Micromechanics with Mathematica (eBook)

Seiichi Nomura, The University of Texas at Arlington, USA Professor Nomura studied his BSc and MSc at the University of Tokyo, Japan before completing his PhD at the University of Delaware. He is now Professor in the Department of Mechanical and Aerospace Engineering at the University of Texas at Arlington, USA. Professor Nomura has undertaken research on the analysis of mechanical and thermal properties of heterogeneous materials including composite materials and functionally graded materials (FGMs) and has published extensively in this area.

Preface

1 Coordinate transformation and tensors 1

1.1 Index notation 1

1.2 Coordinate Transformations (Cartesian tensors) 10

1.3 Definition of tensors 12

References 20

2 Field Equations 21

2.1 Concept of Stress 21

2.1.1 Properties of stress 23

2.1.2 (Stress) Boundary conditions 25

2.1.3 Principal stresses 26

2.1.4 Stress deviator 30

2.1.5 Mohr's Circle 33

2.2 Strain 35

2.2.1 Shear deformation 42

2.3 Compatibility condition 43

2.4 Constitutive Relation, Isotropy, Anisotropy 45

2.4.1 Isotropy 46

2.4.2 Elastic modulus 48

2.4.3 Orthotropy 50

2.5 Constitutive relation for fluids 52

2.6 Derivation of field equations 53

2.6.1 Divergence theorem (Gauss theorem) 53

2.6.2 Material derivative 54

2.6.3 Equation of continuity 56

2.6.4 Equation of motion 56

2.6.5 Equation of energy 57

2.6.6 Isotropic solids 59

2.6.7 Isotropic fluids 59

2.6.8 Thermal effects 60

2.7 General coordinate system 61

2.7.1 Introduction to tensor analysis 61

2.7.2 Definition of tensors in curvilinear systems 62

References 71

3 Inclusions in infinite media 73

3.1 Eshelby's solution for an ellipsoidal inclusion problem 74

3.1.1 Eigenstrain problem 77

3.1.2 Eshelby tensors for an ellipsoidal inclusion 78

3.1.3 Inhomogeneity (inclusion) problem 87

3.2 Multi-layered inclusions 96

3.2.1 Background 96

3.2.2 Implementation of index manipulation in Mathematica 97

3.2.3 General formulation 100

3.2.4 Exact solution for two-phase materials 106

3.2.5 Exact solution for three-phase materials 113

3.2.6 Exact solution for four-phase materials 125

3.2.7 Exact solution for 2-D multi-phase materials 129

3.3 Thermal stress 130

3.3.1 Thermal stress due to heat source 131

3.3.2 Thermal stress due to heat flow 139

3.4 Airy's stress function approach 154

3.4.1 Airy's stress function 154

3.4.2 Mathematica programming of complex numbers 157

3.4.3 Multi-phase inclusion problems using Airy's stree function 161

3.5 Effective properties 173

3.5.1 Upper and lower bounds of effective properties 174

3.5.2 Self-consistent approximation 176

References 180

4 Inclusions in finite matrix 181

4.1 General Approaches for Numerically Solving Boundary Value Problems 182

4.1.1 Method of Weighted Residuals 182

4.1.2 Rayleigh-Ritz Method 194

4.1.3 Sturm-Liouville System 196

4.2 Steady-State Heat Conduction Equations 199

4.2.1 Derivation of permissible functions 200

4.2.2 Finding temperature field using permissible functions 213

4.3 Elastic Fields with Bounded Boundaries 218

4.4 Numerical Examples 224

4.4.1 Homogeneous medium 225

4.4.2 Single inclusion 226

References 235

Appendix