Effective Properties of Heterogeneous Materials (eBook)

Preface 6
Contents 7
1 Non-interaction Approximation in the Problem of Effective Properties 8
1 Introduction 8
2 Quantitative Characterization of Microstructure. Property Contribution Tensors 10
2.1 The Simplest Microstructural Parameters and Their Limitations 11
2.2 Microstructural Parameters are Rooted in the Non-interaction Approximation 13
2.3 Property Contribution Tensors 15
2.4 Hill's Modification (Comparison) Theorem and Its Implications 18
2.5 Benefits of Identifying Proper Microstructural Parameters 22
3 Ellipsoidal Inhomogeneities 23
4 The Non-interaction Approximation and the ``Dilute Limit'' 34
5 Anisotropy Due to Non-random Orientations and Its Approximations 40
5.1 The Concept of Approximate Elastic Symmetry 41
5.2 Best-Fit Approximations of Anisotropies 42
5.3 Elliptic Orthotropy 45
6 The Concept of ``Average Shape'' for a Mixture of Inhomogeneities of Diverse Shapes 48
6.1 Example: 2-D Elliptical Holes of Diverse Eccentricities 50
6.2 Implications for General Shapes 56
6.3 Three Dimensional Cases 57
6.4 Shape Diversity and ``Average Shapes' in the Context of Conductivity 60
7 Approximate Schemes Utilizing the Non-interaction Approximation as a Basic Building Block 61
8 Non-interaction Approximation for Microcracked Materials 68
8.1 General Relations 69
8.2 Circular Cracks: Crack Density Parameters and Overall Anisotropy 72
8.3 Flat Non-circular Cracks: General Considerations 74
8.4 Flat Non-circular Cracks: Rice's Theorem and Other Methods of Estimation of Crack Compliances 77
8.5 Intersected Cracks 79
8.6 Non-flat Cracks 80
9 Explicit Elasticity--Conductivity Connections for Anisotropic Two-Phase Matrix Composites 80
9.1 General Anisotropic Case 81
9.2 Cases of Overall Isotropy and Transverse Isotropy 88
9.3 Materials with Cracks or Rigid Discs 89
9.4 On the Sensitivity of Elasticity--Conductivity Connection to Shapes of Inhomogeneities 90
9.5 Applications of Cross-Property Connections to Specific Materials 92
10 Concluding Remarks 94
References 98
2 Multipole Expansion Method in Micromechanics of Composites 103
1 Homogenization Problem 105
1.1 Conductivity 106
1.2 Elasticity 109
2 Composite with Spherical Inclusions: Conductivity Problem 110
2.1 Background Theory 111
2.2 General Solution for a Single Inclusion 112
2.3 Finite Cluster Model (FCM) 114
2.4 FCM Boundary Value Problem 117
2.5 Representative Unit Cell Model 119
3 Composite with Spherical Inclusions: Elasticity Problem 125
3.1 Background Theory 125
3.2 Single Inclusion Problem 127
3.3 Re-Expansion Formulas 130
3.4 FCM 131
3.5 RUC 134
4 Composite with Spheroidal Inclusions 137
4.1 Scalar Spheroidal Solid Harmonics 137
4.2 Single Inclusion: Conductivity Problem 138
4.3 Re-expansion Formulas 139
4.4 FCM 142
4.5 RUC 143
4.6 Elasticity Problem 147
5 Spherical Particles Reinforced Composite with Transversely-Isotropic Phases 153
5.1 Background Theory 153
5.2 Series Solution 156
5.3 FCM and RUC 158
5.4 Effective Stiffness Tensor 161
6 Fibrous Composite with Interface Cracks 162
6.1 Background Theory 162
6.2 2D RUC Geometry 163
6.3 Model Problem 164
6.4 Displacement Solution 165
6.5 Single Partially Debonded Fiber 166
6.6 Finite Array of Partially Debonded Fibers 167
6.7 RUC Model of Fibrous Composite with Interface Cracks 171
6.8 Effective Stiffness Tensor 173
7 Composite with Elliptic in Cross-Section Fibers 175
7.1 Single Elliptic Inclusion in Non-uniform Far Field 175
7.2 Finite Array of Inclusions 180
8 Fibrous Composite with Anisotropic Constituents 184
8.1 Anti-Plane Shear Problem 185
8.2 Plane Strain Problem 193
8.3 Effective Stiffness Sensor 196
References 199
3 Effective Field Method in the Theory of Heterogeneous Media 204
1 Introduction 204
2 Conductivity Problem 206
2.1 Integral Equations of the Conductivity Problem 206
2.2 Ellipsoidal Inclusions 210
2.3 Numerical Solution of One-Particle Problems 211
2.4 The Homogenization Problem 217
2.5 Integral Equations for Heterogeneous Media 218
2.6 The Effective Conductivity Tensor 219
2.7 The Effective Field Method 220
2.8 The Mori-Tanaka Method 228
2.9 Hybrid Composites 232
2.10 The Maxwell Scheme 236
2.11 Historic Remarks 240
3 Combination of the Effective Field and Numerical Methods 242
3.1 The Effective Field Method for a Complex Composite Cell 242
3.2 The Effective Conductivity Tensor 246
3.3 Discretization of Equations of the Effective Field Method 248
3.4 Conductivity of 3D-Matrix Composite Materials 248
3.5 Conclusions 253
4 Effective Elastic Properties of Composites 253
4.1 One-Particle Problem 253
4.2 Effective Elastic Constants 255
4.3 The Effective Field Method 258
4.4 The Mori-Tanaka Method 260
4.5 The Maxwell Scheme 260
4.6 Numerical Solution of the Homogenization Problem 263
5 Homogenization of Elasto-Plastic Composites 269
5.1 Integral Equations for Heterogeneous Elasto-Plastic Media 269
5.2 The Effective Field Method 275
5.3 Discretization of the Integral Equations of the Effective Field Method 278
5.4 Average Stress-Strain Relations for Elasto-Plastic Composites 279
5.5 Stress-Strain Relations for Elasto-Plastic Composites 280
6 Conclusions 285
References 286
4 Effective Properties of Composite Materials, Reinforced Structures and Smart Composites: Asymptotic Homogenization Approach 288
1 Introduction 289
2 Asymptotic Homogenization Method 290
2.1 Asymptotic Expansion 290
2.2 Multi-Scale Asymptotic Expansion 292
2.3 Asymptotic Homogenization Method 296
3 Three-Dimensional Bulk Composite Materials and Two-Dimensional Thin-Walled Composite Structures 303
3.1 Three-Dimensional Composite Materials 304
3.2 Thin-Walled Composite Reinforced Structures 306
4 Asymptotic Homogenization of Smart Composites 311
4.1 Three-Dimensional Smart Composite Materials 312
4.2 Smart Composite Shells and Plates 316
5 Effective Properties of Thin-Walled Composite Reinforced Structures 320
5.1 Wafer-Reinforced Shells 320
5.2 Rib-Reinforced Shells 322
5.3 Sandwich Composite Shells with Honeycomb Fillers 323
6 Three-Dimensional Smart Grid-Reinforced Composites 324
6.1 Effective Elastic Coefficients 327
6.2 Effective Piezoelectric Coefficients 329
6.3 Effective Thermal Expansion Coefficients 331
7 Smart Grid-Reinforced Composite Shells and Plates 333
7.1 Effective Elastic Coefficients 335
7.2 Effective Piezoelectric Coefficients 340
7.3 Effective Thermal Expansion Coefficients 347
7.4 Examples of the Smart Composite Orthotropic Grid-Reinforced Shells and Plates 351
8 Smart Sandwich Composite Shells with Cellular Cores 360
9 Effective Mechanical Properties of Carbon Nanotubes 362
10 Conclusions 365
References 366
5 Basic Microstructure-Macroproperty Calculations 369
1 Introduction 369
2 Basic Micro-Macro Concepts 371
2.1 Testing Procedures 372
2.2 The Average Strain Theorem 373
2.3 The Average Stress Theorem 374
2.4 Satisfaction of Hill's Energy Condition 374
2.5 The Hill-Reuss-Voigt Bounds 375
2.6 Improved Estimates 377
3 Computational/Statistical Testing Methods 378
3.1 A Boundary Value Formulation 378
3.2 Weak Formulations: The Foundation of Finite Element Methods 379
3.3 Numerical Discretization 380
3.4 Overall Testing Process: Numerical Examples 382
3.5 Increasing Sample Size 385
3.6 A Minimum Principle Interpretation of the Results 387
4 Summary and Closing Comments 388
References 390