Excerpts from Chapter 1: The determination of the transport, electromagnetic and mechanical properties of heterogeneous materials has a long and venerable history, attracting the attention of some of the luminaries of science, including Maxwell (1873), Lord Rayleigh (1892) and Einstein (1906). In his Treatise on Electricity and Magnetism, Maxwell derived an expression for the effective conductivity of a dispersion of spheres that is exact for dilute sphere concentrations. Lord Rayleigh developed a formalism to compute the effective conductivity of regular arrays of spheres that is used to this day. Work on the mechanical properties of heterogeneous materials began with the famous paper by Einstein in which he determined the effective viscosity of a dilute suspension of spheres. Since the early work on the physical properties of heterogeneous materials, there has been an explosion in the literature on this subject because of the rich and challenging fundamental problems it offers and its manifest technological importance. This book is divided into two parts. Part I deals with the quantitative characterization of the microstructure of heterogeneous materials via theoretical, computer-simulation and imaging techniques. Emphasis is placed on theoretical methods. Part II treats a wide variety of effective properties of heterogeneous materials and how they are linked to the microstructure. This is accomplished using rigorous methods. (Readers interested in property prediction can immediately skip to Part II.) Whenever possible, theoretical predictions for the effective properties are compared to available experimental and computer-simulation data. The overall goal of the book is to provide a rigorous means of characterizing the microstructure and properties of heterogeneous materials that can simultaneously yield results of practical utility. A unified treatment of both microstructure and properties is emphasized. In Chapter 2, the various microstructural functions that are essential in determining the effective properties of random heterogeneous materials are defined. Chapter 3 provides a review of the statistical mechanics of particle systems that is particularly germane to the study of random heterogeneous materials. In Chapter 4, a unified approach to characterize the microstructure of a large class of media is developed. This is accomplished via a canonical n-point function H_n from which one can derive exact analytical expressions for any microstructural function of interest. Chapters 5, 6 and 7 apply the formalism of Chapter 4 to the case of identical systems of spheres, spheres with a polydispersivity in size, and anisotropic particle systems (including laminates), respectively. In Chapter 8, the methods of Chapter 4 are extended to quantify the microstructure of cell models. Here the random-field approach is also discussed. Chapter 9 reviews the study of percolation and clustering on a lattice and introduces continuum percolation. Chapter 10 describes specific developments continuum percolation theory. Chapter 11 describes a means to study microstructural fluctuations that occur on local length scales. Finally, Chapter 12 discusses computer-simulation techniques (primarily Monte Carlo methods) to quantify microstructure. Moreover, it is shown how to apply the same methods to compute relevant microstructural functions from two- and three-dimensional images of the material. In Chapter 13, the local governing equations for the relevant field quantities and the method of homogenization leading to the averaged equations for the effective properties are described. The aforementioned four different classes of problems are studied. In Chapter 14, minimum energy principles are derived that lead to variational bounds on all of the effective properties in terms of trial fields. Chapter 15 proves and discusses certain phase-interchange relations for the effective conductivity and elastic moduli. Chapter 16 derives and describes some exact results for each of the effective properties. In Chapter 17, we derive the local fields associated with a single spherical or ellipsoidal inclusion in an infinite medium for all problem classes. Chapter 18 presents derivations of popular effective-medium approximations for all four effective properties. In Chapter 19, cluster expansions of the effective properties of dispersions are described. Chapter 20 presents derivations of so-called strong-contrast expansions for the effective conductivity and elastic moduli of generally anisotropic media of arbitrary microstructure. In Chapter 21, rigorous bounds on the all of the effective properties are derived using the variational principles of Chapter 14 and specific trial fields. Chapter 22 describes the evaluation of the bounds found in Chapter 21 for certain theoretical model microstructures as well as experimental systems using the results of Part I. Finally, cross-property relations between the seemingly different effective properties considered here are discussed and derived in Chapter 23.

Table of Contents

 

Preface vii

 

1 Motivation and Overview 1

1.1 What Is a Heterogeneous Material? . . . . . . . . . . . . . . . . . . . . . 1

1.2 Effective Properties and Applications . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Conductivity and Analogous Properties . . . . . . . . . . . . . . 6

1.2.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.3 Survival Time or Trapping Constant . . . . . . . . . . . . . . . . 8

1.2.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.5 Diffusion and Viscous Relaxation Times . . . . . . . . . . . . . . 9

1.2.6 Definitions of Effective Properties . . . . . . . . . . . . . . . . . . 9

1.3 Importance of Microstructure . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Development of a Systematic Theory . . . . . . . . . . . . . . . . . . . . 12

1.4.1 Microstructural Details . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.2 Multidisciplinary Research Area . . . . . . . . . . . . . . . . . . . 14

1.5 Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5.1 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5.2 Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

 

I Microstructure Characterization 21

 

2 Microstructural Descriptors 23

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 n-Point Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.2 Symmetries and Ergodicity . . . . . . . . . . . . . . . . . . . . . . 28

2.2.3 Geometrical Probability Interpretation . . . . . . . . . . . . . . . 32

2.2.4 Asymptotic Properties and Bounds . . . . . . . . . . . . . . . . . 33

2.2.5 Two-Point Probability Function . . . . . . . . . . . . . . . . . . . 34

2.3 Surface Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4 Lineal-Path Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5 Chord-Length Density Function . . . . . . . . . . . . . . . . . . . . . . . 45

2.6 Pore-Size Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.7 Percolation and Cluster Functions . . . . . . . . . . . . . . . . . . . . . . 50

2.8 Nearest-Neighbor Functions . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.9 Point/q-Particle Correlation Functions . . . . . . . . . . . . . . . . . . . 57

2.10 Surface/Particle Correlation Function . . . . . . . . . . . . . . . . . . . . 58

 

3 Statistical Mechanics of Many-Particle Systems 59

3.1 Many-Particle Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.1.1 n-Particle Probability Densities . . . . . . . . . . . . . . . . . . . 60

3.1.2 Pair Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2 Ornstein–Zernike Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3 Equilibrium Hard-Sphere Systems . . . . . . . . . . . . . . . . . . . . . 75

3.3.1 Low-Density Expansions . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.2 Arbitrary Fluid Densities . . . . . . . . . . . . . . . . . . . . . . . 81

3.4 Random Sequential Addition Processes . . . . . . . . . . . . . . . . . . 83

3.4.1 One-Dimensional Identical Hard Rods . . . . . . . . . . . . . . . 85

3.4.2 Identical Hard Spheres in Higher Dimensions . . . . . . . . . . 87

3.4.3 General Hard-Particle Systems . . . . . . . . . . . . . . . . . . . 88

3.5 Maximally Random Jammed State . . . . . . . . . . . . . . . . . . . . . 88

3.5.1 Random Close Packing Is Ill-Defined . . . . . . . . . . . . . . . . 89

3.5.2 Definition of Maximally Random Jammed State . . . . . . . . . 90

3.5.3 Order Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.5.4 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . 93

3.5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 95

 

4 United Approach to Characterize Microstructure 96

4.1 Volume Fraction and Specific Surface . . . . . . . . . . . . . . . . . . . 97

4.1.1 Bounding Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.1.2 Example Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.2 Canonical Correlation Function Hn . . . . . . . . . . . . . . . . . . . . . 104

4.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.2.2 Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3 Series Representations of Hn . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3.1 Mayer Representation . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.3.2 Kirkwood–Salsburg Representation . . . . . . . . . . . . . . . . 111

4.3.3 Bounding Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.4 Special Cases of Hn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.5 Polydispersivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.6 Other Model Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . 118

 

5 Monodisperse Spheres 119

5.1 Fully Penetrable Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.1.1 n-Point Probability Functions . . . . . . . . . . . . . . . . . . . . 122

5.1.2 Surface Correlation Functions . . . . . . . . . . . . . . . . . . . . 124

5.1.3 Lineal-Path Function . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.1.4 Chord-Length Density Function . . . . . . . . . . . . . . . . . . . 127

5.1.5 Nearest-Neighbor Functions . . . . . . . . . . . . . . . . . . . . . 128

5.1.6 Pore-Size Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.1.7 Point/q-Particle Correlation Functions . . . . . . . . . . . . . . . 129

5.2 Totally Impenetrable Spheres . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.2.1 n-Point Probability Functions . . . . . . . . . . . . . . . . . . . . 130

5.2.2 Surface Correlation Functions . . . . . . . . . . . . . . . . . . . . 134

5.2.3 Lineal-Path Function . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.2.4 Chord-Length Density Function . . . . . . . . . . . . . . . . . . . 137

5.2.5 Nearest-Neighbor Functions . . . . . . . . . . . . . . . . . . . . . 139

5.2.6 Pore-Size Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.2.7 Point/q-Particle Correlation Functions . . . . . . . . . . . . . . . 152

5.3 Interpenetrable Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.3.1 Nearest-Neighbor Functions . . . . . . . . . . . . . . . . . . . . . 154

5.3.2 Volume Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.3.3 Specific Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.3.4 Pore-Size Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.3.5 Other Statistical Descriptors . . . . . . . . . . . . . . . . . . . . . 157

5.4 Statistically Inhomogeneous Systems . . . . . . . . . . . . . . . . . . . . 158

 

6 Polydisperse Spheres 160

6.1 Fully Penetrable Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.1.1 n-Point Probability Functions . . . . . . . . . . . . . . . . . . . . 163

6.1.2 Surface Correlation Functions . . . . . . . . . . . . . . . . . . . . 164

6.1.3 Lineal-Path Function . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.1.4 Chord-Length Density Function . . . . . . . . . . . . . . . . . . . 166

6.1.5 Nearest-Surface Functions . . . . . . . . . . . . . . . . . . . . . . 166

6.1.6 Pore-Size Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.1.7 Point/q-Particle Correlation Functions . . . . . . . . . . . . . . . 167

6.2 Totally Impenetrable Spheres . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.2.1 n-Point Probability Functions . . . . . . . . . . . . . . . . . . . . 169

6.2.2 Surface Correlation Functions . . . . . . . . . . . . . . . . . . . . 170

6.2.3 Lineal-Path Function . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.2.4 Chord-Length Density Function . . . . . . . . . . . . . . . . . . . 171

6.2.5 Nearest-Surface Functions . . . . . . . . . . . . . . . . . . . . . . 172

6.2.6 Pore-Size Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.2.7 Point/q-Particle Correlation Functions . . . . . . . . . . . . . . . 176

 

7 Anisotropic Media 177

7.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.2 Fully Penetrable Oriented Inclusions . . . . . . . . . . . . . . . . . . . . 179

7.3 Impenetrable Oriented Inclusions . . . . . . . . . . . . . . . . . . . . . . 181

7.4 Hierarchical Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

 

8 Cell and Random-Field Models 188

8.1 Cell Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

8.1.1 Voronoi and Delaunay Tessellations . . . . . . . . . . . . . . . . 189

8.1.2 Cell Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

8.1.3 Symmetric-Cell Materials . . . . . . . . . . . . . . . . . . . . . . . 194

8.1.4 Random Checkerboard . . . . . . . . . . . . . . . . . . . . . . . . 199

8.1.5 Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

8.2 Random-Field Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

8.2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . 203

8.2.2 Gaussian Convolved Intensities . . . . . . . . . . . . . . . . . . . 207

 

9 Percolation and Clustering 210

9.1 Lattice Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

9.1.1 Bond and Site Percolation . . . . . . . . . . . . . . . . . . . . . . 211

9.1.2 Percolation Properties . . . . . . . . . . . . . . . . . . . . . . . . . 215

9.1.3 Scaling and Critical Exponents . . . . . . . . . . . . . . . . . . . 217

9.1.4 Infinite Cluster and Fractality . . . . . . . . . . . . . . . . . . . . 222

9.1.5 Finite-Size Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

9.2 Continuum Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

9.2.1 Percolation Properties . . . . . . . . . . . . . . . . . . . . . . . . . 227

9.2.2 Two-Point Cluster Function . . . . . . . . . . . . . . . . . . . . . 230

9.2.3 Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

 

10 Some Continuum Percolation Results 234

10.1 Exact Results for Overlapping Spheres . . . . . . . . . . . . . . . . . . . 234

10.1.1 One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

10.1.2 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

10.1.3 Low-Density Expansions of Cluster Statistics . . . . . . . . . . . 242

10.2 Ornstein–Zernike Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 243

10.3 Percus–Yevick Approximations . . . . . . . . . . . . . . . . . . . . . . . . 245

10.3.1 Permeable-Sphere Model . . . . . . . . . . . . . . . . . . . . . . . 246

10.3.2 Cherry-Pit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

10.3.3 Sticky Hard-Sphere Model . . . . . . . . . . . . . . . . . . . . . . 249

10.4 Beyond Percus–Yevick Approximations . . . . . . . . . . . . . . . . . . . 250

10.5 Two-Point Cluster Function . . . . . . . . . . . . . . . . . . . . . . . . . . 250

10.6 Percolation Threshold Estimates . . . . . . . . . . . . . . . . . . . . . . . 251

10.6.1 Overlapping Disks and Spheres . . . . . . . . . . . . . . . . . . . 252

10.6.2 Nonspherical Overlapping Particles . . . . . . . . . . . . . . . . . 254

10.6.3 Interacting Particle Systems . . . . . . . . . . . . . . . . . . . . . 255

 

11 Local Volume Fraction Fluctuations 257

11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

11.2 Coarseness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

11.2.1 General Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

11.2.2 Asymptotic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 261

11.2.3 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

11.3 Moments of Local Volume Fraction . . . . . . . . . . . . . . . . . . . . . 264

11.4 Evaluations of Full Distribution . . . . . . . . . . . . . . . . . . . . . . . 265

 

12 Computer Simulations, Image Analyses, and Reconstructions 269

12.1 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

12.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

12.1.2 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 271

12.2 Metropolis Method for Gibbs Ensembles . . . . . . . . . . . . . . . . . . 273

12.2.1 Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

12.2.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

12.2.3 Practical Implementation . . . . . . . . . . . . . . . . . . . . . . . 275

12.2.4 Hard Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

12.2.5 Other Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . 278

12.2.6 Cell Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

12.3 Methods for Generating Nonequilibrium Ensembles . . . . . . . . . . . 279

12.4 Sampling in Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . 281

12.4.1 Radial Distribution Function . . . . . . . . . . . . . . . . . . . . . 281

12.4.2 n-point Probability Functions . . . . . . . . . . . . . . . . . . . . 283

12.4.3 Surface Correlation Functions . . . . . . . . . . . . . . . . . . . . 285

12.4.4 Cluster-Type Functions . . . . . . . . . . . . . . . . . . . . . . . . 285

12.4.5 Other Correlation Functions . . . . . . . . . . . . . . . . . . . . . 286

12.5 Sampling Images and Digitized Media . . . . . . . . . . . . . . . . . . . 287

12.5.1 Two-Point Probability Function . . . . . . . . . . . . . . . . . . . 289

12.5.2 Lineal-Path Function . . . . . . . . . . . . . . . . . . . . . . . . . 291

12.5.3 Chord-Length Density Function . . . . . . . . . . . . . . . . . . . 292

12.5.4 Pore-Size Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 292

12.5.5 Two-Point Cluster Function . . . . . . . . . . . . . . . . . . . . . 293

12.6 Reconstructing Heterogeneous Materials . . . . . . . . . . . . . . . . . 294

12.6.1 Reconstruction Procedure . . . . . . . . . . . . . . . . . . . . . . 295

12.6.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . 297

 

 

II Microstructure/Property Connection 303

 

13 Local and Homogenized Equations 305

13.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

13.2 Conduction Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

13.2.1 Local Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

13.2.2 Conduction Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 311

13.2.3 Model One-Dimensional Problem . . . . . . . . . . . . . . . . . . 313

13.2.4 Homogenization of Periodic Problem in _d . . . . . . . . . . . . 315

13.2.5 Homogenization of Random Problem in _d . . . . . . . . . . . 318

13.2.6 Frequency-Dependent Conductivity . . . . . . . . . . . . . . . . . 321

13.3 Elastic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

13.3.1 Local Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

13.3.2 Elastic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

13.3.3 Homogenization of Random Problem in _d . . . . . . . . . . . . 332

13.3.4 Heterogeneous Materials . . . . . . . . . . . . . . . . . . . . . . . 334

13.3.5 Relationship Between Elasticity and Viscous Fluid Theory . . . 337

13.3.6 Viscosity of a Suspension . . . . . . . . . . . . . . . . . . . . . . . 338

13.3.7 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

13.4 Steady-State Trapping Problem . . . . . . . . . . . . . . . . . . . . . . . 339

13.4.1 Local Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

13.4.2 Homogenization of Random Problem in _d . . . . . . . . . . . . 341

13.5 Steady-State Fluid Permeability Problem . . . . . . . . . . . . . . . . . 344

13.5.1 Local Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

13.5.2 Homogenization of Random Problem in _d . . . . . . . . . . . . 346

13.5.3 Relationship to Sedimentation Rate . . . . . . . . . . . . . . . . 348

13.6 Classification of Steady-State Problems . . . . . . . . . . . . . . . . . . . 349

13.7 Time-Dependent Trapping Problem . . . . . . . . . . . . . . . . . . . . . 350

13.7.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

13.7.2 Relationship Between Survival and Relaxation Times . . . . . . 353

13.8 Time-Dependent Flow Problem . . . . . . . . . . . . . . . . . . . . . . . 354

13.8.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

13.8.2 Relationship Between Permeability and Relaxation Times . . . 356

 

14 Variational Principles 357

14.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

14.1.1 Field Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

14.1.2 Energy Representation . . . . . . . . . . . . . . . . . . . . . . . . 361

14.1.3 Minimum Energy Principles . . . . . . . . . . . . . . . . . . . . . 363

14.1.4 Hashin–Shtrikman Principle . . . . . . . . . . . . . . . . . . . . . 367

14.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

14.2.1 Field Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

14.2.2 Energy Representation . . . . . . . . . . . . . . . . . . . . . . . . 370

14.2.3 Minimum Energy Principles . . . . . . . . . . . . . . . . . . . . . 373

14.2.4 Hashin–Shtrikman Principle . . . . . . . . . . . . . . . . . . . . . 377

14.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

14.3.1 Energy Representation . . . . . . . . . . . . . . . . . . . . . . . . 379

14.3.2 Minimum Energy Principles . . . . . . . . . . . . . . . . . . . . . 380

14.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

14.4.1 Energy Representation . . . . . . . . . . . . . . . . . . . . . . . . 383

14.4.2 Minimum Energy Principles . . . . . . . . . . . . . . . . . . . . . 385

 

15 Phase-Interchange Relations 390

15.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

15.1.1 Duality for Two-Dimensional Media . . . . . . . . . . . . . . . . 390

15.1.2 Three-Dimensional Media . . . . . . . . . . . . . . . . . . . . . . 397

15.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

15.2.1 Two-Dimensional Media . . . . . . . . . . . . . . . . . . . . . . . 398

15.2.2 Three-Dimensional Media . . . . . . . . . . . . . . . . . . . . . . 401

15.3 Trapping Constant and Fluid Permeability . . . . . . . . . . . . . . . . . 402

 

16 Exact Results 403

16.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

16.1.1 Coated-Spheres Model . . . . . . . . . . . . . . . . . . . . . . . . . 404

16.1.2 Simple Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

16.1.3 Higher-Order Laminates and Attainability . . . . . . . . . . . . . 410

16.1.4 Fiber-Reinforced Materials . . . . . . . . . . . . . . . . . . . . . . 413

16.1.5 Periodic Arrays of Inclusions . . . . . . . . . . . . . . . . . . . . . 413

16.1.6 Low-Density Cellular Solids . . . . . . . . . . . . . . . . . . . . . 415

16.1.7 Field Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

16.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

16.2.1 Coated-Spheres Model . . . . . . . . . . . . . . . . . . . . . . . . 417

16.2.2 Simple Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

16.2.3 Higher-Order Laminates and Attainability . . . . . . . . . . . . 424

16.2.4 Periodic Arrays of Inclusions . . . . . . . . . . . . . . . . . . . . 426

16.2.5 Low-Density Cellular Solids . . . . . . . . . . . . . . . . . . . . . 428

16.2.6 Equal Phase Shear Moduli . . . . . . . . . . . . . . . . . . . . . 429

16.2.7 Sheets with Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

16.2.8 Dispersions of Particles in a Liquid . . . . . . . . . . . . . . . . 429

16.2.9 Cavities (Bubbles) in an Incompressible Matrix(Liquid) . . . 429

16.2.10 Field Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 430

16.2.11 Link to Two-Dimensional Conductivity . . . . . . . . . . . . . . 430

16.2.12 Link to Thermoelastic Constants . . . . . . . . . . . . . . . . . . 431

16.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

16.3.1 Diffusion Inside Hyperspheres . . . . . . . . . . . . . . . . . . . . 432

16.3.2 Periodic Arrays of Traps . . . . . . . . . . . . . . . . . . . . . . . . 433

16.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

16.4.1 Flow Between Plates and Inside Tubes . . . . . . . . . . . . . . . 434

16.4.2 Periodic Arrays of Obstacles . . . . . . . . . . . . . . . . . . . . . 436

 

17 Single-Inclusion Solutions 437

17.1 Conduction Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

17.1.1 Spherical Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 437

17.1.2 Polarization Within an Ellipsoid . . . . . . . . . . . . . . . . . . . 441

17.2 Elasticity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

17.2.1 Spherical Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 442

17.2.2 Polarization Within an Ellipsoid . . . . . . . . . . . . . . . . . . . 448

17.3 Trapping Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

17.3.1 Spherical Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

17.3.2 Spheroidal Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

17.4 Flow Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

17.4.1 Spherical Obstacle . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

17.4.2 Spheroidal Obstacle . . . . . . . . . . . . . . . . . . . . . . . . . . 457

 

18 Effective-Medium Approximations 459

18.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

18.1.1 Maxwell Approximations . . . . . . . . . . . . . . . . . . . . . . . 460

18.1.2 Self-Consistent Approximations . . . . . . . . . . . . . . . . . . . 462

18.1.3 Differential Effective-Medium Approximations . . . . . . . . . 467

18.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

18.2.1 Maxwell Approximations . . . . . . . . . . . . . . . . . . . . . . . 470

18.2.2 Self-Consistent Approximations . . . . . . . . . . . . . . . . . . 474

18.2.3 Differential Effective-Medium Approximations . . . . . . . . . 477

18.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

18.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

 

19 Cluster Expansions 485

19.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

19.1.1 Dilute Dispersions of Spheres . . . . . . . . . . . . . . . . . . . . 488

19.1.2 Dilute Dispersions of Ellipsoids . . . . . . . . . . . . . . . . . . . 490

19.1.3 Nondilute Concentrations . . . . . . . . . . . . . . . . . . . . . . 491

19.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

19.2.1 Dilute Dispersions of Spheres . . . . . . . . . . . . . . . . . . . . 497

19.2.2 Dilute Dispersions of Ellipsoids . . . . . . . . . . . . . . . . . . . 500

19.2.3 Nondilute Concentrations . . . . . . . . . . . . . . . . . . . . . . 501

19.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

19.3.1 Dilute Dispersions of Spherical Traps . . . . . . . . . . . . . . . 502

19.3.2 Dilute Dispersions of Spheroidal Traps . . . . . . . . . . . . . . . 503

19.3.3 Nondilute Concentrations . . . . . . . . . . . . . . . . . . . . . . 504

19.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

19.4.1 Dilute Beds of Spheres . . . . . . . . . . . . . . . . . . . . . . . . 505

19.4.2 Dilute Beds of Spheroids . . . . . . . . . . . . . . . . . . . . . . . 506

19.4.3 Nondilute Concentrations . . . . . . . . . . . . . . . . . . . . . . 507

 

20 Exact Contrast Expansions 509

20.1 Conductivity Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

20.1.1 Integral Equation for Cavity Electric Field . . . . . . . . . . . . 511

20.1.2 Strong-Contrast Expansions . . . . . . . . . . . . . . . . . . . . . 514

20.1.3 Some Tensor Properties . . . . . . . . . . . . . . . . . . . . . . . . 519

20.1.4 Weak-Contrast Expansions . . . . . . . . . . . . . . . . . . . . . . 520

20.1.5 Expansion of Local Electric Field . . . . . . . . . . . . . . . . . . 521

20.1.6 Isotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

20.2 Stiffness Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

20.2.1 Integral Equation for the Cavity Strain Field . . . . . . . . . . . 530

20.2.2 Strong-Contrast Expansions . . . . . . . . . . . . . . . . . . . . . 534

20.2.3 Weak-Contrast Expansions . . . . . . . . . . . . . . . . . . . . . . 539

20.2.4 Expansion of Local Strain Field . . . . . . . . . . . . . . . . . . . 540

20.2.5 Isotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

 

21 Rigorous Bounds 552

21.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

21.1.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . 554

21.1.2 Contrast Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

21.1.3 Cluster Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

21.1.4 Security-Spheres Bounds . . . . . . . . . . . . . . . . . . . . . . . 564

21.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566

21.2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . 566

21.2.2 Contrast Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

21.2.3 Cluster Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576

21.2.4 Security-Spheres Bounds . . . . . . . . . . . . . . . . . . . . . . . 577

21.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

21.3.1 Interfacial-Surface Lower Bound . . . . . . . . . . . . . . . . . . 579

21.3.2 Void Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

21.3.3 Cluster Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 581

21.3.4 Security-Spheres Upper Bound . . . . . . . . . . . . . . . . . . . 582

21.3.5 Pore-Size Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . 584

21.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

21.4.1 Interfacial-Surface Upper Bound . . . . . . . . . . . . . . . . . . 585

21.4.2 Void Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

21.4.3 Cluster Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 587

21.4.4 Security-Spheres Lower Bound . . . . . . . . . . . . . . . . . . . 589

21.5 Structural Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

21.6 Utility of Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592

 

22 Evaluation of Bounds 593

22.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594

22.1.1 Contrast Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594

22.1.2 Cluster Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

22.1.3 Security-Spheres Bounds . . . . . . . . . . . . . . . . . . . . . . . 610

22.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

22.2.1 Contrast Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

22.2.2 Cluster Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

22.2.3 Security-Spheres Bounds . . . . . . . . . . . . . . . . . . . . . . . 620

22.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

22.3.1 Interfacial-Surface Lower Bound . . . . . . . . . . . . . . . . . . 621

22.3.2 Void Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

22.3.3 Cluster Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 624

22.3.4 Security-Spheres Upper Bound . . . . . . . . . . . . . . . . . . . 625

22.3.5 Pore-Size Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . 625

22.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

22.4.1 Interfacial-Surface Upper Bound . . . . . . . . . . . . . . . . . . 627

22.4.2 Void Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

22.4.3 Cluster Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 630

22.4.4 Security-Spheres Lower Bound . . . . . . . . . . . . . . . . . . . 631

 

23 Cross-Property Relations 632

23.1 Conductivity and Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . 633

23.1.1 Elementary Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 633

23.1.2 Translation Bounds for d _ 2 . . . . . . . . . . . . . . . . . . . . . 636

23.1.3 Translation Bounds for d _ 3 . . . . . . . . . . . . . . . . . . . . . 642

23.2 Flow and Diffusion Parameters . . . . . . . . . . . . . . . . . . . . . . . 647

23.2.1 Permeability and Survival Time . . . . . . . . . . . . . . . . . . . 647

23.2.2 Permeability, Formation Factor, and Viscous Relaxation Times . . . . . . . . . . . . . . . . . . . . 650

23.2.3 Viscous and Diffusion Relaxation Times . . . . . . . . . . . . . . 654

 

A Equilibrium Hard-Disk Program 656

 

B Interrelations Among Two- and Three-Dimensional Moduli 661

 

References 663

 

Index 693

 

See below for the PDF version, or the corrected pages in the 2^nd printing.

See below for the PDF version, or the corrected pages in the 2^nd printing.