Lecture Notes in Appliedand Computational Mechanics

Volume 20

Series Editors

Prof. Dr.-Ing. Friedrich PfeifferProf. Dr.-Ing. Peter Wriggers

Lecture Notes in Applied and Computational Mechanics

Edited by F. Pfeiffer and P. Wriggers

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An Introductionto ComputationalMicromechanics

Corrected Second Printing

Tarek I. Zohdi • Peter Wriggers

With 66 Figures and 9 Tables

Prof. Tarek I. Zohdi Prof. Dr. Peter WriggersUniversity of California Universität HannoverDepartment of Mechanical Fachbereich Bauingenieur- undEngineering VermessungswesenEtcheverry Hall 6195 Institut für Baumechanik undBerkeley, CA 94720-1740 Numerische MechanikUSA Appelstr. 9Azohdi@me.berkeley.edu 30167 Hannover

Germanywriggers@ibnm.uni-hannover.de

ISBN: 978-3-540-77482-2 e-ISBN: 978-3-540-32360-0

Lecture Notes in Applied and Computational Mechanics ISSN 1613-7736

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Preface

Ideally, in an attempt to reduce laboratory costs, one would like to make a predictionof a new material’s behavior by numerical simulation, with the primary goal beingto accelerate trial and error experimental testing. The recent dramatic increase incomputational power available for mathematical modeling and simulation raises thepossibility that modern numerical methods can play a significant role in the analy-sis of heterogeneous microstructures. This fact has motivated the work that will bepresented in this monograph, which contains basic homogenization theory, as wellas introductions to topics such as microstructural optimization and multifield anal-ysis of heterogeneous materials. The text can be viewed as a research monographsuitable for use in a first year graduate course for students in the applied sciences,mechanics and mathematics that have an interest in the computational microme-chanical analysis of new materials.

Berkeley, USA, June 2004 Tarek I. ZohdiHannover, Germany, June 2004 Peter Wriggers

v

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Basic Concepts in Micro–Macro Modeling . . . . . . . . . . . . . . . . . . . . . 31.2 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Objectives of this Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Some Basics of the Mechanics of Solid Continua . . . . . . . . . . . . . . . . . . . 72.1 Kinematics of Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Deformation of Line Elements . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Infinitesimal Strain Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 The Jacobian of the Deformation Gradient . . . . . . . . . . . . . . . 10

2.2 Equilibrium / Kinetics of Solid Continua . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Postulates on Volume and Surface Quantities . . . . . . . . . . . . . 122.2.2 Balance Law Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Referential Descriptions of Balance Laws . . . . . . . . . . . . . . . . . . . . . . 142.4 The First Law of Thermodynamics/An Energy Balance . . . . . . . . . . . 162.5 The Second Law of Thermodynamics/A Restriction . . . . . . . . . . . . . . 172.6 Linearly Elastic Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6.1 The Infinitesimal Strain Case . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6.2 Material Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6.3 Material Constant Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 262.6.4 Consequences of Positive-Definiteness . . . . . . . . . . . . . . . . . . 27

2.7 Hyperelastic Finite Strain Material Laws . . . . . . . . . . . . . . . . . . . . . . . 292.7.1 Basic Requirements for Finite Strain Laws . . . . . . . . . . . . . . . 292.7.2 Determination of Material Constants . . . . . . . . . . . . . . . . . . . . 31

2.8 Moderate Strain Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Fundamental Weak Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1 Direct Weak Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.2 Some Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.3 The Principle of Minimum Potential Energy . . . . . . . . . . . . . . 413.1.4 Complementary Weak Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 42

vii

viii Contents

4 Fundamental Micro–Macro Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Testing Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1.1 The Average Strain Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.2 The Average Stress Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.3 Satisfaction of Hill’s Energy Condition . . . . . . . . . . . . . . . . . . 49

4.2 The Hill-Reuss-Voigt Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4 Classical Micro–Macro Mechanical Approximations . . . . . . . . . . . . . 52

4.4.1 The Asymptotic Hashin-Shtrikman Bounds . . . . . . . . . . . . . . 524.4.2 The Concentration Tensor: Microfield Behavior . . . . . . . . . . . 534.4.3 The Eshelby Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.4.4 Dilute Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.4.5 The Mori-Tanaka Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.4.6 Further Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5 Micro-Geometrical (Manufacturing) Idealizations . . . . . . . . . . . . . . . 594.5.1 Upper and Lower Variational Bounds . . . . . . . . . . . . . . . . . . . 594.5.2 Proof of Energetic Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.5.3 Uses to Approximate the Effective Property . . . . . . . . . . . . . . 62

5 A Basic Finite Element Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1 Finite Element Method Implementation . . . . . . . . . . . . . . . . . . . . . . . . 635.2 FEM Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 Global/local Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 Differential Properties of Shape Functions . . . . . . . . . . . . . . . . . . . . . . 675.5 Differentiation in the Referential Coordinates . . . . . . . . . . . . . . . . . . . 705.6 Post Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.7 Accuracy of the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . 755.8 Local Adaptive Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.8.1 A-Posteriori Recovery Methods . . . . . . . . . . . . . . . . . . . . . . . . 775.8.2 A-Posteriori Residual Methods . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.9 Solution of Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.9.1 Krylov Searches and Minimum Principles . . . . . . . . . . . . . . . 805.9.2 The Method of Steepest Descent . . . . . . . . . . . . . . . . . . . . . . . 805.9.3 The Conjugate Gradient Method . . . . . . . . . . . . . . . . . . . . . . . 825.9.4 Accelerating Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.10 Remark on Penalty Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 Computational/Statistical Testing Methods . . . . . . . . . . . . . . . . . . . . . . . . 856.1 A Boundary Value Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2 Numerical Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2.1 Topological Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3 Elementally Averaged Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.4 Iterative Krylov Solvers/ Microstructural “Correctors” . . . . . . . . . . . . 916.5 Overall Testing Process: Numerical Examples . . . . . . . . . . . . . . . . . . . 92

6.5.1 Successive Sample Enlargement . . . . . . . . . . . . . . . . . . . . . . . . 92

Contents ix

6.5.2 Multiple Sample Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.6 A Minimum Principle Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.6.1 A Proof Based on Partitioning Results . . . . . . . . . . . . . . . . . . . 976.6.2 Relation to the Material Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.7 Dependency on Volume Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.8 Increasing the Number of Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.9 Increasing Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Various Extensions and Further Interpretations of Partitioning . . . . . 1077.1 Partitioning and Traction Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.1.1 Isolating the Subsampling Error/NumericalError Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.2 An Ergodic Interpretation of the Results . . . . . . . . . . . . . . . . . . . . . . . 1097.3 Statistical Shifting Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.4 Partitioning and Ensemble Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.4.1 Primal Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.4.2 Complementary Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.4.3 Homogenized Material Orderings . . . . . . . . . . . . . . . . . . . . . . 1157.4.4 Embedded Orthogonal Monotonicities . . . . . . . . . . . . . . . . . . 115

7.5 Moment Bounds on Population Responses . . . . . . . . . . . . . . . . . . . . . 1177.5.1 First Order (Average) Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.5.2 Second Order (Standard Deviation) Bounds . . . . . . . . . . . . . . 1177.5.3 Third Order (Skewness) Bounds . . . . . . . . . . . . . . . . . . . . . . . . 118

7.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8 Domain Decomposition Analogies and Extensions . . . . . . . . . . . . . . . . . . 1218.1 Boundary Value Problem Formulations . . . . . . . . . . . . . . . . . . . . . . . . 1218.2 Error in the “Broken” Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2.1 Multiscale Proximity Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.2.2 Uses of the Bounds for Domain Decomposition . . . . . . . . . . . 129

8.3 The Connection to Material Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318.4 A “total” Orthogonal Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1348.5 Iterative Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.5.1 Method I: Global/Local CG Iterations . . . . . . . . . . . . . . . . . . . 1368.5.2 Method II: Iterative Equilibration . . . . . . . . . . . . . . . . . . . . . . . 137

9 Nonconvex–Nonderivative Genetic Material Design . . . . . . . . . . . . . . . . 1459.1 Computational Material Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1469.2 Characteristic of Such Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9.2.1 Nonconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1479.2.2 Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

9.3 Introduction of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1509.4 Fatigue-Type Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

9.4.1 Classical Fatigue Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1529.4.2 Construction of a Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

x Contents

9.4.3 Qualitative Behavior of the Fatigue Constraints . . . . . . . . . . . 1549.5 Nonconvex–Nonderivative Genetic Search . . . . . . . . . . . . . . . . . . . . . 1559.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.7 Scope of Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

10 Modeling Coupled Multifield Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 16310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16310.2 A Model Problem Involving Multifield Processes in Multiphase

Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16410.3 Constitutive Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

10.3.1 An Energy Balance Including Growth . . . . . . . . . . . . . . . . . . . 16610.3.2 Mass Transfer and Reaction-Diffusion Models . . . . . . . . . . . . 168

10.4 Staggered Multifield Weak Formulations . . . . . . . . . . . . . . . . . . . . . . . 17010.4.1 A Recursive Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17010.4.2 Convergence and Contraction-Mapping Time Stepping

Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17310.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17510.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

11 Closing Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Chapter 1Introduction

A key to the success of many modern structural components is the tailored behaviorof the material. A relatively inexpensive way to obtain macroscopically desired re-sponses is to enhance a base material’s properties by the addition of microscopicmatter, i.e. to manipulate the microstructure. Accordingly, in many modern engi-neering designs, materials with highly complex microstructures are now in use. Themacroscopic characteristics of modified base materials are the aggregate responseof an assemblage of different “pure” components, for example several particles orfibers suspended in a binding matrix material (Fig. 1.1). Thus, microscale inhomo-geneities are encountered in metal matrix composites, concrete, etc. In the construc-tion of such materials, the basic philosophy is to select material combinations toproduce desired aggregate responses. For example, in structural engineering appli-cations, the classical choice is a harder particulate phase that serves as a stiffeningagent for a ductile, easy to form, base matrix material.

If one were to attempt to perform a direct numerical simulation, for exampleof the mechanical response of a macroscopic engineering structure composed ofa microheterogeneous material, incorporating all of the microscale details, an ex-tremely fine spatial discretization mesh, for example that of a finite element mesh,would be needed to capture the effects of the microscale heterogeneities. The re-sulting system of equations would contain literally billions of numerical unknowns.Such problems are beyond the capacity of computing machines for the foreseeablefuture. Furthermore, the exact subsurface geometry is virtually impossible to ascer-tain throughout the structure. In addition, even if one could solve such a system,the amount of information to process would be of such complexity that it wouldbe difficult to extract any useful information on the desired macroscopic behavior.It is important to realize that solutions to partial differential equations, of even lin-ear material models, at infinitesimal strains, describing the response of small bodiescontaining a few heterogeneities are still open problems. In short, complete solu-tions are virtually impossible.

Because of these facts, the use of regularized or homogenized material models(resulting in smooth coefficients in the partial differential equations) are common-place in practically all branches of the physical sciences. The usual approach is

1

2 1 Introduction

Fig. 1.1 Doping a basematerial with particulateadditives

new "material"

macroscopicstructure

base materialadditives

mixingpropeller

to compute a constitutive “relation between averages”, relating volume averagedfield variables. Thereafter, the regularized properties can be used in a macroscopicanalysis (Fig. 1.2). The volume averaging takes place over a statistically represen-tative sample of material, referred to in the literature as a representative volumeelement (RVE). The internal fields to be volumetrically averaged must be computedby solving a series boundary value problems with test loadings. Such regulariza-tion processes are referred to as “homogenization”, “mean field theories”, “theoriesof effective properties”, etc. For overviews, we refer the interested reader to Jikovet al. [103] for mathematical aspects or to Aboudi [1], Hashin [76], Mura [150] andNemat-Nasser and Hori [151] for mechanically inclined accounts of the subject.

For a sample to be statistically representative it must usually contain a largenumber of heterogeneities (Fig. 1.3) and, therefore, the computations over the RVEare still extremely large, but are of reduced computational effort in comparison witha direct attack on the “real” problem. Historically, most classical analytical meth-ods for estimating the macroscopic response of such engineering materials have

structure witheffective properties

IE(1) IE(2)

Ω

actual structure

effectivepropertiesdetermined

σ = IE∗ : ε

Ω

IE∗

Fig. 1.2 The use of effective properties in structural engineering

1.1 Basic Concepts in Micro–Macro Modeling 3

L1

Ω

L1 >> L2 >> L3 L2

L3

Fig. 1.3 The size requirements of a representative volume element

a strongly phenomenological basis and are, in reality, non-predictive of materialresponses that are unknown a-priori. This is true even in the linearly elastic, infinites-imal strain, range. In plain words, such models require extensive experimental datato “tune” parameters that have little or no physical significance. Criticisms, such asthe one stated, have led to computational approaches which require relatively simpledescriptions on the microscale, containing parameters that are physically meaning-ful. In other words, the phenomenological aspects of the material modeling are re-duced, with the burden of the work being shifted to high performance computationalmethods. Stated clearly, the “mission” of computational micro-macro mechanicsis to determine relationships between the microstructure and the macroscopic re-sponse or “structural property” of a material, using models on the microscale thatare as simple as possible.1

1.1 Basic Concepts in Micro–Macro Modeling

Initially, for illustration purposes, we consider the relatively simple case of linearelasticity. In this context, the mechanical properties of microheterogeneous materi-als are characterized by a spatially variable elasticity tensor IE. Typically, in order tocharacterize the (homogenized) effective macroscopic response of such materials, arelation between averages

〈σ 〉Ω = IE∗ : 〈ε〉Ω (1.1)

is sought, where

〈·〉Ωdef=

1|Ω |

∫Ω·dΩ , (1.2)

1 Appropriately, the following quote, attributed to Albert Einstein: “A model should be as simpleas possible, but not simpler”, is a guiding principle.

4 1 Introduction

and where σ and ε are the stress and strain tensor fields within a statistically repre-sentative volume element (RVE) of volume |Ω |. The quantity, IE∗, is known as theeffective property and is the elasticity tensor used in usual structural scale analyses.Similarly, one can describe other effective quantities such as conductivity or dif-fusivity, in virtually the same manner, relating other volumetrically averaged fieldvariables. However, for the sake of brevity, we initially treat linear elastostatics prob-lems.2 It is emphasized that effective quantities such as IE∗ are not material prop-erties, but relations between averages. A more appropriate term might be “apparentproperty”, which is elaborated upon in Huet [88, 89]. However, to be consistent withthe literature, we continue to refer to IE∗ by the somewhat inaccurate term effective“property”.

1.2 Historical Overview

The analysis of microheterogeneous materials is not a recent development. Withinthe last 150 years, estimates of effective responses have been made under a varietyof assumptions on the internal fields within the microstructure. Works dating backat least to Maxwell [138, 139] and Lord Rayleigh [172] have dealt with determin-ing overall macroscopic transport phenomena of materials consisting of a matrixand distributions of spherical particles. Voigt [212] is usually credited with the firstanalysis of the effective mechanical properties of the microheterogeneous solids,with a complementary contribution given later by Reuss [173]. Voigt assumed thatthe strain field within an aggregate sample of heterogeneous material was uniform,leading to 〈IE〉Ω as an approximation of the effective property, while Reuss approx-imated the stress fields within the aggregate of polycrystalline material as uniform.If the Reuss field is assumed within the RVE, then an approximation for the effectiveproperty is 〈IE−1〉−1

Ω . A fundamental result (Hill [79]) is

〈IE−1〉−1Ω ≤ IE∗ ≤ 〈IE〉Ω . (1.3)

These inequalities mean that the eigenvalues of the tensors IE∗ − 〈IE−1〉−1Ω and

〈IE〉Ω −IE∗ are non-negative. Therefore, one can interpret the Voigt and Reuss fieldsas providing two microfield extremes, since the Voigt stress field is one where thetractions at the phase boundaries cannot be in equilibrium, i.e. statically inadmissi-ble, while the implied Reuss strains are such that the heterogeneities and the matrixcould not be perfectly bonded, i.e. the field is kinematically inadmissible. Typically,the bounds are quite wide and provide only rough qualitative information.

Within the last 50 years improved estimates have been pursued. For example,the Dilute family of methods assume that there is no particle-particle interaction.With this assumption one requires only the solution to a single ellipsoidal particle

2 By direct extension, in the geometrically nonlinear regime, such as 〈S〉Ω0 vs 〈E〉Ω0 or 〈P〉Ω0 vs〈F〉Ω0 , etc ..., where P and S are the First and Second Piola Kirchhoff stresses, E is the Green-Lagrange strain, F is the deformation gradient and where Ω0 is the initial reference domain.

1.3 Objectives of this Monograph 5

in an unbounded domain. This is the primary use of the elegant Eshelby [36] for-malism, based on eigenstrain concepts, which is used to determine the solution tothe problem of a single inclusion embedded in an infinite matrix of material underuniform exterior loading. By itself, this result is of little practical interest, howeverthe solution is relatively compact and easy to use and, thus, has been the founda-tion for the development of many approximation methods based on non-interactingand weakly interacting (particle–particle) assumptions. Non-interaction of parti-cles is an unrealistic assumption for materials with randomly dispersed particulatemicrostructure, even at a few percent volume fraction.

One classical attempt to improve the Dilute family of approximations is throughideas of self-consistency (Budiansky [19], Hill [80]). For example, in the standardSelf-Consistent method, the idea is simply that the particle “sees” the effectivemedium instead of the matrix in the calculations. In other words the “matrix ma-terial” in the Eshelby analysis is simply the effective medium. Unfortunately, theSelf-Consistent method can produce negative effective bulk and shear responses,for voids, with pore volume fractions of 50% and higher. For rigid inclusions it pro-duces infinite effective bulk responses for any volume fraction and infinite effectiveshear responses above 40%. For details, see Aboudi [1]. Attempts have also beenmade to improve these approaches. For example the Generalized Self-Consistentmethod encases the particles in a shell of matrix material surrounded by the effec-tive medium (see Christensen [27]). However, such methods also exhibit problems,primarily due to mixing scales of information in a phenomenological manner, whichare critically discussed at length in Hashin [76]. For a relatively recent and thoroughanalysis of a variety of classical approaches, such as the ones briefly mentionedhere, see Torquato [201, 202, 203, 204].

In addition to ad-hoc assumptions and estimates on the interaction betweenmicroscale constituents, many classical methods of analysis treat the microstruc-ture as being a regular, periodic, infinite array of identical cells. Such an assumptionis not justifiable for virtually any real material. In this monograph we do not con-sider the highly specialized case of periodic microstructures. There exists a largebody of literature for periodic idealizations, and we refer the reader, for example, toAboudi [1] for references.

1.3 Objectives of this Monograph

It is now commonly accepted that some type of numerical simulation is necessaryin order to determine more accurate micro–macro structural responses. Ideally, inan attempt to reduce laboratory expense, one would like to make predictions ofa new material’s behavior by numerical simulations, with the primary goal beingto accelerate the trial and error laboratory development of new high performancematerials. Within the last 20 years, many numerical analyses have been performedtwo-dimensionally, which, in contrast to many structural problems, is relativelymeaningless in micro–macro mechanics. For a statistically representative sample of

6 1 Introduction

microstructural material, three-dimensional numerical simulations are unavoidablefor reliable results. As we have indicated, the analysis of even linearly-elasticmaterials, at infinitesimal strains, composed of a homogeneous matrix filled withmicroscale heterogeneities, is still extremely difficult. However, the recent dramaticincrease in computational power available for mathematical modeling and simula-tion raises the possibility that modern numerical methods can play a significant rolein the analysis of heterogeneous structures, even in the nonlinear range. This fact,among others, has motivated the work that will be presented in this monograph.

Chapter 2Some Basics of the Mechanics of Solid Continua

Throughout this work, boldface symbols imply vectors or tensors. For the innerproduct of two vectors (first order tensors) u and v we have in three dimensions,u ·v = viui = u1v1 +u2v2 +u3v3, where Cartesian bases and Einstein index summa-tion notation are used. At the risk of over simplification, we will ignore the differencebetween second order tensors and matrices. Furthermore, we exclusively employ aCartesian bases. Readers that feel uncomfortable with this approach should consultthe wide range of texts which point out the subtle differences, for example the textsof Malvern [133], Marsden and Hughes [137] and others listed in the references.Accordingly, if we consider the second order tensor A = Aik ei ⊗ ek with its matrixrepresentation

[A] def=

⎡⎣A11 A12 A13

A21 A22 A23

A31 A32 A33

⎤⎦ , (2.1)

then a first order contraction (inner product) of two second order tensors A ·B isdefined by the matrix product [A][B], with components of Ai jB jk = Cik. It is clearthat the range of the inner index j must be the same for [A] and [B]. For threedimensions we have i, j = 1,2,3. The second order inner product of two tensors ormatrices is A : B = Ai jBi j = tr([A]T [B]). The divergence of a vector u, which resultsin a contraction to a scalar, is defined by ∇x · u = ui,i, whereas for a second ordertensor, A, ∇x ·A describes a contraction to a vector with the components Ai j, j. Thegradient of a vector u(a dilation to a second order tensor) is given by ∇xu and hasthe components ui, j, whereas for a second order tensor (a dilation to a third ordertensor) ∇xA has components of Ai j,k. The gradient of a scalar φ (a dilation to avector) is defined by ∇xφ and has the components φ,i. The scalar product of twosecond order tensors, for example the gradients of first order vectors, is defined as

∇xv : ∇xu = ∂vi∂x j

∂ui∂x j

def= vi, jui, j, where ∂ui/∂x j,∂vi/∂x j are partial derivatives of ui

and vi, and where ui,vi are the Cartesian components of u and v. An example forthe product of a tensor with a vector is ∇xu ·n which has components of ui, j n j in aCartesian bases.

7

8 2 Some Basics of the Mechanics of Solid Continua

2.1 Kinematics of Deformations

The term deformation refers to a change in the shape of the continuum betweena reference configuration and current configuration. In the reference configuration,a representative particle of the continuum occupies a point p in space and has theposition vector (Fig. 2.1)

X = X1e1 + X2e2 + X3e3 , (2.2)

where e1,e2,e3 is a Cartesian reference triad, and X1,X2,X3 (with center O) can bethought of as labels for a point. Sometimes the coordinates or labels (X1,X2,X3, t)are called the referential coordinates. In the current configuration the particle orig-inally located at point p is located at point p′ and can be expressed also in termsof another position vector x, with the coordinates (x1,x2,x3,t). These are called thecurrent coordinates. It is obvious with this arrangement, that the displacement isu = x−X for a point originally at X and with final coordinates x.

When a continuum undergoes deformation (or flow), its points move along var-ious paths in space. This motion may be expressed by

x(X1,X2,X3,t) = u(X1,X2,X3,t)+ X(X1,X2,X3,t) , (2.3)

which gives the present location of a point at time t, written in terms of the la-bels X1,X2,X3. The previous position vector may be interpreted as a mapping ofthe initial configuration onto the current configuration. In classical approaches, itis assumed that such a mapping is a one-to-one and continuous, with continuouspartial derivatives to whatever order is required. The description of motion or de-formation expressed previously is known as the Lagrangian formulation. Alterna-tively, if the independent variables are the coordinates x and t, then x(x1,x2,x3, t) =u(x1,x2,x3,t)+X(x1,x2,x3,t), and the formulation is denoted as Eulerian (Fig. 2.1).

2.1.1 Deformation of Line Elements

Partial differentiation of the displacement vector u = x−X, with respect to x andX, produces the following displacement gradients:

Fig. 2.1 Differentdescriptions of a deformingbody

p

upX3, x3

X2, x2

X

x

X + dX dX

u + du

dx

Ω0

ΩX1, x1

e3e1 e2

O

2.1 Kinematics of Deformations 9

∇X u = F−1 and ∇xu = 1−F (2.4)

where

∇X x def=∂x∂X

= F def=

⎡⎢⎢⎢⎢⎢⎢⎣

∂x1

∂X1

∂x1

∂X2

∂x1

∂X3

∂x2

∂X1

∂x2

∂X2

∂x2

∂X3

∂x3

∂X1

∂x3

∂X2

∂x3

∂X3

⎤⎥⎥⎥⎥⎥⎥⎦

, (2.5)

and

∇xX def=∂X∂x

= F (2.6)

with the components Fik = xi,k and Fik = Xi,k. F is known as the material deformationgradient, and F is known as the spatial deformation gradient.

Now, consider the length of a differential element in the reference configura-tion dX and dx in the current configuration, dx = ∇X x · dX = F · dX. Taking thedifference in the magnitudes of these elements yields

dx ·dx−dX ·dX = (∇X x ·dX) · (∇Xx ·dX)−dX ·dX

= dX · (FT ·F−1) ·dX def= 2dX ·E ·dX . (2.7)

Alternatively, we have with dX = ∇xX ·dx = F ·dx and

dx ·dx−dX ·dX = dx ·dx− (∇xX ·dx) · (∇xX ·dx)

= dx · (1−FT ·F) ·dx def= 2dx · e ·dx . (2.8)

Equation (2.7) defines the so-called Lagrangian strain tensor

E def= 12(FT ·F−1) = 1

2 [∇X u+(∇X u)T +(∇X u)T ·∇X u] . (2.9)

Frequently, the Lagrangian strain tensor is defined in terms of the so-called right

Cauchy-Green strain, C = FT ·F leading to E def= 12(C−1). The Eulerian strain tensor

is defined by (2.8) as

e def= 12 (1−F

T ·F) = 12 (∇xu+(∇xu)T − (∇xu)T ·∇xu) . (2.10)

In a similar manner as for the Lagrangian strain tensor, the Eulerian strain tensorcan be defined in terms of the so-called left Cauchy-Green strain b = F ·FT which

then yields e def= 12(1−b−1).

Remark. It should be clear that dx can be reinterpreted as the result of a mappingF · dX → dx, or a change in configuration (reference to current, Fig. 2.2) whileF · dx → dX, maps the current to the reference system. For the deformations to beinvertible, and physically realizable, F · (F ·dX) = dX and F · (F ·dx) = dx. We note

10 2 Some Basics of the Mechanics of Solid Continua

Fig. 2.2 Mapping back andforth with the deformationgradient

F−1

F

reference current

dX dx

Ω0 Ω

that (detF)(detF) = 1 and have the following obvious relation F ·F = 1. It shouldbe clear that F = F−1.

2.1.2 Infinitesimal Strain Measures

In infinitesimal deformation theory, the displacement gradient components being“small” implies that higher order terms like (∇X u)T ·∇X u and (∇xu)T ·∇xu can

be neglected in the strain measures e and E leading to e ≈ εE def= 12 [∇xu +(∇xu)T ]

and E ≈ εL def= 12 [∇X u+(∇X u)T ]. If the displacement gradients are small compared

with unity, εE and εL coincide closely to e and E, respectively. If we assume that,∂

∂X ≈ ∂∂x , we may use εE or εL interchangeably. Usually ε is the symbol used for

infinitesimal strains. Furthermore, to avoid confusion, when using models employ-ing the geometrically linear infinitesimal strain assumption we use the symbol of ∇with no X or x subscript. Hence the infinitesimal strains are defined by

ε =12(∇u+(∇u)T ) . (2.11)

2.1.3 The Jacobian of the Deformation Gradient

The Jacobian of the deformation gradient, F, is defined as

Jdef= detF =

∣∣∣∣∣∣∣∣∣∣∣∣

∂x1

∂X1

∂x1

∂X2

∂x1

∂X3

∂x2

∂X1

∂x2

∂X2

∂x2

∂X3

∂x3

∂X1

∂x3

∂X2

∂x3

∂X3

∣∣∣∣∣∣∣∣∣∣∣∣. (2.12)

To interpret the Jacobian in a physical way, consider a reference differential volumein Fig. 2.3 which is given by dS3 = dω , where dX(1) = dS e1, dX(2) = dS e2 and

2.2 Equilibrium / Kinetics of Solid Continua 11

Fig. 2.3 A differentialvolume element

dx(3)

dx(1)

dX(3)dX(2)

dX(1)

dx(2)

dX(3) = dS e3. The current differential element is described by dx(1) = ∂xk∂X1

dS ek,

dx(2) = ∂xk∂X2

dS ek and dx(3) = ∂xk∂X3

dS ek, where ek is a unit vector, and

dx(1) · (dx(2)×dx(3))︸ ︷︷ ︸def=dω

=

∣∣∣∣∣∣∣∣∣

dx(1)1 dx(1)

2 dx(1)3

dx(2)1 dx(2)

2 dx(2)3

dx(3)1 dx(3)

2 dx(3)3

∣∣∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣∣∣∣∣∣

∂x1

∂X1

∂x2

∂X1

∂x3

∂X1

∂x1

∂X2

∂x2

∂X2

∂x3

∂X2

∂x1

∂X3

∂x2

∂X3

∂x3

∂X3

∣∣∣∣∣∣∣∣∣∣∣∣dS3. (2.13)

Therefore, dω = J dω0. Thus, the Jacobian of the deformation gradient must remainpositive definite, otherwise we obtain physically impossible “negative” volumes.

2.2 Equilibrium / Kinetics of Solid Continua

We start with the following postulated balance law for an arbitrary part ω around apoint P with boundary ∂ω of a body Ω , see Fig. 2.4,

Ω

∂ω

Pω

ω

f

t

P

Fig. 2.4 Newton’s laws applied to a continuum