Intro; Preface; Contents; 1 Background Results on Elasticity and Conductivity; 1.1 Basic Equations of Linear Elasticity. Elastic Symmetries; 1.2 Energy Principles of Elasticity; 1.2.1 Virtual Changes of State; 1.2.2 The Principle of Virtual Displacements; 1.2.3 The Principle of Virtual Forces; 1.2.4 The Principle of Stationarity of Potential Energy of an Elastic Solid; 1.2.5 The Principle of Stationarity of Complementary Energy of an Elastic Solid; 1.3 Approximate Symmetries of the Elastic Properties; 1.4 A Summary of Algebra of Fourth-Rank Tensors; 1.4.1 Isotropic Fourth-Rank Tensors

1.4.2 Anisotropic Fourth-Rank Tensors1.4.3 Transversely Isotropic Tensors; 1.4.4 Averaging of Tensors nn and nnnn Over Orientations in Simplest Cases of Orientation Distribution; 1.4.5 Orthotropic Tensors; 1.5 Thermal and Electric Conductivity: Fourier and Ohm's Laws; 1.6 Green's Tensors in Elasticity and Conductivity and Their Derivatives; 1.6.1 General Representation of Green's Tensor in Elasticity; 1.6.2 Isotropic Elastic Material; 1.6.3 Transversely Isotropic Elastic Material

1.6.4 Green's Tensor for a Monoclinic Material, in the Plane of Elastic Symmetry and in the Direction Normal to It1.6.5 Cubic Symmetry; 1.6.6 Two-Dimensional Anisotropic Elastic Material; 1.6.7 Derivatives of Green's Tensor; 1.6.8 Green's Function in the Conductivity Problem; 1.7 Dipoles, Moments, and Multipole Expansions in Elasticity and Conductivity; 1.7.1 System of Forces Distributed in Small Volume; 1.7.2 Dipole; 1.7.3 Center of Dilatation; 1.7.4 Force Couple; 1.7.5 Center of Rotation; 1.7.6 Multipole Expansion; 1.8 Stress Intensity Factors

1.9 General Thermodynamics Framework for Transition from Microscale to Macroscopic Constitutive Equations (Rice's Formalism)1.10 Mathematical Analogies Between Elastostatics and Steady-State Heat Flux. Conductivity Analogues of Stress Intensity Factors; 1.11 Discontinuities of the Elastic and Thermal Fields at Interfaces of Two Different Materials; 1.11.1 Stress Discontinuities in the Elasticity Problem; 1.11.2 Flux Discontinuities in the Conductivity Problem; 2 Quantitative Characterization of Microstructures in the Context of Effective Properties

2.1 Representative Volume Element (RVE) and Related Issues2.1.1 Hill's Condition. Homogeneous Boundary Conditions; 2.1.2 Averages Over Volume and Their Relation to Quantities Accessible on Its Boundary; 2.1.3 Volumes Smaller than RVE; 2.2 The Concept of Proper Microstructural Parameters; 2.3 The Simplest Microstructural Parameters and Their Limitations; 2.4 Microstructural Parameters Are Rooted in the Non-interaction Approximation; 2.5 Property Contribution Tensors of Inhomogeneities; 2.6 Hill's Comparison (Modification) Theorem and Its Implications