Micromechanics of Heterogeneous Materials

Micromechanics of Heterogeneous Materials

Valeriy A. Buryachenko Unviersity of Dayton Research Institute

Valeriy A. Buryachenko University of Dayton Research Institute 300 College Park Dayton, OH 45469-0168 USA

Library of Congress Control Number: 2007922743

ISBN 978-0-387-36827-6 e-ISBN 978-0-387-68485-7 Printed on acid-free paper. 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com

Elena, Andrey, Alexander, Nina, Irinafor their love and support over the years

Preface

Materials that are either manufactured or occur in nature and used both in indus-try and in our daily lives (metals, rocks, wood, soil, suspensions, and biologicaltissue) are very seldom homogeneous, and have complicated internal structures.Although the combination of two or more constituents to produce materials withcontrolled distinguish properties has been exploited since at least ancient civi-lization, modern composite materials were developed only a few decades ago andhave found intensive application in contemporary life and in all branches of in-dustry. Establishment of a link between the structure and properties in orderto understand which kind of structure provides the necessary properties is anobjective of micromechanics, which exploits information about microtopologyand properties of constituents of the heterogeneous medium for development ofmathematical models predicting the macroproperties. The problem of microme-chanical modeling of the mechanical properties of engineering materials is todaya crucial part of the design process, and sample testing is usually performed onlyduring the nal stage for validation of the virtual design. An accuracy of theclassical trial and error testing method of the new materials and constructionsis no longer be aordable in modern industry and science.

Owing to wide applications of composite materials, their modeling has beendeveloped very intensively over recent decades, as reected in the numerouspapers and books only partially presented in the reference section of this book.A variety of materials and approaches appearing in apparently dierent contextsand among dierent scientic disciplines (solid mechanics, geophysics, solid-statephysics, hydromechanics, biomechanics, chemical technology, etc.) do not allowthe opportunity to investigate adequately the whole eld in a single book. Inlight of this, I was challenged with a natural question as to why it was neces-sary to write another book and what is the dierence between this book andthe ones published earlier. In parallel with this book, there are a few fundamen-tal books combining readily applicable results useful for material scientists witha signicant contribution to progress in theoretical research itself. However, afundamental dierence of this book is a systematic analysis of statistical distri-butions of local microelds rather than only eective properties based on theaverage eld concentrator factors inside the phases.

VIII Preface

The uniqueness of this book consists of the development and expressive rep-resentation of statistical methods quantitatively describing random structureswhich are most adopted for the subsequent evaluation of a wide variety ofmacroscopic transport, electromagnetic, and elastic properties of heterogeneousmedia. The popular methods in micromechanics are essentially one-particle onesthat are invariant with respect to statistical second and higher order quantitiesexamining the association of one particle relative to other particles. This bookexpressively reects the explosive progress of modern micromechanics resultingfrom the development of image analyses and computer-simulation methods onone hand and improved materials processing on the other hand, since processingcontrols the prescribed microstructure. With the appearance of new experimentaltechniques, it is now possible to study the microtopology of disordered materialsmuch more deeply to understand their properties. Modern techniques are alsoavailable to design materials with morphological properties that are suitablefor planned applications. This progress in micromechanics is based on methodsallowing for statistical mechanics of a multiparticle system considering n-pointcorrelation functions and direct multiparticle interaction of inclusions, and thebook presents a universally rigorous scheme for both analyses of microstructuresand prediction of macroscopic properties which leaves room for correction oftheir individual elements if improved methods are utilized for the analysis ofthese individual elements. The book successfully combines advanced numericalmethods for the analysis of a nite number of interacting inhomogeneities ineither the bounded or unbounded domain with analytical methods.

It should be mentioned that there are two coupled classes of micromechan-ical problems for which averaging is critical. Averaging is usually suitable forpredicting eective elastic properties. However, failure and elastoplastic defor-mations will depend on specic details of the local stress elds when uctua-tion is important. In the framework of computational micromechanics in sucha case, one must check the specic observable stress elds for many large sys-tem realizations of the microstructure with the use of an extremal statisticaltechnique. A more eective approach used in analytical mechanics is the esti-mation of the statistical moments of the stress eld at the interface betweenthe matrix and inhomogeneities. The inherent characteristics of this interfaceare critical to understanding the failure mechanisms usually localized at theinterface. In the framework of a unique scheme of the proposed multiparticleeective eld method (MEFM), we attempted to analyze the wide class of sta-tical and dynamical, local and nonlocal, linear and nonlinear micromechanicalproblems of composite materials with deterministic (periodic and nonperiodic),random (statistically homogeneous and inhomogeneous, so-called graded) andmixed (periodic structures with random imperfections) structures in boundedand unbounded domains, containing coated or uncoated inclusions of any shapeand orientation and subjected to coupled or uncoupled, homogeneous or inho-mogeneous external elds of dierent physical natures.

I do not pretend to cover the whole eld of micromechanics in this book(restricted by my interests); there are many other topics in micromechanics,of much industrial and scientic importance, that are either treated schemat-ically or only mentioned. In particular, the homogenization theory of periodic

Preface IX

structures, the geometrically nonlinear problems, ow in porous media, viscoelas-ticity problems, and cross-property relations are not considered at all in this bookwhile stochastic geometry, variation methods, propagation of waves in compos-ites, and multiscale discrete modeling are not treated in depth they deserve andsometimes were only mentioned. Interested readers are referred to the referencescited in the appropriate sections to achieve a deeper understanding of thesetopics. This book nalizes my research in micromechanics that began with thepapers published in 1986. It was written piece by piece at dierent times and indierent countries facing new challenging problems in micromechanics. The bookis suitable as a reference for researchers in dierent disciplines (applied mathe-maticians; physicists; geophysicists; material scientists; and electrical, chemical,civil, and mechanical engineers) working in micromechanics of heterogeneousmedia and providing a rigorous interdisciplinary treatment through experimen-tal investigation. The book is also appropriate as a textbook for an advancedgraduate course.

I gratefully acknowledge the nancial support for 30 years of research in mi-cromechanics of heterogeneous materials by the National Research Council, andthe Air Force Oce of Scientic Research (AFOSR) of the United States, Aus-trian Society for the Promotion of Scientic Research, Max-Planck-Society ofGermany, as well as by the Ministry of Higher Education, National Academy ofScience, and the Ministry of Machine Construction of USSR and Russia. Thisbook would not have been possible without this support. The excellent coopera-tion with the team at Springer is gratefully acknowledged; I owe special thanksto E. Tham, C. Simpson, and C. Womersly for their careful and patient edit-ing of this book. Also, thank you Elsevier for allowing permission to use Figs.3.13.3, 4.12, 4.13, 5.15.4, 8.1, 9.5, 9.6, 11.111.8, 12.2, 13.513.11, 14.114.4,15.115.5, 15.815.11, Tables 11.111.5 originally published in [133], [134], [136],[137], [146], [168], [179], [182], [184], [190], [191], [633]; Birkhauser Verlag AG forpermission to use Figs. 4.10, 4.11, 10.110.6, 15.7, originally published in [145],[152]; SAGE Publication for permission to use Figs. 4.4, 4.64.10, 12.412.6, orig-inally published in [166], [167]; ASME for allowing permission to use Figs. 12.11,15.7, 16.1016.14, originally published in [138], [187]; Begell House Inc. for per-mission to use Figs. 4.14.3, 12.3, 12.12, originally published in [144], [153], [195];Springer Science and Business Media for allowing permission to use Figs. 12.1,12.712.10, 15.6, 16.6, 16.7, 18.718.19 originally published in [131], [155], [185].

My scientic interests go back to my student days in Moscow State Uni-versity, where I had the good fortune of being nurtured by brilliant teach-ers and scientists such as, e.g., V. I. Arnold, G. I. Barenblat, A. A. Ilyushin,A. N. Kolmogorov, S. P. Novikov, A. N. Shiryaev, and my rst research adviserO. A. Oleinik. In later years I met most of the key players in the develop-ment of the subject, many of whom have had a profound inuence on my ownwork. I would like to acknowledge G. J. Dvorak, J. Fish, W. Kreher, E. Kroner,V. I. Kushch, A. M. Lipanov, N. J. Pagano, V. Z. Parton, F. G. Rammerstorfer,T. D. Shermergor, S. Torquato, and G. J. Weng for the helpful and inspiringexchanges of ideas, fantastic collaboration, stimulating discussions, encouragingremarks, and thoughtful criticism I have received over the years. At the riskof forgetting to include some names, I thank the sta of Air Force Research

X Preface

Laboratory, Ohio (and especially MLBC) and University of Dayton ResearchInstitute: K. L. Andersen, J. Baur, V. T. Bechel, J. Braun, S. Chellapilla,A. Crasto, S. Donaldson, T. Gibson, R. Hall, R. Y. Kim, R. J. Kerans, K. Lafdy,S. Lindsay, B. Maruyama, D. B. Miracle, O. O. Ochoa, V. Y. Perel, B. Regland,B. P. Rice, E. Ripberger, R. Roberts, A. Roy, G. Schoeppner, J. E. Spowart,K. Strong, G. P. Tandon, T. Benson Tolle, K. Thorp, M. Tudela, R. A. Vaia,and T. Whitney for the productive collaboration and cordial hospitality.

Last, but certainly not least, my deepest gratitude goes to my wife Elenaand our son Andrey for their love, devotion, care, tolerance, and understandingthat the most benecial scientic insights often come at time that were destinedto be spent with the family. Finally, I extend my greatest thanks to my fatherAlexander, mother Nina, and sister Irina for all the encouragement and supportthey have provided me throughout my life. I am pleased to dedicate this bookto them.

I am aware of the fact that the book may contain controversial statements,too personal or one-sided arguments, inaccuracies, and typographical errors. Anythe remarks and comments regarding the book will be fully appreciated (bury-ach@woh.rr.com).

Valeriy BuryachenkoDayton, Ohio

July 2006

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Classication of Composites and Nanocomposites . . . . . . . . . . . . . . 1

1.1.1 Geometrical Classication of Composite Materials (CM) . 21.1.2 Classication of Mechanical Properties of CM Constituents 51.1.3 Classication of CM Manufacturing . . . . . . . . . . . . . . . . . . . . 8

1.2 Eective Material and Field Characteristics . . . . . . . . . . . . . . . . . . . 81.3 Homogenization of Random Structure CM . . . . . . . . . . . . . . . . . . . . 121.4 Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Foundations of Solid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Elements of Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 The Theory of Strains and Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Basic Equations of Solid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Conservation Laws, Boundary Conditions, andConstitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 The Equations of Linear Elasticity . . . . . . . . . . . . . . . . . . . . . 282.3.3 Extremum Principles of Elastostatic . . . . . . . . . . . . . . . . . . . 29

2.4 Basic Equations of Thermoelasticity and Electroelasticity . . . . . . . 312.4.1 Thermoelasticity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.2 Electroelastic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.3 Matrix Representation of Some Symmetric Tensors . . . . . . 38

2.5 Symmetry of Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.6 Basic Equations of Thermoelastoplastic Deformations . . . . . . . . . . 47

2.6.1 Incremental Theory of Plasticity . . . . . . . . . . . . . . . . . . . . . . 472.6.2 Deformation Theory of Plasticity . . . . . . . . . . . . . . . . . . . . . . 49

3 Greens Functions, Eshelby and Related Tensors . . . . . . . . . . . . 513.1 Static Greens Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 The Second Derivative of Greens Function and Related Problems 54

3.2.1 The Second Derivative of Greens Function . . . . . . . . . . . . . 543.2.2 The Tensors Related to the Greens Function . . . . . . . . . . . 57

3.3 Dynamic Greens and Related Functions . . . . . . . . . . . . . . . . . . . . . . 583.4 Inhomogeneity in an Elastic Medium . . . . . . . . . . . . . . . . . . . . . . . . . 62

XII Contents

3.4.1 General Case of Inhomogeneity in an Elastic Medium . . . . 623.4.2 Interface Boundary Operators . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5 Ellipsoidal Inhomogeneity in the Elastic Medium . . . . . . . . . . . . . . 673.6 Eshelby Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.6.1 Tensor Representation of Eshelby Tensor . . . . . . . . . . . . . . . 713.6.2 Eshelby and Related Tensors in a Special Basis . . . . . . . . . 75

3.7 Coated Ellipsoidal Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.7.1 General Representation for the Concentrator

Factors for Coated Heterogeneity . . . . . . . . . . . . . . . . . . . . . . 793.7.2 Single Ellipsoidal Inclusion with Thin Coating . . . . . . . . . . 813.7.3 Numerical Assessment of Thin-Layer Hypothesis . . . . . . . . 84

3.8 Related Problems for Ellipsoidal Inhomogeneityin an Innite Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.8.1 Conductivity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.8.2 Scattering of Elastic Waves by Ellipsoidal Inclusion

in a Homogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.8.3 Piezoelectric Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4 Multiscale Analysis of the Multiple Interacting InclusionsProblem: Finite Number of Interacting Inclusions . . . . . . . . . . . 954.1 Description of Numerical Approaches Used

for Analyses of Multiple Interacting Heterogeneities . . . . . . . . . . . . 954.2 Basic Equations for Multiple Heterogeneities

and Numerical Solution for One Inclusion . . . . . . . . . . . . . . . . . . . . 984.2.1 Basic Equations for Multiple Heterogeneities . . . . . . . . . . . . 984.2.2 The Heterogeneity vi Inside an Imaginary Ellipsoid v0i . . . 1014.2.3 The Heterogeneity vi Inside a Nonellipsoidal

Imaginary Domain v0i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.2.4 Estimation of Concentrator Factor Tensors by FEA . . . . . . 105

4.3 Volume Integral Equation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.3.1 Regularized Representation of Integral Equations . . . . . . . . 1094.3.2 The Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3.3 Initial Approximation for Interacting Inclusions

in an Innite Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.3.4 First-Order and Subsequent Approximations . . . . . . . . . . . . 1174.3.5 Numerical Results for Two Cylindrical Inclusions

in an Innite Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.4 Hybrid VEE and BIE Method for Multiscale Analysis of

Interacting Inclusions (Macro Problem) . . . . . . . . . . . . . . . . . . . . . . 1204.4.1 Initial Approximation for the Fields Induced

by a Macroinclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.4.2 Initial Approximation in the Micro Problem . . . . . . . . . . . . 1224.4.3 The Subsequent Approximations . . . . . . . . . . . . . . . . . . . . . . 1234.4.4 Some Details of the Iteration Scheme

in Multiscale Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.4.5 Numerical Result for a Small Inclusion Near a Large One . 1264.4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Contents XIII

4.5 Complex Potentials Method for 2-D Problems . . . . . . . . . . . . . . . . . 130

5 Statistical Description of Composite Materials . . . . . . . . . . . . . . 1375.1 Basic Terminology and Properties of Random Variables and

Random Point Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.1.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.1.2 Random Point Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.1.3 Basic Descriptors of Random Point Fields . . . . . . . . . . . . . . 144

5.2 Some Random Point Field Distributions . . . . . . . . . . . . . . . . . . . . . . 1485.2.1 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.2.2 Statistically Homogeneous Clustered Point Fields . . . . . . . . 1505.2.3 Inhomogeneous Poisson Fields . . . . . . . . . . . . . . . . . . . . . . . . 1525.2.4 Gibbs Point Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.3 Ensemble Averaging of Random Structures . . . . . . . . . . . . . . . . . . . 1585.3.1 Ensemble Distribution Functions . . . . . . . . . . . . . . . . . . . . . . 1595.3.2 Statistical Averages of Functions . . . . . . . . . . . . . . . . . . . . . . 1645.3.3 Statistical Description of Indicator Functions . . . . . . . . . . . 1655.3.4 Geometrical Description and Averaging of Doubly

and Triply Periodic Structures . . . . . . . . . . . . . . . . . . . . . . . . 1715.3.5 Representations of ODF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

5.4 Numerical Simulation of Random Structures . . . . . . . . . . . . . . . . . . 1765.4.1 Materials and Image Analysis Procedures . . . . . . . . . . . . . . . 1785.4.2 Hard-Core Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1795.4.3 Hard-Core Shaking Model (HCSM) . . . . . . . . . . . . . . . . . . . . 1805.4.4 Collective Rearrangement Model (CRM) . . . . . . . . . . . . . . . 181

6 Eective Properties and Energy Methodsin Thermoelasticity of Composite Materials . . . . . . . . . . . . . . . . . 1856.1 Eective Thermoelastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.1.1 Hills Condition and Representative Volume Element . . . . . 1856.1.2 Eective Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886.1.3 Overall Thermoelastic Properties . . . . . . . . . . . . . . . . . . . . . . 191

6.2 Eective Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1946.3 Some General Exact Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.3.1 Two-Phase Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.3.2 Polycrystals Composed of Transversally

Isotropic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2076.4 Variational Principle of Hashin and Shtrikman . . . . . . . . . . . . . . . . 2096.5 Bounds of Eective Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

6.5.1 Hills Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2126.5.2 Hashin-Shtrikman Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2176.5.3 Bounds of Higher Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

6.6 Bounds of Eective Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2266.7 Bounds of Eective Eigenstrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

XIV Contents

7 General Integral Equations of Micromechanicsof Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2317.1 General Integral Equations for Matrix Composites

of Any Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2327.2 Random Structure Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

7.2.1 General Integral Equation for Random StructureComposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

7.2.2 Some Particular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2377.2.3 Comparison with Related Equations . . . . . . . . . . . . . . . . . . . 239

7.3 Doubly and Triply Periodical Structure Composites . . . . . . . . . . . . 2417.4 Random Structure Composites with Long-Range Order . . . . . . . . 2447.5 Triply Periodic Particulate Matrix Composites

with Imperfect Unit Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2467.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

8 Multiparticle Eective Field and Related Methods inMicromechanics of Random Structure Composites . . . . . . . . . . . 2498.1 Denitions of Eective Fields and Eective Field Hypotheses . . . 250

8.1.1 Eective Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.1.2 Approximate Eective Field Hypothesis . . . . . . . . . . . . . . . . 2538.1.3 Closing Eective Field Hypothesis . . . . . . . . . . . . . . . . . . . . . 2558.1.4 Eective Field Hypothesis and Composites

with One Sort of Inhomogeneities . . . . . . . . . . . . . . . . . . . . . 2558.2 Analytical Representation of Eective Thermoelastic Properties . 258

8.2.1 Average Stresses in the Components . . . . . . . . . . . . . . . . . . . 2588.2.2 Eective Properties of the Composite . . . . . . . . . . . . . . . . . . 2608.2.3 Some Related Multiparticle Methods . . . . . . . . . . . . . . . . . . . 262

8.3 One-Particle Approximation of the MEFMand Mori-Tanaka Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2648.3.1 One-Particle (QuasiCrystalline) Approximation of

MEFM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2648.3.2 Mori-Tanaka Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2698.3.3 Eective Properties Estimated via the MEF and MTM

at Qi Q0i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2728.3.4 Some Methods Related to the One-Particle

Approximation of the MEFM . . . . . . . . . . . . . . . . . . . . . . . . . 2778.3.5 Some Analytical Representations for Eective Moduli . . . . 280

9 Some Related Methods in Micromechanicsof Random Structure Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2839.1 Related Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

9.1.1 Combined MEFMPerturbation Method . . . . . . . . . . . . . . . 2839.1.2 Perturbation Method for Small Concentrations of

Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2869.1.3 Perturbation Method for Weakly Inhomogeneous Media . . 2879.1.4 Elastically Homogeneous Media with Random Field

of Residual Microstresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

Contents XV

9.2 Eective Medium Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2919.2.1 Application to Composite Materials . . . . . . . . . . . . . . . . . . . 2919.2.2 Analysis of Polycrystal Materials . . . . . . . . . . . . . . . . . . . . . . 296

9.3 Dierential Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2989.3.1 Scheme of the Dierential Method . . . . . . . . . . . . . . . . . . . . . 2989.3.2 One-Particle Dierential Method . . . . . . . . . . . . . . . . . . . . . . 3009.3.3 Multiparticle Dierential Method (Combination with

EMM and with MEFM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3019.4 Estimation of Eective Properties of Composites

with Nonellipsoidal Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3039.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

9.5.1 Composites with Spheroidal Inhomogeneities . . . . . . . . . . . . 3069.5.2 Composites Reinforced by Nonellipsoidal Inhomogeneities

with Ellipsoidal v0i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3119.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

10 Generalization of the MEFM in Random Structure MatrixComposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31510.1 Two Inclusions in an Innite Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 31610.2 Composite Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

10.2.1 General Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31910.2.2 Some Related Integral Equations . . . . . . . . . . . . . . . . . . . . . . 32110.2.3 Closing Assumption and the Eective Properties . . . . . . . . 32210.2.4 Conditional Mean Value of Stresses in the Inclusions . . . . . 323

10.3 First-order Approximation of the Closing Assumption andEective Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32410.3.1 General Equation for the Eective Fields i,j . . . . . . . . . 32410.3.2 Closing Assumptions for the Strain Polarization Tensor

(y)|; vi,xi; vj ,xj(y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32610.3.3 Eective Elastic Properties and Stress Concentrator

Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32910.3.4 Symmetric Closing Assumption . . . . . . . . . . . . . . . . . . . . . . . 32910.3.5 Closing Assumptions for the Eective Fields i,j,k . . . . . 330

10.4 Abandonment from the ApproximativeHypothesis (10.26) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

10.5 Some Particular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33410.5.1 Identical Aligned Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 33410.5.2 Improved Analysis of Composites with Identical Aligned

Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33710.5.3 Eective Field Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33910.5.4 Quasi-crystalline Approximation . . . . . . . . . . . . . . . . . . . . . . 341

10.6 Some Particular Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 342

11 Periodic Structures and Periodic Structures with RandomImperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34711.1 General Analysis of Periodic Structures and Periodic Structures

with Random Imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

XVI Contents

11.2 Triply Periodical Particular Matrix Composites in VaryingExternal Stress Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35111.2.1 Basic Equation and Approximative Eective Field

Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35111.2.2 The Fourier Transform Method . . . . . . . . . . . . . . . . . . . . . . . 35211.2.3 Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35311.2.4 Average Strains in the Components . . . . . . . . . . . . . . . . . . . . 35411.2.5 Eective Properties of Composites . . . . . . . . . . . . . . . . . . . . . 35511.2.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

11.3 Graded Doubly Periodical Particular Matrix Composites inVarying External Stress Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36111.3.1 Local Approximation of Eective Stresses . . . . . . . . . . . . . . 36111.3.2 Estimation of the Nonlocal Operator via the Iteration

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36311.3.3 General Relations for Average Stresses and Eective

Thermoelastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36411.3.4 Some Particular Cases for Eective Properties

Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36411.3.5 Doubly Periodic Inclusion Field in a Finite Stringer . . . . . . 36511.3.6 Numerical Results for Three-Dimensional Fields . . . . . . . . . 36711.3.7 Numerical Results for Two-Dimensional Fields . . . . . . . . . . 36911.3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

11.4 Triply Periodic Particulate Matrix Compositeswith Imperfect Unit Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37111.4.1 Choice of the Homogeneous Comparison Medium . . . . . . . . 37111.4.2 MEFM Accompanied by Monte Carlo Simulation . . . . . . . . 37411.4.3 Choice of the Periodic Comparison Medium.

General Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37611.4.4 The Version of MEFM Using the Periodic Comparison

Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38011.4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

12 Nonlocal Eects in Statistically Homogeneous andInhomogeneous Random Structure composites . . . . . . . . . . . . . . . 38512.1 General Analysis of Approaches in Nonlocal Micromechanics of

Random Structure Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38512.2 The Nonlocal Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39012.3 Methods for the Solution of the Nonlocal Integral Equation . . . . 392

12.3.1 Direct Quadrature Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 39212.3.2 The Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39212.3.3 The Fourier Transform Method for Statistically

Homogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39312.4 Average Stresses in the Components and Eective Properties

for Statistically Homogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . 39612.4.1 Dierential Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 39612.4.2 The Reduction of Integral Overall Constitutive

Equations to Dierential Ones . . . . . . . . . . . . . . . . . . . . . . . . 397

Contents XVII

12.4.3 Quasi-crystalline Approximation . . . . . . . . . . . . . . . . . . . . 39812.4.4 Numerical Analysis of Nonlocal Eects for Statistically

Homogeneous Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39912.5 Eective Properties of Statistically Inhomogeneous Media . . . . . . 403

12.5.1 Local Eective Properties of FGMs . . . . . . . . . . . . . . . . . . . . 40312.5.2 Elastically Homogeneous Composites . . . . . . . . . . . . . . . . . . 40612.5.3 Numerical Results of Estimation of Eective

Properties of FGMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40712.5.4 Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41212.5.5 Combined MEFM-Perturbation Method . . . . . . . . . . . . . . . . 41312.5.6 The MEF Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

12.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

13 Stress Fluctuations in Random Structure Composites . . . . . . . . 41713.1 Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

13.1.1 Exact Representation for First and Second Momentsof Stresses Averaged over the Phase Volumes . . . . . . . . . . . 419

13.1.2 Local Fluctuation of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . 42313.1.3 Correlation Function of Stresses . . . . . . . . . . . . . . . . . . . . . . . 42313.1.4 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . 425

13.2 Method of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42713.2.1 Estimation of the Second Moment of Eective Stresses . . . 42713.2.2 Implicit Representations for the Second Moment

of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42813.2.3 Explicit Estimation of Second Moments of Stresses Inside

the Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43013.2.4 Numerical Estimation of the Second Moments

of Stresses in the Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43113.2.5 Related Method of Estimations of the Second

Moments of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43313.3 Elastically Homogeneous Composites with Randomly

Distributed Residual Microstresses . . . . . . . . . . . . . . . . . . . . . . . . . . . 43413.3.1 The Conditional Average of the Stresses Inside

the Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43513.3.2 The Second Moment Stresses Inside the Phases . . . . . . . . . 43513.3.3 Numerical Evaluation of Statistical Residual Stress

Distribution in Elastically Homogeneous Media . . . . . . . . . 43813.4 Stress Fluctuations Near a Crack Tip in Elastically

Homogeneous Materials with Randomly Distributed ResidualMicrostresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44013.4.1 The Average and Conditional Mean Values of SIF for

Isolated Crack in a Composite Material . . . . . . . . . . . . . . . . 44113.4.2 Conditional Dispersion of SIF for a Crack

in the Composite Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44413.4.3 Crack in a Finite Inclusion Cloud. . . . . . . . . . . . . . . . . . . . . . 44513.4.4 Numerical Estimation of the First and Second Statistical

Moments of Stress Intensity Factors . . . . . . . . . . . . . . . . . . . 447

XVIII Contents

13.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

14 Random Structure Matrix Composites in a Half-Space . . . . . . 45114.1 General Analysis of Approaches in Micromechanics

of Random Structure Composites in a Half-space . . . . . . . . . . . . . . 45114.2 General Integral Equation, Denitions of the Nonlocal Eective

Properties, and Averaging Operations . . . . . . . . . . . . . . . . . . . . . . . . 45514.3 Finite Number of Inclusions in a Half-Space . . . . . . . . . . . . . . . . . . 458

14.3.1 A Single Inclusion Subjected to Inhomogeneous EectiveStress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

14.3.2 Two Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45914.4 Nonlocal Eective Operators of Thermoelastic Properties of

Microinhomogeneous Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46214.4.1 Dilute Concentration of Inclusions . . . . . . . . . . . . . . . . . . . . . 46214.4.2 c2 Order Accurate Estimation of Eective Thermoelastic

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46314.4.3 Quasi-crystalline Approximation . . . . . . . . . . . . . . . . . . . . . . 46514.4.4 Inuence of a Correlation Hole v0ij . . . . . . . . . . . . . . . . . . . . . 468

14.5 Statistical Properties of Local Residual Microstresses inElastically Homogeneous Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . 46914.5.1 First Moment of Stresses in the Inclusions . . . . . . . . . . . . . . 46914.5.2 Limiting Case for a Statistically Homogeneous Medium . . 47114.5.3 Stress Fluctuations Inside the Inclusions . . . . . . . . . . . . . . . . 472

14.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

15 Eective Limiting Surfaces in the Theory of NonlinearComposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48115.1 Local Limiting Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

15.1.1 Local Limiting Surface for Bulk Stresses . . . . . . . . . . . . . . . . 48215.1.2 Local Limiting Surface for Interface Stresses . . . . . . . . . . . . 48315.1.3 Fracture Criterion for an Isolated Crack . . . . . . . . . . . . . . . . 485

15.2 Eective Limiting Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48515.2.1 Utilizing Fluctuations of Bulk Stresses Inside the Phases . 48515.2.2 Utilizing Interface Stress Fluctuations . . . . . . . . . . . . . . . . . . 48715.2.3 Eective Fracture Surface for an Isolated Crack

in the Elastically Homogeneous Medium with RandomResidual Microstresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

15.2.4 Scheme of Simple Probability Modelof Composite Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

15.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49215.3.1 Utilizing Fluctuations of Bulk Stresses Inside the Phases . 49215.3.2 Utilizing Interface Stress Fluctuations . . . . . . . . . . . . . . . . . . 49915.3.3 Eective Energy Release Rate and Fracture Probability . . 501

15.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

Contents XIX

16 Nonlinear Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50516.1 Nonlinear Elastic Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

16.1.1 Popular Linearization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 50616.1.2 Modied Linearization Scheme . . . . . . . . . . . . . . . . . . . . . . . . 510

16.2 Deformation Plasticity Theory of Composite Materials . . . . . . . . . 51316.2.1 General Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51316.2.2 Elastoplastic Deformation of Composites

with an Incompressible Matrix . . . . . . . . . . . . . . . . . . . . . . . . 51616.2.3 General Case of Elastoplastic Deformation . . . . . . . . . . . . . . 517

16.3 Power-Law Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51716.4 ElasticPlastic Behavior of Elastically Homogeneous Materials

with Random Residual Microstresses . . . . . . . . . . . . . . . . . . . . . . . . . 52116.4.1 Leading Equations and Elastoplastic Deformations . . . . . . 52116.4.2 Numerical Results for Temperature-Independent

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52516.5 A Local Theory of Elastoplastic Deformations of Metal Matrix

Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52716.5.1 Geometrical Structure of the Components . . . . . . . . . . . . . . 52716.5.2 Average Stresses Inside the Components and

Overall Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52916.5.3 Elastoplastic Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53016.5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

17 Some related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53717.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

17.1.1 Basic Equations and General Analysis . . . . . . . . . . . . . . . . . 53717.1.2 Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53917.1.3 Self-Consistent Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54317.1.4 Nonlinear and Nonlocal Properties . . . . . . . . . . . . . . . . . . . . . 548

17.2 Thermoelectroelasticity of Composites . . . . . . . . . . . . . . . . . . . . . . . 54917.2.1 General Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54917.2.2 Generalized Hills Conditions and Eective Properties . . . . 55317.2.3 Eective Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55517.2.4 Two-Phase Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55617.2.5 Discontinuities of Generalized Fields at the Interface

Between Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55717.2.6 Phase-Averaged First and Second Moments

of the Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55817.3 Wave Propagation in a Composite Material . . . . . . . . . . . . . . . . . . . 561

17.3.1 General Integral Equations and Eective Fields . . . . . . . . . 56117.3.2 Fourier Transform of Eective Wave Operator . . . . . . . . . . 56417.3.3 Eective Wave Operator for Composites with Spherical

Isotropic Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

XX Contents

18 Multiscale Mechanics of Nanocomposites . . . . . . . . . . . . . . . . . . . . 57118.1 Elements of Molecular Dynamic (MD) Simulation . . . . . . . . . . . . . 571

18.1.1 General Analysis of MD Simulation of Nanocomposites . . . 57118.1.2 Foundations of MD Simulation and Their Use in

Estimation of Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . 57418.1.3 Interface Modeling of NC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576

18.2 Bridging Nanomechanics to Micromechanics in Nanocomposites . 57818.2.1 General Representations for the Local Eective Moduli . . . 57918.2.2 Generalization of Popular Micromechanical Methods to

the Estimations of Eective Moduli of NCs . . . . . . . . . . . . . 58018.3 Modeling of Nanober NCs in the Framework of Continuum

Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58218.3.1 Statistical Description of NCs with Prescribed Random

Orientation of NTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58218.3.2 One Nanober Inside an Innite Matrix . . . . . . . . . . . . . . . . 58318.3.3 Numerical Results for NCs Reinforced with Nanobers . . . 586

18.4 Modeling of Clay NCs in the Frameworkof Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59018.4.1 Existing Modeling of Clustered Materials and Clay NCs . . 59018.4.2 Estimations of Eective Thermoelastic Properties and

Stress Concentrator Factors of Clay NCs via the MEF . . . 59318.4.3 Numerical Solution for a Single Cluster in an Innite

Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59618.4.4 Numerical Estimations of Eective Properties

of Clay NCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59818.5 Some Related Problems in Modeling of NCs Reinforced with

NFs and Nanoplates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

19 Conclusion. Critical Analysis of Some Basic Concepts ofMicromechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611A.1 Parametric Representation of Rotation Matrix . . . . . . . . . . . . . . . . 611A.2 Second and Fourth-Order Tensors of Special Structures . . . . . . . . . 613

A.2.1 E-basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613A.2.2 P-basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614A.2.3 B-basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

A.3 Analytical Representation of Some Tensors . . . . . . . . . . . . . . . . . . . 619A.3.1 Exterior-Point Eshelby Tensor . . . . . . . . . . . . . . . . . . . . . . . . 619A.3.2 Some Tensors Describing Fluctuations

of Residual Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621A.3.3 Integral Representations for Stress Intensity Factors . . . . . 622

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679. . . . .

1Introduction

1.1 Classication of Composites and Nanocomposites

To introduce readers to the subject of composite (or heterogeneous) materials itis necessary to begin with the main notions and denitions which are presentedwith the understanding that any denition and classication scheme is neithercomprehensive nor complete.

The engineering level of composite material denition is given in ASTM D3878-95c: Composite material. A substance consisting of two or more materi-als, insoluble in one another, which are combined to form a useful engineeringmaterial possessing certain properties not possessed by the constituents. Thisdenition is usually added by the notion of length scale when the characteristicsizes of structural elements (the number of which is statistically large) are muchsmaller than characteristic sizes of composite medium but much larger than themolecular dimensions with the result that the mechanical state of structuralelements is described by the equations of continuous mechanics.

The materials used for structural applications are very seldom homogeneous,and because of this, the problem of modeling the mechanical properties of en-gineering materials has to be investigated in the framework of heterogeneousmaterials. The chief purpose of introduction of llers into composite materials(CM) is to create materials with a prescribed set of physico-mechanical proper-ties, which can be controlled, for example, by varying the type of base matrixand llers, and its particle size distribution, shape, and arrangement (see fordetails, e.g., [310], [384], [1135], [1141]). Thanks to considerable advances in CMresearch, their use in a broad range of industries machin building, construction,aerospace technology, etc. has become extensive. Development of an assortmentof CMs, and decisions concerning their possible scope of applicability and serviceconditions involve research into a very wide spectrum of problems: (1) choice ofthe matrix, llers, functional adjuvants, design of the optimal microtopology ofCM for creating materials with prescribed properties; (2) processing of a par-ticular CM into constructions, with optimization of the process parameters toprovide the best tradeo between the properties of each of the constituents of theCM; and (3) prediction of the serviceability of a construction under prescribedservice conditions.

2 Valeriy A. Buryachenko

By now an almost incomprehensible amount of data on the properties andprocessing technology of the individual constituents and CM has been amassed.However, the diversity of dierent materials with dierent geometrical and me-chanical properties of constituents is such that it is useful to have a classicationof these materials to choose the appropriate modeling tool. A number of classi-cation schemes are known that are distinguished by the criteria used for theseclassications: geometry of structural elements, their mechanical properties, andtheir manufacture. The description of CM encountered in materials science canbe useful to mechanical engineers if one avoids excessive intractable details andconsiders at rst the geometrical characteristics and then the mechanical prop-erties of constituent phases. Such a classied description should be favorable to aqualitative understanding of the relation between microstructure and propertieson one hand and of the relation between process parameters and microstructureon the other.

This section outlines a general classication of CM based on the geometry ofstructural elements, their mechanical properties, and their manufacture.

1.1.1 Geometrical Classication of Composite Materials (CM)

Geometrical microstructure of composites can be classied according to the fol-lowing features: (1) nature of structural element connectivity; (2) the charac-teristic relative sizes and shapes of structural elements; and (3) arrangement ofstructural elements.

The nature of structural element connectivity denes three groups of com-posite materials depending on the existence of a phase containing the others.To the rst group, one can assign the composite with one continuous over thevolume constituent called the matrix reinforced by the isolated inhomogeneitiesof the dierent shapes. Such composites are called the matrix ones. The secondgroup of composites forms the skeletal (or mutually penetrated or multiperco-lated) composites where at least two phases make a monolithic frame so theycan be seen as containing the other. The laminate composites as well as opencellular structures such as some metallic foam or porous silicon [383] are partic-ular cases of percolated structures. A more complicated structure is intrinsic toentangled materials such as cellulose-reinforced polymers [384], metallic wools,and glass wool. These materials have both a network connecting them to thepenetrated structure, and also an incomplete network with only one connectionto the network tree. It should be mentioned that variation of the constituent con-centration can lead to changeover from one connectivity group to another. Forexample, closed cell foams [383] can be considered as porous metal matrix com-posites, while the open cell foams have a percolated type of structure. The thirdgroup of composites is characterized by a structure in which there is no contactamong the elements of each constituent and the dierent phases are contiguousbut none can be seen as containing the other. For example, polycrystalline single-phase metal containing the structural elements of the same anisotropic crystalswith dierent orientations of crystallographic axes pertain to the third group ofcomposite materials, which in this case are called the heterogeneous materials.

1 Introduction 3

The grains ll the whole space and no element with one orientation contains theother element with another orientation.

The lack or existence of a continuous matrix in CM is an essential geometricalfeature, but the aspect ratio that may characterize the phases is also important.The matrix composites can be classied by the characteristic relative sizes andshapes of structural elements, as continuously reinforced (typically plies and con-tinuous tapes and bers), short ber (or chopped whiskers) reinforced (aspectratio noticeably larger than 1), particle reinforced (aspect ratio around 1), akeor platelet reinforced (aspect ratio smaller than 1), and hybrid reinforcement. Ifthe ratio of characteristic sizes of the reinforcement and the composite materialsis taken as a dimension of the composite structure, then the particle, ber, andlaminated composites are called zero-dimensional, one-dimensional, and three-dimensional ones, respectively. Polyreinforced (simple and hybrid) CMs containreinforcements from dierent materials. In the simple polyreinforced CM, thereinforcements from dierent materials have the same dimensions. In the hybridCM the reinforcements from dierent materials have dierent dimensions (forexample, bers and particles). The ultimate objective of hybridization is to cre-ate and combine an engineered proportion of material components for a range ofapplications, including the following hybrid systems: ternary alloys, hybrid par-ticulate composites, glass ber-containing toughened plastics, liquid crystallinepolymer hybrid composites, and nanocomposites (for references see [1194]).

The composites can be further structured for identication by noting thearrangement of structural elements. A distinction is made between the periodic(or deterministic) and random structure composites. For example, for wovenber-reinforced composites, one can indicate the following structures: periodic,cylindrical, braided, orthogonal interlock, and angle interlock structures. Sucha perfectly regular structure, of course, does not exist in actual cases, althoughthe periodic modeling can be quite useful, as it provides rigorous estimationswith a priori prescribed accuracy for various material properties. The randomstructure composites, in turn, should be subdivided into statistically homoge-neous structures that are translationally invariant under a shift of the spaceand into statistically inhomogeneous (or functionally graded, FG) compositesin which the arrangement (e.g., the concentration and orientation) of elementsis position-dependent. Statistically homogeneous composites are divided on sta-tistically isotropic one which structure is invariant with respect to the rotationand statistically anisotropic with anisotropy assessing from directional measure-ments, such as, e.g., variation of the chord size distribution with their orientation.A particular sort of FG composites is the clustered (or aggregated) compositesin which the areas with high and small concentration of reinforcements can beseparated. In a polymeric matrix the ller (even if present in minimum quanti-ties) is always more or less agglomerated. The ability of the ller to form thestructural network is largely responsible for its reinforcing action [688]. On theother hand, the structuralization in melt increases the viscosity of the materialand hampers its processability. Microscopic studies have conrmed the existenceof two types of primary structures in CMs, i.e., ller aggregates with particlesbound together rmly enough, and agglomeratessystems of weakly interrelatedaggregates. For a xed ller content, the viscosity and moduli of solid CM based

4 Valeriy A. Buryachenko

on the same matrix with agglomerates is always higher than that of the well-dispersed sample (for references see, e.g., [310], [688]). However, the mentionedbehavior of increasing the eective stiness with the agglomeration of the spher-ical particles changes drastically with variation of the inclusion shape. So, wellknown experimental data point to decrease of eective elastic properties with thegrowth of clustering (when the aligned clusters contain the parallel nanoplateswith a small aspect ratio 0.001), with the maximum eective stiness ofsilicate nanocomposites corresponding to the completely exfoliated structures[992]. At the same time, the strength of CM is usually lower in materials withagglomerates of spherical particles than in CM with uniformly distributed ller.Clustered composites in the simplest case have so-called ideal cluster structurethat contains either nite or innite, deterministic or random ellipsoidal domainscalled particle clouds distributed in the composite matrix. In so doing the con-centration of particles is a piecewise and homogeneous one within the areas ofellipsoidal clouds and composite matrix. An alternative case of clustered struc-tures is a dual scale one existing, e.g., in Al7%Si alloy [264], [384], where theelementary unit of the microstructure is a core of almost pure aluminum, sur-rounding by a thick layer that can be considered as CM consisting of aluminummatrix reinforced by silicon particles. The ratio of linear sizes of the core and thelayer as well as the ratio of the volume fraction of Si particles to the Al matrixin the layer are dened by both the composition of the alloy and technologicalfeatures.

As an example of composites combining random and periodic structure (alsocalled periodic structure with random imperfections), we can indicate the ini-tially periodic structure with randomly distributed damage as well as compos-ites with small random perturbation of inclusion location with respect to theperiodic grid due to manufacturing errors. From another aspect, the laminatedcomposites usually classied at a mesoscale level as either periodic or determin-istic structures can be considered at the microscale level as the FG compositeswith the statistically inhomogeneous packing of bers in each ply. Nanocompos-ites are randomly reinforced by the nanobers which in turn have a periodic ordeterministic structure.

This leads to the notion of coupling of dierent scales. The growing recogni-tion that the properties of materials involve dierent scales expressively reectsthe explosive progress of modern nano-and micromechanics resulting from the de-velopment of image analysis and computer-simulation methods on one hand andboth the advanced experimental technique (e.g., computed X-ray tomographyand electron microscopy) and improved materials processing on the other hand,since processing controls the prescribed structure. One recalls that at the upperlevel, the collective results of events or processes taking place at the lower levelare registered and are measured on a ner scale. Although in material model-ing, one can distinguish four levels of simulation: electronic structure, atomistic,microscale, and macroscale, micromechanics of CMs is usually concentrated juston the micro-macro connection of the multiscale concept when the continuummechanics technique is appropriate.

1 Introduction 5

1.1.2 Classication of Mechanical Properties of CM Constituents

The composites also can be categorized with respect to the mechanical propertiesof constituents according to the following features: polymer matrix composites(PMCs), metal matrix composites (MMCs), ceramic matrix composites (CMCs),carbon-carbon (CC) and hybrids. The polymer resins fall into two groups: ther-mosets and thermoplastics.

The cured thermoset matrix has polymer chains that have become irre-versibly crosslinked in a three-dimensional frame like a space truss. A curedthermoset matrix has the following mechanical properties: elasticity modulus2.510.0 GPa, tensile strength 30120 MPa, compressive strength 80250 MPa,and density 12001400 kg/m3. Among the most often used thermoset resins arepolyester, vinylester, epoxy, phenolic, polyurethane, and polybutadiene. Thermo-plastic resins are processed at high temperatures and solidify on cooling. Thatis, polymer crosslinking does not occur, and thermoplastic resins remain plas-tic, and can be reheated and reshaped. The solidied thermoplastic resins havehigh tensile strength (50100 MPa), high stiness (1.44.2 GPa), and high den-sity 11001700 kg/m3. The most widely used thermoset resins are polyethylene,polystyrene, polypropylene, acrilonitride-butadiene styrene, acetate, polycarbon-ate, polyvinyl chloride, polysulfone, polyphenylene sulde, and nylon.

MMCs have several advantages [1060], [1079]: high strength, elastic mod-uli, fracture toughness, and impact properties; high electrical and thermal con-ductivity; low sensitivity to temperature change or thermal shock; high surfacedurability, and low sensitivity to surface aws.

Reinforcement of the matrix composites is accomplished by the llers car-rying the majority of the loading. Conventional llers are separated into threecategories: reinforcing, active, and inert. The reinforcing class includes mainlyber composites. Disperse llers may also perform the reinforcing function, andthen they are called active. The basic properties of commonly used bers are asfollows: continuous glass bers have a diameter 320106 m, tensile strength 26GPa, stiness 50100 GPa, and 24002600 kg/m3. They are used in fabrication ofcomposite elements in the form of elementary yarns, rovings, and cloth. The ad-vantage of glass bers is in their high tensile strength and compressive strength,low cost of the materials and manufacturing, and good compatibility with poly-meric matrix. The disadvantages are associated with their low stiness and heatstability. The carbon bers have the following characteristics: 1.57.0 GPa tensilestrength, 150800 GPa elasticity modulus, 15002000 kg/m3 density. They havehigher stiness in comparison with glass bers, and are characterized by a neg-ative expansion coecient. Their mechanical properties do not deteriorate withtemperature increases up to 450C, a fact that allows the use of carbon bers incomposites with both polymeric and metal matrixes. Boron bers have 24 GPatensile strength, 370430 GPa modulus of elasticity, and 25002700 kg/m3 den-sity. The main advantage of boron bers is their high stiness and compressionstrength but they are rather expensive, more brittle and less processable thanglass bers. Boron bers are used with both polymeric and metal matrixes in theform of elementary or hybrid threads which are bundles of parallel continuousbers wound with an auxiliary glass thread. Organic bers (e.g., Kevlar) with

6 Valeriy A. Buryachenko

dierent polymer compositions and method of molding can be obtained with24 GPa tensile strength, 70150 GPa modulus of elasticity, 14101450 kg/m3

density. Organic bers preserve their initial characteristics up to 180C. Theyare characterized by high tensile strength and stiness in comparison with glassbers but under compression, composites with organic bers have substantiallylower properties than berglass composites.

It should be emphasized that the interfaces are not always as perfect assometimes assumed in micromechanical society, they may have dierent tensionand shear strength as well as dierent constitutive equations describing theirmechanical properties. The strength must be higher in CMs with llers fea-turing the absolute adhesion to the matrix than that in systems with little orno adhesion. The relative elongation and specic impact strength must, on thecontrary, go up with increasing adhesion. In so doing, the modulus of CMs issignicantly less sensitive to the adhesion properties of the interphase. The mostpromising method of modication of CM is by pretreating the ller or addingto the matrix specic depressants or modiers with the aim of creating chemi-cal bonds at the interphase. It has been convincingly demonstrated that on theller-polymer boundary there exist adsorbed layers of polymer characterized bya density greater than that of the unlled matrix. Macromolecular xation on theller surface, their orientation in the direction coplanar or normal to the surface,and condensation result in hard interphases. Alternatively, selective sorptionby the ller sorption of one of the matrix low molecular components may lead toplasticization of the boundary layers and appearance of soft interphases [310].The role of interfaces increases with reduction of a reinforcement size.

Fiber nanotube composites. Carbon bers are currently considered to be mostpromising. The recent discovery of carbon nanotubes (CNTs) by Iijima [490] hasgained ever-broaded interest due to providing unique properties generated bytheir structural perfection, small size, low density, high strength, and excellentelectronic properties (see, e.g., comprehensive reviews in [907], [1083], [1093]), de-pending on the diameter and chirality [208], [881], [1219]. Indeed, the longitudinalYoungs modulus of CNTs falls between 0.4 and 4.15 TPa, and a tensile strengthapproaching 100 GPa. Carbon nanotubes occur in two distinct forms, single-walled nanotubes (SWNT), which are composed of a graphite sheet rolled intoa cylinder and multi-walled nanotubes (MWNT), which consist of multiple con-centric graphite cylinders. Compared with multi-walled nanotubes, single-wallednanotubes are expensive and dicult to obtain and clean, but they have beenof great interest owing to their expected novel electronic, mechanical, and gasadsorption properties [306]. Nanobers can have a number of dierent internalstructures, wherein graphene layers are arranged as concentric cylinders, nestedtruncated cones, segmented structures, or stacked coins (see, e.g., [738]). Exter-nal morphologies include kinked and branched structures and diameter variation.A high aspect ratio of CNT and extraordinary mechanical properties (strengthand exibility) make them ideal reinforcing bers in nanocomposites, which canproduce advanced nanocomposites with improved stiness and strength [1092],[1099]. Tensile test on composites show that 1 wt% nanotube additions resultin 40 and 30% increase in elastic modulus and strength, respectively, indicatingsignicant load transfer across the nanotube matrix interface. An additional

1 Introduction 7

signicant impact of CNT on the eective properties of nanocomposites is basedon the fact that CNTs provide very high interfacial area if embedded in a matrix,and assemblies with unique architectures might be constructed by interconnectedCNTs. It is emphasized in [864] that the planar graphite surface structure of thesenanotubes provides relatively few sites suitable for facile chemical modication.It is believed that graphitic CNTs are perhaps more attractive additives for fab-ricating polymer/carbon nanober composite materials of enhanced mechanicaland physical properties due to their unusual atomic structure. Depending onthe metal catalyst and thermal syntheses conditions, CNTs can be grown withwidths ranging from 5 to 1000 nm and lengths ranging from 5 to 10 m. Thedesign and fabrication of these materials are performed on the nanometer scalewith the ultimate goal to obtain highly desirable macroscopic properties.

Silicate clay nanocomposites. Polymeric composites reinforced with claycrystals of nanometer scale recently attracted tremendous attention in materialengineering. A particular class of nanocomposites is clay nanocomposites (seeChapter 18 and [351], [721], [992], [1099], [1159], [1160] where additional referencescan be found). In general, layered silicate-reinforced presenting many challengesfor mechanics are conditionally divided into three types (see [198], [375], [486]):conventional composites where the layered silicate is presented as the originalaggregate state of the clay particles (tactoids) with no intercalation of the ma-trix material into the layered silicate; intercalated nanocomposites (modeled inChapter 18) where the matrix material is inserted in the form of thin (a fewnm) layers into the space between the parallel silicate layers of extremely smallthickness ( 1 nm); completely exfoliated or delaminated nanocomposites wherethe individual silicate platelets of 1 nm thickness are randomly dispersed in acontinuous polymer matrix. Single clay layers were proposed to be an ideal re-inforcing agent due to their extremely small aspect ratio ( =0.0010.01), thenanometer ller thickness being comparable to the scale of the polymer chainstructure, as well as to their high stiness (200400 GPa). A doubling of thetensile modulus and strength is achieved, and the heat distortion temperatureincreases by 100C for nylon-layered silicate nanocomposites containing as lit-tle as 2% vol. nanoplates [651], [1217]. However, exfoliation of layered miner-als is seriously hampered by the fact that sheet-like materials exhibit a strongtendency to agglomeration due to their large contact surfaces. Generally, exfo-liated nanocomposites demonstrate better properties than intercalated ones atthe same nanoplate concentration. Vaia and Giannelis [1125] emphasized thatthe presence of many chains at interfaces means that much of the polymer isreally interphase-like instead of having bulk-like properties that modify thethermodynamics of polymer chain conrmations and kinetics of chain motion.Since an interface limits the conrmations that polymer molecules can adopt,the properties of polymer in this interface are fundamentally dierent from thatof polymer far removed from the interface. Consequently, almost all matrixes inCMs with a few volume percent of nanoller can be considered as nanoscopicallyconned interfacial polymer. Thus, many mechanical, physical, and chemical fac-tors could potentially formulate the properties of nanocomposites, and a betterunderstanding of their relative contribution is needed.

8 Valeriy A. Buryachenko

1.1.3 Classication of CM Manufacturing

Classication of composites with respect to the manufacturing route divides theminto articial (i.e., man-made) and natural (i.e., in situ) composites. Arti-cial composites include long and short ber composites, particulate and platecomposites, interpenetrating multiphase composites, cellular and foam solids,and concrete. The manufacturing features can lead to signicant variation ofthe microtopology and mechanical properties of CM. For example, depending onthe thermomechanical treatment, the grains of a polycrystalline material can beisotropic spheroid (after full recrystallization and grain growth) or ake (afterrolling) or elongated cigars (after extrusion). Among the natural composites,one can mention polycrystals, soil, sandstone, ice, wood, bone, cell aggregates,and the Earths crust. Because the mechanical properties and the microtopol-ogy of composites are assumed to be known, their classication based on themanufacturing features will not be considered (see for details [1135], [1141]).

1.2 Eective Material and Field Characteristics

A wide class of phenomena for microinhomogeneous media are described bylinear partial dierential equations with random rapidly oscillating coecientsthat can be reduced to linear integral equations by the use of a Greens functionfor a selected problem for a homogeneous comparison medium in bounded orinnite domains. The replacement of a governing integral equation with rapidlyoscillating coecients by an equation with either constant or smoothly varyingoverall coecients is a main problem of micromechanics.

In classical elasticity, under assumption of geometrical and physical linear-ity, the relations between the local stress (x) and strain (x) tensors of mi-croinhomogeneous media are established by Hookes law: (x) = L(x)(x),(x) = M(x)(x), where L(x) andM(x) = L1 are the known local stiness andcompliance fourth-order tensors. Eective tensors of elastic properties L andM of statistically homogeneous medium are determined as the proportionalityfactors (x) = L(x), (x) = M(x) between the macroscopic stresses(x)and strains (x)averaged over the volume w of a large sample w contain-ing statistically large numbers of inhomogeneities. Thus, at the macroscale sig-nicantly exceeding the length scale of the material inhomogeneity (microscale),the behavior of a microinhomogeneous medium can be described by that of asuitably chosen equivalent homogeneous medium with the macroscopic stressesand strains slowly varying according to the macroscopic geometry and boundaryconditions. In so doing, the local stress and strains, and other values, can besplit into slow and fast (marked by primes) constituents that lead to the averageconstitutive equations presented in the following form: = L + L, = M + M, where f = f f [f = (x),(x),L(x),M(x)]. Obvi-ously, uctuating parts of stresses and strains can be expressed throughthe random inuence functions:

(x) = A(x), (x) = B(x), (1.1)

1 Introduction 9

which exist (but are generally unknown) due to linearity of the problem. Substi-tution of the micro-macro linkages (1.1) into the right-hand sides of the averagedconstitutive equations leads to representations for the eective stiness and com-pliance:

L = L+ LA, M = M+ MB. (1.2)Thus, the problem of eective properties estimation is reduced to nding A

(or B), which is turns out some functional involving all points y surroundingthe considered point x. The mentioned problem of many bodies appearing inmany sections of astronomy, mechanics, and physics can be generally solvedonly approximately.

The problem formulated in the preceding, which is central to our under-standing of micromechanics, looks very trivial. However, this illusion of trivialitydisappears if we recall the rst steps of research in this eld. Voight [1142] pro-posed in 1889 to consider the average of the elastic moduli of the crystallites ina polycrystal as eective elastic moduli of the aggregate L = L. Much laterin 1929, Reuss [934] proposed an analogous operation for obtaining the eectivecompliance M = M. Much later still, Hill [459] proved in 1952 that Voightand Reuss estimations are upper and lower bounds of the true eective elasticmoduli of the polycrystals: M1 L L.

The random values involved in the estimation of eective properties (1.2)can be classied as two sets: the values consisting of the material tensors (e.g.,L,M) and some functions of these tensors, and the values formed by the eldtensors (e.g., ,). The distribution parameters of random material tensors canbe evaluated through the known properties of constitutive phases and their ran-dom topological structure. The distributions of the random eld tensors areexpressed via the random material tensors by the use of the appropriate equa-tions of elasticity theory. The simplest characteristic of a random variable is aone-point probability density whose rst statistical moments dened the meanand dispersion. For example, the Voight and Reuss estimations are based on therst one-point distribution functions such as the phase volume fractions. Sig-nicantly more detailed information is contained in the multipoint probabilitydensities incorporating all points of the medium. So, a knowledge of at leasttwo-point distributions of the material tensors is necessary for estimation of thecorrections introduced the terms LA and MB into the eective propertiesrepresentations (1.2). The progress in micromechanics of random structure com-posites is based on the methods of statistical mechanics of a multiparticle systemconsidering n-point correlation functions and direct multiparticle interactions ofinhomogeneities.

For the sake of brevity, we intentionally sacriced rigorousness for sim-plication of a presentation of basic concepts of micromechanics. For exam-ple, the average can be introduced as either the volume or ensemble averages{f} = (w)1

wf(x) dx), f = n1 ni=1 f i(x), respectively, where f i is an is

realization of a random process in the point x. However, we will demonstratethat the size of RVE depends on the problem being considered and the methodused for solution of this problem, and, because of this, RVE is simply a notationconvenience reecting a nature of nonlocal interaction between microelementsthat interact without touching of elements as in the case of local interaction.

10 Valeriy A. Buryachenko

Probably the rst demonstration of the nonlocality principle was the discoveryof the Newtonian law of gravitation, which implies the existence of interactionbetween bodies remote from one another. A similar law for electric forces wasdemonstrated by Coulomb a century later.

The nonlocal nature of the eective constitutive law is manifested most con-spicuously in the case of the eventual abandonment of the hypothesis of sta-tistically homogeneous elds leading to a nonlocal coupling between statisticalaverages of the stress and strain tensors when the statistical average stress isgiven by an integral of the eld quantity weighted by some tensorial kernel, i.e.,the nonlocal eective elastic operator L: (x) = L(x,y)(y) dy. Per-haps the most challenging issue is how micromechanics can contribute to theunderstanding of the bridging mechanism between the coupled scales, which isdescribed by the nonlocal constitutive equations involving the parameters of arelevant eective nonlocal operator with the microstructure. Considering dis-persed media, this approach makes intuitive sense since the stress at any pointwill depend on the arrangement of the surrounding inclusions. Therefore thevalue of the statistical average eld will locally depend on its value at the otherpoints in its vicinity. This approach is especially appropriate if the inclusionnumber density varies over distances that are comparable to the particle size.

It should be mentioned that a multiscale hierarchy of composite materials istypically investigated via the use of the RVE concept when parameters of largerscale models are measured or calculated on a smaller scale. The RVE conceptserving as a bridge between dierent scales is rigorously justied if a eld scale(eld inhomogeneity) innitely exceeds a material scale (material inhomogene-ity). However, even in this case in a large class of problems, the length scalescannot be separated in this way; the coupling between them is strongly bidirec-tional. It takes place, for example, when the microscopic structure (such as, e.g.,the plastic strain and damage accumulation) develops due to some macroscopicforces. The coupling phenomenon is manifested even more visually in the case ofcomparable scales of the material scale of analyzed structures and the inhomo-geneity scale of internal stresses, which of necessity yields a nonlocal character ofthe constitutive law in the area of coupling of the mentioned scales, and, there-fore, the popular basic assumption acting as a bridge between adjusted scalescan be considered as an approximation of the real nonlocal constitutive law.

We need to emphasize the emphasized an importance of analysis of statisticaldistributions of local microelds rather than only eective properties, which inactuality is a secondary problem deriving from estimation of the average eldconcentrator factors inside the phases. Averaging is usually suitable for predict-ing eective elastic properties. However, failure and elastoplastic deformationswill depend on specic details of the local stress elds when uctuation is im-portant. A determination of failure mechanism usually localized at the interfacesis essentially important for the ceramic and metal matrix composites that haverelatively weak interfaces and multiple interphase regions. Plastic deformationsusually emanating in the vicinity of the interphases lead to loss of the homogene-ity of the mechanical properties of the constituents when the local propertiesof the phases become position dependent. In the framework of computationalmicromechanics in such a case, one must check the specic observable stress

1 Introduction 11

elds for many large system realizations of the microstructure with the use ofan extremal statistical technique. A more eective approach used in analyticalmechanics is the estimation of the statistical moments of the stress eld at theinterface between the matrix and inhomogeneities. The inherent characteristicsof this interface are critical to understanding the failure mechanisms usuallylocalized at the interface.

The widespread accessibility of a unique level of computing power providespromotes the argument that modeling and simulation, properly formulated andperformed, can be a credible partner to the traditional approaches of theory andexperiments. Micromechanical modeling and simulation of random structuresare becoming more and more ambitious as a result of advances in modern com-puter software and hardware. On one hand, some models have been developedwith the goal of minimizing empirical elements and assumptions. Specically,several examples are available in the area of nonlocal and nonlinear problemssuch as localization of the plastic strains and failure based on the estimations oflocal stress concentration factors. In many cases, the resolution of microscopicphenomena has led to improved accuracy, and oers the possibility of solving pre-viously intractable problems. On the other hand, there are ambitions to attackincreasingly large systems. Such methods, usually referred to as computationalmicromechanics, are based on the wide exploitation of Monte Carlo simulationwith forthcoming numerical analysis for each random realization of multiparticleinteractions of microinhomogeneities. However, at the present capacities of com-puter hardware and software, they are practical only for realizations containingno more than a few thousand inhomogeneities. In the case of periodic structures,the asymptotic unit cell homogenization method coupled with the nite elementmethod (FEM) has been justied to be the most successful tool especially forthe establishment of a link between the unit cell microstructure with periodicboundary conditions and overall properties of the heterogeneous media. Sucha perfectly periodic problem, of course, does not exist in actual cases, althoughperiodic modeling can be quite useful, since it provides rigorous estimations witha priori prescribed accuracy for various material properties. However, if the mi-croscopic crack is considered, it should lead to the unrealistic assumption thatthe unit cell model with a crack is repeated. The case of a single microcrack (im-perfection) in a composite material of periodic structure was investigated in theframework of the scheme of matching rules coupling the stresses in the outerand inner regions of the cell containing the imperfection. Implementation of thementioned scheme accompanied by the Monte Carlo simulation has prohibitivecomputation costs for the periodic structures with random eld of imperfections.Because of this, combining opportunities of computational micromechanics withbasic assumptions of analytical micromechanics is very promising and consid-ered in this book. The starting point of any average scheme is the choice of theelastic properties of the comparison medium that usually coincides with the ho-mogeneous matrix. This allows one to restrict the estimation of stress elds tothe elds just inside the inclusions. The choice of the comparison medium in theform of the perfect periodic structure has several benets, which are consideredin this book. These corrections incorporate specially adopted versions of the -nite element analysis (FEA) method of analysis of multiple interacting inclusion

12 Valeriy A. Buryachenko

problems into the modied versions of the MEFM combining a well-establishedtheory of homogenization of periodic media with the methodology of averagingof random structures in the innite domains.

Another barrier limiting Monte Carlo simulation appears in the analysis ofstatistically inhomogeneous random structures. Statistically homogeneous mediaare typically reduced to the averaging over a set of pseudorandom generationsbased on the determination of the exact values of the eective moduli of aperiodic unit cell containing a few tens of particles. Trivialness of this problemis explained by the a priori known fact that the eective elastic moduli is aconstant relating the average stresses and average strains. The case of the statis-tically inhomogeneous random structure is essentially more complicated becausethe eective elastic properties are now described by a nonlocal integral operator,which is a two-point one with a priori unknown structure. However, combiningthe MEFM with the numerical solution for a nite number of inclusions (consid-ered in this book) presents no diculties in the analysis of the indicated problem.In parallel with computational micromechanics mentioned above, classical or an-alytical micromechanics are also widely used and schematically considered in thenext section.

1.3 Homogenization of Random Structure CM

For random structure composites, in general, the problem of estimation of eec-tive properties cannot be solved exactly and is usually treated in an approximatemanner. According to Willis [1187], the numerous methods in micromechanicscan be classied into four broad categories: perturbation methods, self-consistentmethods of truncation of a hierarchy, variational methods, and the model meth-ods among which there are no rigorous boundaries. Perturbation methods areeectively utilized only for the analysis of weakly inhomogeneous media and arebased on the theory of a small parameter having controlled accuracy ([69], [70],[699]). Variational methods generate the bounds of eective properties via sub-stitution of approximate elds into the strict energy bounds obtained by thevolume average of the energy density with the help of lower-order correlationfunctions (see the fundamental works [775], [897], [1103], [1184], [1186] and ref-erences therein). The variational methods represent the most rigorous trend ofmicromechanics of statistically homogeneous media; however, in the case of eitherstatistically inhomogeneous structures or strong inhomogeneity, these bounds arenot instructive. The model methods based on the replacement of the real randomstructure of composites by a specic one (see, e.g., [237]) have intuitive physicalinterpretations; but, majority of model methods are self-contained: usually themodication of either geometrical structure of composites or constitutive equa-tions generates a need for creation of a new model with a priori unknown pre-diction accuracy. The variational and model replacement methods lead to someexact results, however, the analysis of these methods is considered schematicallyin this book.

We will consider in detail the self-consistent methods based on some approx-imate and closing assumptions for truncating an innite system of integral

1 Introduction 13

equations involved and its approximate solution. Solution methods consideredinclude a multiparticle eective eld method (MEFM), eective medium method,Mori-Tanaka method, dierential methods, and some others. The following alsotend to concentrate on methods and concepts and their possible generalizationsand connections of dierent methods rather than explicit results. For linearproblems, the estimation methods of eective properties (compliance, thermalexpansion, stored energy) as well as rst and second statistical moments in thecomponents for the general case of nonhomogeneity of the thermoelastic inclusionproperties are analyzed. Both statistically homogeneous and statistically inho-mogeneous, e.g., functionally graded, composites with a homogeneous matrixare considered, and the dependence of the eective properties (both local andnonlocal) on the concentration of the inclusions is presented. Both the Fouriertransform method and iteration method are analyzed. It is likely that all of themethods discussed for linear elastic static problems can be generalized and inmany cases are generalized to the dynamic case as well as to coupled ph