Micromechanics with Mathematica

Preface ix About the Companion Website xi 1 Coordinate Transformation and Tensors 1 1.1 Index Notation 1 1.1.1 Some Examples of Index Notation in 3-D 3 1.1.2 Mathematica Implementation 3 1.1.3 Kronecker Delta 6 1.1.4 Permutation Symbols 9 1.1.5 Product of Matrices 10 1.2 Coordinate Transformations (Cartesian Tensors) 11 1.3 Definition of Tensors 13 1.3.1 Tensor of Rank 0 (Scalar) 13 1.3.2 Tensor of Rank 1 (Vector) 14 1.3.3 Tensor of Rank 2 15 1.3.4 Tensor of Rank 3 17 1.3.5 Tensor of Rank 4 17 1.3.6 Differentiation 19 1.3.7 Differentiation of Cartesian Tensors 20 1.4 Invariance of Tensor Equations 21 1.5 Quotient Rule 22 1.6 Exercises 23 References 24 2 Field Equations 25 2.1 Concept of Stress 25 2.1.1 Properties of Stress 29 2.1.2 (Stress) Boundary Conditions 30 2.1.3 Principal Stresses 31 2.1.4 Stress Deviator 35 2.1.5 Mohr s Circle 38 2.2 Strain 40 2.2.1 Shear Deformation 47 2.3 Compatibility Condition 49 2.4 Constitutive Relation, Isotropy, Anisotropy 50 2.4.1 Isotropy 52 2.4.2 Elastic Modulus 54 2.4.3 Orthotropy 56 2.4.4 2-D Orthotropic Materials 57 2.4.5 Transverse Isotropy 57 2.5 Constitutive Relation for Fluids 58 2.5.1 Thermal Effect 58 2.6 Derivation of Field Equations 59 2.6.1 Divergence Theorem (Gauss Theorem) 59 2.6.2 Material Derivative 60 2.6.3 Equation of Continuity 62 2.6.4 Equation of Motion 62 2.6.5 Equation of Energy 63 2.6.6 Isotropic Solids 65 2.6.7 Isotropic Fluids 65 2.6.8 Thermal Effects 66 2.7 General Coordinate System 66 2.7.1 Introduction to Tensor Analysis 66 2.7.2 Definition of Tensors in Curvilinear Systems 68 2.7.3 Metric Tensor10, gij 69 2.7.4 Covariant Derivatives 70 2.7.5 Examples 73 2.7.6 Vector Analysis 75 2.8 Exercises 77 References 80 3 Inclusions in Infinite Media 81 3.1 Eshelby s Solution for an Ellipsoidal Inclusion Problem 82 3.1.1 Eigenstrain Problem 85 3.1.2 Eshelby Tensors for an Ellipsoidal Inclusion 87 3.1.3 Inhomogeneity (Inclusion) Problem 95 3.2 Multilayered Inclusions 104 3.2.1 Background 104 3.2.2 Implementation of Index Manipulation in Mathematica 105 3.2.3 General Formulation 108 3.2.4 Exact Solution for Two-Phase Materials 116 3.2.5 Exact Solution for Three-Phase Materials 123 3.2.6 Exact Solution for Four-Phase Materials 132 3.2.7 Exact Solution for 2-D Multiphase Materials 137 3.3 Thermal Stress 137 3.3.1 Thermal Stress Due to Heat Source 138 3.3.2 Thermal Stress Due to Heat Flow 146 3.4 Airy s Stress Function Approach 155 3.4.1 Airy s Stress Function 156 3.4.2 Mathematica Programming of Complex Variables 161 3.4.3 Multiphase Inclusion Problems Using Airy s Stress Function 163 3.5 Effective Properties 172 3.5.1 Upper and Lower Bounds of Effective Properties 173 3.5.2 Self-Consistent Approximation 175 3.5.3 Source Code for micromech.m 178 3.6 Exercises 188 References 189 4 Inclusions in Finite Matrix 191 4.1 General Approaches for Numerically Solving Boundary Value Problems 192 4.1.1 Method of Weighted Residuals 192 4.1.2 Rayleigh Ritz Method 203 4.1.3 Sturm Liouville System 205 4.2 Steady-State Heat Conduction Equations 213 4.2.1 Derivation of Permissible Functions 213 4.2.2 Finding Temperature Field Using Permissible Functions 227 4.3 Elastic Fields with Bounded Boundaries 232 4.4 Numerical Examples 238 4.4.1 Homogeneous Medium 238 4.4.2 Single Inclusion 240 4.5 Exercises 251 References 252 Appendix A Introduction to Mathematica 253 A.1 Essential Commands/Statements 255 A.2 Equations 256 A.3 Differentiation/Integration 260 A.4 Matrices/Vectors/Tensors 260 A.5 Functions 262 A.6 Graphics 263 A.7 Other Useful Functions 265 A.8 Programming in Mathematica 267 A.8.1 Control Statements 268 A.8.2 Tensor Manipulations 270 References 272 Index 273

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