**Applied Analysis of Composite Media: Analytical and Computational Results for Materials Scientists and Engineers**

**Contents **

Biography xi

Preface xiii

Acknowledgments xvii

Nomenclature xix

1 Introduction to computational methods and theory of composites 1

1.1 Traditional approaches 1

1.1.1 Self-consistent methods 2

1.1.2 Series method 4

1.1.3 Getting from traditional approach to neoclassical illustrated by example 4

1.2 Mathematical foundations of constructive homogenization 7

1.2.1 Different approaches to heterogeneity in material sciences and

in statistical mechanics 7

1.2.2 Random geometry of composites 8

1.2.3 The method of structural sums 10

1.2.4 Random homogenization and ergodicity 13

1.3 Asymptotic methods. Critical index 17

1.3.1 Method of self-similar renormalization 20

1.3.2 Critical index. Direct methods 27

1.3.3 Critical indices from self-similar root approximants. Examples 37

1.3.4 Factor approximants and critical index. Example 47

1.3.5 Additive self-similar approximants and DLog additive recursive approximants 51

2 Algorithms, computations and structural information 57

2.1 Computing in modern computational sciences 57

2.2 Symbolic representations of coefficients for the effective conductivity 57

2.2.1 Algorithm 1: Express Aq in terms of structural sums 58

2.2.2 Algorithm 2: Reduction of dependent structural sums in Aq 63

2.3 Algorithms for computing structural sums 67

2.3.1 Simplified formulas for epp and e2222 67

2.3.2 Calculation of multidimensional discrete chain convolutions of functions 68

2.3.3 Efficient computation of structural sums 70

2.3.4 Computing structural sums of a given order 74

2.3.5 Convolutions of functions in case of elastic composites 78

2.4 Symbolic calculations for systems of functional equations 80

2.4.1 Indefinite symbolic sums 80

2.4.2 Symbolic representation of successive approximations 81

2.4.3 Application to the 3D effective conductivity 81

2.5 Structural information and structural features vector 82

2.5.1 Pattern recognition and feature vectors 83

2.5.2 From computational science to information processing 83

2.5.3 Structural information 84

2.5.4 Construction of structural features vector 86

2.5.5 Structural information in three-dimensional space 88

2.6 Classification and analysis of random structures 89

2.6.1 Circular inclusions 90

2.6.2 Non-circular inclusions 94

2.6.3 Irregularity of random structures and data visualization 95

2.7 The Python package for computing structural sums and the effective conductivity 101

3 Elasticity of hexagonal array of cylinders 103

3.1 Method of functional equations for elastic problems 103

3.1.1 Local field. Analytical approximate formulas 110

3.1.2 Cluster approach to elasticity problem 113

3.1.3 Rayleigh’s and Maxwell’s approaches 118

3.2 General formula for the effective shear and bulk moduli 120

3.3 Effective shear modulus for perfectly rigid inclusions embedded into matrix 123

3.3.1 Modified Padé approximations 124

3.3.2 Additive approximants 125

3.3.3 Perfectly rigid inclusions. DLog additive approximants 127

3.3.4 Effective viscosity of 2D suspension 130

3.4 Effective shear modulus for soft fibers 131

3.4.1 Shear modulus in critical regime for arbitrary ν _ 134

3.4.2 DLog additive approximants for soft fibers 138

3.5 Method of contrast expansion for elastic incompressible composites 140

3.5.1 Method of Schwarz 140

3.5.2 Method of functional equations 143

3.5.3 Method of successive approximations in contrast parameter 144

3.5.4 Second order approximation 146

3.5.5 Third order approximation 148

3.5.6 Numerical simulations 153

3.6 Practice of asymptotic methods 156

3.6.1 Glass-resin composite 157

3.6.2 Comparison with numerical FEM results 163

3.6.3 Boron-aluminum composite 165

3.6.4 Boron-epoxy composite 167

4 Random elastic composites with circular inclusions 171

4.1 Introduction 171

4.2 Method of functional equations for local fields 171

4.3 Averaged fields in composites and effective shear modulus 176

4.4 Identical circular inclusions 180

4.5 Numerical examples 183

4.5.1 Symmetric location of inclusions with equal radii 183

4.5.2 Random location of inclusions with equal radii 185

4.5.3 Symmetric location of inclusions with different radii 187

4.5.4 Random location of inclusions with different radii 189

4.6 Critical index for the viscosity of 2D random suspension 190

4.7 Critical behavior of random holes 192

5 Effective conductivity of fibrous composites with cracks on interface 195

5.1 Deterministic and random cracks on interface 195

5.2 Boundary value problem 197

5.3 Maxwell’s approach 201

6 Effective conductivity of a random suspension of highly conducting

spherical particles 205

6.1 Introduction 205

6.2 General formula for highly conducting spheres 207

6.2.1 Statement of the problem 207

6.2.2 High conducting inclusions 209

6.3 Modified Dirichlet problem 211

6.3.1 Finite number of inclusions 211

6.3.2 Computation of undetermined constants 216

6.4 Averaged conductivity 218

6.5 Effective conductivity 220

6.6 Numerical results for random composite 225

6.7 Foams 230

6.8 Slits, platelets 232

7 Permeability of porous media 237

7.1 Models of critical permeability 237

7.2 Permeability of regular arrays 239

7.2.1 Transverse permeability. Square array 239

7.2.2 Transverse permeability. Hexagonal array 241

7.2.3 Longitudinal permeability. Square array 242

7.3 3D periodic arrays of spherical obstacles 244

7.3.1 BCC and SC lattices of spherical obstacles 244

7.3.2 Formula for FCC lattice 247

7.4 Permeability in wavy-walled channels 249

7.4.1 Symmetric sinusoidal two-dimensional channel. Breakdown of

lubrication approximation 252

7.4.2 Symmetric sinusoidal two-dimensional channel. Breakdown

continued 254

7.4.3 Parallel sinusoidal two-dimensional channel. Novel critical

index 257

7.4.4 Symmetric sinusoidal three-dimensional channel. Two-fluid

model 260

8 Simple fluids, suspensions and selected random systems 265

8.1 Compressibility factor of hard-sphere and hard-disk fluids. Index

function 265

8.2 “Sticky” rods and disks. Mapping to Janus swimmers 278

8.3 3D elasticity, or high-frequency viscosity. Critical index 284

8.3.1 Modification with iterated roots. Condition imposed on

thresholds 286

8.3.2 Condition imposed on the critical index 287

8.3.3 Conditions imposed on the amplitude 288

8.3.4 Minimal derivative condition 289

8.4 Diffusion coefficients 292

8.5 Non-local diffusion 296

8.6 Sedimentation. Particle mobility 303

8.7 Polymer coil in 2D and 3D 307

8.8 Factor approximants and critical index for the viscosity of a 3D

suspension 311

Appendix 317

A.1 Equations of viscous flow and elasticity 317

A.2 Eisenstein’s series 319

A.3 Eisenstein–Natanzon–Filshtinsky series 321

A.4 Cluster approach and its limitations 323

A.5 Mathematical pseudo-language, transformation rule sequences and

rule-based programming 329

A.6 Implementations in mathematica 329

A.6.1 Implementation of computations of coefficients Aq 329

A.6.2 Implementation of mirror terms reduction 330

References 333

Index 349