Applied Analysis of Composite Media

Contents......Page 7
Biography......Page 11
Preface......Page 13
Acknowledgments......Page 17
 Conductivity......Page 18
 Other symbols and abbreviations......Page 19
 1.1 Traditional approaches......Page 21
  1.1.1 Self-consistent methods......Page 22
  1.1.3 Getting from traditional approach to neoclassical illustrated by example......Page 24
  1.2.1 Different approaches to heterogeneity in material sciences and in statistical mechanics......Page 27
  1.2.2 Random geometry of composites......Page 28
  1.2.3 The method of structural sums......Page 30
  1.2.4 Random homogenization and ergodicity......Page 33
 1.3 Asymptotic methods. Critical index......Page 37
  1.3.1 Method of self-similar renormalization......Page 40
  1.3.2 Critical index. Direct methods......Page 47
  1.3.3 Critical indices from self-similar root approximants. Examples......Page 57
   1.3.3.3 Permeability of two-dimensional channel......Page 63
   1.3.3.4 Expansion factor of three-dimensional polymer chain......Page 66
  1.3.4 Factor approximants and critical index. Example......Page 67
  1.3.5 Additive self-similar approximants and DLog additive recursive approximants......Page 71
 2.2 Symbolic representations of coefficients for the effective conductivity......Page 77
  2.2.1 Algorithm 1: Express Aq in terms of structural sums......Page 78
   2.2.1.1 Algorithm......Page 79
   2.2.1.2 Pseudocode......Page 82
   2.2.2.1 Reduction of mirror terms in Aq......Page 83
   2.2.2.2 Pseudocode......Page 85
  2.3.1 Simplified formulas for epp and e2222......Page 87
  2.3.2 Calculation of multidimensional discrete chain convolutions of functions......Page 88
  2.3.3 Efficient computation of structural sums......Page 90
   2.3.3.1 Implementation notes......Page 91
  2.3.4 Computing structural sums of a given order......Page 94
   2.3.4.1 Recurrence algorithm......Page 96
   2.3.4.2 Effectiveness of algorithm......Page 97
  2.3.5 Convolutions of functions in case of elastic composites......Page 98
  2.4.1 Indefinite symbolic sums......Page 100
  2.4.3 Application to the 3D effective conductivity......Page 101
 2.5 Structural information and structural features vector......Page 102
  2.5.2 From computational science to information processing......Page 103
  2.5.3 Structural information......Page 104
  2.5.4 Construction of structural features vector......Page 106
  2.5.5 Structural information in three-dimensional space......Page 108
 2.6 Classification and analysis of random structures......Page 109
  2.6.1 Circular inclusions......Page 110
  2.6.2 Non-circular inclusions......Page 114
  2.6.3 Irregularity of random structures and data visualization......Page 115
 2.7 The Python package for computing structural sums and the effective conductivity......Page 121
 3.1 Method of functional equations for elastic problems......Page 123
  3.1.1 Local field. Analytical approximate formulas......Page 130
  3.1.2 Cluster approach to elasticity problem......Page 133
  3.1.3 Rayleigh's and Maxwell's approaches......Page 138
 3.2 General formula for the effective shear and bulk moduli......Page 140
 3.3 Effective shear modulus for perfectly rigid inclusions embedded into matrix......Page 143
  3.3.1 Modified Padé approximations......Page 144
  3.3.2 Additive approximants......Page 145
  3.3.3 Perfectly rigid inclusions. DLog additive approximants......Page 147
  3.3.4 Effective viscosity of 2D suspension......Page 150
 3.4 Effective shear modulus for soft fibers......Page 151
  3.4.1 Shear modulus in critical regime for arbitrary ν'......Page 154
  3.4.2 DLog additive approximants for soft fibers......Page 158
  3.5.1 Method of Schwarz......Page 160
  3.5.2 Method of functional equations......Page 163
  3.5.3 Method of successive approximations in contrast parameter......Page 164
  3.5.4 Second order approximation......Page 166
  3.5.5 Third order approximation......Page 168
  3.5.6 Numerical simulations......Page 173
 3.6 Practice of asymptotic methods......Page 176
  3.6.1 Glass-resin composite......Page 177
  3.6.2 Comparison with numerical FEM results......Page 183
  3.6.3 Boron-aluminum composite......Page 185
  3.6.4 Boron-epoxy composite......Page 187
 4.2 Method of functional equations for local fields......Page 191
 4.3 Averaged fields in composites and effective shear modulus......Page 196
 4.4 Identical circular inclusions......Page 200
  4.5.1 Symmetric location of inclusions with equal radii......Page 203
  4.5.2 Random location of inclusions with equal radii......Page 205
  4.5.3 Symmetric location of inclusions with different radii......Page 207
  4.5.4 Random location of inclusions with different radii......Page 209
 4.6 Critical index for the viscosity of 2D random suspension......Page 210
 4.7 Critical behavior of random holes......Page 212
 5.1 Deterministic and random cracks on interface......Page 215
 5.2 Boundary value problem......Page 217
 5.3 Maxwell's approach......Page 221
 6.1 Introduction......Page 225
  6.2.1 Statement of the problem......Page 227
  6.2.2 High conducting inclusions......Page 229
  6.3.1 Finite number of inclusions......Page 231
  6.3.2 Computation of undetermined constants......Page 236
 6.4 Averaged conductivity......Page 238
 6.5 Effective conductivity......Page 240
 6.6 Numerical results for random composite......Page 245
 6.7 Foams......Page 250
 6.8 Slits, platelets......Page 252
 7.1 Models of critical permeability......Page 257
  7.2.1 Transverse permeability. Square array......Page 259
  7.2.2 Transverse permeability. Hexagonal array......Page 261
  7.2.3 Longitudinal permeability. Square array......Page 262
  7.3.1 BCC and SC lattices of spherical obstacles......Page 264
  7.3.2 Formula for FCC lattice......Page 267
 7.4 Permeability in wavy-walled channels......Page 269
  7.4.1 Symmetric sinusoidal two-dimensional channel. Breakdown of lubrication approximation......Page 272
  7.4.2 Symmetric sinusoidal two-dimensional channel. Breakdown continued......Page 274
  7.4.3 Parallel sinusoidal two-dimensional channel. Novel critical index......Page 277
  7.4.4 Symmetric sinusoidal three-dimensional channel. Two-fluid model......Page 280
   7.4.4.1 Subcritical index......Page 281
 8.1 Compressibility factor of hard-sphere and hard-disk fluids. Index function......Page 285
 8.2 "Sticky" rods and disks. Mapping to Janus swimmers......Page 298
 8.3 3D elasticity, or high-frequency viscosity. Critical index......Page 304
  8.3.1 Modification with iterated roots. Condition imposed on thresholds......Page 306
  8.3.2 Condition imposed on the critical index......Page 307
  8.3.3 Conditions imposed on the amplitude......Page 308
  8.3.4 Minimal derivative condition......Page 309
 8.4 Diffusion coefficients......Page 312
 8.5 Non-local diffusion......Page 316
 8.6 Sedimentation. Particle mobility......Page 323
 8.7 Polymer coil in 2D and 3D......Page 327
 8.8 Factor approximants and critical index for the viscosity of a 3D suspension......Page 331
 A.1 Equations of viscous flow and elasticity......Page 336
 A.2 Eisenstein's series......Page 338
 A.3 Eisenstein-Natanzon-Filshtinsky series......Page 340
 A.4 Cluster approach and its limitations......Page 342
  A.6.1 Implementation of computations of coefficients Aq......Page 348
  A.6.2 Implementation of mirror terms reduction......Page 349
References......Page 352
Index......Page 368