of operators Ψ(t – τ)

Γ ∋ Θ Ξ

gamma-function Rabotnov operator Rzhanitsyn operator Koltunov operator

Indices Upper: × * ′ ″ 0 Lower: M I 0 ∞ E G K

cr

upp low

1,2,3

Brackets 〈 〉

operator viscoelasticity complex number real part imaginary part free strains (shrinkage, physical transitions, etc.) matrix inclusion instantaneous value in creep or relaxation long time value in creep or relaxation related to tension/ compression related to shear related to volumetric deformation related to creep upper estimation lowest estimation related to the directions x1, x2, x3 or order number of element effective characteristics

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6.11 Acknowledgement Part of the material contained in this work was obtained due to the support of ONR Grant N00014-05-1-0532 and Dr Yapa Rajapakse, Solid Mechanics Program.

6.12 References   1. Polymer nanocomposites. Edited by Yiu-Wing Mai and Zhong-Zhen Yu. Woodhead Publishing and CRC Press. 2006. 594 pp.   2. Wing Kam Liu, Eduard G. Karpov, Harold S. Park. Nano Mechanics and Materials. John Wiley & Sons. 2006. 320 pp.   3. Sati N. Bhattacharya, Musa R. Kamal and Rahul K. Gupta. Polymeric Nanocomposites. Springer Verlag, 2007. 390 pp.   4. Beil’, A. I., Brant, I. P., Kalnin, M. M., Karliwan, W. P., Krejtus, A. E., Metra, A. Ja. Besonderheiten der Entwicklung einer Füllstoff- und Verstärkungstheorie für Materialien mit thermoplastischer Matrix. Plaste und Kautschuk, 22. Jahrgang Heft 8/1975, ss. 619–626.   5. Sha X., Beyle A., Ibeh C.C. Effect of Sonication on Mechanical Properties of Nanocomposites. MRS, Boston, 2008.   6. Rabotnov Yu.N. Elements of Hereditary Solid Mechanics. MIR Publishers, Moscow 1977 (In English). 387 pp.   7. Rabotnov Yu. N. Creep Problems in Structural Members. North-Holland Series in Applied Mathematics and Mechanics. V. 7 1969. 822 pp.   8. Rabotnov Yu.N. Equilibrium of elastic medium with consequences. Applied Mathematics and Mechanics XII, 1948, #1, p. 53–62.   9. Vanin G.A. Micro-mechanics of Composite Materials. Naukova Dumka Publ., 1985. 304 pp. (in Russian). 10. Van Fo Fy G.A. Theory of reinforced materials. Kiev, Naukova Dumka Publisher, 1971. 232 pp. (in Russian). 11. Van Fo Fy G.A. Structures from reinforced plastics. Kiev, Technika Publisher, 1971, 220 pp. (in Russian). 12. Van Fo Fy G.A., Grosheva V.M., Karpinos D.M., etc. Fibrous Composite Materials. Kiev, Naukova Dumka Publisher, 1970. 404 pp. (in Russian). 13. Shermergor T.D. Theory of Elasticity of Micrononuniform Media. Nauka Publ., 1977. 400 pp. (in Russian). 14. Christensen R.M. Mechanics of Composite Materials. John Wiley & Sons Publ., 1979, 348 pp. (in Russian.) 15. Christensen R.M. Theory of Viscoelasticity. 2nd edition. 2003. 388 pp. 16. C. P. Chen, R. S. Lakes. Analysis of high loss viscoelastic composites. J. Materials Science, 28, 4299–4304, (1993). 17. Lakes R.S., Wineman A. On Poisson’s Ratio in Linearly Viscoelastic Solids. J. Elasticity (2006) 85: 45–63. 18. Lakes R.S. Viscoelastic solids. CRC Press. 1998. 476 pp. 19. Russel W.B. On the effective moduli of composite materials: effect of fiber length and geometry at dilute concentrations. Z. Angew. Math. Phys., vol. 24, 581 (1973). 20. Beyle, A., Cocke, D. L. Nanoeffect in different types of nanocomposites: possible explanations. ICCE-10 Tenth Annual International conference on composites/nano

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engineering. New Orleans, 2003, pp.55–56. 21. Beyle, A., Cocke, D.L., Juneja, A. Viscoelastic behavior of polymers filled by short carbon fibers and by carbon nanofibers. ICCE-11 Eleventh Annual International conference on composites/nano engineering. Hilton Head, SC, 2004, pp.51–52. 22. Gibson, L.J., Ashby, M.F., Cellular Solids, 2nd edn. Cambridge University Press, Cambridge, UK (1997). 23. Zhou, N., Beyle, A., Ibeh, C.C. Thermal Viscoelastic Analysis of 3D Fabric Reinforced Nanocomposites. MFMS, Hong Kong, 2008. 24. Zhou N., Beyle, A., Ibeh, C.C. Temperature-frequency analogy for vinyl ester reinforced by 3D fabrics. ICCE-16, Kunming, China, 2008. 25. Sun C.T., Mechanical Properties of Unidirectional Composites with Silica Nano­ particle enhanced matrix. Marine Composites and Sandwich Structures, ONR, (2007), 21–28. 26. Onovo, O.B., Beyle, A., Ibeh C.C. Mechanical properties of syntactic foams. ANTEC 2008. 27. Zhou, N., Beyle, A., Ibeh C.C. Viscoelastic properties of epoxy and vinyl-ester nanocomposites. ANTEC, Milwaukee, 2008. 28. Zhou, N., Beyle, A., Ibeh C.C. Viscoelastic Properties Of 3D Composites With Nanofilled Polymer Matrices, SAMPE, Long Beach, CA 2008. 29. Zhou, N., Beyle, A., Ibeh, C.C. Thermal Viscoelastic Analysis of 3D Fabric Nanocomposites. Advanced Materials Research Vols. 47–50 (2008) pp 1133–1136. 30. Ibeh, C. C., Beyle, A. Modeling of Mechanical Behavior of Foam-Filled Honeycombs. In: Marine Composites and Sandwich Structures. ONR, 2006 pp. 204–211. 31. Huang Y., Mogilevskaya S. G., Crouch S. L. Numerical modeling of microand macro-behavior of viscoelastic porous materials. Comput. Mech. (2008) 41:797–816. 32. Hong Mei Yang, Qiang Zheng. The Dynamic Viscoelasticity of Polyethylene Based Montmorillonite Intercalated Nanocomposites. Chinese Chemical Letters Vol. 15, No. 1, pp 74– 76, 2004. 33. Dubenets V. G., Yakovenko O. A. Determination Of Effective Damping Characteristics of Fiber-Reinforced Viscoelastic Composites. Strength of Materials, Vol. 41, No. 4, 2009. 34. Vassileva E., Friedrich K. Epoxy/Alumina Nanoparticle Composites. I. Dynamic Mechanical Behavior. Journal of Applied Polymer Science, Vol. 89, 3774–3785 (2003). 35. Smith G.D., Bedrov D., Li Liwei, and Byutner O. A molecular dynamics simulation study of the viscoelastic properties of polymer nanocomposites. Journal of Chemical Physics, v. 117, No 20 22 (2002). pp. 9478–9491. 36. Levin V., Sevostianov I. Micromechanical Modeling of the Effective Viscoelastic Properties of Inhomogenious Materials Using Fraction-Exponential Operators. International Journal of Fracture (2005) I34: L37–L44. 37. Borodin O., Bedrov D., Smith G.D., Nairn J., Bardenhagen S. Multiscale Modeling of Viscoelastic Properties of Polymer Nanocomposites. Journal of Polymer Science: Part B: Polymer Physics, Vol. 43, 1005–1013 (2005). 38. Bartholomea C., Beyoua E., Bourgeat-Lamib E., Cassagnaua P., Chaumonta P., Davida L., Zydowicza N. Viscoelastic properties and morphological characterization of silica/ polystyrene nanocomposites synthesized by nitroxide-mediated polymerization. Polymer 46 (2005) 9965–9973.

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39. Tong J., Nan C.-W., Fu J., Guan X. Effect of inclusion shape on the effective elastic moduli for composites with imperfect interface. Acta Mechanica 146, 127 134 (2001). 40. Pryamitsyn V., Ganesan V. Origins of Linear Viscoelastic Behavior of PolymerNanoparticle Composites. Macromolecules 2006, 39, 844–856.

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7 Constitutive modeling of viscoplastic deformation of polymer matrix composites M. Kawai , University of Tsukuba, Japan Abstract: This chapter describes a phenomenological approach to constitutive modeling of the viscoplastic behavior of unidirectional polymer matrix composites under off-axis quasi-static loading conditions. The chapter addresses a combined isotropic and kinematic hardening model for the viscoplastic behavior of unidirectional composites under proportional loading conditions. The chapter then deals with a modified viscoplasticity model that allows prediction of different flow stress levels in off-axis tension and compression in addition to description of the rate dependence, nonlinearity and fiber orientation dependence of the inelastic behavior of unidirectional composites under off-axis monotonic loading conditions. The chapter further attempts to elaborate the combined isotropic and kinematic hardening model so that it can adequately predict the transient creep softening due to stress variation. In the respective phases of sophistication, the accuracies of predictions using the associated viscoplasticity models are evaluated by comparing with the experimental results on unidirectional carbon/epoxy laminates at high temperature. Key words: polymer matrix composite, viscoplastic constitutive model, initial orthotropy, combined isotropic and kinematic hardening, tension–compression asymmetry, transient creep softening, off-axis loading, fiber orientation dependence.

7.1

Introduction

Solid polymers, ranging from amorphous to moderately crystalline, which are used as matrix materials in composites have a strong tendency towards timedependent deformation (Vlack, 1970; Nielsen, 1975; Courtney, 1990; Barrett et al., 1973). If solid polymers are not cross-linked, they can easily be deformed permanently by mechanical stresses. Cross-linking introduces anchor points between molecules, and it changes a linear structure into a three-dimensional structure. Thus, the cross-linking restricts the deformation of solid polymers. This explains why large mechanical stress is required to cause permanent deformation in cross-linked solid polymers. At high temperatures, however, the restriction on deformation is loosened, and thus stress-induced permanent deformation more easily occurs even in the solid polymers with cross-linked structures. While the permanent deformation is treated as a viscous part of total deformation and distinguished from recoverable elastic and anelastic parts (Vlack, 1970), this chapter will regard the deformation which is not elastic as the inelastic deformation, as a whole, that may include recoverable and unrecoverable parts. 234 © Woodhead Publishing Limited, 2011



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The strong tendency for solid polymers to exhibit a time- and rate-dependent inelastic behavior (Ishai, 1967; Ha et al., 1991) should have important influences on the mechanical responses of polymer matrix composites (PMCs), especially when they are subjected to transverse load perpendicular to fibers and shear load along fibers; e.g. Fig. 7.1 for AS4/PEEK (Kawai et al., 2001) and Fig. 7.2 for T800H/epoxy (Kawai et al., 2004). The durability of PMC structures that spread over a wide range of applications is affected by the matrix-dominated response of PMCs to external load (Brinson, 1999; Dillard et al., 1982; Kawai et al., 2006; Kawai and Sagawa, 2008). In evaluation of the performance and reliability of PMCs, therefore, it is of crucial importance to quantitatively understand their time- and rate-dependent inelastic behaviors under various loading and temperature conditions. For establishment of a rational design procedure based on the local stress/strain analysis of PMC structures, furthermore, it is an essential prerequisite to develop a constitutive model that can describe the time- and rate-dependent inelastic deformation behavior of PMCs under different loading conditions at different temperatures. Experimental observation and theoretical modeling of the time-dependent deformation behavior of PMCs have been attempted, for example, by the following researchers. Beckwith (1980) has observed the creep and creep strain recovery behaviors of unidirectional and angle-ply laminates of glass fiber reinforced composites (GFRPs), and applied a linear viscoelasticity theory to

7.1  Off-axis loading–unloading behavior of a unidirectional AS4/PEEK laminate at 100°C (Kawai et al., 2001).

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7.2  Off-axis loading–unloading behavior of a unidirectional T800H/ epoxy laminate at 100°C (Kawai et al., 2004).

predict the creep behaviors of those laminates. Sullivan (1990) has examined the off-axis creep behavior of unidirectional GFRP laminates at relatively low stresses. It was also shown by Sullivan (1990) that the Boltzmann superposition principle is applicable to the short-term creep behavior, and the effective time theory (Struik, 1978) is suitable for the long-term creep behavior. Similar results were reported for carbon fiber reinforced composites (CFRPs). Tuttle and Brinson (1986) have examined the creep and creep strain recovery behaviors of unidirectional and symmetric T300/5208 carbon/epoxy laminates, and successfully applied the classical laminated plate theory and a nonlinear viscoelasticity model (Schapery, 1974) for individual plies to description of the creep behaviors observed. Katouzian et al. (1995) have investigated the creep behaviors of the [90] and [±45] laminates made of unidirectional T800/Epoxy and AS4/PEEK systems at various temperatures and stress levels, and showed that the creep behaviors of the CFRP laminates can adequately be described by means of a nonlinear viscoelasticity model (Schapery, 1974). In the range of relatively low stress or small strain, the time-dependent behavior of PMCs is mostly governed by the linear and nonlinear viscoelastic responses of the polymer matrices, and thus viscoelasticity models are favored for description of the time-dependent behavior of PMCs, as seen in the reports quoted above. In the range of relatively high stress or large strain, however, the time-dependent behavior of solid polymers often tends to deviate from the predictions using viscoelasticity models. This is partly because the loading of solid polymers to

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high stress or large strain levels results in time-dependent inelastic deformation (Matsuoka, 1992). Figures 7.1 and 7.2 provide the experimental evidences to support this statement, showing that a certain amount of permanent strain remains in the unidirectional CFRP laminates after complete unloading which is preceded by loading to a relatively high stress level. These figures also demonstrate that an anomalously large hysteretic response appears with a cycle of loading, unloading and reloading in the off-axis direction. Similar results can also be found in literature (Kawai et al., 2001; Kawai et al., 2004; Katouzian et al., 1995; Tuttle et al., 1995; Ha and Springer, 1989). These observations encourage testing an approach in which the time-dependent inelastic deformation of unidirectional PMCs under off-axis loading conditions, especially in the range of high stress or large strain, is approximately identified with viscoplastic deformation. Some researchers have already had an appreciation of the importance of taking into account the viscoplastic response of PMCs. Tuttle et al. (1995), for example, have modified the Schapery nonlinear viscoelasticity model to include a viscoplastic strain term, and successfully applied it to prediction of the creep behaviors of the IM7/5260 CFRP laminates of [90] and [±45] lay-ups. The phenomenological theory of viscoplasticity, which provides a natural framework for the modeling of the rate-dependent inelastic deformation of materials (Jirasek and Bazant, 2002), is described by means of a set of simultaneous differential equations. Pioneering efforts to model the time- and rate-dependent inelastic behavior of unidirectional PMCs by taking advantage of the framework of the phenomenological theory of viscoplasticity have been made by Sun and coworkers (Gates and Sun, 1991; Yoon and Sun, 1991; Wang and Sun, 1997). Using the effective stress and effective plastic strain assumed in the timeindependent plane-stress plasticity model (Sun and Chen, 1989), Gates and Sun (1991) developed a viscoplasticity model based on the concept of overstress (Perzyna, 1966). Yoon and Sun (1991) proposed another plane-stress viscoplasticity model for unidirectional composites by modifying the Bodner-Partom model (Bodner and Partom, 1975) for the viscoplastic behavior of isotropic materials with the aid of the effective stress and effective plastic strain of the Sun-Chen model (Sun and Chen, 1989). Wang and Sun (1997) elaborated the evolution equation of the internal variable of the Gates-Sun model. By comparison with experimental results, it has been shown that that the Gates-Sun model (Gates and Sun, 1991), the Yoon-Sun model (Yoon and Sun, 1991) and the Wang-Sun model (Wang and Sun, 1997) can describe the rate-dependent inelastic behavior of unidirectional PMCs under off-axis loading conditions. A great success achieved in those attempts has therefore proved that the modeling of the time- and ratedependent inelastic behavior of PMCs by making use of the framework of the phenomenological theory of viscoplasticity is a viable alternative. In order to establish a general viscoplastic constitutive model for PMCs that is applicable over a wide range of loading conditions, however, further consideration is required in regard to the following important respects.

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(1) Deformation-induced anisotropy: If metallic materials which are initially isotropic undergo a certain amount of plastic deformation in a certain direction, they often yield at a lower level of stress under the subsequent reversed loading. This phenomenon is called the Bauschinger effect. The Bauschinger effect in the inelastic deformation of metal matrix composites has been found by Yanagisawa and Yano (1987) and Taya et al. (1990), and theoretically predicted by Dvorak and Rao (1976) and Povirk et al. (1992). A decrease in creep strain with time just after partial unloading of a prior creep stress, which is called creep strain recovery, is a similar phenomenon, and it has been observed in a unidirectional PMC laminate at elevated temperature (Kawai, 2001). The Bauschinger effect and creep strain recovery in a material suggest that a certain magnitude of internal stress develops in the material with prior inelastic deformation and it promotes the subsequent inelastic deformation in the opposite direction. These facts probe into the constitutive assumption made by most existing viscoplasticity models for fiber-reinforced composites that the ratedependent inelastic behavior can be described using a single scalar internal variable. (2) Inherent anisotropy: Most solid polymers have different yield strengths in tension and compression (Caddell et al., 1973; Tuttle et al., 1992; Bekhet et al., 1994). This fact strongly suggests that PMCs should also exhibit different inelastic flow behaviors under tensile and compressive loading conditions. However, such an inelastic flow differential effect in PMCs has not systematically been quantified, and thus there are few studies on the constitutive modeling of the viscoplastic behavior of PMCs that take into account the inelastic flow differential effect. It is therefore suggested that a more detailed consideration of the initial anisotropy of composites is required in order to develop a viscoplastic constitutive model that allows accurately describing the tension–compression asymmetry as well as the nonlinearity, rate dependence and fiber orientation dependence in the inelastic flow behavior of composites under off-axis loading conditions. (3) History dependence: The structural components made of PMCs are often subjected to cyclic load during service. The change in the internal state of a composite under cyclic loading is different from the change under monotonic loading in general, since the irreversible events that take place in the composite and resultant inelastic deformation depend on the history of loading or deformation. For accurate and reliable design by analysis of PMC structures, therefore, the inelastic constitutive model for the PMC employed should be formulated so that it can distinguish between the changes in the internal state of PMC under monotonic and variable loading conditions. This chapter will focus on phenomenological modeling of the rate-dependent inelastic behavior of unidirectional PMCs under off-axis loading conditions and experimental validation. The emphasis is on the development of a general constitutive model that can predict the viscoplastic behavior under complicated off-axis loading conditions. The chapter addresses a general framework for the phenomenological theory of viscoplasticity for unidirectional composites in which a combined isotropic and kinematic hardening is taken into account. The

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chapter then discusses improved description of the initial anisotropy of unidirectional PMCs, and presents a viscoplastic constitutive model which allows prediction of the different flow stress levels in off-axis tension and compression besides description of the rate dependence, nonlinearity and fiber orientation dependence of the inelastic behavior under off-axis loading conditions. The chapter closes with another attempt to extend the combined isotropic and kinematic hardening viscoplasticity model to allow accurately describing the temporal creep softening in unidirectional PMCs due to stress variation. In the extension attempt, a modified nonlinear kinematic hardening rule for orthotropic fiber composites is formulated which is characterized by an accelerated evolution of kinematic hardening that occurs while viscoplastic strain lies in a certain history-dependent region of the viscoplastic strain space. Associated with the respective stages of sophistication, the accuracies of predictions using the viscoplasticity models are evaluated by comparing with the experimental results on unidirectional carbon/ epoxy laminates for different off-axis loading conditions at high temperature.

7.2

Framework for constitutive modeling of the viscoplastic deformation of anisotropic materials

In the macromechanical viscoplasticity models (Gates and Sun, 1991; Yoon and Sun, 1991; Wang and Sun, 1997) that were developed for describing the ratedependent nonlinear inelastic behavior of PMCs, a single scalar internal variable is used. The formulation of those viscoplasticity models is thus based on the premise that the internal state of homogenized PMCs isotropically changes with inelastic deformation. This assumption, however, is not always applicable, since some experimental results (Yanagisawa and Yano, 1987; Taya et al., 1990; Kawai, 2001) have suggested that directional hardening occurs in PMCs. This fact reveals that the applicability of those macromechanical viscoplasticity models to prediction of the time- and rate-dependent inelastic behavior of PMCs has been validated only for monotonic proportional loading conditions. For properly describing the actual internal state of PMCs that changes with inelastic deformation, therefore, we need a framework for the phenomenological theory of viscoplasticity that involves not only scalar internal variables but also higher-rank tensor internal variables. An attempt to develop a viscoplasticity model for orthotropic fiber composites that incorporates a second-rank tensor variable has been made, for example, by Kawai (1993, 1994). It has further been developed into a combined isotropic and kinematic hardening model with a more theoretically firm structure (Kawai and Masuko, 2003). They also discussed the pure isotropic hardening model and pure kinematic hardening model that can be reduced from the combined isotropic and kinematic hardening model. The isotropic and kinematic hardening models are alike in mathematical structure, and the isotropic hardening model is similar to a special case of the existing macromechanics models for unidirectional composites. Therefore, the combined isotropic and kinematic hardening viscoplasticity model is encouraged to be used as

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a base for developing a more sophisticated model for orthotropic fiber composites that takes into account the vital factors mentioned above and allows application to more general loading conditions. Modeling of the viscoplastic behavior of anisotropic media that entails not only an isotropic hardening variable but also a kinematic hardening variable has also been attempted by Robinson (1984), Lee and Krempl (1991), and Yeh and Krempl (1992) for metal matrix composites, and by Choi and Krempl (1989) and Nouailhas and Freed (1992) for single crystals. To the best of the present author’s knowledge, however, those different forms of combined isotropic and kinematic hardening viscoplasticity models have not been tested for the time- and rate-dependent inelastic behavior of PMCs. This section reviews the combined isotropic and kinematic hardening viscoplasticity model for orthotropic fiber composites that was developed in the previous studies (Kawai, 1993; Kawai, 1994; Kawai and Masuko, 2003). An emphasis is placed on a general description of the internal state of homogenized unidirectional composites by means of not only a scalar variable but also a secondrank tensor variable. The pure isotropic hardening and kinematic hardening models which can be derived from the combined isotropic and kinematic hardening model as special cases are also discussed. The viscoplasticity models presented in this section will be used as bases for developing elaborated versions in the succeeding sections. Unidirectional composites are treated as homogeneous transversely isotropic media that harden with inelastic deformation. We limit our discussion to a quasiisothermal inelastic process in which temperature and temperature gradient terms are negligible. An infinitesimal deformation is assumed, and thus the total strain is expressed by the sum of the elastic and inelastic components. The trace of the product xy of two second-rank tensors x and y is denoted by x · y, and the image of x under a linear transformation A (a fourth-rank tensor) by Ax.

7.2.1 Combined isotropic and kinematic hardening model Internal variables and associated thermodynamic forces The internal variable approach based on irreversible thermodynamics (Landau and Lifshitz, 1970; Ziegler, 1983; Lemaitre and Chaboche, 1985; Maugin, 1999; Chaboche, 2003) is used to establish a basic constitutive model for describing the elastic and viscoplastic behavior of orthotropic fiber composites. An advantage of this approach over the other empirical ones is that the constitutive model derived typically in a local form automatically satisfies the principle of irreversible thermodynamics. The thermodynamic formalism adopted here requires two scalar potential functions, a free energy function and a dissipation function. The free energy function is assumed to be a function of the internal variables that are chosen for describing the phenomena of interest, and it is used to define the thermodynamic forces that are energy conjugate with the internal variables. The

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dissipation function is a function of the thermodynamic forces, and it is used to define the rates of viscoplastic strain and internal variables. Introduce two internal variables, a scalar ρ and a symmetric second-rank deviatoric tensor , in order to describe the internal state of homogenized orthotropic media. The Helmholtz free energy function , by which the thermodynamic forces associated with the assumed internal variables are defined, is assumed to take the form:

[7.1]

where is the viscoplastic strain, C and H are the elastic and inelastic tensors of the fourth rank, respectively, and h(ρ) is a positive scalar function of the scalar internal variable ρ. Using the assumed free energy, we can define the thermodynamic forces , p and r associated with , and ρ, respectively, as: [7.2]



[7.3] [7.4] where the inelastic tensor H is assumed to take the form H = HI in order to simplify the model; H is a material constant and I is a unit tensor of the fourth rank. The thermodynamic force p is a deviatoric symmetric tensor of the second rank, and it works as a kinematic hardening variable; alternatively, it may be called internal stress or back stress. Evolution equations The dissipation potential to define the evolution equations of the assumed internal variables is assumed to take the form:

[7.5]

where [7.6]

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[7.7] [7.8]

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The quantity σ0 in Eq. 7.6 is the size of the initial elastic region that is described as U(A, s) = σ0 in which s is the deviatoric part of the Cauchy stress , and A is a fourth-rank tensor that prescribes the initial anisotropy (transverse isotropy) (Boehler, 1987). The angular brackets < > stand for the singular function defined as < x > = x (x ≥ 0); < x > = 0 (x < 0). In Eq. 7.6, it is defined as s* = s – p, and s* is called the effective stress tensor. The scalar invariant function U(A, s*), which is defined by Eq. 7.8, represents the magnitude of the effective stress tensor s* = s – p, and it is identified with the effective stress for a current state of stress. The coefficients K, L and m are material constants. Note that W0 in Eq. 7.6 is a virtual term, and W0 = 0 under the state relation prescribed by Eq. 7.3. The inclusion of this term allows us to derive the evolution equation of which is of the nonlinear kinematic hardening format (Lemaitre and Chaboche, 1985; Chaboche, 1977; Armstrong and Frederick, 1966). The time rate of viscoplastic strain can be derived from Eq. 7.5 as: [7.9]

where

[7.10]

. Note that P stands for the magnitude of the viscoplastic strain rate tensor, and it is identified with the effective viscoplastic strain rate. Taking the gradients of the dissipation function with respect to the thermodynamic forces, we can obtain the time rates of the assumed internal variables as: [7.11] [7.12] Using the state relations with Eq. 7.3 and 7.4, we can express the time rates of the kinematic and isotropic hardening variables, respectively, as:

[7.13] [7.14]

Consequently, the combined isotropic and kinematic hardening model can be described using the rate equations 7.9, 7.13 and 7.14 in conjunction with Eq. 7.8 and 7.10. The accuracy of description of the nonlinearity in the viscoplastic behavior of homogenized composites can be improved with the aid of the superposition of the isomorphic variables characterized by the evolution equations

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with different nonlinearities (Lemaitre and Chaboche, 1985); when this extension is needed, Eq. 7.13 and 7.14 are applied in the following forms: [7.15]



[7.16]



7.2.2 Particular cases Isotropic hardening model Removing the kinematic hardening variable from the combined hardening viscoplasticity model (i.e., p = 0, s* = s), we can obtain a pure isotropic hardening model: [7.17]



[7.18]

where

[7.19]

Two kinds of representation of the evolution equation of the isotropic hardening variable are considered. First, assume the scalar function h(ρ) of the form (Lemaitre and Chaboche, 1985): [7.20] where µ is a material constant. This scalar function yields the time rate of r that can be expressed as: [7.21] Equation 7.21 can be rewritten in the form: Integrating Eq. 7.22, we can obtain the power-law relation:

[7.22]

[7.23] For the case of plane stress with σ0 = 0, the isotropic hardening variable r specified by Eq. 7.23 may be identified with the quasi-static stress of the Wang-Sun model

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(or a modified Gates-Sun model) (Wang and Sun, 1997). Therefore, the isotropic hardening model described above, which is a special case of the combined isotropic and kinematic hardening model for orthotropic fiber composites, has a predictive capability similar to that of the plane-stress Wang-Sun model that has succeeded in adequately describing the viscoplastic behavior of PMCs under monotonic loading conditions. An alternative form of the evolution equation of the isotropic hardening variable can be derived from the following scalar function (Lemaitre and Chaboche, 1985): By use of Eq. 7.24, we can obtain the following evolution equation of r: Integration of Eq. 7.25 yields the following relation:

[7.24]

[7.25]

[7.26] where the constant Q characterizes the saturation value of r, and the constant b determines how quickly the isotropic hardening saturates with viscoplastic deformation. Equations 7.22 and 7.25 can be extended, respectively, to the following forms: [7.27] [7.28] The use of the multicomponent representations given by Eq. 7.27 and 7.28 allows an enhanced description of the viscopalstic behavior of orthotropic fiber composites under monotonic loading conditions. Kinematic hardening model Removing the isotropic hardening variable from the combined isotropic and kinematic hardening model, we obtain the pure kinematic hardening model which can be expressed in the form: [7.29]

where

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The evolution equation of the kinematic hardening variable, which is defined by Eq. 7.30, may be modified into a multicomponent form: [7.32] which is in line with Eq. 7.27 and 7.28. The pure kinematic hardening model described above is similar to a special case of the Nouailhas-Freed model (Nouailhas and Freed, 1992) for anisotropic materials.

7.2.3 Plane stress representation Essentially no difference appears between the predictions using the three kinds of viscoplasticity models that assume the combined isotropic and kinematic hardening, the pure isotropic hardening and the pure kinematic hardening, respectively, as far as monotonic proportional loading is concerned. To that type of loading, therefore, we can apply the isotropic hardening model without loss of generality. As validated by Sun and coworkers (Gates and Sun, 1991; Yoon and Sun, 1991; Wang and Sun, 1997), it is reasonable to assume that no viscoplastic strain develops in the fiber direction of unidirectional high modulus fiber composites. This condition can be accommodated by the viscoplasticity models described above, regardless of the kind of hardening assumed, with restrictions on the coefficients involved by the anisotropic tensor A. For the case of plane stress, the constitutive equations of the isotropic hardening model for unidirectional composites that satisfies the condition of no viscoplastic strain in the fiber direction can be expressed as: [7.33] in the principal axes of material anisotropy. The effective stress U and the effective . viscoplastic strain rate P, which were defined by Eq. 7.8 and 7.10, respectively, can be written in exactly the same forms as those derived by Sun and coworkers (Gates and Sun, 1991; Yoon and Sun, 1991; Wang and Sun, 1997):

[7.34] [7.35]



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Equation 7.33 is used in conjunction with the master relationship with Eq. 7.19, . i.e. P = K U – r – σ0 m, and the evolution equation of the isotropic hardening variable r which is described using either Eq. 7.27 or Eq. 7.28. The material constant a66 characterizes the in-plane anisotropy of a given unidirectional composite, and it is determined in such a way that a single master relationship . P = K U – r – σ0 m is established between the effective stress U calculated using . Eq. 7.34 and the effective viscoplastic strain rate P calculated using Eq. 7.35. The detail of the procedure for identifying a66 can be found, for example, in Kawai and Masuko (2003, 2004), Masuko and Kawai (2004) and Takeuchi et al. (2008).

7.2.4 Comparison with experimental results Off-axis tensile stress–strain relationships for unidirectional carbon/epoxy laminates Figure 7.3 illustrates the fiber orientation dependence of the off-axis tensile stress–strain relationship for a unidirectional T800H/2500 carbon/epoxy laminate. This figure includes the experimental data for different fiber orientations θ = 0, 10, 15, 30, 45 and 90° which were obtained from monotonic tension tests at 100°C at a constant strain rate of 10%/min (Kawai et al., 2009a). It is clearly seen that the off-axis stress–strain relationships are in order of fiber orientation angle, indicating that the flow stress required to continue the off-axis deformation becomes lower

7.3  Off-axis stress–strain curves for a unidirectional T800H/epoxy laminate at 100°C in tensile loading at a constant strain rate of 10%/min (Kawai et al., 2009a).

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with increasing fiber orientation angle. It can be confirmed that the tensile stress– strain relationship in the longitudinal direction θ = 0° is almost linear and smooth to final failure, in contrast to remarkable nonlinearity in the responses for the offaxis fiber orientations θ = 10, 15, 45°. Similar fiber orientation dependence has been observed in the off-axis stress–strain relationships at a lower constant strain rate of 0.1%/min (Kawai et al., 2009a). Figures 7.4 (a)–(e) show comparisons between the predicted and observed offaxis tensile stress–strain curves for five different fiber orientations θ = 10, 15, 30, 45 and 90°, respectively. Each figure includes the results associated with two different strain rates of 10%/min and 0.1%/min. The solid lines in these figures indicate the predictions using the isotropic hardening viscoplasticity model. These comparisons demonstrate that that the isotropic hardening viscoplasticity model can adequately predict not only the fiber orientation dependence but also the strain rate dependence of the off-axis nonlinear stress–strain relationship for the unidirectional CFRP laminate exposed to monotonic tensile loading at high temperature. Off-axis tensile creep curves for unidirectional carbon/epoxy laminates Figures 7.5 (a)–(c) demonstrate the time and stress dependence of the off-axis tensile creep behavior of a unidirectional T800H/3631 carbon/epoxy laminate at

(a)

7.4  Comparison of the off-axis tensile stress–strain curves at different strain rates of 10%/min and 0.1%/min for a unidirectional T800H/epoxy laminate at 100°C (Kawai et al., 2009b): (a) θ = 10°, (b) θ = 15°, (c) θ = 30°, (d) θ = 45°, (e) θ = 90°. (Continued)

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(b)

(c)

7.4  Continued.

100°C for different fiber orientations θ = 30, 45 and 90°, respectively (Kawai and Masuko, 2004). It can be seen that creep deformation develops with time in the unidirectional laminate for all of these fiber orientations, and the creep strain rate tends to rapidly disappear as creep strain increases regardless of the creep stress level and of the fiber orientation. The latter observation suggests that the transient © Woodhead Publishing Limited, 2011



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(d)

(e)

7.4  Continued.

(or primary) creep response is dominant in the matrix-dominated creep behavior of the unidirectional carbon/epoxy laminate at high temperature. Similar features have been discussed in an earlier study (Kawai, 2001). The solid lines in Fig. 7.5 (a)–(c) indicate the off-axis creep curves predicted using the isotropic hardening viscoplasticity model. The total strain is plotted

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(a)

(b)

7.5  Off-axis tensile creep curves for a unidirectional T800H/epoxy laminate at 100°C (Kawai and Masuko, 2004): (a) θ = 30°, (b) θ = 45°, (c) θ = 90°.

in these figures, since the entire loading path that is composed of the prior static loading to a specified creep stress level and the subsequent constant stress creep loading has been simulated. Considering that the scatter involved by the creep data was about 2–8%, we may conclude that the isotropic hardening

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(c)

7.5  Continued.

viscoplasticity model has succeeded in adequately predicting the off-axis creep behavior as well as the instantaneous elastoviscoplastic behavior of the unidirectional CFRP laminate, regardless of the fiber orientation and creep stress level.

7.3

Modeling of tension–compression asymmetry in initial anisotropy

The solid polymers used as the matrix materials of composites often exhibit not only nonlinear inelastic deformation that depends on the time and rate of loading over a wide range of temperature (Vlack, 1970; Nielsen, 1975; Courtney, 1990; Barrett et al., 1973; Ishai, 1967; Ha et al., 1991), but also yielding at different stress levels in tension and compression (Caddell et al., 1973; Tuttle et al., 1992; Bekhet et al., 1994). These facts about the rate-dependent and asymmetric yielding behavior of solid polymers suggest that a similar rate-dependent flow differential effect should appear in the inelastic deformation behavior of unidirectional PMCs under off-axis loading conditions. Recently, Kawai et al. (2009a) have elucidated the effect of loading direction (tension or compression) on the magnitude of flow stress, rate dependence and nonlinearity in the stress–strain relationship for a unidirectional CFRP laminate. They compared the off-axis tensile and compressive stress–strain curves obtained at an equal strain rate for different fiber orientations. The experimental results

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demonstrated that while the off-axis tensile and compressive behaviors were similar in nonlinearity, fiber orientation dependence and rate dependence, the flow stress level in compression was about 50% higher than that in tension at the same value of total strain for each of the relatively large off-axis angles of 30°, 45° and 90°. A similar off-axis flow stress differential effect was also shown to appear at a different strain rate. The macromechanical constitutive models developed, for example, by Sun and coworkers (Gates and Sun, 1991; Yoon and Sun, 1991; Wang and Sun, 1997) and Kawai and Masuko (2003) have succeeded in accurately predicting the ratedependent inelastic behavior of unidirectional CFRPs under off-axis monotonic proportional loading conditions, as illustrated in the previous section. However, most of those viscoplasticity models developed for unidirectional PMCs have assumed a quadratic form of effective stress in the formulation for the sake of simplicity, and thus they have not been furnished with the capability to describe the flow stress differential effect in the rate-dependent inelastic behavior of PMCs under off-axis loading conditions. The optimum design of PMC structures, on the other hand, requires the reliable application of PMCs to components that can support not only predominant tensile load but also compressive load or combined tensile and compressive load in bending. These facts show the importance of establishing a viscoplastic constitutive model that can accurately predict the ratedependent inelastic behavior of PMCs under both off-axis tensile and compressive loading conditions for any fiber orientations. This section devotes attention to the tension–compression asymmetry in the initial anisotropy of unidirectional composites, and discusses a modified viscoplasticity model that takes into account the flow differential effect in the inelastic behavior of unidirectional PMCs under off-axis tension and compression (Kawai et al., 2009b). It includes attempts to model the difference between the shear flow stress levels under transverse tension and compression, i.e. a shear flow differential effect, as well as the difference between the transverse tensile and compressive flow stress levels, i.e. a transverse flow differential effect.

7.3.1 Pressure-sensitive effective stress For developing a viscoplasticity model based on the concept of overstress (Perzyna, 1966), a static yield function that becomes a base for time-independent plasticity is required to define the effective overstress which determines the magnitude of viscoplastic strain rate. For the present purpose, the static yield function should meet the condition that it can describe the plastic flow behavior of orthotropic media with different yield strengths in tension and compression. Here we will adopt the pressure-modified Hill’s yield function defined as (Caddell et al., 1973; Stassi-D’Alia, 1969):

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where Ki represents the material constants related to the pressure sensitivity of yielding, and Φ*Hill is the non-dimensional Hill’s yield function that takes the following form (Hill, 1950): [7.37] The stress components in Eq. 7.37 are taken with respect to the principal axes of anisotropy. The coefficients H, F, G, L, M and N are called the anisotropic parameters, and they characterize the orthotropic nature of yielding in a given material. The anisotropic parameters involved by the pressure-modified Hill’s yield function can be related to the yield strengths associated with the principal axes of material anisotropy (Hill, 1950). Equation 7.36 can be reduced to the Drucker-Prager yield function (Drucker and Prager, 1952) for the isotropic materials that are sensitive to pressure. The Hill’s yield function Φ *Hill does not allow description of different yield strengths in tension and compression for any loading direction, since it is quadratic with respect to the principal stress components. However, a sensitivity of yielding to pressure returns by adding the linear terms K1σ11 + K2σ22 + K3σ33 to Φ *Hill as in Eq. 7.36. Note that the modified Hill’s yield function defined by Eq. 7.36 allows not only pressure-sensitive yielding but also non-vanishing plastic dilatation. Accordingly, a modified viscoplasticity model to be developed on the basis of the pressure-modified Hill’s yield function also satisfies both pressure-sensitive viscoplastic flow and non-vanishing viscoplastic dilatation. It is important to note that the relationships between the anisotropic parameters involved by Eq. 7.36 and the yield strengths associated with the principal axes of material anisotropy are different from those identified using the original Hill’s yield function with Eq. 7.37. Based on the pressure-modified Hill’s yield function, a pressure-sensitive effective stress σ– can be defined as: – where Y denotes the average yield strength.

[7.38]

7.3.2 Plane-stress pressure-sensitive effective stress A general expression of pressure-sensitive effective stress given by Eq. 7.38 suggests considering the following modified effective stress (Kawai et al., 2009b): [7.39] for the plane state of stress. It is important to note that the modified effective stress – Σ* has different values depending on the sign of the transverse stress σ22. – Two additional parameters β and Λ are involved by the definition of Σ* with Eq. 7.39, unlike the definition of U with Eq. 7.8. The parameter β is introduced to

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quantify the degree of tension–compression asymmetry in the transverse direction, and it can be defined as: [7.40] where T2 and C2 denote the tensile and compressive yield strengths in the transverse direction, respectively. The parameter Λ in Eq. 7.39 plays a role to distinguish between the shear flow stress levels under off-axis tension and compression, and it should be modeled as a function of the transverse stress component in general. We call the parameters β and Λ a transverse flow differential (TFD) parameter and a shear flow differential (SFD) parameter, respectively (Kawai et al., 2009b). Assuming Λ = 1, we can simulate the viscoplastic behavior of unidirectional composites that involves no shear flow differential effect. Taking as β = 1 further, – we can reduce the modified effective stress Σ* (Eq. 7.39) to the basic form U (Eq. 7.33) that agrees with the definition by Sun and Chen (Sun and Chen, 1989).

7.3.3 Shear flow differential parameter Shin and Pae (1992) have performed torsion tests on a unidirectional composite under different magnitudes of pressure, and found that the initial shear yield strength increases with increasing pressure, and accordingly the shear flow stress in the subsequent torsion becomes higher with increasing pressure. This observation suggests that the shear flow behavior is affected by the compressive normal stress acting on the shear plane. It is the SFD parameter that reflects such an interaction between the normal and shear stresses acting on the same plane in regard to yielding. A simple model that assumes a constant SFD effect has been proposed by Kawai et al. (2009b), and it can be described by means of the following function: [7.41] where γ is a material constant, and χ+ and χ– are the characteristic functions to deal with the sign of transverse stress and they are defined as: [7.42a] [7.42b]

7.3.4 Pressure-modified viscoplasticity model For the case of plane stress, the modified viscoplasticity model for unidirectional composites that takes into account the inelastic flow differential effect can be expressed in the matrix form as:

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[7.43] where

[7.44] [7.45]

The evolution equation of the isotropic hardening variable r is specified by either Eq. 7.27 or Eq. 7.28. It is emphasized that the pressure-modified viscoplasticity model is characterized by the modified effective stress defined by Eq. 7.39 and the effective viscoplastic strain rate defined by Eq. 7.45 that take into account both the TFD and SFD effects.

7.3.5 Comparison with experimental results The off-axis tensile and compressive stress–strain relationships for a unidirectional T800H/2500 carbon/epoxy laminate (Kawai et al., 2009a) are compared in Fig. 7.6 (a) and (b) for two different constant nominal strain rates of 10%/min and

(a)

7.6  Comparison of off-axis stress–strain curves in tension and compression (Kawai et al., 2009a): (a) 10%/min, (b) 0.1%/min. (Continued)

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(b)

7.6  Continued.

0.1%/min, respectively, for the case of a representative fiber orientation θ = 45°. It can be seen that the off-axis flow stress level in compression is higher than that in tension, regardless of the strain rate. Similar flow differential effects can be seen in the results for the other fiber orientations θ = 10, 15, 30 and 90° which are presented in Fig. 7.7 (a)–(e). Figures 7.7 (a)–(e) show the tensile and compressive stress–strain curves predicted using the pressure-modified viscoplasticity model for different fiber orientations θ = 10°, 15°, 30°, 45° and 90°, respectively (Kawai et al., 2009a), along with the experimental data (Kawai et al., 2009b). We can see that good agreements between the predictions and the test data have been achieved, regardless of the fiber orientation and strain rate. These comparisons verify that the pressuremodified viscoplasticity model, which takes into account both the TFD and SFD effects, can successfully predict the tension–compression asymmetry, fiber orientation dependence, and rate-dependence in the nonlinear behavior of the unidirectional composite under off-axis loading conditions. The dashed lines in Fig. 7.7 (a)–(e) indicate the predictions using the prototype viscoplasticity model that considers only a TFD effect and corresponds to the particular case of assuming Λ = 1 in the pressure-modified viscoplasticity model. It is obvious that the off-axis compressive stress–strain curves (dashed lines) predicted by the prototype viscoplasticity model are less accurate than those (solid lines) by the pressure-modified viscoplasticity model that takes into account both the TFD and SFD effects. Note that the tensile stress–strain curves predicted by

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7.7  Comparison of the off-axis tensile and compressive stress–strain curves at different strain rates of 10%/min and 0.1%/min for a unidirectional T800H/epoxy laminate at 100°C (Kawai et al., 2009b): (a) θ = 10°, (b) θ = 15°, (c) θ = 30°, (d) θ = 45°, (e) θ = 90°. (Continued)

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(c)

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7.7  Continued.

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(e)

7.7  Continued.

the prototype viscoplasticity model agree with the solid lines, regardless of the fiber orientation and strain rate, since the two kinds of viscoplasticity models coincide with each other for the case of tensile loading conditions. Also note that no difference appears between the transverse compressive stress–strain curves for θ = 90°, because of no SFD effect involved. These comparisons reveal that accurate prediction of the viscoplastic deformation of unidirectional composites under off-axis tensile and compressive loading conditions requires properly taking into account the SFD effect as well as the TFD effect.

7.4

Modeling of transient creep softening due to stress variation

Creep deformation of a material that develops under constant stress is sensitive to the current internal state of the material, and thus the observation of creep response just after stress variation is helpful to identify the characteristic feature of hardening in the material. Figure 7.8 shows the off-axis creep and creep recovery behavior of a unidirectional T800H/3631 carbon/epoxy laminate at 100°C for the fiber orientation θ = 15° under a square waveform of intermittent loading (Kawai and Kamioka, 2005). From this figure, we can see that (1) the creep recovery rate

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7.8  Off-axis creep curve for a unidirectional T800H/epoxy laminate subjected to repeated loading and unloading at 100°C (Kawai and Kamioka, 2005).

just after the first stress removal is as high as the initial creep rate just after the first loading to the prescribed creep stress level, and (2) the subsequent creep rate just after reloading up to the same stress level as before is again as high as the initial creep rate just after the first loading. This observation suggests that material softening is temporarily induced by a stepped change in stress. Moreover, we can see that the amount of creep strain recovery ∆ε C1 x during the first stress-free period during the second. This indicates that material hardening has is larger than ∆ε C2 x also been involved in the combined creep and creep recovery behavior of the unidirectional CFRP laminate. It is considered that the extent of such temporal softening and overall hardening in PMCs depends on the history of loading in general. For developing a constitutive model that allows accurately predicting the time- and rate-dependent inelastic behavior of PMCs under variable loading conditions, therefore, we need to carefully take into account the effect of load history on the hardening and softening in PMCs. An attempt at modeling a transient softening behavior that appears under unloading followed by reversed loading has been made by Ishii et al. (2001). They demonstrated (1) the usefulness of considering an inelastic strain region that memorizes the prior maximum inelastic strain, and (2) the validity of assuming that the dynamic recovery rate involved by the evolution equation of a kinematic hardening variable is transiently enhanced while inelastic strain changes in the inelastic strain memory region. Kawai and coworkers (Kawai, 2004; Kawai et al.,

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2007; Kawai and Zhang, 2007) have also attempted to develop a similar yet more general constitutive model for describing the transient softening behavior due to stress variation. They applied the idea of an inelastic strain memory region that was proposed by Murakami and Ohno (1982) for isotropic materials to the modeling of the history-dependent transient creep softening behavior of anisotropic materials. In those attempts (Kawai, 2004; Kawai et al., 2007; Kawai and Zhang, 2007), they took advantage of the basic framework of the combined isotropic and kinematic hardening viscoplasticity model and elaborated the evolution equation of the kinematic hardening variable by means of a history-dependent anisotropic memory region that can move and expand in the viscoplastic strain space. This section discusses a sophisticated viscoplasticity model that allows accurately describing the creep behavior affected by momentary softening due to stress variation. The combined isotropic and kinematic hardening viscoplasticity model that has been described earlier in this chapter is used as a base for this attempt, and the evolution equation of the kinematic hardening variable is elaborated to enhance the accuracy of prediction of the transient creep softening due to stress variation. Age hardening is also taken into account in the modeling.

7.4.1 Combined isotropic–kinematic-age hardening model To develop a viscoplasticity model that can predict the transient softening due to stress variation, we will use the framework of the combined isotropic and kinematic hardening model for transversely isotropic media that has been described earlier in Section 7.2. Here, it is put in the following form: [7.46]



[7.47] [7.48] where K, m, H(a), L(a)p, L(a)r, and c are material constants. The additional coefficient c is a mixed hardening parameter that controls the proportion of isotropic hardening to total hardening, and it is assumed to have a value in the range 0 ≤ c ≤ 1. Pure kinematic hardening behavior is predicted by assuming c = 0, while pure isotropic hardening behavior is predicted by assuming c = 1. The evolution equation of the isotropic hardening variable that was defined by Eq. 7.28 is modified into the form given by Eq. 7.48 that is similar to the evolution equation of the kinematic hardening variable. The modified expressions of the evolution equations, Eq. 7.47 and 7.48, facilitate identification of the material constants involved. It is important to note that no distinction can be drawn between isotropic

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hardening and kinematic hardening as far as monotonic proportional loading is concerned. . The effective viscoplastic strain rate P is redefined as: [7.49] where U is the effective stress defined by Eq. 7.33. Equation 7.49 includes an additional scalar internal variable r* that represents the effect of physical aging. The physical aging of polymers is related to the change in free volume with time (Matsuoka, 1992), and it often results in material hardening that is called age hardening. While it should be related to the rate of the change in free volume (Kawai, 2001), the rate of age hardening variable is assumed here to take the following time hardening format: [7.50] where b(i) and Q(i) are material constants, and they are identified by fitting Eq. 7.50 to the yield strength versus aging time plot for a given material.

7.4.2 Modified kinematic hardening rule The creep strain recovery due to partial or complete unloading is of an anisotropic nature in itself. Therefore, it is reasonable to start with modeling the transient softening behavior of anisotropic materials due to stress variation by means of modification of the evolution equation of the kinematic hardening variable. The modified kinematic hardening rule discussed by Kawai (2004) and Kawai and Zhang (2007) is characterized by an accelerated evolution of kinematic hardening while viscoplastic strain varies in a certain history-dependent region of the viscoplastic strain space. It is assumed that the inelastic strain memory region by which the transient softening behavior of the material is controlled can be described as: [7.51] where * is an effective inelastic strain tensor defined as The variables π and κ stand for the center location and size of the inelastic strain memory region, respectively. The evolution equations of these additional internal variables are described by means of the following formulae:

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[7.53] with [7.54] The coefficient λ takes a value in the interval 0 < λ ≤ 1, and it is determined so as to reproduce the creep behavior under reversed loading. The transient softening after load variation is assumed to occur while inelastic strain changes inside the inelastic strain memory region. For manipulation of the inelastic strain memory region, it is convenient to introduce a scalar index, called a softening index, defined as: [7.55] Another softening promotion parameter associated with the softening index is further defined as: [7.56] where qmax = κ/λ, and it represents the softening index associated with the state just on the surface of the inelastic strain memory region. Note that 0 ≤ χ ≤ 1. It is more clearly indicated by the softening promotion parameter that material softening occurs while the point indicating the current state of inelastic strain moves inside the inelastic strain memory region. By using the softening promotion parameter, the hardening coefficients H(a) involved by the evolution equation of the kinematic hardening variable is modified as: [7.57] Note that in the modified kinematic hardening viscoplasticity model the modified hardening coefficients Hˆ (a) are used in place of H(a).

7.4.3 Comparison with experimental results Figures 7.9 (a) and (b) show comparisons between the predicted and observed off-axis creep and creep recovery behaviors for the fiber orientations θ = 15 and 30°, respectively. Note that the simulation using the modified kinematic hardening model was performed for the entire loading path which was composed of the

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7.9  Comparison of the predicted and observed off-axis creep curves for a unidirectional T800H/epoxy laminate subjected to repeated loading and unloading at 100°C: (a) θ = 15°, (b) θ = 30°. KH, kinematic hardening.

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instantaneous loading up to a constant creep stress level, the dwell of stress at a constant level, the complete unloading of the creep stress, and the repetition of these loading and unloading steps. These comparisons demonstrate that the creep and creep recovery behavior of the composite under off-axis repeated loading and unloading, which is a complicated load history, can adequately be predicted by the modified kinematic hardening model. The dashed lines in Fig. 7.9 (a) and (b) indicate the predictions using the original kinematic hardening model that does not consider the temporal softening due to stress variation; i.e. χ ≡ 0. The comparisons of these predictions with the experimental results reveal that the accuracy of prediction of the creep recovery behavior after complete unloading can be improved by the modified kinematic hardening controlled by the memory of prior viscoplastic strain. Also for the other off-axis fiber orientations, it has been confirmed that the creep recovery behavior after complete unloading can adequately be predicted using the modified kinematic hardening model.

7.5

Conclusions

For the reliable design-by-analysis of composite structures, it is an essential prerequisite to establish an engineering constitutive model that can accurately predict the time- and rate-dependent behavior of the composites employed. This chapter has focused on a phenomenological anisotropic theory of viscoplasticity for orthotropic fiber composites, and presented a framework for a combined isotropic and kinematic hardening model along with its generalized forms that allow describing different flow stress levels in tension and compression and transient creep softening due to stress variation. The assessment of these attempts on the basis of the experimental results on unidirectional polymer matrix composites has also been given. Macromechanical modeling of the overall inelastic behavior of composites requires introducing the internal variables that can represent the dissipative processes taking place in the composites. The inelastic behavior of composites cannot always satisfactorily be described using a single scalar internal variable. The problem arises when composites are subjected to external load that is varied and non-proportional. This suggests that not only scalar but also tensor internal variables need to be introduced for developing a more general viscoplasticity model. The chapter has addressed a three-dimensional phenomenological viscoplasticity model for homogenized unidirectional composites that takes into account combined isotropic and kinematic hardening. It is formulated by assuming two kinds of internal variables, a scalar and a second-rank tensor, along with a transversely isotropic tensor of the fourth rank that represents the initial anisotropy of a given unidirectional composite. A pure isotropic hardening model which can be reduced as a particular case from the combined isotropic and kinematic hardening model is shown to be similar in structure to existing macromechanics models for unidirectional

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composites. It is demonstrated that the isotropic hardening viscoplasticity model can adequately predict not only the rate-dependent tensile stress–strain behavior but also the time- and stress-dependent tensile creep behavior of a unidirectional carbon/ epoxy laminate for any off-axis fiber orientations. This also suggests a conclusion that the combined isotropic and kinematic hardening model and the pure knematic hardening model as a particular case are also applicable as far as monotonic proportional loading conditions are concerned. The inelastic constitutive model for composites should be furnished with the capability to predict the time- and rate-dependent inelastic behavior not only under tensile loading conditions but also under compressive loading conditions, since the composite structures to be analyzed with the inelastic constitutive model are locally subjected to tensile and compressive stresses in general. This requirement should be taken into account carefully, since most practical orthotropic fiber composites exhibit different inelastic flow behaviors under tensile and compressive loading conditions. The chapter, therefore, has developed into the discussion on a viscoplasticity model that can take account of the difference between the ratedependent flow behaviors of unidirectional composites under off-axis tensile and compressive loading conditions. A modified viscoplasticity model that can cope with the tension–compression asymmetry in the initial anisotropy of unidirectional composites has successfully been formulated. It can be reduced to the form that coincides with a particular case of the combined isotropic and kinematic hardening model mentioned above. This feature facilitates identification of material constants and comparison with other existing phenomenological constitutive theories. The comparison between the predicted and experimental results reveals that consideration of both the transverse and shear flow differential effects is crucial for accurate prediction of the different viscoplastic behaviors of unidirectional composites under off-axis tensile and compressive loading conditions. The external load withstood by composite structures varies with time during their service, usually in a very complicated manner. Therefore, the inelastic constitutive model that is applied to the local analysis of composite structures should be furnished with the capability to accurately predict the inelastic behavior of composites under not only monotonic loading conditions but also variable loading conditions. The essence of this requirement is a guarantee of accurately predicting the history dependence of the inelastic behavior of composites. The chapter has further discussed the subject, and formulated a modified kinematic hardening model for unidirectional composites that can accurately predict the creep behavior affected by temporal softening due to stress variation on the basis of the constitutive framework that takes into account combined isotropic and kinematic hardening. The method for elaborating the evolution equation of the kinematic hardening variable is characterized by an accelerated evolution of the kinematic hardening variable while viscoplastic strain changes in a certain memory range, and it is demonstrated to be useful for predicting the off-axis creep behavior of unidirectional composites under variable loading conditions.

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Future trends

Further attempts to develop a more sophisticated constitutive model for composites require consideration of the effect of damage that grows with inelastic deformation. Ladeveze and Dantec (1992) have developed an isotropic hardening plasticity model for unidirectional composites that considers the anisotropic reduction in stiffness due to damage. In the plasticity-damage modeling, the nonlinearity in the stress–strain response of a unidirectional composite under off-axis loading conditions is ascribed to the reduction in stiffness due to the nonlinear growth of damage as well as the nonlinear hardening of the composite. The Ladevese-Dantec plasticity-damage model was favorably applied, for example, to the progressive failure analysis of composite laminates with fiber rotation by Herakovich et al. (2000) and to the modeling of damage under cyclic loading by Payan and Hochard (2002). Kawai (1997) has also developed a viscoplasticity-damage model that assumes the combined isotropic and kinematic hardening coupled with isotropic damage. It has then been developed into a model that can account for the history dependence of the growth of damage (Kawai, 1998) and a model that can describe the different damage-affected behaviors under tension and compression (Kawai, 2002). Accurate prediction of the ultimate failure in a given composite laminate that follows development of inelastic deformation and accumulation of damage is a goal of design-by-analysis. In view of this, an attempt has been made to predict the creep rupture lives of unidirectional composites under off-axis loading conditions by means of a formula that comes from a simple viscoplasticity-damage model (Kawai et al., 2006). While it is more computationally efficient when applied to the analysis of composite structures, the macromechanical approach to constitutive modeling of the inelastic behavior of composites smears the details of physical events that occur in the composites. Therefore, we always need to take care of the range of applications in which the overall behavior of a given composite can adequately be modeled by the macromechanical approach. An alternative micromechanical approach allows more detailed consideration of the dissipation processes taking place in composites, since in this approach they are treated as the mixtures of fibers and matrices, and the elastic-inelastic-damage behaviors of the constituents are averaged by means of either analytical or numerical homogenization technique. An analytical homogenization technique allows us to derive a set of micromechanicsbased constitutive equations for the overall inelastic behavior of composites. Many attempts can be found in literature. For example, the mean field theory developed by Mori and Tanaka (1973) has successfully been applied to predicting the creep behavior (Chun and Daniel, 1997) and the plastic behavior (Tszeng, 1994) of unidirectional metal matrix composites. Kawai et al. (2001) have derived a set of combined isotropic and kinematic hardening viscoplasticity model for a unidirectional polymer matrix composite by means of the method of cell that was proposed by Aboudi (1991). The same analytical approach has also been applied to

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a unidirectional composite that consists of shape memory alloy fibers and an elastic-viscoplastic matrix material (Kawai, 2000). A numerical homogenization technique, which has comprehensively been described in Chapter 4 of this volume, allows more detailed consideration of the geometry of microstructure in composites along with suitable boundary conditions and thus more accurate analysis of the local and global behavior of composites, although it is numerically more intensive. In the micromechanical approach, whichever homogenization technique we adopt, we need a constitutive model for describing the inelastic behavior of matrix materials. The inelastic constitutive models developed from a macromechanical point of view are often applied to the matrix materials embedded in composites. Therefore, development of a sophisticated macroscopic inelastic constitutive model for homogenized media serves not only to predict the overall inelastic behavior of composites but also to establish the micromechanical inelastic constitutive model for composites.

7.7

Acknowledgements

The research on which this chapter is based has been carried out with my students at the University of Tsukuba, mainly with the financial support of the University of Tsukuba and the Ministry of Education, Culture, Sports, Science and Technology of Japan. The author is grateful to all the members in the lab who have contributed to the work. This chapter has been written in the course of the research work supported in part by the Ministry of Education, Culture, Sports, Science and Technology of Japan under a Grant-in-Aid for Scientific Research (No. 20360050).

7.8

References

Aboudi J (1991), Mechanics of composite materials, Elsevier Science Publishers. Armstrong P J and Frederick C O (1966), ‘A mathematical representation of the multiaxial Bauschinger effect’, CEGB Report RD/B/N731, Berkeley Nuclear Laboratories. Barrett C R, Nix W D and Tetelman A S (1973), The principles of engineering materials, Prentice-Hall, USA. Beckwith S W (1980), ‘Creep evaluation of a glass/epoxy composite’, SAMPE Quarterly, 11, 8–15. Bekhet N E, Barton D C and Craggs G (1994), ‘Biaxial yielding behavior of highly oriented polypropylene’, J Mater Sci, 29, 4953–4961. Bodner S R and Partom Y (1975), ‘Constitutive equations for elastic-viscoplastic strainhardening materials’, ASME J Appl Mech, 42, 385–389. Boehler J P (ed) (1987), Applications of tensor functions in solid mechanics, CISM No. 292, Springer-Verlag. Brinson H F (1999), ‘Matrix dominated time dependent failure predictions in polymer matrix composites’, Compos Struct, 47, 445–456. Caddell R M, Raghava R S and Atkins A G (1973), ‘A yield criterion for anisotropic and pressure dependent solids such as oriented polymers’, J Mater Sci, 8, 1641–1646.

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Matsuoka S (1992), Relaxation phenomena in polymers, Carl Hanser Verlag. Maugin G A (1999), The thermomechanics of nonlinear irreversible behaviors – an introduction, World Scientific Publishing Co. Pte. Ltd., printed in Singapore. Mori T and Tanaka K (1973), ‘Average stress in matrix and average energy of materials with misfitting inclusions’, Acta Metallurgica et Materialia, 21, 571–574. Murakami S and Ohno N (1982), ‘A constitutive equation of creep based on the concept of a creep-hardening surface’, Int J Solids Struct, 18, 597–609. Nielsen L E (1975), Mechanical properties of polymers and composites, Marcel Dekker, Inc., New York, USA. Nouailhas D and Freed A D (1992), ‘A viscoplastic theory for anisotropic materials’, ASME J Eng Mater Technol, 114, 97–104. Payan J and Hochard C (2002), ‘Damage modeling of laminated carbon/epoxy composites under static and fatigue loadings’, Int J Fatigue, 24, 299–306. Perzyna P (1966), ‘Fundamental problems in viscoplasticity’, in Kuerti G (ed), Advances in Applied Mechanics, 9, pp. 243–377, Academic Press. Povirk G L, Stout M G, Bourke M, Goldstone J A, Lawson A C, Lovato M, Macewen S R, Nutt S R and Needleman A (1992), ‘Thermally and mechanically induced residual strains in Al-SiC composites’, Acta Metallurgica et Materia, 40, 2391–2412. Robinson D N (1984), ‘Constitutive relationships for anisotropic high-temperature alloys’, Nuclear Engineering and Design, 83, 389–396. Schapery R A (1974), ‘Viscoelastic behavior and analysis of composite materials’, Mech Compos Mater, pp. 85–168, Academic Press. Shin E S and Pae K D (1992), Effects of hydrostatic pressure on the torsional shear behavior of graphite/epoxy composites, J Compos Mater, 26(4), 462–485. Stassi-D’Alia F (1969), ‘Limiting conditions of yielding for anisotropic materials’, Meccanica, 4, 349–363. Struik L C E (1978), Physical aging in amorphous polymers and other materials, Elsevier, Amsterdam. Sullivan J L (1990), ‘Creep and physical aging of composites’, Compos Sci Technol, 39, 207–232. Sun C T and Chen J L (1989), ‘A simple flow rule for characterizing nonlinear behavior of fiber composites’, J Compos Mater, 23, 1009–1020. Takeuchi F, Kawai M, Zhang J Q and Matsuda T (2008), ‘Rate-dependence of off-axis tensile behavior of cross-ply CFRP laminates at elevated temperature and its simulation’, Adv Compos Mater, 17, 57–73. Taya M, Lulay K E, Wakashima K and Lloyd D J (1990), ‘Bauschinger effect in particulate SiC-6061 aluminum composites’, Mater Sci Engng, A124, 103–111. Theocaris P S (1986), ‘A general yield criterion for engineering materials, depending on void growth’, Meccanica, 21, 97–105. Tszeng T C (1994), ‘Micromechanics characterization of unidirectional composites during multiaxial plastic deformation’, J Compos Mater, 28(9), 800–820. Tuttle M E and Brinson H F (1986), ‘Prediction of the long-term creep compliance of general composites’, Exp Mech, 26(1), 89–102. Tuttle M E, Pasricha A, and Emery A F (1995), ‘The nonlinear viscoelastic-viscoplastic behavior of IM7/5260 composites’, J Compos Mater, 29(15), 2025–2046. Tuttle M E, Semeliss M and Wong R (1992), ‘The elastic and yield behavior of polyethylene tube subjected to biaxial loadings’, Exp Mech, 32 (1), 1–10. Vlack L H V (1970), Materials science for engineers, Addison-Wesley Publishing Company, USA.

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8 Creep analysis of polymer matrix composites using viscoplastic models E. Kontou, National Technical University of Athens, Greece Abstract: This chapter discusses creep behavior of polymer matrix composites in the framework of viscoplasticity. The chapter first reviews the main aspects of viscoplastic creep modeling, the state of the art, with emphasis on the tensile creep response of a glass-fiber polymer composite, considering it as a thermally activated rate process. The viscoplastic part of strain is calculated separately assuming that viscoplasticity arises mainly from the polymeric matrix. Both tensile creep and monotonic loading under the same theoretical analysis in finite deformation regime are also presented. Kinematic and constitutive descriptions are presented for a three-dimensional problem. Recent developments such as nanocomposites and future trends in creep analysis are also discussed. Key words: viscoplasticity, creep, thermal activation, composites, modeling.

8.1

Introduction

Polymeric matrix composites can be tailored to meet the design requirements, which may include low density, high strength, high stiffness, corrosion and chemical resistance. As a consequence, these materials are widely used in aerospace structures, automotive parts, marine structures, etc. Moreover, learning about their properties over a range of temperature and strain rates has been an interesting task through the last few decades. Linear viscoelastic behavior can offer a frame of describing these properties (Eyring and Halsey 1946, Ward and Handley 1993, Guedes et al. 2004). However, polymer matrix composites generally exhibit non-linear and rate-dependent behavior. This non-linearity can be treated as a kinematic or physical effect. Due to the small strain these materials usually attain, physical non-linearity is mainly taken into account, which is treated as a plastic response. Therefore, as far as fiber-reinforced composites are concerned, attempts to characterize their mechanical response in the frame of linear elasticity are not adequate. A lot of theories were developed to formulate this non-linear stress– strain relationship (Hahn and Tsai 1973, Sun et al. 1974, Dvorak and Balei-El-Din 1982, Sun and Chen 1989, Van Paepegem et al. 2006). Apart from the need to consider the general non-linear viscoelastic-viscoplastic response of polymeric composites, their creep behavior has always been a topic of particular interest, due to their use in structural applications, at both ambient and elevated temperatures. Therefore, in order to obtain a correct design of structural elements, it is important to predict the long-term mechanical behavior of these materials. 273 © Woodhead Publishing Limited, 2011

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During the last few decades, several studies have been undertaken to improve the mechanical and thermal properties, as well as flame resistance, barrier properties and electrical conductivity of polymers and conventional type polymer composites. Lately, this property upgrade has been achieved by dispersing fillers of various aspect ratios on nanoscale (Fu et al. 1992, Novak 1993, Giannelis 1996, and Zhou et al. 2007, Wilkinson et al. 2006, Chow et al. 2007, Karger-Kocsis 2008, Siengchin and Karger-Kocsis 2009). The improved dispersion of nanoparticles in a polymer matrix, as well as the interfacial interaction between nanoparticles and matrix, plays a crucial role for simultaneous enhancement of a variety of properties at low contents, not met by microcomposites. In spite of the fact that the mechanical properties of nanocomposites have been widely investigated in recent years (Zheng and Ning 2003, Ash et al. 2002, Someya and Shibata 2004), there is still a lack of information regarding their time-dependent response. Referring to creep behavior, it has been shown that polymer nanocomposites exhibit substantial creep resistance due to the restricted molecular motion by the presence of nanoparticles, but their response is still nonlinear (Pegoretti et al. 2004, Zhang et al. 2004, Shaito et al. 2006). Non-linear viscoelastic or viscoplastic behavior of polymer composites can be manifested in terms of monotonic loading, creep under constant load, stress relaxation under constant deformation, time-dependent creep rupture, and timedependent strain recovery after load removal. These effects have been extensively studied by Schapery (1997) and Drozdov (1998). In the case of a small strain regime, time-dependent response of the composite can be described by linear and non-linear viscoelasticity. However, at high strain due to high stresses, yielding or alternatively time-dependent inelastic response takes place, which needs viscoplasticity models to be introduced. According to the well-known Schapery’s formulation, the non-linear viscoelastic response of any material is controlled by four stress and temperature dependent parameters, which reflect the deviation from linear viscoelasticity. In a series of works by Zaoutsos et al. (1998) and Papanicolaou et al. (1999), a new methodology for the separate estimation of the viscoelastic parameters was developed, and a further development of this methodology has been performed. For this trend, Schapery’s non-linear viscoelastic and viscoplastic model was used to describe the time-dependent response of unidirectional glass fiber reinforced epoxy matrix composites to load in the work by Megnis and Varna (2003). The non-linear time and stress dependence of viscoplastic strain was determined experimentally in off-axis creep tests as the difference between measured creep strain and predicted linear viscoelastic response. In a recent work by Vinet and Gamby (2008), a model is developed that is applicable at the scale of a quasi-isotropic visco-elastoplastic laminate during compression creep tests. The non-linear asymmetric/anisotropic viscoplastic response of fiber composites has been analytically studied in Weeks and Sun (1998), where two rate-dependent models are introduced to model the corresponding response of a fiber-reinforced

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thermoplastic matrix. Both models were developed by using a one-parameter plastic potential function to describe the non-linear behavior. In the work of Thiruppukuzhi and Sun (2001), two viscoplasticity models were examined to characterize the rate-dependent non-linear behavior of two polymeric composite material systems, namely unidirectional fiber glass and woven E-glass. Model predictions were proved to describe well the stress–strain experimental results of off-axis composite specimens at various strain rates. In a work by Kawai et al. (2007), creep and creep-recovery tests were performed on plain coupon specimens of carbon/epoxy composite laminate at various fiber orientations. Creep simulations were attempted using the modified kinematic hardening model for homogenized anisotropic inelastic composites, in which an accelerated change in kinematic hardening over a certain range of viscoplastic strain is considered. It has been shown that the proposed model can adequately describe the off-axis creep and creep recovery behavior of the unidirectional composite system. In the frame of anisotropic viscoplasticity, a model is also introduced in the work by Saleeb et al. (2003), where typical experimental data (strain-controlled tensile) and constantstress creep tests are simulated by applying the developed algorithms. Al-Haik et al. (2001), based on an elastic-viscoplastic constitutive model proposed by Gates (1991, 1992, 1993), studied the viscoplasticity of carbon fiber/polymer composite, using load relaxation and creep data. The theoretical creep curves, generated through the material constants found from stress relaxation tests, showed a close agreement with experimental data. Therefore, the model can be a design tool for durability prediction of composites for infrastructure industry applications. From another point of view, it is interesting to mention a viscoplasticity theory based on the concept of overstress in the work of Dusunceli and Colak (2008). The small strain, isotropic, viscoplasticity theory is modified so that the influences of crystallinity content on mechanical properties of semicrystalline properties are included. Considering the semicrystalline polymers somewhat as a composite, since they consist of amorphous and crystalline phases, with different resistance to deformation, this theory can be successfully extended to isotropic polymer composites. Viscoplasticity models were also introduced by Gates and Sun (1991), and Colak (2005). A plastic model based on a one-parameter potential function, by Sun and Chen (1991), was able to describe the orthotropic rate-independent behavior of polymeric composites, while another viscoplastic model, of the same trend, was proposed by Yoon and Sun (1991), for thermoplastic matrix composites, applying the concept of effective stress and effective inelastic strain. In these models, the unidirectional fiber-reinforced composites were considered as homogeneous anisotropic continua, and a single internal variable was assumed to formulate the rate effect during deformation of off-axis specimens. Tension and stress relaxation experiments were used for the estimation of model parameters, and then the prediction of stress–strain, creep, relaxation and cyclic loading was possible at various rates and temperatures. However, as mentioned by Al-Haik

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et al. (2006), these viscoplastic models fail to predict the creep response under high thermomechanical conditions, when the material parameters are derived from stress-relaxation tests. This specific issue, of describing all aspects of inelastic deformation behavior of polymeric systems in terms of a unified model, has been an interesting topic (Krempl 1998, Hasan and Boyce 1995, Spathis and Kontou 2001). In the work by Hasan and Boyce (1995), a model was presented that can describe both types of experimental results, i.e. stress–strain at compression at various temperatures, as well as creep tests at various stress levels and temperatures. The distributed nature of the microstructural state and the thermally activated evolution of the glassy state are the main assumptions of their constitutive model. Later, to the same trend, Spathis and Kontou (2001) introduced a functional form of the rate of plastic deformation in glassy polymers, based on a mechanism of plastic deformation introduced by Oleinik et al. (1993, 1995). As was shown in this work, the deformation mechanism is the same under a constant crosshead speed experiment, as well as in a creep test, with varying stress levels over a wide range. The model was proved to be capable of predicting the nonlinear viscoelastic-viscoplastic response at both types of deformation. Information on long-term deformation and strength are normally obtained by extrapolation of short-term test data, obtained under accelerated testing conditions such as higher temperature, stress and humidity to service conditions by using a prediction model (Raghavan 1997). The accuracy of prediction depends on both the model’s accuracy and its validity in extrapolation. Most of the creep models that have been introduced to model the creep of polymer composites are mechanical analogs, hereditary integrals, Schapery’s model (1997), Findley’s approach (1949), Findley and Kholsla (1955) and thermal activation theory (Eyring and Halsey 1946, Krausz and Eyring 1975). The time-temperature superposition principle, used in most of the above models, assumes that the compliance function retains its shape with respect to the logarithmic time scale. This may not be the case for many polymeric matrices of a variety of polymer composites. Apart from this shortcoming, it may be summarized that viscoplastic creep modeling in polymer composites depends on a series of factors, such as kinematic description (small and finite strain), constitutive analysis, procedure of homogeneity in the case of anisotropy, kinematic hardening rules and a micromechanics model for linking the externally imposed strain rate with the intrinsic material response. All these aspects of analysis, with the requirement of a unified description of viscoplastic deformation (creep, stress relaxation, monotonic-cycling loading), constitute an issue of high interest.

8.2

Viscoplastic creep modeling for polymer composites

One of the most common assumptions in a material’s viscoelasticity/viscoplasticity is that strain accumulation in a creep procedure follows a thermally activated

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mechanism. Thermal activation theory assumes that the macroscopic creep rate is related to the product of the number of flow units, the average strain increment per jump over an energy barrier and the probability that each unit would undergo such a jump (Conrad 1964, Krausz and Eyring 1975). However, this approach was proven to be successful in a limited strain-rate/time scale, due to the assumption that a constant number of flow units were available for transition during creep. In the work by Raghavan and Meshii (1994), a spectrum of activation processes has been used to model a creep procedure in a wider time range. In Raghavan and Meshii (1997), the creep of unidirectional carbon fiber reinforced polymer composites, as well as the epoxy matrix, both characterized as thermo-rheologically complex, was successfully modeled in a wide range of temperature and strain. The creep procedure of these composites in a temperature range of 295–433 K and at various stress levels up to 80% of the material’s tensile strength could be described. Good qualitative agreement regarding the shape of viscoelastic functions, as well as satisfactory predictive model capability has been verified. It was found that this creep mechanism is not altered by the presence of fibers; however, creep rate and magnitude is substantially affected by increasing the resistance to cooperative chain segmental mobility of the polymer chain segments and stiffening the matrix. In the following section, 8.2.1, and referring to the small strain regime, it will be shown that thermal activation theory can be a successful tool for the description of viscoplastic response during creep for polymeric composites when it is combined with a proper mechanism for calculating the rate of plastic deformation accumulating with time. Finite deformation analysis, on the other hand, is needed for many engineering applications of polymers and polymeric composites. A lot of works have dealt with finite elastic-plastic or viscoplastic deformation of polymers, but little attention has been paid to polymeric composites, in spite of the fact that they can attain high strain values, especially at high temperatures. In Section 8.2.2, the basic principles regarding kinematic formulation and constitutive modeling in the case of finite plastic deformation are considered. The implementation of this analysis on monotonic loading and creep behavior of a fiber composite is presented.

8.2.1 Small strain framework: constitutive analysis In this section, the viscoplastic creep behavior of polymer composites is studied at various temperatures and several stress levels. It is assumed, as mentioned earlier, that strain accumulation in a creep procedure follows a thermally activated mechanism. The plastic part of the strain is separately calculated following a specific function, which is based on the distributed nature of microstructural state (Spathis and Kontou 2001, Kontou 2005). The study is made in the frame of small strain, so the additive decomposition of strain is taken into account.

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Generally, assuming uniaxial loading in the small strain regime, the total strain ε may be represented in terms of the following components:

ε = εel + εR + εp

[8.1]

where εel is the instantaneous part of strain, εp is the plastic part of strain that develops during creep procedure and εR is the recoverable strain. Following Mindel and Brown (1973), creep results can be represented by an equation of the general form: . ε = f1(T) f2 (σ/ T) f3 (ε) [8.2] where f1(T), f2(σ/T), f3(ε) are separate functions of the variables temperature T, stress σ and strain ε. According to the data of Sherby and Dorn (1956), f1(T) has the form derived for a thermally activated process: f1(T) = const × exp [–Q/kT]

[8.3]

where k is Boltzmann’s constant, Q is the activation energy, and f2(σ/T) is expressed as: f2 (σ/T) = exp[συp /3kT] exp[συs /kT]

[8.4]

with υp, υs being the pressure and shear activation volume, respectively. Ward (1971) has shown that υp is much smaller than υs, so that the measured activation volume is mostly υs. The functional form of f3(ε) is as follows: f3 (ε) = α εR

[8.5]

where α is a constant and εR is the recoverable strain. Following Eq. 8.2–8.5, creep strain rate can be written as: . ε = A exp[συp/3kT] exp[(σ – σint)υs/kT]

[8.6]

where A is a constant and σint is given by:

σint = K2 εR

[8.7]

where K2 is a constant proportional to the temperature. The quantity σint has the character of an internal stress, which opposes the applied stress. Since σint increases with strain, then the effective stress σ - σint decreases with strain and therefore creep rate reaches a minimum value. In a quite similar way, in the work by Ma and Tjong (2001), the stress dependence of creep rate has been formulated as follows:

[8.8]

where A′ is a constant, σ0 is the threshold stress, denoting that the observed deformation is not driven by the applied stress σ but rather by an effective stress

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σ – σ0. G is the shear modulus, n is the stress exponent, Q is the activation energy, R the gas constant and T the temperature. Regarding plastic strain εp, it is manifested through the non-linear creep behavior exhibited by the materials. This non-linearity is accelerated when creep tests are performed at higher temperatures. The development and growth of εp was assumed to follow a mechanism, introduced by Oleinik et al. (1993, 1995), and formulated by Spathis and Kontou (2001), which will be presented below. As in the case of monotonic loading during creep, strain is accumulated around specific regions. This accumulation takes place around a large number of voids or defects, randomly distributed into the volume of the deformed material. The total evolved deformation will consequently be distributed inhomogeneously around specific regions, in a way related with the special features of each anomaly. When the distributed elastic energy, associated with strain, around each region reaches a critical value, a non-reversible transition takes place, which manages the emergence of plastic deformations. If we accept that each one of these transitions proceeds at a certain rate, then the macroscopic plastic deformations will come out with a rate proportional to the number of simultaneously appeared localized transformations. Ongoing, it is assumed that the necessary strain accumulated around the i-region randomly selected from the statistical ensample obeys a normal Gaussian distribution determined from a mean equivalent strain µ˜ , and a standard deviation ˜s . Then, the distribution density function having the strain εi as a random variable will be given by:

[8.9]

The fraction of such processes that have enough activation energy to attain a new non-reversible state is given by the probability:

[8.10]

where the lower limit (– ∞) of integration is substituted by zero, because the standard deviation is considered to be.very small. Making the further assumption that the rate of plastic deformation Γp is proportional to the fraction of plastic transformations that have achieved . a non-reversible state, and that this transition takes place with an average rate k for every plastic transformation, then we have:

[8.11]

. The value of k can be estimated, assuming that at the onset of plastic deformation, that is, at the moment where the strain ε is equal to the mean value µ˜ , the

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. rate of plastic deformation Γ py will become equal to the evolved creep strain . rate ε :

[8.12]

then:

[8.13]

The functional form of the rate of plastic deformation, presented above, was developed for the plastic deformation of polymeric materials. This concept is now extended in the case of plastic creep strain of fiber-reinforced polymer composites, denoting this way that plastic behavior exhibited by those materials arises mainly from the viscoplastic response of the polymeric matrix. The set of equations 8.1, 8.6, 8.7 and 8.13 can now be combined for the calculation of the total creep strain ε. Elastic strain εel can be obtained from the experimental data, while the plastic strain εp is calculated via a numerical . integration of Γp. In Fig. 8.1 to 8.5 the experimentally obtained tensile creep compliance of an epoxy matrix, and the corresponding off-axis epoxy-fiber glass composites, is plotted, for various stress levels at two different temperatures (Kontou 2005). The simulated values are also presented for comparison.

8.1  Tensile creep compliance of the epoxy matrix at temperatures of 333 and 353 K and at a stress level of 2 MPa. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou (2005).

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8.2  Total creep compliance of the 15° off-axis composite, at two different temperatures 333 and 353 K at a stress level of 8.6 MPa. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou (2005).

8.3  Total creep compliance of the 30° off-axis composite, at two different temperatures 333 and 353 K at a stress level of 3.68 MPa. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou (2005).

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8.4  Total creep compliance of the 60° off-axis composite, at two different temperatures 333 and 353 K at a stress level of 2 MPa. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou (2005).

8.5  Total compliance of the 90° off-axis composite, at two different temperatures 333 and 353 K at a stress level of 2.2 MPa. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou (2005).

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Comparing the creep curves of the epoxy matrix with those of the composites, it is observed that the magnitude of strain and the way it is accelerated due to the rise in temperature in the case of the epoxy is much higher. The corresponding model parameters are shown in Table 8.1. From Table 8.1 it can be observed that the pressure activation volume remains constant for all material types, while the shear activation volume appears to have a slight decrement with increasing temperature. The quite similar values for the shear activation volume for pure matrix and the 90° off-axis specimen indicate that this response arises from the viscoplastic behavior of epoxy matrix. The mean value µ˜ of the probability density function expresses an upper strain limit, which is related with a saturated material state evolved during creep procedure. The creep compliance tensor of the unidirectional fiber composites, if the tensile creep of the off-axis composites is treated as a two-dimensional problem, is given by:



[8.14]

The experimental creep results obtained from specimens oriented at 90° and 15° to the loading axis were analyzed (Kontou 2005) with the creep model to give the analytical form of compliances S22(t) and Sθ(t). The compliance of 0° samples, i.e. S11 was taken to be a constant, equal to 0.0002 MPa–1, since no creep strain emerges for those samples, even at higher temperatures. Then the compliance S66(t) could be calculated after Hyer (1998) according to the following transformation equation:

[8.15]

Table 8.1  Creep model parameter values (reproduced from Kontou 2005) Sample

Temperature (K)

υp/3kT υs/kT A (s–1) (MPa–1) (MPa–1)

Pure resin Off-axis [15°] Off-axis [30°] Off-axis [60°] Off-axis [90°]

333 353 333 353 333 353 333 353 333 353

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

1.0 0.8 0.3 0.2 0.3 0.2 0.35 0.1 0.85 0.80

1.8 10–4 1.8 10–4 2 10–6 2 10–5 2 10–6 3 10–5 2 10–6 3 10–5 2 10–6 3 10–5

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K2

εel

µ˜

700 500 800 700 1000 800 1500 1000 1500 1000

0.0055 0.0065 0.001 0.002 0.001 0.0065 0.0015 0.0022 0.002 0.003

0.015 0.016 0.02 0.025 0.02 0.01 0.019 0.025 0.02 0.025

(MPa)

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8.6  Total creep compliance of the 60° off-axis composite, at 333 K. Thick lines: experimental data. Thin lines: model predicted results. Reproduced from Kontou (2005).

where ν12 is the Poisson’s ratio, which was considered to be constant with time, equal to 0.31. However, it must be mentioned that according to the fundamental theory of viscoelasticity (Christensen 1981), Poisson’s ratios are described as time-dependent functions. Especially during polymer matrix processing, where thermal expansion takes place, the assumption of time-independent Poisson’s ratios is unacceptable, as it is pointed out by Hilton (1998, 2001). Such an assumption may lead to a non-correct characterization of real viscoelastic materials, regarding the true behavior of anisotropic moduli or compliances and relaxation/or creep functions. Only if anisotropic or isotropic viscoelastic moduli and relaxation or creep functions are characterized by identical time functions, the corresponding Poisson’s ratios must be time-independent. On the other hand, experimental measurements of Poisson’s ratios of fiber-reinforced composites, under structural conditions, have indicated that for these materials Poisson’s ratios do not vary significantly with loading frequency, as it is reported by Mead and Ioannides (1991) and Melo and Radford (2003). Comparison between experimental results and simulated ones for laminates, based on time-independent Poisson’s ratios, has shown that this assumption does not noticeably affect the predictions. The model validity can be tested through the calculation of the creep compliance of an off-axis specimen with fibers oriented at an arbitrary angle θ to the loading axis, applying equations 8.14 and 8.15. The results obtained by this equation (Kontou 2005) are presented in Fig. 8.6 for 60° off-axis at 333 K, in comparison with the corresponding experimental data. A satisfactory agreement is observed for this off-axis sample. The same procedure has been followed to check the

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8.7  Total creep compliance of the 30° off-axis composite, at 353 K. Thick lines: experimental data. Thin lines: model predicted results. Reproduced from Kontou (2005).

model validity at 353 K. The corresponding results are plotted in Fig. 8.7 for a 30° off-axis specimen. In this case, a deviation of the order of 23% is obtained, while the shape of the calculated creep compliance is retained parallel to the experimental time scale examined.

8.2.2 Finite strain viscoplasticity The inelastic material behavior under finite strain and rotation has been the subject of many works, including Onat (1982), Loret (1983), Dafalias (1984, 1985), and Anand (1985). The kinematic description of the deformed body is based on the deformation gradient F = ∇xX, where X is the reference position of a material point, and x the current position. The multiplicative decomposition of the deformation gradient tensor F into elastic and plastic components was first proposed by Lee (1969), who introduced the concept of a relaxed intermediate configuration, represented by the plastic deformation gradient. Later, the physical significance of this intermediate configuration was debated, as this is not uniquely determined, since an arbitrary rigid rotation can be superimposed on it and leave it unstressed. The basic principles of plasticity theory for finite deformation are systematically presented by Khan and Huang (1995). Hill and Rice (1972) and Asaro (1983) formulated the case of single crystal plasticity, while Loret (1983) and Asaro (1983) have extended these ideas for the single crystal to polycrystalline materials, utilizing Mandel’s approach (1971) that introduced a triad of director vectors to monitor material orientation. The difficulty

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of assigning these vectors is overcome in the case of a single crystal, whereas the vectors are naturally defined by the lattice. Haupt and Kersten (2003) and Tsakmakis (2004) attempted to represent spatial orientation of the crystal lattice by the same internal variable, which has been introduced into the strain energy function. Through this analysis, it was possible to keep the formulation compatible with the postulate of full invariance. Anisotropic kinematic hardening, on the other hand, for yield condition of slip systems has been described, among others, by Han et al. (2004). Moreover, in a recent work by Saleeb and Arnold (2004), a specific form for the polymeric material hardening function is evaluated, having a saturation value (or limit state) associated with it. This viscoplastic formulation accounts for both non-linear kinematic hardening and static recovery mechanisms. Considering the analogies in crystal anisotropy, plastic deformation of anisotropic polymer composites can be treated in terms of crystal plasticity, remaining in the frame of finite strain viscoplasticity. The basic assumption in the constitutive formulation of finite strain elastoplasticity is the multiplicative decomposition of the deformation gradient F into elastic and plastic parts introduced by Lee (1969): F = Fe Fp

[8.16]

The plastic deformation gradient tensor Fp maps a material point from a reference configuration to a relaxed configuration, which is obtained from the reference state by purely plastic deformation and rotation. Subsequently, the elastic deformation gradient tensor Fe maps the material point from the relaxed to the current configuration, which is obtained from the relaxed by purely elastic deformation and rotation. The relaxed configuration is arbitrarily defined, since an arbitrary rigid rotation can be superimposed on it and leave it unstressed. The velocity gradient tensor L in the current configuration is defined by: L = F˙ F– 1 = D + W

[8.17]

where D, W are the tensors of the rate of deformation and material spin, respectively. To overcome the problem of arbitrariness of the intermediate configuration, Mandel (1971) proposed a triad of orthonormal vectors that represent the material substructure and follow different kinematics than the continuum. The kinematic quantities Fe and Fp are then defined in terms of the corotational rates as follows:

[8.18]

where ω is the spin of the substructure, related with the rotation of the three vectors. Following this consideration, and taking into account Eq. 8.18, the kinematics in the current configuration are given by:

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287 [8.19]

where De is the elastic part of the rate deformation tensor, and Dp the plastic part of it. All these quantities are now invariant upon superposed rigid body rotation. Regarding spin tensor, Dafalias (1985) showed that the total spin of the continuum, or material spin W is expressed by: W = ω + Wp

[8.20]

where Wp is the material plastic spin tensor and ω is the substructural spin. Through this relation, which is analogous to Eq. 8.19 obtained from the additive decomposition of the deformation rate tensor, the arbitrariness of Wp can be overcome with a proper value of ω. Therefore, Dafalias (1998) introduced different ω’s associated with each internal variable. The quantities Wp and Dp of Eq. 8.19 and 8.20 will be further constitutively described. Our analysis will be applied on the tensile stress–strain behavior of off-axis composite materials, as well as on the tensile creep response, assuming that it is a plane-stress problem. Given the material anisotropy, a material axis slip system can be identified by a triad of vectors s, m, n where s is a unit vector along the fiber direction, and m is a vector perpendicular to s and n = sxm. Vector s follows the materials axis, forming an angle θ with the direction x of tension (see Fig 8.8). The plastic rate of deformation tensor Dp can then be defined by the dyadics in respect to vectors s, m, n as follows: Assuming that there is a simultaneous two-dimensional shearing deformation along the fiber axis 1, and a longitudinal deformation normal to this axis, Dp can be given additively by a combination of these two deformation procedures:

[8.21]

p with D33 being the out-of-plane plastic strain rate component of tensor Dp. p It has been assumed that D11 is equal to zero, taking into account that there is no plasticity along the fiber direction, as has also been mentioned by Sun and . . Chen (1989). The multiplicative factors γ 1p, γ 2p will be defined below, and the p p isovolume condition will be valid by taking D33 = –D22 . Accordingly, the plastic spin tensor can be obtained by the following expression:



[8.22]

where  is the plastic spin coefficient, introduced by Dafalias (1998). . . The rates of plastic shear deformation γ 1p along the fiber direction and γ 2p normal to it, are assumed to be given by the expressions:

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8.8  Schematic presentation of material axes, the direction of tension, and the relative position of unit vectors s and m. Reproduced from Kontou and Spathis (2006).

[8.23]

where c is a constant, τ is the resolved shear stress equal to (σxx/2) sin 2θ, with σxx the applied tensile stress and h is a hardening modulus, which is a material parameter. Modulus h represents the internal structure of the material and may . evolve with strain hardening. In our case it was treated as a constant. A(r) is a function that is taken to be dependent on strain rate, and σ22 is the normal stress along the axis 2. To formulate the strain rate effect, which is strongly exhibited by the polymeric . fiber composites, function A(r) will be expressed in terms of a scaling rule, valid in viscoelasticity and introduced by Matsuoka (1992). Following this rule, a . stress–strain curve at a strain rate r can be obtained from a corresponding curve at

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. a rate r0 by multiplying the stress and strain with the scaling factors

289 and

correspondingly. Therefore, function A can be scaled as follows:

[8.24]

. where k, n are constants and r0 is a reference strain rate. To this trend, the anisotropic hardening and rate-sensitive behaviors of fiber-reinforced composites has been formulated in the work by Chung and Ryou (2009a, b) by a power law type, isotropic hardening law as:

[8.25]

where σ, ε are the tensile yield stress and yield strain (effective) respectively, and the other quantities are material parameters. In the same approach, the anisotropic kinematic hardening has been formulated on a Chaboche (1986) type back stress α evolution rule:

[8.26]

where 1, 2 are the fourth-order tensors containing parameters to be experimentally determined and dε is the effective strain increment. In finite deformation, the elastic deformation is generally assumed to be smaller than plastic one. The constitutive law for Dp is the flow rule at finite deformation, while the constitutive law for De is written in the frame of hypoelasticity theory. The general form of the objective stress-rate and strain-rate relation, according to Rivlin (1955) and Truesdell (1955), is given by:

[8.27]

where  is the Gauchy stress tensor, De is the elastic part of the deformation rate o tensor D, and  is the objective rate of stress tensor. The type of objective rate that is selected may lead to different results, and remains still a problem, as is also mentioned by Khan and Huang (1995). In the following analysis, the objective rate introduced by Dafalias (1985) will be applied. Therefore, the objective rate of the Gauchy stress tensor will be given by:

[8.28]

where the substructural spin is the difference between the material plastic spin and the total spin of the continuum (Eq. 8.20). Hereafter, a constitutive equation for Wp, which appears in the objective rate, is required.

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A typical form of constitutive equations of hypoelasticity is:

[8.29]

and consequently given by:

[8.30]

where C is a fourth-order tensor, which in the case of an orthotropic material and two-dimensional problem is given by:



[8.31]

– The matrix coefficients Qij are the transformed reduced stiffnesses to x-y coordinate system, with x being the axis of tension, and are defined by Hyer (1998):



[8.32]

where the matrix coefficients Qij can be written in terms of the engineering constants as:



[8.33]

Since, in the uniaxial tension experiment, the specimen is restrained by the grips of the loading machine, the total rotation of the continuum is zero. Therefore, the total material spin W is taken to be zero, assuming that for the off-axis specimens

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examined, the end-constraint effect becomes less significant. Then, according to Eq. 8.20, we obtain: p

ω  = - W

[8.34]

Taking also into account that unit vector s remains always along axis 1, it obeys the evolution law:

[8.35]

Then, given that s = (cosθ, sinθ) in respect to the x-y coordinates, we have:

[8.36]

The constitutive laws for Dp, Wp in combination with the hardening rules and the above-mentioned constitutive equations, after applying the transformation law, lead to the calculation of stress–strain in the x-y axis loading system. For similar approaches, a kinematic description, developed by Rubin (1994a, b) is presented in the work by Spathis and Kontou (2004), while the one based on the multiplicative decomposition by Lee (1969) is used in the work by Kontou and Kallimanis (2006). The analysis presented above was proven to be capable of describing a quite broad region of strain and strain rates, starting from low to moderate strain rates up to rates three orders of magnitude higher, in a variety of fiber polymer composites, as is shown in Fig. 8.9 to 8.11 (Kontou and Spathis 2006). The model capability of simulating experimental data at high rates (400–700 s–1) by Tsai and

8.9  Stress–strain curves for [15°] at three strain rates, 10–1, 10–3, 10–5 s–1. Points: experimental data after Weeks and Sun (1998). Lines: calculated results. Reproduced from Kontou and Spathis (2006).

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8.10  Stress–strain curves for [30°] at three strain rates, 10–1, 10–3, 10–5 s–1. Points: experimental data after Weeks and Sun (1998). Lines: calculated results. Reproduced from Kontou and Spathis (2006).

8.11  Stress–strain curves for [45°] at three strain rates, 10–1, 10–3, 10–5 s–1. Points: experimental data after Weeks and Sun (1998). Lines: calculated results. Reproduced from Kontou and Spathis (2006).

Sun (2002), is depicted in Fig. 8.12 and 8.13. The plastic spin coefficient η was proven to play an important role in this non-linear finite description, even in the low strains, as it is depicted in Fig. 8.14. Moreover, the model capability has also been checked in a higher strain range, implementing this analysis on unidirectional glass-fiber-epoxy matrix composites at a high temperature of 80°C, where the

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8.12  Stress–strain curves for [15°] at 400 s–1. Points: experimental data after Tsai and Sun (2002). Line: calculated results. Reproduced from Kontou and Spathis (2006).

8.13  Stress–strain curves for [30°] at 700 s–1. Points: experimental data after Tsai and Sun (2002). Line: calculated results. Reproduced from Kontou and Spathis (2006).

contribution of Wp due to angle rotation is expected to be more essential. These results are depicted in Fig. 8.15. To further demonstrate the model capability at a wider strain range, as well as its applicability to creep data, the above equations can be applied in a creep procedure. The tensile creep experiments were performed at 80°C at a stress of © Woodhead Publishing Limited, 2011

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8.14  Stress–strain curves for [15°] at a strain rate of 10–1 s–1. Points: experimental data after Weeks and Sun (1998). Lines: calculated data with the plastic spin coefficient η = 0 (linearized model) and with η = 4. Reproduced from Kontou and Spathis (2006).

8.15  Stress–strain curves for [15°] at three strain rates and at a temperature of 80°C. Thick lines: experimental data. Thin lines: calculated results. Reproduced from Kontou and Spathis (2006).

8.66 MPa, for a period of 4 hours. The specific temperature was selected as that where a significant creep rate is exhibited by the materials, while it is still under the Tg of the materials examined (95°C). A typical creep curve from the off-axis specimen of 15° was obtained and plotted in Fig. 8.16 in a logarithmic time scale. Some further definitions are © Woodhead Publishing Limited, 2011



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8.16  Tensile creep compliance of [15°] off-axis composite at 80 °C and at a stress level of 8.6 MPa. Points: experimental data. Thick line: calculated results. Reproduced from Kontou and Spathis (2006).

necessary for the formulation of creep. It was assumed that there is an internal stress σint, which opposes the applied stress, expressed by:

[8.37]

where Cr is a constant proportional to temperature and εxx is the axial strain. . . Therefore, the quantities γ 1p, γ 2p are now given by:

[8.38.a]



[8.38.b]

. where r, h, c are constants. The resolved shear stress τ * and the normal stress σ *22 are accordingly given by:

[8.39.a]

[8.39.b)

Applying the whole set of the above analysis, with the proper changes for creep deformation, the axial creep strain was calculated and plotted in terms of total

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creep compliance in Fig. 8.16. The main features of the experimental creep compliance curve were captured by the model simulations.

8.3

Concluding remarks

The long-term mechanical behavior of polymer matrix composites, due to their use in structural applications has always been a topic of great importance. Viscoplastic creep modeling of polymer composites, depending on a series of factors like constitutive analysis, kinematics, micromechanical models that link the externally imposed strain rate with the material response, as well as kinematic hardening rules and anisotropy, constitutes an issue of high research interest. The non-linear viscoplastic behavior exhibited by polymer composites was mainly attributed to the polymeric matrix, and analyzed in terms of a thermally activated rate process. The creep strain was expressed by the additive form of three components: elastic, plastic and recoverable. Plastic strain was separately formulated following a functional form of the rate of plastic deformation accumulated in a creep procedure. This formalism is based on a mechanism related with the distributed nature of polymeric glassy state. The model was proved to be capable of describing the creep compliance of off-axis unidirectional glass-fiber composites at various stress levels and temperatures. A theoretical procedure was also presented, dealing with the joint study of tensile response and tensile creep behavior of polymer matrix fiber composites, in the finite deformation regime. An anisotropic model of finite viscoplasticity is described, with constitutive laws for Dp and Wp written in accordance with material anisotropy, while the constitutive law of hypoelasticity is applied in its objective form.

8.4

Future trends

The prediction of long-term creep response of polymeric composites, especially under high thermomechanical conditions, is not always possible in terms of various viscoplastic models. This inadequacy is more intense when other parameters, such as humidity, UV radiation, and physical and thermal aging are taken into account. This is mainly due to the thermorheological complexity of the polymeric matrix and its highly non-linear/viscoplastic features. In the previous paragraphs, an effort has been made to present some aspects of the viscoplastic deformation of polymeric composites, selecting the class of unidirectional fiber composites, given that the inherent anisotropy renders the problem of more interest. In the preceding analysis, both the monotonic deformation and creep response have been described with the same procedure. A possible subject of future research could incorporate other modes of deformation, such as stress-relaxation and cyclic loading. In spite of the fact that the main problems of such a description still remain, a new generation of polymeric composites, that of polymeric nanocomposites, emerges. The nanofillers

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usually result in a slightly decreased creep compliance and stable creep rate at long creep periods. On the other hand, as already has been mentioned, the non-linear/viscoelastic/ viscoplastic creep response of polymer composites arises mainly from the polymeric matrix, and under certain conditions (high stress level, elevated temperature, long-term loading) is expected to be accelerated, independently of the presence of nanofillers. Moreover, the improved creep resistance is not always systematic, due to the agglomerates of nanoparticles and poor filler/ matrix interaction. To overcome this, Zhou et al. (2007) proved that in situ grafting and crosslinking of nanosilica in the course of melt and mixing with polypropylene is an effective way to improve interfacial interaction in the nanocomposites. This procedure can further be applied in manufacturing other creep-resistant thermoplastics. In a recent work by Anthoulis and Kontou (2008), a procedure is presented which provides a better understanding of the relationship between nanoclay content/structure and final nanocomposite properties. The analysis is based on the effective particle concept analyzed by Brune and Bicerano (2002) and Sheng et al. (2004), initially presented for separate clay sheets, which was extended to involve nanosized particles or agglomerates surrounded by matrix material attached to it that possesses different properties than that of the pure matrix. This concept was satisfactorily applied on tensile tests, with the further assumption that this region around a nanoparticle undergoes viscoplastic deformation (Kontou 2007) when the imposed stress field is high enough. As a future work, this analysis could be adapted for modeling of the viscoplastic creep response of polymer nanocomposites. To this trend, some recent works deal with the creep response of polymer nanocomposites such as the work of Starkova et al. (2007), where the tensile behavior and the long-term creep of PA66 and its nanocomposites filled with TiO2 nanoparticles have been investigated. Non-linear viscoelastic models and a power law have been applied, and proved to be valid in a limited time domain, while it was shown that the smaller the nanoparticles, the higher the creep resistance. These results support the need for further exploration of the fundamental mechanism for mechanical enhancement of polymer nanocomposites. Considering all these factors, namely the quality of nanofiller dispersion, the exact features of the reinforcing mechanism, and the way polymeric matrix is affected by the nanofiller’s presence, the development of a model that can predict the viscoplastic creep performance at longer times, taking into account all kinds of structural features of nanocomposites, has been a challenging topic of research.

8.5

References

Al-Haik M, Vaghar MR, Garmestani H, Shahawy M (2001), ‘Viscoplastic analysis of structural polymer composites using stress relaxation and creep data’, Composites: Part B, 32, 165–170.

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Gates TS (1993), ‘Effects of elevated temperature on the viscoplastic modeling of graphite/ polymer composites’, ASTM (Special Technical Publication), 1174, 201–221. Gates TS, Sun CT (1991), ‘Elastic/viscoplastic constitutive model for fiber reinforced thermoplastic composites’, AIAA J, 29(3), 457–463. Gates TS, Sun CT (1991) ‘Elastic/viscoplastic constitutive model for fiber reinforced thermoplastic composites’, AIAA J, 29, 457–463. Giannelis EP (1996), ‘Polymer layered silicate nanocomposites’, Adv Mater, 8, 29–35. Guedes RM, Tavares CML, Ferreira AJM (2004), ‘Experimental and theoretical study of the creep behavior of GFRP-reinforced polymer concrete’, Comp Sci Techn, 64, 1250–1259. Hahn T, Tsai SW (1973), ‘Nonlinear elastic behavior of unidirectional composite laminate’, J Comp Mat, 7, 102–118. Han CS, Wagoner RH, Barlat F (2004), ‘On orecipitate-induced hardening in crystal plasticity theory’, Int J Plast, 20, 477–494. Hasan OA, Boyce MC (1995), ‘A constitutive model for the non-linear viscoelastic viscoplastic behavior of glassy polymers’, Polym Eng Sci, 35(4), 331–344. Haupt P, Kersten T (2003), ‘On the modelling of anisotropic material behaviour in viscoplasticity’, Int J Plast, 19(11), 1885–1915. Hill R, Rice JR (1972), ‘Constitutive analysis of elastoplastic crystals at arbitrary strain’, J Mech Phys Sol, 20, 401–416. Hilton H (1998), ‘The significance of (an)isotropic viscoelastic Poisson’s ratio stress and time dependencies’, Int J Solids and Structures, 35(23) 3081–3095. Hilton H (2001), ‘Implications and constraints of time-dependent Poisson ratios in linear isotropic and anisotropic viscoelasticity’, J of Elasticity, 63, 221–251. Hyer MW (1998), Stress Analysis of Fiber-reinforced Composite Material, Boston, WCB McGraw Hill. Karger-Koscis J (2008), ‘Water-mediated dispersion of “nanofillers” in thermoplastics: is it the right way?’, EXPRESS Polym Lett, 2, 312. Kawai M, Kamioka H, Zhang JQ, Matsuda T (2007), ‘Off-axis creep recovery of unidirectional carbon/epoxy composites at high temperatures’, Key Engineering Materials, 334–335 I, 65–68. Khan AS, Huang S (1995), Continuum Theory of Plasticity, New York, Wiley. Kontou E (2005), ‘Tensile creep behavior of unidirectional glass-fiber polymer composites’, Polym Comp, 26(3), 287–292. Kontou E (2007), ‘Micromechanics model for particulate composites’, Mech Mater, 39, 702–709. Kontou E, Kallimanis A (2006), ‘Thermo-visco-plastic behavior of fibre-reinforced polymer composites’, Comp Sci Techn, 66, 1588–1596. Kontou E, Spathis G (2006), ‘Application of finite strain viscoplasticity to polymeric fiber composites’, Int J Plast, 22, 1287–1303. Krausz AS, Eyring H (1975), Deformation Kinetics, New York, John Wiley and Sons. Krempl E (1998), ‘Some general properties of solid polymers inelastic deformation behavior and their application to a class of clock models’ J Rheology, 42(4),713–725. Lee EH (1969), ‘Elastic-plastic deformation at finite strains’ ASME, J Appl Mech, 36, 1–6. Loret B (1983), ‘On the effect of plastic rotation on the finite deformation of anisotropic elastoplastic materials’, Mech Mater, 2, 287–304. Ma ZY, Tjong SC (2001), ‘Creep deformation characteristics of discontinuously reinforced aluminum-matrix composites’, Comp Sci Techn, 61, 771–786. Mandel J (1971), ‘Plasticité classique et viscoplasticité’. In: Courses and Lectures, International Center for Mechanical Sciences/Springer, Udine/Wien-New York.

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Matsuoka S (1992), Relaxation phenomena in polymers, Munich, Hanser Publishers. Mead DJ, Joannides RJ (1991), ‘Measurement of the dynamic moduli and Poisson’s ratios of a transversely isotropic fibre-reinforced plastic’, Composites, 22(1) 15–29. Megnis M, Varna J (2003), ‘Micromechanics based modeling of nonlinear viscoplastic response of unidirectional composite’, Comp Sci Techn, 63, 19–31. Melo JDD, Radford DW (2003), ‘ Viscoelastic characterization of transversely isotropic composite laminae’, J Comp Mater, 37(2), 129–145. Mindel MJ, Brown N (1973), ‘Creep and recovery of polycarbonate’, J Mat Sci, 8, 863–870. Novak BM (1993), ‘Hybrid nanocomposite materials – between inorganic glasses and organic polymers’, Adv Mater, 5, 422–433. Oleinik EF, Salamatina OB, Rudnev SN, Shenogin SV (1993), Polym Sci, 35(11),1532. Oleinik EF, Salamatina OB, Rudnev SN, Shenogin SV (1995), ‘Plastic deformation and performance of engineering polymer materials’, Polymers for Advanced Technologies, 6, 1–9. Onat ET (1982), Recent advances in creep and fracture of engineering materials and structures, Pineridge Press, Swansea, ch.5. Papanicolaou GC, Zaoutsos SP, Cardon AH (1999), ‘Prediction of the non-linear viscoelastic response of unidirectional fiber composites’, Comp Sci Techn, 59(9), 1311–1319. Pegoretti A, Kolarik J, Peroni C, Migliaresi C (2004), ‘Recycled polyethylene terephthalate/ layered silicate nanocomposites: morphology and tensile mechanical properties’, Polymer, 45, 2751–2759. Raghavan J, Meshii M (1994), ‘Activation theory for creep of matrix resin and carbon fibre-reinforced of polymer composite’, J Mat Sci, 29, 5078–5084. Raghavan J, Meshii M (1997), ‘Creep of polymer composites’, Comp Sci Techn, 57, 1673–1688. Rivlin PS (1955), ‘Further remarks on the stress-deformation relations for isotropic materials’, J Rat Mech Anal, 4,681. Rubin MB (1994a), ‘Plasticity theory formulated in terms of physically based microstructural variables: Part I. Theory’, Int J Solids Struct, 31(19), 2615–2634. Rubin MB (1994b), ‘Plasticity theory formulated in terms of physically based microstructural variables: Part II. Theory’, Int J Solids Struct, 31(19), 2635–2652. Saleeb AF, Arnold SM (2004), ‘ Specific hardening function definition and characterization of a multimechanism generalized potential-based viscoelastoplasticity model’, Int J Plast, 20, 2111–2142. Saleeb AF, Wilt TE, Al-Zoubi NR, Gendy AS (2003), ‘An anisotropic viscoelastoplastic model for composites-sensitivity analysis and parameter estimation’, Composites: Part B, 34, 21–39. Schapery RA (1997), ‘Nonlinear viscoelastic and viscoplastic constitutive equations based on thermodynamics’, Mechanics of Time Dependent Materials, 1, 209–240. Shaito AA, D’Souza NA, Fairbrother P, Sterling J (2006), ‘Non-linear stress and temperature creep relations in polymer nanocomposites’, ASME, IMECE, Chicago IL. Sheng N, Boyce MC, Parks DM, Rutledge GC, Abes JI, Cohen RE (2004), ‘Multiscale micromechanical modeling of polymer/clay nanocomposites and the effective clay particle’, Polymer, 45, 487–506. Sherby OD, Dorn JE (1956), J Mech Phys Solids, 6, 145. Siengchin S, Karger-Koscis J (2009), ‘Structure and creep response of toughened and nanoreinforced polyamides produced via the latex route: Effect of nanofiller type’, Comp Sci Techn, 69(5), 677–683.

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Someya Y, Shibata M (2004), ‘Morphology, thermal, and mechanical properties of vinylester resin nanocomposites with various organo-modified montmorillonites’, Polym Eng Sci, 44(11), 2041–2046. Spathis G, Kontou E (2001), ‘Nonlinear viscoelastic and viscoplastic response of glassy polymers’, Polym Eng Sci, 41(8), 1337–1344. Spathis G, Kontou E (2004), ‘Non-linear viscoplastic behavior of fiber-reinforced polymer composites’, Comp Sci Techn, 64(15), 2333–2340. Starkova O, Yang J, Zhang Z (2007), ‘Application of time-stress superposition to nonlinear creep of polyamide 66 filled with nanoparticles of various sizes’, Comp Sci Techn, 67(13), 2691–2698. Sun CT, Chen JL (1989), ‘A simple flow rule for characterizing non-linear behavior of fiber composites’, J Comp Mat, 23, 1009–20. Sun CT, Chen JL (1991), ‘A micromechanical model for plastic behavior of fibrous composites’, Comp Sci Techn, 40, 115–129. Sun CT, Feng WH, Koh SL(1974), ‘A theory for physically non-linear elastic fiberreinforced composites’, Int J Eng Sci, 12(11), 919–935. Thiruppukuzhi SV, Sun CT (2001), ‘Models for the strain-rate-dependent behavior of polymer composites’, Comp Sci Techn, 61(1), 1–12. Truesdell C (1955), ‘The simplest rate theory of pure elasticity’, Comm Pure Appl Math, 8, 123. Tsai J, Sun CT (2002), ‘Constitutive model for high strain rate response of polymeric composites’, Comp Sci Techn, 62, 1289–1297. Tsakmakis Ch (2004), ‘Description of plastic anisotropy effects at large deformations. Part I: restrictions imposed by the second law and the postulate of Il’iushin’, Int J Plasticity, 20(2), 167–198. Van Paepegem W, De Baere I, Degrieck J (2006), ‘Modeling the non-linear shear stress– strain response of glass fibre-reinforced composites. Part I: Experimental results’, Comp Sci Techn, 66, 1455–1464. Vinet A, Gamby D (2008), ‘Prediction of long-term mechanical behavior of fibre composites from the observation of micro-buckling appearing during creep compression tests’, Comp Sci Techn, 68, 526–536. Ward IM (1971), ‘Review: The yield behaviour of polymers’, J Mech Phys Solids, 6, 1397. Ward IM, Handley DW (1993), An Introduction to the Mechanical Properties of Solid Polymers, New York, John Wiley and Sons. Weeks CA, Sun CT (1998), ‘Modeling non-linear rate-dependent behavior in fiberreinforced composites’, Comp Sci Techn, 58(3–4), 603–611. Wilkinson AN, Man Z, Stanford JL, Matikainen P, Clemens ML, Lees GC, Liauw CM (2006), ‘Structure and dynamic mechanical properties of melt intercalated polyamide 6-montmorillonite nanocomposites’, Macromol Mater Eng, 291(8), 917–928. Yoon KJ, Sun CT (1991), ‘Characterization of elastic-viscoplastic properties of an AS4/ PEEK thermoplastic composite’, J Comp Mat, 25(10), 1277–1296. Zaoutsos SP, Papanicolaou GC, Cardon AH (1998), ‘On the non-linear viscoelastic behaviour of polymer matrix composites’, Comp Sci Techn, 58(6), 883–889. Zhang Z, Yang J-L, Friedrich K (2004), ‘Creep resistant polymeric nanocomposites’, Polymer, 45(10), 3481–3485. Zheng Y, Ning R (2003), ‘Effects of nanoparticles SiO2 on the performance of nanocomposites’, Mater Lett, 57(19), 2940–2944. Zhou TH, Ruan WH, Yang JL, Rong MZ, Zhang MQ, Zhang Z (2007), ‘A novel route for improving creep resistance of polymers using nanoparticles’, Comp Sci Techn, 67(11–12), 2297–2302.

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9 Micromechanical modeling of viscoelastic behavior of polymer matrix composites undergoing large deformations J. A boudi, Tel Aviv University, Israel Abstract: In the present investigation, a comprehensive finite strain micromechanical analysis is offered for the prediction of the viscoelastic behavior of polymer matrix composites undergoing large deformations. The finite viscoelasticity of the matrix governs the rate-dependent response of the composite, its hysteresis in cyclic loading–unloading, as well as its creep and relaxation behavior. The finite viscoelasticity of the polymeric constituent of the composite allows, in particular, large deviations away from the thermodynamic equilibrium. Finite linear viscoelasticity (where the deviations are small) and linear viscoelasticity are obtained as special cases. In addition, perfectly elastic behavior of the polymeric matrix (hyperelasticity) forms another special case of the present theory. Furthermore, the possibility of evolving damage in the polymeric finite viscoelastic matrix is accounted for. This is expressed by an evolution law according to which the damage accumulates depending on the maximum strain history. As a result, the Mullins damage effects can be modeled and observed. The micromechanical modeling is based on the homogenization technique for periodic microstructure, which establishes, in conjunction with its instantaneous tangent and viscoelasticdamage tensors, the macroscopic (global) constitutive equations of the viscoelastic composite undergoing large deformations. Results are given which exhibit the response of the composite to cyclic loading as well as its creep and relaxation behavior in various circumstances. Key words: periodic composites, finite viscoelasticity, large deformations, polymer matrix composites, evolving damage, finite strain high-fidelity generalized method of cells.

9.1

Introduction

Polymeric materials exhibit viscoelastic effects. These dissipative effects can be represented by linear viscoelasticity where the deformations are infinitesimal, finite linear viscoelasticity where the deformations are large but the deviations from the equilibrium state are small, or by finite viscoelasticity where both deformations and deviations from equilibrium are large. A comprehensive discussion of the first two types of viscoelasticity can be found in the monographs by Christensen (1982) and Lockett (1972), for example. Finite viscoelasticity was presented by Haupt (1993), and Reese and Govindjee (1998). Examples for the micromechanical modelings of viscoelastic composites in which one of the polymeric phases behaves as a viscoelastic material were 302 © Woodhead Publishing Limited, 2011



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presented by Aboudi (1991), Haj-Ali and Muliana (2003) and Muliana and Kim (2007). In these investigations the viscoelastic polymeric constituent was assumed to behave either as a linear or nonlinear viscoelastic material but the strains were still assumed to be infinitesimal. The present investigation is concerned with the micromechanical modeling of viscoelastic composites reinforced by unidirectional continuous fibers undergoing large deformations. Any polymeric phase in the composite is assumed to behave as a finite viscoelastic material, which is modeled by the finite strain constitutive relations of Reese and Govindjee (1998) where the strains are large and the evolution equations of the internal variables are nonlinear. This is in contrast to finite linear viscoelasticity theories; see Simo (1987) and Holzapfel (2000) for example and references cited in Reese and Govindjee (1998), which, although they account for large deformations, restrict the formulation to states close to thermodynamic equilibrium by choosing linear evolution laws for the internal variables. Finite linear viscoelastic and hyperelastic behaviors can be obtained as special cases of the general theory. In the finite viscoelasticity theory of Reese and Govindjee (1998) damage effects are not taken into account. Loss of stiffness in polymers, however, occurs at strain levels below the maximum value of the previously applied strain. This softening effect is referred to as the Mullins effect (Mullins and Tobin, 1957). In the present investigation, evolving damage in the finite viscoelastic polymeric material is included by adopting, in the framework of continuum damage mechanics, the derivation of Lin and Schomburg (2003) and Miehe and Keck (2000), according to which the rate of damage depends upon the kinematic arc-length. By neglecting the viscous effects, the special case of a hyperelastic material with evolving damage is obtained. The micromechanical analysis is based on the homogenization technique for periodic composites, which is capable of predicting the finite strain viscoelastic composite’s behavior in conjunction with the known properties of its constituents, their constitutive relations, detailed interaction and volume ratios. This analysis, referred to as the High-Fidelity Generalized Method of Cells (HFGMC), provides the instantaneous concentration tensors that relate the local induced deformation gradient in the phase to the current externally applied deformation gradient. In addition, it yields the macroscopic constitutive equations of the multiphase composite in terms of its instantaneous stiffness (tangent) and viscoelastic tensors. A review of the finite strain HFGMC model and its applications for the prediction of the response of various types of inelastic composites has been presented by Aboudi (2008). A preliminary investigation of finite strain viscoelastic composites, where the analysis was limited to monotonic loading only, was reported by Aboudi (2009a), but the present work provides a generalization to any type of applied loading. This chapter is organized as follows. The governing equations of the finite strain viscoelasticity theory coupled with damage are presented. This is followed by a

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brief review of the finite strain micromechanical analysis which is based on the homogenization technique for periodic composites reinforced by continuous fibers for which the instantaneous concentration, tangent and viscoelastic tensors are established. Applications are given which exhibit the behavior of the monolithic (unreinforced) matrix and the polymer matrix composite that are subjected to loading and unloading in the presence and absence of damage. This is followed by studying the response of the unidirectional viscoelastic composite that is subjected to a cyclic loading-unloading applied at different rates and amplitudes. Next, the creep behaviors of the composite and its unreinforced matrix are studied in various circumstances. Finally, the relaxations of the composite and its unreinforced matrix are shown when they are subjected to constant deformation gradients. This chapter is concluded by suggestions for further extensions and generalizations.

9.2

Finite strain viscoelasticity coupled with damage model of monolithic materials

In the present section we briefly present the constitutive behavior of finite strain viscoelastic polymeric materials that exhibit evolving damage. The presentation follows the papers of Reese and Govindjee (1998) where no damage is accounted for, and Lin and Schomburg (2003) where evolving damage is included. As mentioned in the introduction, the present viscoelastic modeling allows finite strain and large deviations from the thermodynamic equilibrium state and, therefore, is referred to as finite viscoelasticity (in contrast to finite linear viscoelasticity where small deviations from the equilibrium state are assumed). Let X and x denote the location of a point in the material with respect to the initial (Lagrangian) and current systems of coordinates, respectively, and t is the time. In terms of the local deformation gradient tensor F(X, t), dx = F(X, t)dX. The deformation gradient F is expressed by the multiplicative decomposition: F(X, t) = Fe(X, t) Fυ (X, t)

[9.1]

where Fe and Fυ are the elastic and viscous parts. The modeling that is presented herein is based on single Maxwell and elastic elements, but it can be extended to include several Maxwell elements. The total free energy per unit reference volume is decomposed into equilibrium (EQ), which represents the strain energy of the elastic element, and a nonequilibrium (NEQ) part that accounts for the Maxwell element: NEQ ψ = ψ EQ + ψ NEQ ≡ (1 – D)ψ EQ 0 + (1 – D)ψ 0

[9.2]

where ψ 0EQ and ψ0NEQ are referred to as the effective free energy of the undamaged material, and D denotes the amount of damage such that 0 ≤ D ≤ 1. The resulting Kirchhoff stresses are given by:

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[9.3]



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where C = FT, F is the right Cauchy-Green deformation tensor, and

[9.4]

where Ce = FeTFe and τ0EQ, τ0NEQ correspond to the effective Kirchhoff stresses of the undamaged material. Let the left Cauchy-Green tensor B = F FT be represented in terms of its eigenvalues: B = diag [b1, b2, b3]

[9.5]

¯ = J–2/3 B can be With J = det F = , the volume preserving tensor B accordingly represented in the form:

[9.6]

The finite strain elastic contribution can be modeled by the Ogden’s compressible material representation (Ogden, 1984; Holzapfel, 2000) as follows:

[9.7]

where Ke is the elastic bulk modulus and µ ep and α pe are material parameters of the elastic element. For the Maxwell element, the free energy is represented by Reese and Govindjee (1998): [9.8] where:

[9.9]

and Je = , b¯ Ae = (Je)–2/3bAe , and µυp, αυp, Kυ are material parameters. By employing Eq. 9.2–9.4 the following expression for the principal values of τ0EQ and τ0NEQ are obtained: [9.10]    A, B, C, = 1 , 2, 3 [9.11]   

A, B, C = 1, 2, 3

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The evolution equation for the internal variables is given by (Reese and Govindjee, 1998):

[9.12]

where ηD and ηV are the deviatoric and volumetric viscosities, respectively, and Lυ [Be], being the Lie derivative of Be, can be expressed as:

[9.13]

with Cυ = FυT Fυ. For elastic bulk behavior, 1/ηV = 0 and for infinitesimal strains the relaxation time is given by ξ = ηD/µ where µ is the small strain shear modulus of the Maxwell element (the nonequilibrium part). The integration of the evolution equation (Eq. 9.12) is performed by means of the return mapping algorithm in conjunction with the logarithmic strain and the backward exponential approximation, which were developed in the framework of elastoplasticity; see Weber and Anand (1990), Eterovic and Bathe (1990), Cuitino and Ortiz (1992) and Simo (1992). Thus, by employing the exponential mapping algorithm, Eq. (9.12) is reduced to:

[9.14]

where the principal values of the elastic logarithmic strain ε eA are given by ε Ae = 1/2log(b Ae ) and ∆t is the time increment between the current and previous step. In e trial Eq. 9.14, the trial values of ε n+1,A can be expressed in terms of the eigenvalues of e Bn as will be discussed in the sequel. Equation 9.14 forms a system of coupled e nonlinear equations in the three unknowns: ε n+1,A , A = 1,2,3. It can be rewritten in terms of the elastic logarithmic strain increments in the form:

[9.15]

The rate of damage evolution is given according to Lin and Schomburg (2003) and Miehe and Keck (2000) by:

[9.16]

where the rate of kinematic arc length is defined by:

[9.17]

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and:

[9.19]

In these relations, ηdam, D∞0 and αdam are material parameters. The incremental form of the constitutive equations of the finite viscoelastic material is determined as follows. From Eq. 9.4 the following expression can be established:

[9.20]

Let the second-order tensor M be defined by:

[9.21]

with:

[9.22]

the explicit components of M are given by Reese and Govindjee (1998):

[9.23]



[9.24]

and:

[9.25]

In conjunction with Eq. 9.15, we obtain from Eq. 9.20 that:



[9.26]

Let ∆ε Ae and ∆εAυed be defined by:

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[9.27]

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Therefore Eq. 9.26 can be represented by:

[9.28]

where the components ∆W ANEQ involve the viscoelastic and damage effects. The fourt-order tangent tensor d NEQ is defined by:

[9.29]

where SNEQ is the second Piola-Kirchhoff stress tensor:

[9.30]

whose principal values are given by:

[9.31]

where bυA = bA/beA being the principal values of Bυ = Fυ[Fυ]T = diag [bυ1, bυ2, bυ3]. The principal values of d NEQ can be determined from the following expression (Holzapfel, 2000):

[9.32]

where:

[9.33]

with λA = √bA and NA denoting the principal referential orthonormal directions. It should be noted that for λA = λB, a Taylor expansion shows that:

[9.34]

The fourth-order first tangent tensor R NEQ which is defined by:

[9.35]

where T NEQ is the first Piola-Kirchhoff stress tensor, which can be determined from:

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[9.36]



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with I denoting the unit second-order tensor. Thus, the rate form of the nonequilibrium portion of the constitutive equations of the finite viscoelastic material is given by:

[9.37]

Taking into account the relation between the Kirchhoff τ and the first PiolaKirchhoff T stress tensors τ = FT, the following expression for the viscousdamage term V˙ NEQ can be established:

[9.38]

The same procedure can be followed for the establishment of the first tangent ˙  EQ involves this time the tensor REQ of equilibrium elastic element where W υ damage effects only and F = I. It yields:

[9.39]

The final total form of the finite viscoelastic material is as follows:

[9.40]

where T˙  = T˙  NEQ + T˙  EQ, R = R NEQ + REQ and V˙ = V˙ NEQ + V˙ EQ. Constitutive equations can be obtained from Eq. 9.40 as special cases in the presence/absence of damage and viscous effects.

9.3

Finite strain micromechanical analysis

Finite strain HFGMC micromechanical analyses for the establishment of the macroscopic constitutive equations of various types of composites with doubly periodic microstructure undergoing large deformations have been previously reviewed by Aboudi (2008). These micromechanical analyses are based on the homogenization technique in which a repeating unit cell of the periodic composite can identified. This repeating unit cell represents the periodic composite in which the double periodicity is taken in the transverse 2–3 plane, so that the axial 1-direction corresponds to the continuous direction; see Fig. 9.1 (for a fiberreinforced material, for example, the 1-direction coincides with the fibers’ orientation). In the framework of these HFGMC micromechanical models, the displacements are asymptotically expanded and the repeating unit cell is discretized. This is followed by imposing the equilibrium equations, the displacement and traction interfacial conditions as well as the conditions that ensure that the displacements and tractions are periodic across the repeating unit cell. In particular, the imposition of the equilibrium equations provide the strong form of the Lagrangian equilibrium conditions of the homogenization theory that must be satisfied. The resulting homogenization derivation establishes the deformation concentration tensor A(Y), where Y is the local Lagrangian system of coordinates with respect to which field variables in the repeating unit cell are

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9.1  A multiphase composite with doubly periodic microstructures defined with respect to global initial coordinates of the plane X2–X3. The repeating unit cell is defined with respect to local initial coordinates of the plane Y2–Y3.

characterized. This tensor relates the rate of the local deformation gradient gradient F˙ (Y) at a material point Y within the repeating unit cell to the externally ¯˙ in the form: applied deformation gradient rate F

[9.41]

˙υd is the viscous-damage contribution which can be determined at every where A increment when no mechanical loading is applied. It follows from Eq. 9.41 in conjunction with Eq. 9.40 that the local stress rate at this point is given by:

[9.42]

Hence the resulting macroscopic constitutive equation for the multiphase composite undergoing large deformation is given by:

[9.43]

where R* is the instantaneous effective stiffness (first tangent) tensor of the multiphase composite. It can be expressed in terms of the first tangent tensors of the constituents R(Y) and the established deformation concentration tensor A(Y) in the form:

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where SY is the area of the repeating unit cell. The viscous-damage contribution to the macroscopic constitutive equations (9.43) is given by:

[9.45]

More details can be found in the aforementioned reference (Aboudi 2008). It ˙ should be noted that the current values of R* and V¯ of the composite are affected by the current value of damage variable through the instantaneous value of tangent ˙ (Y) of the finite strain constituents. tensors R(Y) and V The finite strain HFGMC micromechanical model predictions were assessed and verified by comparison with analytical and numerical large deformation solutions by Aboudi and Pindera (2004) and Aboudi (2009b) for composites with hyperelastic constituents. In the latter reference, the Mullins damage effect was incorporated with the hyperelastic constituents.

9.4

Computational procedure

In the following, the computational procedure for the determination of the finite viscoelastic composite’s response is described. At time step tn, the local deformation gradient Fn and the left Cauchy-Green deformation tensors Bne have already been established, and Fn+1 is assumed to be known at time tn+1 = tn + ∆t. 1. From Fn and Fn–1, the local right Cauchy-Green deformation tensors Cn and Cn–1 can be readily determined. Hence z˙ in Eq. 9.17 can be computed, from which D˙  that is given by Eq. 9.16 can be integrated to provide the damage variable at time step n + 1:

[9.46]

Equation 9.19 yields:



[9.47]

e trial 2. The local trial elastic left Cauchy-Green deformation tensor Bn+1 can be computed from:



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[9.48]

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where:

[9.49]

e trial the principal values be trial and the logarithmic elastic strains 3. From Bn+1 A e trial ε A = 1/2 log(beA trial ) can be determined at time step n + 1. The solution of the nonlinear equations (14) provides bAe , ε Ae and Be at time step n + 1. 4. From the established bAe , the Kirchhoff stresses τANEQ and tensor M can be calculated from Eq. 9.11 and 9.21, respectively. Hence the principal components of the viscoelastic tensor ∆WANEQ can be determined (see Eq. 9.28) from which ˙  NEQ and V˙ NEQ are obtained. Tensor V˙ EQ can be similarly established. W 5. The local tangent tensors RNEQ and REQ are determined from the procedure that leads to Eq. 9.36. 6. With the established local values of R = RNEQ + REQ and V˙ = V˙ NEQ + V˙ EQ, it is possible to proceed with the micromechanical analysis to compute the concentration tensor A and A˙υd, see Aboudi (2008). Consequently, Eq. 9.44 and ˙ 9.45 can be employed to determine the tangent tensor R* and V¯ . ¯˙ is determined in 7. The rate of the externally applied deformation gradient F accordance with the prescribed type of loading. For a uniaxial deformation in ¯˙ are the 1-direction, for example, F¯˙11 is known while all other components of F ˙ ¯ zero. Hence, it is possible to compute the stress rate T using Eq. 9.43. In addition, by integrating the resulting global stress rate, T¯ is obtained. ¯˙ has been determined, it is possible to compute the rates of the local 8. Once F deformation gradients from Eq. 9.41. The integration of the latter would yield the local deformation gradients to be used in the next time step. 9. If, on the other hand, a uniaxial stress loading is applied on the composite such ˙ that F¯11 only is known together with T¯, the components of which are zero in the other directions, an iterative procedure is needed to determine the other components of F¯˙ from these conditions.

9.5

Applications

In the present section, applications are given which exhibit under various circumstances the response of a composite undergoing large deformations, which consists of a viscoelastic rubber-like material reinforced by continuous linearly elastic fibers. The viscoelastic matrix is characterized by the free energy functions Eq. (9.7 and 9.8) that represent elastic and Maxwell elements, respectively. The parameters in these functions are given in Tables 9.1 and 9.2 together with ηD and 1/ηv = 0 (assuming elastic bulk deformations). The continuous linearly elastic fibers are oriented in the 1-direction and their Young’s and Poisson’s ratios are 2GPa and 0.4, respectively, which correspond to the properties of nylon fibers. It is worth mentioning that the elastic nylon fibers could alternatively be characterized by a suitable hyperelastic energy function depending on whether such a function and its corresponding material parameters are known. The volume fraction of the

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Table 9.1  The material parameters in function ψ0EQ, Eq. 9.7 (Reese and Govindjee, 1998)

µe1 (MPa)

µe2 (MPa)

µe3 (MPa)

α e1

α e2

α e3

Ke (MPa)

0.13790

–0.04827

0.01034

1.8

–2

7

50

The parameters µ pe and α pe, p = 1,2,3 are the Ogden’s material constants and Ke is the bulk modulus. In the small strain domain, the shear modulus of this material is 0.208MPa. Table 9.2  The material parameters in function ψ NEQ , Eq. 9.8 (Reese and Govindjee, 0 1998)

µυ1 (MPa) µυ2 (MPa) µυ3 (MPa) αυ1

αυ2

αυ3

Kυ (MPa)

ηD (MPa s)

ηV (MPa s)

0.3544

–2

7

50

9.38105

∞

–0.1240

0.0266

1.8

The parameters µυp and αυp, p = 1,2,3 are the Ogden’s material constants, and K υ is the bulk modulus. ηD and ηV are the viscoelastic constants. In the small strain domain, the shear modulus of this material is 0.536MPa.

fibers is υf = 0.05 which is characteristic for a rubber-like material reinforced by nylon fibers. The damage parameters in Eq. 9.16–9.18 are: ηdam = 0.1, D∞0 = 1 and αdam = 1. The effect of damage can be totally neglected by choosing 1/ηdam = 0. Figure 9.2 shows the behavior of the homogeneous (H) (unreinforced) viscoelastic matrix that is subjected to a uniaxial stress loading in the 1-direction (i.e., all stress components of T are zero except T11) followed by unloading, both of which are applied at a rate of F˙11 = 0.01s–1. Fig. 9.2(a) exhibits a comparison between the response of the viscoelastic (VE) material and the corresponding elastic (EL) one in which 1/ηD = 0. In both cases the effect of the evolving damage can be clearly observed (Mullins effect). The damage evolutions in both cases are shown in Fig. 9.2(b). In this figure and its counterparts that are given in the following, D denotes the maximum of all damage variables that evolve at all locations (Y2, Y3) of the matrix constituent. Figure 9.2(b) shows that the damage evolution in the viscoelastic case coincides with the elastic one. It should be noted that the damage remains almost constant during unloading, which is a characteristic observation of the Mullins effect. A comparison between the response of the viscoelastic and elastic responses is shown in Fig. 9.2(c) where the effect of damage is neglected. Whereas the loading and unloading responses in the elastic case are, as expected, coincident, different behavior in loading and unloading is exhibited by the viscoelastic material which is solely caused by the existence of dissipative mechanism. The three parts of Fig. 9.2 well exhibit the effects of viscoelasticity and damage in the homogeneous material.

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9.2  Comparisons between the responses to a uniaxial stress loading and unloading, applied in the 1-direction at a rate of F˙11 = 0.01s-1, of the homogeneous (H) unreinforced viscoelastic (VE) and elastic (EL) materials. (a) Stress vs. deformation gradient, (b) damage vs. deformation gradient, (c) stress vs. deformation gradient in the absence of damage.

The behavior of the nylon/rubber-like viscoelastic composite to a unixial stress loading and unloading in the 2-direction (i.e., all stress components of T¯ are zero ˙ except T¯22) at a rate of F¯22 = 0.01s–1 is shown in Fig. 9.3 in the presence and absence of damage in the finite viscoelastic matrix constituent. The loading is applied in the 2-direction (transverse loading) since a loading in the fiber 1-direction will be dominated by the elastic fibers whose modulus is much higher than that of the matrix. In the presence of damage in the polymeric matrix phase, its evolution is exhibited in Fig. 9.3(b). It can be observed from this figure that the effect of viscoelasticity and damage in the matrix phase is pronounced although the volume fraction of the fibers is quite low.

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9.3  Comparisons between the responses to uniaxial transverse stress loading and unloading, applied at a rate of F¯˙ 22 = 0.01s-1, of the nylon/ rubber-like composite in the presence and absence of damage in the viscoelastic matrix phase. (a) Global stress vs. the externally applied deformation gradient, (b) damage evolution in the matrix phase vs. the externally applied deformation gradient.

In the next four figures, Fig. 9.4–9.7, the uniaxial stress responses of the viscoelastic nylon/rubber-like composite to five cycles of loading-unloading (linearly increasing and decreasing) applied at different rates and amplitudes in the transverse direction are studied. Figure 9.4(a) shows the composite response ˙ to five cycles of transverse loading-unloading (0.5 ≤ F¯22 ≤ 1.5) applied at a rate of ˙ F¯22 = 1s–1 in the presence of evolving damage in the polymeric matrix phase. This response is compared to the response of the homogeneous (H) viscoelastic matrix. The corresponding damage evolutions are shown in Fig. 9.4(b) which indicates that the values of damage generated in the matrix phase are higher than those induced in the unreinforced matrix. This observation can be attributed to the fact that the presence of the fibers gives rise to higher values of accumulated strains in the polymeric matrix constituent. It should be noted, however, that the damage retains its value after the completion of the first cycle. Figure 9.4(c), which is the counterpart of Fig. 9.4(a), exhibits the corresponding composite and its homogeneous matrix behaviors in the absence of damage. A comparison between Fig. 9.4(a) and (c) shows that the effect of damage evolution in the matrix is significant (note that the scale of the ordinate of Fig. 9.4(c) is three times the ordinate of Fig. 9.4(a)). Furthermore, the effect of dissipation is clearly observed by the attenuated amplitudes in Fig. 9.4(a) and (c). In all cases, the fifth cycle ends 10 s after the initiation of applied cyclic loading. It should be interesting to compare the response shown in Fig. 9.4 with that caused by applying the same five cycles of transverse loading-unloading but at a

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9.4  The response to uniaxial transverse stress cyclic loading-unloading (0.5 ≤ F¯ 22 ≤ 1.5) applied at a rate of F¯˙ 22 = 1s-1 of the nylon/rubber-like composite. Also shown is the corresponding response of the homogeneous (H) unreinforced finite viscoelastic matrix. (a) Global stress vs. the number cycles, (b) damage vs. number of cycles, (c) global stress vs. number of cycles in the absence of damage.

˙ rate of F¯22 = 0.01s–1. The results are shown in Fig. 9.5. As expected, lower values of induced stresses are presently obtained, but the effect of rate on the evolving damage appears to be negligible. Presently, the fifth cycle ends 1000 s after the initiation of applied loading-unloading. Figures 9.4 and 9.5 exhibited the global transverse stress response T¯22 against the number of loading-unloading cycles. It should be interesting to display the stress T¯22 against the externally applied deformation gradient F¯22. This is shown in Fig. 9.6, which corresponds to Fig. 9.5 where the deformation gradient

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9.5  The response to uniaxial transverse stress cyclic loading–unloading (0.5 ≤ F¯ 22 ≤ 1.5) applied at a rate of F¯˙ 22 = 0.01s-1 of the nylon/rubber-like composite. Also shown is the corresponding response of the homogeneous (H) unreinforced finite viscoelastic matrix. (a) Global stress vs. the number cycles, (b) damage vs. number of cycles, (c) global stress vs. number of cycles in the absence of damage.

˙ (0.5 ≤ F¯22 ≤ 1.5) is applied at a rate of F¯22 = 0.01s–1. Figures 9.6(a) and (b) show the degradation of the global stress of the nylon/rubber-like viscoelastic composite with the externally applied cyclic deformation gradient in the presence and absence of damage, respectively. Figures 9.6(c) and (d), on the other hand, show the corresponding degradation of the stress in the homogeneous (H) finite viscoelastic matrix with the applied cyclic deformation gradient in the presence and absence of damage, respectively. It is interesting that the degradation of the ¯ = 1.5, exhibiting a stress in the composite is pronounced in the vicinity of F 22

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9.6  The response to uniaxial transverse stress cyclic loading–unloading (0.5 ≤ F¯ 22 ≤ 1.5) applied at a rate of F¯˙ 22 = 0.01s-1 of the nylon/rubber-like composite. (a) Global stress vs. applied deformation gradient, (b) global stress vs. applied deformation gradient in the absence of evolving damage in the viscoelastic matrix phase, (c) stress vs. applied deformation gradient in the homogeneous (H) unreinforced finite viscoelastic matrix, (d) stress vs. applied deformation gradient in the homogeneous (H) unreinforced finite viscoelastic matrix in the absence of evolving damage.

decrease of the tensile stress with the increase of number of cycles, whereas in the homogeneous material it is pronounced in the vicinity of F22 = 0.5, exhibiting an increase of the compressive stress with cycles. Figures 9.4 and 9.5 displayed the effect of applying cyclic loading at different rates. Presently, the effect of applying cyclic loading of different amplitudes but at the same rate is investigated. Figure 9.7 shows the transverse stress response and damage evolution in the viscoelastic matrix of the nylon/rubber-like composite that is subjected to five cycles of uniaxial transverse stress loading applied at a

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9.7  The response of the nylon/rubber-like composite to two uniaxial transverse stress cyclic loading–unloading: 0.7 ≤ F¯ 22 ≤ 1.3 and 0.4 ≤ F¯ 22 ≤ 1.6, both of which are applied at a rate of F¯˙ 22 = 0.01s-1. (a) Global stress vs. the number of cycles, (b) damage in the matrix phase vs. number of cycles.

˙ rate of F¯22 = 0.01s–1 but amplitudes of 0.7 ≤ F¯22 ≤ 1.3 and 0.4 ≤ F¯22 ≤ 1.6 (i.e., doubling the amplitude). It can be readily concluded that the effect of nonlinearity on the transverse stress response shown in Fig. 9.7(a) is tremendous, since the response caused by the first applied cyclic loading is much lower than the second one although the amplitude of the second is just twice the first one. Doubling the amplitude causes the maximum value of damage to increase to about 1.3 times; see Fig. 9.7 (b). The creep behavior of the nylon/rubber-like composite and its homogeneous unreinforced matrix can be investigated by applying a constant transverse stress T¯22 with all other stress components equal to zero. The resulting variation of the deformation gradients F¯22 against time t are shown in Fig. 9.8 for an applied transverse stress of T¯22 = 1MPa. Figure 9.8 (a) shows the creep behavior of the composite, which is compared with the creep of the homogeneous (H) viscoelastic matrix. Obviously, the existence of the fibers decreases the resulting deformation gradient. Figure 9.8(b) exhibits the evolving damage in the viscoelastic matrix of the composite as well as the damage evolution in the unreinforced matrix, which is observed to be lower than the former. Figure 9.8(c) shows the deformation gradient variation of the composite and its homogeneous (H) matrix in the absence of any damage effects. As expected, lower deformation values are obtained in this case. It is interesting to observe the creep behavior of the nylon/rubber-like composite by doubling the applied transverse loading to T¯22 = 2MPa. Figure 9.9

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9.8  Creep behavior of the nylon/rubber-like composite which is subjected to a transverse stress loading T¯22 = 1MPa. Also shown is the corresponding creep behavior of the homogeneous (H) unreinforced viscoelastic matrix. (a) Global deformation gradient vs. time, (b) damage evolution vs. time, (c) global deformation gradient vs. time in the absence of damage effects.

presents a comparison between the resulting deformation gradients and damage evolutions caused by the application of T¯22 = 1MPa and 2MPa. Here too, the significant nonlinear behavior of the viscoelastic matrix is well displayed. By doubling the applied transverse stress, the maximum value of damage increases to about 1.05 times. When the nylon/rubber-like composite and its homogeneous (H) unreinforced matrix are subjected to a deformation gradient F¯22 in the transverse direction (with the global stress components in all other directions are zero) relaxation behavior is obtained. This is shown in Fig. 9.10(a) for an applied global deformation ¯ = 1.5, whereas the resulting damage evolutions are exhibited in gradient F 22 Fig. 9.10(b). As in the creep behavior which was shown in Fig. 9.8(b), here too

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9.9  Creep behavior of the nylon/rubber-like composite which is subjected to transverse stress loading of T¯22 = 1MPa and 2MPa. (a) Global deformation gradient vs. time, (b) damage evolution vs. time.

9.10  Relaxation behavior of the nylon/rubber-like composite which is subjected to a transverse deformation gradient F¯22 = 1.5. Also shown is the corresponding relaxation behavior of the homogeneous (H) unreinforced viscoelastic matrix. (a) Global stress vs. time, (b) damage evolution vs. time, (c) global stress vs. time in the absence of damage effects.

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9.11  Relaxation behavior of the nylon/rubber-like composite which is subjected to transverse deformation gradients of F¯ 22 = 1.5, 2, 2.5 and 3. (a) Global stress vs. time, (b) damage evolution vs. time.

the amount of damage which is induced in the viscoelastic matrix phase of the composite is higher than the resulting damage in the unreinforced matrix that is subjected to the same value of loading. Figure 9.10(c) shows the relaxation behavior of the composite and its homogeneous matrix in the absence of any damage effects. As expected, the existence of the fibers causes a smaller amount of relaxation as compared to that of the unreinforced matrix, and the absence of damage increases the values of the relaxing stresses. Finally, Fig. 9.11 compares the relaxation behavior and the resulting damage evolutions in the nylon/rubber-like composite for applied transverse deformation gradients of F¯22 = 1.5, 2, 2.5 and 3. This figure well exhibits the severe effects of viscoelasticity and nonlinearity of the matrix. It is noted that with the applied loading of F¯22 = 1.5 and 3 there is a load increase of 4 times, but the damage increase shows just about 1.3 times.

9.6

Conclusions

A finite strain micromechanical analysis has been employed to investigate the behavior of unidirectional composites that consist of finite viscoelastic polymeric matrix phase. The micomechanical analysis is based on the homogenization technique for periodic microstructure and establishes the instantaneous concentration and effective stiffness tensors of the composite. The finite viscoelastic model is based on the multiplicative decomposition of the deformation gradient and allows severe deviations from thermodynamic equilibrium by adopting a nonlinear evolution law for the internal variables. The integration of this law is based on the return mapping algorithm and exponential

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approximation that were previously developed in the context of elastoplasticity. In addition, damage evolution effects in the viscoelastic matrix have been incorporated according to which the damage is proportional to the maximum accumulated strain in the material. In particular, this damage model provides the Mullins effect observed in hyperelastic material. The present finite strain micromechanical model with finite viscoelastic coupled with damage constituents is quite comprehensive since the prediction of the behavior of composites with finite linear viscoelastic phases where the deformations are large but the deviations from equilibrium are small (i.e., Be ≈ I, implying that the dependence on the strain is nonlinear but the dependence on the strain rate is linear) is obtained merely as a special case. In addition, the micromechanical analysis of hyperelastic polymer matrix composites forms another special case of the present model. Applications have been presented that exhibit the response of the composite under loading-unloading, cyclic loading and its creep and relaxation behavior. Comparisons with the response, creep and relaxation behavior of the homogeneous (unreinforced) finite viscoelastic matrix in the presence and absence of damage were presented. The present finite strain micromechanical model can be employed as constitutive equations in a finite element software to investigate the response, creep and relaxation behavior of viscoelastic composite structures (e.g., polymer matrix composite beams, plates and shells). Here, every point on the structure is governed by the established constitutive equations (Eq. 9.43) and has its own instantaneous stiffness tensor (Eq. 9.44) and nonlinear evolution equation (Eq. 9.12). Such an investigation will extend the multiscale micro-macrostructural analysis of Haj-Ali and Aboudi (2009) which, in the framework of infinitesimal strains, linked the HFGMC micromechanical model for nonlinearly elastic and viscoplastic composites to ABAQUS finite element software, to obtain the corresponding behavior of viscoelastic composite structures undergoing large deformations.

9.7

References

Aboudi J (1991), Mechanics of Composite Materials – A Unified Micromechanical Approach, Amsterdam, Elsevier. Aboudi J (2008), ‘Finite strain micromechanical modeling of multiphase composites’, Int J Multiscale Comput Engrg, 6, 411–434. Aboudi J (2009a), ‘The effect of evolving damage on the finite strain response of inelastic and viscoelastic composites’, Materials, 2, 1858–1894. Aboudi J (2009b), ‘Finite strain micromechanical analysis of rubber-like matrix composites incorporating the Mullins damage effect’, Int J Damage Mech, 18, 5–29. Aboudi J and Pindera M-J (2004), ‘High-fidelity micromechanical modeling of continuously reinforced elastic multiphase materials undergoing finite deformation’, Math Mech of Solids, 9, 599–628.

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Christensen R M (1982), Theory of Viscoelasticity, New York, Academic Press. Cuitino A and Ortiz M (1992), ‘A material-independent method for extending stress update algorithms from small-strain plasticity with multiplicative kinematics’, Engng Comp, 9, 437–451. Eterovic A L and Bathe K-J (1990), ‘A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures’, Int J Num Meth Engrg, 30, 1099–1114. Haj-Ali R and Aboudi J (2009), ‘Nonlinear micromechanical formulation of the high-fidelity generalized method of cells’, Int J Solids Struct, 46, 2577–2592. Haj-Ali R and Muliana A H (2003), ‘Micromechanical models for the nonlinear viscoelastic behavior of pultruded composite materials’, Int J Solids Struct, 40, 1037–1057. Haupt P (1993) ‘On the mathematical modelling of material behaviour in continuum mechanics’, Acta Mech, 100, 129–154. Holzapfel G A (2000), Nonlinear Solid Mechanics, New York, John Wiley. Lin R C and Schomburg U (2003), ‘A finite elastic-viscoelastic-elastoplastic material law with damage: theoretical and numerical aspects’, Comput Methods Appl Mech Engrg, 192, 1591–1627. Lockett F J (1972), Nonlinear Viscoelastic Solids, New York, Academic Press. Miehe C and Keck J (2000), ‘Superimposed finite elastic-viscoelastic stress response with damage in filled rubbery polymers. Experiments, modelling and algorithmic implementation’, J Mech Phys Solids, 48, 323–365. Muliana A H and Kim J S (2007), ‘A concurrent micromechanical model for nonlinear viscoelastic behaviors of composites reinforced with solid spherical particles’, Int J Solids Struct, 44, 6891–6913. Mullins L and Tobin N R (1957), ‘Theoretical model for the elastic behavior of filledreinforced vulcanized rubbers’, Rubber Chem Tech, 30, 555–571. Ogden R W (1984), Non-linear Elastic Deformations, Chichester, Ellis Horwood. Reese S and Govindjee S (1998), ‘A theory of finite viscoelasticity and numerical aspects’, Int J Solids Struct, 35, 3455–3482. Simo J C (1987), ‘On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects’, Comp Meth Appl Mech Engrg, 60, 153–173. Simo J C (1992), ‘Algorithms for static and dynamic multiplicative plasticity that presuurve the classical return mapping schemes of the infinitesimal theory’, Comp Meth Appl Mech Engrg, 99, 61–112. Weber G and Anand L (1990), ‘Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids’, Comp Meth Appl Mech Engrg, 79, 173–202.

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10 Fibre bundle models for creep rupture analysis of polymer matrix composites F. K un, University of Debrecen, Hungary Abstract: Fibre bundle models are one of the most important theoretical approaches to the investigation of the failure of fibre reinforced composites under various loading conditions. The chapter first presents the basic concept of fibre bundle models. It then reviews extensions of the modelling approach to capture physical mechanisms responsible for the time-dependent response and creep rupture of composites. The chapter discusses the macroscopic time evolution of the fibre bundle, the microscopic process of rupture, and finally outlines methods to estimate the rupture life of composites. Key words: creep rupture, fibre bundle model, global load sharing.

10.1 Introduction Under a constant external load materials typically exhibit a time-dependent deformation and fail after a finite time. This creep rupture process also substantially affects the technological application of fibre-reinforced composites which calls for a thorough theoretical understanding. Fibre bundle models (FBM) are one of the most important theoretical approaches to the damage and fracture of fibrereinforced composite materials which have also been applied to investigate creep rupture phenomena. The basic concept of fibre bundle modelling was introduced by Peirce (1926) to understand the strength of cotton yarns. Cotton threads were represented by fibres with different load-bearing capacity organized in a bundle. In his pioneering work, Daniels (1945) provided the sound probabilistic formulation of the model and carried out a comprehensive study of bundles of threads assuming equal load sharing after subsequent failures. Already this basic setup of the model provided a surprisingly deep insight into the microscopic dynamics of failure of fibrous systems. The first attempt to capture fatigue and creep effects in FBMs was made by Coleman (1958), who proposed a time-dependent formulation of the model, assuming that the strength of loaded fibres is a decreasing function of time. Later on these early works initiated an intense research in both the engineering (Harlow and Phoenix, 1978; Harlow and Phoenix, 1991; Phoenix, et al., 1997) and physics (Alava et al., 2006; Herrmann and Roux, 1990; Chakrabarti and Benguigui, 1997; Curtin, 1998; Sornette, 1989; Andersen et al., 1997; Pradhan and Chakrabarti, 2003) communities making fibre bundle models successful not only for the study of composites but also for the understanding of the damage and fracture of the broader class of disordered materials. 327 © Woodhead Publishing Limited, 2011

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During the past few decades, the development of fibre bundle models encountered two kinds of challenges: on the one hand, it is important to work out realistic models of materials failure which have a detailed representation of the microstructure of the material, the local stress fields and their complicated interaction. Applications of materials in construction components require the development of analytical and numerical models which are able to predict the damage histories of loaded specimens in terms of the characteristic microscopic parameters of the constituents. In this context, fibre bundle models served as a starting point to develop more realistic micromechanical models of the failure of fibre-reinforced composites including also polymer matrix composites (PMC). Analytical methods and numerical techniques have been developed making possible realistic treatment of even large-scale fibrous structures (Evans and Zok, 1994; Du and McMeeking, 1995; Curtin and Scher, 1997; Curtin, 1998; Chudoba et al., 2006; Phoenix and Beyerlein, 2000; Kun and Herrmann, 2000; Hidalgo et al., 2002a; Kun et al., 2003; Kovacs et al., 2008). On the other hand, it is important to reveal universal aspects of the fracture of composites which are independent of specific material properties relevant on the micro level. Such universal quantities can help to extract the relevant information from measured data and make it possible to design monitoring techniques of the gradual degradation of composites’ strength and construct methods to forecast catastrophic failure events (Alava et al., 2006; Garcimartin et al., 1997; Vujosevic and Krajcinovic, 1997; Johansen and Sornette, 2000; Pradhan and Chakrabarti, 2003; Pradhan et al., 2005; Kun and Herrmann 2000; Kun et al., 2003a, 2003b; Nechad et al., 2005, 2006). The damage and fracture of disordered materials addresses several interesting problems also for statistical physics. It is challenging to embed the failure and breakdown of materials into the general framework of statistical physics clarifying its analogy to phase transitions and critical phenomena. For this purpose fibre bundle models provide an excellent testing ground of ideas offering also the possibility of analytic solutions (Alava et al., 2006; Baxevanis and Katsaunis, 2006; Baxevanis and Katsaunis, 2007; Baxevanis, 2008; Pradhan and Chakrabarti, 2003; Pradhan et al., 2005; Kun et al., 2000, 2003a, 2003b; Kovacs et al., 2008). Fibre-reinforced composites (FRCs), where long parallel fibres are embedded in a matrix material, provide an improved mechanical performance compared to their constituents. Subjecting the composite to an external load, the matrix material typically suffers multiple cracking or yields at load levels much below the strength of the fibres. It has the consequence that practically all the load is kept by the fibres and the matrix material mainly affects the stress transfer following fibre breakings (Jones, 1999; Evans and Zok, 1994). Motivated by these observations, fibre bundle models consist of a set of parallel fibres having statistically distributed strengths. The sample is loaded parallel to the fibres’ direction, and the fibres fail if the load on them exceeds their threshold value. In stress-controlled experiments, after each fibre failure the load carried by the broken fibre is redistributed among the intact ones. In basic FBMs the matrix material only affects the load transfer from the broken to the

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intact fibres. The behaviour of a fibre bundle under external loading strongly depends on the range of interaction, i.e. on the range of load sharing among fibres. Exact analytic results on FBM have been achieved in the framework of the mean field approach, or global load sharing, which means that after each fibre breaking the stress is equally distributed on the intact fibres, implying an infinite range of interaction and a neglect of stress enhancement in the vicinity of failed regions (Daniels, 1945; Sornette, 1989; Hemmer and Hansen, 1992; Andersen et al., 1997; Garcimartin et al., 1997; Kloster et al., 1997; Kun and Herrmann, 2000; Moreno et al., 2000; Hidalgo et al., 2002b; Kun et al., 2003a, 2003b; Kovacs et al., 2008; Pradhan and Chakrabarti, 2003). In spite of their simplicity, FBMs capture the most important aspects of material damage and they provide a deep insight into the fracture process. Based on their success, FBMs have served as a starting point for more complex models like the micromechanical models of the failure of fibrereinforced composites (Harlow and Phoenix, 1991; Phoenix et al., 1997; Curtin and Scher, 1997; Du and McMeeking, 1995; Kun and Herrmann, 2000; Hidalgo et al., 2002a, 2002b). Over the past few years several extensions of FBM have been carried out by considering stress localization (local load transfer) (Harlow and Phoenix, 1991; Phoenix et al., 1997; Curtin, 1998; Hidalgo et al., 2002b), the effect of matrix material between fibres (Harlow and Phoenix, 1991; Phoenix et al., 1997; Hidalgo et al., 2002a, 2002b), possible non-linear behaviour of fibres (Kun et al., 2000; Hidalgo et al., 2001), coupling to an elastic block (Roux et al., 1999), and thermally activated breakdown (Phoenix and Tierney, 1983; Newman and Phoenix, 2001; Curtin and Scher, 1997; Roux, 2000; Scorretti et al., 2001; Yoshioka et al., 2008). Modelling the creep rupture of fibre-reinforced composites requires a substantial extension of FBMs to capture the physical mechanisms which lead to timedependent response of the composite under a constant external load. In this chapter we first present the basic formulation of the classical fibre bundle model and briefly summarize the most important recent results obtained on the macroscopic response and microscopic damage process in the framework of FBMs. For the investigation of creep rupture of fibre composites two modelling approaches will be discussed. First we consider a bundle of visco-elastic fibres which have an instantaneous breaking, then we construct a bundle of brittle fibres which undergo a slow relaxation process after failure. Assuming global load sharing, the time evolution of the deformation and of the gradual damaging of the creeping system will be analyzed analytically, while the microscopic process of failure will be investigated by means of computer simulations. Based on fibre bundle models, reliable methods will be deduced which make it possible to estimate the rupture life or to forecast the imminent catastrophic failure event of composites.

10.2 Fibre bundle model In the framework of the fibre bundle model, the composite is represented as a discrete set of parallel fibres of number N organized on a regular lattice (square,

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triangular, . . .; see Fig. 10.1a. The fibres can solely support longitudinal deformation which make it possible to study only loading of the bundle parallel to the fibre axis. When the bundle is subjected to an increasing external load, the fibres are assumed to have perfectly brittle response, i.e. they have a linearly elastic behaviour until they break at a failure load σ ith, i = 1, . . ., N as illustrated in Fig. 10.1b. The elastic behaviour of fibres is characterized by the Young’s modulus E, which is identical for all fibres. The breaking of fibres is assumed to be instantaneous and irreversible such that the load on broken fibres drops down to zero immediately at the instant of failure (see Fig. 10.1b); furthermore, broken fibres are never restored (no healing). The strength of fibres σ ith, i = 1, . . ., N, i.e. the value of the local load at which they break, is an independent identically distributed random variable with the probability density p(σth) and distribution function . The randomness of breaking thresholds is assumed to represent the disorder of fibre strength, and hence it is practically the only component of the classical FBM where material-dependent features (e.g. amount of disorder) can be taken into account. After a fibre fails its load has to be shared by the remaining intact fibres. The range and form of interaction of fibres, also called the load sharing rule, is a crucial component of the model which has a substantial effect on the micro and macro behaviour of the bundle. Most of the studies in the literature are restricted to two extreme forms of the load sharing rule: in the case of global load sharing (GLS), also called equal load sharing (ELS), the load is equally redistributed over all intact fibres in the bundle irrespective of their distance from the failed one (Daniels, 1945; Coleman, 1958; Sornette, 1989; Curtin and Scher, 1997; Phoenix, 2000; Kloster et al., 1997; Pradhan et al., 2005). The GLS rule corresponds to the mean field approximation of FBM where the topology of the fibre bundle becomes

10.1  (a) Basic setup of the fibre bundle model. The system is represented as a bundle of parallel fibres. (b) Single fibres have a linearly elastic behaviour with an identical Young’s modulus E up to failure at a random threshold value σth or εth (brittle breaking).

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irrelevant. Such a loading condition naturally arises when parallel fibres are loaded between perfectly rigid platens, like for the wire cable of an elevator. FBM with global load sharing is a usual starting point for more complex investigations since it makes it possible to obtain the most important characteristic quantities of the bundle in closed analytic forms (Phoenix, 2000; Phoenix and Beyerlein, 2000; Kloster et al., 1997; Curtin, 1997; Hemmer and Hansen, 1992; Pradhan et al., 2005; Pradhan and Hansen, 2003; Andersen et al., 1997; Sornette, 1989; Hidalgo et al., 2001). In the other extreme of the local load sharing (LLS), the entire load of the failed fibre is redistributed equally over its local neighbourhood (usually nearest neighbours) in the lattice considered, leading to stress concentrations along failed regions. Due to the non-trivial spatial correlations, the analytic treatment of LLS bundles has serious limitations (Gomez et al., 1993; Phoenix et al., 1997; Harlow and Phoenix, 1978; Harlow and Phoenix, 1991) most of the studies here rely on large-scale computer simulations (Hansen and Hemmer, 1994; Curtin, 1998; Harlow and Phoenix, 1978; Hidalgo et al., 2002a, 2002b). Such localized load sharing occurs when a bundle of fibres is loaded between plates of finite compliance. In the following we briefly summarize the most important results of FBMs on the microscopic failure process and macroscopic response of disordered materials under quasi-static loading conditions focusing on the case of equal load sharing. Loading of a parallel bundle of fibres can be performed in two substantially different ways: when the deformation ε of the bundle is controlled externally, the load on single fibres σi, i = 1, . . ., N is always determined by the externally imposed deformation ε as σi = Eε, i.e. no load sharing occurs and consequently the fibres break one by one in the increasing order of their breaking thresholds. At a given deformation ε the fibres with breaking thresholds σ ith < Eε are broken; furthermore, all intact fibres keep the equal load Eε. Hence, the macroscopic constitutive behaviour σ (ε) of the bundle can be cast in the analytic form

[10.1]

where the term 1 – P(ε) provides the fraction of intact fibres at the deformation ε. For a broad class of threshold distributions the constitutive curve Eq. 10.1 has a linearly increasing behaviour for low deformations followed by a quadratic maximum and a softening regime. The stochastic strength of fibres is well described by the Weibull distribution

[10.2]

where the parameters ρ and λ depend on material properties. The value of λ sets the scale of fibres’ strength, while the Weibull exponent ρ controls the amount disorder in the system. Figure 10.2 presents the constitutive curve Eq. 10.1 of FBMs with Weibull distributed thresholds for different values of the Weibull exponent ρ. In stress-controlled experiments gradually increasing the external

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10.2  Constitutive curves of fibre bundles with Weibull distributed fibre strength for different values of the Weibull exponent ρ. The value of the scale parameter λ and of Young’s modulus E was fixed λ = 1, E = 1.

load σ, a more complex failure process arises: breaking events are followed by the redistribution of load over the remaining intact fibres which might induce further breakings. It follows that the breaking of a single fibre can trigger an entire avalanche of breakings which either stops and the bundle stabilizes, or the avalanche destroys the entire bundle leading to macroscopic failure. The catastrophic avalanche appears when the maximum of σ(ε) is reached, hence the position εc and value σc of the maximum define the critical strain and stress of the bundle. It can be observed in Fig. 10.2 that decreasing the amount of disorder, i.e. increasing the Weibull exponent ρ, the macroscopic failure of the bundle becomes more and more brittle. Analytic calculations revealed that the macroscopic strength of fibre bundles depends on the size of the bundle σc(N), i.e. assuming equal load sharing over the fibres σc(N) rapidly converges to a finite number with increasing N. However, when the load redistribution gets localized to a small neighbourhood of failed fibres, the bundle becomes more brittle with a lower strength than its equal load sharing counterpart, and in the limit of N → ∞ the strength of the bundle tends to zero with the logarithm of the number of fibres σc(N) ∝ 1/ln N (Kloster et al., 1997; Phoenix and Beyerlein, 2000). As the external load approaches the ultimate tensile strength σc, damaging of the bundle proceeds in larger and larger breaking bursts. Kloster et al. (1997) and Pradhan et al. (2005) pointed out that the distribution P of burst sizes ∆ in FBMs has a power law

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functional form P(∆) ∝ ∆–κ with an exponent κ = 5/2 which is universal for a broad class of fibre strength distribution.

10.3 Fibre bundle models for creep rupture In order to model creep rupture of fibre-reinforced composites the classical fibre bundle model outlined above has to be complemented with physical mechanisms which capture the time-dependent response of composite materials. In this chapter we consider two physical origins of time-dependent behaviour which are of high importance to model polymer composites. First, we assume that the fibres are visco-elastic and exhibit a time-dependent response under a constant load. Then we consider the case when the fibres are brittle without any time dependence; however, due to the plastic response of the surrounding matrix material broken fibres undergo a slow relaxation process. The slow redistribution of the load after failure events gives rise to an overall time-dependent response of the entire system. In the literature, time-dependent fibre bundles have also been introduced by assuming that brittle fibres have a finite (stochastically distributed) lifetime when subject to a constant load (Coleman, 1958; Newman and Phoenix, 2001; Roux, 2000; Scorretti et al., 2001; Yoshioka et al., 2008). This breaking mechanism may arise due to thermally activated crack nucleation and damage accumulation processes. These types of modelling approaches will not be presented here.

10.3.1  Bundle of viscoelastic fibres One of the simplest physical mechanisms which results in a time-dependent response of a system under a constant external load is the visco-elastic behaviour of the constituents. Following the general presentation of FBMs in Section 10.2, the visco-elastic fibre bundle model of creep rupture of composites consists of N parallel fibres having visco-elastic constitutive behaviour. For simplicity, the pure linearly visco-elastic fibres are modelled by a Kelvin-Voigt element which consists of a spring and a dashpot in parallel, which is illustrated in Fig. 10.3a (Du and McMeeking, 1995; Hidalgo et al., 2002a, 2002b). The constitutive equation of . fibres can be cast into the form σ0 = βε + Eε, where σ0 is the constant external load, β denotes the damping coefficient, and E is the Young’s modulus of fibres. In order to capture failure in the model a strain-controlled breaking criterion is imposed, i.e. a fibre fails during the time evolution of the system when its strain exceeds a i , i = 1, . . ., N drawn from a probability distribution P(ε). For breaking threshold εth the stress transfer between fibres following local failure events, equal load sharing is assumed, i.e. the excess load is equally shared by all the remaining intact fibres, which provides a satisfactory description of load redistribution in unidirectional long fibre reinforced composites (Phoenix et al., 1997; Curtin, 1997; Jones, 1999). The construction of the model is illustrated in Figure 10.3a. In the framework of global load sharing, most of the quantities describing the behaviour of the fibre

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10.3  (a) The viscoelastic fibre bundle model. Intact fibres are modelled by Kelvin-Voigt elements. (b) Deformation–time diagram ε(t) of the model obtained analytically at different external loads σ below and above σc.

bundle can be obtained analytically. In this case the time evolution of the system under a steady external load σ0 is described by the first-order differential equation

[10.3]

where the visco-elastic behaviour is coupled to the failure of fibres (Hidalgo et al., 2002a; Kun et al., 2003). Note that the above equation of motion has a similar structure to the constitutive equation of simple fibre bundles Eq. 10.1; the viscoelasticity just introduces a delay term in the equation. To describe the creeping process of the fibre bundle, Eq. 10.3 has to be solved for the deformation ε(t) at a constant external load σ0. For the behaviour of the solutions ε(t), two distinct regimes can be distinguished depending on the load level σ0: when σ0 falls below a critical value σc, Eq. 10.3 has a stationary solution εs, which can be obtained by . setting ε = 0 so that

[10.4]

It means that as long as this equation can be solved for εs at a given external load σ0, the solution ε(t) of Eq. 10.3 converges to the stationary value ε(t) → εs when t → ∞, and the system suffers only a partial failure. However, when σ0 exceeds . the critical value σc no stationary solution exists; furthermore, the derivative ε remains always positive, which implies that for σ0 > σc the strain of the system ε(t) monotonically increases until the system fails globally at a finite time tf (Hidalgo et al., 2002b; Kun et al., 2003a, 2003b). The behaviour of ε(t) is presented in Figure 10.3b for several values of σ0 below and above σc for Weibull

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distributed breaking thresholds with the parameters λ = 1, ρ = 2. It follows from the above argument of stationary solutions that the critical value of the load σc is the ultimate tensile strength (UTS) of the bundle. The creep rupture of the viscoelastic bundle can be interpreted so that for σo ≤ σc the bundle is partially damaged implying an infinite lifetime tf = ∞ and the emergence of a stationary macroscopic state, while above the critical load σ0 > σc global failure occurs at a finite time tf, but in the vicinity of σc the global failure is preceded by a long-lived steady state. The nature of the transition occurring at σc can be characterized by analyzing how the creeping system behaves when approaching the critical load both from below and above. For σ0 ≤ σc the bundle relaxes to the stationary deformation εs through a gradually decreasing breaking rate of fibres. It can be shown analytically that ε(t) has an exponential relaxation to εs with a characteristic time scale τ that depends on the external load σ0 as τ ∝ (σc – σ0)–1/2 for σ0 < σc, i.e. when approaching the critical point from below the characteristic time of the relaxation to the stationary state diverges according to a universal power law with an exponent –1/2 independent of the form of the disorder distribution P. Above the critical point the lifetime tf defines the characteristic time scale of the bundle which can be cast in the analytic form tf ∝ (σ0 – σc)–1/2 for σ0 > σc, so that tf also has a power law divergence at σc with a universal exponent –1/2 like τ below the critical point. Hence, for global load sharing the system exhibits scaling behaviour on both sides of the critical point indicating a continuous transition at the critical load σc. It has been discussed in the previous section that even in the case of equal load sharing, the macroscopic strength of fibre bundles exhibits a size effect, which gets more pronounced when the load sharing is localized. We have shown analytically that the lifetime of creeping fibre bundles shows a similar size effect. Fixing the external load above the critical point σ0 > σc, the lifetime tf (N) of a finite bundle of N fibres can be cast into the analytic form

[10.5]

which means that tf (N) exhibits a universal scaling tf (N) – tf (∞) ∝ 1 / N with respect to the number N of fibres of the bundle (Hidalgo et al., 2002a; Kun et al., 2003). Here tf (∞) denotes the lifetime of the infinite system N → ∞. It has to be emphasized that the precise form of the threshold distribution affects only the multiplication factor of the N dependence.

10.3.2  Bundle of slowly relaxing fibres Another important microscopic mechanism which can lead to macroscopic creep is the slow relaxation of fibres after breaking. In this case, fibres of the composite are linearly elastic until they break; however, after breaking they undergo a slow relaxation process, which can be caused, for instance, by the sliding of broken fibres with respect to the matrix material or by the creeping matrix. To take into

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account this effect, our approach is based on the model introduced by Du and McMeeking (1995), Fabeny and Curtin (1996), Hidalgo et al. (2002b), and Kun et al. (2003), where the responses of a visco-elastic-plastic matrix reinforced with elastic and also visco-elastic fibres have been studied. The model consists of N parallel fibres, which break in a stress-controlled way, i.e. by subjecting a bundle to a constant external load, fibres break during the time evolution of the system when the local load on them exceeds a stochastically distributed breaking i , i = 1, . . ., N. Intact fibres are assumed to be linearly elastic i.e. threshold σ th σ = Ef εf holds until they break, and hence for the deformation rate it applies

[10.6]

Here εf denotes the strain and Ef the Young’s modulus of intact fibres, respectively. The main assumption of the model is that when a fibre breaks its load does not drop down to zero instantaneously; instead it undergoes a slow relaxation process introducing a time scale into the system. In order to capture this effect, the broken fibres with the surrounding matrix material are modelled by Maxwell elements as illustrated in Figure 10.4a, i.e. they are conceived as serial couplings of a spring and a dashpot which result in a non-linear response

[10.7]

Here σb and εb denote the time dependent load and deformation of a broken fibre, respectively. The relaxation process of broken fibres is characterized by three

10.4  (a) Bundle of brittle fibres which undergo a slow relaxation process after failure. Broken fibres are modelled by Maxwell elements which are serial couplings of a spring and a dashpot. (b) Deformation–time diagrams of the model obtained at different load levels below and above the critical load σc.

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parameters Eb, B and m, where Eb is the effective stiffness of a broken fibre, and the exponent m characterizes the strength of non-linearity of the element. We study the behaviour of the system for the region m ≥ 1. Assuming global load sharing for the load redistribution, the constitutive equation describing the macroscopic elastic behaviour of the composite reads as

[10.8]

The second term on the right-hand side of the above equation takes into account that broken fibres carry also a certain amount of load σb(t), furthermore, P(σ (t)) and 1 – P(σ (t)) denote the fraction of broken and intact fibres at time t, respectively (Du and McMeeking, 1995; Fabeny and Curtin, 1996; Kun, 2003; Kovacs, 2008). It can be seen from Eq. 10.8 that under a constant external load σ0, the load of intact fibres σ will also be time dependent due to the slow relaxation of the broken ones. Due to the boundary condition illustrated in Fig. 10.4a, the two time derivatives . . have to be always equal: εf = εb. The differential equation governing the time evolution of the system can be obtained by expressing σb in terms of σ using the boundary condition and Eq. 10.7, and substituting it finally into Eq. 10.8 [10.9] In order to determine the initial condition for the integration of Eq. 10.9 we note that upon subjecting the undamaged specimen to an external stress σ0 all the fibres attain this stress value immediately due to the linear elastic response. Hence the time evolution of the system can be obtained by integrating Eq. 10.9 with the initial condition σ (t = 0) = σ0. Since intact fibres are linearly elastic, the deformation-time history ε(t) of the model can be deduced as ε (t) = σ (t) / Ef , which has an initial jump to ε0 = σ0 / Ef . It follows that those fibres which have breaking thresholds σ thi smaller than the externally imposed σ0 immediately break. To characterize the macroscopic behaviour of the composite the solutions σ(t) of Eq. 10.9 have to be analyzed at different values of the external load σ 0. Similarly to the previous model, two different regimes of σ(t) can be distinguished depending on the value of σ 0: if the external load falls below the critical load σ c a stationary . solution σs of the governing equation exists which can be obtained by setting σ = 0 in Eq. 10.9

[10.10]

This means that until the above equation can be solved for σs the solution σ(t) of Eq. 10.9 converges asymptotically to σs resulting in an infinite lifetime tf of the composite. Note that Eq. 10.10 also provides the asymptotic constitutive behaviour of the model which can be measured by quasi-static loading. If the . . external load falls above the critical value the deformation rate ε = σ / Ef always

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remains positive resulting in a macroscopic rupture in a finite time tf (Kun et al., 2003; Kovacs et al., 2008). Representative examples of the solution ε(t) = σ(t) / Ef can be seen in Figure 10.4b for Weibull distributed breaking thresholds with the parameters λ = 1, ρ = 2. The behaviour of the system shows again universal aspects in the vicinity of the critical load σc. Similarly to the previous model, it can also be shown that the lifetime tf of the bundle has a power law divergence when the external load approaches the critical point from above

[10.11]

for σ0 > σc. It is important to emphasize that the exponent is universal in the sense that it is independent of the disorder distribution of the breaking thresholds; however, it depends on the stress exponent m of broken fibres (Kun et al., 2003; Kovacs et al., 2008). In order to numerically justify the analytically deduced behaviour of the time to failure tf as a function of the distance from the critical point, computer simulations were performed for several different values of the creep exponent m. In Fig. 10.5a the results are presented for m = 1.5 and m = 2.5. The slope of the fitted straight lines agrees very well with the analytic predictions of Eq. 10.11. The size scaling of the time to failure tf was analyzed by simulating the creep rupture of bundles of size N = 5 × 102–107 setting a Weibull distribution with λ = 1, ρ = 3 for the breaking thresholds. We found that tf (N) converges to the lifetime of the infinite system tf (∞) according to the universal law Eq. 10.5 independently of the value of the exponent m. In Fig. 10.5b the best fit was obtained for both curves with slope 1 ± 0.05 for both m values in excellent agreement with the analytic predictions.

10.5  (a) Lifetime of the fibre bundle as a function of the distance from the critical load σ0 – σc. Results of computer simulations (symbols) are compared to the analytic results (continuous lines) for two different values of the stress exponent m. (b) The lifetime of finite bundles exhibits a universal scaling which is independent even of the value of m.

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10.3.3  Microscopic process of failure One of the main advantages of fibre bundle studies of creep rupture of composites is that FBMs also provide a detailed physical picture of the microscopic process of rupture (Du and McMeeking, 1995; Fabeny and Curtin, 1996; Kun et al., 2003; Kovacs et al., 2008). In the framework of FBMs, assuming equal load sharing after fibre breakings the macroscopic time evolution of a creeping system can be analyzed in details by analytical means; however, the microscopic process of creep rupture is accessible only by means of computer simulations. On the micro level, during the creep process the fibres break in a single avalanche which either stops and the bundle stabilizes after the breaking of a finite fraction of fibres (σ0 < σc), or the avalanche continues until macroscopic failure occurs (σ0 > σc). Inside this avalanche, due to the disordered breaking thresholds, fibres may break in faster or slower sequences leading to fluctuations of the breaking rate. The process of fibre breaking on the micro level can easily be monitored experimentally by means of acoustic emission techniques. Except for the primary creep regime, where a large amount of fibres break in a relatively short time, the time of individual fibre failures can be recorded with high precision. The microscopic evolution of the rupture process can be characterized by the waiting times ∆t between consecutive fibre breakings and their distribution providing information also on the cascading nature of breakings. By analyzing waiting times based on the acoustic emission techniques in experiments, valuable diagnostic tools can be designed. Representative examples of waiting times ∆t obtained by computer simulations of the slowly relaxing fibre bundle are shown in Fig. 10.6 for loads below (Fig. 10.6a) and above σc (Fig. 10.6b). It can be seen that in both cases at the beginning of the creep process a large number of fibres break, which results in short waiting times, i.e. all the ∆ts are small at the beginning. Below the critical load σ0 < σc, at the macro level a stationary state is attained after a finite fraction of fibres break. Approaching the stationary state ∆t becomes larger (Fig. 10.6a) and reaches a maximum value on the plateau of ε(t) (compare to Fig. 10.4b). Above the critical load σ0 > σc, however, the slow plateau regime with long waiting times is followed by a strain acceleration (Fig. 10.4b) accompanied by a large number of breakings resulting again in small ∆t values (Fig. 10.6b). Varying the stress exponent m the qualitative behaviour of ∆t in Fig. 10.6 does not change. Besides the overall tendencies described above, the waiting times ∆t show quite an irregular local pattern with large fluctuations and have a non-trivial distribution. We determined the distribution function f (∆t) on both sides of the critical point σc varying the stress exponent m within a broad range. Examples of f (∆t) are presented in Fig. 10.7 for a bundle of 106 fibres with the stress exponent m = 2.0, where a power law form of f (∆t) can be observed on both sides of the critical load σc. For σ0 < σc no cutoff function can be identified, the power law prevails over six to seven orders of magnitude in ∆t up to the largest

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10.6  Waiting times ∆t between consecutive fibre breakings as a function of time t for a bundle of N = 105 fibres at a load level σ0 below (a) and above (b) the critical load σc.

values. For σ0 > σc the power law regime is followed by an exponential cutoff which shifts to higher ∆t values when approaching the critical load from above. Hence, the distribution function f(∆t) can be cast into the functional form

[10.12]

where the cutoff waiting time ∆t0 is a decreasing function of the external load σ0 for σ0 > σc. The value of the exponent α is different on the two sides of σc but inside one regime it is independent of the actual value of σ0. It is interesting that computer simulations revealed a strong dependence of α on the stress exponent m. The inset of Fig. 10.7 demonstrates that with increasing m the value of α decreases both in the under-critical and over-critical cases and tends to the same limit value α → 1 at large m. It was shown analytically (Kun et al., 2003; Kovacs et al., 2008)

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10.7  Distribution of waiting times f (∆t) for several different external loads below and above the critical value. The inset presents the exponent α of the distribution as a function of the creep exponent m.

that approaching the critical load from above, ∆t0 rapidly increases and has a power law divergence as a function of σ0 – σc

[10.13]

This behaviour implies that in the limit σ0 → σc the exponential cutoff disappears and f(∆t) becomes a pure power law. The cutoff exponent β can be calculated analytically as a function of the stress exponent m of the Maxwell elements: since .  the cutoff ∆t0 is proportional to the inverse of the minimum strain rate ε –1m, it follows that β = m. Below the critical point σc the distribution of waiting times does not have a cutoff function; furthermore, changing the load in this regime σ0 < σc not only reduces the statistics of the results (at lower loads less fibres break) but the functional form of f(∆t) does not change (see Fig. 10.7). The behaviour of ∆t0 as a function of σ0 in Eq. 10.13 shows that for σ0 < σc the system is always in the state of ∆t0 → ∞ so that in Eq. 10.12 a pure power law remains. We note that a similar power law functional form of the waiting time distribution has recently been found by Baxevanis and Katsaunis (2007) and by Baxevanis (2008) in FBMs using a different type of rheological element to describe the response of fibres and of the surrounding matrix material. These studies showed that the power law form of the waiting time distribution is universal; however, the value of the exponent depends on the precise rheology of elements.

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10.3.4  Predicting the lifetime of fibre bundles One of the most challenging problems for theoretical studies on creep rupture of composites is to derive observables and scaling laws which allow for an accurate prediction of the lifetime of samples from short-term measurements (Monkman and Grant, 1956; Vujosevic and Krajcinovic, 1997; Johansen and Sornette, 2000; Guedes, 2006; Brinson et al., 1981; Binienda et al., 2003). Computer simulations of the fibre bundle model of slowly relaxing fibres showed that the rupture life tf of finite bundles has large sample-to-sample fluctuations due to the quenched disorder of fibre strength. In order to characterize these fluctuations, the probability distribution of lifetime p(tf) was determined numerically, which proved to have a log-normal form, i.e., the logarithm of tf has a normal distribution with the form

[10.14]

where 〈lntf〉 and s(lntf) denote the mean and standard deviation of the logarithmic lifetime lntf (Kovacs et al., 2008). In order to demonstrate the validity of Eq. 10.14, in Fig. 10.8a the standardized distribution is presented, namely, s(lntf)p(lntf) is plotted as a function of (lntf – 〈lntf〉) / s(lntf) together with the standard Gaussian . An excellent agreement can be seen between the numerical results and the standard Gaussian for all the stress exponents m considered. The inset of Fig. 10.8a demonstrates that both the mean 〈lntf〉 and standard deviation s(lntf) of the rupture life increase exponentially with the stress exponent of the material (Kovacs et al., 2008). The result implies that in the case of high m values relevant for experiments, large fluctuations of tf arise. Consequently, the lifetime estimation for finite samples requires the development of methods which provide reliable results for single samples without averaging (Binienda et al., 2003; Brinson et al., 1981; Guedes, 2006; Vujosevic and Krajcinovic, 1997). . Based on the evolution of the rate of deformation ε(t), the creep rupture process . can be divided into three regimes. In the primary creep regime, ε(t) rapidly decreases with time. The secondary creep is characterized by a slowly varying, almost steady deformation rate, which is then followed by strain acceleration in . the tertiary regime for high-enough external loads. Therefore, the strain rate ε(t) . attains a minimum with a value εm at the so-called transition time tm between the secondary and tertiary creep regimes (see Fig. 10.8b for representative examples obtained by numerical solution of the bundle of slowly relaxing fibres). In laboratory experiments, the failure time tf of the specimen is usually estimated . from the variation of the deformation rate ε(t) based on the Monkman-Grant (MG) relationship (Monkman and Grant, 1956). The Monkman-Grant relation is a semiempirical formula which states that the time-to-failure of the system tf is uniquely . related to the minimum creep rate εm in the form of a power law , where the MG exponent ς depends on material properties. The advantage of the MG relation

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10.8  (a) The probability distribution of the lifetime has a universal lognormal functional form. The inset shows that both the average lifetime and its standard deviation increase exponentially with increasing stress exponent. (b) Strain rate of the bundle of slowly relaxing fibres for three different load values above the critical load. The inset demonstrates the validity of the Monkman-Grant relation for the model for the case of m = 2.

is that once the relation is established from short-term tests, the rupture life tf can be . determined just following the system until the minimum of ε(t) is reached (Brinson et al., 1981; Binienda et al., 2003; Guedes, 2006; Vujosevic and Krajcinovic, 1997). For the bundle of slowly relaxing fibres one can obtain analytically the MonkmanGrant relation

[10.15]

which shows that the MG exponent of the model depends on the stress exponent of relaxing fibres ς = 1 – 1/2m. The inset of Fig. 10.8b presents results of computer simulations which justify the validity of the Monkman-Grant relation and the m dependence of the MG exponent for the model (Kovacs et al., 2008). It has recently been pointed out that besides the MG relation, the lifetime tf can directly be related also to the transition time tm between the secondary and tertiary creep regimes (Nechad et al., 2005, 2006). Experiments on different types of composites revealed that tf = 3/2tm holds from which tf can be obtained from a measurement of tm with a significantly shorter duration (Nechad et al., 2005, 2006). For the model of slowly relaxing fibres, Fig. 10.9a presents tf obtained by computer simulations as a function of tm for a uniform distribution between 0 and 1 and for a Weibull distribution of breaking thresholds varying the stress exponent m. We emphasize that each symbol in Fig. 10.9a stands for a single sample with different realizations of the disorder and different values of the external load above the corresponding critical point σ0. It can be observed that all the points fall on the same straight line with relatively small deviations implying a linear relationship tf = atm, where the parameter a has a universal value a = 2.05

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10.9  (a) Lifetime tf of the bundle as a function of the transition time tm for two different values of the ratio Eb / Ef of the Young’s modulus of broken and intact fibres. The linear relation can be nicely observed. (b) The inset shows that in the tertiary creep regime a power law acceleration of the strain rate is obtained with an exponent γ, which depends on m. The main panel of the figure presents the m dependence of the exponent γ.

independent of the type of disorder and of the stress exponent m. The value of a ≈ 2 implies a symmetry of the time evolution of the system with respect to the transition time, which might not be valid for certain type of materials as it is indicated also by the experiments of Nechad et al. (2005, 2006). It can be observed in Fig. 10.9b that during damage-enhanced creep processes in the tertiary creep regime the macroscopic failure of the specimen is approached by an acceleration of the strain rate and breaking rate of fibres. An analogous effect has been observed in rupture experiments on disordered materials with increasing external load, where the acoustic emission rate has a power law singularity at catastrophic failure which also allowed for the possibility of predicting imminent failure events (Andersen et al., 1997; Alava et al., 2006; Johansen and Sornette, 2000; Garcimartin et al., 1997). In the framework of FBM with slowly relaxing fibres, computer simulations were carried out analyzing the functional form of . strain rate ε(t) in the vicinity of the failure time tf . The numerical calculations . revealed a power law divergence of ε as a function of the distance from tf

[10.16]

which is illustrated in Fig. 10.9b Extensive simulations showed that the value of the exponent γ does not depend on the external load σ0 and on the disorder distribution, but is a decreasing function of the stress exponent m governing the relaxation of broken fibres (see the inset of Fig. 10.9b). Similar power law acceleration was revealed analytically by the model of Nechad et al. (2005, 2006), which was also confirmed experimentally. Since Eq. 10.16 contains the rupture

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life of the system, it makes it possible to forecast the imminent failure event of the composite under a steady external load.

10.4 Summary and outlook Under high steady stresses fibre composites may exhibit time-dependent failure called creep rupture, which limits their lifetime and consequently has a high impact on their applicability for construction components. Fibre bundle models, as one of the most important modelling approaches to the failure of fibre-reinforced composites, provide a rich theoretical framework in which creep rupture phenomena of FRCs can also be investigated. In this chapter we have given an overview of the basic concepts of fibre bundle modelling and presented extensions of FBMs which capture physical mechanisms relevant for the time-dependent response and creep rupture of polymer matrix composites. We presented a fibre bundle model of viscoelastic fibres, where the rheological behaviour of single fibres was modelled by Maxwell elements. Assuming equal load sharing over intact fibres, we determined analytically the macroscopic time evolution of the bundle characterized by the deformation-time diagram at different load levels. It was discussed how the time evolution of the accumulating damage changes when the external load approaches the ultimate tensile strength of the system both from below and above. As a consequence of long-range load sharing, a universal power law behaviour of the characteristic time scale of the bundle was revealed where the exponent was independent of any materials details. A more detailed investigation was devoted to the bundle of slowly relaxing fibres where intact fibres exhibit brittle breaking; however, they undergo a slow stress relaxation after failure. It was demonstrated that on the macro level the model provides realistic deformation-time diagrams and in the vicinity of the ultimate tensile strength of the system the lifetime shows again a universal power law behaviour. The exponent of the power law is only influenced by the non-linearity exponent of stress relaxation. The chapter presented a detailed analysis of the microscopic damage accumulation process focusing on the breaking sequence of fibres. The disordered strength of fibres results in large fluctuations of the waiting times elapsed between consecutive fibre breakings so that the damage accumulation process can only be characterized by the probability distribution of waiting times. The waiting time distribution proved to have a power law decay with an exponent which has different values below and above the critical load but only depends on the stress exponent of the relaxation process. Another important consequence of the disordered fibre strength is that the rupture life of the system has a lognormal distribution with a relatively large standard deviation. Special emphasis was put on methods which can be used to predict the rupture life of composites based on fibre bundles. Besides the Monkman-Grant relation it was demonstrated that a unique relation can be established between the lifetime of

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creeping fibre bundles and the transition time between the secondary and tertiary creep regimes which can also be exploited for lifetime estimates based on shortterm measurements. The acceleration of damage accumulation has been found in heterogeneous materials to show universal behaviour as the critical time of failure is approached. A similar universal power law divergence of the creep rate was obtained, which provides possibilities to forecast the imminent catastrophic failure event of structural components subject to a steady load. When analysing the creep rupture of fibre-reinforced composites a number of challenges have to be overcome, among which the correct representation of the load transfer and redistribution among fibres, as well as between fibres and the embedding matrix, taking into account the multiple fibre cracks and debonding mechanisms of the fibre-matrix interface, are of outmost importance. Theoretical approaches designed at the mesoscopic structural level of composites, such as the shear load, fibre bundle, micromechanical, and continuum damage mechanics models, proved to be very successful during the past decade in modelling various aspects of the damage and fracture of composites under different loading conditions. These modelling approaches provide important alternatives to continuum mechanics based numerical models. Modelling approaches which also require large-scale computer simulations can benefit from the rapid development of computational power in recent years. Powerful computers make it possible to work out more detailed models, allowing for a more realistic description of the materials’ mesostructures and of the relevant physical mechanisms underlying the degradation process. A major outcome of this development is the possibility of introducing hybrid approaches which blend the advantageous features of mesoscopic and continuum approaches. Along this line, fibre bundle models of the creep rupture of polymer matrix composite materials are expected to become more realistic in the future, providing a more detailed description of the creep rupture process and also making possible the analysis of large-scale structures. The predictive power of such advanced models also makes it possible to complement the experimentally available information on the failure process by reliable numerical results of computer simulations. Over the last decade, applications of fibre-reinforced composites using polymer matrices have seen tremendous growth. In spite of the complexity of their behaviour and the unconventional nature of fabrication, the usage of such composites, even for primary load-bearing structures in military fighters and transport aircraft, as well as satellites and space vehicles, has been beneficially realised. Most of such usage constituted structural applications where service temperatures are not expected to be beyond 120°C. Research efforts are now focused on expanding the usage of such composites to other areas where temperatures could be considerably higher. The intended applications are structural and non-structural parts around the aero-engines and airframe components of aircraft. The development of reliable FRPs for these novel applications requires adequate modelling approaches where advanced fibre bundle models in combination with hybrid modelling strategies can play an important role in the future.

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10.5 Acknowledgement The author is grateful for the financial support of the Bolyai Janos research fellowship of the Hungarian Academy of Sciences.

10.6 References Alava M J, Nukala P K, and Zapperi S (2006), ‘Statistical models of fracture’, Adv Phys, 55, 349. Andersen J V, Sornette D and Leung K T (1997), ‘Tricritical behaviour in rupture induced by disorder’, Phys Rev Lett, 78, 2140. Baxevanis T (2008), ‘A coarse-grained model of thermally activated damage in heterogenous media: Time evolution of the creep rate’, Europhys Lett, 83, 46004. Baxevanis T and Katsaounis T (2007), ‘Load capacity and rupture displacement in viscoelastic fibre bundles’, Phys Rev E, 75, 046104. Baxevanis T and Katsaounis T (2008), ‘Scaling of the size and temporal occurrence of burst sequences in creep rupture of fibre bundles’, Eur Phys J B, 61, 153. Binienda W K, Robinson D N, and Ruggles M R (2003), ‘Creep of polymer matrix com­ posites. II: Monkman-Grant failure relationship for transverse isotropy’, J Engrg Mech, 129, 318. Brinson H F, Griffith W I, and Morris D H (1981), ‘Creep rupture of polymer-matrix composites’, Experimental Mechanics, 21, 392. Chakrabarti B K and Benguigui L G (1997), Statistical physics of fracture and breakdown in disordered systems, Oxford, Clarendon Press. Chudoba R, Vorechovsky M, and Konrad M (2006), ‘Stochastic modelling of multifilament yarns. I. Random properties within the cross-section and size effect’, Int J Solids Struct, 43, 413. Coleman B D (1958), ‘On the strength of classical fibres and fibre bundles’, Journal of the Mechanics and Physics of Solids, 7, 60–70. Curtin W A (1998), ‘Size scaling of strength in heterogeneous materials’, Phys Rev Lett, 80, 1445, 1998. Curtin and Scher H (1997), ‘Time-dependent damage evolution and failure in materials: I. Theory’, Phys Rev B, 55, 12 038. Daniels H E (1945), ‘The statistical theory of the strength of bundles of threads’, Proc Royal Soc London A, 183, 405. Du Z Z and McMeeking R M (1995), ‘Creep models for metal matrix composites with long brittle fibres’, J Mech Phys Solids, 43, 701. Evans A G and Zok F W (1994), ‘The physics and mechanics of fibre-reinforced brittle matrix composites’, J Mat Sci, 29, 3857. Fabeny B and Curtin W A (1996), ‘Damage-enhanced creep and rupture in fibre-reinforced composites’, Acta Mater, 44, 3439. Garcimartin A, Guarino A, Bellon L and Ciliberto S (1997), ‘Statistical Properties of Fracture Precursors’, Phys Rev Lett, 79, 3202. Gomez J B, Iniguez D and Pacheco A F (1993), Solvable fracture model with local load transfer’, Phys Rev Lett, 71, 380. Guedes R M (2006), ‘Lifetime predictions of polymer matrix composites under constant or monotonic load’, Composites: Part A, 37, 703. Hansen A and Hemmer P C (1994), ‘Burst avalanches in bundles of fibres: Local versus global load-sharing’, Phys Lett A, 184, 394. © Woodhead Publishing Limited, 2011

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Harlow D G and Phoenix S L (1978), ‘The chain-of-bundles probability model for the strength of fibrous materials: analysis and conjectures’, J Comp Mat, 12, 195. Harlow D G and Phoenix S L (1991), ‘Local load transfer models of fracture’, J Mech Phys Solids, 39, 173. Hemmer P C and Hansen A (1992), ‘The distribution of simultaneous fibre failures in fibre bundles’, J Appl Mech, 59, 909. Herrmann H J and Roux S (1990), Statistical models for the fracture of disordered media, Amsterdam, North-Holland. Hidalgo R C, Kun F and Herrmann H J (2001), ‘Bursts in a fibre bundle model with continuous damage’, Phys Rev E, 64, 066122. Hidalgo R C, Kun F and Herrmann H J (2002a), ‘Creep rupture of visco-elastic fibre bundles’, Phys Rev E, 65, 032502. Hidalgo R C, Moreno Y, Kun F and Herrmann H J (2002b), ‘Fracture model with variable range of interaction’, Phys Rev E, 65, 046148. Johansen A and Sornette D (2000), ‘Critical ruptures’, Eur Phys J B, 18, 163. Jones R M (1999), Mechanics of Composite Materials, 2nd ed, Philadelphia/London, Taylor & Francis. Kloster M, Hansen A and Hemmer P C (1997), ‘Burst avalanches in solvable models of fibrous materials’, Phys Rev E, 56, 2615. Kovacs K, Nagy S, Hidalgo R C, Kun F, Herrmann H J and Pagonabarraga I (2008), ‘Critical ruptures in a bundle of slowly relaxing fibres’, Phys. Rev. E, 77, 036102. Kun F and Herrmann H J (2000), ‘Damage development under gradual loading of composites’, J Mat Sci, 35, 4685. Kun F, Hidalgo R C and Herrmann H J (2003a), ‘Scaling laws of creep rupture’, Phys Rev E, 67, 061802. Kun F, Moreno Y, Hidalgo R C and Herrmann H J (2003b), ‘Creep rupture has two universality classes’, Europhys Lett, 63, 347. Kun F, Zapperi S and Herrmann H J (2000), ‘Damage in fibre bundle models’, Eur Phys J B, 17, 269. Monkman F C and Grant N J (1956), ‘An empirical relationship between rupture life and creep rate in creep-rupture tests’, Proc ASTM, 56, 593. Moreno Y, Gomez J B and Pacheco A F (2000), ‘Fracture and second-order phase transitions’, Phys Rev Lett, 85, 2865. Nechad H, Helmstetter A, Guerjouma R El and Sornette D (2005), ‘Andrade and critical time-to-failure laws in fibre matrix composites: experiments and model’, J Mech Phys Solids, 53, 1099. Nechad H, Helmstetter A, El Guerjouma R, and Sornette D (2006), ‘Creep ruptures in heterogeneous materials’, Phys Rev Lett, 94, 045501. Newman W I and Phoenix S L (2001), ‘Time-dependent fibre bundles with local load sharing’, Phys Rev E, 63, 021507. Peirce F T (1926), ‘Tensile tests for cotton yarns – the weakest link – theorems on the strength of long and of composite specimens’, J Text Ind, 17, 355. Phoenix S L (2000), ‘Modeling the statistical lifetime of glass-fiber/polymer matrix composites in tension’, Comp Struct, 48, 19. Phoenix S L, Ibnabdeljalil M and Hui C Y (1997), ‘Failure process of composite materials’, Int J Solids Struct, 34, 545. Phoenix S L and Beyerlein I J (2000), ‘Statistical strength theory for fibrous composite materials’, volume 1 of Comprehensive Composite Materials, chapter 1.19, pages 1–81. Pergamon (Elsevier Science).

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Phoenix S L and Tierney L J (1983), ‘A statistical model for the time-dependent failure of unidirectional composite materials under local elastic load sharing among fibres’, Eng Fract Mech, 18, 193. Pradhan S and Chakrabarti B K (2003), ‘Failure properties of fibre bundle models’, Int J Mod Phys B, 39, 637. Pradhan S, Hansen A and Hemmer P C (2005), ‘Crossover behaviour in burst avalanches: Signature of imminent failure’, Phys Rev Lett, 95, 125501. Roux S (2000), ‘Thermally activated breakdown of disordered materials’, Phys Rev E, 62, 6164. Roux S, Delaplace A, and Pijaudier-Cabot G (1999), ‘Damage at heterogeneous interfaces’, Physica A, 270, 35. Roux S and Hild F (2002), ‘On the relevance of mean field to continuum damage mechanics’, Int J Fract, 116, 219. Scorretti R, Ciliberto S, and Guarino A (2001), ‘Disorder enhances the effect of thermal of thermal noise in the fibre bundle model’, Europhys Lett, 55, 626. Sornette D (1989), ‘Elasticity and failure of a set of elements loaded in parallel’, J Phys A, 22, L243. Vujosevic M and Krajcinovic D (1997), ‘Creep rupture of polymers: a statistical model’, Int J Solid Structures, 34, 1105. Yoshioka N, Kun F, and Ito N (2008), ‘Size scaling and bursting activity in thermally activated breakdown of fibre bundles’, Phys Rev Lett, 101, 145502. Zapperi S, Ray P, Stanley H E, and Vespignani A (1997), ‘First-order transition in the breakdown of disordered media’, Phys Rev Lett, 78, 1408.

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11 Micromechanical modeling of time-dependent failure in off-axis polymer matrix composites J. Koyanagi, Institute of Space and Astronautical Science, Japan Abstract: This chapter describes a micromechanical models approach to time-dependent transverse failure in unidirectional composites. One of the dominant factors determining time-dependent transverse strength is time-dependent interfacial strength. This is extracted from the results of transverse tensile tests with various loading rates and their fractography in the unidirectional composite. The results show that whilst material failure is most commonly caused by an interface failure under a relatively high loading rate, matrix failure is more common under a relatively low loading rate. In light of the results, it is found that the time dependence of interfacial strength might be negligible, or at least could be less significant than that of matrix strength. It is concluded that it is typically enough to focus on only matrix time-dependent failure in order to assess the long-term reliability of the transverse strength of composite materials. Key words: off-axis failure, micromechanics of failure, finite element analysis, fractography, time-dependent transverse strength.

11.1 Introduction Long-term reliability is one of the most critical problems in fiber-reinforced polymeric matrix composites (PMC). The space elevator and space colony are typical examples of structures which require a certain guaranteed long-term durability. The space elevator, which is synchronized with the earth’s rotation, is designed to transport materials from the earth’s surface into space. Its weight induces a huge tensile stress in the elevator cable. The space colony is cylindrical in shape, and filled with air. In the space colony, a centrifugal force is used to produce artificial gravity, allowing human beings to live as they would on the earth. Both of these structures are subjected to severe creep loads over long time periods and cannot be allowed to fail. One of the candidate materials for these structures is carbon fiber reinforced polymer matrix composite. The mechanical properties of PMC are time dependent due to the viscoelastic characteristic of the polymeric matrix, which means that the materials can fail in creep rupture at loads that are lower than those which could be tolerated at their static strength. Therefore, it is important to understand the time-dependent behaviors of PMC in order to ensure their long-term reliability. There have been many articles on micromechanics in which the authors predict static and time-dependent unidirectional composite longitudinal tensile strengths (Rosen 1964, Curtin 1991, Harlow 1978a,b, Okabe et al. 2005, Koyanagi et al. 350 © Woodhead Publishing Limited, 2011



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2004, 2007, 2009a), which should govern the composite ultimate tensile strength. These micromechanics approaches have received so much attention because the failure of PMC results from an accumulation of micro-failures, hence it is essential to consider micro-failures in order to predict a macro-failure. Koyanagi et al. (2009a) have established a comprehensive model for determining tensile strength in various composites, and formulated a generalized time-dependent stress–strain relationship (Koyanagi 2009). It is implied that by combining these two results it is possible to make realistic creep rupture predictions. That is to say, if the time-dependent properties of matrix, fiber and the fiber/matrix interface are known, the long-term strength of the unidirectional composite can be predicted. Thus, micromechanical models approaches to time-dependent failure of composites in the longitudinal direction have been widely studied. On the other hand, the studies of the time-dependent properties of off-axis PMC should also be important for such applications as the design of flywheels made from PMC, and a leak issue induced by transverse cracking. However, there has been little micromechanical modeling work done on the time-dependent failure of off-axis composites (Koyanagi et al. 2010b). Based on Ha’s work (Ha et al. 2008), which is called MMF (micromechanics of failure), both stress–strain relationship and strength can be reasonably represented by a unit cell model. This chapter describes a micromechanical model approach to the time-dependent stress–strain relationship and strength of off-axis PMC, considering MMF. In the scheme, the transverse strength of a unidirectional composite is determined by the lower strength of the matrix or interface. The time-dependent strength of the fiber/ matrix interface plays a very important role in discussions about time-dependent off-axis strengths. Yet the number of studies conducted in this area, which would also be useful for predicting time-dependent composite strength in the longitudinal direction, has been extremely limited. Unless the time-dependent strength of the interface is clarified, it is impossible to accurately predict the time-dependent failure of off-axis PMC. The scarcity of works related to interfacial time dependency is due to the fact that it is hard to evaluate interfacial properties, even in a static condition. Following on from a number of works done in the field of interfacial properties, the use of a single-fiber composite has become common in methods such as the fiber fragmentation test, micro-droplet test, push-out test, pull-out test, Broutman test and transverse tensile test. The first four tests, which have been widely implemented (Chua and Piggott 1985, Keran and Parthasarathy 1991, Zhandarov and Pisanova 1997, Zhou et al. 1999, Kimura et al. 2006), are primarily used for the evaluation of interfacial shear properties. The remaining two tests are applicable to the evaluation of tensile properties of the interface (Ageorges et al. 1999, Tandon et al. 2000, Ogihara et al. 2009, Koyanagi et al. 2009b, Ogihara and Koyanagi 2010). These publications do not deal with the time-dependent properties of interfaces. In terms of time dependency, although viscoelastic behavior near the interface (e.g. interphase) has been studied (Fisher and Brinson 2001, Fink and

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McCullough 1999, Dzenis 1997, Gosz et al. 1991, Hashin 1991), there has rarely been any research into the time dependency of interfacial strength itself. There are some existing works on the time dependency of interface strength (Beyerlein et al. 2003, Zhou et al. 2002, 2003, Koyanagi et al. 2007) but the values they present are quite strongly dependent upon the assumptions made in the derivation. Therefore, a proper representation of the time dependency of interface strength is still under discussion. In this chapter, the time dependence of the interfacial strength, which is a critical factor affecting time-dependent failure of off-axis PMC, was mainly investigated on the basis of a simplified scheme presented by Ha et al. (2008). The paper developed micromechanics of failure (MMF) to predict the short- and long-term behavior of composite structures. The MMF consists of two major parts: (1) a calculation of micro-stresses from the macro-stresses of direct 3D finite element method (FEM), covering a wide range of the material properties of fiber and matrix by random fiber model, and (2) fiber and matrix failure criteria, including the effects of residual stresses on the failures and master curves for life predictions. In the article, they also explained that the transverse strength of a unidirectional composite is determined by the weaker strength of the matrix or interface. The transverse tensile test was performed for a unidirectional carbon fiber reinforced polymeric composite with various loading speeds. Following that, the fractured surface of the specimen was observed using a scanning electron microscope to determine whether the failure derived mainly from interfacial or matrix failure. It is expected that the failure mode, whether interface or matrix dominant, will vary with the loading rate, provided that the time dependencies of the matrix and interface are different. Here, time dependency refers to the phenomenon whereby the strength of components increases with the tensile testing speed. In other words, there will be a transition point at which the order of the two strengths will change with the loading rate. Otherwise, i.e., if the two time dependencies are identical, the failure mode should remain the same for all loading rates, e.g., the weaker component will always dominate the transverse composite failure. Although this study is very simple, by integrating the test results at various testing speeds with the fractography results, the two timedependent strengths of the interface and matrix can be qualitatively compared. Moreover, on the basis of Ha et al.’s work (2008), the weaker strength can be obtained quantitatively by multiplying the specimen’s macro-stress by the stress-concentration factor at the tip of interface (the critical point, where the stress-concentration factor is at its maximum). It should be noted that the stressconcentration factor may vary with time. Hence, in this study, viscoelastic finite element analysis was also performed to determine the stress-concentration factor as a function of time. Similarly to the work of Ha et al. (2008), a unit-cell model was used to perform the finite element analysis. A very popular viscoelastic constitutive equation, which contains the generalized Voigt model, was applied to the matrix mechanical properties. The viscoelastic parameter of the matrix was

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determined by fitting the analytical stress–strain curves to the experimental results. Using the viscoelastic parameter, the time-dependent stress-concentration factor is determined by the stress at the critical point, divided by the specimen average stress. By considering the results from the tensile test, fractography and finite element analysis, the time dependency of the interfacial strength was extracted. The effect of the time-dependent interfacial strength on the time-dependent failure of off-axis composite material is discussed on the basis of these results.

11.2 Experiments 11.2.1  Experimental procedure A rectangular strip specimen, 250 mm long (including 50 mm in tapered tabs for both sides of the edges) 20 mm wide and 2 mm thick was prepared, as shown in Fig. 11.1. The tab thickness was 2 mm. T300 carbon fiber was used for reinforcement, and the matrix was 180°C curing type epoxy resin (Hyej17HX1, Mitsubishi). The specimen was fabricated using an autoclave system. The fiber volume fraction of the specimen was approximately 55%. Various loading rates, between approximately 0.002 and 10 mm/min, were applied to the specimen, thus varying the strain rates. The longest test lasted for approximately one whole

11.1  Specimen geometry and dimensions.

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day. The test was a displacement-controlled tensile test. The tests were conducted at a constant room temperature of 24°C, and at around 50% relative humidity (RH). The ambient condition was maintained during all the tests. Strain gages were used to measure the strains on the surface of the specimen. The strain gages used were manufactured by Kyowa Dengyo, and the model of the testing machine used was Shimazu AUTO GRAPH. The fractured surfaces were observed using a scanning electron microscope (SEM), following platinumsurface treatment (Koyanagi et al. 2010a).

11.2.2  Experimental results and fractography Figure 11.2 shows the experimental results obtained under various strain rates, and failure modes based on the fractography results, as shown in Fig. 11.3. Typically the matrix-failure dominant mode is observed under relatively low loading rate conditions (Fig. 11.3(a)), whereas relatively high loading rate conditions are dominated by interface failure (Fig. 11.3(c)). At intermediate strain speed (Fig. 11.3(b)), the combined failure of interface and matrix is observed. This implies that the matrix becomes weaker than the interface in the lower strain rate range. The interfacial and matrix stresses in the loading direction at the location, where the stress-concentration factor is maximum (critical point as

11.2  Results of transverse tensile failure stress for various strain rates.

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11.3  Failure surfaces for various tensile strain rates: (a) 1.0 × 10–7/sec (slow); (b) 4.0 × 10–7/sec (middle); (c) 4.0 × 10–5/sec (rapid).

shown in Fig. 11.4), remain equal to each other, independently of any viscoelastic behavior. Therefore, if both the interface and matrix are assumed to have normal qualitative time dependencies i.e., the strength decreases with a decrease in the tensile testing speed, it is possible to suggest here that the time dependency of the matrix is more pronounced than that of the interface. In other words, as the strain rate decreases, the matrix strength decreases more significantly than the interface strength. In the case of the intermediate strain rate, both strengths can be expected to be similar to each other. Thus, the failure dominant mode transition was observed in this test. Of course, the strain rate at which the failure mode shifts from being matrix dominant to interface dominant would be different for individual materials. Nevertheless, for other material systems, the transition point can presumably be observed if the tests are conducted over a wide range of strain rates. The only situation in which the transition point does not exist is if both strengths show the same strain rate dependence; the probability of this situation occurring is extremely low. The approach used in this work is very simple, yet it enables us to find the transition point where the failure mode shifts from being matrix dominant to

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11.4  Geometry and boundary condition of numerical model.

interface dominant, so that the time dependencies of interface and matrix strengths can be studied separately. However, since the stress-concentration factor is still unknown, it is still impossible to estimate the time dependency of the interfacial strength quantitatively. Furthermore, depending on viscoelastic behavior around the critical point, there is a possibility that the critical point might move. For this reason, a viscoelastic finite element analysis was performed to determine the time-dependent stress-concentration factor, as explained in the following section.

11.3 Finite element analysis 11.3.1  Stress–strain curves The calculation of the time-dependent stress-concentration factor is indispensable in terms of extracting the interfacial strength from the experimental results. If the stress-concentration factor decreases with time, it might be expected that the location of the critical point also moves with time. Whether or not the critical point moves with time should also be verified. Following Ha et al.’s work (2008), a viscoelastic analysis was performed using a square unit-cell model, as shown in

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Fig. 11.4. In the model, the left and bottom edges were constrained symmetrically, and the right edge was constrained to remain straight and vertical, i.e. periodical boundary condition. A tensile displacement was applied to the upper edge in the vertical direction. The analysis was performed using commercial software ABAQUS 6.7.1, and 3D solid elements were used to model the problem. The range of the applied strain rate was from 2.0 × 10–7 to 4.0 × 10–4/sec. In this analysis, the fiber diameter under consideration was 6 μm and the length of each side of the square was determined with a fiber volume fraction of 55%. The fiber is assumed to be elastic and isotropic in this two-dimensional plain; the elastic modulus used corresponds to fiber radial modulus of 19 GPa and Poisson’s ratio is 0.4. For the viscoelastic matrix, the instantaneous elastic modulus 4500 MPa, Poisson’s ratio 0.34 (which is not time dependent), and the viscoelastic relaxation modulus based on the power law compliance model, were specified, which has conventionally been used well (Koyanagi 2009, Koyanagi et al. 2007).

[11.1]

Here, J0 is instantaneous (initial) compliance, T0 is characteristic relaxation time, and n is the power law component. T0 and n are regarded as viscoelastic parameters. In the present study, J0=(1/4.5=) 0.222 1/GPa, T0 = 500 000 sec and n = 0.3 are employed. These viscoelastic parameters, T0 = 500 000 sec and n = 0.3, were determined by fitting the analytical stress–strain curves to those of the experimental results as shown in Fig. 11.5. Here, the stress is defined by the average unit-cell boundary stress at the upper nodes, where the tensile displacement is given. As Fig. 11.5 shows, for a relatively rapid test speed result, neither the analytical nor the experimental stress–strain curve shows significant viscoelastic behavior. The viscoelastic behavior becomes significant as the testing speed decreases. Since every analytical curve shows reasonable agreement with the experimental result, it can be used to confirm the validity of the employed unit-cell model approach and the material constants.

11.3.2  Time-dependent stress-concentration factor By applying the unit-cell model mentioned in the previous section, the stressconcentration factor can be obtained. Plate IV, in the color section (between pages 288 and 289), shows the stress distribution in the vertical direction, i.e. in the tensile direction, under a given tensile strain of 0.44%, which is the typical rupture strain of the specimen when strain rates are (a) 2.0 × 10–7 and (b) 4.0 × 10–4. The figures verify that the maximum tensile stress occurs at the critical point. As the applied strain rate increases, the maximum stress increases, due to viscoelastic behavior of the matrix. Figure 11.6 shows the stress-concentration factor as a function of time, which was

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11.5  Stress–strain curves of experiments and numerical results.

11.6  Stress-concentration factor at interface with time.

obtained by: the interface (matrix) stress at the critical point, divided by macro-stress (as used in the stress–strain curves mentioned above). This result is for the 4.0 × 10–7 strain rate test. The analysis performed here is of linear viscoelastic behavior, which can be described by the following convolution integral form. © Woodhead Publishing Limited, 2011



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359 [11.2]

Here, if the applied strain rate is, for instance, 0.1 times that of an arbitrary rate, the stress will always be 0.1 times the value it was in the case of the arbitrary strain rate. Both the average stress of the unit-cell boundary and stress of the interface (matrix) at the critical point vary with time independently of each other, but vary simultaneously against the strain rate variation. Hence, the ratio between the average stress and interface stress, which is the stress-concentration factor, depends on only time, not on the strain rate. The ratio of every strain rate test is represented by the identical curve drawn in Fig. 11.6. As shown in the figure, although there is a difference between stress–strain curves as time progresses, the time-dependent stress-concentration factor is almost constant in the range of this analysis. Here, it is assumed that the stress-concentration factor is always 1.4, and that the time-dependent variation of the stress-concentration factor is negligible. Regarding the location of the critical point, since the stress-concentration factor increases slightly with time, it can be assumed that the location of the critical point does not move with time. In other words, the product values of the specimen stresses in Fig. 11.2 and 11.4 approximately correspond to the weaker strength of either the matrix or interface. By using this value, the time-dependent strengths of matrix and interface can be discussed quantitatively.

11.4 Discussion The findings from this study are summarized in Fig. 11.7. In this figure a bi-linear curve represents the strength, of either the matrix or the interface, as a function of the strain rate, based on the following discussion. First of all, since the stressconcentration factor at the critical point is assumed to have no time dependency, specimen stress multiplied by 1.4 corresponds to the approximated weaker strength of either the interface or the matrix. There is a transition point, which separates the interfacial failure dominant and matrix failure dominant region around the strain rate value of 4.0 × 10–7/sec, as discussed above with the fractography results. In Fig. 11.7, the right-hand side of the transition represents the interfacial failure dominant region, and the left-hand side, the matrix failure dominant region. The strength of a polymer material generally shows a tendency to decrease with a decrease in tensile strain rate, and the experimental result shows a qualitatively similar tendency in the matrix failure dominant region, since the left-hand curve is consistent with this phenomenon. For the region at the righthand side of the transition point, a horizontal curve is drawn. From the experimental results, it cannot be observed that the strength increases with an increase in the strain rate. Instead, the experimental results seem to decrease slightly with strain rate, which is a behavior completely opposite to that of the general time dependency of viscoelastic materials. However, since the scattering is relatively large, it is

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11.7  Approximated strengths of matrix and interface as a function of strain rate with transition of specimen-failure mode.

assumed the experimental results can be approximately represented by the horizontal curve. Thus, based on the range of the strain rates considered, this result indicates that the interface has no considerable time dependency or might have no time dependency. The test employed in the present study is expected to produce scattered results; it is hard to conclude more explicitly from this test. Since the failure mode of the specimen changes from interface dominant to matrix dominant with decreasing strain rate, even if there is a time dependency for interfacial strength, it is less significant when compared with the time dependency for matrix strength. In other words, the interface less remarkably degrades with time than the matrix, and the long-term transverse strength of unidirectional composites is typically related to the time-dependent strength of the matrix only. Thus, a composite in which the matrix is weaker than the interface at a static condition will always fail in the matrix failure dominant mode. Even a composite in which the matrix is stronger than the interface at a static condition will fail in the matrix failure dominant mode after enough time has elapsed. As time passes, the matrix strength will be weaker than the interface strength. At the same time, it can be suggested for any other unidirectional CFRP that a relatively rapid test can potentially lead to interface failure dominant failure and a relatively slower test can cause matrix failure dominant failure. Taniguchi et al. (2008) presented the dynamic strength of a unidirectional CFRP using the Split Hopkinson Bar method. In their work, it has been reported that static and dynamic tests demonstrate no remarkable difference in the strength of a 90-degree specimen

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(transverse strength), and interfacial failure dominant mode was observed in both cases. This indicates that the time dependency of the interface strength is also unremarkable under a dynamic condition. It is implied that an interfacial property, which cannot be measured due to having greater strength than the matrix, can be measured by a rapid transverse tensile test. Here, it is assumed that interface strength is not time dependent. As mentioned in the introduction to this chapter, several articles argued that the interfacial shear strength decreases with time (Beyerlein et al. 2003, Zhou et al. 2002, 2003). That seems to be inconsistent with the findings of this chapter. On the other hand, some researchers have reported that the interfacial shear strength is enhanced by compressive stress at the interface (Koyanagi et al. 2007, Park et al. 2002). Considering the inconsistent results mentioned above, and this enhancement effect, it can be assumed that interfacial compressive stress can be time dependent, hence the shear strength enhancement can be time dependent, so the shear strength being subjected to shear and compressive stress can be also time dependent. In other words, experimentally obtained apparent interfacial shear strength is a superposition of the ‘pure’ interfacial shear strength and enhancement due to interfacial compressive stress. ‘Pure’ shear strength refers to interfacial strength when the compressive stress at the interface vanishes, and it is presumably not time dependent. This has been partly studied by Koyanagi et al. (2010b). This discussion correlates with the fact that ‘Parabolic criterion’ (Eq. 11.3) for interface failure criterion under combined stress-state maybe more accurate than ‘Quadratic criterion’ (Eq. 11.4), which is conventionally assumed to be valid. This is an interesting aspect to be investigated in the near future.

[11.3]



[11.4]

Here, tn is interfacial normal stress, ts the interfacial shear stress, Yn the interfacial normal strength, Ys the interfacial shear strength, and      is the McAuley bracket with its usual definition,

. For the quadratic criterion, Eq. 11.4,

interfacial shear stress at debonding is constant, regardless of normal stress when the normal stress is negative; and the failure surface plot is elliptical in shape when the normal stress is positive. For the parabolic criterion, Eq. 11.3, the failure surface plot is parabolic in shape, and interfacial shear stress at debonding is enhanced by normal compressive stress. The parabolic criterion was originally intended for isotropic material, which does not fail under a compressive load. Both failure surface plots are shown in Fig. 11.8. According to the parabolic criterion, Eq. 11.3, all of the above results, regarding apparent time-dependent

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11.8  Interfacial failure surface plots in normalized normal-shear stress field for the quadratic failure criterion and the parabolic failure criterion.

and actual time-independent interfacial strengths, can be explained qualitatively without any inconsistencies.

11.5 Conclusion This chapter discusses the micromechanical models approach to time-dependent transverse tensile failure in unidirectional composites. Tensile tests with various testing speeds are conducted in order to investigate the time-dependent failure of the composites. On the basis of the MMF scheme presented by Ha et al. (2008), it is assumed that the transverse strength is determined by the weaker strength of either the matrix or the interface. From the fractography results, the tensile test results are separated into either matrix or interface failure dominant modes. In the case of relatively low testing speeds, the matrix failure dominant mode is observed. On the other hand, in the case of relatively high testing speeds, the interface failure dominant mode is observed. A viscoelastic numerical simulation is then performed to assess the time dependence of the stress-concentration factor, regarding the assumed unit-cell micromechanical model, which is a critical factor in the discussion of time-dependent failure. The viscoelastic parameters are determined by a comparison of the experimental and analytical stress–strain curves. It is found that the stress-concentration factor is almost constant in the range of this analysis. By combining the experimental, analytical and fractography results, it is found that the time dependence of interfacial strength might be negligible, or at least less significant than that of the matrix. This chapter concludes

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with the following points: (1) transverse failure is time dependent in the long term and time independent in the short term; (2) this is because matrix strength is time dependent but interfacial strength is not; (3) the transition time between short and long term is when the interface and matrix strengths are equal; and (4) the timedependent property of the matrix is the only factor that needs to be understood in order to guarantee the long-term durability of transverse strength in unidirectional composite materials.

11.6 Future trends The finding of this chapter, that there is no time-dependent interface strength, also has an impact in predicting time-dependent longitudinal composite strength. Since the practical test of unidirectional composite creep rupture is extremely difficult, it is much more common to use the accelerated testing method at an elevated temperature, although this has not been verified theoretically in terms of micromechanics (Miyano et al. 1999, 2004, 2005). The problem was that the unknown time- and temperature-dependent interfacial property prevented the formation of valid long-term strength predictions. Although the temperature dependence of interface strength has not yet been clarified, if the interfacial failure criterion is assumed to have no temperature-dependent characteristic (excluding the effect of the apparent interfacial strength variation induced by a variation of thermal residual stress), the micromechanical validity of the accelerated testing method can be verified theoretically. Whether or not interface strength is temperature dependent is an important issue. This can be clarified by the same examination at various temperature conditions. If the matrix strength decreases more significantly than the interfacial strength with an increase in temperature, for a consideration of thermal residual stress effect, more rapid tensile testing is required to make the specimen fail in the interface failure dominant mode; the transition between interface and matrix failure dominant modes moves to the rapid testing speed side as the temperature increases. Otherwise, the matrix strength decreases less significantly than the interface strength with an increase in temperature; relatively low-speed testing can make the specimen fail in interface failure dominant mode; the transition point moves to the slow testing speed side as temperature increases. Thus, the temperaturedependent interface strength can be obtained. The obtained interfacial strengths include the thermal residual stress effects; the actual temperature-dependent interface strength can be calculated considering thermal residual stress. This result will certainly be helpful for discussing the long-term durability of composite strength, and this issue should be investigated in the near future.

11.7 References Ageorges C, Friedrich K, Schüller T, Lauke B. (1999) ‘Single-fibre Broutman test (fiber-matrix interface transverse debonding).’ Composites Part A 30: 1423–1434.

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Beyerlein IJ, Zhou CH, Schadler LS. (2003) ‘A time-dependent micro-mechanical fiber composite model for inelastic zone growth in viscoelastic matrices.’ International Journal of Solids and Structure 40: 1–24. Chua PS, Piggott MR. (1985) ‘The glass fibre-polymer interface: II – Work of fracture and shear stresses.’ Composites Science and Technology 22: 107–119. Curtin W. (1991) ‘Theory of mechanical properties of ceramic-matrix composites.’ J Amer Ceram Soc 74: 2837–2845. Dzenis YA. (1997) ‘Effective thermo-viscoelastic properties of fibrous composite with fractal interfaces and an interphase.’ Composite Science and Technology 57: 1057–1063. Fink BK, McCullough RL. (1999) ‘Interphase research issues.’ Composites Part A 30: 1–2. Fisher FT, Brinson LC. (2001) ‘Viscoelastic interphases in polymer-matrix composites: theoretical models and finite-element analysis.’ Composite Science and Technology 61: 731–748. Gosz M, Moran B, Achenbach JD. (1991) ‘Effect of a viscoelastic interface on the transverse behavior of fiber-reinforced composites.’ International Journal of Solids and Structure 27: 1757–1771. Ha SK, Jin KK, Huang Y. (2008) ‘Micro-mechanics of failure (MMF) for continuous fiber reinforced composites.’ Journal of Composite Materials 42: 1873–1895. Harlow D, Phoenix S. (1978a) ‘The chain-of-bundles probability model for the strength of fibrous composites: I: Analysis and conjectures.’ J Comp Mater 12: 195–214. Harlow D, Phoenix S. (1978b) ‘The chain-of-bundles probability model for the strength of fibrous composites: II: A numerical study of convergence.’ J Comp Mater 12: 314–334. Hashin Z. (1991) ‘Composite materials with viscoelastic interphase: creep and relaxation.’ Mechanics of Materials 11: 135–148. Keran RJ, Parthasarathy TA. (1991) ‘Theoretical analysis of the fiber pull-out and pushout tests.’ Journal of American Ceramics Society 74: 1585–1596. Kimura S, Koyanagi J, Kawada H. (2006) ‘Evaluation of initiation of the interfacial debonding in single-fiber composites (Energy balance method considering an energy dissipation of the plastic deformation).’ JSME International Journal Series A 49 (3): 451–457. Koyanagi J. (2009) ‘Comparison of a viscoelastic frictional interface theory with a kinetic crack growth theory in unidirectional composites.’ Composite Science and Technology 69: 2158–2162. Koyanagi J, Kiyota G, Kamiya T, Kawada H. (2004) ‘Prediction of creep rupture in unidirectional composite (creep rupture model with interfacial debonding around broken fibers).’ Adv Comp Mater 13: 199–213. Koyanagi J, Kotani M, Hatta H, Kawada H. (2009a) ‘A comprehensive model for determining tensile strengths of various unidirectional composites.’ Journal of Composite Materials 43: 1901–1914. Koyanagi J, Ogawa F, Kawada H, Hatta H. (2007) ‘Time-dependent reduction of tensile strength caused by interfacial degradation under constant strain duration in UD-CFRP.’ Journal of Composite Materials 41: 3007–3026. Koyanagi J, Yoneyama S, Eri K, Shah P. (2010a) ‘Time dependency of carbon/epoxy interface strength.’ Composite Structure 92: 150–154. Koyanagi J, Yoshimura A, Kawada H, Aoki Y. (2010b) ‘A numerical simulation of timedependent interface failure under shear and compressive loads in single-fiber composite.’ Applied Composite Materials, 17: 31–41. Koyanagi J, Shah P, Kimura S, Ha S, Kawada H. (2009b) ‘Mixed-mode interfacial debonding simulation in single fiber composite under transverse load.’ Journal of Solid Mechanics and Materials Engineering 3: 796–806.

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Miyano Y, Nakada M, Muki R. (1999) ‘Applicability of fatigue life prediction method to polymer composites.’ Mech Time Depen Mater 3: 141–157. Miyano Y, Nakada M, Sekine N. (2004) ‘Accelerated testing for long-term durability of GFRP laminates for marine use.’ Composites Part B 35: 497–502. Miyano Y, Nakada M, Sekine N. (2005) ‘Accelerated testing for long-term durability of FRP laminates for marine use.’ J Comp Mater 39: 5–20. Ogihara S, Koyanagi J. (2010) ‘Investigation of combined stress-state failure-criterion for glass-fiber/epoxy interface by cruciform specimen test.’ Composite Science and Technology 70: 143–150. Ogihara S, Sakamoto Y, Koyanagi J. (2009) ‘Evaluation of interfacial tensile strength in glass-fiber/epoxy resin interface using the cruciform specimen method.’ Journal of Solid Mechanics and Materials Engineering 3: 1071–1080. Okabe T, Sekine H, Ishii K, Nishikawa M, Takeda N. (2005) ‘Numerical method for failure simulation of unidirectional fiber-reinforced composites with spring element model.’ Comp Sci Tech 65: 921–933. Park JM, Kim JW, Yoon DJ (2002). ‘Comparison of interfacial properties of electrodeposited single carbon fiber/epoxy composites using tensile and compressive fragmentation tests and acoustic emission.’ Journal of Colloid and Interface Science, 247: 231–245. Rosen B. (1964) ‘Tensile failure of fibrous composites.’ AIAA Journal 2: 1982–1991. Taniguchi N, Nishiwaki T, Kawada H. (2008) ‘Experimental characterization of dynamic tensile strength in unidirectional carbon/epoxy composite.’ Advanced Composite Materials 17: 139–156. Tandon GP, Kim RY, Bechel VT. (2000) ‘Evaluation of interfacial normal strength in a SCS-0/Epoxy Composite with cruciform specimens.’ Composites Science and Technology 60: 2281–2295. Zhandarov SF, Pisanova EV. (1997) ‘The local bond strength and its determination by fragmentation and pull-out tests.’ Composites Science and Technology 57: 957–964. Zhou XF, Nairn JA, Wagner HD. (1999) ‘Fiber-matrix adhesion from the single-fiber composite test: nucleation of interfacial debonding.’ Composites Part A 30: 1387–1400. Zhou CH, Schadler LS, Beyerlein IJ. (2002) ‘Time-dependent micromechanical behavior in graphite/epoxy composites under constant load: a combined experimental and theoretical study.’ Acta Materialia 50: 365–377. Zhou CH, Beyerlein IJ, Schadler LS. (2003) ‘Time-dependent micromechanical behavior in graphite/epoxy composites under constant load at elevated temperatures.’ Journal of Materials Science 38: 877–884.

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Plate IV Tensile stress distribution (MPa) at applied strain 0.44% when strain rates are (a) 2.0 × 10–7 and (b) 4.0 × 10–4.

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12 Time-dependent failure criteria for lifetime prediction of polymer matrix composite structures R. M. G uedes , Faculdade de Engenharia da Universidade do Porto, Portugal Abstract: The use of fibre-reinforced polymers in civil construction applications originated structures with a high specific stiffness and strength. Although these structures usually present a high mechanical performance, their strength and stiffness may decay significantly over time. This is mainly due to the viscoelastic nature of the matrix, damage accumulation and propagation within the matrix and fibre breaking. One serious consequence, as a result of static fatigue (creep failure), is a premature failure which is usually catastrophic. However, in civil engineering applications, the structural components are supposed to remain in service for 50 years or more in safe conditions.   One argument used to replace steel by polymer matrix composites is its superior corrosion resistance. Yet stress corrosion of glass fibres takes place as soon as moisture reaches the fibre by absorption. This phenomenon accelerates fibre breaking. In most civil engineering applications, glass fibre reinforced polymers (GRP) are the most common, especially because the raw material is less expensive.   The lack of full understanding of the fundamental parameters controlling long-term materials performance necessarily leads to over-design and, furthermore, inhibits greater utilization. In this context, lifetime prediction of these structures is an important issue to be solved before wider dissemination of civil engineering applications can take place. As an example, standards dealing with certification of GRP pipes require at least 10 000 hours of testing for a high number of specimens. Even though these strong requirements may be foreseen as reasonable, concerning the safety of civil engineering applications, they severely restrict the improvement and innovation of new products.   The present chapter reviews some theoretical approaches for long-term failure criteria. Time-dependent failure criteria will be presented and developed for practical applications and illustrated with experimental cases. Key words: creep, polymer matrix composites (PMCs), fatigue, durability, damage accumulation.

12.1 Introduction The structural applications of composite materials in civil construction are becoming more important. One major application is on rehabilitation in renewal of the structural inventory, to repair or strengthen. The success of these applications has promoted the development of new solutions based on FRP (fibre-reinforced polymers). Although these new products may promise a better mechanical 366 © Woodhead Publishing Limited, 2011



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performance on a long-term basis, the lack of a historical record prevents an immediate structural optimization. The main reason is because durability factors depend on material systems; usually long-term experimental tests must be performed for each system. Further on, structural designers do not have full access to all the available data since some databases are restricted. On the other hand, the full comprehension of internal material changes from the microscopic scale up to the full structure length is far from being known. The interaction between different mechanisms acting at different scale levels is extremely complex and not yet fully understood. Furthermore, many processes affect the durability of a material/structure, defined by Karbhari et al.1 as ‘its ability to resist cracking, oxidation, chemical degradation, delamination, wear, and/or the effects of foreign object damage for a specified period of time, under the appropriate load conditions, under specified environmental conditions’. In this chapter we will discuss time-dependent failure criteria used to predict the lifetime of polymer matrix dominated composites. Although in many structural applications the reinforced fibres are aligned with principal loading directions, it was verified that for glass fibre reinforced composites with fibre volume fraction content of 0.6, these criteria can be applied successfully to predict creep lifetime (Guedes et al.2). Obviously the matrix must play an important role in those cases. In the beginning of the seventies Gotham,3 starting from an idealized failure, tested several polymers at continuous load in tension at 20°C to obtain their creep rupture curves and determine a possible ductile-brittle transition. After the experimental results, Gotham3 concluded that none of the failure curves resemble the idealized curve and most of the failures were ductile. Further, he noted that the scatter of data was low and the curves were smooth without discontinuities. Today the existing experimental results for creep rupture of ductile homogeneous materials are very extensive. As for the PMCs, the published experimental results and theoretical description of creep rupture are not so well documented as for the polymers. This could be explained, in part, by the extremely large variety of material systems offered by the manufacturing industry. The other reason is because the systems are naturally very complex, with a great number of boundaries between the constituents, giving origin to a large number of local defects, such as debonds and cracks. According to Reifsnider et al.,4 all creep rupture analyses could be divided into two major categories: the local and direct analysis of the growth of the defects and the global and homogeneous analysis. The late analysis concerns only the summation of all micro-process effects acting concomitantly and often designated as accumulation or quasi-homogeneous damage models. Of course this category is more promising for practical applications due to the inherent complexity of the former analysis. Brüller5,6,7 investigated profoundly the applicability of an energy criterion to model the creep rupture of thermoplastics. He applied the Reiner-Weissenberg (R-W) theory8 successfully to determine the linear viscoleastic limit. For the

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prediction of fracture and crazes, Brüller proposed a modification of the theory: only the total time-dependent energy contributed to the creep rupture; provided that the stress level was lower than the static rupture stress, then failure occurs when the energy reaches a limit value, considered a material property. After this, Brüller established the conditions for creep and stress relaxation failure, showing some experimental evidence. More recently Theo9,10 and Boey11 presented an application of the R-W criterion to a three-element mechanical model to describe the creep rupture of polymers. The predictions produced by this creep rupture model were in good agreement with the experimental data for several tested polymers. Moreover, the model appears to be able to determine the upper stress limit (instantaneous failure) and lower stress limit (no creep failure). The drawbacks can be related with a difficult mathematical manipulation of the model, with some eventual difficulties to fit the model into the experimental data and with the uncoupled descriptions of creep and creep rupture. Griffith et al.12 applied the Zhurkov relationship to predict the time to rupture of continuous fibre-reinforced plastics with a reasonable success. Dillard13,14 determined the creep rupture curves by fitting experimental data using the classical Larson and Miller and Dorn theories. Dillard developed a numerical model, based on the classical laminate theory, to predict the creep of general laminates coupled with a lamina failure model based on a modification of the Tsai-Hill theory to include creep rupture curves. After this, failure data at elevated temperatures for several general laminates of T300/934, a carbon fibre/epoxy resin, was compared favourably with numerical predictions; i.e. the predictions were within the same magnitude of the experimental data. Hiel15 applied the Reiner-Weissenberg criterion to the very same experimental data with promising results. Later Raghavan and Meshii16,17 presented a creep-rupture model, based on a creep model and a critical fracture criterion, and applied it to a high Tg epoxy and its carbon fibre reinforced composites with good results. These authors found out that the critical fracture energy was dependent on strain rate and temperature. For elevated temperatures or very long times, a constant value for the critical fracture energy was considered suitable, in this case in accordance with the ReinerWeissenberg criterion. Miyano et al.18,19,20,21 showed experimental evidence that flexural and tensile strength of CRFP (carbon fibre reinforced composite) depend on rate loading and temperature, even near room temperature. It was also proved that the timetemperature superposition principle was applicable to obtain master curves for strength of CFRP. Moreover, the same time-temperature superposition principle was applicable for static, creep and fatigue strengths, which presented the same failure mechanism over a wide range of time and temperature. Since materials are far from being homogeneous, disorder plays an important role in strength of materials. Size effects on sample strength are one consequence of this phenomenon. Today, the importance of a statistical treatment in determining the strength of materials is widely recognized. Statistical theories for fracture

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models are able to describe damage progression and failure in a qualitative manner of real (brittle and quasi-brittle) materials.22 The designated fibre bundle models (FBM) are one of the simplest approaches used to analyse fracture in disordered media.22 Time-dependent fracture was approached by generalizations of FBM. In these models each fibre obeys a time-dependent constitutive law.23 These models were also extended to interface creep failure.24 Furthermore, experimental results proved that these simple models with a large heterogeneity, based on interaction of representative elements with a simple nonlinear rheology, were sufficient to explain qualitatively and quasi-quantitatively the creep failure of composite materials.25,26 This research revealed the existence of a stress threshold, under which an infinite lifetime is expected, and that there is a strong correlation between the primary creep and rupture time. Also a strong correlation was found between the time of the minimum of the strain rate in the secondary creep regime and the failure time. A related relationship was found by Little27 for random continuous fibre mat reinforced polypropylene composite. It was found that the estimated secondary creep rate was inversely related to the observed creep rupture response time.27 It is to be noted that this is quite similar to the relationship found for metals by Monkman and Grant.28 Dealing directly with engineering aspects of the failure problem, statistical models combined with the micromechanical analysis have been used, with success, to model the strength and creep rupture of fibre composites. Wagner et al.29 conducted an experimental study on the creep rupture of single Kevlar 49 filaments and concluded that the lifetimes follow a two-parameter Weibull distribution, as predicted by the theory based on a statistical micromechanical model. Later Phoenix et al.30 developed a statistical model for the strength and creep rupture of idealized carbon fibre composites, combining creep in the matrix and the statistics of fibre strength. The experiments verified the theoretical predictions for the strength but were not very conclusive for the creep rupture, pointing the need for more reliable characterization of the fibre strength, matrix creep and the time-dependent debonding at the fibre matrix interface. More recently Vujosevic and Krajcinovic31 developed a micromechanical statistical model to predict the creep and creep rupture of epoxy resins, using a 2D lattice to describe the microstructure and a probabilistic kinetic theory of rupture of the molecular chains to characterize the creep deformation evolution. The time to creep failure is defined as the state at which the lattice stiffness reaches a zero value. All these local and direct analyses of the growth of the defects have shown promising results. This type of analysis has the advantage of allowing a deeper understanding of the mechanisms responsible for the rupture and creep rupture. Nevertheless the global and homogeneous analysis, simpler to formulate and solve, is more convenient for practical applications. Hiel15,32 in fact states that the failure should be a part of a complete constitutive description of the material. Brinson33 argued that this approach could simplify the procedure to predict the

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delayed failure in structural polymers without losing the necessary accuracy. Failure being part of the complete constitutive description of the viscolastic material, it is easy to demonstrate that failure criterion benefits from the TTSP and TSSP procedures. These feature are present in the Schapery viscoelastic model,34,35 for example, which allows the accelerated tests, certainly in accordance with Miyano et al.’s20 experimental evidence. Abdel-Tawab and Weitzman36 developed a model that couples viscoelastic behaviour with damage in which the thermodynamic force conjugate to damage depends on the viscoelastic internal state variables. It appears to be a good theoretical framework to understand the applicability of TTSP obtained from the viscoelastic properties to the strength. Usually the strength theories do not include the creep to yield or creep to rupture process. Since the stress-strain analysis is based on continuum mechanics it presents a difficulty with predicting failure in general and creep failure in particular of polymers and polymer matrix based composites. Fracture mechanics and damage mechanics include the distribution of defects into continuum models, which allow time-dependent failure prediction. Energy-based failure criteria provide another possible approach; for example, establishing that the energy accumulated in the springs of the viscoelastic mechanical model, designated as free energy, has a limit value. This limit can be either constant or not. Earlier approaches to the prediction of time-dependent failures provide explicit elementary equations to predict lifetime. In the context of the present chapter a global and homogeneous analysis was chosen because it is more convenient for practical engineering applications.

12.2 Energy-based failure criteria One of the first theoretical attempts to include time on a material strength formulation was developed by Reiner and Weissenberg,8 for viscoelastic materials. Briefly, the Reiner-Weissenberg Criterion8 states that the work done during the loading by external forces on a viscoelastic material is converted into a stored part (potential energy) and a dissipated part (loss energy). In summary, the criterion says that the instant of failure depends on a conjunction between distortional free energy and dissipated energy; a threshold value of the distortional energy is the governing quantity. Let us assume that the unidirectional strain response of a linear viscoelastic material, under arbitrarily stress σ(t), is given by the power law as,

[12.1]

where D0, D1, n are material constants and τ0 represents the time unity (equal to one second or one hour or one day, etc.).

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The free stored energy, using Hunter’s37 formulation, is given by

[12.2]  The total energy is defined as

[12.3]

Accordingly these time-dependent failure criteria38 predict the lifetime under constant load, as a function of the applied load σ0 and the strength under instantaneous condition σR: Reiner-Weissenberg Criterion (R-W), states that ,  

[12.4]

Maximum Work Stress Criterion (MWS), states that

[12.5]

Maximum Strain Criterion (MS), states that ε(t)  D0σR,

[12.6]

Modified Reiner-Weissenberg Criterion (MR-W), states that



[12.7]

where γ = σ02/σR2. If we consider the normalized rupture free energy as

[12.8]

and the normalized applied stress as

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[12.9] then it is possible to rewrite all the previous criteria in an non-dimensional form. Reiner-Weissenberg Criterion (R-W):

[12.10]

Maximum Work Stress Criterion (MWS):

[12.11]

Maximum Strain Criterion (MS):

[12.12]

Modified Reiner-Weissenberg Criterion (MR-W): [12.13]



These relationships, computed for a linear viscoelastic material with n = 0.3, are plotted in Fig. 12.1. In summary, these approaches establish a relationship between time to failure, viscoelastic properties and static stress failure throughout a stored elastic energy limit concept. As an approximation it is not difficult to conclude that we can take t~σ –2/n for the R-W and the MSW criteria and t~σ –1/n for the MS and the MR-W criteria. Similar results were obtained by using fracture mechanics concepts.39 In fact these concepts established a relationship between time to failure, viscoelastic properties and strength properties,40,41 which are similar to the present approach. The main difference in these failure criteria is the interpretation of the physical

12.1  Comparison of failure criteria for a linear viscoelastic material with n = 0.3.

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constants. According to Song et al.42 there are three major phenomena, which frequently occur simultaneously, responsible for creep failure of viscoelastic materials: (1) time-dependent constitutive equations; (2) time to the formation of overstressed polymer chains in a localized plastic area, i.e. fracture initiation mechanism; (3) the kinetics of molecular flow and bond rupture of the overstressed polymer chains. The facture mechanics approach assumes the existence of defects from the start and develops a theory about the kinetic crack growth, i.e. the fracture initiation process is neglected. In the present approach the stored energy in the material, i.e. the energy stored by all springs in the viscoelastic model, can be compared with energy necessary to stretch the polymer chains and promote their bond rupture. In fact it is possible to visualize the polymer chains as linear springs acting as energy accumulators. Nevertheless, these energy accumulators have a limited capacity above which bond rupture occurs. Therefore the stored energy limit, denominated critical energy, can be related with energy involved in all microscale bond ruptures that lead to creep rupture. Most probably this critical energy depends on the internal state. In reality, there are some experimental indications that this critical energy is temperature and strain rate dependent,16 at least for temperatures lower than the glass transition temperature Tg (or shorter times). This is in accordance with results obtained in this work, i.e. the R-W criterion is not universal. On the other hand there is some experimental evidence, for polymers and composite polymers, that change in fracture mode is a result of change in critical energy with temperature and strain rate.16 Finally, it is not difficult to accept that creep rupture is strongly related with creep compliance or relaxation modulus. This relationship comes out naturally from theoretical approaches like fracture mechanics and energy criteria. Furthermore, creep rupture and relaxation modulus variations with time, measured experimentally, resemble one another in an extraordinary manner. Most probably this signifies that a change in the relaxation modulus corresponds to a change in the strength.

12.3 Creep rupture based on simple micromechanical models In this section we describe two simple micromechanical models used to simulate creep rupture of unidirectional composites by assuming that the matrix is a viscoelastic material and the fibre is an elastic material or by assuming that both constituents are viscoelastic solids. The results for both models are compared with energy failure criteria for illustration purposes. Du and McMeeking43 predicted the creep rupture time in unidirectional composites under tensile loads. The model assumed that when the composite strain (or stress) of the McLean44 model had reached the rupture strain (or stress) of the Curtin45 model, the composite failed. Later on, Koyanagi et al.46 proposed a modified version applied to a unidirectional glass fibre/vinylester composite, which was in part experimentally validated.

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This simple model can be used to illustrate some aspects of creep failure. Following Du and McMeeking43 and Koyanagi et al.46 the Curtin-McLean model (CML) lifetime expressions are deduced for creep loading condition. The McLean model was derived considering that the fibre was elastic and the matrix was viscoleastic. The fibre strain is (elastic),

[12.14]

and the matrix strain (viscoelastic),

[12.15]

where ε is the total strain, B and n the creep coefficients, σm the matrix stress, σf the fibre stress, and Em, Ef are the matrix and fibre modulus, respectively. The composite stress is given by the rule of mixtures,

[12.16]

where Vf is the fibre volume fraction. From the above equations the composite strain under creep load σ = σ0 is derived.

[12.17]

 The initial creep strain is

[12.18]

In this model the creep strain is limited to a defined value,

[12.19]

Assuming the allowable maximum fibre stress as

[12.20]

the creep lifetime expression is obtained.

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[12.21]



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Now it is possible to determine the limit stress level, σS, under which the condition for an infinite lifetime under creep is verified, which is given by

[12.22]

The general behavior of polymers under the effect of time, changing from glassy to rubbery state, is to deform in an asymptotic way as Brinson33 recently discussed in detail. This response is in accordance with the features displayed by the CML model. The creep master curve of typical polymers reaches a plateau after a certain temperature level or equivalent time, by applying the time–temperature superposition principle (TTSP). Therefore the simple power law cannot capture the long-term range creep behavior. Furthermore, it is expected that the rupture stress becomes constant in the rubbery plateau region of the polymer. Following Yang’s47 research, a variation of the usual generalized power based on the ColeCole function is used to model the creep strain under arbitrary stress σ (t) as,

[12.23]

where D0, D∞, n are material constants and τ0 = f (σ). For a constant applied stress, i.e. a creep load, the equation becomes

[12.24]

It is very easy to demonstrate that if we apply the maximum strain as failure criterion, i.e. εmax = D0σR, the following lifetime expression is obtained

[12.25]

As before, it is possible to determine the limit stress level, σS, under which the condition for an infinite lifetime under creep is verified, which is given by

[12.26]

Since in the CML model we have the following relationships,

[12.27]

By applying Eq. 12.26 we obtain the following

This coincides with the previous result given by Eq. 12.22.

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[12.28]

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In order to illustrate the previous results a numerical example is given. Table 12.1 gives the CML parameters used. The creep compliance for the CML is depicted in Fig. 12.2. The nonlinearity effect is very pronounced in this case. From creep data obtained from the CML model, a nonlinear viscoelastic power law was fitted. The parameters are presented in Table 12.2. Using the viscoelastic parameters in the lifetime expressions presented before, its predictions were compared with CML calculated lifetime. The results are depicted in Fig. 12.3. As would be expected, the MS criterion give the best approach to the CML lifetime results, since the model uses the fibre maximum strain as failure criterion. Nevertheless the other time-dependent failure criteria are capable of predicting lifetimes quite close to the CML model and even predict a stress limit for infinite lifetime. Let us now consider a variation of the previous model designated, from now on, as the Modified Curtin-McLeen model (MCML). In this case the fibre and the matrix are considered viscoelastic. Table 12.1  Elastic and viscoelastic CML parameters used to simulate a unidirectional composite Ef (MPa)

Em Vf (MPa)

Ec (MPa)

B n 1/hour

Smax (MPa)

70200

10000

13010

2.50E-10

1100

0.05

2.5

12.2  Creep compliance obtained for the simulated unidirectional composite using the CML model.

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Table 12.2  General power law parameters fitted to the simulated unidirectional composite D0 (1/GPa)

D∞ n (1/GPa)

σR (MPa)

0.0769

0.287

204

0.800

τ0 = 2.54 x 106 σ –1.47.

12.3  Calculated lifetime for the simulated unidirectional composite using the CML model.

The fibre strain is (viscoelastic),

[12.29]

and the matrix strain (viscoelastic),

[12.30]

where ε is the total strain, B, H, n and p the creep coefficients, σm the matrix stress, σf the fibre stress, and Em, Ef are the matrix and fibre modulus, respectively. The composite stress is given by the rule of mixtures,

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[12.31]

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where Vf is the fibre volume fraction. From the above equations the following differential equation is derived, considering the creep loading condition σ0 = constant,

[12.32]

The differential equation can be analytically integrated provided that n = p = 2. In that condition the solution is readily obtained with the initial condition [12.33] Integrating [12.34]

where [12.35]

with Assuming the allowable maximum fibre stress as,

[12.36]

and assuming that B >> H the creep lifetime is obtained,

[12.37]

with In this case, for tR → ∞ we must have

[12.38]

Therefore the creep stress under which the condition of an infinite lifetime is verified is obtained

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[12.39] In conclusion, the applied stress range which provokes creep failure σS < σ0 < σR is

[12.40]

Therefore it can be concluded that although the MCML model exhibits an unbound creep strain, the creep failure stress is bounded due to the fibre relaxation. In order to illustrate the previous results a numerical example follows. Table 12.3 gives the MCML parameters used. The creep compliance for the MCML is depicted in Fig. 12.4. The nonlinearity effect is also very pronounced in this case. From creep data obtained from the MCML model, a nonlinear viscoelastic power law was fitted. The parameters are presented in Table 12.4. Table 12.3  Elastic and viscoelastic MCML parameters used to simulate a unidirectional composite Ef (MPa)

Em Vf (MPa)

Ec (MPa)

B n 1/hour

Smax (MPa)

H 1/hour

p

70200

10000

13010

2.50E-10

1100

2.50E-13

2

0.05

2

12.4  Creep compliance obtained for the simulated unidirectional composite using the MCML model.

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Creep and fatigue in polymer matrix composites Table 12.4  General power parameters fitted to the simulated unidirectional composite D0 (1/GPa)

C n (1/GPa)

σR (MPa)

0.0769

0.0000440

204

0.790

τ0 = –3.52 x 10–3 σ + 1.18.

12.5  Calculated lifetime for the simulated unidirectional composite using the MCML model.

As done previously, using the viscoelastic parameters in the lifetime expressions presented before, the resulting predictions were compared with MCML calculated lifetime. The results are depicted in Fig. 12.5. Again, as would be expected, the MS criterion give the best approach the MCML lifetime results, since the model also uses the fibre maximum strain as failure criterion. The other failure criterion that is quite close the MCML lifetime is the MSW criterion. The other failure criteria fail largely to predict the MCML lifetime. However, all failure criteria fail to predict the stress limit for infinite lifetime. The reason lies in the fact that the creep compliance evolution does not tend to a plateau (limit value); consequently, since all failure criteria depend on the compliance, they are not capable of predicting that the creep failure stress reaches a plateau. Although both micromechanical models are very simple to formulate and solve they give us an insight of what are the possible mechanics which lead to creep failure of unidirectional fibre-reinforced composites.

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12.4 Experimental cases The purpose of this section is to give several practical examples that show the applicability of the previous time-dependent failure criteria. These examples are divided into three distinct groups. The first one gathers two thermoplastic polymers: polyamide (or nylon) and polycarbonate used for technical applications. The second gathers representative thermoset composite systems used to produce large structural automotive components. Finally, the last group uses creep lifetime experimental data of thermoset composite systems obtained from an accelerated methodology based on the time–temperature superposition principle (TTSP).

12.4.1  Thermoplastics The experimental results for thermoplastics, taken from the technical literature,48,49 are used to illustrate the present discussion. The materials examined are polyamide 66 (Nylon 66 A100) unfilled, polyamide 66 filled with 30% of glass fibre (Nylon 66 A190)48 at room temperature and polycarbonate (PC Lexan 141) at room temperature.49 These materials were chosen for two reasons: firstly they are already characterized in terms of creep and creep rupture; secondly, as Gotham3 demonstrated experimentally, the Nylon 66 A100 and PC Lexan 141 present a ductile failure at room temperature and the Nylon 66 A190 a brittle failure. Considering that the materials behavior is linear viscoelastic, which is an approximation for the higher stress levels, the material parameters were determined for the general power law model and are presented in Table 12.5. The creep compliance of polyamide 66 displays a linear behavior for times lower than 100 hours. For longer times there is no information for the higher stress levels but we should expect a certain level of nonlinearity. Nevertheless, the impact of higher stress on creep failure prediction for longer times decreases as time increases, which minimizes the possible influence of the nonlinear behavior. Table 12.5  Viscoelastic and rupture parameters of the polymer materials Material

T

Ref.



Creep compliance (1/MPa)

(°C) D0

D1

n

Static failure σR (MPa)

τ0

Nylon 66 Moulding 23° 3.36E-04 2.31E-05 0.399 1 hour A100   Powders   Group   (1974)48 Nylon 66 Moulding 23° 1.05E-04 6.77E-06 0.294 1 hour A190   Powders   Group   (1974)48 PC Lexan Challa 23° 3.12E-04 1.44E-04 0.085 1 hour 141   (1995)49

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R-W MR-W MWS MS 77

77

77

77

95

95

95

95

75

89

89

86

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The same could be said about the other two materials. The instantaneous rupture stresses, σR, were also determined for each material and criterion. This can be done by backward calculation over the experimental results for the higher stress levels and lowest times of the failure data. In practical cases these rupture stresses are not necessarily coincident with static rupture stresses. Sometimes they are not even equal for all the criteria, as in the case of polycarbonate, depending on the creep compliance. The results are depicted in Fig. 12.6–12.8. In general, the predictions are close to the experimental data; however, as it can be observed, each material follows a different time-dependent failure criteria.

12.6  Experimental and calculated creep lifetime for Nylon 66 (A100) at room temperature.

12.7  Experimental and calculated creep lifetime for Nylon 66 (A190) at room temperature.

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12.8  Experimental and calculated creep lifetime for PC Lexan 141.

As discussed before, it is not difficult to accept that creep rupture of polymers is strongly related to creep compliance or relaxation modulus. In the present analysis it was assumed that effect of the nonlinear viscoelastic behaviour was negligible and apparently the experimental results confirmed that assumption. Nevertheless, a deeper analysis of the impact of non-linear viscoelastic behaviour on the creep rupture predictions should be performed. For extrapolation purposes a methodology is necessary to determine, from the available creep rupture data, which approach is appropriated and how far that extrapolation could go. From the present cases it appears that a minimum of 24 to 48 hours of creep rupture data is necessary to choose the appropriate theoretical model in order to extrapolate data to almost two decades with a good accuracy. In the present work, one crystalline polymer (polyamide 66) and another non-crystalline polymer (polycarbonate) were used to access theoretical models. In all cases, except for glass fibre reinforced polyamide 66, ductile failure was observed. Apparently these energy-based criteria can be applied to different types of polymers with ductile or brittle failures.

12.4.2  Thermosetting polymer-based composites Experimental results obtained from the literature, for thermosetting polymerbased composites, are compared with theoretical lifetime predictions. The viscoelastic properties of these materials are displayed in Table 12.6. The experimental data was obtained from the project conduct by the Oak Ridge National laboratory50–55 to develop experimentally based, durability-driven design guidelines to assure the long-term integrity of representative thermoset composite systems to be used to produce large structural automotive components. These examples include different types of reinforcements; E glass fibre randomly orientated chopped strands and continuous strand, swirl-mat fibre and continuous T300 carbon fibre.

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Chopped-glass-fibre/urethane Glass-fibre/urethane Glass-fibre/urethane Crossply carbon-fibre ±45° Crossply carbon-fibre 90°/0° Quasi-isotropic carbon-fibre Quasi-isotropic carbon-fibre

Material (2001a)51   8.47E-05 1.06E-04 1.06E-04 8.85E-05 2.14E-05 3.09E-05 3.36E-05

(°C) D0

T

Corum 23 Corum (1998)50 Ren (2002a54 2002b)55   23 Corum (1998)50 Ren (2002a54 2002b)55 120 Corum (2001b)52   23 Corum (2001b)52   23 Corum (2002)53   23 Corum (2002)53 120

Ref. 9.59E-06 7.35E-06 1.37E-05 2.04E-08 8.30E-07 7.96E-07 6.68E-06

D1

0.141 0.196 0.196 0.200 0.112 0.161 0.192

n

Creep compliance (1/MPa)

σR 1hour 160 1hour 120 1hour   85 1hour 140 1hour 455 1hour 295 1hour 270

τ0

Static failure (MPa)

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Table 12.6  Viscoelastic and rupture parameters of the composite materials

384

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385

Therefore a total of seven experimental cases of creep rupture were used to test energy-based failure criteria. The first two cases are E glass fibre/urethane matrix composites with two different reinforcements; randomly orientated chopped strands and continuous strand, swirl-mat fibres. The other cases are crossply and quasi-isotropic composites reinforced with continuous T300 carbon fibres in a urethane matrix. The reinforcement was in the form [±45°]3S crossply composite and [0°/90°/±45°]S quasi-isotropic composite. The results are plotted throughout Fig. 12.9–12.15. In all cases MWS and MR-W failure criteria predict similar lifetimes and R-W predicts systemically higher lifetimes for each stress level. In most cases experimental data fall between R-W and MR-W failure criteria. The

12.9  Experimental and calculated creep lifetime for chopped-glass-fibre composite.

12.10  Experimental and calculated creep lifetime for glass-fibre/ urethane composite at 23°C.

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12.11  Experimental and calculated creep lifetime for glass-fibre/ urethane composite at 120°C.

12.12  Experimental and calculated creep lifetime for crossply carbonfibre 45°/45° at 23°C.

12.13  Experimental and calculated creep lifetime for crossply carbonfibre 90°/0° composite at 23°C. © Woodhead Publishing Limited, 2011



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12.14  Experimental and calculated creep lifetime for quasi-isotropic carbon-fibre composite at 23°C.

12.15  Experimental and calculated creep lifetime for quasi-isotropic carbon-fibre composite at 120°C.

exception is for the crossply carbon fibre [+45°/–45°] composite at 23°C. In this case experimental data lifetime is higher, but close to R-W predictions. In these cases, it appears that the energy-based failure criteria present a remarkable potential to extrapolate experimental creep rupture data with a high degree of confidence.

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12.4.3  Time–temperature superposition principle Creep strength for three different composites is used to illustrate the potential applicability of the previous approaches in an extended time scale by using the time–temperature superposition principle (TTPS). One composite system is the CFRP laminate consisting of nine layers of plain woven cloth of carbon fibre and matrix vinylester resin. These CFRP laminates (T300/VE) were moulded by the resin transfer moulding (RTM) method and cured for 48 hours at room temperature and for 2 hours at 150°C. The volume fraction of the fibre in the CFRP is approximately 52%.21 Another composite system is a carbon fibre reinforced polymer, UT500/135, which consists of twill-woven UT500 carbon fibre and 135 epoxy resin.56 The last one is the T800S/3900-2B composite which consists of unidirectional T800 carbon fibre and 3900 epoxy resin with toughened interlayer.56 The input data for the lifetime models, i.e., the viscoelastic properties and the strength under instantaneous conditions, are presented in Table 12.7. In many cases the TTSP applied to creep compliance D(t;T) holds valid for static and creep strength σR(tf ;T ) with the same shift factors aT(T),

[12.41]

where T represents the temperature and T0 the reference temperature. From this a reduced time to failure is defined as

[12.42]

As an example, the shift factors for the T300/VE composite were determined as,21

[12.43]

where is G the gas constant 8.314E-3 KJ/(K mol).

Table 12.7  Viscoelastic and rupture parameters of the composite materials Creep compliance (1/MPa) Material Ref. D1 n D0

Instantaneous failure stress (MPa)

τ0

σR

T300/VE Miyano (2005)21 3.60E-04 1.69E-05 0.209 1 min 700 T800S/3900-2B Miyano (2006)56 1.85E-05 6.33E-07 0.119 1 min 830 UT500/135 Miyano (2006)56 2.60E-05 2.55E-08 0.281 1 min 500

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Using the shift factors in the time-dependent failure criteria applied for the T300/VE it was possible to predict the lifetime for each temperature with good agreement with experimental data, as depicted in Fig. 12.16. Consequently it become possible to predict the creep lifetime for an enlarged time scale, using the TTSP and the concept of reduced time. This can be observed in Fig. 12.17–12.19). It can be concluded that the time-dependent failure criteria is in good agreement with experimental data. In all cases the MS and MR-W lifetime predictions match quite well the experimental data.

12.16  Experimental and calculated creep lifetime for T300/VE composite at different temperatures.

12.17  Experimental and calculated creep reduced lifetime for T300/VE composite.

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12.18  Experimental and calculated creep reduced lifetime for T800/3900-2B composite.

12.19  Experimental and calculated creep reduced lifetime for UT500/135 composite.

12.5 The Crochet model (time-dependent yielding model) A multi-axial yield/failure model for viscoelastic/plastic materials was developed by Naghdi and Murch57 and later extended and refined by Crochet.58 This approach was recently revised by Brinson.33 In this theory, the total strain is assumed to be the sum of the viscoelastic and plastic strains. Stresses and strains are separated into elastic and viscoelastic deviatoric and dilatational components. © Woodhead Publishing Limited, 2011



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The yield function is given as

[12.44]

Crochet gave specific form to the function χij such that

[12.45]

and defined a time-dependent uniaxial yield function as (empirical equation)

[12.46]

where A, B and C are material constants. No additional explanation for this empirical equation was given by Crochet.58 Assuming a linear viscoelastic law given by the power law, the creep strain, under constant stress σ(t) = σ0, can be calculated as

[12.47]

The difference between viscoelastic and elastic strains in creep loading conditions becomes

[12.48]

and the lateral strains become, upon assuming a constant Poisson’s ratio υ,

[12.49]

The time factor, χ, in Crochet’s time-dependent yield criteria for uniaxial tension becomes

[12.50]

Using the previous formulations an equation for the time to yield, tc, for a linear viscoelastic material can be found

[12.51]

where the symbol σf is used, instead of σy, to indicate that the process may be used as well for creep rupture. Since the relationship between yield (or failure) stress and χ is empirical, the suggested form given by Crochet can be questioned. For instance, the timedependent uniaxial yield (or failure) function can be given by a simple linear relationship, i.e., © Woodhead Publishing Limited, 2011

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where α and β are material constants. Following the previous developments the lifetime expression becomes

[12.53]

This expression remarkably resembles the lifetime formulations obtained for energy-based time-dependent failure criteria. For uniaxial cases it is an interesting exercise to plot the evolution of creep failure stress versus the χ function. In this manner it becomes possible to conclude about the best empirical approach to relate both parameters. For this purpose the experimental data previously given for the thermoplastics was used: polyamide 66 (Nylon 66 A100) unfilled, polyamide 66 filled with 30% of glass fibre (Nylon 66 A190)48 at room temperature and polycarbonate (PC Lexan 141)49 at room temperature. The evolution of creep failure stress with χ function was plotted and linear and exponential relationships were fitted considering just the initial 24h experimental data. Using these fitted equations the long-term creep failure was predicted by extrapolation. The results are depicted in Fig. 12.20–12.25. These results show that the linear relationship between the

12.20  Experimental and fitted curves for the creep failure stress as function of χ function for Nylon 66 (A100).

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12.21  Experimental and predicted creep lifetime using Crochet model for Nylon 66 (A100).

12.22  Experimental and fitted curves for the creep failure stress as function of χ function for Nylon 66 (A190).

creep stress and the function χ could be sufficient to fit the experimental data. Furthermore, for the present cases, the extrapolation for the long term predicts lifetimes in good agreement with experimental data. Expanding this exercise to larger time scales, by applying the TTSP, the experimental data published for the CFRP laminates (T300/VE)21 and for the

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12.23  Experimental and predicted creep lifetime using Crochet model for Nylon 66 (A190).

12.24  Experimental and fitted curves for the creep failure stress as function of χ function for PC Lexan 141.

T800S/3900-2B laminate56 was used. As before, the evolution of creep failure stress with χ function was plotted and linear relationships were fitted, in this case, just using experimental data obtained at the lower temperature. The results are depicted in Fig. 12.26–12.29.

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12.25  Experimental and predicted creep lifetime using Crochet model for PC Lexan 141.

12.26  Experimental and fitted curves for the creep failure stress as function of χ function for the composite T300/VE.

Again, these results show that the simple linear relationship between the creep stress and the function χ is sufficient to obtain good predictions. However, the experimental data reveal that this relationship, for large time scales, appears to be of exponential type.

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12.27  Experimental and predicted creep lifetime using Crochet model for the composite T300/VE.

12.28  Experimental and fitted curves for the creep failure stress as function of χ function for the composite T800S/3900-2B.

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12.29  Experimental predicted creep lifetime using Crochet model for the composite T800S/3900-2B.

Nevertheless, for the present cases, the model extrapolation for the long term, based on limited data, computes lifetime predictions in good agreement with experimental data.

12.6 Kinetic rate theory The rate theory of fracture is based on a molecular approach, i.e. on the kinetics of molecular flow and bond rupture of the polymer chains. Based on these approaches Zhurkov59 presented the first models to predict materials lifetime tf (except for very small stresses) in terms of a constant stress level σ,

[12.54]

where t0 is a constant in the order of the molecular oscillation period of 10–13s, k is the Boltzmann constant, T is the absolute temperature, U0 is a constant for each material regardless of its structure and treatment and γ depends on the previous treatments of the material and varies over a wide range for different materials. Griffith et al.12 applied a modified version of Zurkov’s equation,

[12.55]

known as the modified rate equation, to predict the time to rupture of continuous fibre-reinforced plastics with reasonable success.

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12.7 Fracture mechanics extended to viscoelastic materials The fracture mechanics analysis was extended to viscoelastic media to predict the time-dependent growth of flaws or cracks. Several authors developed extensive work on this area.40,41,60–63 Schapery62,63 developed a theory of crack growth which was used to predict the crack speed and lifetime for an elastomer under uniaxial and biaxial stress states.40 For a centrally cracked viscoelastic plate with a creep compliance given by Eq. 12.1 under constant load, Schapery40 deduced, after some simplifications, a simple relation between stress and failure time,

[12.56]

where n is the exponent of the creep compliance power law and B a parameter which depends on the geometry and properties of the material. Leon and Weitsman64 and Corum et al.51 used this approach, where B was considered an experimental constant, to fit creep rupture data with considerable success. Recently Christensen65 developed a kinetic crack formulation to predict the creep rupture lifetime for polymers. The lifetime was found from the time needed for an initial crack to grow to a size large enough to cause instantaneous further propagation. The method assumed quasi-static conditions and only applies to the central crack problem. The polymeric material was taken to be in the glassy elastic state, as would be normal in most applications. For general stress, σ (t) we have,

[12.57]

where γ = σ 20 / σ 2R, m is the exponent of the power law relaxation function and α is a parameter governed by the geometry and viscoelastic properties. For constant stress, σ = σ0, the lifetime is given as,

[12.58]

12.8 Continuum damage mechanics A classical approach to consider the degradation of mechanical properties is provided by the method of continuum damage mechanics (CDM). Following the original ideas of Kachanov,66 the net stress, defined as the remaining load-bearing cross-section of the material, is given67 by,

[12.59]

where 0 ≤ ω ≤ 1 is the damage variable. At rupture no load-bearing area remains and the net stress tends to infinity when ω → 1.

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Kachanov66 assumes the following damage growth law

[12.60]

where C and ν are material constants. This equation leads to a separable differential equation for ω (t), assuming ω (0) = 0

[12.61]

The damage growth law is given as

[12.62]

Assuming failure when ω = 1 then the following expression is obtained:

[12.63]

From the previous relationship, the time to failure for creep is readily obtained assuming σ (t) = σ0,

[12.64]

Clearly this result is equivalent to that obtained previously by using the Schapery theory.40 Therefore the creep lifetime expressions obtained for both theoretical approaches are directly comparable and are, in fact, equivalent, i.e.



[12.65]

even though the parameters have distinct physical interpretations.

12.9 Damage accumulation models for static (creep) and dynamic fatigue The damage evolution depends strongly on different factors acting simultaneously, i.e. temperature, moisture, stress, viscoelasticity, viscoplasticity, etc. In their turn these factors are time dependent. In practice its influence on long-term failure is measured independently, i.e. under constant conditions. Afterwards it is a necessary methodology to evaluate its combined effects. However, this is this out of the scope of this chapter.

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One crucial question remaining to be solved completely is how to predict damage accumulation, or the remaining strength, after a fatigue or creep cycle at multiple stress levels, based on the fatigue and creep master curves. Miner’s Rule68 is an example of a simple way to account for damage accumulation due to different fatigue cycles. This damage fraction rule is also designated as the Linear Cumulative Damage law (LCD). For the fatigue, it states that failure occurs when the following condition is verified

[12.66]

where nf is the number of cycles to failure at stress level σi and ∆ni is the number of cycles applied at each stress level σi of the fatigue cycle. Hence Eq. 12.66 provides a failure criterion for fatigue. The corresponding form for creep conditions is given by

[12.67]

where tc is the creep rupture lifetime at stress level σi and ∆ti is time applied at each stress level σi. Once more Eq. 12.67 specifies a criterion for lifetime at multiple stress levels. Later, Bowman and Barker69 suggested a combination of both damage fraction rules to analyse experimental data, for thermoplastics tested until failure under a trapezoidal loading profile, which combined fatigue with creep. Although the Miner’s rule can predict accurately failure of fibre-reinforced polymers under certain combined stress levels, it proved to be inappropriate in many other cases. However, due to its simplicity, it is still used nowadays in engineering design. More sophisticated models, concerning fatigue of fibre-reinforced polymers, have appeared in recent years.70–74  These approaches are either empirical or semiempirical. Yet these models aim to capture the inherent nonlinearity of damage accumulation, allowing prediction of the fatigue behavior using a well-defined minimum number of tests. The conditions of applicability of Miner’s rule were discussed by Christensen.75 It provided a theoretical validation of the use of LCD law when relating creep failure conditions to constant stress (or strain) rate failure conditions. For that purpose Christensen75 developed a very simple kinetic crack growth theory following the Schapery40 formalism and based on a generalization to viscoelastic material of the Griffith result for elastic material. Guedes39,76 extended this validation by using other existing theoretical frameworks. A cumulative damage theory developed to address various applied problems in which time, temperature and cyclic loading are given explicitly, was developed by Reifsnider et al.77–79 The basic form of the strength evolution equation calculates the remaining strength Fr

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401 [12.68]

where, z = t/τ, t is the time variable, τ is a characteristic time associated with the process, Fa is the normalized failure function that applies to a specific controlling failure mode and j is a material parameter. This material parameter influences the damage progression; if j<1 the rate of degradation is greatest at beginning but if j>1 the rate of degradation increases as a function of time, while if j = 1 there is no explicit time dependence in the rate of degradation. The failure criterion is given by Fr = Fa. The Strength Evolution Integral (SEI) has not been developed from first principles but based on few postulates combined with a kinetic theory of solids. Despite this shortcoming, SEI has been used successfully for almost 20 years by Reifsnider et al.,77–79 to solve various applied problems in which time, temperature and cyclic loading were explicit influences.

12.10 Conclusions Several time-dependent failure criteria applied to polymers and polymer-based matrix composites were presented in this chapter and illustrated using experimental data of two different polymers and composites. All criteria, excluding the energybased failure criteria and the MS criterion, must have some or all the parameters obtained by curve-fitting the experimental lifetime data. This happens because the background theories lead to relatively simple expressions recurring to empirical laws and simplifications, which ultimately enclose several variables related to the basic properties, with physical meaning, into few parameters. Consequently it becomes impossible to determine these parameters based on the basic properties, i.e. geometry, elastic and viscoelastic parameters and instantaneous strength. However, in this form, these lifetime expressions possess a more general character which can be used to represent creep failure data for a large time scale. One important characteristic of these criteria is their forecast lifetime capability, based on short-time experimental data. In that respect all the present approaches appear to possess this capability. This review also shows the remarkable ability of energy-based failure and MS criteria to forecast lifetime based on viscoelastic parameters and instantaneous strength.

12.11 References   1 Karbhari V.M., Chin JW, Hunston D, Benmokrane B., Juska T., Morgan R., Lesko J.J., Sorathia U. and Reynaud D. Durability gap analysis for fibre-reinforced polymer composites in civil infrastructure. Journal of Composites for Construction (3):238– 247, 2003.

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65 Christensen R.M., An evaluation of linear cumulative damage (Miner’s Law) using kinetic crack growth theory. Mech Time-Dependent Mater 6(4):363–77, 2002. 66 Kachanov, L.M., On the Time to Failure Under Creep Conditions. Izv. Akad. Nauk SSR, Otd. Tekhn. Nauk 8:26–31, 1958. 67 Stigh U. Continuum Damage Mechanics and the Life-Fraction Rule. Journal of Applied Mechanics 73(4):702–704, 2006. 68 Miner M.A. Cumulative damage in fatigue. J. Appl. Mech. 12:A159–A164, 1945. 69 Bowman J., Barker M.B.A. Methodology for describing creep fatigue interactions in thermoplastic components. Polym Eng Sci 26(22):1582–90, 1986. 70 Yao W.X. and Himmel N. A new cumulative fatigue damage model for fibre-reinforced plastics. Composites Science and Technology 60:59–64, 2000. 71 Epaarachchi J.A., Clausen P.D. An empirical model for fatigue behavior prediction of glass fibre-reinforced plastic composites for various stress ratios and test frequencies, Composites Part A 34(4):313–326, 2003. 72 Epaarachchi J.A. Effects of static-fatigue (tension) on the tension-tension fatigue life of glass fibre reinforced plastic composites. Compos Struct 74(4):419–25, 2006. 73 Epaarachchi J.A. A study on estimation of damage accumulation of glass fibre reinforce plastic (GFRP) composites under a block loading situation. Composite Structures 75:88–89, 2006. 74 Sutherland H.J., Madell J.F. The Effect of Mean Stress on Damage Predictions for Spectral Loading of Fibreglass Composite Coupons. Wind Energ. 8:93–108, 2005. 75 Christensen R.M. An evaluation of linear cumulative damage (Miner’s Law) using kinetic crack growth theory. Mech Time-Depend Mater 6(4):363–77, 2002. 76 Guedes R.M. Durability of polymer matrix composites: Viscoelastic effect on static and fatigue loading. Composites Science and Technology 67 (11–12):2574–2583, 2007. 77 Reifsnider K.L, Stinchcomb W.W. A critical element model of the residual strength and life of fatigue-loaded composite coupons. In: Hahn HT, editor. Composite materials: fatigue and fracture (ASTM STP 907). Philadelphia (PA): American Society for Testing and Materials; p. 298–313, 1986. 78 Reifsnider K., Case S., Duthoi J. The mechanics of composite strength evolution. Composite Sci Technol 60:2539–46, 2000. 79 Reifsnider K., Case S. Damage tolerance and durability in material systems. Wiley-Interscience; 2002.

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13 Testing the fatigue strength of fibers used in fiber-reinforced composites using fiber bundle tests P. K. Mallick, University of Michigan-Dearborn, USA Abstract: High-strength fibers, such as glass and carbon fibers, used in fiber-reinforced composites are known to exhibit wide strength variation under static tensile tests. The fiber strength variation not only affects the strength of the composite, but also its failure mode. Under high-fatigue loading where the fatigue failure is controlled by fiber failure, the fiber strength variation may even have a larger role than under static loading. This chapter considers the fatigue strength determination of fibers under fatigue cycling. It first reviews the static strength variation and the test methods to determine the static strength of high-strength fibers. It then describes the recent efforts to determine their fatigue strength and its variation using fiber bundle tests. Key words: high-strength fibers, fiber bundle tests, fatigue cycling, static strength variation.

13.1 Introduction Fibers used in fiber-reinforced composites have very high tensile modulus and tensile strength, but their strain-to-failure is very small and they exhibit brittle failure mode. The high tensile strengths of the reinforcing fibers are generally attributed to their filamentary form in which there are statistically fewer surface flaws than in the bulk form. However, as with other brittle materials, their tensile strength data exhibit a wide variation. An example is shown in Fig. 13.1. The strength variation of brittle filaments is modeled using the following twoparameter Weibull strength distribution function (Coleman, 1958): [13.1]  where f (σfu) = probability of filament failure at a stress level equal to σfu σfu = filament (fiber) strength L f = filament length Lo = reference length β = shape parameter σo = scale parameter (which is also the filament strength at Lf = Lo) 409 © Woodhead Publishing Limited, 2011

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13.1  Strength variation of two types of carbon fibers.

The cumulative distribution of strength is given by the following equation: 

[13.2]

where Pf represents the cumulative probability of filament failure at a stress level lower than or equal to σfu. The parameters β and σo in Eq. 13.1 and 13.2 are called the Weibull parameters, and are determined using the experimental data. β can be regarded as an inverse measure of the coefficient of variation. The higher the value of β, the narrower is the distribution of filament strength. The scale parameter σo may be regarded as a reference stress level. The mean filament strength   is given by: 

[13.3]

where Γ represents a gamma function. Equation 13.3 clearly shows that the mean strength of a brittle filament decreases with increasing length. This is also demonstrated in Fig. 13.2. Since individual filament diameter is very small, fibers in composites are used in the form of bundles. Each bundle contains many parallel filaments. It has been observed that even though the tensile strength distribution of individual filaments

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13.2  Mean fiber strength vs. fiber length.

follows the Weibull distribution, the tensile strength distribution of fiber bundles containing a large number of parallel filaments follows a normal distribution (Coleman, 1958). The maximum tensile strength, σfm, that the filaments in the - , can be bundle will exhibit and the mean tensile strength of the bundle, σ b expressed in terms of the Weibull parameters determined for individual filaments. They are given as follows: [13.4]  [13.5] 

13.2 Determination of fiber strength distribution parameters The two most common test methods for determining the Weibull parameters for fiber strength distribution are the single fiber test and the fiber bundle test (Manders and Chou, 1983; Andersons et al., 2002). They are briefly described below.

13.2.1  Single fiber test In this test, a single fiber (filament) is mounted along the centerline of a slotted paper or cardboard tab using a quick-setting adhesive (Fig. 13.3). The length of the slot is equal to the gage length selected for the test. After clamping the tab ends

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13.3  Schematic of the single-fiber test arrangement.

in the grips of a tension testing machine, the midsection of the slotted paper is either cut or burnt away. The tension test is then carried out at a constant loading rate until the fiber fails. The fiber tensile strength is calculated by dividing the force at failure with the average cross-sectional area of the fiber. Two different procedures can be used to determine the Weibull parameters using the single fiber test (Manders and Chou, 1983). Multiple single fiber tests with different fiber lengths It can be seen from Eq. 13.3 that a plot of ln( ) against ln(Lf) is linear with a negative slope of β1– . Thus, the shape parameter β can be determined by conducting multiple single fiber tensile tests with different fiber gage lengths. For each gage length, at least ten specimens are tested and the average tensile strength is calculated. A plot of ln( ) vs. ln(Lf) is then produced and using a linear regression method the slope of the plot is determined. β is determined from the slope. One such plot is shown in Fig. 13.4. Multiple single fiber tests with the same fiber length In this case, the fiber gage length is maintained the same for each single fiber specimen. After the tensile tests, the strength data are arranged from the lowest to the highest value and a probability of failure is assigned to each strength value using the following equation:

[13.6]

where s is the total number of specimens tested and i represents the rank order of the specimen (1 ≤ i ≤ s).

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13.4  Determination of the shape parameter β using multiple single fiber tests with different gage lengths (after Manders and Chou, 1983).

Rearranging Eq. 13.2 and taking logarithms on both sides of the equation, one can write: [13.7]  Taking logarithms of both sides of Eq. 13.7, it can be rewritten as: [13.8]  Equation 13.8 shows that a plot of ln{ln[1/(1 – Pf )]} vs. ln(σfu) is linear with a slope of β. Thus, a linear regression plot of ln{ln[1/(1 – Pf )]} against ln(σfu) will produce both β and σo. While the single fiber test is a simple test, there are several sources of error that must be recognized. The fiber axis alignment with the tensile load direction and fiber failure within the tab are two of these sources of error. The third source of error is related to the fiber extraction from the fiber bundle (also called strand or tow). During fiber extraction, some of the weak fibers may break and may be

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eliminated from the fiber population, which, in effect, is equivalent to censoring the fiber sample before the tests are performed.

13.2.2  Fiber bundle test In the fiber bundle test (Chi et al., 1984), a fiber bundle is subjected to tensile loading and the tensile stress–strain diagram of the fiber bundle is obtained. If it is assumed that (a) the strength distribution of each fiber in the fiber bundle obeys the Weibull distribution function given by Eq. 13.2, (b) the stress–strain relationship for the fibers follows Hooke’s law, i.e., σ = Ef ε, where σ is fiber stress, ε is the corresponding fiber strain and Ef is the fiber modulus, (c) each fiber in the bundle experiences the same strain, i.e., an iso-strain condition exists at all stress levels, and (d) the applied tensile load is equally distributed among the fibers at any instant during the fiber bundle test, then the tensile load F on the fiber bundle can be written in terms of fiber strain, ε, as: [13.9] where A is the fiber cross-sectional area, L is the bundle gage length, No is the number of fibers in the bundle before any tensile load is applied, and n is the number of unbroken fibers in the bundle at any instant. Initially, n = No, but it decreases as fibers in the bundle start to fail one or more at a time. In Eq. 13.9, εo is the scale parameter expressed in terms of strain and εfu is the fiber failure strain. They are given by:

[13.10]  where σo is the scale parameter in terms of stress and σfu is the fiber strength (see Eq. 13.1). The maximum load that the fiber bundle will produce is given by: [13.11]  and the strain corresponding to the maximum load is:  The shape parameter β can be determined using the following equation:

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[13.12]



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 where So is the initial slope of the fiber bundle stress–strain diagram (i.e., at ε = 0). Knowing the value of β, the other Weibull strength distribution parameter, σo, can be determined using Hooke’s law, i.e., σo = Ef εo and Eq. 13.12. The stress–strain diagram of an E-glass fiber bundle is shown in Fig. 13.5. It is worth noting that even though fibers in a fiber bundle exhibit a linear stress–strain diagram in tensile loading, the stress–strain diagram of the fiber bundle is nonlinear and the load drop after the maximum load is gradual. Both nonlinearity and gradual load drop are due to the strength distribution of the fibers in the bundle. The fiber bundle test provides a more realistic evaluation of the failure of fibers in a composite. Single fibers are not used in composites; instead they are bundled together for easier processing and handling. The fiber bundle test to determine the Weibull parameters is also easier to conduct than the single fiber test, but there may also be errors introduced in the fiber bundle test results for the following reasons: (1) the individual fiber lengths in the bundle may vary, and as a result, there may be some slack or loose fibers in the bundle;

13.5  Tensile stress–strain diagram of an E-glass fiber bundle.

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(2) there may be fiber-to-fiber interactions due to friction and sticking, which means equal load sharing assumption may not be valid.

13.3 Fiber bundle model for fatigue In the fiber bundle model for fatigue (Zhou and Mallick, 2004), a fiber bundle, initially containing No parallel fibers, each of length L and cross-sectional area A, is rigidly fixed at both ends (Fig. 13.6). The fiber bundle is subjected to a tensile force P in the fiber direction, which creates a normal stress σ in each fiber. The following assumptions are made in the fiber bundle model. (1) The tensile stress–strain curve for each fiber is linear and elastic until the fiber breaks. (2) The interaction between the fibers is negligible. As n fibers break, the load carried by them is instantaneously distributed equally among the surviving (No – n) fibers so that the stress in the surviving fibers can be written as: [13.14]  where Ef is the fiber modulus, σ is the stress and ε is the corresponding strain. (3) The fatigue life of each fiber fits the following relationship: 

13.6  Fiber bundle model.

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[13.15]



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where σmax is the maximum cyclic stress applied on the fiber during fatigue cycling, σf is the fatigue strength coefficient and Nf is the number of cycles to failure or fatigue life at stress σ. (4) The fatigue strength coefficient σf of each fiber follows a two-parameter Weibull distribution: 

[13.16]

where P is the probability that a fiber will survive stress σf , and σo and β are the Weibull scale parameter and the Weibull shape parameter in fatigue, respectively.

From Eq. 13.15 and 13.16, the relationship between the survival probability in fatigue, the maximum cyclic stress and the fatigue life can be written as: [13.17]  In Eq. 13.17, P represents the cumulative probability of surviving Nf cycles at the maximum cyclic stress σmax. To determine the parameters b, β and σo in Eq. 13.17, strain-controlled fatigue tests are conducted on fiber bundle specimens. In a strain-controlled fatigue test, the maximum cyclic strain applied on each fiber is a constant. Using σmax = Ef εmax, Eq. 13.17 can be modified to write: [13.18]  It can be noted from Eq. 13.14 that: [13.19]

Taking double logarithms on both sides of Eq. 13.18, one can obtain:

[13.20] Equation 13.20 exhibits a linear relationship between ln[–ln(P)] and ln(2Nf ). The slope of the linear plot of ln[–ln(P)] against ln(2Nf) is –bβ and the intercept with the ln[–ln(P)] axis is [β ln(Ef εmax) – β ln(σo)]. To determine the fatigue strength exponent b, strain-controlled fatigue tests can be conducted at two different maximum strain levels, say for example at ε1 and ε2, so that  © Woodhead Publishing Limited, 2011

[13.21]

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and [13.22]  At the same survival probability, i.e., for P1 = P2, Eq. 13.21 is set equal to Eq. 13.22, from which: [13.23]  After b is determined, β and σo can be determined from the intercept and slope of the linear plot represented by Eq. 13.20. Zhou and Mallick (2004) developed an experimental strategy to determine the fatigue life parameters of fibers using strain-controlled fatigue tests on fiber bundles. In these tests, fiber bundle samples are tested in cyclic fatigue at different maximum cyclic strain levels. During the fatigue tests, the load on the fiber bundles is recorded and the maximum stress, calculated by dividing it with the average fiber bundle diameter, is plotted against the number of cycles to failure. One such plot is shown in Fig. 13.7. It can be observed from this figure that the maximum

13.7  Relationship between maximum stress and the number of fatigue cycles in a strain-controlled fatigue test of E-glass fiber bundles.

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13.8  P– ε –N curves for E-glass fiber bundles (maximum cyclic strains are indicated in the figure).

stress level on the fiber bundle decreases with increasing number of cycles, initially at a relatively slow rate; however, as increasing number of fibers in the fiber bundle starts to fail, a rapid decrease in the maximum stress level occurs. The probability of survival P can be calculated using the stress decrease plot and Eq. 13.19 and then plotted against the number of cycles, as shown in Fig. 13.8.

13.4 Stress-life diagram of fiber bundles A stress-life diagram in which the maximum cyclic stress is plotted against the number of fatigue cycles to failure may be of interest in many applications. The stress-life diagram is usually obtained by conducting stress-controlled fatigue tests in which the maximum stress applied on the fiber bundle is maintained at a constant value until failure. The stress-controlled fatigue failure process of a fiber bundle can be simulated using Monte-Carlo simulation. For this simulation, the fatigue strength coefficient, σfi, of each fiber in the bundle must be known. A random array ηi (1 ≤ i ≤ n), equally distributed between 0 and 1, is produced, where ηi is defined as:  © Woodhead Publishing Limited, 2011

[13.24]

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In Eq. 13.24, P(σfi) represents the probability distribution function of the fatigue strength coefficient σfi of each fiber, which from Eq. 13.16 can be written as: [13.25]  Assuming that the interaction between the fibers is negligible, if m fibers break after j fatigue cycles, the load carried by them before fracture is instantaneously distributed equally among the surviving (No – m) fibers. The stress on each surviving fiber in the bundle can be expressed as:

[13.26]

where σ* is the maximum cyclic stress applied on the fiber bundle, subscript i is the fiber number in the fiber bundle, and subscript j is the number of fatigue cycles. According to Eq. 13.15, the theoretical fatigue life of the i-th fiber in the fiber bundle at the stress level σ ij can be calculated as: 

[13.27]

Assuming a linear damage model, the total damage on the i-th fiber can be defined as: [13.28] For the i-th fiber to fail, Di = 1. Based on the fiber damage model, the stress-controlled fatigue failure process can be simulated using the following procedure (Fig. 13.9). (a) Randomly assign a statistical fatigue coefficient, σfi, for each fiber element. (b) Calculate the theoretical fatigue life of each fiber element. (c) Determine whether the fiber element has failed and determine the number of surviving fibers. (d) Determine the cycle number, strain, stress and damage on the bundle. (e) Increase cycle number, calculate the actual stress level on each surviving fiber element, and repeat steps (b) and (c) until each fiber element in the fiber bundle has failed. It has been shown that to obtain a smooth damage against number of fatigue cycles using Monte-Carlo simulation, the number of fiber elements in the fiber bundle should be greater than 3000. This is shown in Fig. 13.10. The stress-life diagram obtained by Monte-Carlo simulation shows a very good correlation with the experimental data, as shown in Fig. 13.11.

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13.9  Flow diagram for the simulation of stress-controlled fatigue S–N plot.

13.5 Conclusion This chapter has described a methodology developed by Zhou and Mallick (2004) and later used by Zhou et al. (2006) to determine the fatigue strength of fibers using strain-controlled fatigue tests on fiber bundles. It should be mentioned that the assumptions made in the fiber bundle model are somewhat idealistic. Although glass fibers exhibit linear stress–strain diagrams, carbon and other fibers may exhibit nonlinearity when subjected to tensile stress. Some of the organic fibers may also exhibit time-dependent creep during fatigue cycling. The assumption regarding no fiber-fiber interaction may not be completely valid, since it is possible that friction between fibers in a fiber bundle may lead to load sharing between neighboring fibers after a fiber in the fiber bundle fails.

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13.10  Effect of number of fibers in the fiber bundle on the simulated total damage.

13.11  S–N diagram of fiber bundles: comparison of simulated and experimental data. © Woodhead Publishing Limited, 2011



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13.6 References Andersons J, Joffe R, Hojo M and Ochiai S (2002), Glass fibre strength distribution determined by common experimental methods, Composites Science and Technology, 62, 131–145. Chi Z, Chou T-W and Shen G (1984), Determination of single fibre strength distribution from fiber bundle testings, J. Materials Science, 19, 3319–3324. Coleman B D (1958), Statistics and time dependence of mechanical breakdown in fibers, J. Applied Physics, 29, 968–983. Manders P W and Chou T-W (1983), Variability of carbon and glass fibers, and the strength of aligned composites, J. Reinforced Plastics and Composites, 2, 43–59. Zhou Y and Mallick P K (2004), Fatigue strength characterization of E-glass fibers using fiber bundle tests, J. Composite Materials, 38, 22, 2025–2036. Zhou Y, Baseer M A, Mahfuz H and Jeelani S (2006), Statistical analysis on the fatigue strength distribution of T700 carbon fibers, Composites Science and Technology, 66, 2100–2106.

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14 Continuum damage mechanical modeling of creep damage and fatigue in polymer matrix composites D. Perreux and F. T hiebaud, MaHyTec Ltd, France Abstract: This chapter deals with a general modeling of laminate behavior for long-term application. The model is a mesomodel and then describes the layer behavior. The viscoelastic behavior is first discussed. The effect of microcracking on the behavior is analyzed by using a continuum damage theory. The description of damage at the layer level is introduced in the modeling as well as plastic or viscoplastic strain induced by the damage. The damage kinetics is investigated. Instantaneous damage and time-dependent damage are both discussed and conclude the description of the mesomodel. The non-linear laminate theory is then used to provide the behavior of a laminate. Examples of simulation of this model are compared with experimental tests to illustrate the prediction of the model. Key words: mesomodel, viscoelasticity, viscoplasticity, plasticity, damage, time-dependent damage.

14.1 Introduction The long-term behavior of laminates is a topic which has received a lot of attention, mainly during the two last decades. The cycle of Duracosys conferences in the 1990s, under the leadership of Albert Cardon, was one of the places where the problem of durability was the subject of intensive discussions. Among the phenomena investigated, the viscoelastic strain – linear or not – was the subject of much research. Damage was also widely discussed. The description of damage at the mesoscopic level (the layer level) seems to be one of the best levels for modeling. Laurin et al.1 describe the role of progressive microcracking at the mesoscopic scales. The mesoscopic failure criterion, based on Hashin’s assumptions, introduces a distinction between the fiber failure and the interfiber failure modes. As with most of the materials, some of the approach focuses investigations of composite laminates on instantaneous damage phenomena. Damage growth in laminates with polymer matrix is most of the time described by using Continuum Damage Mechanics. For instance, Liu and Zheng2 explain the progressive failure analysis of carbon fiber/epoxy composite laminates using this methodology. They describe the failure of a gas tank by introducing damage kinetics derived from a damage criterion. 424 © Woodhead Publishing Limited, 2011



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For other materials, viscous damage has been considered by some authors. For instance, Barbat et al.3 have introduced a damage constitutive model based on Kachanov’s theory. They used the model within a finite element frame and applied it to Timoshenko beam elements. Their model takes into account viscous effects. For other composite materials such as concrete, for instance, Faria et al.4 propose a model which considers the viscous damage. For metal matrix composite, Kruch and Arnold5 describe the coupling between creep damage and creep-fatigue damage interactions. However, the extension of continuum damage mechanics for the prediction of damage under creep is still an issue. Reifsnider et al.6 divide the creep-rupture analyses into two major categories: the local and direct analysis of the defects’ growth and the global and homogeneous analysis. The late analysis concerns only the summation of all micro-processes effects acting concomitantly and often designated as accumulation or quasi-homogeneous damage models. Guedes7 reviews two existing theoretical approaches for creep failure criteria of viscoelastic materials. One approach is based on continuum damage mechanics (CDM) and the other is based on fracture mechanics extended to viscoelastic materials. He concludes that, although both theoretical frameworks are based on different physical concepts, the deduced lifetime expressions turn out to be equivalent even though their parameters have different physical interpretation. This chapter is devoted to a model based on continuum damage mechanics taking into account the viscous damage at the layer scale. Firstly, the viscoelastic behavior of the material without damage is discussed. Then the effect of damage due to microcracking is described by a homogenization method. In the last part, no linear strain due to damage is introduced and, finally the damage kinetics is proposed.

14.2 Mesomodel: viscoelastic strain, damage and viscoplastic strain of a layer Let us consider a lamina which has an assemblage of unidirectional fibers embedded in a polymeric thermosetting matrix. The axes’ orientation is shown in Fig. 14.1. The material at this mesoscopic scale is transversely isotropic. By considering the thermal expansion phenomenon, the relation between reversible strain, stress and temperature is:

[14.1]

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14.1  Layer reference axes.

where T0 is a reference temperature. We call reversible strain ε r the strain which vanishes with time when the stress vanishes and the temperature is T0. The transversally isotopic condition and the classical notation provide the following relation for the components of the compliance tensor = S and for the thermal expansion coefficient vector α_:

[14.2] In the reversible strain the part of the elastic strain is defined by: [14.3] Those first relations refer to the thermoelastic behavior. However, the matrix, which is a polymer, exhibits a viscoelastic behavior. The viscoelastic strains can be linear or non-linear depending onf the polymer nature as discussed in many chapters of this book. However, the viscoelastic strain at usual room temperature is most of the time smaller than the elastic strain because the fibers limit the stress in the polymer. Moreover, as it will be shown later in this chapter, other viscous strains will be involved after damage occurs. Then, the visoelasticity due to the matrix behavior can be introduced by assuming a linear viscoelastic strain defined with a time spectrum of relaxation. The free energy density is written as a combination of the elastic and viscoelastic components: [14.4] The ζi are a set of elementary viscoelastic tensor strains, each one associated with a relaxation time τi and a weight µi. is the relaxed viscoelastic compliance tensor,

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[14.5]

where βij are material constants. Note that ’s definition assumes the absence of viscoelastic strains in the fibers’ direction. As the dual variable of the strain is the stress defined by: 

[14.6]

The dual variable of the elementary viscoelastic strain is given by: [14.7]

and then viscoelastic dissipation is given by:

[14.8]

The elementary viscoelastic strain rate can be assessed by using the following relation: [14.9] The total viscoelactic strain is the sum of the elementary viscoelastic strains: [14.10]



The form of the relaxation time spectrum can be chosen with various shapes depending on the type of polymer; for epoxy matrix, a triangular form seems convenient.8 In the absence of damage, thermoviscoelastic strain rate is obtained by:

[14.11]

However, in the stress space, the zone where damage is not observed is usually small with respect to the whole behavior of a composite before failure. Then Eq. 14.11 is limited by a damage criterion which is a small part of the stress

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space. To illustrate this assumption, an example of a comparison between experimental damage criterion and failure criterion is provided for a glass-epoxy laminate [+55,–55]3 in Fig. 14.2. The first type of damage which occurs in a layer and then which defines the damage criterion is due to microcracking parallel to fibers and fiber-matrix debonding. Even if the causes of both type of microcracking are different, their effects are the same and can be considered from a mechanical point of view as the same damage. Figures 14.3(a) and 14.3(b) illustrate this damage and the model used in its mechanical description.

14.2  First damage criterion with respect to the failure criterion of glass-epoxy laminate [+55,–55]3.

14.3  (a) Debonding and microcracking; (b) damage material model.

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The damage can be defined as a second phase in the composite. In this approach, the damaged material is analysed in the same way as undamaged homogeneous material and by a phase of flaws. This hypothesis is not concerned with the physical nature of the damage, but simply with the mechanical effect of microcracking. This damaged material is a model of the real material and different methods can be used to model the role of defects on the previous thermoviscoelastic behavior of the material. The objective of this method is to describe the change in Eq. 14.11 when the defects exist in the virgin material. First of all, we discussed how such damage modifies the elastic behavior. Laws et al.9 were among the first to show that one parameter, the microcrack density, is sufficient to model the variations of the compliance tensor component between the undamaged and damaged materials. With a similar idea, by extending the work of Gottesman et al.,10 Perreux and Oytana11 presented the development of a damaged compliance tensor  , of a layer in the following form: [14.12]

 with:

[14.13]

 where:

[14.14]



 

 

 

with σ2 = 0 if σ2 < 0 and σ2 =1 if σ2 ≥ 0. The presence of σ2 is to take into account the unilateral damage effect that is due to the non-symmetric effect of the microcracks on the elastic stiffness, when they are opened by traction tests or when they are closed by compression tests. More complex description of the unilateral effect could be examined, but for most of the usual applications and for multiaxial proportional loadings the assumption used in Eq. 14.14 is convenient. Equation 14.14 is the result of a long process of homogenization method close to a self-consistent model. This equation provides the variation of all the

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compliances tensor when the variable DI, the variation of transverse Young’s modulus is known: [14.15]  In continuum damage mechanics, it is assumed that the elastic strain of the damaged material loaded with stress tensor σ_ is the same as the elastic strain of virgin material loaded with effective stress tensor σ_˜ :  From Eq. 14.12 and 14.16  can be obtained:

[14.16]

[14.17]  From this definition the effect of damage on the magnitude of the viscoelastic strain can be taken into account by introducing the effective stress in Eq. 14.9: [14.18]  The last part of Eq. 14.11 addresses the thermal expansion of the material. It has been shown (D. Perreux, D. Lazuardi12), that the thermal expansion is not affected by the damage and so:  Finally, when damage occurs the reversible strain is:

[14.19]

 [14.20] But damage also involves irreversible strain. We call it irreversible strain because if the stresses vanish, the irreversible strain does not. These strains can be associated to friction of the lips of the microcracks, and then include instantaneous strain similar to plastic strain and retarded strain similar to viscoplastic strain   : [14.21]  In contrast to the reversible strain, which involves possible deformation in all directions, the irreversible strains are only strains associated with shear deformation with respect to the microcrack’s plane. Then, because not all the tensors of stress are able to develop such strain, a plastic criterion is proposed by using the theory of the tensorial invariant: [14.22]  is a plastic anisotropy tensor. In order to be consistent with the hypothesis of wall friction, which involves plastic strain only for shear stress, has the following expression: © Woodhead Publishing Limited, 2011



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[14.23]

 Zc is the yield point in respect to the plane shear (σ6), Zc,4 (respectively Zc,5) is the yield point in respect to the out-of-plane shear (σ4) (respectively σ5). In fact, it is useful to assume that Zc,4 = Zc and Zc,5 = 0 (no plasticity for out-of-plane shear stress σ5). X is a kinematic hardening variable. The plastic strain kinetic is obtained by –– the normality rule applied to the plastic criterion: [14.24]  λp is a Lagrange multiplier that can be obtained from the consistency equation of the plastic criterion. In order to describe the non-linear plastic strain obtained during the experimental test, the hardening variable must be chosen as a sum of two hardening variables: a linear one, , and a non-linear one,  :

[14.25] 

δ1, δ2, γ2 are material parameters which must be identified and plastic variable, which can be proposed as: 

the accumulated

[14.26]

is the pseudo-inverse . To complete the instantaneous irreversible strain, a second cause of plasticity must be presented. As already discussed, this plastic strain depends on the microcracking density. Nevertheless, in view of the damage modeling already discussed, damage variable DI itself can be used in place of microcracking density. It is clear that a similar method to the previous one for this particular plastic strain could be developed. However, due to damage kinetic formalism, which is very

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similar to classical plastic formalism, it is much easier to use a relation between the plastic strain and the damage variable to model the second cause of plastic behavior than to introduce other plastic criteria and hardening variables. Thus the transverse plastic strain can be presented as: [14.27] δ3 is a plastic parameter. Due to the damage kinetic discussed later, this plastic strain exhibits isotropic hardening, which is consistent with the physical cause of this plastic strain. Similar to the instantaneous irreversible strain, a retarded one can be defined by using the same thermomechanical method. The viscoplastic strain is defined by the set of the following equation:

[14.28]

 Finally, from Eq. 14.20, 14.24 and 14.27 the total strain of the material obtained whether the material is damaged or not:

is

[14.29]  To complete this model, the damage kinetics has to be proposed. This part is the most important point to predict the durability of the layer and then of the laminate. As for the irreversible strain rate, to obtain the damage rate, the thermodynamics of the irreversible process is an appropriate framework. This method is based on the definition of two potentials, one for free energy density and the other for dissipation. When the material is damaged Eq. 14.4 is modified and the free energy density can be defined as:

[14.30]

∼ S is defined by Eq. 14.12 and Ψ´ completes the definition of free energy density = used to describe all the viscoelastic or irreversible strain already discussed. We assume in the following that Ψ´ is not depending on the damage – which is formally not true but is a simplification. By using free energy density, the driving force of the damage can be obtained. This variable is similar to G, the energy release rate used in fracture mechanics: [14.31] 

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After the definition of the driving force of a dissipation potential, ϕ has to be proposed: [14.32]  ϕD is the part of the dissipation potential affected by damage and function of YI and then, to ensure a positive dissipation during the damage process, it can be proposed: [14.33]  It is useful to share in the damage rate the instantaneous growth and the retarded one. Then Eq. 14.33 can be written again by introducing a damage criterion fD (already discussed; see Fig. 14.2) and a purely retarded dissipation potential ϕ *D such that the damage rate is: [14.34]  The first part of Eq. 14.34 is able to describe instantaneous damages encountered when the material is loaded for the first time. Let us consider in a first stage that ϕ *D = 0. From previous work,12 it has been shown that: [14.35]  where Yc is a damage threshold. The value of this material parameter depends on the internal stresses. RD is a hardening variable, which maintains the maximum value of YI reached during loading. That means the damage grows only if, during loading, YI is higher than the maximum value of YI reached during the previous loading. The dual variable of RD is  , the accumulated damage variable, which is defined as: 

[14.36]

It can be shown that:

[14.37]

RD must be consistent with the experimental test and then for lamina; if in a first stage we consider ϕ D* = 0, then it can be written as: [14.38] In Eq. 14.34 or 14.37, λD is a Lagrange multiplier, which can be obtained from the damage criterion consistency equation: [14.39]



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14.4  Damage criterion of a glass-epoxy laminate.

Figure 14.4 shows this theoretical damage criterion for a laminate; from this damage and by using the laminate theory the damage criterion of a laminate such as the one presented in Fig. 14.2 can be deduced. The second part of Eq. 14.34, where ϕ D* is concerned, is rarely discussed by authors, but it is probably the most important part for long-term application. ϕ *D affects all the loadings including fatigue and creep. However, the damage due to creep or fatigue is often too complex to be merged in one equation, and most of the authors prefer to share the damage kinetics by introducing phenomenological law where, for instance in fatigue, the rate of the damage is proposed as a function of the stress or the strain. In fact, the driving force of damage is never stress or strain but YI. Then, during fatigue loading, Eq. 14.34 provides the damage rate by cycle (other than for the first cycle) as: 

[14.40]

Contrary to purely phenomenological damage fatigue laws, Eq. 14.40 is able to describe the role of the frequency and can take into account the coupling between fatigue and creep. The main difficulty, of course, is to propose the function ϕ D*. The present authors have proposed a first function13 as the following:



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[14.41]



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where Dc is the critical damage and Z is the equilibrium driving force of the damage and has the following rate:

[14.42]  With this assumption, the retarded damage kinetics is connected with the ‘distance’ between the driving force and the equilibrium variable. This definition allows a simulation of the creep and fatigue for glass-epoxy material in accordance to the experimental tests. Equation 14.41 is derived from Norton Laws used in viscoplasticity; more recently Treasurer et al.14 proposed another ϕ D* for carbon-epoxy laminate derived from the viscoelastic dissipation potential. It is obvious than the expression of ϕ D* is still an open issue and is probably a complex function.

14.3 From meso- to macroscopic behavior To consider non-linear behavior in composite laminates an extension of classical lamination theory is one of the best options for the volume element. Based on the Love-Kirchoff hypothesis, it assumes that a section perpendicular to the middle plane of the laminate remains perpendicular under loading. In a linear analysis, the classical matrices A, B, and D relate the middle plane strain ε and curvature ρ to the generalized force N and moment M: [14.43]  Here, A, B, and D are constant. The extension to non-linear behavior is made by writing Eq. 14.43 in an incremental form: [14.44]  ∆M and ∆N are increments of the generalized force and moment for a given time step, and ∆ε and ∆ρ are the incremental mid-plane strain and curvature. Matrices A*, B* and D* are the laminate tangent stiffness dependent on loading history. Matrices A*, B* and D* can be assessed using the Newton-Raphson method. Another method is to implement this model in a finite element code. This option is of course powerful in designing a structure when the state of stress is not homogeneous.

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14.3.1  Example of simulation in comparison with tests As an example of the capability of the model, we propose some results obtained on glass-epoxy laminates [+55,–55]3. Tests are performed on tube specimen made by winding filament. Special attention was paid to the manufacturing in order to get pure laminates and not crossing structures [+/–55]. These tubes were loaded by internal pressure (axial stress=0=σzz) or internal pressure with end effect (hoop stress=σθθ = 2axial stress==2 σzz). Axial (εzz) and hoop (εθθ ) strains were recorded during the tests. The tests are repeated progressive loadings. The tests are performed on several phases of loading with an increasing level of maximal stress between each phase. Each phase consists of a loading and unloading up to a certain level of stress, and afterwards a pure tensile test with low axial stress is performed in order to measure the variation of the axial modulus of the sample. Figures 14.5 and 14.6 show the results and the good agreement with the simulation obtained by using the previous model.

14.5  Experiment and simulation of internal pressure test with closeended effect for [+55,–55]3: (a) simulation; (b) experiment.

14.6    Experiment and simulation of pure internal pressure test for [+55,–55]3: (a) simulation; (b) experiment.

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14.4 Conclusion With Eq. 14.44, or by using a finite element code taking into account the mesoscopic model, the assessment of the non-linear time-dependent behavior of a composite laminate or a component made of laminate can be obtained and then prediction of long-term behavior can be considered. Depending on the application, this model, or a restriction of it to instantaneous phenomena, was used for many applications including tubes for fluid transportation or tank for gas storage. This model involves the determination of several parameters. The number of parameters is usually low with respect to the number of phenomena described by the model. Each parameter has a value which depends on the scattering of the material properties, mainly due to the manufacturing process. Therefore each parameter can be associated to a statistical function. The average response of a laminate due to a long time loading can be predicted by the model by using mean values of the parameters. However, the average behavior is often not sufficient to design a laminate; a safety factor has to be determined to ensure that a large number of components made of laminate will support the loading without failure. Among the applications of this model, its use combined with a reliability method can help to determine the safety factor.

14.5 References   1 Laurin F., Carrere N., Maire J.-F., A multiscale progressive failure approach for composite laminates based on thermodynamical viscoelastic and damage models, Composites: Part A 38 (2007) 198–209.   2 Liu P.F., Zheng J.Y., Progressive failure analysis of carbon fiber/epoxy composite laminates using continuum damage mechanics, Materials Science and Engineering A 485 (2008) 711–717.   3 Barbat A.H., Oller S., Onate E. and Hanganu A., Viscous damage model for timoshenko beam structures, Inf. J. Solids Structures 34, no. 30 (1997) 3953–3976.   4 Faria R., Oliver J. and Cervera M., A strain-based plastic viscous-damage model for massive concrete structures, Int. J. Solids Structures 35, no. 14 (1998) 1533–1558.   5 Kruch S. and Arnold S.M. Creep damage and creep-fatigue damage interaction model for unidirectional metal-matrix composites. Application of continuum mechanics, Fatigue and Fracture, ASTM STP 1315 D.L. McDowell, ed ASTM (1997), pp. 7–28.   6 Reifsnider K., Stinchcomb W and Osiroff R. Modeling of creep rupture mechanisms in composite material systems. ASME AD (1992); 27: 51–8.   7 Guedes R.M., Relationship between lifetime under creep and constant stress rate for polymer-matrix composites, Composites Science and Technology 69 (2009) 1200–1205.   8 Thiebaud F. and Perreux D., Overall mechanical behaviour modelling of composite laminate. European Journal of Mechanics-SOLIDS 15, no. 3 (1997) 423–445.   9 Laws N., Dvorack G.C., Hejazi M., Stiffness changes in unidirectional composites caused by cracks systems. In: Mechanics and Materials, vol. 2. Amsterdam; NorthHolland, (1976), pp. 123–37.

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10 Gottesman T., Hashin Z., Brull M.A., Effective elastic moduli of cracked composites. In: Bunsell AR, et al., editors. Advances in Composite Materials, 3rd International Conf. on Composite Materials, vol 1. Oxford: Pergamon Press (1980), pp. 749–58. 11 Perreux D., Oytana C., Continuum damage mechanics for microcracked composites. Composites Engineering 3 no. 2 (1993) 115–122. 12 Perreux D., Lazuardi D., Residual stress effect on the non-linear behaviour of composite laminates Part II: Layer, laminate non-linear models and residual stress effect on the model parameters, Composite Science and Technology 61, no. 2 (2001) 177–191. 13 Perreux D., Joseph E., Frequency effect on the fatigue performance of filament wound pipes under biaxial loading: Experimental results and damage modelling. Composites Science and Technology 57 (1997) 353–364. 14 Treasurer P., Perreux D., Poirette Y., Thiebaud F., Characterization and analysis of carbon fiber/epoxy tubes, Proceedings of 17 ICCM, 27–31 July 2009, Edinburgh.

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15 Accelerated testing methodology for predicting long-term creep and fatigue in polymer matrix composites M. Nakada, Kanazawa Institute of Technology, Japan Abstract: Accelerated testing methodology (ATM) for the long-term creep and fatigue life prediction of various polymer matrix composites and their structures is summarized. Firstly, the ATM will be explained in detail as the foundation of the long-term creep and fatigue life prediction of polymer matrix composites. We then describe the detailed procedures for generating the master curves of creep and fatigue strengths based on the ATM. Key words: polymer composite, creep, fatigue, life prediction, accelerated testing, time–temperature superposition principle.

15.1 Introduction Recently, polymer matrix composites reinforced with carbon fibers have been used in the primary structures of airplanes, ships, spacecraft, etc., which require highly reliable long-term operation. Therefore, it is essential to assess the reliability of the composite structures subject to long-term creep and fatigue loadings and environmental conditions (temperature, moisture absorption, etc.). Polymer matrix composites exhibit mechanical behaviors significantly dependent on time and temperature under operation because of the viscoelastic behavior of polymer matrix resin. The viscoelastic behavior has been observed not only above glass transition temperature Tg but also below Tg.1–5 With service lifetimes measured in years, it is almost unthinkable to do realtime testing under a variety of conditions. Accelerated testing methodologies for metals have been studied for some time. One of the most popular tools used to predict the fatigue life of metals is the S-N curve, which is based on the assumption that fatigue life depends on cycles, but not on time. Therefore, the cyclic loads can be applied at much higher frequencies than the actual loading to accelerate the fatigue test. Unlike metals, polymer composites are viscoelastic and their properties exhibit strong time and temperature dependencies as mentioned above. Since time plays an important role in the fatigue and creep of polymer composites, simply applying the S-N curve to polymer composites will not provide accurate prediction of the fatigue life. Previously, we have developed an accelerated testing methodology (ATM) to predict the long-term creep and fatigue life of the polymer matrix composites based on the time–temperature superposition principle (TTSP) held for the 439 © Woodhead Publishing Limited, 2011

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viscoelastic behavior of matrix resin.6–10 The ATM using the TTSP enables us to describe the long-term life by means of master curves covering wide ranges of loading and environmental conditions, including load duration, temperature, frequency of load cycles, stress amplitude ratios, etc. The ATM has been applied for various composites and their joint structures, and demonstrated great success as a robust and powerful methodology for long-term life prediction. Some theoretical explanations were made to back up their experimental observations.11–14 In this chapter, we revisit and summarize the earlier efforts of the ATM for the long-term creep and fatigue life prediction of various polymer matrix composites and their structures. Firstly, the ATM will be explained in detail as the foundation of the long-term creep and fatigue life prediction of polymer matrix composites. We then describe the detailed procedures for generating the master curves of creep and fatigue strengths based on the ATM.

15.2 Accelerated testing methodology 15.2.1  Procedure of ATM The ATM has been established with the three following conditions: (A) the same TTSP is applicable for both non-destructive viscoelatic behavior and destructive strength properties of matrix resin and their composites; (B) linear cumulative damage (LCD) law is applicable to the strength by the monotonic loading; and (C) the fatigue strengths exhibit linear dependence on the stress ratio of the cyclic loadings. A key component of the ATM is the empirical observation (A). The applicability of ATM for various polymeric composite materials and their structures was demonstrated by the author as mentioned later. With the condition (B), it is possible to calculate the creep-strength master curve from the constant strain rate (CSR) strength master curve determined from relatively easy tests under CSR loading at several elevated temperatures and a fixed strain rate. With the condition (C), the fatigue strength at any stress ratio can be interpolated with those between zero and unity, which are obtained by the creep- and the fatigue-strength master curves, respectively. Therefore, the fatigue strength under any arbitrary combinations of frequency, temperature, and stress ratio can be determined with the two test results for obtaining (i) CSR strength measured at a single strain rate and several elevated temperatures and (ii) fatigue strength measured at zero stress ratio, a single frequency, several stress levels and several elevated temperatures. The detailed procedure of ATM is illustrated in Fig. 15.1. Firstly, change in modulus of the viscoelastic matrix resin is measured over time at a constant temperature. The tests repeat for several elevated temperatures, which results in several modulus curves with the function of time. Time–temperature shift factors (TTSFs) are then determined by shifting the viscoelastic modulus curves at the several temperatures into time scale to form a master curve of the modulus at a reference temperature. The TTSF is thus the measure of the acceleration of the

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15.1  Accelerated testing methodology (ATM).

life of matrix resin by means of the elevated temperatures. The next step is to obtain the creep strength master curves. This step consists of two parts. The first part is to determine the CSR strength master curve of the composites from the CSR loading tests conducted at a single strain rate and several elevated temperatures by using condition (A) and the second part is to convert the CSR strength master curve to the creep strength master curve of the composites by using condition (B). Thirdly, the master curves of fatigue strength of the composites at zero stress ratio are determined by conducting the fatigue tests at several stress levels, a single frequency, a single stress ratio (zero stress ratio) and several elevated temperatures by using condition (A). In this step, the CSR strength master curve is used as the fatigue strength master curve at the number of cycles to failure Nf = 1/2. Finally, the creep and fatigue strengths at any arbitrary frequency, stress ratio and temperature are obtained from the master curves of creep strength and fatigue strength at zero stress ratio by using condition (C). The detail of the method will be presented in respective subsections.

15.2.2  Master curve of CSR strength For the polymer resin, the time-dependent mechanical responses at a temperature T can be mapped onto the responses at the reference temperature T0 as shown in Fig. 15.2 based on the TTSP by using the time–temperature shift factor aT (T) 0 defined by 

[15.1]

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15.2  Master curves of viscoelastic modulus for matrix resin and CSR strength for FRP.

For the polymer composites, the CSR test is performed for several strain rates over a wide range of temperature. As shown in Fig. 15.2, the CSR strength σs versus time to failure ts at various temperatures T can be measured from this test and then can be used to create a master curve of CSR strength versus reduced failure time ts' at a reference temperature T0 by shifting these strengths horizontally. The TTSP holds for the CSR strength of CFRP composites if a smooth curve is produced.

15.2.3  Master curve of creep strength We propose here a prediction method of creep strength σc from the master curve of CSR strength using the linear cumulative damage law. Let ts(σ) and tc(σ) be the CSR and creep failure time for the stress σ. Suppose that the material experiences a monotone stress history σ (t) for 0 ≤ t ≤ t* where t* is the failure time under this stress history. The linear cumulative damage law states [15.2] © Woodhead Publishing Limited, 2011

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When σ (t) = σ0, the above formula implies t* = tc(σ0). Choose an equally-spaced increasing sequence of stress σi = i∆σ (i = 1, 2, 3, . . .) and denote the associated CSR and creep failure time by ts(i) = ts(σi) and tc(i) = tc(σi), respectively. In the CSR test, the strain rate is kept constant and the force-deflection curves are found linear up to just before the failure. We consider, therefore, that the stress increases linearly during the CSR test. Further, we approximate the linear stress history by a staircase function with steps σ1, σ3, σ5, . . . as shown in Fig. 15.3. Thus, the linear stress history up to the stress level σ4 is replaced by σ1 for 0 < σ < σ2 and σ3 for σ2 < σ < σ4. Applying the linear cumulative damage law successively to the staircase loading history, the failure time tc(2i-1) (i = 1, 2, 3, . . .) is expressed by ts(2i) as follows: [15.3] Equation 15.3 implies that 

[15.4]

For the determination of tc from Eq. 15.3 over the entire stress range, we must have the master curve of CSR strength over the entire stress range. However, the portion of the curve below a certain stress cannot be determined due to the experimental limitations. We therefore extrapolate that portion of the master curve by a decaying exponential curve with the same slope at the lowest available stress level.

15.3  Linear cumulative damage law for monotonic loading (condition B).

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15.2.4 Master curve of fatigue strength for zero stress ratio We regard the fatigue strength σf either as a function of the number of cycles to failure Nf or of the time to failure tf = Nf / f for a combination of f, R, T and denote them by σf (Nf ; f, R, T) or σf  (tf ; f, R, T). Further, we consider that the CSR strength σs(ts; T) is equal to the fatigue strength at Nf = 1/2 and R = 0 by choosing ts = (2f)–1. At this point, we introduce special symbols for fatigue strength at zero and unit stress ratio by σf:0 and σf:1. To describe the master curve of σf:0, we need the reduced frequency f ′ in addition to the reduced time tf′, each defined by [15.5] Thus, the master curve has the form, σf:0(tf′; f ′, T0). An alternative form of the master curve is possible by suppressing the explicit dependence on frequency in favor of Nf as σf:0(tf′; Nf, T0). Recall that the master curve of fatigue strength at Nf = 1/2 represents approximately the master curve of CSR strength. We describe here the steps to draw the master curves σf:0(tf′; f ′, T0) from S-N curves, σf:0(Nf; f0, T) for a single frequency f0 and various temperatures as shown in the upper graph of Fig. 15.4 where the CSR strength, σf:0(1/2; f0, T) is represented by a vertical dotted line while the strengths for constant temperatures are displayed by solid curves. This graph may also be viewed as the graph for σf:0(tf; f0, T). We redraw the curves in the upper graph against the reduced time for the reference temperature T0 as sketched in the bottom graph where the dotted line for Nf = 1/2 in the upper graph becomes the master curve for CSR strength while the curves for constant temperature in the upper graph are mapped onto the master curves of constant reduced frequency as apparent from the first part of Eq. 15.5 with f replaced by f0. On the other hand, each point on the master curves of constant reduced frequency represents a number of cycles to failure. Connecting the points of the same number of cycles to failure on these curves, the master curves of constant number of cycles to failure are constructed as shown in solid curves in the bottom graph. Once the master curve for fatigue strength of zero stress ratio is established for Nf, the fatigue strength for tf under arbitrary combination of f and T is determined as

[15.6]

where

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[15.7]

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15.4  Master curves of fatigue strength for FRP (R = 0).

15.2.5 Prediction of fatigue strength for arbitrary frequency, stress ratio and temperature We have the master curves for creep strength σc(tc′; T0) from which follows the creep strength at any temperature T. The creep strength, in turn, may be regarded the fatigue strength σf:1(tf; f, T) at unit stress ratio R = 1 and arbitrary frequency f with tc = tf as shown in the upper graph of Fig. 15.5.

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15.5  Linear dependence of fatigue strength upon stress ratio (condition C).

Further, from the master curve for fatigue strength at zero stress ratio, we can deduce the fatigue strength σf:0(tf; f, T) at zero stress ratio for any frequency f and temperature T. Invoking the condition (C), we propose a formula to estimate the fatigue strength σf(tf; f, R, T) at an arbitrary combination of f, R, T by [15.8] The fatigue strength σf (tf ; f, R, T ) can be obtained by substituting the σf:1(tf ; f, T ) and σf:0(tf; f, T) at R = 1 and 0 in Eq. 15.8 as shown in the bottom graph of Fig. 15.5.

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15.3 Experimental verification for ATM Satin-woven CFRP laminates6 were chosen as specimens to show the validity of the proposed prediction method based on the condition (A), (B) and (C).

15.3.1  Specimen and testing method The three-point bending tests for CSR, creep and fatigue loadings were conducted for various temperatures on satin-woven CFRP laminates referred to as T400/3601. The satin-woven CFRP laminates are made from carbon fibers T400 and a matrix epoxy resin 3601 with a high glass transition temperature of Tg = 236°C. Eight prepreg sheets were stacked symmetrically about the midplane. The CFRP laminates T400/3601 were formed by hot pressing these prepreg sheets into 2.7 mm-thick plates. The volume fraction of the fibers in the composites is approximately 65.5%. Three-point bending tests for CSR, creep and fatigue for T400/3601 were conducted where the span, width and the thickness used in the tests are L = 50 mm, b = 15 mm and h = 2.7 mm, respectively. The CSR tests were carried out at four deflection rates and ten uniform temperatures on an Instron-type testing machine. A creep testing machine with a constant temperature chamber was used to perform the creep tests at three uniform temperatures. The fatigue tests were performed by using an electroservo-controlled hydro testing machine with a constant temperature chamber at various frequencies, temperatures, and stress ratios.

15.3.2  Flexural CSR strength On the left side of Fig. 15.6, the dependence of flexural CSR strength σs for the CFRP laminates T400/3601 upon the time to failure ts is presented for various temperatures T where each point for a temperature corresponds to one of the four deflection rates V = 0.2, 2, 20, 200 mm/min. Shifting the curves in this figure horizontally, the smooth master curve for the flexural CSR strength against the reduced time ts′ at the reference temperature T0 = 50°C is obtained as shown in the right side of this figure. The shift factors for T400/3601 used are shown by the open circles in Fig. 15.7 and the solid lines represent Arrhenius equation. The experimental shift factors are very close to Arrhenius equation with ∆ H1 = 200 kJ/mol for T < 239°C and with ∆H2 = 797 kJ/mol for T > 239°C. The solid circles in the figure represent the shift factors for the storage modulus of matrix 3601. The two shift factors, one for CFRP composites T400/3601 and another for matrix 3601, agree well each other and also with Arrhenius equations. This remarkable agreement stems from the fact that the failure is triggered by the microbuckling of fibers on the compressive surface of the specimen and time– temperature dependency of microbuckling is controlled by the viscoelastic behavior of matrix resin.

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15.6  Master curve of flexural CSR strength of T400/3601.

15.3.3  Flexural creep strength The three-point bending creep tests were carried out for T400/3601 with several stress levels at temperatures T = 50, 150, and 230°C. The deflection increases slowly with time until the sudden and instantaneous failure. Figure 15.8 displays the flexural creep strength σc versus time to failure tc; the left side shows the experimental data at the temperatures T = 50, 150, 230°C, while the right side exhibits the data shifted to the reference temperature T0 = 50°C using the shift factors for the CSR strength. The right side of this figure also displays the master curve of CSR strength in solid curve and that of creep strength in dashed curve which is calculated by Eq. 15.3 using the master curve of CSR strength. The predicted creep strength captures the experimental points shifted using the shift factor of the CSR strength for the reference temperature T0 = 50°C. Thus, the validity of condition (A), same time–temperature superposition principle for all strength, is confirmed for CSR and creep strength together with that of condition (B), the linear cumulative damage law.

15.3.4  Flexural fatigue strength for zero stress ratio Fatigue tests were carried out for combinations of frequency and temperature at stress ratio R = 0.05 which is the least value of R possible to get reliable results in our testing method. We consider the results thus obtained as the data for R = 0 and use these in the estimate of fatigue strength.

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15.7  Time–temperature shift factors for 3601 and T400/3601.

Curves of the flexural fatigue strength σf versus the number of cycles to failure Nf, S-N curves, are shown in Fig. 15.9 where σf at Nf = 1/2 is represented by σs at ts = (2f)–1. The top of the graph is for f = 2 Hz, while the bottom is for f = 0.02 Hz in which the curves for f = 2 Hz are included in the dotted curves. The S-N curves are found to be linear with the same slope for all frequencies and temperatures tested over the six decades of Nf . Furthermore, the condition (A) is valid for fatigue strength since frequency-temperature dependency of S-N curves is controlled by σf at Nf = 1/2 which is equal to σs{(2f)–1; T} obeying the condition (A). We thus confirmed that tests at a single frequency are sufficient to construct the master curve of σf:0.

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15.8  Master curve of flexural creep strength of T400/3601.

The master curves of fatigue strength σf:0(tf ; Nf , T0) are constructed for each number of cycles to failure Nf based on S-N curves, Fig. 15.9, following the steps outlined in Section 15.2 as exhibited by the solid curves in Fig. 15.10. This figure also includes the master curves of fatigue strength for each reduced frequency f ′ plotted by the dotted lines against the reduced time to failure tf′. It should be emphasized that one can infer the fatigue strength at Nf or tf for any combination of f and T from the master curves for Nf or f ′ noting Nf = f ′ tf′ = ftf.

15.3.5  Flexural fatigue strength for arbitrary stress ratio Figure 15.11 shows experimental data of σf - tf for f = 2Hz, R = 0.05, 0.5, 0.8, 1.0 and T = 50, 150, 230°C. The solid lines represent the least squares fit for R = 0.05 which we consider the curves for R = 0, while the dashed lines are the creep strength obtained from master curve of CSR strength using the linear cumulative damage law. The dash-dotted and dash-double-dotted lines are calculated from Eq. 15.8 on the basis of the curves for R = 0 and R = 1. As can be seen, the predictions correspond well with the experimental data for all temperatures tested. We can predict σf- tf relation for given R, f and T from Eq. 15.8 if σf:1(tf; f, T) and σf:0(tf ; f, T ) are known.

15.4 Applicability of ATM The ATM has successfully been applied to many different kinds of composite materials and their structures subject to various loading conditions. Table 15.1

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15.9  S-N curves of T400/3601 for frequencies f = 2, 0.02 Hz and stress ratio R = 0.05 at various temperatures.

lists the materials and loading types as well as their reference article numbers for the condition (A).6,8,11,15–29 Table 15.2 lists the same for the conditions (B) and (C).6–8,16,18,20,22–25 Table 15.3 lists the applicability of the conditions (A), (B) and (C) for various composite joint structures subject to tensile loading.30–34 As the tables show, the conditions are satisfied for almost all composites made of PAN-based carbon fibers and thermosetting resin matrix regardless of fiber

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15.10  Master curves of flexural fatigue strength of T400/3601 for stress ratio R = 0.05.

architectures (unitape and fabric). Furthermore, they are satisfied for tensile fatigue strengths of adhesively bonded and bolted joints. Therefore, the ATM can be used for the long-term life prediction of these materials and their joint structures using the creep and fatigue strength master curves. It should be noted that not all materials satisfy the conditions. For example, it was observed that the composites with PEEK resin and some pitch-based carbon fibers failed to satisfy the conditions.7, 20, 27 The reasons for failure might be due to crystallization of the PEEK resin during loading and viscoelastic behaviors of the high-modulus pitch-based carbon fibers. Therefore, the present ATM cannot be used for the life prediction for these materials. Excluding these few cases, the ATM can be used for many available composite materials in general.

15.5 Theoretical verification of ATM The failure of composites is dependent on the type of loads. Therefore, the applicability of the condition (A) for the TTSP can be explained differently for the different types of loads. For example, the time and temperature dependence of failure of unidirectional FRP subjected to longitudinal tensile and compressive loading in the direction parallel to the fiber alignment is caused by the nondestructive viscoelastic behavior of the polymer matrix resin, which has been observed for both CSR loading and constantly applied creep loading.11–12 The former can further be explained by cumulative damage of fibers, which can be

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15.11  Prediction of flexural fatigue strength σf(tf; f, R, T) of T400/3601 for various stress ratios at frequency f = 2Hz and various temperatures.

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G40-800/5260

Vinylester PW T300/VE – NCF T700/VE



LB LB LB

LB

–

BMI



IM600/PIXA-M MR50K/PETI-5

LB LB

QIL QIL

– –

PI



LT LT LT LC LT LB LB TB LB LB LB LB LB

Deformation Type Fiber/matrix Dc E’

UD a T400/828 Carbon: Epoxy – PAN HR40/828 T300/828 – UD b T400/828 – UD Fortafil510/Cape2002 – UD T300/PEEK – SW T400/3601 – PW T300/828 – QIL T800S/3900-2B UT500/#135 T800S/TR-A33

Fiber Matrix







× ×

Loading     direction CSR

– –

–

– –

– – × × –



–

– –

– – – × × –

Conditions (A) Creep Fatigue

25 21 21

24

24 24

8, 11 15 16 17 18 19 7 20 6 21 22 22 23

Reference

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Table 15.1  Applicability of condition A for FRP

454

– –



PW PW

E-glass/VE WE18W/VE

XN05/828 XN50/828 XN40/25C XN70/25C YS15/25P XN05/25P LB LB

LT LT LB LB LB LB

× × –

– – × – – – –

– – × ×

UD, unidirectional;  UD a, strand;  UD b , ring;  SW, satin woven;  PW, plain woven;  QIL, quasi-isotropic laminates; NCF, non crimp fabric;  LT, longitudinal tension;  LB, longitudinal bending;  LC, longitudinal compression;  TB, transverse bending.

Glass Vinylester

Carbon: Epoxy – UD a pitch – UD 25 29

26 26 7 27 28 28



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Table 15.2  Applicability of conditions (B) and (C) for FRP Fiber Matrix Type Fiber/matrix

Loading Conditions Reference direction (B) (C)

Carbon: Epoxy UD a T400/828 LT PAN T300/828 LT UD b T400/828 LT UD T300/PEEK LB TB QIL T800S/3900-2B LB UT500/#135 LB T800S/TR-A33 SW T400/3601 LB

Vinylester PW

T300/VE

LB



UD

XN40/25C

LB

×

Vinylester PW

E-glass/VE

LB



Carbon: pitch Epoxy Glass

× ×

  8 – 16 18 ×   7 × 20 22 22 – 23   6

25

  7

25

UD, unidirectional;  UD a , strand;  UD b, ring;  SW, satin woven;  PW, plain woven;  QIL, quasi-isotropic laminates;  LT, longitudinal tension LB, longitudinal bending;  TB, transverse bending.

Table 15.3  Applicability of conditions (A), (B) and (C) for FRP joints FRP joint system    Conditions (A) (B) (C)

Reference

Conical shaped joint of GFRP/metal Brittle adhesive joint of GFRP/metal Ductile adhesive joint of GFRP/metal Bolted joint of GFRP/metal Bolted joint of CFRP/metal

30 31 32 33 34



–

–

explained by a Rosen’s model,35 while the latter can be explained by microbuckling of fibers. Meanwhile, the failure of the unidirectional composites subjected to transverse tensile loading in the transverse direction to the fiber alignment is caused by the failure of the polymer matrix formed by microcracks.14 In all cases, the failure is caused by both non-destructive viscoelatic behavior and destructive strength properties of the matrix material. Table 15.4 summarizes the various failure mechanisms under different loading types.

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Table 15.4  Failure mechanisms of unidirectional composites subject to various loading types Loading direction Failure mechanism Longitudinal tensile

Controller of time and temperature dependence

Cumulative damage Viscoelastic behavior of fibers (Rosen’s model) of polymer matrix

TTSP Yes

Longitudinal Microbuckling of fibers compressive

Viscoelastic behavior of polymer matrix

Yes

Transverse tensile

Failure of polymer matrix

Yes

Matrix crack

As stated earlier, the condition (B) for the LCD law is used in obtaining the creep strength master curves from the CSR strength master curve. The applicability of the condition (B) can also be explained by a crack kinetics theory by Christensen and Miyano.36, 37 The crack kinetics theory showed that the rate of decrease in the creep strength over time is the same as that of CSR loading strength, so that the master curve of the former can be obtained by horizontal shifting of the latter. It was independently checked that the amount of the horizontal shifting is quantitatively equal to the shifting amount by the LCD law.6 The condition (C), the fatigue strengths exhibit linear dependence on the stress ratio, is used to obtain the fatigue strength at any stress ratios in the range between 0 and 1 by using the creep- and the fatigue-strength master curves. The applicability of condition (C) has not yet been verified theoretically.

15.6 Future trends and research To design the composite structures which require highly reliable long-term operation under actual loading and environmental conditions, we point out some researches based on ATM as mentioned below:

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(1) Applicability of ATM to the long-term life prediction under water absorption conditions. (2) Applicability of ATM to the polymer composites with thermoplastic resin as the matrices. (3)  Extension and verification of the condition to the full range of stress ratios. (4) Long-term life prediction under variable stress amplitude, frequency, and temperature. (5) Development of life prediction tool for composite structures based on the micromechanics of failure.38

15.7 Conclusions In this chapter, the accelerated testing methodology (ATM) for the long-term creep and fatigue life prediction of various polymer matrix composites and its structures has been summarized. Firstly, the ATM was explained in detail as the foundation of the long-term creep and fatigue life prediction of the polymer matrix composites. Second, the detailed procedure for generating the master curves of creep and fatigue strengths based on the ATM was explained by using the test data. The applicability and theoretical verification of ATM were discussed. Finally, the future trends and researches for ATM were pointed out.

15.8 References   1. Aboudi J and Cederbaum G, ‘Analysis of Viscoelastic Laminated Composite Plates’, Composite Structure, 1989, 12, 243–256.   2. Sullivan J, ‘Creep and Physical Aging of Composites’, Composite Science and Technology, 1990, 39, 207–232.   3. Gates T, ‘Experimental Characterization of Nonlinear, Rate-Dependent Behavior in Advanced Polymer Matrix Composites’, Experimental Mechanics, 1992, 32, 68–73.   4. Rotem A and Nelson H G, ‘Fatigue Behavior of Graphite-Epoxy Laminates at Elevated Temperatures, In: ‘Fatigue of Fibrous Composite Materials’, ASTM STP, 1981, 723, 152–173.   5. Kharrazi M R and Sarkani S, ‘Frequency-Dependent Fatigue Damage Accumulation in Fiber-Reinforced Plastics’, Journal of Composite Materials, 2001, 35, 1924–1953.   6. Miyano Y, Nakada M, McMurray M K and Muki R, ‘Prediction of Flexural Fatigue Strength of CFRP Composites under Arbitrary Frequency, Stress Ratio and Temperature’, Journal of Composite Materials, 1997, 31, 619–638.   7. Miyano Y, Nakada M and Muki R, ‘Applicability of Fatigue Life Prediction Method to Polymer Composites’, Mechanics of Time-Dependent Materials, 1999, 3, 141–157.   8. Miyano Y, Nakada M, Kudoh H and Muki R, ‘Prediction of Tensile Fatigue Life under Temperature Environment for Unidirectional CFRP’, Advanced Composite Materials, 1999, 8, 235–246.   9. Miyano Y, Tsai S W, Christensen R M and Kuraishi A, ‘Accelerated Testing for the Durability of Composite Materials and Structures’, Long Term Durability of Structural Materials (Durability 2000), Berkeley, Elsevier, 265–276, 2001.

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10. Miyano Y, Tsai S W, Christensen R M and Muki R, ‘Accelerated Testing Methodology for the Durability of Composite Materials and Structures’, Proceedings of the 5th Composites Durability Workshop, Paris, 2002, 1–14. 11. Nakada M, Miyano Y, Kinoshita M, Koga R, Okuya T and Muki R, ‘Time-Temperature Dependence of Tensile Strength of Unidirectional CFRP’, Journal of Composite Materials, 2002, 36, 2567–2581. 12. Miyano Y, Nakada M, Kinoshita M, Koga R and Okuya T, ‘Time-Temperature Dependence of Tensile Strength of Unidirectional CFRP’, Proceedings of the 5th International Conference on Durability Analysis of Composite Systems (Duracosys 2001), Tokyo, 2001, 169–173. 13. Nakada M and Miyano Y, ‘Accelerated Testing for Long-Term Durability of Various FRP Laminates for Marine Use’, Proceedings of the 16th International Conference on Composite Materials (ICCM-16), Kyoto, 2007, WeFM1-02. 14. Miyano Y, Kanemitsu M, Kunio T and Kuhn A H, ‘Role of Matrix Resin on Fracture Strengths of Unidirectional CFRP’, Journal of Composite Materials, 1986, 20, 520–538. 15. Muki R, Nakada M, Watanabe N and Miyano Y, ‘Influence of Fiber Stiffness on TimeTemperature Dependent Tensile Strength of Unidirectional CFRP’, Proceedings of 2004 SEM X International Congress and Exposition on Experimental and Applied Mechanics (SEM 2004), Costa Mesa, 2004, 32. 16. Nakada M, Yoshioka K and Miyano Y, ‘Prediction of Long-Term Creep Life for Unidirectional CFRP, Proceedings of the 6th International Conference on Mechanics of Time Dependent Materials (MTDM 2008), Monterey, 2008, 88. 17. Miyano Y, Nakada M, Watanabe N, Murase T and Muki R, ‘Time-Temperature Superposition Principle for Tensile and Compressive Strengths of Unidirectional CFRP’, Proceedings of 2003 SEM Annual Conference & Exposition on Experimental and Applied Mechanics (SEM 2003), Charlotte, 2003, 147. 18. Miyano Y, Nakada M, Kudoh H and Muki R, ‘Prediction of Tensile Fatigue Life for Unidirectional CFRP’, Journal of Composite Materials, 2000, 34, 538–550. 19. Miyano Y, Sekine N, Ichimura J and Nakada M, ‘Fatigue Life Prediction of CFRP Laminates under Temperature and Moisture Environments’, Proceedings of 2004 SEM X International Congress and Exposition on Experimental and Applied Mechanics (SEM 2004), Costa Mesa, 2004, 36. 20. Nakada M, Maeda M, Hirohata T, Morita M and Miyano Y, ‘Time and Temperature Dependencies on The Flexural Fatigue Strength in Transverse Direction of Unidirectional CFRP’, Proceedings of International Conference on Experimental Mechanics, Singapore, 1996, 492–497. 21. Nakada M and Miyano Y, ‘Accelerated Testing for Long-Term Fatigue Strength of Various FRP Laminates for Marine Use’, Composites Science and Technology, 2009, 69, 805–813. 22. Miyano Y, Nakada M and Nishigaki K, ‘Prediction of Long-term Fatigue Life of Quasi-isotropic CFRP Laminates for Aircraft Use’, International Journal of Fatigue, 2006, 28, 1217–1225. 23. Nakada M, Hamagami Y, Sekine N and Miyano Y, ‘Time-Temperature Dependence of Flexural Behavior of CFRP Laminates for Aircraft Use’, Proceedings of the 8th Japan International SAMPE Symposium, Tokyo, 2003, 2, 1299–1302. 24. Hamagami Y, Sekine N, Nakada M and Miyano Y, ‘Time and Temperature Dependence Flexural Strength of Heat-resistant CFRP Laminates’, JSME International Journal, 2003, Series A, 46, 437–440.

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25. Miyano Y, Nakada M and Sekine N, ‘Accelerated Testing for Long-term Durability of FRP Laminates for Marine Use’, Journal of Composite Materials, 2005, 39, 5–20. 26. Watanabe N, Koga R, Nakada M, Miyano Y and Muki R, ‘Time-Temperature Dependent Tensile Behavior of Unidirectional CFRP’, Proceedings of the 14th International Conference on Composite Materials (ICCM-14), San Diego, 2003, 842. 27. Nakada M, Miyano Y, Daicho N Takemura S, ‘Time and Temperature Dependence on The Flexural Fatigue Behavior of Unidirectional Pitch-Based Carbon Fiber Reinforced Plastics’, Proceedings of the 1st Asian-Australasian Conference on Composite Materials (ACCM-1), Osaka, 1998, 442-1–442-4. 28. Nakada M, Miyano Y, Ikeda M and Takemura S, ‘Time and Temperature Dependence of Flexural Fatigue Strength for Pitch-based CFRP’, Proceedings of the 12th International Conference on Composite Materials (ICCM-12), Paris, 1999, 257. 29. Nakada M, Kosho S and Miyano Y, ‘Time-Temperature Dependence on Flexural Behavior of GFRP with Different Surface Treatment for Glass Fiber’, Proceedings of the 13th International Conference on Composite Materials (ICCM-13), Beijing, 2001, ID 1473. 30. Miyano Y, Nakada M and Muki R, ‘Prediction of Fatigue Life of a Conical Shaped Joint System for Fiber-Reinforced Plastics under Arbitrary Frequency, Load Ratio and Temperature’, Mechanics of Time-Dependent Materials, 1997, 1, 143–159. 31. Miyano Y, Tsai S W, Nakada M, Sihn S and Imai T, ‘Prediction of Tensile Fatigue Life for GFRP Adhesive Joint’, Proceedings of the 11th International Conference on Composite Materials (ICCM-11), Gold Coast, 1997, VI, 26–35. 32. Miyano Y, Nakada M, Yonemori T, Sihn S and Tsai S W, ‘Time and Temperature Dependence of Static, Creep, and Fatigue Behavior for FRP Adhesive Joints’, Proceedings of the 12th International Conference on Composite Materials (ICCM-12), Paris, 1999, 259. 33. Sekine N, Nakada M, Miyano Y and Tsai S W, ‘Time-Temperature Dependence of Tensile Fatigue Strength for GFRP/Metal and CFRP/Metal Bolted Joints’, Proceedings of the 13th International Conference on Composite Materials (ICCM-13), Beijing, 2001, ID 1610. 34. Sekine N, Nakada M, Miyano Y, Kuraishi A and Tsai S W, ‘Prediction of Fatigue Life for CFRP/Metal Bolted Joint under Temperature Conditions’, JSME International Journal, 2003, Series A, 46, 484–489. 35. Rosen B W, ‘Tensile Failure of Fibrous Composites’, AIAA Journal, 1964, 2, 1985–1991. 36. Christensen R and Miyano Y, ‘Stress Intensity Controlled Kinetic Crack Growth and Stress History Dependent Life Prediction with Statistical Variability’, International Journal of Fracture, 2006, 137, 77–87. 37. Christensen R and Miyano Y, ‘Deterministic and Probabilistic Lifetimes from Kinetic Crack Growth – Generalized Forms’, International Journal of Fracture, 2007, 143, 35–39. 38. Cai, H., Miyano, Y., Nakada, M. and Ha, S. K., ‘Long-term Fatigue Strength Prediction of CFRP Structure Based on Micromechanics of Failure’, Journal of Composite Materials, 2008, 42, 825–844.

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16 Fatigue testing methods for polymer matrix composites W. Van Paepegem, Ghent University, Belgium Abstract: This chapter discusses the fatigue testing of polymer matrix composites. Different testing methodologies in uniaxial and multiaxial testing are discussed, including the effects of gripping and boundary conditions. Next, the typical fatigue mechanisms in polymeric composites are presented, together with the common inspection methods for detection of the different fatigue damage types. Key words: composites, fatigue, testing, damage mechanisms.

16.1 Introduction This chapter discusses the fatigue testing of structural polymeric composites. It gives an overview of the most common testing methods in tension, bending, and shear, and possible online monitoring methods. Further it discusses the inspection techniques for visualization of fatigue damage and the typically observed fatigue damage mechanisms. The overview is limited to structural composites with a polymeric matrix and continuous fibre reinforcement. Composites with a metal or ceramic matrix and/or short fibre or particulate reinforcement can show quite different fatigue behaviour and fatigue mechanisms, although the discussions in this chapter about common fatigue testing methods and monitoring and inspection techniques also hold for most of these materials.

16.2 Fatigue testing methods For fibre-reinforced composites, a variety of fatigue tests can be done, because the number of parameters is very large: (i) the amplitude control (stress or strain), (ii) the testing frequency, (iii) the loading direction (axial, bending, biaxial), (iv) the load ratio (tension–tension, tension–compression, compression–compression). However, only one of these tests has been standardized in both ASTM and ISO standards: that is the tension–tension fatigue test with a constant-amplitude load (EN ISO 13003:2003: ‘Fibre-reinforced plastics – Determination of fatigue properties under cyclic loading conditions’ and ASTM D3479/D3479M-96(2007) ‘Standard Test Method for Tension–Tension Fatigue of Polymer Matrix Composite Materials’). The EN ISO 13003:2003 standard gives general principles for fatigue testing that can be applied, with care, to all modes of testing. The general aspects are generally applicable to all testing modes and types of composite 461 © Woodhead Publishing Limited, 2011

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materials. It is recommended that, when available, the equivalent static test methods should be used (Harris, 2003). The number of parameters that affect the result of fatigue experiment is very large. They can be classified in two categories: parameters that are inherent to the material used, and parameters that are related with the fatigue loading conditions. Parameters inherent to the composite specimen • The matrix can be a thermosetting or a thermoplastic matrix. Due to hysteretic heating, the load frequency effect is much more pronounced in the case of thermoplastic matrices. Temperature rises up to 100 °C are not unusual for higher test frequencies (5–10 Hz) (Xiao, 1999). The thermal conductivity of the matrix is also important for temperature rises in the material. When the conductivity is low, the hysteresis losses accumulate locally and the material degrades faster due to the hysteretic heating. The frequency effect is also very pronounced for elastomer matrices, which are used in nylon fibre-reinforced elastomers for the carcass of bias aircraft tires (Lee and Liu, 1994). • The stacking sequence of the laminate has a very important effect on the fatigue behaviour of the composite specimen (Adali, 1985). Kim (1980) showed that the onset of delaminations can be delayed by changing the stacking sequence of graphite/epoxy specimens. For the [0°/±45°/90°]s layup, delaminations initiated at N = 200 cycles, while for the [0°/90°/±45°]s stacking sequence, delamination onset was only observed at N = 20 000 cycles. Further, a dependency of the maximum crack density state on the onset of delamination was noticed. • The mechanical properties of the constituents (fibres and matrix) and the fibre volume fraction, as well as the bond strength between fibres and matrix, affect the fatigue performance. Adali (1985) has shown that the fibre volume content is an important variable for design against fatigue. • Defects during fabrication can be a cause of deteriorated fatigue performance. For example defects can be due to draping of the fibre reinforcement. McBride and Chen (1997) studied the evolution of microstructure in dry plain-weave fabric during large shear deformation. For instance, in various resin-transfer moulding techniques, dry fabric is shaped around complex mould shapes. The fabric can be in a state of large in-plane shear deformation which significantly alters the spatial orientation of the constituent yarns and fibres. These deformations can lead to shear wrinkling in the fabric and reductions of structural integrity, due to varying fibre volume content. Lomov et al. (2000) have shown that each textile geometry model should take this effect into account. • Discontinuities, like cuts or holes, or previously initiated damage (due to impact damage (Jones et al., 1987) for instance) act like stress concentrators and can alter the fatigue behaviour of the composite specimen significantly.

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• Residual stresses can significantly affect the time to first damage. Bassam et al. (1998) have performed quasi-static tensile tests on cross-ply E-glass/ epoxy specimens. They observed a residual strain on unloading which was due to matrix cracking. In their opinion, matrix cracks give rise to the relief of thermal curing stresses, and any damage which locally reduces the balanced macroscopic curing stresses can lead to a change in laminate dimensions. The residual strain could be reasonably well predicted based on shear-lag or variational mechanics. Parameters of the fatigue setup • In composite materials, the inhomogeneity resulting from the fibre distribution may be so great as to influence the fatigue response of a sample, the size of which is comparable with the scale of the inhomogeneity. Thus, in wovenroving laminates, the width of a test piece should be sufficiently large to include several repeats of the weaving pattern (Harris, 1981). • The choice of an appropriate specimen shape is strongly influenced by the characteristic nature of the composite being tested and has given rise to some difficulties. Early attempts to use test specimens similar to metallic designs led to unrepresentative modes of failure in composites that were (i) highly anisotropic and (ii) had relatively low in-plane shear resistance, such as unidirectionally reinforced carbon fibre plastics containing high-modulus untreated carbon fibres. Much early work was carried out on waisted samples of conventional kind, and designs were adapted for unidirectional carbon fibre reinforced composites by using long samples with very large radii and relatively little change in crosssection in the gauge length. An alternative approach that has been used to prevent the familiar shear splitting in composites of low shear resistance was to reduce the sample thickness rather than the width. For tensile fatigue testing, this has now been almost universally superceded by the technique of using parallel-sided strips with end-tabs, either of glass fibre reinforced polymer or soft aluminium, bonded onto the samples for gripping. Carefully done, this eliminates the risk of grip damage (and resultant premature failure) without introducing significant stress concentrations at the ends of the test length, and failures can usually be expected to occur in the test section (Harris, 1981). However, for tension–compression and compression–compression fatigue testing, various sorts of tabs are used (see for example Gathercole et al., 1994; Walsh et al., 1982; Ramkumar, 1982). • The nature of the fatigue stress (tension/compression/shear). • The stress ratio

is a very important parameter in fatigue testing. The

stress levels σmin and σmax are evaluated with their algebraic sign, so a negative stress ratio (–∞ < R < 0) refers to tension–compression loading, while tension– tension loading comprises the range 0 < R < 1 and compression–compression loading comprises the range 1 < R < +∞ (see Fig. 16.1). The remaining cases

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16.1  Constant-life diagram showing lines of constant stress ratio R (designated in the graph as Rs) (Schaff and Davidson, 1997b).

are zero-tension loading (R = 0) and zero-compression loading (R = –∞). For most composite materials, the worst fatigue loading condition is fully reversed axial fatigue, or tension–compression loading (R = –1) (Curtis, 1989; Brocker and Woithe, 1991). Indeed, in compression, although the fibres remain the principal load-bearing elements, they must be prevented from becoming locally unstable and undergoing a micro buckling type of failure. This is the task of the matrix and the fibre/matrix interface, the integrity of both being of far greater importance in compressive loading than in tensile loading. Under tensile fatigue loading, many of the laminate plies without fibres in the test direction develop intraply damage and this causes local layer delamination at relatively short lifetimes. In compression, this tensile-induced damage can lead to local layer instability and layer buckling, perhaps before resin and interfacial damage within the layers has initiated fibre micro buckling. Thus fatigue lives in reversed axial loading are usually shorter than for zero-compression or zero-tension loading. • The fatigue experiments can be load-controlled or strain-controlled. • When the fatigue stress amplitude is low compared to the static strength, it is called ‘high-cycle fatigue’, because the specimen will sustain the load during a large number of cycles. When, on the other hand, the fatigue stress amplitude is a large percentage of the static strength, it is called ‘low-cycle fatigue’. Damage types can be quite different, since static failure mechanisms are often involved in low-cycle fatigue, while typical fatigue damage mechanisms rather occur in high-cycle fatigue (Harik et al., 2000). • The frequency is a very important parameter. Ellyin and Kujawski (1992) clearly showed the influence of frequency in fatigue tests on E-glass/epoxy

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specimens. At lower stress amplitudes they found that the cyclic creep increased and the fatigue life decreased with reduced cyclic frequency. In contrast, at higher stress amplitudes, the effect of loading frequency on both cyclic creep and fatigue life was opposite to that at lower stresses, in that the fatigue life decreased with higher frequency. • Environmental conditions can play an important role. Bach (1996) has proved that moisture has a significant effect on the fatigue performance of glass fibre reinforced plastics. • The in-service fatigue loadings as they act on a real composite structure are rarely reproduced in laboratory tests, mainly because of two reasons: (i) most fatigue testing machines are not equipped for such complex loading conditions; and (ii) the in-service fatigue loadings are often known insufficiently. Therefore, the in-service fatigue loadings are replaced by representative, standardized load spectra for fatigue testing purposes. Typical examples are: (i) the WISPER spectrum for wind loads on the rotor blades of wind turbines (WInd SPEctrum Reference) (ten Have, 1991; Brøndsted et al., 1997; Bond, 1999); and (ii) the FALSTAFF spectrum for the load on the wing root area of fighter aircraft (Fighter Aircraft Load STAndard For Fatigue evaluation) (Schutz, 1981; Schaff and Davidson, 1997a, 1997b). Another option is the use of block loading tests, where loading blocks with high load amplitude and low load amplitude are applied in different order, to assess the effect of low-high and high-low load transitions on the fatigue life and the damage evolution. Van Paepegem and Degrieck (2002b) have shown that there is no consensus in international literature on which sequence is the most severe: a low-high load sequence or a high-low load sequence. It strongly depends on the type of material, stacking sequence and amplitude levels. Recent studies in block loading have been reported by Found and Quaresimin (2003), Bourchak et al. (2007) and Epaarachchi (2006). Naturally, the number of these variable amplitude tests should be limited due to their expensive and time-consuming nature. Therefore the fatigue life of the composite component under variable amplitude loads is often estimated, based on the fatigue testing results under constant amplitude loading. In the following paragraphs, the main fatigue testing methods are discussed: • • • • •

tension–tension fatigue tension–compression and compression–compression fatigue bending fatigue shear-dominated fatigue multiaxial fatigue.

Attention will not only be given to the test procedure, but also to the instrumentation methods. The importance of online monitoring techniques cannot be stressed enough, because fatigue damage leads to measurable degradation of macroscopic (elastic) properties in almost all types of polymeric composites.

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16.2.1  Tension–tension fatigue The uniaxial tension–tension fatigue test is the most widely used fatigue test. The coupon geometry is a parallel-sided specimen, instrumented with tabs. The choice of the tabbing material differs among the testing laboratories. Some prefer steel or aluminium tabs, but most of them use glass/epoxy tabs, where the glass reinforcement has a [+45°/–45°]ns stacking sequence. In most cases, the tabs are straight-sided non-tapered tabs. A fatigue test is usually conducted with a servo-hydraulic testing machine, equipped with grips that clamp the specimen. The alignment of the specimen is very important. No bending loads must be induced in the specimen due to misalignment. The load is recorded by the load cell, while the extensometer records the axial strain. The grip displacement is recorded as well, but is not very useful. The transverse strain can be measured by a strain gauge or a biaxial extensometer. A thermocouple can be read out to monitor the surface temperature of the composite. The test frequency is always chosen as high as possible to limit the duration of the test and minimize the cost, but the fatigue response of some composites strongly depends on the frequency (especially in case of fibre-reinforced thermoplastics). In tension–tension fatigue tests, the stress ratio R is often chosen to be 0.1. Nevertheless the stress ratio (or the mean stress) also has a clear effect on the damage growth rate in tension–tension fatigue. Many authors have shown that if the maximum stress is kept constant, the damage growth is reduced for increasing mean strength. Wevers et al. (1987, 1990) studied the fatigue damage development in carbon/epoxy laminates. They reported that the number of matrix cracks in the cross-ply laminates was considerably smaller if the stress ratio for tension–tension fatigue was increased from R = 0.03 to R = 0.5. According to Wevers et al. (1987, 1990), this phenomenon could be due to crack closure. During the growth of 90° cracks, material debris is formed between the crack faces. When the crack is closing down, this excess material causes (i) compressive forces in the 90° plies, for perpendicular cracks, or (ii) sliding forces, for inclined matrix cracks. For the fatigue tests with increased stress ratio, the cracks will stay open and no additional cracks can be formed at the low stress level. Several authors have reported similar results. Mallick (1997) presented representative fatigue data, in the form of a Goodman diagram, for several unidirectional composites at 107 cycles. If the stress ratio R is increased, the maximum stress can be larger for the same fatigue life. Beaumont (1987) studied the damage behaviour of quasi-isotropic carbon/ epoxy laminates. He did not compare the damage growth rates for constant maximum stress σmax, but for constant stress amplitude ∆σ. When recalculating his results it can be clearly seen that when the stress ratio R is increasing for constant maximum stress, the damage growth rate is decreasing, in agreement with the reported results by other researchers.

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In the international standards, the number of cycles to failure is considered as the main outcome of the tension–tension fatigue test. Yet it is worth the effort to use online instrumentation methods. The most simple and effective online measurement is the axial stiffness evolution. The axial stiffness can be directly calculated from the axial stress (load cell) and the axial strain (extensometer). The axial strain must never be calculated from the axial displacement and the gauge length, because the inevitable slip in the clamps can lead to serious errors in the strain calculation. Depending on the fibre and matrix type and the stacking sequence, the stiffness degradation can range from a few percent to several tens of percent (Hashin, 1985; Whitworth, 2000; Highsmith and Reifsnider, 1982; Yang et al., 1990, 1992; Kedward and Beaumont, 1992). If the transverse strain is measured, the Poisson’s ratio νxy can be followed up as well. It has been recently showed by Van Paepegem et al. (2007) that the evolution of the Poisson’s ratio is a very sensitive parameter for fatigue damage. Figure 16.2 shows the evolution of the Poisson’s ratio for a unidirectional glass fabric/epoxy composite in tension–tension fatigue. The νxy – εxx curves in straincontrolled fatigue between 0.0006 (0.06%) and 0.006 (0.6%) show a highly nonlinear behaviour and are upper-bounded by the static degradation of the Poisson’s ratio. Another simple measurement is surface temperature evolution. Neubert et al. (1987) demonstrated a fairly good agreement between temperature change and stiffness degradation for carbon/epoxy laminates. Quaresimin (2002) showed that

16.2  Evolution of Poisson’s ratio for a unidirectional glass fabric/epoxy composite in tension–tension fatigue (Van Paepegem et al., 2007).

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the high cycle fatigue strength of woven carbon/epoxy composites can be estimated, based on the infrared thermal analysis of the specimen temperature. One of the popular online monitoring techniques is acoustic emission. The acoustic emission technique basically involves the detection of low-intensity stress waves generated by failure events as they occur, such as delamination and fibre fracture. The stress waves are formed from the strain energy released during a failure event, and propagate to the surfaces where they are detected by acoustic sensors (Mouritz, 2003). The reliability of the acoustic emission technique to accurately monitor the progression of fatigue damage has been proven in numerous studies (for review see Wevers, 1987; Mouritz, 2003). Recently, Bourchak et al. (2007) used acoustic emission energy as a fatigue damage parameter for unidirectional and woven fabric carbon composites. They suggested that the arbitrary choice of fatigue stress levels at a high percentage of the static ultimate tensile strength should be reconsidered, given the significant acoustic emission energy at low stress levels in the first cycles of fatigue loading. Another online technique is the use of embedded optical fibre sensors with a Bragg grating. The Bragg grating is a periodical variation of the optical refractive index that is written in the core of the glass fibre and is typically a few millimetres in length. When broadband light is transmitted into the optical fibre, the Bragg grating acts as a wavelength selective mirror. For each grating only one wavelength, the Bragg wavelength, λB is reflected with a full width at half maximum (FWHM) of typically 100 pm, while all other wavelengths are transmitted. The Bragg wavelength is directly proportional with the period of the Bragg grating. If the optical fibre sensor is embedded in a composite laminate, the strain in the loaded laminate will cause the period of the Bragg grating to change, and hence the value of the reflected Bragg wavelength. The advantages are numerous: • the measurement is absolute and does not drift in time, • fibre optic sensors are rugged passive components resulting in a high lifetime (>20 years) and are insensitive to electromagnetic interference, • the fibre Bragg grating forms an intrinsic part of the optical fibre and has very small dimensions which makes it very suitable for embedding in composite plates, • many fibre Bragg gratings can be multiplexed employing only one optical line so more sensing points can be read out at the same time. Doyle et al. (1998) experimented on the use of fibre optic sensors for tracking the cure reaction of a fibre-reinforced epoxy, with success. They also successfully demonstrated the feasibility of these sensors for monitoring the stiffness reduction due to fatigue damage, for thermosetting matrix. De Baere et al. (2007b) have shown that the optical fibre sensors also survive the production process for carbon fabric thermoplastics (both autoclave and compression moulding) and that the correspondence between the axial strain

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measurements from the extensometer and the optical fibre sensor were identical in tension–tension fatigue tests. That means that the adhesion of the embedded optical fibre sensor to the surrounding thermoplastic material is very good. Resistance measurement is a well-established damage detection technique for unidirectional carbon composites (Abry et al., 1999). For a long time, there has been disagreement between researchers on whether the resistance should increase or decrease when local fibre fractures occur (Angelidis et al., 2004; Chung and Wang, 2006; Angelidis et al. 2006). In a recent series of articles, it has been clearly demonstrated that the resistance must increase with increasing damage, but a lot of researchers observe a decrease of resistance, due to bad contact of the electrodes. Recently, De Baere et al. (2007a) showed that resistance measurement also works very well for monitoring damage in carbon fabric reinforced thermoplastics under tension–tension fatigue loading. Figure 16.3 shows the evolution of relative resistance change ρ and axial fatigue stress σxx during fatigue cycles 4025–4030.

16.2.2 Tension–compression and compression– compression fatigue In general the damage growth rate in compression is smaller when two restrictions are made: (i) there are no delaminations; (ii) the stress ratio R is not negative (Mallick, 1997).

16.3  Evolution of relative resistance change ρ and axial fatigue stress σxx during fatigue cycles 4025 to 4030 in a five-harness satin weave carbon/PPS laminate (De Baere et al., 2007a).

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On the other hand, it is well known that the stress ratio R = –1 is very detrimental for stacking sequences which can develop delaminations. It has been often reported in the literature that this sign reversal of the applied stress has a detrimental effect on fatigue life (Bartley-Cho et al., 1998; Gathercole et al., 1994; Adam et al., 1994; Badaliance et al., 1982; Gamstedt and Sjogren, 1999). This is because the plies, with and without fibres in the loading direction, develop intraply damage, and this causes local delamination at relatively short lifetimes. In tensile loading this is less serious, as the plies containing fibres aligned with the loading direction continue to support the majority of the applied load. In compression, however, tensile-induced damage can lead to local instability and buckling, perhaps before resin and interfacial damage within the plies initiate fibre microbuckling. Thus fatigue lives in reversed tension–compression loading are usually shorter than for tension–tension loading. In tension–compression and compression–compression fatigue, the alignment of the specimen is very crucial. Bending induced by misalignment can cause premature failure of the specimen. ASTM E1012 is the only standard which specifically addresses the procedure for measurement of misalignment-induced bending (ASTM E1012-05 Standard Practice for Verification of Test Frame and Specimen Alignment Under Tensile and Compressive Axial Force Application). A more extensive description of this procedure can be found in the Code of practice for the measurement of misalignment-induced bending in uniaxially loaded tension–compression test pieces, published by the Institute for Advanced Materials of the European Commission (Bressers, 1995). Further, buckling of the specimen must be avoided. There are two options: (i) decrease the unsupported gauge length; or (ii) use anti-buckling guides for longer gauge lengths. Matondang and Schutz (1984) studied the influence of the anti-buckling guide design on the compression fatigue behaviour of carbon fibre reinforced composites. Quaresimin (2002) showed that the static compressive strength of woven carbon composites increases as the unsupported length decreases and decided to carry out the tension–compression fatigue tests on short unsupported specimens. Gagel et al. (2006a) studied the tension– compression fatigue behaviour of E-glass multi-axial non-crimp fabric/epoxy laminates with the use of an anti-buckling guide and a PTFE-coated paper to minimize friction.

16.2.3  Bending fatigue Uni-axial fatigue experiments in tension/compression are most often used in fatigue research (Fujii et al., 1993; Schulte et al., 1987; Hansen, 1997) and accepted as a standard fatigue test, while bending fatigue experiments are scarcely used to study the fatigue behaviour of composites (Ferry et al., 1997; Herrington and Doucet, 1992; Chen and Matthews, 1993b). Bending fatigue tests differ in several aspects:

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• The bending moment is (piecewise) linear along the length of the specimen (three-point bending, four-point bending, cantilever beam bending). Hence stresses, strains and damage distribution vary along the gauge length of the specimen. On the contrary, with tension–compression fatigue experiments, the stresses, strains and damage are assumed to be equal in each cross-section of the specimen. • Due to the continuous stress redistribution, the neutral fibre (as defined in the classic beam theory) is moving in the cross-section because of changing damage distributions. Once a small area inside the composite material has moved, for example from the compressive side to the tensile side, the damage behaviour of that area is altered considerably. • The finite element implementation of related damage models gives rise to several complications, because each material point is loaded with a different stress, strain and possibly stress ratio, so that damage growth can be different for each material point. In tension/compression fatigue tests, the stress- or strain-amplitude is constant during fatigue life and differential equations describing decrease of stiffness or strength can often be simply integrated over the considered number of loading cycles. • Smaller forces and larger displacements in bending allow a more slender design of the fatigue testing facility. Basically, three types of bending fatigue tests can be distinguished: (i) three-point bending (Sidoroff and Subagio, 1987; El Mahi et al., 2002); (ii) four-point bending (Caprino and D’Amore, 1998); and (iii) cantilever bending (Herrington and Doucet, 1992; Van Paepegem and Degrieck, 2001, 2002a, 2002b, 2002c, 2005). The success of these tests for fatigue of (textile) composites is quite limited, because the interpretation of the results is more difficult and, in the case of stiffness degradation, stress redistribution across the specimen height comes into play. Moreover, as long as the bending stiffness of the laminate is high enough (e.g. sandwich composites), the deflections are small and linear beam theory still applies, but once the bending stiffness of the composite decreases (e.g. thin laminates), the deflections are large and geometric nonlinearities and friction at the roller supports affect the fatigue results. Van Paepegem et al. (2006c) have shown that in three-point bending fatigue of woven carbon thermoplastics, the friction at the supporting rolls (due to large deflections) results in a ‘hysteresis-like’ behaviour of the stress-strain curve, which would normally be attributed to damage, but is entirely due to friction. The majority of flexural fatigue tests is performed on sandwich composites (El Mahi et al., 2004; Abbadi et al., 2006; Judawisastra et al., 1997; Farooq et al., 2002). Only a limited number of papers have been published on flexural fatigue testing of woven fabric composites. Miyano et al. (1994) studied the effect of loading rate and temperature on the flexural fatigue behaviour of a satin woven carbon/epoxy laminate. A test set-up

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in three-point bending was used. They showed that the flexural fatigue strength of the laminate is affected by temperature, even at temperatures far below the glass transition temperature. The effect of seawater on the bending fatigue behaviour of glass/polyester composites with chopped strand mats and woven fabrics was investigated by Hasan et al. (1998). Caprino and Giorleo (1999) investigated the fatigue behaviour of plain weave glass/epoxy composites in four-point bending. A statistical fatigue model was presented based on the hypothesis of a two-parameter Weibull distribution of the static strength. Mahfuz et al. (2000, 2001) described the fatigue modulus degradation for a thick-section plain weave S2-glass fabric/ vinylester composite loaded in three-point bending fatigue. Van Paepegem and Degrieck (2001, 2002a, 2002b, 2005) investigated the fatigue behaviour of plain weave glass/epoxy composites in cantilever bending. A phenomenological damage model was developed for fatigue loading in warp/weft and bias direction.

16.2.4  Shear-dominated fatigue Fatigue testing in pure shear is very difficult. Lessard et al. (1995) modified the static three-rail shear test (ASTM D 4255/D 4255M – 01) to do fatigue testing on carbon/epoxy plates. A much more common method is the tension–tension fatigue tests on a [+45°/– 45°]ns laminate. This test is based on the ASTM D3518/D3518M-94(2007) Standard Test Method for In-Plane Shear Response of Polymer Matrix Composite Materials by Tensile Test of a ±45° Laminate. This standard explains how the shear stress-strain curve can be derived from a static tensile test on a ±45° laminate, by measuring the longitudinal and transverse strain. The test is also called a bias tension test, because the bias (or cross-grain) direction is the 45° direction between warp and weft direction in the case of fabric-reinforced composites. In both pure shear and shear-dominated fatigue, the test frequency is a very important parameter. The shear stresses can lead to significant autogeneous heating and once the temperature exceeds the glass transition temperature, the deformations can be very large. A little-studied phenomenon is the accumulation of permanent strain during shear dominated fatigue loading. For composite materials with a thermoplastic matrix, creep effects seem to be dominant, while in the case of thermosetting materials, permanent strain is simply neglected in most reported literature. Moreover, for both types of material, the phenomenon is not well understood . Van Paepegem et al. (2006a, 2006b) studied the accumulation of permanent shear strain in [+45°/–45°]2s glass/epoxy laminates under cyclic loading. They showed that the shear modulus significantly degrades, but that the accumulation of permanent shear strain is even more important. Figure 16.4 shows the accumulation of permanent shear strain in cyclic loading of unidirectional glass fabric/epoxy composites.

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16.4  Accumulation of permanent shear strain in cyclic [+45°/–45°]2s loading of unidirectional glass fabric/epoxy composites (Van Paepegem et al., 2006a, 2006b).

16.2.5  Multiaxial fatigue As opposed to metals, where there exists an extensive amount of research on biaxial/multiaxial fatigue, research in the same field on composite materials is far less complete. Literature reviews on multiaxial/biaxial fatigue of composites can be found in Shokrieh and Lessard, 2003; Chen and Matthews, 1993a; Quaresimin and Susmel, 2002; and Philippidis and Vassilopoulos, 1999. The four main types of multiaxial fatigue set-ups described in literature are: (i) tension/torsion, (ii) internal pressure/tension, (iii) planar biaxial set-ups, and (iv) bending/torsion: • In the case of tension/torsion set-ups (Lee and Hwang, 2001; Fujii et al., 1992; Inoue et al., 2000), composite tubes are clamped in a servo-hydraulic machine with two actuators. A combination of uniaxial load (tension/compression) and torsional moment is applied to the composite tube and realizes a multiaxial stress state in the tube. The main problem with these tests is the clamping of the tubes. The design of the tube ends must be such that the effect of the stress concentration at the grips is minimized and does not cause premature fatigue failure of the tube. • Internal pressure/tension (Ellyin and Martens, 2001; Perreux et al., 2005; Perreux and Joseph, 1997) set-ups realize a combination of axial stress and hoop stress in composite tubes. The ASTM D2992 Standard practice for

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obtaining hydrostatic or pressure design basis for fiberglass (Glass-FiberReinforced Thermosetting-Resin) pipe and fittings can be used as a guideline. • For multiaxial fatigue testing on flat specimens, planar biaxial set-ups are most commonly used. The specimen typically has a cruciform shape, but the precise geometry of the specimen is extremely important to generate the most critical stress/strain state in the centre of the specimen, and not in the loading arms (Smits et al., 2006). Chen and Matthews (1993b) clamped all edges of flat rectangular composite plates and applied fatigue load by a central indenter to obtain biaxial bending fatigue. • Bending/torsion set-ups can be used to apply three- or four-point bending and torsion simultaneously to flat rectangular specimens. More information can be found in the papers by Ferry et al. (1997, 1999). Only a few of the cited investigations deal with textile composites. Fujii et al. (1992) and Inoue et al. (2000) reported the results of tension/torsion biaxial cyclic loading on plain weave glass/polyester tubes. They found that the modulus decay in shear is affected by the loading path, while the modulus decay in tension is not. Also fibre misalignment and fibre reorientation during testing appeared to have a substantial influence on the results.

16.3 Effect of boundary conditions and specimen geometry 16.3.1 Stress state near tabbed regions in uniaxial fatigue loading When performing fatigue experiments, there is always discussion about the gripping method: bonded/non-bonded tabs, aluminium tabs or non-woven E-glass tabs, tapered/non-tapered tabs, etc. Even the international standards are not fully compatible. The ISO 527–4 standard Plastics – Determination of tensile properties – Part 4 leaves the choice between gripping without end tabs or with non-tapered bonded end tabs. The ISO-specification even allows the use of ‘. . . alternative tabs made from the material under test, mechanically fastened tabs, unbonded tabs made of rough materials (such as emery paper or sandpaper), and the use of roughened grip faces’. On the other hand, the ASTM D 3039 standard Tensile properties of fibre-resin composites concludes that ‘The most consistently used bonded tab material has been continuous E-glass fiber-reinforced polymer matrix materials (woven or unwoven) in a [0/90]ns laminate configuration. The tab material is commonly applied at 45° to the loading direction to provide a soft interface’, although other gripping methods are also allowed. It is well known that fatigue failure near or inside the tabbing region occurs frequently, especially in the case of specimens with prismatic rectangular crosssection and fibre-dominated stacking sequences.

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In open literature, there are very few articles on the subject. In most cases, end tab design is just a small remark when discussing experimental results, such as in Lavoie et al. (2000) and Odegard and Kumosa (2000). Kulakov et al. (2004) and Portnov et al. (2006, 2007) have done a study on gripping of unidirectional reinforced epoxies. In their study, they consider different tab geometries, the effect of the adhesive and a few tab materials. They also present two loading schemes, based on a normal and tangential force and a friction coefficient, but no analytical formula to calculate the grip pressure for a given clamp geometry is given. De Baere et al. (2008) have investigated the 3D stress state near and inside the tabbing region. First a model of hydraulic wedge-operated grips has been developed, yielding a simplified analytical formula to calculate grip pressure. This analytical formula has been validated with finite element models (Fig. 16.5 left). This model proved excellent correspondence between the simulated and the predicted contact pressure. Based on this conclusion, a detailed finite element model of the tabbing region was developed to investigate which end tab geometry gives best results for tension tests on a carbon fabric reinforced thermoplastic, namely polyphenylene sulphide (PPS) (Fig. 16.5 right). The resulting contact pressure from the grips is calculated with the proposed analytical formula, depending on the geometry of the used grips. Four different geometries and four different tabbing material combinations were examined. From these simulations, it could be concluded that a chamfered glass-epoxy or carbon-PPS with a [+45°,–45°]ns stacking sequence gave lowest stress concentration factors, which is in line with the recommendation of the ASTM D 3039 standard.

16.3.2  Topology optimization in biaxially loaded specimens In uni-axial fatigue loading, the geometry of the specimen is normally prescribed by the international standards. However, in biaxial fatigue loading, the geometry

16.5  Finite element model of the hydraulic wedge-operated grips (left) and detailed 3D model of the tabbed region of the composite specimen (right).

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of the specimen is much less standardized, and the geometry interferes with the gripping method and other boundary conditions. In biaxial fatigue testing of composites, there are two major approaches: (i) the use of composite tubes, that are loaded in tension/compression, torsion and/or internal pressure; or (ii) the use of planar cruciform geometries that are loaded in-plane with one or more independent actuators. Reviews of biaxial test methods can be found in Chen and Matthews (1993a) and Makris et al. (2007). Tubes have been used for a very long time to create biaxial or multiaxial stresses in materials. The first tests were performed on metals, but later on this test method was adapted to investigate composite materials. Since the 1970s, polymer composites in the form of tubes have been tested under biaxial conditions (Protasov and Georgievskii, 1967; Halpin et al., 1969). In the beginning, the applied loads were mainly torsion or a combination of torsion and axial loads. Later on, tubes were subjected to a combination of axial load and pressure, which produces tensile hoop stress in the tube. From the early beginnings till the present time, many researchers have used this type of specimen. There are a few studies on the gripping of tubular specimens (Kim and Hahn, 1998; Miyano et al., 2000; Liu et al., 2005). The use of planar cruciform geometries was only initiated later on. The cruciform design was favoured mainly because it allows the application of in-plane biaxial loading without any out-of-plane stress. Furthermore, real construction components are often flat or gently curved and differ a lot from tubular specimens (with large curvature), and the planar specimen geometry to obtain uniform biaxial stress states is free from the radial stress influence that appears in tubular testing. An early investigation on glass/epoxy laminates with a planar cross-shaped geometry subjected to biaxial loading was published by Bert et al. (1969) from Oklahoma University, USA in 1969. Most tests before 1990 were limited to tension–tension loading and the use of four independent servo-hydraulic actuators was not widely known. Up until today, most biaxial data on planar cruciform geometries are limited to static loading. Very few data exist on biaxial fatigue loading. The main reason is that all sorts of edge effects and stress concentrations have a much stronger effect in fatigue loading than in static loading, and cause the fatigue failure to be triggered by boundary effects. It is very difficult to obtain a uniform fatigue damage in the centre of the specimen, and to maintain the maximum stress state there, because due to fatigue damage, stress will be redistributed in the specimen. Lamkanfi et al. (2008, 2009) have used evolutionary strategies to optimize a planar cruciform geometry for in-plane biaxial loading. Evolutionary strategies (ES) are stochastic optimization methods based on the theory of Charles Darwin. These strategies, developed by Rechenberg (1973) and Schwefel (1981) in the 1970s, were in the beginning commonly applied for continuous optimization problems. In the nineties Thierauf and Cai (1995) proposed a modified ES

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algorithm to handle also discrete problems. These stochastic methods differ from the conventional optimization algorithms by using randomized operators such as the mutation and the recombination operator which are crucial in the optimization of the cruciform specimen. Plate V (in the colour section between pages 288 and 289) shows an example of a non-optimized starting geometry (left) loaded in biaxial tension (load ratio 3.85/1), and depending on the evolutionary strategies, evolving to two different geometries. The middle one is very similar to the shape published by Rochdi et al. (2006), while the right one is very similar to the geometry that is used by Qinetiq (Williamson and Clarke, 2004; Williamson et al., 2007). For the geometry in the middle, a deep milling in the centre area is necessary to get the maximum stress state in the middle. The geometry in the middle is also very sensitive to slight changes in the shape of the loading arms. Plate VII shows three slightly different geometries of the loading arms, and the corresponding strain concentrations. Below, these strain concentrations have been validated with Digital Image Correlation on carefully milled specimens. This means that the utmost care is needed when preparing the specimens, because slight changes in the geometry have a large effect. It has been proved by finite element simulations that these unwanted strain concentrations depend much more on slight changes in geometry than on changes in stacking sequences. The geometry in Plate V at the right is much less sensitive to edge effects, but requires much larger forces to get an acceptable stress state in the centre of the specimen.

16.4 Typical fatigue damage in structural composites 16.4.1 Inspection techniques for visualization of fatigue damage The easiest inspection technique is visual inspection. Depending on the difference in optical refraction index of the matrix and fibre materials, the transparency of the composite laminate can be very high. Gagel et al. (2006a, 2006b) reported an extraordinary high transparency of E-glass multiaxial non-crimp fabric epoxy laminates. Matrix cracks, voids and inclusions could be detected easily by transmitted light. Optical or light microscopy provides a direct path from observations made with the naked eye, to what is visible at magnifications up to about 1000 times (Hull, 1999). Fracture surfaces are embedded in resin and polished before observation. Figure 16.6 shows a microscopic image of the damage in plain weave glass/epoxy composites loaded in bending fatigue (Van Paepegem, 2002). Scanning Electron Microscopy (SEM) is by far the most popular technique for fractographic studies. The fracture surface is scanned, on a rectangular or square raster, by a finely focused beam of high-energy electrons (typically 5–40 keV).

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16.6  Microscopic image of a plain weave glass/epoxy composite loaded in bending fatigue (Van Paepegem, 2002).

The electrons penetrate the surface of the specimen and interact with the atoms of the material in a variety of elastic and inelastic scattering processes. By selectively collecting and measuring the scattered signals, information is obtained about the surface and near-surface properties and characteristics (Hull, 1999). Typically, for conventional modern instruments, the maximum resolution for SEM images is a few nanometres and the range of possible magnifications is very wide, from about 5 times to 20 000 times. The depth of field is much greater than the corresponding values for light microscopy (Hull, 1999). Edge replication of composite specimens has been used with considerable success in the study of ply cracking and edge delamination (Masters and Reifsnider, 1982; Wevers, 1987). The specimen edge is polished and a plastic replica of the surface is taken. This is achieved either by painting a film of plastic, such as cellulose acetate dissolved in acetone, on the surface, or by pressing a thin sheet of a gel of the cellulose acetate onto the surface, so that it completely wets the surface and follows the intricate contours. When the plastic has fully dried, it is removed from the fracture surface by peeling and forms a negative replica of the original surface. This replica of the edge can then be studied by a variety of microscopic techniques: light microscopy, SEM and TEM (Hull, 1999). A very common inspection technique for fatigue damage in polymeric composites is ultrasonics. Ultrasonics can be performed in various modes of operation, but the most common for fatigue damage detection is the throughtransmission (C-scan) technique. Through-transmission ultrasonics basically consists of a transducer for emitting ultrasonic pulses that is placed at or near one surface and a receiver sensor that is located at the opposite surface. The technique applies to relatively low frequency sound beams, typically 0.5 MHz to 15 MHz, having a small aperture. The transducer and receiver are coupled to the surfaces or they are immersed in water together with the composite. The ultrasound waves are attenuated by defects in the composite and the acoustic attenuation is monitored using the receiver (Mouritz, 2003).

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In classical C-scans, the size of the applied bounded beam and the applied wavelength are too big to characterize microscopic defects with high spatial resolution. Contrary to low-frequency ultrasound, high-frequency ultrasound, as used in acoustic microscopy, can detect such defects with very high in-depth resolution, if a high-frequency focused beam with relatively low aperture is used (Declercq, 2005). Another well-known inspection technique is radiography. During passage of the radiation through the material, the rate of energy absorption can be changed by defects that have a different absorption coefficient to the parent material. Certain types of defects in composites are not easily detected using standard X-ray radiography because of poor contrast on the radiograph. In order to enhance the contrast between defects and the parent material, a variety of X-ray sensitive penetrants can be used (Mouritz, 2003). High-resolution 3D X-ray micro-tomography or micro-CT is a relatively new technique which allows scientists to investigate the internal structure of their samples without actually opening or cutting them (Cnudde et al., 2006). Without any form of sample preparation, 3D computer models of the sample and its internal features can be produced with this technique. In order to perform tomography, digital radiographs of the sample are made from different orientations by rotating the sample along the scan axis from 0 to 360 degrees. After collecting all the projection data, the reconstruction process produces 2D horizontal crosssections of the scanned sample. These 2D images can then be rendered into 3D models, which enable the researcher to virtually look into the object. Figure 16.7 shows the micro-tomography images of plain woven glass/epoxy composite (left) and damaged five-harness satin weave carbon/PPS (right). Another technique is thermography, which registers the infrared radiation and estimates the surface temperature of the specimen. Hansen (1999) successfully applied this method to the tension–tension fatigue of woven glass fabric/epoxy composites. Toubal et al. (2006) used infrared thermography for their study of [±45°]2s woven carbon fabric/epoxy composites in tension–tension fatigue loading. They observed temperature increases up to 80 °C, which can be attributed

16.7  Micro-tomography images of plain woven glass/epoxy composite (left) and damaged five-harness satin weave carbon/PPS (right) (De Baere et al., 2007b).

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to the very high testing frequency of 10 Hz. Gagel et al. (2006b) observed similar temperature rises for E-glass multi-axial non-crimp fabric/epoxy composites in tension–tension fatigue at 6 Hz. Other inspection techniques that have not been mentioned so far are the leaky Lamb wave technique (Chimenti and Nayfeh, 1985; Seale et al., 1993), acoustic polar scans (Declercq, 2005; Maes, 1998), acoustography, vibrothermography or SPATE (Stress Patterns Analysis by the measurement of Thermal Emissions) and Moiré interferometry (Mouritz, 2003). These techniques are rarely used to study fatigue damage in composites.

16.4.2  Typical fatigue damage mechanisms Fibre-reinforced polymer composites have quite a good rating as regards lifetime in fatigue. The same does not apply to the number of cycles to initial damage. Although the fatigue behaviour of fibre-reinforced polymer composites has been studied for many years, it is so diverse and complex that present knowledge is far from complete. There are a number of important differences between the fatigue behaviour of metals and of continuous fibre-reinforced polymer composites. In metals the stage of gradual and invisible deterioration takes a relatively large part of the total life. No significant reduction of stiffness is observed in metals during the fatigue process. The final stage of the process starts with the formation of small cracks, which are the only form of macroscopically observable damage. Gradual growth and coalescence of these cracks quickly produce a large crack and final failure of the structural component. Continuous fibre-reinforced polymers are made of long, reinforcing fibres embedded in a polymer matrix. This makes them heterogeneous and anisotropic. The first stage of deterioration by fatigue is observable by the formation of ‘damage zones’, which contain a multitude of microscopic cracks and other forms of damage, such as fibre/matrix interface debonding and pull-out of fibres from the matrix. It is important to observe that damage starts very early, after only a few or a few hundred loading cycles. This early damage is followed by a second stage of very gradual degradation of the material, characterized by a gradual reduction of the stiffness. More serious types of damage appear in the third stage, such as fibre breakage and unstable delamination growth, leading to an accelerated decline and finally catastrophic failure. This three-stage stiffness reduction was first reported by Schulte et al. (1984, 1985, 1987) but has since then been observed for many different types of composite materials, and also for textile composites. Fujii et al. (1993) have performed fatigue tests on plain woven glass fibrereinforced composites with a polyester matrix. The tests were performed in uniaxial zero-tension loading along the warp direction. They reported a modulus degradation which was very similar to the one described by Schulte (1984). According to the experiments, the weft tows first debond from the matrix (see Fig. 16.8).

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16.8  Microscopic fatigue processes in a plain woven glass/epoxy laminate (Fujii et al., 1993).

Simultaneously matrix cracks occur in the resin-rich regions between two woven sheets. In a second stage, the debonds propagate in the weft along the fibre and connect with other debonds that are present near the adjacent cross-over points. Small delaminations (so-called meta-delaminations) occur between warp and weft fibre bundles near the fabric cross-over points. Finally fibre fracture occurs. These observations have been confirmed by Pandita et al. (2001). Tension– tension fatigue tests were performed on plain woven glass/epoxy specimens and damage development was monitored by means of acoustic emission and SEM. Later on, Pandita and Verpoest (2004) conducted similar experiments on E-glass knitted fabric epoxy laminates. There the matrix cracks and yarn-matrix debonds initiated from the part of the knitting loop that was perpendicular to the loading direction. The fatigue damage then propagated following the knitting structure. Gagel et al. (2006b) described the damage evolution in E-glass multi-axial noncrimp fabric epoxy laminates under tension–tension fatigue loading. The gradual development of matrix cracks in the 45° and 90° directions was observed with an optical scanner in transmitted light mode. The authors did not report whether or not the polyester stitching yarn affected the fatigue damage onset and propagation. Quaresimin (2002) reported large scale inter-layer delaminations in tension– compression fatigue of twill carbon fabric epoxy laminates, where a considerable

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amount of 45° layers were present in the stacking sequence. Later on, Found and Quaresimin (2003) reported similar observations for five-harness satin weave carbon epoxy laminates in tension–tension fatigue. If textile composites with a central hole are tested in fatigue, the hole acts as a stress concentrator and damage concentrates around the hole. Recent investigations for woven carbon fabric laminates are discussed by Toubal et al. (2006), Hochard et al. (2006) and Bourchak et al. (2007).

16.5 Future trends At least two major trends and challenges can be observed for fatigue testing of composites: (i) the trend towards better testing and instrumentation methods; and (ii) the challenge for faster assessment of the fatigue performance of new composite material combinations.

16.5.1  Better testing and instrumentation methods For a long time, fatigue testing of composites was only focused on providing the S-N fatigue life data. No efforts were made to gather additional data from the same test by using more advanced instrumentation methods. The development of methods such as digital image correlation (strain mapping) and optical fibre sensing allows for much better instrumentation, combined with traditional equipment such as extensometers, thermocouples and resistance measurement. Validation with finite element simulations of the realistic boundary conditions and loading conditions in the experimental set-up must maximize the generated data from one single fatigue test. Plate VI (in the colour section) shows an example of the comparison between experimentally measured and numerically simulated strain fields in a cruciform specimen for biaxial fatigue loading (Lamkanfi et al., 2006). Currently a lot of research effort is spent on the integration of smart sensors into composite structures. Such embedded sensors must monitor the deformation and local strains in the composite component under in-service loading conditions and must be capable of detecting damage at an early stage. In that way structural health monitoring would lead to (i) smaller safety factors at design (because the loads are better known), (ii) early repair of fatigue damage, and (iii) extended fatigue lives (because the load history is well known).

16.5.2 Faster assessment of fatigue performance of new composite materials The constituent materials for polymeric composites are developing very rapidly: new resins are developed, toughening fillers are added, new geometrical arrangements of fibre reinforcement are created, and new fibre types are explored

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(silk, banana, bamboo, etc.). This contrasts sharply with the very time-consuming and expensive characterization of the fatigue performance of only one fibre/ matrix combination. Therefore faster methods for assessment of the fatigue performance of new composite materials should be developed.

16.6 Sources of further information and advice The textbook Fatigue in composites. Science and technology of the fatigue response of fibre-reinforced plastics, edited by Bryan Harris (2003), is one of the best reference books for further information about fatigue of composite materials. The three-yearly International Conference on Fatigue of Composites provides a forum for all researchers active in this field. The first session was held in 1997 (Paris, France), followed by the conferences in 2000 (Williamsburg, USA), 2004 (Kyoto, Japan) and 2007 (Kaiserslautern, Germany). The next conference will be held in 2010 in China. Relevant testing standards for fatigue of structural composites are: • ASTM D3479/D3479M-96(2007) ‘Standard Test Method for Tension– Tension Fatigue of Polymer Matrix Composite Materials’ • ASTM D6115-97(2004) ‘Standard Test Method for Mode I Fatigue Delamination Growth Onset of Unidirectional Fiber-Reinforced Polymer Matrix Composites’ • EN ISO 13003:2003 ‘Fibre-reinforced plastics – Determination of fatigue properties under cyclic loading conditions’ • EN ISO 14692 ‘Petroleum and natural gas industries – Glass-reinforced plastics (GRP) piping’ • EN 12245 ‘Transportable gas cylinders – Fully wrapped composite cylinders’

16.7 References Abbadi, A., Azari, Z., Belouettar, S., Gilgert, J. and Pluvinage, G. (2006). Static and fatigue characterization of sandwich panels. Materials Science Forum, 518, 555–562. Abry, J.C., Bochard, S., Chateauminois, A., Salvia, M. and Giraud, G. (1999). In situ detection of damage in CFRP laminates by electrical resistance measurements. Composites Science and Technology, 59, 925–935. Adali, S. (1985). Optimization of fibre reinforced composite laminates subject to fatigue loading. In: Marshall, I.H. (ed.). Composite Structures 3. Proceedings of the third International Conference on Composite Structures (ICCS-3), 9–11 September 1985, Scotland, Elsevier, pp. 43–55. Adam, T., Gathercole, N., Reiter, H. and Harris, B. (1994). Life prediction for fatigue of T800/5245 carbonpfibre composites: II. Variable amplitude loading. International Journal of Fatigue, 16(8), 533–547. Angelidis, N., Wei, C.Y. and Irving, P.E. (2004). The electrical resistance response of continuous carbon fibre composite laminates to mechanical strain. Composites Part A, 35, 1135–47.

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Angelidis, N., Wei, C.Y. and Irving, P.E. (2006). Short communication – Response to discussion of paper: The electrical resistance response of continuous carbon fibre composite laminates to mechanical strain. Composites Part A, 37, 1495–1499. Bach, P.W. (1996). The effect of moisture on the fatigue performance of glass fibre reinforced polyester coupons and bolted joints. Energieonderzoek Centrum Nederland (ECN), http://www.ecn.nl/publications/default.aspx?nr=ECN-C-96-015, contract report ECNC-96-015, 34 pp. Badaliance, R., Dill, H.D. and Potter, J.M. (1982). Effects of spectrum variations on fatigue life of composites. In: Daniel, I.M. (ed.). Composite materials: Testing and design (Sixth Conference). ASTM STP 787. Philadelphia, American Society for Testing and Materials, pp. 274–286. Bartley-Cho, J., Lim, S.G., Hahn, H.T. and Shyprykevich, P. (1998). Damage accumulation in quasi-isotropic graphite/epoxy laminates under constant-amplitude fatigue and block loading. Composites Science and Technology, 58, 1535–1547. Bassam, F., Boniface, L., Jones, K. and Ogin, S.L. (1998). On the behaviour of the residual strain produced by matrix cracking in cross-ply laminates. Composites Part A, 29(11), 1425–1432. Beaumont, P.W.R. (1987). The fatigue damage mechanics of composite laminates. In: Wang, A.S.D. and Haritos, G.K. (eds.). Damage mechanics in composites. Presented at the Winter Annual Meeting of the ASME, 13–18 December 1987, Boston, Massachusetts, ASME, pp. 53–63. Bert, C.W., Mayberry, B.L. and Ray, J.D. (1969). Behavior of fiber-reinforced plastic laminate under biaxial loading, ASTM STP. Bond, I.P. (1999). Fatigue life prediction for GRP subjected to variable amplitude loading. Composites Part A, 30(8), 961–970. Bourchak, M., Farrow, I.R., Bond, I.P., Rowland, C.W. and Menan, F. (2007). Acoustic emission energy as a fatigue damage parameter for CFRP composites. International Journal of Fatigue, 29(2007), 457–470. Bressers, J. (ed.) (1995). A code of practice for the measurement of misalignment induced bending in uniaxially loaded tension–compression test pieces. European Commission, Institute of Advanced Materials, ISBN 92-826-9681-2, 52 pp. Brocker, W. and Woithe, K. (1991). Fatigue life behaviour of composite structures. Report for Industrieanlagen-Betriebsgesellschaft MBH Ottobrunn (Germany). Brøndsted, P., Andersen, S.I. and Lilholt, H. (1997). Fatigue damage accumulation and lifetime prediction of GFRP materials under block loading and stochastic loading. In: Andersen, S.I., Brøndsted, P., Lilholt, H., Lystrup, Aa., Rheinländer, J.T., Sørensen, B.F. and Toftegaard, H. (eds.). Polymeric Composites – Expanding the Limits. Proceedings of the 18th Risø International Symposium on Materials Science, 1–5 September 1997, Roskilde, Denmark, Risø International Laboratory, pp. 269–278. Caprino, G. and D’Amore, A. (1998). Flexural fatigue behaviour of random continuous-fibrereinforced thermoplastic composites. Composites Science and Technology, 58, 957–965. Caprino, G. and Giorleo, G. (1999). Fatigue lifetime of glass fabric/epoxy composites. Composites Part A, 30(3), 299–304. Chen, A.S. and Matthews, F.L. (1993a). A review of multiaxial/biaxial loading tests for composite materials. Composites, 24(5), 395–406. Chen, A.S. and Matthews, F.L. (1993b). Biaxial flexural fatigue of composite plates. In: Miravete, A. (ed.). ICCM/9 Composites: properties and applications. Volume VI. Proceedings of the Ninth International Conference on Composite Materials, 12–16 July 1993, Madrid, Spain, Woodhead Publishing Limited, pp. 899–906.

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Chimenti, D.E. and Nayfeh, A.H. (1985). Leaky lamb waves in fibrous composite laminates. Journal of Applied Physics, 58(12), 4531–4538. Chung, D.D.L. and Wang, S. (2006). Discussion on paper ‘The electrical resistance response of continuous carbon fibre composite laminates to mechanical strain’ by Angelidis N, Wei CY, Irving PE, Composites Part A 2004; 35: 1135–47. Composites Part A, 37, 1490–1494. Cnudde, V., Masschaele, B., Dierick, M., Vlassenbroeck, J., Van Hoorebeke, L. and Jacobs, P. (2006). Recent progress in X-ray CT as a geosciences tool. Applied Geochemistry, 21(5), 826–832. Curtis, P.T. (1989). The fatigue of organic matrix composite materials. In: Partridge, I.K. (ed.). Advanced composites. London, Elsevier, pp. 331–367. De Baere, I., Van Paepegem, W. and Degrieck, J. (2007a). The use of rivets for electrical resistance measurement on carbon fibre-reinforced thermoplastics. Smart Materials and Structures, 16(5), 1821–1828. De Baere, I., Van Paepegem, W. and Degrieck, J. (2008). On the design of end tabs for quasi-static and fatigue testing of fibre-reinforced composites. Polymer Composites, 30(4), 381–390. De Baere, I., Voet, E., Van Paepegem, W., Vlekken, J., Cnudde, V., Masschaele, B. and Declercq, N. (2005). The interaction of complex harmonic elastic waves with periodically corrugated surfaces and with anisotropic viscoelastic or piezoelectric layered media. Chapter XIII. Acoustic microscopy. PhD thesis, Ghent University, Ghent, 2005, pp. 595–620. Degrieck, J. (2007b). Strain monitoring in thermoplastic composites with optical fibre sensors: embedding process, visualization with micro-tomography and fatigue results. Journal of Thermoplastic Composite Materials, 20(5), 453–472. Doyle, C., Martin, A., Liu, T., Wu, M., Hayes, S., Crosby, P.A., Powell, G.R., Brooks, D. and Fernando, G.F. (1998). In-situ process and condition monitoring of advanced fibrereinforced composite materials using optical fibre sensors. Smart Materials & Structures, 7(2), 145–158. El Mahi, A., Berthelot, J.-M. and Bezazi, A. (2002). The fatigue behaviour and damage development in cross-ply laminates in flexural tests. Proceedings of the Tenth European Conference on Composite Materials (ECCM-10), Brugge, Belgium, 3–7 June 2002. El Mahi, A., Farooq, M.K., Sahraoui, S. and Bezazi, A. (2004). Modelling the flexural behaviour of sandwich composite materials under cyclic fatigue. Materials & Design, 25(3), 199–208. Ellyin, F. and Kujawski, D. (1992). Fatigue testing and life prediction of fiberglass-reinforced composites. In: Neale, K.W. and Labossière, P. (eds.). First International Conference on Advanced Composite Materials in Bridges and Structures (ACMBS–I), 1992, Sherbrooke, Quebec, Canada, Canadian Society for Civil Engineering, pp. 111–118. Ellyin, F. and Martens, M. (2001). Biaxial fatigue behaviour of a multidirectional filamentwound glass-fiber/epoxy pipe. Composites Science and Technology, 61, 491–502. Epaarachchi, J.A. (2006). A study on estimation of damage accumulation of glass fibre reinforced plastic (GFRP) composites under a block loading situation. Composite Structures, 75(1–4), 88–92. Farooq, M.H., Sahraoui, S. and El Mahi, A. (2002). Evaluation of fatigue behaviour in sandwich composite materials. Proceedings of the Tenth European Conference on Composite Materials (ECCM-10), Brugge, Belgium, 3–7 June 2002. Ferry, L., Gabory, D., Sicot, N., Berard, J.Y., Perreux, D. and Varchon, D. (1997). Experimental study of glass-epoxy composite bars loaded in combined bending and

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torsion loads. Fatigue and characterisation of the damage growth. In: Degallaix, S., Bathias, C. and Fougères, R. (eds.). International Conference on Fatigue of Composites. Proceedings, 3–5 June 1997, Paris, France, La Société Française de Métallurgie et de Matériaux, pp. 266–273. Ferry, L., Perreux, D., Varchon, D. and Sicot, N. (1999). Fatigue behaviour of composite bars subjected to bending and torsion. Composites Science and Technology, 59, 575– 582. Found, M.S. and Quaresimin, M. (2003). Two-stage fatigue loading of woven carbon fibre reinforced laminates. Fatigue & Fracture of Engineering Materials & Structures, 26(1), 17–26. Fujii, T., Amijima, S. and Okubo, K. (1993). Microscopic fatigue processes in a plainweave glass-fibre composite. Composites Science and Technology, 49, 327–333. Fujii, T., Amijima, S., Lin, F. and Sagami, T. (1992). Study on strength and nonlinear stress-strain response of plain woven glass-fiber laminates under biaxial loading. Journal of Composite Materials, 26(17), 2493–2510. Gagel, A., Fiedler, B. and Schulte, K. (2006a). On modelling the mechanical degradation of fatigue loaded glass-fibre non-crimp fabric reinforced epoxy laminates. Composites Science and Technology, 66(5), 657–664. Gagel, A., Lange, D. and Schulte, K. (2006b). On the relation between crack densities, stiffness degradation, and surface temperature distribution of tensile fatigue loaded glass-fibre non-crimp-fabric reinforced epoxy. Composites Part A – Applied science and manufacturing, 37(2), 222–228. Gamstedt, E.K. and Sjogren, B.A. (1999). Micromechanisms in tension–compression fatigue of composite laminates containing transverse plies. Composites Science and Technology, 59(2), 167–178. Gathercole, N., Reiter, H., Adam, T. and Harris, B. (1994). Life prediction for fatigue of T800/5245 carbon-fibre composites: I. Constant-amplitude loading. International Journal of Fatigue, 16(8), 523–532. Halpin, J.C., Pagano, N.J., Whitney, J.M., Wu, E.M. (1969). Characterization of anisotropic composite materials, Composite Materials: Testing and Design, ASTM STP 460, 37–4. Hansen, U. (1997). Damage development in woven fabric composites during tension– tension fatigue. In: Andersen, S.I., Brøndsted, P., Lilholt, H., Lystrup, Aa., Rheinländer, J.T., Sørensen, B.F. and Toftegaard, H. (eds.). Polymeric Composites – Expanding the Limits. Proceedings of the 18th Risø International Symposium on Materials Science, 1–5 September 1997, Roskilde, Denmark, Risø International Laboratory, pp. 345–351. Hansen, U. (1999). Damage development in woven fabric composites during tension– tension fatigue. Journal of Composite Materials, 33(7), 614–639. Harik, V.M., Klinger, J.R. and Bogetti, T.A. (2000). Low cycle fatigue of unidirectional laminates: stress ratio effects. Journal of Engineering Materials and Technology, 122, 415–419. Harris, B. (1981). Fatigue testing of fibre composites. In: Advisory Group for Aerospace Research & Development (AGARD). Fatigue test methodology. AGARD Lecture Series No. 118 (LS-118), pp. 6.1–6.18. Harris, B. (ed.) (2003). Fatigue in composites. Science and technology of the fatigue response of fibre-reinforced plastics. Cambridge, Woodhead Publishing Ltd., 742 pp. Hasan, E.H., Abdel Hakeem, H.M. and El-Sayed, A.A.M. (1998). Environmental rotating bending fatigue behavior of fiber glass/polyester composites. In: Crivelli Visconti, I. (ed.). ECCM–8: Science, technologies and applications. Volume 3. Proceedings of the

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Schulte, K., Baron, Ch., Neubert, H., Bader, M.G., Boniface, L., Wevers, M., Verpoest, I. and de Charentenay, F.X. (1985). Damage development in carbon fibre epoxy laminates: cyclic loading. In: Proceedings of the MRS-symposium ‘Advanced Materials for Transport’, November 1985, Strassbourg, 8 p. Schulte, K., Reese, E. and Chou, T.-W. (1987). Fatigue behaviour and damage development in woven fabric and hybrid fabric composites. In: Matthews, F.L., Buskell, N.C.R., Hodgkinson, J.M. and Morton, J. (eds.). Sixth International Conference on Composite Materials (ICCM-VI) & Second European Conference on Composite Materials (ECCM-II): Volume 4. Proceedings, 20–24 July 1987, London, UK, Elsevier, pp. 4.89–4.99. Schutz, D. (1981). Variable amplitude fatigue testing. In: AGARD Lecture Series No. 118. Fatigue test methodology, pp. 4.1–4.31. Schwefel, H.-P. (1981). Numerical optimization for computer models, Wiley & Sons, Chichester, UK. Seale, M.D., Smith, B.T., Prosser, W.H. and Masters, J.E. (1993). Lamb wave response of fatigued composite samples. Technical Report conf-rpqnde-93-p1261, NASA Langley Technical Report Server. Shokrieh, M.M. and Lessard, L.B. (2003). Fatigue under multiaxial stress systems. In: Harris, B. (ed.). Fatigue in Composites. Cambridge, Woodhead Publishing and CRC Press, 2003, pp. 63–113. Sidoroff, F. and Subagio, B. (1987). Fatigue damage modelling of composite materials from bending tests. In: Matthews, F.L., Buskell, N.C.R., Hodgkinson, J.M. and Morton, J. (eds.). Sixth International Conference on Composite Materials (ICCM-VI) & Second European Conference on Composite Materials (ECCM-II): Volume 4. Proceedings, 20–24 July 1987, London, UK, Elsevier, pp. 4.32–4.39. Smits, A., Van Hemelrijck, D., Philippidis, T.P. and Cardon, A. (2006). Design of a cruciform specimen for biaxial testing of fibre reinforced composite laminates. Composites Science and Technology, 66, 964–975. ten Have, A.A. (1991). Wisper and WisperX final definition of two standardised fatigue loading sequences for wind turbine blades. NLR report NLR TP91476. Thierauf, G. and Cai, J. (1995). A two level parallel evolution strategy for solving mixeddiscrete structural optimization problems, The 21th ASME Design Automation Conference, 17–221. Toubal, L., Karama, M. and Lorrain, B. (2006). Damage evolution and infrared thermography in woven composite laminates under fatigue loading. International Journal of Fatigue, 28(12), 1867–1872. Van Paepegem, W. (2002). Development and finite element implementation of a damage model for fatigue of fibre-reinforced polymers. Ph.D. thesis. Gent, Belgium, Ghent University Architectural and Engineering Press (ISBN 90-76714-13-4), 403 p. Van Paepegem, W. and Degrieck, J. (2001). Fatigue Degradation Modelling of Plain Woven Glass/epoxy Composites. Composites Part A, 32(10), 1433–1441. Van Paepegem, W. and Degrieck, J. (2002a). A New Coupled Approach of Residual Stiffness and Strength for Fatigue of Fibre-reinforced Composites. International Journal of Fatigue, 24(7), 747–762. Van Paepegem, W. and Degrieck, J. (2002b). Effects of Load Sequence and Block Loading on the Fatigue Response of Fibre-reinforced Composites. Mechanics of Advanced Materials and Structures, 9(1), 19–35. Van Paepegem, W. and Degrieck, J. (2002c). Modelling damage and permanent strain in fibre-reinforced composites under in-plane fatigue loading. Composites Science and Technology, 63(5), 677–694.

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Van Paepegem, W. and Degrieck, J. (2005). Simulating Damage and Permanent Strain in Composites under In-plane Fatigue Loading. Computers & Structures, 83(23–24), 1930–1942. Van Paepegem, W., De Baere, I. and Degrieck, J. (2006a). Modelling the nonlinear shear stress-strain response of glass fibre-reinforced composites. Part I: Experimental results. Composites Science and Technology, 66(10), 1455–1464. Van Paepegem, W., De Baere, I. and Degrieck, J. (2006b). Modelling the nonlinear shear stress-strain response of glass fibre-reinforced composites. Part II: Model development and finite element simulations. Composites Science and Technology, 66(10), 1465–1478. Van Paepegem, W., De Baere, I., Lamkanfi, E. and Degrieck, J. (2007). Poisson’s ratio as a sensitive indicator of (fatigue) damage in fibre-reinforced plastics. Fatigue and Fracture of Engineering Materials & Structures, 30, 269–276. Van Paepegem, W., De Geyter, K., Vanhooymissen, P. and Degrieck, J. (2006c). Effect of friction on the hysteresis loops from three-point bending fatigue tests of fibre-reinforced composites. Composite Structures, 72(2), 212–217. Walsh, R.M. and Pipes, R.B. (1982). Compression fatigue behaviour of notched composite laminates. Journal of Materials Science, 17, 2567–2576. Wevers, M. (1987). Identification of fatigue failure modes in carbon fibre reinforced composites. Part I: Text. Master thesis, Leuven, Belgium, Faculteit Toegepaste Wetenschappen, 210 pp. Wevers, M., Verpoest, I. and De Meester, P. (1990). Is crack closure due to fatigue loading causing more damage in carbon fibre reinforced epoxy composites? In: Füller, J., Grüninger, G., Schulte, K., Bunsell, A.R. and Massiah, A. (eds.). Developments in the science and technology of composite materials. Proceedings of the Fourth European Conference on Composite Materials (ECCM/4), 25–28 September 1990, Stuttgart, Elsevier Applied Science, pp. 181–188. Wevers, M., Verpoest, I., Aernoudt, E. and De Meester, P. (1987). Fatigue damage development in carbon fibre reinforced epoxy composites: correlation between the stiffness degradation and the growth of different damage types. In: Matthews, F.L., Buskell, N.C.R., Hodgkinson, J.M. and Morton, J. (eds.). Sixth International Conference on Composite Materials (ICCM-VI) & Second European Conference on Composite Materials (ECCM-II): Volume 4. Proceedings, 20–24 July 1987, London, UK, Elsevier, pp. 4.114–4.128. Whitworth, H.A. (2000). Evaluation of the residual strength degradation in composite laminates under fatigue loading. Composite Structures, 48(4), 261–264. Williamson, C., Cook, J. and Clarke, A.B. (2007). Investigation into the failure of open and filled holes in CFRP laminates under biaxial loading conditions. Proceedings of the 13th International Conference on Experimental Mechanics (ICEM13), Alexandroupolis, Greece, 1–6 July. Williamson, C. and Clarke, A. (2004). Biaxial testing – a route to faster certification. Proceedings of the Aerospace Testing Expo 2004, 30, 31 March & 1 April 2004, Hamburg, Germany. Xiao, X.R. (1999). Modeling of load frequency effect on fatigue life of thermoplastic composites. Journal of Composite Materials, 33(12), 1141–1158. Yang, J.N., Jones, D.L., Yang, S.H. and Meskini, A. (1990). A stiffness degradation model for graphite/epoxy laminates. Journal of Composite Materials, 24, 753–769. Yang, J.N., Lee, L.J. and Sheu, D.Y. (1992). Modulus reduction and fatigue damage of matrix dominated composite laminates. Composite Structures, 21, 91–100.

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Plate VI Comparison of experimentally measured (courtesy of VUB) and numerically simulated strain fields in a cruciform specimen for biaxial fatigue loading (Lamkanfi et al., 2006).

Plate V Evolutionary strategies for biaxially loaded composite cruciform specimens: starting geometry (left), first optimized cruciform shape (middle) and second optimized cruciform shape (right).

Plate VII Dependence of the strain concentration location on the exact geometry of the load introducing arms of the cruciform specimen.

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17 The effect of viscoelasticity on fatigue behaviour of polymer matrix composites J. A. Epaarachchi, University of Southern Queensland, Australia Abstract: This chapter details the effect of viscoelasticity on fatigue behaviour of polymeric matrix composites. The viscoelastic effects on composite materials under static and dynamic loading are explained using linear viscoelastic analysis. The details of the governing parameters of the fatigue process, such as stress ratio, temperature and the loading frequency are also presented with fatigue life prediction models. Finally, three fatigue life prediction models that have included fatigue and static-fatigue processes in the predictions will be discussed and compared with some experimental data. Key words: composites, fatigue, creep, static-fatigue, viscoelasticity.

17.1 Introduction Polymer matrix composite materials comprise a combination of two or more materials, differing in form or composition on a macro-scale. The constituents retain their characteristics, that is, they do not break up or combine completely into one another although they behave as a single entity. Normally, the components can be physically identified and exhibit an interface between one another. Fibrereinforced plastics (FRP) are good examples of composites. Fibre-reinforced composite materials, for example, contain high-strength and high-modulus fibres in a matrix material. In these composites, fibres are the principal load-carrying members, and the matrix material keeps the fibres together, acts as a load-transfer medium between fibres, and protects fibres from being exposed to the environment. Since the polymeric matrices and the many types of fibres used in advance composite materials possess their own distinct inherent viscoelastic properties, these are all governing factors of the properties of resultant composite materials. This time-dependent viscoelastic nature of polymeric materials has become increasingly important for designers and engineers. The viscoelastic behaviour of polymeric matrices used in advanced structural composites can significantly influence their deformation, strength and failure response under operational loading and various environmental factors. The timedependent nature of the viscoelastic properties of the matrix and fibres are in fact the most desirable factors which greatly influence the gross properties of the resulting composite materials. Unlike a solid material, a viscoelastic material does not stay at a constant deformation when it is subjected to a constant load. The viscoelastic material continues to flow with time; this phenomenon is called creep. 492 © Woodhead Publishing Limited, 2011



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Immediately, on removal of the applied load, the specimen relaxes from the strain at that time until it reaches the magnitude which depends on the loading time and the load level. It is possible that there is no residual strain and, in this instance, the specimen regains its original geometry after some time. The particular natures discussed here are the critical characteristics that originate due to the inherit viscoelastic properties of the polymeric materials. Fatigue of composites has attracted the attention of material researchers since composite materials started revolutionizing many engineering fields, such as aerospace and automobile industries, which were eagerly looking for more efficient materials. Components and structures manufactured from composite materials are used increasingly in situations where high fatigue loading is present, for example wind turbine blades and in aircraft components. Understanding the damage mechanics associated with fatigue failure of composite materials has been a major objective of the researchers. The crack initiation, saturation and propagation until final failure were investigated in detail and the results were published in numerous research papers and symposia. However, there are still number of unanswered problems existing at the root level of damage mechanics, such as interactions of viscoelastic nature of the material during the crack propagation process. The time-dependent nature of the viscoelastic property of the polymeric materials causes many undesirable phenomena under dynamic loading (i.e. cyclic loads). For example, there is a phase difference between stress and strain of a material under cyclic loading. As such, viscoelastic materials have a property called passive resistance, which is in contrast to pure isotropic materials. This phenomenon was also called the ‘hereditary response’ of the material. These materials’ structural response to an applied dynamic load depends not only on the present state of loading input but also on previous states. As a consequence there are major implications for the fatigue of polymeric materials such as the dependence of loading sequence and the so-called ‘memory effect’. Therefore the complete fatigue process of the fibre-reinforced polymeric composite system depends on independent viscoelastic properties whose exact involvement in the fatigue process are not completely understood yet. When polymeric composite materials experience small stress levels while undergoing certain fatigue processes, the linear viscoelastic modelling approach is a basic tool that may be used for the investigation of the effects of the materials’ viscoelastic nature. However, in the presence of higher stress levels, the constitutive relations will be inherently non-linear when damage occurs and therefore non-linear viscoelastic analysis has to be employed to analyse such a system for accurate predictions. At the beginning of this chapter the fundamental concepts of linear viscoelasticity will be described for easy understanding of the time-dependent processes of viscoelastic materials. Later sections of this chapter contain a detailed discussion of the interaction of cyclic fatigue and static fatigue. The next section will brief

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the reader on a linear viscoelastic approach to understand the viscoelastic properties and the behaviour of polymer matrix composite materials under dynamic loading. The discussion of the influence of viscoelastic properties on the fatigue life of composites will be presented in Section 17.3. However, the effects on polymeric materials of individual fatigue governing factors such as stress ratio, load sequencing, temperature, etc., are not explained in depth. This is because they behave in a completely different manner in composites. Section 17.3 also presents a few fatigue prediction models which have addressed the complex viscoelastic effects on the fatigue life of polymer matrix composites. However, the discussion is limited to a brief outline of the topics as discussions on viscoelasticity and long-term life prediction of composites are available in other chapters of this book. The reader is advised to refer to the references in this chapter for more specific details related to the topics contained herein.

17.2 Linear viscoelastic analysis of the characteristics of viscoelastic materials under static and dynamic loading 17.2.1  Effects of static loading on viscoelastic properties Viscoelastic materials exhibit both elastic and viscous properties. Within the scope of the theory of linear viscoelasticity, the elastic property follows Hooke’s law while viscous properties follow Newton’s laws of viscosity. Creep and stress relaxation are the major characteristics of the viscous nature of polymeric composites which will describe the viscous behaviour under static/ fatigue loading. Figure 17.1 shows a typical behaviour of a viscoelastic material under a constant stress. When a viscoelastic material sample is loaded with a constant stress σ0 for the period of time t1, it immediately strains elastically up to ε0 and then non-linearly increases its strain level until the release of the load at time t1. Within the nonlinear portion of the strain curve, there are two types of independent components, namely the delayed elastic strain εd and the viscous flow εv. Delayed elastic strain increases in a monotonically decreasing rate while viscous flow increases nonlinearly. The delayed elastic strain is also known as ‘primary creep’ and it is fully reversible. However, it takes a long time to return to its original geometry. The viscous flow is the irreversible part of the strain. The other phenomenon called ‘stress relaxation’ is shown in Fig. 17.2. When a viscoelastic material sample is loaded with a constant strain elastically up to ε0 and the initial stress level σ0 monotonically decreases with time. The stress level may decompose to zero at substantially long time. Readers should note that the details given in this chapter are limited to fundamentals of the viscoelastic nature of the polymeric materials and more comprehensive details are available in the listed references.

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17.1  Creep and recovery of a viscoelastic material sample under a constant stress for time t1.

17.2  Stress relaxation of a viscoelastic material sample under constant strain.

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17.2.2  Effects of dynamic loading on viscoelastic properties This section details the application of linear viscoelastic theories on polymeric materials to investigate the effects of viscoelasticity on dynamically loaded viscoelastic material. The reader should be aware that the mathematical working has been omitted in the derivation of the formulae for clarity. These missing details can be found in Haddad (1995). In the case of linear viscoelastic material, the stress and strain exhibit a time delay between the response to each other under static loading. Similarly, in the case of dynamic load application on a viscoelastic material and when equilibrium is reached, both the stress and strain vary sinusoidally, but the strain lags behind the stress. Due to this reason the terms stress and strain would be complex numbers. 

[17.1] [17.2]

  E1/E2

[17.3] [17.4]

where,

σ is the stress, σ0 initial stress, ω is

the loading frequency and E1 and E2 are modulus. It is clear that the stress has two components which are the modulus E1 in phase with the strain and E2 which is 90° out of phase with the strain. Alternatively using complex representation of stress and strain: [17.5]



[17.6]



[17.7]  The linear viscoelastic properties under dynamic loading can be expressed approximately by a complex modulus E*(iω) which is a function of the loading frequency ω, for small stress and strain values. Therefore:  The complex modulus has real and imaginary parts: 

[17.8]

[17.9]

Similarly the complex compliance can be decomposed to:  where J, J1, and J2 are compliance.

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[17.10]



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The Maxwell model The Maxwell model is one of the simple idealizations of the viscoelastic characteristics of a real material. This model is comprised of a linear spring and a dashpot as shown in Fig. 17.3. Now consider when an applied stress σo is applied to the viscoelastic material sample, the spring immediately extends and the piston moves through the viscous fluid in the dashpot. Therefore the following relationships can be obtained:

17.3  The Maxwell model.

[17.11]



[17.12]

Since the spring and the dashpot are in series:

[17.13]

After some mathematical treatments, the ‘creep response’ can be obtained as:

[17.14]

where λ = η/E and η is the dashpot constant. Also the relaxation response can be obtained as: [17.15] There are disadvantages in the Maxwell model when applying to real viscoelastic materials. The model shows an unlimited deformation of the viscoelastic material under a constant stress level, as time increases (Eq. 17.15), which is not the case for a real viscoelastic material. Furthermore, the model is not able to describe the behaviour when the applied stress level ceases to zero. Once the applied stress is removed, the spring will contract and recover elastic strain. However, there would be no force available to retract the dashpot and recover the time-dependent strain. An additional limitation of this model becomes apparent from examining Eq. 17.15, which shows the applied stress level decay to zero at infinite time. This is again not the case for a real viscoelastic material. Application of the Maxwell model to the case of dynamic loading of a viscoelastic material sample results:

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and

[17.17]  and the phase lag: [17.18]  Figure 17.4 depicts the variation of E1, E2 and tan(θ) with the frequency of loading ω. At small frequencies E2, which is a measure of energy loss, would be small. At small displacements and small frequencies the moduli E1 and E2 would be small according to the Maxwell model. With the increase of frequency up to moderate and higher levels, both modului E1 and E2 will increase. At very high frequencies, E1 becomes larger and most of the applied force is balanced by the spring action. The phase lag tan(θ) decreases monotonically with the increase of frequency. The Kelvin-Voigt model The Kelvin-Voigt model comprises a parallel arrangement of a linear spring and a dashpot as shown in Fig. 17.5. This parallel arrangement of the spring and dashpot causes this model to exhibit the primary creep phenomenon as there is no possibility for the spring and dashpot to expand independently. Therefore this model is not able to demonstrate steady state creep and steady state stress relaxation. After releasing the stress (t > t1) the spring would not be able to contract

17.4  Variation of E1, E2 and phase lag with loading frequency in the Maxwell model (after Haddad 1995).

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17.5  The Kelvin-Voigt model.

its length back to its original position immediately. Instead the spring will apply a force on the dashpot and thus compressive creep under external stress will occur. Eventually, after a finite time, all creep strain will be recovered. This phenomenon is called ‘viscoelastic contraction’ which is significant in real viscoelastic materials. For the Kelvin-Voigt model, the stress is given as: [17.19]

  and after some mathematical treatment the creep reponse:



[17.20]

[17.21]

In the case of dynamic loading, the Kelvin-Voigt model gives:

 

[17.22] [17.23]

and [17.24]



Figure 17.6 shows the response of the Kelvin-Voigt model to a dynamic loading case. At very small frequencies the compliance J1 become larger and J2 becomes smaller and thus the energy loss would be minimal. At moderate to higher frequencies, the compliance J1 decreases while the compliance J2 increases. At the higher frequencies, both compliances J1 and J2 approach zero as the displacements become very small.

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17.6  Variation of E1, E2 and phase lag with loading frequency in the Kelvin-Voigt Model (after Haddad 1995).

17.2.3  Summary In this section the viscoelastic nature of viscoelastic materials under static and dynamic loads is presented using simple linear viscoelastic theory. Two simple one-dimensional models, namely Maxwell and Kelvin-Voigt, were used in this section to analyse the viscoelastic nature of composites quantitatively and qualitatively. There are limitations on these models which cause some restrictions to application for real viscoelastic materials. However, the reader can grasp sufficient knowledge about the viscoelastic nature of composites under various loading conditions. There is a wealth of reading materials on advanced viscoelastic analysis available and some of them are listed in the references for those who are interested in investigating more about viscoelasticity.

17.3 Fatigue behaviour of composite materials About half a century ago, Boller (1957) studied the fatigue properties of plastics, now the most common matrix material in glass fibre reinforced plastics (GFRP). In the early development of GFRP composites, Owen and Bishop (1974) used the ‘Paris Law’ to calculate fatigue crack growth rate in chopped strand mat and glass fabric reinforced polyester composites and they have proposed that this criterion be used in safe life design methods. With the increased demand for composites in the field, many issues associated with composites, such as the viscoelastic nature of resins and their effects on the composite’s properties, have been continuously under investigation. Schapery (1975) and Hertzberg and Manson (1980) studied the viscoelastic properties associated with polymers and composites. They have reviewed the influence of the governing fatigue parameters such as mean stress, temperature and loading frequency on both polymers and composites. They also

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formulated the fatigue life under constant amplitude loading by using linear crack propagation mechanics. Dillard (1991) extensively reviewed the issues within viscoelastic behavior in composite materials and he discussed the application of non-linear viscoelasticity to explain some practical problems such as stress concentration around fibres. He has investigated the factors influencing viscoelastic creep, relaxation, damping and damage, which can accumulate to induce delayed failure. He stressed the importance of considering the inclusion of time varying loads in cumulative damage rules. Reifsnider (1982, 1991) reviewed the fatigue behaviour of composite materials. Following many representative physical observations, he proposed a model for fatigue behaviour. He has shown that crack propagation in composite materials is influenced by their anisotropy. The in-plane and through-thickness cracks cause a degradation in the material strength and stiffness which leads to final failure. Through experimental observation, he has shown that the number of throughthickness cracks of a fixed length becomes stable after a given number of stress cycles. In this situation the spacing between cracks becomes fixed. Reifsnider has named this phenomenon the ‘Characteristic Damage State’ (CDS) of the laminate. He proposed that the CDS of a laminate controls the state of stress and the state of strength of an unnotched laminate and provides a well-defined damage state for analysis in the same way a single crack serves that purpose for homogeneous materials. The fatigue damage mechanism of composites is schematically reproduced in Fig. 17.7 including the stages of CDS.

17.7  Schematic representation of the development of damage during the fatigue life of composite laminate (adopted from Reifsnider, 1990).

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Importantly, Reifsnider (1991) described the strength degradation mechanism associated with composite materials: the redistribution of stress levels at crack propagation in the composites. He has presented a concept called the ‘critical element’: elements which cause the final fatigue failure of the composites. The strength and other property degradation are dependent on the properties associated with the critical and sub-critical elements in the material. He has made a few suggestions about the modelling of strength degradation using the critical element. These include the incorporation of multi-axiality of the stress field into the model, the inclusion of non-uniform stress states due to geometric variations and their suitability for any composite configuration. He was of the opinion that the fatigue process is a time-dependent process and separation of cycle-dependent processes (fatigue) should not be promoted as the long-term objective of the modelling process. Fatigue failure of a material is a result of a progressive accumulation of damage due to applied cyclic loading and hence the subsequent degradation of materials, integrity, strength and stiffness. The process of fatigue failure is caused by a combination of micro events which are complex and intricately related to a variety of failure modes in different circumstances, progressing within the material due to applied dynamic loading. There are two major processes governing the fatigue process, namely cycle-dependent (discussed in detail in the previous paragraph) and time-dependent processes. However, the distinct mechanisms associated with the latter process would make the micro damage mechanics happening in the fatigue process of composites more complicated. As a consequence, the complete understanding of fatigue processes of composite materials is difficult to achieve yet. The fatigue life of a composite material must be described as a combination of static fatigue that is strength degradation under the action of the static load, a creep mechanism, and dynamic degradation of its strength due to the cyclic component of the load. The long-term behaviour has been a critical issue for composites with the growing demand for these materials in various engineering fields. Two decades ago, Dumpleton and Bucknall (1987) and Crowther et al. (1989) successfully isolated the static fatigue component in polymeric composites operating in a wide range of conditions. Recently, Guedes (2007) compiled the history of the significant research work carried out on viscoelastic effects on composites under static and fatigue loading and proposed a comprehensive model for total fatigue life prediction of composite materials.

17.3.1  Fatigue and static fatigue The fatigue behaviour of composites has been shown to be highly dependent on the stress ratio, R, and the frequency of applied cyclic load, f. Mandell and Meier (1983) and El-Kadi and Elyn (1994) have discussed the effects of R on the fatigue life of composites and have shown that for a given maximum stress in a tension-tension case, the fatigue life of the composite increases with increasing magnitude of R (–∞ < R < 1). In compression-compression loading, increasing the magnitude of

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R (1 < R < ∞) reduces the fatigue life of the composite. Also, Schapery (1975), Sun and Chan (1979), Mandell and Meier (1983) and Saff (1983) have shown that by increasing f, the fatigue crack propagation rate will decrease and in so doing increase the fatigue life of the polymer provided any increase in temperature is small. Various fatigue theories have been proposed for correlating constant amplitude fatigue behavior of composite materials. The available theories can be categorized as empirical, residual strength degradation, stiffness degradation and damage mechanism based. Even though models based on strength degradation have a major drawback of not being directly related to the damage mechanism of the composite, they are the currently accepted life prediction method for composites based on the residual strength as the damage metric. Many fatigue models have been proposed that are based on strength and stiffness degradation of composites. For example, the model proposed by D’Amore et al. (1996) for flexural fatigue of continuous strand mat reinforced plastics has R as a variable. They conclude that this model could be used for continuous strand mat reinforced thermoset plastics. Caprino and D’Amore (1998) later validated the same kind of model for flexural fatigue of random-continuous fibre-reinforced thermoplastics and concluded that this model is applicable to thermoset and thermoplastic matrix composites, provided the reinforcement is in the form of continuous randomly oriented fibres. Their model showed that by using one characteristic curve, the fatigue behavior of a composite may be expressed for various values of R. Ellyn and El Kadi (1990; El Kadi and Ellyn, 1994) proposed a model that uses a strain energy concept to describe the fatigue life of a composite for positive values of R. Composite damage models have been proposed by Hwang and Han (1989) based on fatigue modulus degradation. It is important, however, that a comprehensive fatigue model be developed to suit a range of composite materials and must reflect the governing viscoelastic effects on the fatigue life. According to our fundamental discussion on linear viscoelastic properties of composites, static and dynamic loads cause creep and the dissipation of energy in the composite respectively. As shown in Section 17.2.2, the loading frequency plays an important role in the fatigue process. According to the Kelvin-Voigt model, the fatigue damage is minimal at the very high frequencies whilst the Maxwell model shows a higher damage at very high frequencies. However, both models have close agreement of the predictions at the lower to higher frequency levels. Furthermore the ‘hereditary response’ of the composite material will cause more discrepancies, such as load sequencing effects and memory effects, which render theories of linear fracture mechanics unsuitable to evaluate the progressive development of cracks in composite materials. For an example, many researchers have reported that the Miner’s damage sum at fatigue failure is always smaller than or equal to unity for low-high load sequence whereas it is always greater than or equal to unity for high-low load sequences (Yang and Jones 1980, Epaarachchi and Clausen 2005). High-low and low-high load sequences are the application of high load first followed by low load until failure, and low load first followed by

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high load until failure, respectively. Figure 17.8 shows a schematic representation of high-low and low-high load sequence (source: Yang and Jones 1983). Dark lines show the degradation of strength. Yang and Jones (1980) have classified the factors contributing to load sequencing as: 1. The difference in the residual strength level when fatigue fracture occurs referred to as ‘boundary effect’ and 2. The memory effect of materials with respect to previously experienced load histories. Figure 17.8 illustrates two sequences of two-step fatigue loading spectra such that one material sample is under a high-low sequence and the other under lowhigh sequence. Both spectra have the same number of low cycles and the same number of high cycles. Due to the boundary effect, the sample under high-low loading has a longer fatigue life than the sample under low-high load sequence. The ‘memory effect’ influences the strength/fatigue properties of the composite by an accumulation of the previous load history, since the matrix is viscoelastic. Therefore the sequence of the load application for a spectrum loading situation results in significantly different fatigue lives for the same composite material. As such, an integral approach to these micro events of damage accumulation in composite materials is an exceptionally difficult task. Due to this reason there is yet to be a complete mathematical algorithm to resolve this problem. Alternatively, most of the research work on the long-term behaviour of composite materials has been performed independently on fatigue, static fatigue and aging of composites.

17.8  Schematic representation of high-low and low-high load sequence (adopted from Yang and Jones, 1983). Dark lines show the degradation of strength until failure.

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According to the discussion so far it is obvious that at the high load levels and the higher stress ratios the fatigue lives of composites should exhibit the integral effects of cyclic and static-fatigue effects. This is when the material is exposed to a significantly higher mean load which causes considerable creep damage. Figure 17.9 shows a fatigue data set obtained for glass/polyester [90°/0°/±45°/0°]s for various stress ratios. Close inspection of Fig. 17.9 reveals that the fatigue data at high stress levels are clustered together irrespective of the stress ratio. Such an observation is a result of the apparent interaction of fatigue and creep damage at those load levels. Interestingly, with the decrease of load levels the fatigue lives at various stress levels move apart. This is a clear indication of the influence of stress level and stress ratios in fatigue and creep interaction. A few existing fatigue models of composite material have included some dependent fatigue variables such as temperature, stress ratio, loading frequency and environmental factors. Most of these models partially satisfy the damage mechanics associated with both time-dependent and cycle-dependent fatigue processes. The majority of these models are empirical relationships which suffer from various critical issues such as memory effects and load sequencing effects and are not representative of the progressive damage accumulation of the composite materials. Some of the dynamic fatigue prediction models are residual strength based and others are based on the degradation of modulus. Most of the

17.9  Fatigue curves for glass/polyester [90°/0°/±45°/0°]s – experimental and calculated (Epaarachchi, 2003) for R = 0.1 and R = 0.5 to R = 0.9.

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static fatigue or creep prediction models are based on energy criteria and have performed to reasonably accurate levels (Kaminskii and Selivanov 2000; Guedes 2004, 2006). Here the interest is in the fatigue models which were developed with the inclusion of static fatigue effects. Three noticeable models are presented here in order to evaluate the co-existence of fatigue and static fatigue under cyclic loading. These three models have included the variables critical to the viscoelastic nature of the composite materials.

17.3.2  Fatigue and static fatigue interaction models Miyano et al. (1997, 2000, 2004, 2005) have proposed a formulation for fatigue strength. Since the creep loading is considered as fatigue loading with stress ratio R(σmin/σmax) = 1 at an arbitrary frequency, the fatigue strength σf (tf ;R, f,T ) at an arbitrary combination of frequency (f), stress ratio (R) and temperature (T) is: [17.25]  where R is the stress ratio and σf:1, σf:0 the fatigue strength for R = 1 and 0, respectively. This relationship was built on several assumptions: strength independence of loading rate, the linear cumulative damage law for monotonic and the loading, and linear dependence of fatigue strength on stress ratio. Guedes (2007, 2008) has thoroughly investigated this relationship and concluded that the assumption of the linear damage accumulation law for monotonic (increasing) loading is inappropriate to estimate the life under complex loading, by adding up the damages for individual load steps. He has proposed a model for fatigue strength based on the Strength Evolution Integral (SEI) concept to predict fatigue strength for an arbitrary load ratio. He has assumed that the static-fatigue (R = 1) and fatigue (R = 0) effects on strength degradation have a linear dependence upon stress ratio (R): 

[17.26]

where z1 = t/τ1, z2 = t/τ2, t is the time variable, τ1 and τ2 are characteristic times associated with the static-fatigue (R = 1) and fatigue (R = 0) processes, while j1 and j2 are material parameters for static-fatigue and fatigue respectively. Full details of this model are available in Guedes (2007). The other model considered here is proposed by Epaarachchi (2006) as an expanded version of his fatigue prediction model presented in 2003. He has postulated an empirical model to predict the residual strength of a composite which has undergone a constant amplitude cyclic stress until failure. [17.27] 

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where f frequency (Hz), N fatigue life (cycles) of the material at σmax, R stress ratio, α , and β are material constants, and θ the smallest angle of fibres between the loading direction and the fibre direction, σmax is the maximum applied stress in the loading direction and σu the ultimate stress of the virgin material in the loading direction. Later he added an additional formulation to compensate for the creep phenomenon. The additional formulation was postulated after the work of Barbero and Damiani (2003). The proposed formulation has combined the rate of strength degradation and the creep of the material, in a power law type relationship

[17.28]  where tf is the time to failure, σr is residual strength and λ and ξ are material constants related to static-fatigue. Further, he proposed that if the applied cyclic stress levels are less than half of the ultimate strength then the final failure would be decided by the static fatigue. For complete details readers are directed to Epaarachchi (2006). Figures 17.10–17.13 show the fatigue data of glass/polyester [90°/0°/±45°/0°]s and the predicted fatigue curves using the formulations proposed by Guedes (2007) and Miyano et al. (1997, 2000, 2004, 2005). Figures 17.14–17.17 show the predicted fatigue curves using the formulation proposed by Epaarachchi (2003, 2006) for the same material. The predicted fatigue curves show a significant improvement on previous fatigue models. The obvious conclusion is that the inclusion of governing fatigue parameters such as stress ratio, frequency etc., do

17.10  Fatigue curves for R = 0.5 for glass/polyester [90°/0°/±45°/0°]s (adopted from Guedes, 2007).

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17.11  Fatigue curves for R = 0.7 for glass/polyester [90°/0°/±45°/0°]s (adopted from Guedes, 2007).

17.12  Fatigue curves for R = 0.8 for glass/polyester [90°/0°/±45°/0°]s (adopted from Guedes, 2007).

not completely address the effects of viscoelasticity on the fatigue life of composite materials.

17.4  Concluding remarks The fatigue life of polymeric composite materials shows a complex behaviour due to their inherent viscoelastic properties. The time-dependent fatigue process (static fatigue) is as important as the governing parameters of cyclic fatigue process such as stress ratio, temperature and the loading frequency. In this chapter, the main focus was on the time-dependent effects on the fatigue life of composite

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17.13  Fatigue curves for R = 0.9 for glass/polyester [90°/0°/±45°/0°]s (adopted from Guedes 2007).

17.14  Fatigue curves for R = 0.5 for glass/polyester [90°/0°/±45°/0°]s using the method of Epaarachchi (2006).

materials because the cycle-dependent fatigue process has been widely investigated and a wealth of research outcomes are available. The future trend in this context should be the further development of fatigue and static fatigue interaction models of the composite material though this topic is not a major focus of research at present.

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17.15  Fatigue curves for R = 0.7 for glass/polyester [90°/0°/±45°/0°]s using the method of Epaarachchi (2006).

17.16  Fatigue curves for R = 0.8 for glass/polyester [90°/0°/±45°/0°]s using the method of Epaarachchi (2006).

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17.17  Fatigue curves for R = 0.9 for glass/polyester [90°/0°/±45°/0°]s using the method of Epaarachchi (2006).

17.5 Acknowledgements The author wishes to express his gratitude to Dr Andrew Wandel (USQ) and Dr Hao Wang (USQ) for reading the manuscript and their valuable comments.

17.6 References Barbero E J, Damiani T M (2003), Interaction between static fatigue and zero-stress aging in E-Glass fibre composites, ASCE J Comp for Const, 7(1), 3–9. Boller K H (1957), Fatigue properties of fibrous glass-reinforced plastics laminates subjected to various conditions, Modern Plastics, 19(3), 241–245. Caprino G, D’Amore A (1998), Flexural fatigue behaviour of random continous-fibrereinforced thermoplastic composites, Comp Sci Tech, 58, 957–965. Crowther M F, Wyatt R C, Phillips M G (1989), Creep-fatigue interaction in glass fibre/ polyester composites, Comp Sci Tech, 36, 191–210. D’Amore A, Caprino G, Stupak P, Zhou J (1996), Effect of stress ratio on the flexural fatigue behaviour of continous strand mat reinforced plastics, Sci and Eng Comp Matrl, 4(1), 1–8. Dillard D (1991), Viscoelastic behaviour of laminated composite materials, in Reifsnider K L, Fatigue of Composite Materials, Amsterdam, Elsevier, BV, pp. 339–363.

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Dumpleton F, Bucknall C K (1987), Comparison of static and dynamic fatigue crack growth rates in high-density polyethylene, Intl J Fat, 9(03), 151–155. El-Kadi H, Ellyn F (1994), Effect of stress ratio on the fatigue of unidirectional glass fibre/ epoxy composite laminae, Composites, 25(10), 917–924. Ellyn F, Asad Y (1991), Time-dependent fatigue failure: the creep-fatigue interaction, J Fatigue, 13(2), 157–164. Ellyn F, El-Kadi H (1990), A fatigue failure criterion for fiber-reinforced composite laminae, Comp Struct, 15, 61–74. Epaarachchi J A (2006), Effects of static-fatigue (tension) on the tension-tension fatigue life of a glass fibre reinforced plastic composites, Comp Struc, 74(4), 419–425. Epaarachchi J A, Clausen P D (2005), On predicting the cumulative fatigue damage in glass fibre reinforced plastic (GFRP) composites under step/spectrum loading, Composites A, 36(9), 1236–1245. Epaarachchi J A and Clausen P D (2003), A model for fatigue behavior prediction of glass fibre-reinforced plastic (GFRP) composites for various stress ratios and test frequencies, Composites A, 34, 313–326. Guedes R M (2004), An energy criterion to predict delayed failure of multi-directional polymer matrix composites based on a non-linear viscoelastic model, Composites A, 35, 559–571. Guedes R M (2006), Lifetime predictions of polymer matrix composites under constant or monotonic load, Composites A, 37, 703–715. Guedes R M (2007), Durability of polymer matrix composites: Viscoelastic effects on static and fatigue loading, Comp Sci Tech, 67, 2574–2583. Guedes R M (2008), Creep and fatigue lifetime prediction of polymer matrix composites based on simple cumulative damage lawas, Composites A, 39, 1716–1725. Haddad Y M (1995), Viscoelasticity of Engineering Materials, London, Chapman & Hall. Hertzberg R W, Manson J A (1980), Fatigue of Engineering Plastics, New York, Academic Press. Hwang W, Han K S (1989), Fatigue of composite materials – damage model and life prediction, ASTM STP 1012, 87–102. Kaminskii A A and Selivanov M F (2000), Long-term failure of layered viscoelastic composite material with a crack under time-dependent load, Mech Comp Mat, 36(4), 327–336. Mandell J F and Meier U (1983), Effects of stress ratio, frequency, and loading time on the tensile fatigue of glass-reinforced epoxy, ASTM STP 813, 55–77. Miyano Y, Nakada M, McMurray M K, Muki R (1997), Prediction of flexural fatigue strength of CRFP composites under arbitrary frequency, stress ratio and temperature. J Compos Mater, 31(6), 619–638. Miyano Y, Nakada M, Kudoh H, Muki R (2000), Prediction of tensile fatigue life for unidirectional CFRP. J Compos Mater, 34(7), 538–550. Miyano Y, Nakada M, Sekine N (2004), Accelerated testing for long-term durability of GFRP laminates for marine use. Composites B, 35, 497–502. Miyano Y, Nakada M, Sekine N (2005), Accelerated testing for long-term durability of FRP laminates for marine use. J Compos Mater, 39(1), 5–20. Owen M J, Bishop T (1974), Crack-growth relationships for glass-reinforced plastics and their application to design, J Appl Phy, 7, 1214–1224. Reifsnider K L, Schulte K, Duke J C (1982), Long-term fatigue behaviour of composite materials, ASTM symposium on high modulus fibres and their composites, ASTM 813, Philadelphia.

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Reifsnider K L (1991), Damage and damage mechanics, in Reifsnider K L, Fatigue of Composite Materials, Amsterdam, Elsevier, BV, pp. 11–77. Saff C R (1983), Effect of load frequency and lay-up on fatigue life of composites, ASTM STP 813, 564–595. Sauer J A, Richardson G C (1980), Fatigue of polymers, Intl J Fract, 16(6), 499–532. Schapery R A (1975), Deformation and failure analysis of viscoelastic composite materials, AMD-vol 13, ASME, 127–156. Sun C T, Chan W S (1979), Frequency effect on the fatigue life of a laminated composite, ASTM STP 674, 418–430. Yang J N, Jones D L (1983), Load sequence effects on graphite/epoxy [± 35]2s laminates, ASTM STP 813, 246–262. Yang J N, Jones D L (1980), Effect of load sequence on the statistical fatigue of composites, AIAA Journal, 18(12), 1525–1531.

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18 Characterization of viscoelasticity, viscoplasticity and damage in composites J. Varna , Lulea University of Technology, Sweden Abstract: Empirical material models for short fiber composites with inelastic stress–strain response and methodology for parameter determination in these models are analyzed. The main sources of inelastic behavior of short fiber composites are identified: (a) damage related reduction of thermo-mechanical properties; (b) nonlinear viscoplastic strains developing with time at high stresses; (c) nonlinear viscoelasticity. These phenomena are included in the material model. The necessary tests are tensile quasi-static loading-unloading tests and creep tests with following strain recovery after load removal. The methodology is demonstrated on short fiber composites: (a) SMC with glass fiber bundles; (b) flax/lignin composites; (c) paper/phenol-formaldehyde composites; (d) flax/ starch composites. Key words: fiber composites, stiffness reduction, viscoelasticity, viscoplasticity, creep.

18.1 Introduction Due to polymer resin properties and microdamage development, polymeric composites in general are inelastic materials. For unidirectional (UD) long fiber composites the stress–strain relationship in the longitudinal direction is mostly linear elastic with very limited amount of viscoelasticity, which is usually in the linear region, and almost zero viscoplasticity. In the transverse direction the behavior is mostly related to the matrix performance and therefore is expected to be time dependent and nonlinear. However, since the transverse strain to UD ply failure is low the specimen usually breaks before large viscoelasticity or viscoplasticity is observed. So, in shear loading the time-dependent phenomena are observed best. UD composites have been extensively studied and experimental methodology for linear viscoelastic and nonlinear viscoplastic material characterization was given by Megnis and Varna, 2003. The experimentally confirmed assumption that in UD composites the viscoelasticity is linear with respect to the stress level used in creep tests significantly simplified the testing routines for material model identification. In short fiber composites inelastic behavior is very usual. It is especially typical for composites based on biofibers and bioresins. They show a very nonlinear response to mechanical loading: stress–strain curves are nonlinear; in loading-unloading cycles hysteresis loops are typical (see Fig. 18.1a) 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. 514 © Woodhead Publishing Limited, 2011



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18.1  Mechanical behavior of flax/starch composite with 40w% fiber content at room temperature (RT) and relative humidity (RH) 34%: (a) hysteresis in high stress cycling; (b) elastic modulus determination in loading and unloading.

The main possible sources for this behavior are: (a) microdamage development with increasing load leading to elastic properties degradation; (b) viscoelastic effects which would lead to time dependence and loading rate dependence; (c) viscoplastic strains which develop with time and depend on the whole loading history. The viscoelasticity (as well as viscoplasticity) may be nonlinear with respect to stress. The most consistent and general material model for nonlinear viscoelastic materials was presented by Lou and Schapery, 1971 and Schapery, 1997 and this is the model used in this chapter. Since the presented data mostly correspond to uniaxial loading, the discussion of the model is also performed for this case. Nonlinear viscoplasticity is also included in Schapery’s model (Schapery, 1997). It accounts for viscoplasticity 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 viscoplastic strain, εVP(t, σ). The viscoplastic term was described by a nonlinear functional by Zapas and Crissman, 1984 and used by Tuttle et al., 1993. According to this model the viscoplastic strain developing in creep tests is a product of stress-dependent and time-dependent functions, both being power functions determined by master curves obtained in tests. This representation is used in the following. The constitutive equation was further modified to account for microdamage in Marklund et al., 2008. In this chapter the material model will be analyzed and efficient tests methodology will be suggested for complete material characterization to identify

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the stress-dependent parameters in the material model. It will be demonstrated 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 following strain recovery; (b) elastic properties reduction tests after loading to certain stress level.

18.2 Material model Building the material model we assume that the viscoelastic and viscoplastic responses may be decoupled. A general thermodynamically consistent theory of nonlinear viscoelastic materials developed by Lou and Schapery, 1971 and Schapery, 1997 is used. The test results used here are from uniaxial loading cases (in most of cases even Poisson’s effect related strains were not recorded) and the viscoelastic model contains three stress-dependent functions which characterize the nonlinearity. In addition a function d(εmax) with values dependent on the maximum (most damaging) strain/stress state experienced during the previous service life was incorporated to account for microdamage. The basic assumption of the material model is that strain decomposition is possible: the microdamage influenced viscoelastic strain response can be separated from viscoplastic response which is also affected by damage: [18.1] As demonstrated below the physical meaning of the damage function is the elastic compliance of the composite which is degrading due to experienced high stress/ strain. Certainly the form of interaction between d and viscoplastic strain (VP-strain) εVP used in Eq. 18.1 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 composite than in undamaged composite. It has to be noted that microdamage itself without any viscoplasticity can introduce irreversible strains (for example, the thermal stress release in layers of laminates due to intralaminar cracking leads to elongation of the whole laminate). The final form of the material model written for one-dimensional case is:

[18.2]

In Eq. 18.2 integration is over ‘reduced time’ introduced as:

[18.3]

ε0 represents the elastic strain which, generally speaking, may be nonlinear with respect to stress. ∆S(ψ) is the transient component of the linear viscoelastic creep compliance. g1 and g2 are stress-dependent material properties. aσ is the shift

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factor, which in fixed conditions is a function of stress only. Actually, these functions also depend on temperature and humidity, but under fixed environmental conditions they are functions of stress only. For sufficiently small stresses (linear region), g1 = g2 = aσ = 1, and thus Eq. 18.2 turns into the strain–stress relationship for linear viscoelastic nonlinear viscoplastic materials. It was shown by Schapery (1997) that the viscoelastic creep compliance can be written in the form of Prony series:

[18.4]

Ci are constants and τi are called retardation times. From Eq. 18.2 the elastic response of the damaged material is written as:

[18.5]

The physical meaning of the damage-related function d(εmax) can be revealed by comparing the elastic strains in damaged and undamaged composite at given stress σ : and



[18.6]

From here: [18.7] Obviously d(εmax) has the physical meaning of normalized elastic compliance – it is inverse to the normalized elastic tangent modulus dependence on the ‘worst’ experienced strain. In cases where large irreversible strains are present, the use of strain can be confusing and it may be more proper to present the damage function as dependent on the ‘worst’ stress. The optimal set of experiments needed to determine the stress-dependent functions in the material model and development of reliable methodology for data reduction is still a debatable issue. Most researchers agree that it is important to obtain as many characteristics as possible from one and the same ‘representative’ specimen. Since for the considered materials the properties variation is very large, averaging has to be performed, but it is better to average the final results than to use all the available data pool at once to define the model ‘in average’. The development of viscoplastic strains has been previously successfully described by a model presented in Zapas and Crissman, 1984, Marklund et al., 2008, Marklund et al., 2006, and Nordin and Varna, 2006. In this model:



[18.8]

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CVP, M and m are constants to be determined, τ = t/t* where t* is a characteristic time constant; for example, 3600 seconds, if the time interval of interest, is about one hour. Actually t* and C are introduced to have dimensionless strain expression. According to Eq. 18.8

[18.9]

A generalization of expression 18.8, if necessary, is possible by replacing σ(τ)M by a more general function f (σ (τ)) to be identified.

18.3 Microdamage effect on stiffness 18.3.1  Elastic modulus The elastic modulus of the composite may degrade as the result of microdamage developing at high stresses. Therefore initial elastic properties have to be determined in a relatively low strain region ε ∈ [ε1, ε2] where we expect that damage and irreversible phenomena will not develop. The most common method is to use linear fit to relevant stress–strain data in a loading-unloading cycle. For any of the short fiber composites analyzed in this chapter the maximum strain value in modulus determination was fixed: it was in the region 0.2 < ε2 < 0.3%, whereas ε1 was 0.05%. As demonstrated in Fig. 18.1b, for flax/starch composite even in this low strain region the loading and unloading curves, both being rather linear, may have different slopes. Hence three different values of the modulus may be determined (in loading, in unloading and the average of both). The composite in Fig. 18.1b has the loading modulus 4.48 GPa and the unloading modulus 4.71 GPa. Detailed description of this material and material properties is given in Sparnins et al., 2010a. If ε2 is too high for the given material, the unloading slope in elastic modulus measurements may be slightly lower than the loading slope due to damage accumulated in the loading part. However, in Fig. 18.1b 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. 18.1b, where strain reaches maximum value at the end of the region where the modulus (slope) is determined. Simple simulation of this type of loading ramp for linear viscoelastic materials shows that locally at the ‘turning point’ the unloading slope may be very high and definitely higher than the elastic modulus given by input data. 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 the experimental data in Fig. 18.1a. A practical suggestion based on these observations is to have the strain at the turning point at least 0.05% above ε2.

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Since none of these slopes represent the elastic response, the strain rate in the tests has to be increased or at least kept constant and the average of loading and unloading modulus used as an ‘apparent modulus’.

18.3.2  Stiffness reduction measurements The elastic modulus after loading to high strain/stress levels has to be measured to examine the significance of the modulus degradation and the amount of microdamage accumulation during the loading history. The loading ramp in this test is shown in Fig. 18.2 and the methodology is as follows. The test consists of blocks each containing a sequence of following steps: (a) loading to certain (high) strain level εi and unloading to ‘almost zero’ stress; (b) waiting at ‘almost zero’ stress for decay of all viscoelastic effects for a time at least five times longer than the length of the previous step (some irreversible (viscoplastic) strains called ε iresid in Fig. 18.2 can be present at the end of the ‘waiting time’); (c) the elastic modulus is determined applying high strain rate and low level loading-unloading;

18.2  Example of the applied strain steps during one block of the test.

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(d) recovery after modulus measurement for the time that is at least five times longer than the modulus measurement time interval. Then the same sequence can be repeated for 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 may be still present. Therefore it is not suitable to keep in the following modulus determination step the strain region defined in Section 18.3.1. Instead the strain region should be 0.05 to 0.2% in addition to the irreversible residual strain after each loading cycle, ε ∈ [ε1 + εiresid, ε2 + εiresid]. As an alternative, a fixed stress interval can be used for elastic properties determination. In that case the stress interval to use can be determined in the first elastic modulus test. It is the stress interval corresponding to ε ∈ [ε1, ε2] for an undamaged composite.

18.3.3 Microdamage and stiffness degradation in short fiber composites The main reason for elastic properties degradation in composites with increasing load is microcracks. They open due to loading (especially if the loading leads to tensile stress component transverse to the crack surface) thus 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 strain in the damaged composite is lower, which means that the elastic modulus is reduced due to damage (Lundmark and Varna, 2005). 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, fiber breaks, 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 optical microscopy observation, acoustic emission during failure events, edge replicas taken from loaded specimens, or X-rays (the last is possible only for damage which is linked to the specimen surface). The difficulty in the case of short fiber composites is that the variety of damage modes is much larger and even damage entities of the same mode may have very different sizes. Large differences in damage modes have been observed dependent on composite mesoscale architecture: in composites where fibers have bundle meso-structure (or just composites with fiber clusters) the damage entities are larger (comparable with the bundle size and larger) and more stiffness reduction than in dispersed fiber composites is observed before the final failure. If the dispersion is good, the size of defects at moderate loads is of the length scale of fiber diameter (for example, fiber/matrix debonds at fiber breaks). These damage entities have a very small opening during loading and therefore the stiffness reduction is marginal.

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For example, the in situ micrograph of poorly impregnated regions in a hemp/ lignin composite specimen shown in Marklund et al., 2008 reveals only a few well-defined microcracks. Instead there was a large amount of almost randomly oriented ‘line-like’ defects on the specimen edge but it seems that they are surface defects and they do not grow inside the composite (in a direction which is transverse to the surface of observation). Micrographs were taken on loaded specimens using the MIMIMAT testing machine of Polymer Laboratories and an optical microscope. It was observed that these defects were not opening with increasing tensile load and were not growing. Because of that their effect on elastic modulus is negligible. In contrast to many of these surface defects, only a few microcracks were found during in situ optical microscopy. They showed an increase of the opening with increasing loading transverse to the crack plane. Damage modes are different in composites with bundle structure of fibers. For example, SMC composites are made of E-glass fibers in polyester matrix and fibers which have an average fiber volume content of 20–22%. Fibers are in relatively small bundles of 180–400 fibers with volume fraction of fibers inside bundles about 50% (Hull, 1981, Kabelka et al., 1996). An edge view of a damaged SMC composite is shown in Fig. 18.3 (Oldenbo and Varna, 2005). In addition to fiber breaks in longitudinal bundle one can observe large intrabundle cracks in bundles with off-axis orientation with respect to the loading direction. These cracks run parallel to fibers and are roughly transverse to the composite plane. Often a bundle works as a thick fiber and debonding of the bundle from the matrix can be observed. At large loads large matrix cracks appear transverse to the load. They are bridged by partially debonded longitudinal and off-axis bundles. The described damage modes as shown in Fig. 18.4 lead to much larger stiffness reduction than damage modes related to disperse fibers. The presented examples show that in short fiber composites (a) microdamage quantification is difficult and (b) models for stiffness prediction would be very complex.

18.3  Micrograph of an edge of a damaged SMC specimen (adapted from Oldenbo and Varna, 2005).

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18.4  Elastic modulus reduction in SMC composite after loading to high strain.

Hence, evaluation of the damage state by back-calculation from the level of elastic properties reduction seems to be more practical than microscopy-based microdamage quantification. In this chapter, where the constitutive models are one-dimensional and empirical, the damage effect can be represented by one scalar parameter accounting in an integrated way for all damage modes. In more general cases, certainly, tensorial description of damage entities is necessary. The above discussion on the link between composite micro-meso structure and the microdamage modes and their development has strong experimental evidence. For example composite made from phenol-formaldehyde impregnated kraftliner paper (with good dispersion of fibers) (Nordin and Varna, 2005), in tensile tests shows nonlinear behavior and hysteresis loops if the loading is slow. As one possible cause of inelasticity, stiffness degradation in tension was studied by comparing the initial modulus with modulus after loading specimen to 100 MPa (Nordin and Varna, 2005). The same was done in compression tests (Nordin and Varna, 2006) by loading the material up to a maximum stress level of 80 MPa and then after unloading measuring Young’s modulus in the strain region 0.1–0.3%. However, the results of these tests, presented in Table 18.1, do not indicate any stiffness reduction and, hence, damage-related stiffness reduction could be excluded from consideration in the constitutive model (Nordin and Varna, 2005, 2006). The elastic properties reduction due to damage development in SMC composite with bundle structure (Oldenbo and Varna, 2005) shown in Fig. 18.3 is considerable. The elastic modulus normalized with respect to the initial modulus

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Table 18.1  Stiffness of composites before and after tension and compression to high stress Material Initial modulus (GPa)

Modulus after 100 MPa Modulus after 80 MPa tension (GPa) compression (GPa)

L-direction T-direction T-direction

20.7 10.7 –

20.6 10.8 10.9

– – 10.8

as a function of the maximum value of the previously applied strain level is shown in Fig. 18.4. The reduction starts at about 0.3% strain and at 1% strain may reach 25–30%. Another example of stiffness degradation is shown in Fig 18.5, where elastic modulus change in flax/starch composites with two different fiber contents (20 and 40% weight fraction) is compared (Sparnins et al., 2010a). The test was performed at room temperature and relative humidity RH = 34%. Certainly, temperature and moisture which makes the matrix more ductile may affect the microdamage development, shifting it to larger strains. The elastic modulus normalized with respect to its initial value is presented in Fig. 18.5. The elastic modulus of the Wf = 20% composite is not changing (the slight increase should be considered as an artifact), whereas for Wf = 40% composite the dependence on the applied strain is rather smooth and at the end the modulus is reduced by almost 20%. In the Wf = 20% composite the fibers are well separated and the damage starts with small fiber/matrix debonds which have very small effect on modulus. For Wf = 40% composite fibers and as a consequence also debonds are closer to each other and are able to coalesce with increasing load. Increasing probability of large fiber cluster formation increases the probability of appearance of large (bundle size) cracks. It has to be noted that the final strain values applied in this test were rather close to the maximum stress position in the stress–strain curve and therefore the strain in Fig. 18.5a contains elastic as well as large viscoelastic and viscoplastic components. The stress in this region is not changing much and therefore the elastic modulus reduction with stress presented in Fig. 18.5b at the end of the test is much faster. It should be noted that the last modulus measurement in Fig. 18.5b is after reaching the maximum stress level in the stress–strain curve for the specimen. In the high-stress region of the stress–strain curves a strong interaction between viscoelastic, viscoplastic and damage mechanisms may be expected and the decomposition assumed in this work may not be valid. For this material group the modulus reduction is relatively small and cannot be the main reason for the large nonlinearity of stress–strain curves shown in Fig. 18.1a. Hence, in the first approximation the damage term was neglected in the constitutive model (Sparnins et al., 2010b).

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18.5  Elastic modulus reduction in flax/starch composites due to presumed microdamage development with increasing strain and stress: (a) strain dependence; (b) stress dependence.

In contrast, for the hemp/lignin composite (Marklund et al., 2008) the stiffness reduction, even if it is moderate as shown in Fig. 18.6, was included in the material model. The initial modulus E0 was determined from the first loading step with strains growing to 0.20%. The initial Young’s modulus of this composite is in the range 2.3–2.8 GPa. Tensile strength is 14–16 MPa. Figure 18.6 shows the reduction of the modulus normalized with respect to the initial modulus. No stiffness degradation could be seen for strain values lower than 0.3% and consequently this strain level was set as the threshold for onset of © Woodhead Publishing Limited, 2011



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18.6  Normalized elastic modulus of hemp/lignin composite.

stiffness reduction. At the strain level of 0.9% the stiffness reduction was roughly 7–8%. The regression line in Fig. 18.6 is used to define the damage function required in Eq. 18.2, which depends on the ‘worst’ strain-state during the previous loading.

[18.10]

In monotonously increasing loading the ‘worst’ is always the current strain state.

18.4 Viscoplasticity 18.4.1  Viscoplasticity modeling in creep test The development of viscoplastic strains (VP-strain) has been previously successfully described by a functional presented by Zapas and Crissman, 1984 (and used in Sparnins et al., 2010a, Marklund et al., 2008, Nordin and Varna, 2006, and Marklund et al., 2006). Assuming that the law in Eq. 18.8 for VP-strain development is valid we can design test procedures needed for parameter CVP, M and m determination. Certainly, the validity of Eq. 18.8 has to be checked with experimental data. First we consider a sequence of creep tests at fixed stress σ (t) = σ0. In this case the integration in Eq.18.8 is trivial and the VP-strain accumulated during the time interval t ∈ [0; t1] is: © Woodhead Publishing Limited, 2011

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If the creep test is performed for a longer time interval t1 + t2 the same rule applies and the accumulated viscoplastic strain will be:

[18.12]

According to Eq. 18.8 the interruption of the constant stress test for an arbitrary time ta has no effect on VP-strain development: since stress during the unloaded state is zero, only the total time under loading is of importance:

[18.13] As a consequence, instead of testing at stress σ0 for time tΣ = t1 + t2 continuously, one could perform the testing in two steps: (1) creep at stress σ0 for time t1; unloading the specimen and measuring the permanent strain ε 1VP developed during this step (since during the creep test viscoelastic strains are also developing simultaneously with VP-strains, the VP-strains cannot be measured directly after the load removal. It has to be done after the recovery of viscoelastic strains); (2) now the stress σ0 is applied again for interval t2; after strain recovery measuring the new VP-strain ε 2VP developed during the second test of length t2. The total VP-strain after these two creep tests is found by addition and is denoted as:

[18.14]

Here and in the following the upper index Σ (k) is used to indicate that the specimen k is used to denote VP-strain has been subjected to k creep loading tests. Symbol ε VP developed in the k-th creep test of length tk. According to Eq. 18.12 and 18.13 the sum of two VP-strains corresponding to two tests of length t1 and t2 will be equal to the VP-strain that would develop in one creep experiment with the length tΣ = t1 + t2 at the same stress:

[18.15]

The assumption that the creep test interruption during the strain recovery period does not affect the VP-strain development (hypothesis of additivity) has to be checked experimentally for each material. It is an important step in validation of the VP-strain model (Eq. 18.8). Generalizing the above discussion, the summary VP-strain after k creep tests with total length tΣ k = t1 + t2 + . . . tk at the fixed stress level σ0 will be independent on the prolongation of interruptions between them and can be written as:

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[18.16]

From Eq. 18.16, the new VP-strain stress is σ0 is:

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developed during the k+1 creep test at

[18.17] From the above discussion it follows that the time dependence of VP-strains at fixed stress level can be determined by performing a sequence of creep-strain recovery tests on the same specimen. After strain recovery in each test the new irreversible strain corresponding to the loading period is measured and identified as the new-developed VP-strain. Several creep tests with different lengths are performed and the developed VP-strains are summed and presented as a function of the total creep time. According to the theory, the VP-strain after k steps of creep loading at the same stress level σ0 is given by Eq. 18.16. Hence the time dependence of VP-strains should be fitted (if the model is applicable) with a power function with respect to time:

[18.18]

The constant m is determined as the best fit by power function to test data (standard option in Excel). According to the model Eq. 18.8, the exponent m has to be independent of the stress level used in creep test (it has to be proved experimentally). Hence, only a few representative specimens are necessary to obtain the time dependence of VP-strains and to determine m. As it follows from Eq. 18.11, the stress dependence of VP-strains in creep tests also has to obey the power law. Performing creep tests of fixed length tσ at several stress levels σ on specimens without any previous loading history and measuring the VP-strains after strain recovery we can plot the VP-strain εVP(tσ) versus stress. These data have to be fitted with the theoretical relationship for this loading case:

[18.19]

From the fitting procedure M and CVP are obtained.

18.4.2  Experimental procedure The test program to identify the time dependence and stress dependence of viscoplastic (irreversible) strains is based on the discussion in Section 18.4.1 and is as follows. Tensile creep tests have to be performed, for example using a creep

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rig with dead weights and measuring (using extensometer or strain gauges) and recording the development of strains during the creep and during the following period of strain recovery. Compressive creep tests are usually performed using hydraulic, electronically steered testing machines. To identify the time dependence of VP-strains a sequence of steps is selected at fixed level of stress, each consisting of creep and strain recovery. The test sequence performed on the same specimen is schematically shown in Fig. 18.7. Since often the VP-strain rate decreases with time, the length of the creep loading time tk in each following step can be increased (proportionally increasing also the strain recovery time). The time scale depends on the time region to be covered. For example, in Giannadakis et al., 2010, t1 = 5, t2 = 15, t3 = 30, and t4 = 60 min were used. The strain recovery time after the load application step has to be at least five to eight times the length of the loading step. The whole creep-recovery curve can be recorded but, actually, for viscoplasticity only the final recovery strain value is of importance: the developed VP-strain is defined as the remaining strain at the end of the strain recovery. As explained above, according to the model the VP-strains are not developing during the strain recovery part (zero applied load) and, when the same load is applied again in the next step, the viscoplastic process continues without any effect of the test interruption time interval. An example of the VP-strain development in recycled CF/recycled PP matrix composite (Giannadakis et al., 2010) is shown in Fig. 18.8. Four data points for the specimen were obtained from Fig. 18.8 to establish the VP-strain dependence on time for the used stress level. This experiment has to be performed for several stress levels (using other specimens) to make sure that the shape of the time dependence is not changing and if it is, how the parameters depend on the stress level. Often the VP-strain developed during the first loading step is larger than the VP-strain created during the following step. It can be seen in Fig. 18.8 that even if the second loading step is significantly longer than the first the additional VP-strain during this step is smaller. For materials with a decreasing rate of VP-strain development, this feature can be used to ‘condition’ specimens before viscoelastic analysis: first several high-stress creep steps are performed until the

18.7  The sequence of creep and recovery steps to identify the time dependence of VP-strains.

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18.8  Viscoplastic strain accumulation in recycled carbon fiber/recycled polypropylene specimen 5 during the sequence of creep (16 MPa) and strain recovery tests.

VP-strain rate slows down. Then the same specimen is used in viscoelasticity tests, preferably at lower stress than during conditioning. It is expected that the viscoplasticity during the test will be negligible. Certainly, one has to be very careful with the conditioning procedure to make sure that damage is not introduced and stiffness is not degraded.

18.4.3  Experimental results for short fiber composites First we consider composite laminates made of phenol-formaldehyde impregnated kraftliner paper investigated in Nordin and Varna, 2006. The development of viscoplastic strains in compression was performed as described in Section 18.4.2. The time dependence of VP-strains was studied at 80 MPa. As shown in Table 18.1, elastic modulus degradation was not observed at this stress level. For creep loading intervals 15 min 1 h, 2 h and 2 h the VP-strains are shown in Fig. 18.9. The slightly lower data point at 18 000 sec is for a different specimen loaded continuously for 5 h without interruptions (no unloading and strain recovery after 15 min, 1 h etc). The results are very consistent showing that summation of VP-strains as described in section 4.1 is valid and that strain recovery intervals do not add anything to the accumulated VP strain. The small difference between two data points at t = 18 000 sec is just a variation between specimens, which may be rather large. As follows from Fig. 18.9a, fitting of these data with power function (Excel option) is good, showing that Eq. 18.18 for this composite is applicable. The value © Woodhead Publishing Limited, 2011

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18.9  Viscoplastic strains in paper/phenol-formaldehyde composite: (a) time dependence; (b) stress dependence.

m = 0.323 was determined as a result of fitting. To obtain stress dependence of VPstrains, 5h creep tests with following strain recovery were performed at several stress levels. Only one specimen was used for each stress level. The dependence of VP-strains on stress is presented in Fig. 18.9b. The relationship can be fitted by power function as a trendline, confirming the stress dependence given by Eq. 18.19. From Fig. 18.9b we determine M · n = 3.03. Using results in Fig. 18.9b also, the value of CVP can be determined, thus finishing the parameter identification in the viscoplasticity model (Eq. 18.8). Finally the value of M = 9.357 was calculated. After identification of these parameters the VP-strain model was validated in a test where two specimens (T7 and T8 in Fig. 18.10) were subjected to a sequence of five creep and strain recovery steps of different length and at different stresses. The stress levels and time intervals were increased from step to step: Step 1: σ10 = 25 MPa, t1 = 5h Step 2: σ20 = 50 MPa, t2 = 5h Step 3: σ30 = 50 MPa, t3 = 3h Step 4: σ40 = 65 MPa, t4 = 5h Step 5: σ50 = 80 MPa, t5 = 5h Viscoplastic strains accumulating in each loading step were measured as irreversible strains after strain recovery. The results (summary VP-strain after k loading steps) for two specimens (T7 and T8) are shown in Fig. 18.10. Simulation of this loading sequence using the VP-strain law (Eq. 18.8) with the identified viscoplastic constants was also performed. The value of the total VP-strain after the k-th step according to the model is:

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18.10  Viscoplastic strain development in a sequence of five creep tests in paper/phenol-formaldehyde composite (adapted from Nordin and Varna, 2006).



[18.20]

Predictions are also presented in Fig. 18.10 and they are in excellent agreement with the test data. The scatter between two specimens is about 10% and the predictions have a comparable accuracy. The agreement is better than expected: the power law for time dependence was established using data at 80 MPa only and the stress dependence was obtained using data for five-hour tests whereas now the predictions cover a total creep test length of 23 hours. In the next example we analyze viscoplasticity of hemp fiber/lignin matrix composite. The material and its behavior are described in detail in Marklund et al., 2008. The open symbols (three for each stress level) in Fig. 18.11 are experimental points at 9 and 10 MPa stress respectively. Solid lines are the best fit to data by power function leading to two different values of m = 0.4 for 9 MPa and m = 0.57 for 10 Mpa. However according to the theoretical model m cannot be dependent on stress level. Therefore a new attempt was made to find one m value that would fit data sets for both stress levels. Certainly the quality of the fitting is reduced but the model is built to be consistent with the theory. The best fit for both experimental data sets taken together is obtained when m = 0.49. As one can see in Fig. 18.11 the fitting for 9 MPa is very good and for 10 MPa it is with an acceptable accuracy. One can conclude that even if in reality the exponent in the power function slightly increases with stress, the fitting is still good if this dependence is ignored. The benefit of using one stress independent value is that the viscoplasticity theory

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given in this chapter is applicable. From the fitting procedure at different stress levels values of M and CVP are also found. Finally we present some viscoplasticity data for flax fiber/starch composites analyzed in Sparnins et al., 2010b. The viscoplastic behavior of this composite is rather different: as shown in Fig. 18.12 the dependence on time is linear (m = 1).

18.11  Viscoplastic strain development with time in hemp/lignin composites. Experimental data (open symbols) at 9 MPa and 10 MPa stress and fitting by power functions (fitting with one exponent is shown as symbols).

18.12  Development of viscoplastic strains in flax/starch composites: (a) with increasing time the VP-strains are larger; (b) with increasing stress the slope is getting steeper.

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This means that the specimen ‘conditioning’ before viscoelastic testing is not possible because the development of VP-strain does not decay with time. An advantage is that the VP-model in this particular case is much simpler and easier to implement in calculation schemes. As will be shown in Section 18.5.2, the linearity simplifies also the nonlinear viscoelastic characterization. The VP-strains are also much larger than in two previously discussed composites. The dependence of VP-strains on stress is shown in Fig. 18.12b for several values of test length: it follows power law.

18.5 Nonlinear viscoelasticity 18.5.1  Viscoelasticity in creep and strain recovery test In a creep test the stress is applied at t = 0 and is kept constant until some time instant t1 whereby the stress is removed and the strain recovery period begins according to σ = σ [H(t) – H(t – t1)] where H(t) is the Heaviside step function. The material model (Eq. 18.2) may therefore be applied separately to the creep interval t ∈ [0, t1] and to the strain recovery strain in a creep test t > t1. We obtain the following expressions for creep strain εcreep and recovery strain εrec respectively: [18.21] The recovery strain after removing the load is: [18.22] where:

[18.23]

Using the Prony series (Eq. 18.4) for creep compliance in Eq. 18.21 we obtain the following expression to describe the strain development during the creep test:

[18.24]

Substituting the compliance in Eq. 18.4 in Eq. 18.212 to obtain strain expression during the recovery interval, we have:

[18.25]

Expressions 18.24 and 18.25 have to be used to fit the experimental creep and strain recovery data. Constants Ci i = l,. . . .I, (stress independent) and ε0, aσ , g1

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and g2 (stress dependent) are found as a result of fitting. The retardation times τi in 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 ten between them. Optimization of the τi selection can be made comparing the accuracy of the obtained fit for different selections using least square method as described below. Even if it is not a necessary condition, the additional requirement Ci > 0 usually leads to improved fitting of test results. If stiffness reduction effect can be ignored then Eq. 18.24 and 18.25 are reduced to:

[18.26]



[18.27]

The stress- and time-dependent VP-strains enter Eq. 18.24 and 18.25. The term εVP (σ, t1) is the VP-strain developed during the current 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. 18.24) the time dependence of the VP-strain during this test is required. If it were known, then it could be subtracted from the total strain during the creep test with the rest being a pure nonlinear viscoelastic strain. Generally speaking there are three alternatives to account for viscoplasticity when viscoelasticity is analyzed using Eq. 18.24 and 18.25: (a) using low stress levels where viscoplasticity is almost not present (then the viscoelasticity is usually also linear and this method is not applicable in the nonlinear viscoelasticity region); (b) performing specimen ‘conditioning’ by subjecting it first to high stress creep. If the viscoplasticity development rate decays with time (as it does for many materials: Megnis and Varna, 2003, Marklund et al., 2008, Nordin and Varna, 2006, Marklund et al., 2006) and damage does not develop during this test, then this specimen after conditioning can be used for nonlinear viscoelasticity analysis performing creep tests at any level below the conditioning stress level. The viscoplastic strain development in these tests 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) viscoplasticity will be very small. It can be checked from the curves after the strain recovery: the irreversible strain should be small; (c) if the viscoplasticity is understood, described by equations and the previous loading history for the given specimen is known, the time-dependent viscoplastic strain during the current creep test can be calculated theoretically and subtracted from

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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 below. 1. The damage related function d(εmax) is found as described in Section 18.3. 2. Viscoplastic strains (if present) are subtracted from creep strain using expression A6. 3. Creep and strain recovery data in low stress region (expected linear response) where g1 = g2 = aσ = 1 are used to determine Ci by fitting simultaneously the reduced creep and strain recovery data (method of least squares (LSQ)) using expressions 18.26 and 18.27. Retardation times τi are selected rather arbitrarily with about one decade step and adjusted to ensure that all Ci are positive. 4. Data at higher stress (possible nonlinear viscoelastic region) are fitted by Eq. 18.24 and 18.25 using previously obtained Cm, τm. 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 that gives the minimum of the misfit function is considered as the correct set. Certainly other available numerical nonlinear minimization routines may be more efficient. 5. The procedure described in point 4 have to be repeated for all available stress levels to obtain the stress dependence of aσ, g1, g2 and ε0.

18.5.2  Examples of nonlinear viscoelastic behavior Creep and strain recovery tests have to be performed at different stress levels and time intervals, which is very time consuming. It is therefore crucial for the characterization that the specimen is representative for the analyzed material. The specimen could be singled out on the basis that its elastic properties were intermediate in this group of specimens. It can also be suggested that each specimen in the nonlinear viscoelastic analysis is analyzed separately, 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. For hemp/lignin composite (Marklund et al., 2008) the elastic modulus reduction is given in Fig. 18.6 and viscoplastic strains are presented in Fig. 18.11. The values of the stress-dependent functions in the nonlinear viscolelastic model were found for each stress level as described in Section 18.5.1 and the results are presented in Fig. 18.13.

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An interesting observation is that g1 is almost not changing with stress (it is slightly increasing), whereas aσ and g2 are increasing. Linearity of viscoelasticity holds until 3 MPa only. As a final validation, the developed material model was used to simulate the composite strain response in stress controlled test with low stress rate. The stress was increased and then reduced and then increased to failure again with a constant stress rate. Simulation results together with experimental curves for two specimens are shown in Fig. 18.14. The good agreement confirms that the developed material model is adequate for this material.

18.13  Stress-dependent functions in Schapery’s model for hemp/lignin composite.

18.14  Simulated and experimental loading-unloading curves for hemp/ lignin composite.

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In some cases however, the stiffness reduction could be neglected in the constitutive model, for example for the paper/phenol-formaldehyde composite (in both tension and compression; Nordin and Varna, 2005, 2006) and for the hemp/ starch composite in tension (Sparnins et al., 2010b). For flax/starch composite the subtraction of VP-strains (step 2 in Section 18.5.1) is simple because Eq. A6 is significantly simplified in a particular case when m = 1, which appears to be the case for the analyzed material; see Section 18.4, Fig. 18.12(a). The time dependence of VP-strains is given by

[18.28]

In this case we need only the ‘new’ developed VP-strain value after the current c which comes from strain recovery data, the time instant t when the test test ε VP k starts and the time instant tk+1 when the load is removed. The viscoelastic compliance, defined as viscoelastic strain divided by stress level in creep test, is shown for several stress levels in Fig. 18.15. The material is getting more and more compliant with increasing stress in the creep test. These curves and the corresponding strain recovery curves were used to determine the stress dependent functions in the viscoelastic material model. The shift factor aσ (see Fig. 18.16), is increasing much more than for the hemp/ lignin composite shown in Fig. 18.13. The dependence is fitted by linear function (ignoring the very low and unexplainable value at 11 MPa) but nonlinear fit could possibly be more appropriate.

18.15  Stress-dependent viscoelastic creep compliance of flax/starch composite in tension.

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18.16  Shift factor aσ dependence on stress for flax/starch composite.

18.17  Parameters g1 and g2 for flax/starch composite with Wf = 40%. Symbols: calculation results; solid line: approximation with the power function.

The nonlinearity functions g1 and g2 are shown in Fig. 18.17. The trends are as for the hemp/lignin composite in Fig. 18.13: parameter g2 is increasing whereas g1 is almost not changing. However, the magnitude of the g2 change is much larger for flax/starch composite. Parameter g1 is slightly decreasing in contrast to hemp/lignin where it was slightly increasing. The elastic response (see Fig. 18.18) obtained from fitting the viscoelastic strain in the considered stress region is rather © Woodhead Publishing Limited, 2011



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18.18  Initial strain for flax/starch composite obtained at different stresses. Solid line: linear fit to the data.

linear which justifies the assumption used for this material that elastic modulus reduction is not significantly affecting the composite behavior.

18.6 Conclusions In this chapter the main sources of inelastic behavior of short fiber composites were analyzed. They are identified as: (a) damage-related reduction of thermomechanical properties; (b) nonlinear VP-strains developing with time at high stresses; (c) nonlinear viscoelasticity. A methodology was suggested and validated in experiments for determination of parameters and stress-dependent functions in the material model. The necessary tests are tensile quasi-static loading-unloading tests and creep tests with following strain recovery after load removal. The methodology was demonstrated on the following short fiber composites: (a) SMC with glass fiber bundles; (b) flax/lignin composites; (c) paper/phenolformaldehyde composites; (d) flax/starch composites. It was shown that for these materials viscoelasticity and viscoplasticity have more effect on the macroscopic inelastic behavior than microdamage accumulation.

18.7 References Giannadakis K, Szpieg M and J Varna (2010), Mechanical performance of recycled Carbon Fibre/PP, Experimental Mechanics, submitted. Hull D (1981), An introduction to composite materials, Cambridge solid state science.

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Kabelka J, Hoffman L, Ehrenstein GW (1996), Damage process modeling in SMC, J. Appl. Pol. Sci, 62, 181–198. Lou YC, Schapery RA (1971), Viscoelastic characterization of a nonlinear fiber-reinforced plastic, Journal of Composite Materials, 5, 208–234. Lundmark, P, Varna J (2005), Constitutive relationships for damaged laminate in in-plane loading, Int. J. Dam. Mech., 14(3), 235–259. Marklund E, Eitzenberger J, Varna J (2008), Nonlinear viscoelastic viscoplastic material model including stiffness degradation for hemp/lignin composites, Composites science and Technology, 68, 2156–2162. Marklund E, Varna J, Wallström L (2006), Nonlinear viscoelasticity and viscoplasticity of flax/polypropylene composites, JEMT, 128, 527–536. Megnis M, Varna J (2003), ‘Nonlinear Viscoelastic, viscoplastic characterization of unidirectional GF/EP composite’, Mechanics of Time-dependent Materials, 7, 269–290. Nordin L-O, Varna J (2005), ‘Nonlinear viscoelastic behavior of paper fiber composites’, Composites Science and Technology, 65, 1609–1625. Nordin L-O, Varna J (2006), ‘Nonlinear viscoplastic and nonlinear viscoelastic material model for paper fiber composites in compression’, Composites Part A, 37, 344–355. Oldenbo M, Varna J (2005), ‘A constitutive model for nonlinear behavior of SMC accounting for linear viscoelasticity and microdamage’, Polymer Composites, 84–97. Schapery RA (1997), ‘Nonlinear viscoelastic and viscoplastic constitutive equations based on thermodynamics’, Mechanics of Time-Dependent Materials, 1, 209–240. Sparnins E, Pupurs A, Varna J, Joffe R, Nattinen K, Lampinen J (2010a), ‘The moisture and temperature effect on mechanical performance of flax/starch composites in quasi-static tension’, Polymer Composites, submitted. Sparnins E, Varna J, Joffe R, Nattinen K, Lampinen J (2010b), ‘Time-dependent behavior of flax/starch composites’, Mechanics of Time-Dependent Materials, submitted. Tuttle ME, Pasricha A, Emery AF (1993), ‘Time-temperature behavior of IM7/5260 composites subjected to cyclic loads and temperatures’, Mechanics of Composite Materials: Nonlinear Effects, AMD, 159, ASME, 343–357. Zapas LJ, Crissman JM (1984), ‘Creep and recovery behavior of ultra-high molecular weight polyethylene in the region of small uniaxial deformations’, Polymer, 25(1), 57–62.

18.8 Appendix: time-dependence of VP-strain in a creep test VP-strains develop on a similar timescale and at similar stresses to 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 VP-strain development depends on the previous loading history the approach is as described below. We consider the current creep test at stress σk+1 in time interval t ∈ [tk, tk+1] (the k + 1 creep test). 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 loaded condition. The VP-strains can be correctly subtracted only if the previous loading history in terms of accumulated VP-strains for this specimen is known and accounted for. The previous loading in our case

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consists of k creep tests with different length and stress levels. We assume that the ‘history’ is all that happened over the loading time in tk = tΣ(k). The viscoplasticity developed during the ‘history’ we denote as (for example it can be the sum of VP-strains developed in all earlier creep tests). According to Eq. 18.8 the further development of the total VP-strain during the current creep test is described by:

[A1]

The ‘new’ VP-strain developing in the current step is:

[A2]

The upper index in the first term indicates that this is new VP-strain developing during the k + 1 creep test. The first term in Eq. A1 is equal to

and therefore:

 t > tk [A3] Using Eq. A2:

[A4]

Expression A4 can be used directly for VP-strain in the k + 1 creep test if 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 viscoplasticity for a particular specimen in a particular creep test. Much more accurate is 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 below. If from experiment (strain recovery) we know the ‘new’ , we can use it in Eq. VP-strain at the end of the current creep step, A4 to express:

[A5]

Substituting it back into Eq. A4 we obtain the time-dependence of VP-strain in the current step to be used to subtract from the creep strain before performing viscoelastic analysis:

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[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 VP-strain during previous c . loading steps, εΣVP(k); (b) the ‘new’ VP-strain during the current strain ε VP During the first step ε ΣVP(0) = 0, t0 = 0 and Eq. A6 simplifies:

[A7]

In the particular case when m = 1 Eq. A6 simplifies:

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[A8]

19 Structural health monitoring of composite structures for durability S. Alampalli, New York State Department of Transportation, USA Abstract: During the last decade, fiber-reinforced polymer composite materials have been increasingly considered by infrastructure owners for extending the life of existing structures and increasing the durability of new structures due to the advantages they offer over conventional materials. Considering the wide variability of application, this chapter focuses on application of composites in a relatively new arena, i.e., bridge structure applications. As owners start considering the life-cycle costs for effective bridge management, these materials have potential for increased use in the future. At the same time, structural health monitoring (SHM) is emerging as another tool that bridge owners are more and more comfortable using to make decisions related to bridge management activities. Given that composites are relatively new to the bridge industry, SHM has great potential for use in a complementary fashion with composite materials. This chapter briefly describes this complementary relationship with case studies. Key words: structural health monitoring (SHM), composites, bridge engineering, fiber-reinforced polymers (FRP), bridge durability.

19.1 Introduction With increased use and constrained resources, the transportation industry is facing severe challenges due to rapidly deteriorating infrastructure. Due to an emphasis on uninterrupted mobility and high reliability from the transportation infrastructure, reduction in the 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 costeffective construction methodologies, and new design procedures are under consideration by bridge owners. Composite materials are increasingly becoming popular due to the advantages they offer that include their light weight, better corrosion resistance, shop fabrication capabilities, and high strength. During the last ten years, several bridge superstructures and decks were built around the world (Alampalli et al. 2002; Triandafilou and O’Connor 2009). There has also been an increased use of external wrapping to protect bridge components for strengthening and protection from adverse environments (Hag-Elsafi et al. 2002; Sen 2003). Considering these materials are still new to the industry and the harsh in-service environments bridge structures are subjected to, more data on their in-service structural behavior, durability, maintainability, and serviceability are 543 © Woodhead Publishing Limited, 2011

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required to use them appropriately and cost-effectively. This chapter will explore the use of structural health monitoring of composite materials in bridge structural applications. The next two sections of this chapter will discuss the use of FRP structures in bridge applications and structural health monitoring (SHM) issues. The fourth section discusses how SHM can benefit understanding of FRP structures, help improve durability, and assist in implementation issues. This section will be followed by case studies to further illustrate the use of SHM in FRP applications. Finally, the last section will present some recommendations for future research and practice.

19.2 FRP structures in the bridge industry Fiber-reinforced polymer (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 industry due to their light weight and non-corrosive properties. Compared to these industries, 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 light weight 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 lightweight characteristics. These included higher strengths, non-corrosive properties, engineerable characteristics, water-resistance, easy transportation, shop fabrication, ease of erection, and perceived long-term durability. Capabilities to shop fabricate these components, coupled with short erection times compared to conventional materials, are an attractive feature to prevent long traffic interruptions and costly work zone control required during the construction. In general, use of FRP materials in bridge applications can be broadly divided into four areas: superstructures/decks (see Fig. 19.1–19.5), external reinforcement/wrapping for strengthening (bond-critical) applications (see Fig. 19.6), maintenance/temporary (non-bond-critical) applications (see Fig. 19.7 and 19.8), and internal reinforcement (see Fig. 19.9). Each application differs significantly in both structural and non-structural requirements and thus requires very different structural and material characteristics, workmanship, quality control and quality assurance methods, and inspection requirements. For example, fatigue properties are extremely important for a composite superstructure or deck, whereas creep is more important when used as an internal reinforcement in prestressed

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19.1  An FRP slab bridge during construction: Bennetts Creek Bridge in the USA.

19.2  Bennetts Creek Bridge after construction during the proof load testing.

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; Triandafilou and O’Connor 2009). There have also been hundreds of column/beam wrappings,

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19.3  An FRP slab bridge after construction: Troups Creek Bridge in the USA.

19.4  An FRP bridge deck during construction: Bentley Creek Bridge in the USA.

mostly for maintenance applications with a few for strengthening (Hag-Elsafi 2002; Sen 2003). Internal reinforcement is also used in several applications, but is still not yet used widely (e.g. Chen et al. 2008). Standard specifications are emerging in recent years, but are still in their infancy due to the limited long-term

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19.5  An FRP bridge deck after construction: Schroon River Bridge in the USA.

data available on their behavior in the harsh in-service conditions in which these bridge structures operate.

19.3 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 costs 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, occupants’ 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

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19.6  External FRP reinforcement for strengthening application.

19.7  External reinforcement for maintenance application: during construction.

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19.8  Columns and pier-cap wrapped with CFRP: Everett Road, New York in the USA.

19.9  Bridge deck with FRP rebars (courtesy of Dr GangaRao, West Virginia University).

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loads it is intended to carry with uninterrupted mobility during operational hours. In order to ensure required safety 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 that 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 of these reasons 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, 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, 2011b) 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, ‘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. 19.10). Full treatment of the subject with detailed applications can be found in Ettouney and Alampalli (2011a, 2011b).

19.10  Structural health in civil engineering concept.

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19.4 FRP structures and SHM As noted earlier, even though there have been numerous applications of FRP materials in infrastructure and bridge applications, a majority of these applications were funded by special grants or research funding on an experimental basis and are still considered by many as relatively new to civil engineering. There has been considerable research on their behavior through analysis, load testing, and monitoring (Alampalli 2006; Reising et al. 2004; Farhey 2005; Reay and Pantelides 2006). 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: • 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 training materials, • 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. The behavior of FRP materials for mechanical and environmental demands has been extensively researched in the aerospace and automotive industries and is relatively well understood. But the knowledge gained cannot be used directly in civil applications due to the following differences: • 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 two years. Most inspections are still visual based with 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.

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• Corrective and preventive maintenance plans are also very different. In civil applications, most of these plans are reactive not pro-active. 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 conventional materials and thus require pro-active 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 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 the above complexities and differences with 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 costeffective 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 following areas for 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.

19.5 Case studies The New York State Department of Transportation has used FRP materials for several applications in the last two decades. Realizing that FRP structures and the structures retrofitted with these materials should be monitored to ensure their adequate in-service 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, 2005b; Alampalli 2005, 2006; Alnahhal et al. 2007). 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.

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19.5.1 Bridge deck One of the primary applications of the FRP decks has been its 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 (Dutta 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 this needs careful evaluation. This case study gives one such example where SHM that includes 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). 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. 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 State in the United States. The floor system was made up of steel transverse floor-beams at 4.27 m center-tocenter spacing with longitudinal steel stringers. It was originally built as a single simple-span, steel truss bridge with a reinforced concrete slab. 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 the reinforced concrete slab with a fiber-reinforced polymer (FRP) deck to prolong the structure’s service life as well as satisfying new load rating requirements (see Fig. 19.11 and 19.12). 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 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

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19.11  FRP bridge deck on a truss bridge that replaced an old concrete deck with asphalt overlays (elevation view).

19.12  FRP bridge deck on a truss bridge that replaced an old concrete deck with asphalt overlays (plan view).

150 mm × 300 mm × 350 mm isocycrinate foam blocks used as stay-in-place forms. The deck was designed using finite element 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. © Woodhead Publishing Limited, 2011

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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 floor-beam flange. A structural bolt secured with a locking nut attached the deck to the superstructure. The drilled holes were then filled with a non-shrink 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. 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 19.13). The strain gages were bonded to steel and the FRP deck with adhesive and then waterproofed. A total of 18 strain gages were used, six placed on a steel floor-beam and 12 placed on the FRP deck. The data was collected using a computerized data acquisition system. 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 finite element models that would be developed for further analysis (see Fig 19.14). Truck configuration and 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 finite element model. A finite element model of the entire bridge was developed, according to the construction drawings, using commercial finite element modeling software and

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19.13  Strain gage locations on FRP deck during load testing for measuring joint effectiveness and data for calibration of finite element models.

required analysis was performed using a general purpose finite element analysis package. This model with the FRP deck system was validated against load test results obtained from the field testing. The FRP deck was also replaced by a generic reinforced concrete deck in a model to simulate a pre-rehabilitated 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

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19.14  Load testing of the FRP deck.

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). 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, in 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 finite element 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 (ADTT) and new construction assumption. The results indicated that the fatigue life of the FRP deck system almost doubles when compared with the pre-rehabilitated 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 thus imply 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 in one of the trusses of the FRP deck was found to be more than 1000 years as illustrated in Fig. 19.15.

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19.15  Estimated fatigue life comparison (in years) of critical truss members after deck replacement. (Symbol 8 = infinity)

19.5.2 Bridge wrapping (non-bond-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 with conventional concrete repairs. Lightweight characteristics, resistance to expansive tendencies of the corrosion products, relatively easy assessment with visual and simple non-destructive evaluation (NDE) methods, and relatively minor changes to the original structural geometry and dimensions makes these wrappings attractive besides their cost-effectiveness. Rajan Sen (2003) provides a good overview of 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

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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 on 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 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 the SHM has been very useful to evaluate the surface preparation options, available to bridge owners on the durability of FRP repairs for short-term applications. More details on this case study can be found in Alampalli (2005). Reason for SHM Different surface preparation (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 was investigated. 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 the above costs do not include costs for FRP wrapping (material or labor), pressure washing the concrete, and sand blasting 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. 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. 19.16). 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 non-structural and was repaired using concrete patch work in 1991–92. These repairs failed quickly and hence,

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19.16  The bridge intended for FRP wrapping for investigation of surface preparation effects on durability.

further non-structural (cosmetic) repairs were again needed in 1999 to slow or avoid future deterioration. 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 section above with one layer of FRP materials to see their long-term influence on the durability of the repair in terms of rebar corrosion rate and bond between the FRP materials and concrete surface. Repairs The North column was repaired using repair strategy 1, South column was repaired using strategy 2, and 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, anti-carbonation coating. The repair work was conducted in August and September of 1999.

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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 the corrosion probes from Concorr, Inc., which were installed inside the column during the concrete repair (see Fig. 19.17). PR500 data acquisition 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 which 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. 19.18).

19.17  Corrosion sensor installation during the concrete repairs.

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 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. 19.19.

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19.18  Humidity sensor after installation.

19.19  Typical time history of corrosion rates in column.

Conclusions The corrosion rates initially went up, then gradually slowed down and decreased with time. After about two 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 irrespective 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. 19.20). Thus, results indicate that FRP wrapping

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19.20  The columns after seven years in-service.

was effective, for temporary (five to seven years) repairs, in confining the repaired/ delaminated concrete columns and that concrete removal strategies did not influence the durability during the four-year monitoring duration.

19.5.3  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 costeffective rehabilitation of deteriorated reinforced concrete bridges to improve capacity and extend service life. This case study briefly describes this project and use of SHM to evaluate the durability of the FRP strengthening system after two years in service. More details on this case study can be found in Hag-Elsafi et al. (2001, 2004). 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 and NDE-based SHM techniques are often required to ensure desired quality of installation and assess bond effectiveness. Coin tapping and thermographic imaging are generally used respectively for local and global assessment of bond quality and

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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. Structure The 12.19 m long and 36.58 m wide reinforced-concrete structure with 26 simplysupported T-beams was built in 1932 and carries Route 378 over the Wynantskill Creek in the City of South Troy, New York (see Fig. 19.21). 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.

19.21  T-beam bridge before strengthening.

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 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

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19.22  T-beam bridge strengthened with FRP laminates for additional flexure and shear strength.

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. 19.22). SHM instrumentation The initial instrumentation and loading was 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 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. 19.23). 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. Monitoring and 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

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19.23  Typical strain gage scheme for load testing and data for calibration of finite element models.

with known loads were used for the load testing (see Fig. 19.24). The load tests were repeated after two 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. 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 two years in service (see Fig. 19.25).

19.6 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 and the 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

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19.24  Load testing of the T-beam bridge.

19.25  Effectiveness of repairs, using rebar strain, after two years in service.

some bridge applications using FRP materials in New York State 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.

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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. 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 on-route 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 pro-active 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

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et al. 2006; Alnahhal et al. 2006) and integrating SHM with these applications could help develop such procedures. Durability Durability issues have been investigated by several researchers in the literature (Singhvi and Mirmiran 2002; Harries 2005; Aref and Alampalli 2001; El-Ragaby 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 this knowledge that can further lead to better durable systems.

19.7 References Alampalli, S. (2005) ‘Effectiveness of FRP Materials with Alternative Concrete Removal Strategies for Reinforced Concrete Bridge Column Wrapping.’ International Journal of Materials and Product Technology, Inderscience Publishers, 23(3/4), 338–347. Alampalli, S. (2006) ‘Field Performance of an FRP Slab Bridge.’ Journal of Composite Structures, Elsevier Science, 72(4), 494–502. Alampalli, S., and Ettouney, M.M. (2006) ‘Long-Term Issues Related to Structural Health of FRP Bridge Decks.’ Journal of Bridge Structures: Assessment, Design and Construction, Taylor and Francis, 2(1), 1–11. Alampalli, S., and Jalinoos, F. (2009) ‘Use of NDT Technologies in US Bridge Inspection Practice,’ Materials Evaluation, Journal in Nondestructive Testing/Evaluation/ Inspection, 67(11), 1236–1246. Alampalli, S., and Kunin, J. (2002) ‘Rehabilitation and Field Testing of an FRP Bridge Deck on a Truss Bridge.’ Journal of Composite Structures, Elsevier Science, 57(1–4), 373–375. Alampalli, S., and Kunin, J. (2003) ‘Load Testing of an FRP Bridge Deck on a Truss Bridge.’ Journal of Applied Composite Materials, Kluwer Academic Publishers, 10(2), 85–102. Alampalli, S., O’Connor, J., and Yannotti, A. (2002) ‘Fiber-Reinforced Composites for the Superstructure of a Short-Span Rural Bridge.’ Journal of Composite Structures, Elsevier Science, Vol. 58, No. 1, pp. 21–27, September 2002. Alnahhal, W.I. and Aref, A.J. (2008) ‘Structural Performance of Hybrid Fiber Reinforced Polymer-Concrete Bridge Superstructure Systems.’ Composite Structures, Elsevier Science, 84, 319–336. Alnahhal, W.I., Chiewanichakorn, M., Aref, A.J., and Alampalli, A. (2006) ‘Temporal Thermal Behavior and Damage Simulations of FRP Deck.’ Journal of Bridge Engineering, 11(4), 452–464. Alnahhal, W.I., Chiewanichakorn, M., Aref, A.J., Kitane, Y., and Alampalli, S. (2007) ‘Simulations of Structural Behavior of Fiber-Reinforced Polymer Bridge Deck Under Thermal Effects.’ International Journal of Materials and Product Technology, Interscience Publishers, 28(1/2), 122–140. Amano, M., Okabe, Y.O., Takeda, N., and Ozaki, T. (2007) ‘Structural Health Monitoring of an Advanced Grid Structure with Embedded Fiber Bragg Grating Sensors.’ Structural Health Monitoring, Sage Publications, 6(4), 309–316.

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Creep and fatigue in polymer matrix composites

Aref, A., and Alampalli, S. (2001) ‘Vibration Characteristics of a Fiber-Reinforced Polymer Bridge Superstructure.’ Journal of Composite Structures, Elsevier Science, 52(3–4), 467–474, 2001. Aref, A.J., Alampalli, S., and He, Y. (2001) ‘A Ritz-Based Static Analysis Method for Fiber-Reinforced Plastic Rib Core Skew Bridge Superstructure.’ Journal of Engineering Mechanics, 127(5), 450–458. Aref, A.J., Alampalli, S., and He, Y. (2005a) ‘Performance of a Fiber-Reinforced Polymer Web Core Skew Bridge Superstructure: Field Testing and Finite Element Simulations.’ Journal of Composite Structures, Elsevier Science, 69(4), 491–499. Aref, A.J., Alampalli, S., and He, Y. (2005b) ‘Performance of a Fiber-Reinforced Polymer Web Core Skew Bridge Superstructure: Failure Modes and Parametric Study.’ Journal of Composite Structures, Elsevier Science, 69(4), 500–509. Brown, D.L. and Berman, J.W. (2010) ‘Fatigue and Strength Evaluation of Two Glass FiberReinforced Polymer Bridge Decks.’ Journal of Bridge Engineering, 15(3), 290–301. Chiewanichakorn, M., Aref, A., and Alampalli, S. (2003) ‘Failure Analysis of Fiber-Reinforced Polymer Bridge Deck System.’ Journal of Composites Technology and Research, 25(2), 119–128. Chiewanichakorn, M., Aref, A.J., and Alampalli, S. (2006) ‘Dynamic and Fatigue Response of a Truss Bridge with Fiber-Reinforced Polymer Deck.’ International Journal of Fatigue, Elsevier Science, 29(8), 1475–1489. Chen, R.H.L., Choi, J-H., GangaRao, H.V., and Kopac, P.A. (2008) ‘Steel Versus GFRP Rebars?’ Public Roads, 72(2). Dutta, P.K., Lopez-Anido, R., and Kwon, S.C. (2007) ‘Fatigue Durability of FRP Composite Bridge Decks at Extreme Temperatures.’ International Journal of Materials and Product Technology, 28(1/2), 198–216. El-Ragaby, A., El-Salakawy, E., and Benmokrane, B. (2007) ‘Fatigue Life Evaluation of Concrete Bridge Deck Slabs Reinforced with Glass FRP Composite Bars.’ Journal of Composites for Construction, 1193, 258–268. Ettouney, M., and Alampalli, S. (2011a) ‘Infrastructure Health in Civil Engineering: Theory and Components.’ CRC Press (to be published) Ettouney, M., and Alampalli, S. (2011b) ‘Infrastructure Health in Civil Engineering: Applications and Management.’ CRC Press (to be published) Farhey, D.N. (2005) ‘Long-Term Performance of Tech 21 All-Composite Bridge.’ Journal of Bridge Engineering, 9(3), 255–262. Hag-Elsafi, O., Alampalli, S., and Kunin, J. (2001) ‘Applications of FRP Laminates for Strengthening a Reinforced Concrete T-Beam Bridge Structure.’ Journal of Composite Structures, Elsevier Science, 52(3–4), 453–466. Hag-Elsafi, O., Alampalli, S., and Kunin, J. (2004) ‘In-Service Evaluation of a Reinforced Concrete T-Beam Bridge FRP Strengthening System.’ Journal of Composite Structures, Elsevier Science, 64(2), 179–188. Hag-Elsafi, O., Alampalli, S., Kunin, J., and Lund, R. (2000) ‘Application of FRP Materials in Bridge Retrofit,’ Seventh Annual International Conference on composites Engineering, Denver, CO, 305–306. Hag-Elsafi, O., Lund, R., and Alampalli, S. (2002) ‘Strengthening of a Bridge Pier Capbeam Using Bonded FRP Composite Plates.’ Journal of Composite Structures, Elsevier Science, 57(1–4), 393–403. Halstead, J.P., O’Connor, J.S., Luu, K.T., Alampalli, S., and Minser, A. (2000) ‘FiberReinforced Polymer Wrapping of Deteriorated Concrete Columns.’ Transportation Research Record 1696, National Research Council, Washington, D.C., 2, 124–130.

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Harries, K. (2005) ‘Fatigue Behavior of Bonded FRP Used for Flexural Reinforcement,’ International Symposium on Bond Behavior of FRP in Structures, International Institute for FRP in Construction, 547–552. Kalny, O., Peterman, R.J., and Ramirez, G. (2004) ‘Performance Evaluation of Repair Technique for Damaged Fiber-Reinforced Polymer Honeycomb Bridge Deck Panels.’ Journal of Bridge Engineering, 9(1), 75–86. Kitane, Y., Aref, A.J., and Lee, G.C. (2004) ‘Static and Fatigue Testing of Hybrid FiberReinforced Polymer-Concrete Bridge Superstructure.’ Journal of Composites for Construction, 8(2), 182–190. Reay, J.T. and Pantelides, C.P. (2006) ‘Long-Term Durability of State Bridge on Interstate 80.’ Journal of Bridge Engineering, 11(2), 205–216. Reising, R.M., Shahrooz, B.M., Hunt, V.J., Neumann, A.R., and Helmicki, A.J. (2004) ‘Performance Comparison of Four Fiber-Reinforced Polymer Deck Panels.’ Journal of Composites for Construction, 8(3), 265–274. Sen, R. (2003) ‘Advances in the Application of FRP for Repairing Corrosion Damage,’ Progress in Structural Engineering and Materials, John Wiley Publications, 5(2), 99–113. Singhvi, A., and Mirmiran, A. (2002) ‘Creep and Durability of Environmentally Conditioned FRP-RC Beams Using Fiber Optic Sensors.’ Journal of Reinforced Plastics and Composites, 21(4), 2002. Triandafilou, L., and O’Connor, J. (2009) ‘FRP Composites for Bridge Decks and Superstructures: State of the Practice in the U.S.’ International Conference on Fiber Reinforced Polymer (FRP) Composites for Infrastructure Applications, University of Pacific, Stockton, CA. Wattanadechachan, P., Aboutaha, R., Hag-Elsafi, O., and Alampalli, S. (2006) ‘Thermal Compatibility and Durability of Wearing Surfaces on GFRP Bridge Decks.’ Journal of Bridge Engineering, 11(4), 465–473.

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Index

ABAQUS software, 85, 86, 87 Abel operator, 193 accelerated testing methodology, 52, 440–6 applicability, 450–2, 454–6 condition (A) for fibre-reinforced plastic, 454–5 conditions (A), (B) and (C) for fibrereinforced plastic, 456 conditions (B) and (C) for fibre-reinforced plastic, 456 constant strain rate strength master curve, 441–2 creep strength master curve, 442–3 detailed procedure illustration, 441 experimental and verification, 447–50, 451, 452, 453 flexural constant strain rate strength, 447 flexural creep strength, 448 flexural fatigue strength for arbitrary stress ratio, 450 flexural fatigue strength for zero stress ratio, 448–50 specimen and testing method, 447 fatigue strength master curve for zero stress ratio, 444 fatigue strength prediction for arbitrary frequency, stress ratio and temperature, 445–6 future trends, 457–8 long-term creep and fatigue in polymer matrix composites, 439–58 fatigue strength linear dependence on stress ratio, 446 fatigue strength master curve for FRP, 445 linear cumulative damage law for monotonic loading, 443 unidirectional composites failure mechanisms, 457 viscoelastic modulus master curves, 442 procedure, 440–1 T400/3601 flexural constant strain rate strength master curve, 448 flexural creep strength master curve, 450 flexural fatigue strength master curves for stress ratio, 452 flexural fatigue strength prediction, 453

S–N curves, 451 time–temperature shift factors, 449 theoretical verification, 452, 456–7 accumulated damage variable, 433 acoustic emission, 468 acoustic polar scans, 480 acoustography, 480 acrylic, 176 active fibre composites, 71 microstructures with PZT-5A and epoxy matrix, 83 active polymer matrix composites active fibre and macro fibre composites electric fields, 104 long-term elastic responses, 100–1 long-term piezoelectric responses, 102–3 time-dependent long-term compliance and PZT constant, 105 Von Mises stress contour, 102 application as actuators, 104–8, 109 accumulated electric field in the active fibre composites, 108 cantilever beam with active fibre composites actuator, 106 finite element mesh for the smart beam, 106 lateral deflections in the smart cantilever beam, 107 normal and shear stresses in the deformed smart beam, 107 tip-deflection in the smart cantilever beam, 109 effective electromechanical and piezoelectric properties, 87–104 time-dependent properties, 95–104 homogenised composites simplified micromechanical model, 75–83 constant stress time-dependent strain and electrical displacement, 76 simplified microstructures, 79 strain and electrical displacement subject to linear ramp electric field, 77 strain, electrical displacement subject to ramp electric field, 78 LaRC-Si matrix creep compliance, 96 Prony parameters, 96

572 © Woodhead Publishing Limited, 2011



Index

linear electromechanical and piezoelectric properties, 87–94 epoxy electromechanical properties, 88 PZT-5A electromechanical and PZT properties, 88 PZT-5A/epoxy composites effective properties, 88–90 PZT-7A electromechanical and piezoelectric properties, 91 piezoelectric and dielectric properties, 97 PZT-7A/LaRC-Si composites effective dielectric properties, 94 effective elastic properties, 91–2 effective piezoelectric constants, 93 time-dependent elastic constants, 97–8 time-dependent piezoelastic constants, 99 representative volume elements finite element models, 83–7 fibre dimensions, 84 five-fibre models, 84 prescribed boundary conditions, 85 single-fibre models, 84 time-dependent behaviour, 70–110 linearised time-dependent model for materials with electromechanical coupling, 73–5 active structure fibre (ASF), 71 analytical homogenisation technique, 267 Arrhenius equation, 12–13, 447 aspect ratio, 220 ASTM D 4255/D 4255M–01, 472 ASTM D2992, 473–4 ASTM D3039, 474–5 ASTM D3479/D3479M-96 (2007), 461 ASTM D3518/D3518M-94(2007), 472 ASTM E1012-05, 470 Bauschinger effect, 238 bending moment, 471 Bodner-Partom model, 237 Boltzmann superposition integral, 9–10, 31 Boltzmann superposition principle, 8, 31, 36 illustration, 9 Boltzmann theory, 32, 34 Boltzmann’s superposition principle, 49 boundary effect, 504 Bragg grating, 468 Brueller’s model, 35 buckyballs, 220–4 bulk modulus, 217 complex effective, 203 carbon fibre-reinforced plastic (CFRP), 114, 118 carbon fibre-reinforced plastic laminates analysis conditions, 123–5, 126 carbon fibres and epoxy matrix material constants, 125 laminae microstructure, 124 macroscopic tensile curves, 126 elastic-viscoplastic analysis and experimental verification, 118–32 elastic-viscoplastic constitutive equation, 119–22 laminae homogenisation, 120–1 laminate modelling and basic assumptions, 119–20

573

laminates in-plane equation, 121–2 long fibre-reinforced laminate structure with Cartesian co-ordinates, 119 experimental procedure, 122, 123 coupon specimen diagram, 123 off-axis angle for unidirectional, crossply and quasi-isotropic laminates, 123 experiments vs predictions, 126–9 macroscopic stress vs strain relations, 127–8 fibre distribution randomness effects in laminae transverse fibre distributions in laminae, 130 unit cell and finite element mesh, 131 unit cell arrangements, 130 fibre distribution randomness effects on laminae, 129–32 unidirectional creep analysis at elevated temperature, 142–4, 145 analysis conditions, 142–3, 144 analysis results, 144, 145 glass fibre and epoxy matrix material constant, 144 macroscopic creep curves, 145 macroscopic tensile curves, 143 unit cell and finite element mesh, 143 carbon/epoxy laminates off-axis tensile creep curves, 247–51 off-axis tensile stress–strain relationships, 246–7 unidirectional T800H/epoxy laminate off-axis creep curve, 260 off-axis loading–unloading behaviour, 236 off-axis stress–strain curves, 246 off-axis tensile and compressive stress–strain curves, 257–9 off-axis tensile creep curves, 250–1 off-axis tensile stress–strain curves, 247–9 predicted vs observed off-axis creep curves, 264 Cartesian co-ordinates, 115, 119 Cauchy-Green deformation tensor, 305, 311–12 ceramics, 95 Characteristic Damage State (CDS), 501 classic beam theory, 471 Cole-Cole function, 375 complex Poisson’s ratio, 196, 200 composite made from polymeric matrix filled by rigid spheres imaginary part dependence on frequency, 215 real part dependence on frequency, 214 polymer matrix viscoelasticity imaginary part dependence on frequency, 200 real part dependence on frequency, 199 complex shear modulus, 210 complex Young’s modulus, 212 composite stress, 374, 377–8 compressive creep test, 528 constant strain rate (CSR), 440 continuum damage mechanics (CDM), 398–9, 424, 425 polymer matrix composites creep damage and fatigue modelling, 424–37 crack density, 520

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574 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 43X

Index

crack kinetics theory, 457 creep, 4–7, 259, 492 damage accumulation models, 399–401 model parameter values, 283 nylon/rubber-like composites, 319–22 transverse stress loading T22 = 1MPa, 320 transverse stress loading T22 = 1MPa and 2MPa, 321 polymer matrix composites continuum damage mechanical modelling, 424–37 stress dependence, 278 viscoelastic constitutive modelling in polymers and polymer matrix composites, 3–45 constant strain application and stress relaxation, 7 constant stress application and strain response, 5 recovery, 7 relaxation, 7 stages, 6 creep analysis and elastic-viscoplastic of polymer matrix composites, 113–46 polymer matrix composites using viscoplastic models, 273–97 future trends, 296–7 viscoplastic creep modelling, 276–96 unidirectional CFRP laminates at elevated temperature, 142–4, 145 analysis conditions, 142–3, 144 analysis results, 144, 145 glass fibre and epoxy matrix material constant, 144 macroscopic creep curves, 145 macroscopic tensile curves, 143 unit cell and finite element mesh, 143 creep compliance, 5 creep compliance tensor, 283 creep curves, 197 instantaneous and long-time moduli determination, 198 malein-epoxy matrix at different levels of stresses, 197 creep lifetime, 374, 399 creep rupture, 342, 367–8 analysis in polymer matrix composites, 327–46 outlook, 345–6 fibre bundle models, 329–45 microscopic process of failure, 339–41 predicting lifetime, 342–5 slowly relaxing fibres, 335–8 viscoelastic fibres, 333–5 micromechanical models, 373–80 creep strain, 277–85, 280, 296, 374, 391 creep stress, 378–9 critical damage, 435 critical element, 502 Crochet model, 390–7 experimental and fitted curves for creep failure stress Nylon 66 (A100), 392 Nylon 66 (A190), 393 PC Lexan 141, 394 T300/VE composite, 395 T800S/3900-2B composite, 396

experimental and predicted creep lifetime Nylon 66 (A100), 393 Nylon 66 (A190), 394 PC Lexan 141, 395 T300/VE composite, 396 T800S/3900-2B composite, 397 cumulative damage theory, 400 Curtin-McLean (CML) model, 374 see also modified Curtin-McLean model simulated unidirectional composite calculated lifetime, 377 creep compliance, 376 elastic and viscoelastic parameters, 376 general power law parameters, 377 cycle-dependent processes, 502 damage, 424, 425–35 characterisation in composites, 514–39 evolutions, 315, 316, 318 rate, 306–7 damage accumulation models, 399–401 damage growth law, 399 deformation concentration tensor, 309–10 deformation gradient, 285, 286, 304 deformation-induced anisotropy, 238 Digital Image Correlation, 477 Dirac delta function, 16 dissipation function, 241 DMA see Dynamic Mechanical Analyser Doolittle equation, 13 Durometer hardness D, 168, 170, 175 Dynamic Mechanical Analyser, 62 dynamic mechanical thermal analysis (DMTA), 189 E-glass fibre bundle maximum stress and number of fatigue cycles, 418 probability of survival, fibre strain and number of cycles, 419 tensile stress–strain diagram, 415 effective shear modulus, 207–8 effective stress sensor, 242 effective Young’s modulus dependencies aspect ratio for inclusions with different relative stiffness, 222 parallel to plane of platelet inclusions, 218 relative stiffness for different aspect ratio, 221 rigid inclusions concentration, 217 solid phase concentration, 224 volume concentration of inclusions, 223 formula cubic lattice of platelet inclusions, 218 randomly oriented platelet inclusions, 219 elastic bulk behaviour, 306 elastic deformation, 286 elastic logarithmic strain, 306 elastic strain, 280 elastic-viscoplastic behaviour creep analysis of polymer matrix composites, 113–46 using homogenisation theory and experimental verification, 118–41 CFRP laminates, 118–32 plain-woven GFRP laminates, 132–41

© Woodhead Publishing Limited, 2011



Index

electroservo-controlled hydro testing machine, 447 EN ISO 13003:2003, 461 Epotherm materials, 564 epoxy matrix, 125 epoxy resins, 6–7 equal load sharing see global load sharing equilibrium driving force, 435 equilibrium equation, 161 FALSTAFF spectrum (Fighter Aircraft Load STAndard For Fatigue evaluation), 465 fatigue composite materials behaviour, 500–8, 509–11 fatigue and static fatigue, 502–6 fatigue and static fatigue interaction models, 506–8 damage accumulation models, 399–401 fibre bundle model, 416–19 polymer matrix composites continuum damage mechanical modelling, 424–37 testing fibre strength by fibre bundle tests, 409–22 fibre bundles stress-life diagram, 419–22 fibre strength distribution parameters determination, 411–16 viscoelasticity effect on polymer matrix composite, 492–511 linear viscoelastic analysis, 494–500 fatigue testing methods bending fatigue, 470–2 effect of boundary conditions and specimen geometry, 474–7 stress state near tabbed regions in uniaxial fatigue loading, 474–5 topology optimisation in biaxially loaded specimens, 475–7 future trends, 482–3 new composite materials fatigue performance assessment, 482–3 testing and instrumentation methods, 482 multiaxial fatigue, 473–4 plain weave glass/epoxy composite micro-tomography images, 479 microscopic fatigue processes, 481 microscopic image, 478 polymer matrix composites, 461–83 constant-life diagram, 464 fatigue set-up parameters, 463–5 hydraulic wedge-operated grips and tabbed region of composite specimen, 475 parameters inherent to the composite specimen, 462–3 permanent shear strain accumulation, 473 Poisson’s ratio evolution, 467 relative resistance change and axial fatigue stress, 469 shear-dominated fatigue, 472 tension–compression and compression– compression fatigue, 469–70 tension–tension fatigue, 466–9 fibre bundle models, 329–33, 345 basic set-up, 330 creep rupture, 333–45 brittle fibres undergoing slow relaxation, 336

575

microscopic process of failure, 339–41 predicting lifetime, 342–5 slowly relaxing fibres, 335–8 viscoelastic fibres, 333–5 fibre bundle test, 414–16 fibre bundles, 413 constitutive curves, 332 lifetime, 335, 342–5 as a function of distance from critical load, 338 as a function of transition time, 344 probability distribution, 343 number of fibres on simulated total damage, 422 relaxation, 335–8 S–N diagram, 422 stress-life diagram, 419–22 waiting times, 340 distribution for external loads, 341 fibre optic sensors, 468–9 fibre reinforced plastics (FRP), 492 structural health monitoring, 543–69, 551–2 structures in bridge industry, 544–7, 548–9 fibre-reinforced composites (FRCs), 328 repeating unit cell, 310 fibres fatigue strength testing, 409–22 carbon fibres strength variation, 410 fibre bundle model for fatigue, 416–19 mean fibre strength vs fibre length, 411 strength distribution parameters determination, 411–16 fibre bundle test, 414–16 single fibre test, 411–14 stress-life diagram of fibre bundles, 419–22 effect of number of fibres in fibre bundle on simulated total damage, 422 simulated vs experimental data, 422 simulation of stress-controlled fatigue S–N plot, 421 finite deformation analysis, 277 finite element analysis, 356–9 stress–strain curves, 358 time-dependent stress-concentration factor, 357–9 finite element method, 72, 117 finite strain viscoelasticity, 304–9 finite strain viscoplasticity, 285–96 finite viscoelasticity, 303, 304 fracture mechanics, 398 free energy density, 426, 432 free energy functions, 240, 305, 312 free stored energy, 371 free volume theory, 13 Gates-Sun model, 237 Gauchy stress sensor, 289 generalised Kelvin model, 49 glass fibre reinforced plastics (GFRP), 114, 500 global load sharing (GLS), 330, 337 Goodman diagram, 466 Green, Rivlin and Spencer model, 31–4 Heaviside step function, 533 Helmholtz free energy function, 241 Hereditary integral, 10 hereditary response, 493, 503

© Woodhead Publishing Limited, 2011

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576 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 43X

Index

hexagonal fibre array, 123–4, 129 high-cycle fatigue, 464 High-Fidelity Generalised Method of Cells (HFGMC), 303, 309–11 high-resolution 3D X-ray micro-tomography (micro-CT), 479 Hill’s yield function, 252–3 homogenisation theory non-linear time-dependent composites, 115–18 basic equations, 115–16 body with periodic internal structure, 115 computational procedure, 117–18 microscopic stress evolution equation and macroscopic constitutive relation, 117 solution for perturbed velocity field, 116–17 polymer matrix composites elasticviscoplastic and creep behaviour, 113–46 CFRP laminates elastic-viscoplastic analysis and experimental verification, 118–32 plain-woven GFRP laminates elasticviscoplastic analysis and experimental verification, 132–41 unidirectional CFRP laminates creep analysis at elevated temperature, 142–4, 145 Hooke’s law, 414, 415, 494 hypoelasticity, 290 inherent anisotropy, 238 initial anisotropy tension-compression asymmetry modelling comparison with experimental results, 255–9 plane-stress pressure-sensitive effective stress, 253–4 pressure-modified viscoplasticity model, 254–5 pressure-sensitive effective stress, 252–3 shear flow differential parameter, 254 tension–compression asymmetry modelling, 251–9 Interdigitated Electrode (IDE) actuator, 72 ISO 527–4, 474 isotropic hardening law, 289 isotropic hardening model, 243–4 Kachanov’s theory, 425 Kelvin–Voigt model, 19–24, 188, 192, 498–500, 503 creep, 20–1 E1, E2 and phase lag variation with loading frequency, 500 generalised, 29, 30 parallel connection, 29 series connection, 30 illustration, 20 recovery, 21 relaxation, 21–4 schematic, 499 typical stress/time trace, 23 kinematic hardening model, 244–5, 275 kinematic hardening variable, 263 kinetic rate theory, 397 Kirchoff stresses, 304–5

Kohlrausch model, 50 Kolrausch-William-Watts (KWW) model, 95 Koltunov’s operator, 194 Ladevese-Dantec plasticity-damage model, 267 Lagrange multiplier, 431, 433 Laplace transform, 49 Leaderman’s model, 34–5 leaky Lamb wave technique, 480 lifetime prediction time-dependent failure criteria for polymer matrix composites, 366–401 continuum damage mechanics, 398–9 creep rupture, 373–80 Crochet model, 390–7 damage accumulation models for static and dynamic fatigue, 399–401 energy-based failure criteria, 370–3 experimental cases, 381–90 fracture mechanics, 398 kinetic rate theory, 397 limit stress level, 375 linear beam theory, 471 linear cumulative damage (LCD), 440, 442–3 linear cumulative damage law, 400, 443, 457 linear viscoelastic materials time–temperature–age superposition principle, 48–68 effective time theory, 64–6 procedure to predict long-term creep, 66–7 short-term data correlation, 48–50 temperature compensation, 67–8 time–age superposition, 58–64 time–temperature superposition, 50–8 linear viscoelastic models, 14–29 four-element model, 25–7 creep-recovery behaviour, 26 parameters determination, 27 Kelvin–Voigt model, 19–24 linear spring, 14–15 response to constant stress, 15 linear viscous dashpot, 15–16, 17 response to constant strain, 17 response to constant stress, 16 Maxwell model, 17–19, 28 three-element solid, 24–5 illustration, 24 linear viscoelastic strain, 426 load sharing rule, 330 local load sharing, 331 Lorentzian function, 155 Love-Kirchoff hypothesis, 435 low-cycle fatigue, 464 macro fibre composite, 71 Massachusetts Institute of Technology, 71 MATLAB algorithm, 50, 56, 59 matrix viscosity, 149 maximum work stress criterion, 371, 372 Maxwell elements, 304, 305 Maxwell model, 17–19, 24, 28, 188, 497–8, 503 creep, 17 E1, E2 and phase lag variation with loading frequency, 498 generalised model, 28 illustration, 18 recovery, 18

© Woodhead Publishing Limited, 2011



Index

relaxation, 18–19 relaxation response, 19 schematic, 497 mean field theory, 267 memory effect, 493, 504 meta-delaminations, 481 micro-Raman spectrometer, 154–5 microcrack density, 429 microcracks, 520 micromechanical modelling, 71–2 micromechanics of failure (MMF), 351, 352, 362 MIMIMAT testing machine, 521 Miner’s rule, 400 Mitsubishi Rayon Co. Ltd., 122, 125 modified Curtin-McLean model, 376 simulated unidirectional composite calculated lifetime, 380 creep compliance, 379 elastic and viscoelastic parameters, 379 general power law parameters, 380 modified kinematic hardening rule, 262–3 modified rate equation, 397 modified Reiner-Weissenberg criterion, 371, 372 Moiré interferometry, 480 momentary curves, 53 Monkman-Grant relation, 342–3 Monte-Carlo simulation, 419, 420 Mullins effect, 303, 313, 314 multiaxial fatigue, 473–4 bending/torsion set-ups, 474 internal pressure/tension set-ups, 473–4 planar biaxial set-ups, 474 tension/torsion set-ups, 473 nanocomposites, 184–7 nanofibres, 219–20 nanoparticles, 184–7 nanotubes, 220–4 NASA Langley Research Centre, 71 neutral fibre, 471 Newton-Raphson method, 82, 435 Nitto Shinko Corporation, 138 non-linear superposition theory, 34 non-linear viscoelastic behaviour, 29, 31–45 different materials applications, 37–45 parameters characteristic surfaces for carbon/epoxy composite, 40 linearity limits, 29–31 multiple integral representations, 31–4 Green, Rivlin and Spencer model, 31–4 linear kernel function vs time, 32 Pipkin and Rogers model, 34 second-order kernel functions vs time parameters, 33 non-linear parameters experimental values vs model predictions, 38 experimental vs numerical values vs model predictions, 42 variation for aluminium/epoxy composite, 43–4 parameters determination, 36–7 single integral representations, 34–6 Brueller’s model, 35 Leaderman’s model, 34–5 Robotnov’s model, 35

577

Schapery’s constitutive equation, 35–6 stress shift factor characteristic surfaces for carbon/epoxy composite, 41 experimental values vs model predictions, 39 experimental vs numerical values vs model predictions, 43 variation for aluminium/epoxy composite, 44–5 Norton Laws, 435 numerical homogenisation technique, 268 nylon/rubber-like viscoelastic composites creep behaviour, 319–22 transverse stress loading T22 = 1MPa, 320 transverse stress loading T22 = 1MPa and 2MPa, 321 relaxation behaviour transverse deformation gradient F22 = 1.5, 321 transverse deformation gradient F22 = 1.5, 2, 2.5 and 3, 322 two uniaxial transverse stress-cyclic loading–unloading, 319 uniaxial transverse stress cyclic loading and unloading (0.5 ≤ F22 ≤ 1.5) applied at rate of F22 = 0.1s1, 317 (0.5 ≤ F22 ≤ 1.5) applied at rate of F22 = 1s1, 316 global stress degradation, 318 uniaxial transverse stress loading and unloading, 315 off-axis failure see time-dependent off-axis failure off-axis tensile creep behaviour, 247–51 off-axis tensile stress–strain, 246–7 optical microscopy, 477 parabolic criterion, 361 Paris Law, 500 passive resistance, 493 piezoceramic fibres, 70–110 Pipkin and Rogers model, 34 plain-woven GFRP laminates analysis conditions, 138–40 basic cell and finite element mesh, 139 glass fibres and epoxy matrix material constant, 140 macroscopic tensile curves, 140 elastic-viscoplastic analysis and experimental verification, 132–41 homogenisation theory, 137 laminates with two types of plain fabrics laminate configurations, 133 experimental procedure, 137–8 coupon specimen diagram, 138 experiments vs predictions, 140–1 macroscopic stress vs strain relations, 141 laminate configurations and domains of analysis, 134–7 unit cells and basic cells, 135–6 plane stress, 245–6 pressure-sensitive effective stress, 253–4 plastic deformation, 279–80 gradient tensor, 286

© Woodhead Publishing Limited, 2011

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578 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 43X

Index

plastic rate of deformation tensor, 287 plastic shear deformation, 287–8 plastic spin coefficient, 292, 294 plastic spin tensor, 287 plastic strain, 279, 280, 296 plasticity-damage modelling, 267 platelet-shape nanoparticles, 217–19 PMCs see polymer matrix composites Poisson’s ratio, 85, 178, 194, 201, 284, 467 see also complex Poisson’s ratio change with time during creep malein-epoxy matrix, 198 on tension, 199 polymeric matrix filled by glass solid microspheres, 215 polymer matrix generalised Maxwell and Kelvin-Voigt models, 188 malein-epoxy matrix creep curves at different levels of stresses, 197 Poisson’s ratio change with time during creep, 198 viscoelasticity, 188–200 polymer matrix composites see also active polymer matrix composites active fibres composites time-dependent behaviour, 70–110 applications as actuators, 104–8, 109 effective electromechanical and piezoelectric properties, 87–104 homogenised active PMCs simplified micromechanical model, 75–83 linearised time-dependent model for materials with electromechanical coupling, 73–5 representative volume elements finite element models, 83–7 creep analysis using viscoplastic models, 273–97 future trends, 296–7 viscoplastic creep modelling, 276–96 creep and stress relaxation viscoelastic constitutive modelling, 3–45 creep damage and fatigue continuum damage mechanical modelling, 424–37 elastic-viscoplastic and creep behaviour using homogenisation theory, 113–46 CFRP laminates elastic-viscoplastic analysis and experimental verification, 118–32 non-linear time-dependent composites, 115–18 plain-woven GFRP laminates elasticviscoplastic analysis and experimental verification, 132–41 unidirectional CFRP laminates creep analysis at elevated temperature, 142–4, 145 fatigue curves for glass/polyester experimental and calculated, 505 R = 0.5, 507 R = 0.7, 508 R = 0.8, 508 R = 0.9, 509

fatigue curves for glass/polyester using the method of Epaarachchi R = 0.5, 509 R = 0.7, 510 R = 0.8, 510 R = 0.9, 511 fatigue testing methods, 461–83 effect of boundary conditions and specimen geometry, 474–7 fatigue damage in structural composites, 477–82 fatigue set-up parameters, 463–5 future trends, 482–3 main testing methods, 465–74 parameters inherent to the composite specimen, 462–3 fibre bundle models for creep rupture analysis, 327–46 creep rupture, 333–45 fibre bundle model, 329–33 outlook, 345–6 from meso- to macroscopic behaviour, 435–6 internal pressure test with close-ended effect, 436 pure internal pressure test, 436 long-term creep and fatigue prediction by accelerated testing methodology, 439–58 accelerated testing methodology, 440–6 applicability, 450–2, 454–6 experimental verification, 447–50, 451, 452, 453 future trends, 457–8 theoretical verification, 452, 456–7 measuring fibre strain and creep behaviour using Raman spectroscopy, 149–82 stress or strain measurement, 152–6 stress relaxation in broken fibres, 157–64 time-dependent variation in fibre stress during pull-out tests, 164–81 unidirectional composites reinforced with long fibres creep mechanism, 149–52 time-dependent failure criteria for lifetime prediction, 366–401 continuum damage mechanics, 398–9 creep rupture, 373–80 Crochet model, 390–7 damage accumulation models for static and dynamic fatigue, 399–401 energy-based failure criteria, 370–3 experimental cases, 381–90 fracture mechanics, 398 kinetic rate theory, 397 time-dependent off-axis failure micromechanical modelling, 350–63 experiments, 353–6 finite element analysis, 356–9 future trends, 363 viscoelastic behaviour micromechanical modelling, 302–23 applications, 312–22 computational procedure, 311–12 finite strain micromechanical analysis, 309–11 finite strain viscoelasticity, 304–9 viscoelastic strain, damage and viscoplastic strain mesomodel, 425–35 debonding and microcracking, 428

© Woodhead Publishing Limited, 2011



Index

first damage criterion, 428 glass-epoxy laminate damage criterion, 434 layer reference axes, 426 viscoelasticity effect on fatigue behaviour, 492–511 composite materials fatigue behaviour, 500–8, 509–11 linear viscoelastic analysis, 494–500 viscoplastic deformation constitutive modelling, 234–68 framework, 239–51 future trends, 267–8 tension–compression asymmetry in initial anisotropy, 251–9 transient creep softening, 259–65 polymer nanocomposites predicting viscoelastic behaviour, 184–231 fibrous composites with nano-filled matrices, 227, 229 nanoparticles and nanocomposites, 184–7 nanoporous polymers, 224–9 notations, 230 polymer matrix, 188–200 polymers filled by buckyballs and nanotubes, 220–4 polymers filled by nanofibres, 219–20 polymers filled by platelet-shape nanoparticles, 217–19 polymers filled by quasi-spherical nanoparticles, 200–16 polymers creep and stress relaxation viscoelastic constitutive modelling, 3–45 creep, 4–7 linearity, 7–11 non-linear viscoelastic behaviour, 29–45 time–stress superposition principle, 13 time–temperature superposition principle, 11–13 time–temperature–stress superposition principle, 13–14 power law, 10, 37, 59, 60 PR500 data acquisition equipment, 561 pressure-modified viscoplasticity model, 254–5 pressure-sensitive effective stress, 252–3 primary creep, 494 Prony series, 10, 95, 517, 533–4 pure shear strength, 361 Q9100, 553 QM6408, 553 quadratic criterion, 361 quasi-spherical nanoparticles, 200–16 Rabotnov algebra of resolvent operators, 191–2, 193 Rabotnov operators, 193, 194 radiography, 479 Raman bands, 152, 156 Raman microprobe, 156, 160 Raman spectrometer, 152, 154–7 Raman spectroscopy interfacial bonding and slippage effect, 177–81 experimental vs analytical solution with and without interfacial debonding, 180 interfacial shear stress distribution, 180

579

pull-out tests, 178–9 stress relaxation tests, 177–8 theoretical vs experimental results on interfacial shear stress, 179 matrix creep influence on time-dependent change in fibre stress, 175–7 creep tests for matrix resins, 175–6 matrix creep influence, 176–7 matrix resins tensile creep curves, 177 measuring temporal stress change in polymer matrix composites, 149–82 Raman spectra free standing carbon fibre and epoxy resin, 156 free standing carbon fibre with and without tensile stress, 153 stress or strain measurement, 152–6, 157 2700 cm–1 band peak position dependence, 153 fibre strain vs specimen overall strain in single-fibre model composite, 154 micro-Raman spectroscopy system, 155 Raman spectrometer and measurement, 154–7 Raman spectrum for 2700 cm–1 of carbon fibre, 157 stress measurement fundamentals, 152–4 stress relaxation at 0.7% constant strain, 161–3 axial stress profile in broken fibre, 161 interfacial shear stress profile in broken fibre, 162 normal stress relaxation in matrix, 162 stress relaxation at 1.4% constant strain, 163–4 axial stress profile in broken fibre, 163 interfacial shear stress profile in broken fibre, 164 stress relaxation in broken fibres, 157–64 carbon fibre and epoxy resin mechanical properties, 158 loading device for stress relaxation tests, 159 materials and specimens, 158 measurement location in broken fibre, 160 single-fibre model composite specimen shape, 159 stress relaxation tests and Raman spectroscopy measurement, 158–60 time-dependent variation in fibre stress during pull-out tests, 164–75 constant-load pull-out tests, 169 fibre axial stress profiles in pull-out test specimens, 171–2 fibre axial stress profiles in pull-out tests, 170 fibre pull-out test conditions for model composites, 169 fibre stress profile, 166 interfacial shear stress profiles in pull-out test specimens, 173–4 long-term pull-out tests at constant loads, 168–9 materials, 167 matrix resins stress relaxation curves at tensile strain, 175 normal stress relaxation in matrix, 165

© Woodhead Publishing Limited, 2011

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 43X

580 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 43X

Index

pull-out test specimens preparation, 167 pull-out test specimens shape, 168 pull-out tests at constant loads, 170–4 resins mechanical properties for pull-out composites, 167 single-fibre pull-out specimens preparation, 167–8 stress relaxation tests for matrix resins, 170, 174–5 time-dependent change in fibre stress profiles in constant-load pull-out test, 166 unidirectional composites reinforced with long fibres creep mechanism, 149–52 SCS-6/Beta21S creep curves, 150 stress profile and stress relaxation in broken fibres, 151 recursive method, 74 Reiner-Weissenberg criterion, 370–2 Reiner-Weissenberg theory, 367–8 relaxed viscoelastic compliance tensor, 426–7 Replark 30 unidirectional carbon fibres, 564 Replark laminate system, 564 representative volume element, 72, 83–7 resistance measurement, 469 reversible strain, 426 Robotnov’s model, 35 Rosen’s model, 456 RVE see representative volume element Rzhanitsin’s operator, 194 scalar invariant function, 242 scanning electron microscopy (SEM), 477–8 Schapery’s constitutive equation, 35–6 Schapery’s model, 515 servo-hydraulic testing machine, 466 shape parameter, 413, 414 shear compliance, 208 shear flow differential parameter, 254 shear loss modulus, 210 shear storage modulus, 209 short fibre composites elastic modulus, 518–19 normalised elastic modulus, 525 reduction after loading to high strain, 522 reduction due to presumed microdamage development, 524 microdamage effect on stiffness, 518–25 applied strain steps during one block of the test, 519 damaged composite edge view, 521 microdamage and stiffness degradation, 520–5 stiffness before and after tension and compression to high stress, 523 stiffness reduction measurements, 519–20 non-linear viscoelasticity, 533–9 examples of behaviour, 535–9 initial strain for flax/starch composite, 539 parameters for flax/starch composite, 538 shift factor dependence on stress, 538 simulated and experimental loadingunloading curves, 536 stress-dependent functions in Schapery’s model, 536 stress-dependent viscoelastic creep compliance, 537

viscoelasticity in creep and strain recovery test, 533–5 viscoelasticity, viscoplasticity, and damage characterisation, 514–39 flax/starch composite with 40% fibre content mechanical behaviour, 515 material model, 516–18 viscoplastic strain accumulation in recycled carbon fibre/ recycled polypropylene, 529 time- and stress-dependence in paper/ phenol-formaldehyde composite, 530 time-dependence in creep test, 540–2 viscoplasticity, 525–33 creep and recovery steps to identify the time-dependence of VP-strain, 528 experimental procedure, 527–9 modelling in creep test, 525–7 Sika Wrap Hex 106G, 560 Sikadur 330, 560 Sikadur 670W, 560 single fibre test, 411–14 arrangement, 412 multiple fibre tests with different fibre lengths, 412 multiple fibre tests with the same fibre length, 412–14 shape parameter determination, 413 SLS model see standard linear solid model snapshot condition, 53 softening index, 263 SPATE see Stress Patterns Analysis by the measurement of Thermal Emissions spin tensor, 287 Split Hopkinson Bar method, 360–1 standard linear solid model, 48 static fatigue, 508 stochastic fibre fracture, 149 strain rate, 288–9 strand see fibre bundle Strength Evolution Integral (SEI), 401, 506 Stress Patterns Analysis by the measurement of Thermal Emissions, 480 stress ratio, 463–4, 466 stress relaxation, 494 viscoelastic constitutive modelling in polymers and polymer matrix composites, 3–45 stress-life diagram, 419–22 stress-strain relationship, 31 stress–strain curves, 356–7, 358 structural adhesives, 6 structural health monitoring, 547, 550 bridge deck, 553–8 elevation view, 554 estimated fatigue life comparisons, 558 load testing, 557 plan view, 554 repairs, 553–5 SHM instrumentation, 555 strain gage locations during load testing, 556 structure, 553 testing and analysis, 555–7 bridge wrapping, 558–63 columns after seven years in-service, 563 corrosion rates time history, 562

© Woodhead Publishing Limited, 2011



Index

corrosion sensor installation, 561 humidity sensor, 562 monitoring, 561 repairs, 560 SHM instrumentation, 561 structure, 559–60 surface preparation effect on durability, 560 composite structures for durability, 543–69 case studies, 552–66 FRP structures, 551–2 external reinforcement, 563–6 effectiveness of repairs, 567 FRP-strengthened T-beam bridge, 565 monitoring and testing, 565–6 repairs, 564–5 SHM instrumentation, 565 strain gage scheme for load testing, 566 structure, 564 T-beam bridge before strengthening, 564 T-beam bridge load testing, 567 FRP structures in bridge industry, 544–7, 548–9 Bennetts Creek Bridge during construction, 545 Bennetts Creek Bridge during proof load testing, 545 Bentley Creek Bridge, 546 bridge deck with FRP rebars, 549 columns and pier-cap wrapped with CFRP, 549 external FRP reinforcement for strengthening application, 548 external reinforcement for maintenance application, 548 Schroon River Bridge, 547 Troups Creek Bridge, 546 potential use in FRP materials use various stages, 567–9 construction, 568 design and analysis, 568 durability, 569 in-service issues, 568–9 planning, 568 structural health in civil engineering concept, 550 Sun-Chen model, 237 System 6000, 565 T-Beam bridge, 563 tensile creep test, 527–8 tension–compression asymmetry modelling in initial anisotropy, 251–9 comparison with experimental results, 255–9 plane-stress pressure-sensitive effective stress, 253–4 pressure-modified viscoplasticity model, 254–5 pressure-sensitive effective stress, 252–3 shear flow differential parameter, 254 thermal activation theory, 277 thermal expansion phenomenon, 425–6 thermodynamic force, 241 thermodynamically consistent theory, 516 thermography, 479–80 thermoplastics, 381–3

581

experimental and calculated creep lifetime Nylon 66 (A100), 382 Nylon 66 (A190), 382 PC Lexan 141, 383 viscoelastic and rupture parameters, 381 thermosetting polymer-based composites, 383–7 experimental and calculated creep lifetime chopped-glass-fibre composite, 385 crossply carbon-fibre 45°/45°, 386 crossply carbon-fibre 90°/0° composite, 386 glass-fibre/urethane composite at 120°C, 386 glass-fibre/urethane composite at 23°C, 385 quasi-isotropic carbon-fibre composite at 120°C, 387 quasi-isotropic carbon-fibre composite at 23°C, 387 viscoelastic and rupture parameters, 384 through-transmission ultrasonics, 478–9 time-dependent behaviour active PMCs incorporating piezoceramic fibres, 70–110 applications as actuators, 104–8, 109 effective electromechanical and piezoelectric properties, 87–104 homogenised active PMCs simplified micromechanical model, 75–83 linearised time-dependent model for materials with electromechanical coupling, 73–5 representative volume elements finite element models, 83–7 time-dependent failure criteria continuum damage mechanics, 398–9 damage accumulation models for static and dynamic fatigue, 399–401 energy-based failure criteria, 370–3 linear viscoelastic material, 372 experimental cases, 381–90 thermoplastics, 381–3 thermosetting polymer-based composites, 383–7 time–temperature superposition principle, 388–90 lifetime prediction of polymer matrix composites, 366–401 continuum damage mechanics, 398–9 creep rupture, 373–80 Crochet model, 390–7 fracture mechanics, 398 kinetic rate theory, 397 time-dependent off-axis failure experiments, 353–6 failure surfaces for tensile strain rates, 355 numerical model geometry and boundary condition, 356 procedure, 353–4 results and fractography, 354–6 specimen geometry and dimensions, 353 finite element analysis, 356–9 stress–strain curves, 356–7 time-dependent stress-concentration factor, 357–9 micromechanical modelling in polymer matrix composites, 350–63 future trends, 363 interfacial failure surface plots, 362

© Woodhead Publishing Limited, 2011

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 43X

582 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 43X

Index

strengths of matrix and interface as a function of strain rate, 360 time-dependent processes, 502, 508 time-dependent stress-concentration factor, 357–9 stress-concentration factor as a function of time, 358 time–stress superposition principle (TSSP), 13 time–temperature shift factors (TTSF), 440–1 time–temperature superposition principle (TTSP), 11–13, 276, 375, 388–90, 439–40 estimated and calculated creep lifetime T300/VE composite, 389 estimated and calculated creep reduced lifetime T300/VE composite, 389 T800/3900-2B composite, 390 UT500/135 composite, 390 viscoelastic and rupture parameters, 388 time–temperature–age superposition principle linear viscoelastic materials long-term response, 48–68 effective time theory, 64–76 procedure to predict long-term creep, 66–7 short-term data correlation, 48–50 temperature compensation, 67–8 time–age superposition, 58–64 ageing shift factor plot for creep data, 63 compliance vs time at constant temperature and various ages, 59 initial regression parameters for ageing study, 60 power law model for creep data, 61 regression parameters and shift factors for ageing study, 60 temperature and ageing studies momentary master curve, 64 time–temperature superposition, 50–8 material compliance and retardation time values, 51 material compliance double logarithmic plot, 52 momentary curves at various temperatures, 54 momentary master curve vs long-term creep, 56 regression parameters and shift factors, 55 temperature shift factor plot, 57 time span over two momentary curves superpose, 54 time–temperature–stress superposition principle (TTSSP), 13–14 Timoshenko beam elements, 425 Toray Industries, Inc., 142 tow see fibre bundle transient creep softening, 259–65 transmission emission microscopy (TEM), 478 transverse flow differential (TFD) parameter, 254 U-jacket laminates, 565 ultrasonics, 478 unidirectional composites failure mechanisms, 457 reinforced with long fibres creep mechanism, 149–52 SCS-6/Beta21S creep curves, 150 stress profile and stress relaxation in broken fibres, 151

Vaisala HMP44 humidity/temperature probes, 561 velocity gradient tensor, 286 vibrothermography, 480 viscoelastic behaviour see viscoelasticity viscoelastic compliance, 537 viscoelastic contraction, 499 viscoelastic fibre bundle, 333–5 model, 334 viscoelasticity, 3, 188, 424, 425–35 applications, 312–22 material parameters for elastic element, 313 material parameters for Maxwell element, 313 responses to uniaxial stress loading and unloading, 314 basic concepts on polymers and polymeric composites, 3–45 time–stress superposition principle, 13 time–temperature superposition principle, 11–13 time–temperature–stress superposition principle, 13–14 characterisation in composites, 514–39 complex Poisson’s ratio imaginary part dependence on frequency, 200, 215 real part dependence on frequency, 199, 216 composite materials fatigue behaviour, 500–8, 509–11 composites fatigue damage mechanism, 501 fatigue and static fatigue, 502–6 fatigue and static fatigue interaction models, 506–8 two-step fatigue loading spectra, 504 creep, 4–7 dynamic loading effect, 496–500 Kelvin-Voigt model, 498–500 Maxwell model, 497–8 effect on polymer matrix composite fatigue behaviour, 492–511 linear viscoelastic analysis under static and dynamic loading, 494–500 effective Young’s modulus dependencies aspect ratio for inclusions with different relative stiffness, 222 parallel to plane of platelet inclusions, 218 relative stiffness for different aspect ratio, 221 rigid inclusions concentration, 217 solid phase concentration, 224 volume concentration of inclusions, 223 fibrous composites with nano-filled matrices, 227, 229 mechanical losses at glass transition temperature, 229 linear viscoelastic models, 14–29 four-element model, 25–7 generalised Kelvin–Voigt model, 29, 30 generalised Maxwell model, 28 Kelvin–Voigt model, 19–24 linear spring, 14–15 linear viscous dashpot, 15–16, 17 Maxwell model, 17–19 three-element solid, 24–5

© Woodhead Publishing Limited, 2011



Index linearity, 7–11 Boltzmann superposition principle, 9 illustration, 8 micromechanical modelling in polymer matrix composites, 302–23 computational procedure, 311–12 finite strain micromechanical analysis, 309–11 finite strain viscoelasticity, 304–9 nanoporous polymers, 224–9 change of Poisson’s ratio, 224, 227 effective bulk creep function, 228 effective bulk modulus relative drop, 225 shear modulus relative drop, 226 Young’s modulus relative drop, 226 non-linear viscoelastic behaviour, 29, 31–45 different materials applications, 37–45 linearity limits, 29–31 multiple integral representations, 31–4 parameters determination, 36–7 single integral representations, 34–6 nylon/rubber-like composites creep behaviour at transverse stress loading T22 = 1MPa, 320 creep behaviour at transverse stress loading T22 = 1MPa and 2MPa, 321 relaxation behaviour at transverse deformation gradient F22 = 1.5, 321 relaxation behaviour at transverse deformation gradient F22 = 1.5, 2, 2.5 and 3, 322 responses to uniaxial transverse stress cyclic loading and unloading, 315 two uniaxial transverse stress-cyclic loading–unloading, 319 uniaxial transverse stress cyclic loading and unloading, 316, 317, 318 operators, 193–6 Poisson’s ratio change with time during creep malein-epoxy matrix, 198 on tension, 199 polymer matrix, 188–200 generalised Maxwell and Kelvin-Voigt models, 188 instantaneous and long-time moduli from creep curve, 198 malein-epoxy matrix creep curves, 197 polymer nanocomposites, 184–231 form factors for inclusions, 186 nanoparticles and nanocomposites, 184–7 polymers filled by nanofibres, 219–20 polymers filled by platelet-shape nanoparticles, 217–19 polymers filled by buckyballs and nanotubes, 220–4 effective bulk modulus dependencies, 223 polymers filled by quasi-spherical nanoparticles, 200–16 bulk loss modulus dependencies, 206 bulk storage modulus dependencies, 206 change of Poisson’s ratio, 215 dependencies of effective loss modulus, 215 dependencies of effective shear loss modulus, 211 dependencies of effective shear storage modulus, 210

583

dependencies of effective storage modulus, 214 effective bulk creep function, 204–5 effective bulk modulus increase, 202 effective shear creep function, 209 shear modulus increase, 207 tension/compression creep function, 214 Young’s modulus increase, 213 properties shear, 195, 196 tension/compression, 195, 196 static loading effect, 494–5 creep and recovery under constant stress, 495 stress relaxation under constant strain, 495 viscoplastic creep modelling finite strain viscoplasticity, 285–96 material axes, 288 tensile creep compliance, 295 polymer matrix composites, 276–96 future trends, 296–7 small strain framework constitutive analysis, 277–85 creep model parameter values, 283 tensile creep compliance of epoxy matrix, 280 stress–strain curves 15° at 10–1, 10–3, 10–5s–1, 291 15° at 400 s–1, 293 15° at strain rate of 10–1 s–1, 294 15° at three strain rates and at 80°C, 294 30° at 10–1, 10–3, 10–5s–1, 292 30° at 700 s–1, 293 45° at 10–1, 10–3, 10–5s–1, 292 total creep compliance 15° off-axis composite at 333 and 353K, 281 30° off-axis composite at 333 and 353 K, 281 30° off-axis composite at 353 K, 285 60° off-axis composite at 333 and 353 K, 282 60° off-axis composite at 333K, 284 90° off-axis composite at 333 and 353 K, 282 viscoplastic deformation combined isotropic and kinematic hardening model, 240–3 evolution equations, 241–3 internal variables and thermodynamic forces, 240–1 constitutive modelling in polymer matrix composites, 234–68 framework for constitutive modelling, 239–51 comparison with experimental results, 246–51 isotropic and kinematic hardening model, 243–5 plane stress representation, 245–6 off-axis loading–unloading behaviour unidirectional AS4/PEEK laminate, 235 unidirectional T800H/epoxy laminate, 236 tension-compression asymmetry in initial anisotropy comparison with experimental results, 255–9 off-axis stress–strain curves, 255–6

© Woodhead Publishing Limited, 2011

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 43X

584 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 43X

Index

off-axis tensile and compressive stress– strain curves, 257–9 plane-stress pressure-sensitive effective stress, 253–4 pressure-modified viscoplasticity model, 254–5 pressure-sensitive effective stress, 252–3 shear flow differential parameter, 254 tension–compression asymmetry in initial anisotropy, 251–9 transient creep softening, 259–65 combined isotropic–kinematic-age hardening model, 261–2 comparison with experimental results, 263–5 modified kinematic hardening rule, 262–3 off-axis creep curve, 260 predicted vs observed off-axis creep curves, 264 viscoplastic strain (VP-strain), 242, 425–35, 525 development flax/starch composites, 532 hemp/lignin composites, 532 paper/phenol-formaldehyde composite, 531 rate, 242, 245, 262

viscoplasticity, 237, 275, 435 see also viscoplastic deformation characterisation in composites, 514–39 damage model, 267 models, 237, 239–45, 275–6 pressure-modified model, 254–5 Volterra integral, 10 Volterra integral equation, 190 Volterra integral operator, 190 volume creep, 203 Wang-Sun model, 237, 243–4 Weibull distribution, 331–2 Weibull strength distribution function, 409–10 Williams-Landel-Ferry equation (WLF equation), 12, 57–8, 62 WISPER spectrum (WInd SPEctrum Reference), 465 Y-periodicity, 116, 129 Yoon-Sun model, 237 Young’s modulus, 178, 189, 211–12, 217, 430 see also complex Young’s modulus; effective Young’s modulus

© Woodhead Publishing Limited, 2011