Preface ........................................................ xv
CHAPTER 1 Introduction ......................................... 1
1.1 Motivation for the Work .................................... 1
1.2 Geometry Models ............................................ 3
1.2.1 Single Inclusion .................................... 3
1.2.2 Finite Arrays of Inclusions ......................... 4
1.2.3 Composite Band and Layer ............................ 4
1.2.4 Representative Unit Cell (RUC) Model ................ 6
1.3 Method of Solution ......................................... 8
1.4 Homogenization Problem: Volume vs. Surface Averaging ...... 10
1.4.1 Conductivity ....................................... 11
1.4.2 Elasticity ......................................... 14
1.5 Scope and Structure of the Book ........................... 17
PART I Particulate Composites .................................. 19
CHAPTER 2 Potential Fields of Interacting Spherical
Inclusions ..................................................... 21
2.1 Background Theory ......................................... 21
2.1.1 Scalar Spherical Harmonics ......................... 21
2.1.2 Selected Properties of Solid Spherical Harmonics ... 24
2.1.3 Spherical Harmonics vs. Multipole Potentials ....... 26
2.2 General Solution for a Single Inclusion ................... 26
2.2.1 Multipole Expansion Solution ....................... 27
2.2.2 Far Field Expansion ................................ 27
2.2.3 Resolving Equations ................................ 28
2.3 Particle Coating vs. Imperfect Interface .................. 29
2.4 Re-Expansion Formulas for the Solid Spherical Harmonics ... 30
2.4.1 Equally Oriented Coordinate Systems ................ 31
2.4.2 Multipole Expansion Theorem ........................ 32
2.4.3 Arbitrarily Oriented Coordinate Systems ............ 33
2.5 Finite Cluster Model (FCM) ................................ 35
2.5.1 Superposition Principle ............................ 35
2.5.2 FCM Boundary-Value Problem ......................... 36
2.5.3 Convergence Proof .................................. 37
2.5.4 Modified Maxwell Method for Effective Conductivity .. 40
2.6 Composite Sphere .......................................... 42
2.6.1 Outer Boundary Condition ........................... 43
2.6.2 Interface Conditions ............................... 43
2.6.3 RSV and Effective Conductivity of Composite ........ 44
2.7 Half-Space FCM ............................................ 45
2.7.1 Double Fourier Transform of Solid Spherical
Harmonics .......................................... 45
2.7.2 Homogeneous Half-Space ............................. 47
2.7.3 Superposition Sum .................................. 47
2.7.4 Half-Space Boundary Condition ...................... 48
2.7.5 Interface Conditions ............................... 48
CHAPTER 3 Periodic Multipoles: Application to Composites ...... 51
3.1 Composite Layer ........................................... 51
3.1.1 2P Fundamental Solution of Laplace Equation ........ 51
3.1.2 2P Solid Harmonics ................................. 54
3.1.3 Heat Flux Through the Composite Layer .............. 55
3.2 Periodic Composite as a Sandwich of Composite Layers ...... 57
3.3 Representative Unit Cell Model ............................ 59
3.4 3P Scalar Solid Harmonics ................................. 61
3.4.1 Direct Summation ................................... 61
3.4.2 Hasimoto's Approach ................................ 61
3.4.3 2P Harmonics-Based Approach ........................ 63
3.5 Local Temperature Field ................................... 64
3.6 Effective Conductivity of Composite ....................... 65
CHAPTER 4 Elastic Solids with Spherical Inclusions ............ 69
4.1 Vector Spherical Harmonics ................................ 70
4.1.1 Vector Surface Harmonics ........................... 70
4.1.2 Vector Solid Harmonics ............................. 71
4.2 Scalar and Vector Solid Spherical Biharmonics ............. 74
4.3 Partial Solutions of Lame Equation ........................ 76
4.3.1 Definition ......................................... 76
4.3.2 Properties of Spherical Lame Solutions ............. 78
4.4 Single Inclusion in Unbounded Solid ....................... 81
4.4.1 Far Field Expansion ................................ 82
4.4.2 Resolving Set of Linear Equations .................. 83
4.4.3 Single Inclusion in Viscous Fluid (Stokes's
Problem) ........................................... 84
4.5 Application to Nanocomposite: Gurtin & Murdoch Theory ..... 86
4.5.1 Imperfect Interface Conditions ..................... 86
4.5.2 Formal Solution .................................... 88
4.5.1 Single Cavity Under Hydrostatic Far Field Load ..... 89
4.5.4 Single Cavity Under Uniaxial Far Field Load ........ 90
4.6 Re-Expansion Formulas for the Vector Harmonics and
Biharmonics ............................................... 91
4.6.1 Translation of Scalar Biharmonics .................. 91
4.6.2 Translation of Vector Harmonics .................... 94
4.6.3 Translation of Vector Biharmonics .................. 96
4.6.4 Translation of Lame Solutions ...................... 97
4.6.5 Re-Expansion Due to Rotation ....................... 99
4.7 Finite Array of Inclusions (FCM) .......................... 99
4.7.1 Direct (Superposition) Sum ........................ 100
4.7.2 Local Expansion Sum ............................... 101
4.7.3 Infinite System of Linear Equations ............... 101
4.7.4 Two Cavities Under Uniaxial Far Tension ........... 102
4.7.5 Interface-Induced Stress Concentration in
Nanostructured Solid .............................. 103
4.7.6 Stress Concentration Factors of Interacting
Inclusions ........................................ 104
4.8 Isotropic Solid with Anisotropic Inclusion ............... 105
4.8.1 Formal Solution ................................... 106
4.8.2 Resolving Set of Equations ........................ 107
4.9 Effective Stiffness of Composite: Modified Maxwell
Approach ................................................. 109
4.9.1 Cubic Symmetry .................................... 110
4.9.2 Bulk Modulus k* ................................... 111
4.9.3 Shear Modulus μ1* ................................. 112
4.9.4 Shear Modulus μ2* ................................. 112
4.10 Elastic Composite Sphere ................................. 113
4.11 RSV and Effective Elastic Moduli ......................... 114
4.11.1 Macroscopic Strain and Stress Tensors ............. 114
4.11.2 Effective Bulk Modulus ............................ 115
4.11.3 Effective Shear Modulus ........................... 116
CHAPTER 5 Elasticity of Composite Half-Space, Layer, and
Bulk .......................................................... 119
5.1 Vector Harmonics and Biharmonics for Half-Space .......... 119
5.1.1 Definition ........................................ 119
5.1.2 Integral Transforms ............................... 121
5.1.3 Series Expansions ................................. 123
5.2 Vector Lame Solutions for Half-Space ..................... 124
5.2.1 Definition ........................................ 124
5.2.2 Properties of Lame Solutions hαβ(i)± ............... 125
5.2.3 Integral Transforms and Series Expansions .......... 127
5.3 FCM for Elastic Half-Space ............................... 127
5.3.1 Problem Statement ................................. 127
5.3.2 Solution for Homogeneous Half-Space ............... 128
5.3.3 Heterogeneous Half-Space .......................... 129
5.4 Doubly Periodic Models ................................... 131
5.4.1 2P Lame Solutions ................................. 132
5.4.2 Composite Layer ................................... 134
5.4.3 Periodic Composite as a Sandwich of Composite
Layers ............................................ 137
5.5 Triply Periodic Vector Multipoles ........................ 137
5.5.1 Scalar Biharmonics ................................ 137
5.5.2 Periodic Solutions of Lame Equation ............... 139
5.6 RUC Model of Elastic Spherical Particle Composite ........ 140
5.6.1 Formal Solution ................................... 140
5.6.2 Effective Stiffness Tensor ........................ 142
5.7 Numerical Study .......................................... 145
5.7.1 Local Stress Fields ............................... 145
5.7.2 Effective Stiffness Tensor ........................ 148
CHAPTER 6 Conductivity of a Solid with Spheroidal
Inclusions .................................................... 155
6.1 Scalar Spheroidal Solid Harmonics ........................ 155
6.1.1 Laplace Equation in Spheroidal Coordinates ........ 155
6.1.2 Spheroidal Solid Harmonics: Definition and
Properties ........................................ 157
6.1.3 Relationships Between the Spherical and
Spheroidal Harmonics .............................. 159
6.1.4 Alternate Set of Spheroidal Harmonics ............. 161
6.2 Single Inclusion: Conductivity Problem ................... 162
6.2.1 Series Solution ................................... 163
6.2.2 Resolving Equations ............................... 163
6.2.3 Limiting Cases: Spherical, Penny-Shaped, and
Needle-Like Inclusions ............................ 164
6.3 Re-Expansion Formulas for Spheroidal Solid Harmonics ..... 168
6.3.1 Formal Series Expansion ........................... 169
6.3.2 Translation: Integral Form of Expansion
Coefficients ...................................... 170
6.3.3 Translation: Rational Form of Expansion
Coefficients ...................................... 172
6.3.4 Translation: General Formula ...................... 174
6.3.5 Rotation .......................................... 177
6.4 Finite Cluster Model of Spheroidal Particle Composite .... 179
6.4.1 Formal Solution ................................... 179
6.4.2 Modified Maxwell Method for Effective
Conductivity ...................................... 180
6.5 Double Fourier Integral Transform of Spheroidal
Harmonics ................................................ 183
6.6 Doubly Periodic Harmonics ................................ 185
6.7 Triply Periodic Harmonics ................................ 187
6.8 Heat Conduction in Periodic Composite .................... 188
6.8.1 Problem Statement ................................. 188
6.8.2 Temperature Field in Periodic Composite: 3P
Approach .......................................... 188
6.8.3 Temperature Field in Periodic Composite: 2P
Approach .......................................... 191
6.8.4 Multiple Inclusion RUC Model ...................... 192
6.8.5 Effective Conductivity ............................ 193
6.9 Numerical Examples ....................................... 195
6.9.1 Spheroidal Cavities and Inclusions ................ 195
6.9.2 Penny-Shaped Cracks ............................... 196
6.9.3 Superconducting Hakes ............................. 200
CHAPTER 7 Elastic Solid with Spheroidal Inclusions ........... 203
7.1 Background Theory ........................................ 203
7.1.1 Vector Solid Harmonics in Spheroidal Coordinates .. 203
7.1.2 Scalar and Vector Biharmonics. Spheroidal Lame
Solutions ......................................... 206
7.1.3 Selected Properties of Spheroidal Lame
Solutions ......................................... 209
7.2 Single-Inclusion Problem ................................. 215
7.2.1 Single Particle in Unbounded Solid ................ 215
7.2.2 Single Particle in an Unbounded Fluid ............. 218
7.2.3 Stress Intensity Factors for the Penny-Shaped
Crack ............................................. 219
7.3 Re-Expansion Formulas for the Spheroidal Lame
Solutions ................................................ 222
7.3.1 Translation ....................................... 224
7.3.2 Rotation .......................................... 227
7.4 Finite Cluster Model of Composite with Spheroidal
Inclusions ............................................... 228
7.4.1 Problem Statement ................................. 228
7.4.2 Formal Solution ................................... 229
7.4.3 Local Expansion ................................... 230
7.4.4 Numerical Example: Penny-Shaped Crack
Interacting with Another Crack or Inclusion ....... 231
7.5 Half-Space Problem ....................................... 236
7.5.1 Integral Transforms of the Spheroidal Lame
Solutions ......................................... 236
7.5.2 Elastic Half-Space Containing a Finite Array of
Spheroidal Inclusions ............................. 238
7.6 RUC Model of Elastic Spheroidal Particle Composite ....... 239
7.6.1 Periodic Solutions of the Lame Equation ........... 239
7.6.2 Formal Solution ................................... 240
7.6.3 Effective Stiffness Tensor of the Spheroidal
Particle Composite ................................ 242
7.6.4 Numerical Study ................................... 242
CHAPTER 8 Composites with Transversely Isotropic
Constituents .................................................. 251
8.1 Transversely Isotropic Conductivity ...................... 251
8.1.1 Partial Solutions ................................. 252
8.1.2 Problem Statement ................................. 253
8.1.3 Series Solution ................................... 253
8.1.4 Effective Conductivity Tensor ..................... 256
8.2 Transversely Isotropic Elastic Solid with Spherical
Inclusions ............................................... 258
8.2.1 Partial Vector Solutions .......................... 258
8.2.2 Single Inclusion Problem .......................... 263
8.2.3 Finite Array of Inclusions ........................ 266
8.3 RUC Model ................................................ 269
8.3.1 Formal Solution ................................... 269
8.3.2 Effective Stiffness Tensor ........................ 270
8.4 Numerical Examples ....................................... 271
8.4.1 Stress Concentration .............................. 271
8.4.2 Effective Stiffness ............................... 275
PART II Fibrous Composites: Two-Dimensional Models ............ 281
CHAPTER 9 Circular Fiber Composite with Perfect Interfaces ... 283
9.1 In-Plane Conductivity and Out-of-Plane Shear: The
Governing Equations ...................................... 284
9.1.1 Conductivity ...................................... 284
9.1.2 Out-of-Plane Shear ................................ 284
9.2 Finite Array of Circular Inclusions ...................... 285
9.2.1 General Solution for a Single Inclusion ........... 285
9.2.2 Finite Array of Inclusions in Unbounded Plane ..... 286
9.2.3 Convergence Proof ................................. 290
9.2.1 Half Plane with Circular Inclusions ............... 291
9.4 Infinite Arrays of Circular Inclusions ................... 295
9.4.1 Periodic Complex Potentials ....................... 295
9.4.2 Composite Band .................................... 296
9.4.3 Composite Layer ................................... 298
9.5 Representative Unit Cell Model ........................... 301
9.5.1 Problem Statement ................................. 301
9.5.2 Local Thermal Fields: 1P Approach ................. 302
9.5.3 Local Thermal Fields: 2P Approach ................. 304
9.5.4 Averaged Fields and Effective Conductivity ........ 306
9.6 Finite Array of Circular Inclusions: In-Plane
Elasticity Problem ....................................... 307
9.6.1 Basic Equations in Terms of Complex Potentials .... 307
9.6.2 Solution for an Unbounded Plane ................... 309
9.7 Circular Inclusions in Half-Plane ........................ 312
9.7.1 Problem Statement ................................. 312
9.7.2 Determination of the Integral Densities p(β) and
q(β) .............................................. 314
9.7.3 Resolving Linear System: Integrals vs. Rational
Expressions ....................................... 315
9.8 RUC Model of Fibrous Composite: Elasticity ............... 318
9.8.1 The Problem Statement ............................. 318
9.8.2 Displacement Solution ............................. 319
9.8.3 Transverse Effective Stiffness of Fibrous
Composite ......................................... 321
9.9 Statistics of MicroStructure, Peak Stress and Interface
Damage in Fibrous Composite .............................. 323
9.9.1 MicroStructure Statistics: Nearest Neighbor
Distance .......................................... 324
9.9.2 Peak Interface Stress and Statistics of
Extremes .......................................... 325
9.9.3 Stress Concentration vs. Nearest Neighbor
Distance .......................................... 326
9.9.4 Micro Damage Model of FRC ......................... 329
CHAPTER 10 Fibrous Composite with Interface Cracks ............ 331
10.1 General Solution for a Single, Partially Debonded
Inclusion ................................................ 331
10.1.1 The Problem Statement ............................. 331
10.1.2 The R(ζ) Function ................................. 332
10.1.3 Formal Solution ................................... 334
10.1.4 Heat Flux Intensity Factor ........................ 337
10.2 Finite Array of Partially Debonded Inclusions ............ 337
10.3 Conductivity of Fibrous Composite with Interface Damage .. 340
10.3.1 Formal Solution ................................... 340
10.3.2 Evaluation of the Lattice Sums .................... 342
10.3.1 Effective Conductivity Tensor ..................... 343
10.3.4 Numerical Examples ................................ 344
10.4 In-Plane Elasticity: General Form of the Displacement
Solution ................................................. 346
10.5 Displacement Solution for the Partially Debonded
Inclusion ................................................ 348
10.5.1 Problem Statement ................................. 348
10.5.2 General Form of Potentials ........................ 349
10.5.3 The Rλ(ζ) and Хλ(ζ) Functions ..................... 350
10.5.4 Analytical Solution ............................... 352
10.5.5 Stress Intensity Factor ........................... 354
10.6 A Finite Number of Interacting Inclusions with
Interface Cracks ......................................... 355
10.6.1 Problem Statement and Iterative Solution
Procedure ......................................... 355
10.6.2 Evaluation of Integrals in Eq. (10.87) ............ 357
10.7 RUC Model of Fibrous Composite with Interface Cracks .... 359
10.7.1 Formal Solution ................................... 359
10.7.2 Effective Stiffness Tensor ........................ 360
10.7.3 Numerical Study: Stiffness Reduction
vs. Interface Crack Density ....................... 362
CHAPTER 11 Solids with Elliptic Inclusions .................... 367
11.1 Single Elliptic Inclusion in an Inhomogeneous Far
Field .................................................... 367
11.1.1 Problem Statement and Form of Solution ............ 367
11.1.2 Displacement and Traction at the Elliptic
Interface ......................................... 369
11.1.3 Formal Solution ................................... 371
11.1.4 Stress Intensity Factor ........................... 374
11.2 Re-Expansion Formulas for the Elliptic Solid Harmonics ... 374
11.3 Finite Array of Inclusions ............................... 378
11.4Half-Space Containing a Finite Array of Elliptic Fibers ... 382
11.4.1 Integral Transforms for Elliptic Harmonics ........ 382
11.4.2 Half-Plane with Elliptic Hole: Out-of-Plane
Elasticity /Conductivity Problem .................. 383
11.4.3 Half-Plane with Elliptic Inclusion: Plane
Elasticity Problem ................................ 386
11.4.4 FCM in Half-Plane ................................. 389
11.5 Periodic Complex Potentials .............................. 390
11.6 Micrornechanical Model of Cracked Solid .................. 391
11.6.1 Geometry .......................................... 392
11.6.2 Boundary-Value Problem ............................ 393
11.6.3 Out-of-Plane Shear ................................ 394
11.6.4 Plane Strain ...................................... 395
11.6.1 Effective Stiffness Tensor ........................ 399
11.6.6 Stress Intensity Factors .......................... 400
11.7 Numerical Examples ....................................... 401
11.7.1 Geometry with Pre-Defined Crack Orientation
Statistics ........................................ 401
11.7.2 Effective Stiffness vs. Crack Density and
Orientation ....................................... 402
11.7.3 SIF Statistics .................................... 406
CHAPTER 12 Fibrous Composite with Anisotropic Constituents .... 411
12.1 Out-of-Plane Shear ....................................... 411
12.1.1 Outline of the Approach and Basic Formulas ........ 411
12.1.2 Single Inclusion Problem .......................... 414
12.1.3 Finite Array of Inclusions ........................ 416
12.2 Periodic Complex Potentials .............................. 418
12.2.1 RUC Model ......................................... 419
12.3 Plane Strain ............................................. 420
12.3.1 General Solution .................................. 421
12.3.2 Single Inclusion Problem .......................... 421
12.3.3 Array of Inclusions ............................... 423
12.4 Effective Stiffness Tensor ............................... 424
APPENDIX A Sample Fortran Codes ............................... 427
A.l FCM Conductivity Problem (Chapter 2) ..................... 428
A.2 RUC Conductivity Problem for Spherical Particle
Composite (Chapter 3) .................................... 436
A.3 FCM Elasticity Problem (Chapter 4) ....................... 440
A.4 RUC Elasticity Problem for Spherical Particle Composite
(Chapter 5) .............................................. 453
A.5 RUC Conductivity and Elasticity Problems for Fibrous
Composite (Chapter 9) .................................... 459
A.6 Standard Lattice Sums .................................... 465
A.6.1 Triple Harmonic and Biharmonic Sums for Simple
Cubic Lattice ...................................... 465
A.6.2 Double Harmonic and Biharmonic Sums for Square
Lattice ............................................ 469
Bibliography .................................................. 471
Index ......................................................... 485
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