Self-Consistent Methods for Composites

1. Introduction2. An elastic medium with sources of external and internal stresses 2.1 Medium with sources of external stresses 2.2 Medium with sources of internal stresses2.3 Discontinuities of elastic fields in a medium with sources of external and internal stresses 2.4 Elastic fields far from the sources 2.5 Notes 3. Equilibrium of a homogeneous elastic medium with an isolated inclusion 3.1 Integral equations for a medium with an isolated inhomogeneity 3.2 Conditions on the interface between two media3.3 Ellipsoidal inhomogeneity3.4 Ellipsoidal inhomogeneity in a constant external field 3.5 Inclusion in the form of a plane layer 3.6 Spheroidal inclusion in a transversely isotropic medium 3.7 Crack in an elastic medium3.8 Elliptical crack 3.9 Radially heterogeneous inclusion 3.9.1 Elastic fields in a medium with a radially heterogeneous inclusion 3.9.2 Thermoelastic problem for a medium with a radially heterogeneous inclusion 3.10 Multi-layered spherical inclusion 3.11 Axially symmetric inhomogeneity in an elastic medium 3.12 Multi-layered cylindrical inclusion 3.13 Notes 4. Thin inclusion in a homogeneous elastic medium 4.1 External expansions of elastic fields 4.2 Properties of potentials (4.4) and (4.5) 4.3 External limit problems for a thin inclusion 4.3.1 Thin soft inclusion 4.3.2 Thin hard inclusion 4.4 Internal limiting problems and the matching procedure 4.5 Singular models of thin inclusions 4.6 Thin ellipsoidal inclusions 4.7 Notes 5. Hard fiber in a homogeneous elastic medium 5.1 External and internal limiting solutions 5.2 Principal terms of the stress field inside a hard fiber 5.3 Stress fields inside fibers of various forms 5.3.1 Cylindrical fiber 5.3.2 Prolate ellipsoidal fiber 5.3.3 Fiber in the form of a double cone 5.4 Curvilinear fiber 5.5 Notes 6. Thermal and electric fields in a medium with an isolated inclusion 6.1 Fields with scalar potentials in a homogeneous medium with an isolated inclusion6.2 Ellipsoidal inhomogeneity6.2.1 Constant external field 6.2.2 Linear external field 6.2.3 Spheroidal inhomogeneity in a transversely isotropic medium 6.3 Multi-layered spherical inclusion in a homogeneous medium6.4 Thin inclusion in a homogeneous medium 6.5 Axisymmetric fiber in a homogeneous media 7. Homogeneous elastic medium with a set of isolated inclusion 7.1 The homogenization problem 7.2 Integral equations for the elastic fields in a medium with isolated inclusions 7.3 Tensor of the effective elastic moduli7.4 The effective medium method and its versions7.4.1 Differential effective medium method 7.5 The effective field method 7.5.1 Homogeneous elastic medium with a set of ellipsoidal inclusions 7.5.2 Elastic medium with a set of spherically layered inclusion 7.6 The Mon-Tanaka method 7.7 Regular lattices 7.8 Thin inclusions in a homogeneous elastic medium 7.9 Elastic medium reinforced with hard thin flakes or bands 7.9.1 Elastic medium with thin hard spheroids (flakes) of the same orientation 7.9.2 Elastic medium with thin hard spheroids homoge neousl distributed over the orientations7.9.3 Elastic medium with thin hard unidirected bands of the same orientation 7.10 Elastic media with thin soft inclusions and cracks7.10.1 Thin soft inclusions of the same orientation7.10.2 Homogeneous distribution of thin soft inclusions over the orientations 7.10.3 Elastic medium with regular lattices of thin inclusions 7.11 Plane problem for a medium with a set of thin inclusions7.11.1 A set of thin soft elliptical inclusions of the same orientation 7.11.2 Homogeneous distribution of thin inclusions over the orientations 7.11.3 Regular lattices of thin inclusions in plane 7.11.4 A triangular lattice of cracks 7.11.5 Col