Introduction to the Micromechanics of Composite Materials

234 Pages·2016·6.81 MB·English

by  Yin, Huiming, Zhao, Yingtao

Introduction to the
Micromechanics of
Composite Materials
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Introduction to the
Micromechanics of
Composite Materials
Huiming Yin
Columbia University, New York
New York, USA
Yingtao Zhao
Beijing Institute of Technology
PR of China
A SPON PRESS BOOK
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2016 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20150904
International Standard Book Number-13: 978-1-4987-0730-5 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been
made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid-
ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright
holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this
form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may
rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti-
lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy-
ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the
publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://
www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,
978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For
organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface,xi
Acknowledgments,xiii
CHAPTER 1 (cid:2) Introduction 1
1.1 COMPOSITE MATERIALS 1
1.2 HISTORY OF MICROMECHANICS 5
1.3 A BIG PICTURE OF MICROMECHANICS-BASED MODELING 6
1.4 BASIC CONCEPTS OF MICROMECHANICS 7
1.4.1 RepresentativeVolumeElement 7
1.4.2 InclusionandInhomogeneity 8
1.4.3 Eigenstrain 9
1.4.4 Eshelby’sEquivalentInclusionMethod 10
1.5 CASE STUDY: HOLES SPARSELY DISTRIBUTED IN A PLATE 11
1.5.1 TheExactSolutiontoanInfinitePlateContainingaCircular
Hole 11
1.5.2 PredictionoftheEquivalentPropertyofanInfinitePlate
ContainingPeriodicHoles 13
1.6 EXERCISES 15
CHAPTER 2 (cid:2) Vectors and Tensors 17
2.1 CARTESIAN VECTORS AND TENSORS 17
2.1.1 SummationConventionintheIndexNotation 17
2.1.2 Vector 18
2.1.3 Tensor 19
2.1.4 SpecialTensors 20
2.2 OPERATIONS OF VECTORS AND TENSORS 20
2.2.1 MultiplicationofVectors 20
2.2.2 MultiplicationofTensors 22
2.2.3 IsotropicTensorsandStiffnessTensor 23
v
vi (cid:2) Contents
2.3 CALCULUS OF VECTOR AND TENSOR FIELDS 25
2.3.1 DelOperatorandOperations 25
2.3.1.1 Gradient 25
2.3.1.2 Divergence 26
2.3.1.3 Curl 26
2.3.2 Examples 27
2.3.3 TheGaussTheorem 27
2.3.4 Green’sTheoremandStokes’Theorem 28
2.4 POTENTIAL THEORY AND HELMHOLTZ’S DECOMPOSITION
THEOREM 28
2.4.1 ScalarandVectorPotentials 28
2.4.2 Helmholtz’sDecompositionTheorem 29
2.5 GREEN’S IDENTITIES AND GREEN’S FUNCTIONS 30
2.5.1 Green’sFirstandSecondIdentities 30
2.5.2 Green’sFunctionfortheLaplacian 30
2.5.3 Green’sFunctionintheSpaceofLowerDimensions 32
2.5.4 Example 34
2.6 ELASTIC EQUATIONS 35
2.6.1 StrainandCompatibility 35
2.6.2 ConstitutiveLaw 37
2.6.3 EquilibriumEquation 37
2.6.4 GoverningEquations 38
2.6.5 BoundaryValueProblem 39
2.7 GENERAL SOLUTION AND THE ELASTIC GREEN’S FUNCTION 40
2.7.1 Papkovich–Neuber’sGeneralSolution 40
2.7.2 Kelvin’sParticularSolution 42
2.7.3 ElasticGreen’sFunction 44
2.8 EXERCISES 46
CHAPTER 3 (cid:2) Spherical Inclusion and Inhomogeneity 49
3.1 SPHERICAL INCLUSION PROBLEM 49
3.2 INTRODUCTION TO THE EQUIVALENT INCLUSION METHOD 52
3.3 SPHERICAL INHOMOGENEITY PROBLEM 55
3.3.1 Eshelby’sEquivalentInclusionMethod 56
3.3.2 GeneralCasesofInhomogeneitywithaPrescribedEigenstrain 57
3.3.3 InterfaceConditionandtheUniquenessoftheSolution 59
Contents (cid:2) vii
3.4 INTEGRALS OF φ, ψ, φ , ψ AND THEIR DERIVATIVES IN 3D
P P
DOMAIN 60
3.5 EXERCISES 62
CHAPTER 4 (cid:2) Ellipsoidal Inclusion and Inhomogeneity 63
4.1 GENERAL ELASTIC SOLUTION CAUSED BY AN EIGENSTRAIN
THROUGH FOURIER INTEGRAL 63
4.1.1 AnEigenstrainintheFormofaSingleWave 63
4.1.2 AnEigenstrainintheFormofFourierSeriesandFourier
Integral 65
4.1.3 Green’sFunctionforIsotropicMaterials 66
4.2 ELLIPSOIDAL INCLUSION PROBLEMS 69
4.2.1 EllipsoidalInclusionwithaUniformEigenstrain 69
4.2.2 EllipsoidalInclusionwithaPolynomialEigenstrain 75
4.2.3 EllipsoidalInclusionwithaBodyForce 77
4.3 EQUIVALENT INCLUSION METHOD FOR ELLIPSOIDAL
INHOMOGENEITIES 79
4.3.1 ElasticSolutionforaPairofEllipsoidalInhomogeneitiesin
theInfiniteDomain 79
4.3.2 EquivalentInclusionMethodforPotentialProblemsof
EllipsoidalInhomogeneities 81
4.4 EXERCISES 83
CHAPTER 5 (cid:2) Volume Integrals and Averages in Inclusion and
Inhomogeneity Problems 85
5.1 VOLUME AVERAGES OF STRESS AND STRAIN 85
5.1.1 AverageStressandStrainforanInclusionProblem 85
5.1.2 AverageStressandStrainforanInhomogeneityProblem 86
5.1.3 Tanaka–Mori’sTheorem 88
5.1.4 ImageStressandStrainforaFiniteDomain 89
5.2 VOLUME AVERAGES IN POTENTIAL PROBLEMS 92
5.2.1 AverageMagneticFieldandFluxforanInclusionProblem 92
5.2.2 AverageMagneticFieldandFluxforanInhomogeneity
Problem 93
5.3 STRAIN ENERGY IN INCLUSION AND INHOMOGENEITY
PROBLEMS 94
5.3.1 StrainEnergyforanInclusioninanInfiniteDomain 94
viii (cid:2) Contents
5.3.2 StrainEnergyforanInclusioninaFiniteSolid 96
5.3.3 StrainEnergyforanInclusionwithBothanEigenstrainand
anAppliedLoad 98
5.3.4 StrainEnergyforanInhomogeneityProblem 100
5.4 EXERCISES 102
CHAPTER 6 (cid:2) Homogenization for Effective Elasticity Based on the
Energy Methods 103
6.1 HILL’S THEOREM 103
6.2 HILL’S BOUNDS 105
6.3 CLASSICAL VARIATIONAL PRINCIPLES 108
6.4 HASHIN–SHTRIKMAN’S VARIATIONAL PRINCIPLE 110
6.5 HASHIN–SHTRIKMAN’S BOUNDS 117
6.5.1 TheLowerBound 119
6.5.2 TheUpperBound 120
6.6 EXERCISES 120
CHAPTER 7 (cid:2) Homogenization for Effective Elasticity Based on the
Vectorial Methods 123
7.1 EFFECTIVE MATERIAL BEHAVIOR AND MATERIAL PHASES 123
7.2 MICROMECHANICS-BASED MODELS FOR TWO-PHASE
COMPOSITES 125
7.2.1 TheVoigtModel 125
7.2.2 TheReussModel 126
7.2.3 TheDiluteModel 127
7.2.4 TheMori–TanakaModel 128
7.2.5 TheSelf-ConsistentModel 129
7.2.6 TheDifferentialScheme 130
7.3 EXERCISES 132
CHAPTER 8 (cid:2) Homogenization for Effective Elasticity Based on the
Perturbation Method 133
8.1 INTRODUCTION 133
8.2 ONE-DIMENSIONAL ASYMPTOTIC HOMOGENIZATION 135
8.3 HOMOGENIZATION OF A PERIODIC COMPOSITE 139
8.4 EXCERCISES 143
Contents (cid:2) ix
CHAPTER 9 (cid:2) Defects in Materials: Void, Microcrack, Dislocation,
and Damage 145
9.1 VOIDS 145
9.2 MICROCRACKS 147
9.2.1 Penny-ShapeCrack 147
9.2.2 Slit-LikeCrack 150
9.2.3 FlatEllipsoidalCrack 152
9.2.4 CrackOpeningDisplacement,StressIntensityFactor,and
J-Integral 154
9.3 DISLOCATION 160
9.3.1 Introduction 160
9.3.2 BurgersVectorandBurgersCircuit 161
9.3.3 ContinuumModelforDislocation 161
9.4 DAMAGE 163
9.4.1 Category1σ > σ > σ > σ 167
1 cri 2 3
9.4.2 Category2σ > σ > σ > σ 168
1 2 cri 3
9.4.3 Category3σ > σ > σ > σ 168
1 2 3 cri
9.5 EXERCISES 170
CHAPTER 10 (cid:2) Boundary Effects on Particulate Composites 171
10.1 FUNDAMENTAL SOLUTION FOR SEMI-INFINITE DOMAINS 172
10.2 EQUIVALENT INCLUSION METHOD FOR ONE PARTICLE IN A
SEMI-INFINITE DOMAIN 174
10.3 ELASTIC SOLUTION FOR MULTIPLE PARTICLES IN A
SEMI-INFINITE DOMAIN 189
10.4 BOUNDARY EFFECTS ON EFFECTIVE ELASTICITY OF A
PERIODIC COMPOSITE 192
10.4.1 UniaxialTensileLoadingontheBoundary 194
10.4.2 UniformSimpleShearLoadingontheBoundary 197
10.5 INCLUSION-BASED BOUNDARY ELEMENT METHOD FOR
VIRTUAL EXPERIMENTS OF A COMPOSITE SAMPLE 201
10.6 EXERCISES 213
References,215
Index,219