Self-Consistent Methods for Composites

1. Introduction 2. 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 stresses 2.3 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 media 3.3 Ellipsoidal inhomogeneity 3.4 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 medium 3.8 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 inclusion 6.2 Ellipsoidal inhomogeneity 6.2.1 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 medium 6.4 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 moduli 7.4 The effective medium method and its versions 7.4.1 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 orientations 7.9.3 Elastic medium with thin hard unidirected bands of the same orientation 7.10 Elastic media with thin soft inclusions and cracks 7.10.1 Thin soft inclusions of the same orientation 7.10.2 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 inclusions 7.11.1 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 Collinear cracks 7.11.6 Vertical row of parallel cracks 7.12 Matrix composites reinforced by short axisymmetric fibers 7. 13 Elastic medium reinforced with unidirectional multi-layered fibers 7.14 Thermoelastic deformation of composites with multi-layered spherical or cylindrical inclusions 7.15 The point defect model in the theory of composite materials 7.16 Effective elastic properties of hybrid composites 7.16.1 Two different populations of inclusions in a homogeneous matrix (hybrid composite) 7.16.2 Two-point correlation functions for a hybrid composite with sets of cylindrical and spheroidal inclusions 7.16.3 Overall elastic moduli of three phase composites 7.17 Conclusions 7.18 Notes 8. Multi-particle interactions in composites 8.1 The effective field method beyond the quasicrystalline approximation 8.2 Mean values of some homogeneous random fields 8.3 General scheme for constructing multi-point statistical moments 8.4 The operator of the effective properties 8.5 Pair interactions between inclusions 8.6 Notes 9. Thermo and electroconductive properties of composites 9.1 Integral equations for a medium with isolated inclusions 9.2 The effective medium method 9.2.1 Differential effective medium method 9.3 The effective field method 9.3.1 Random set of thin inclusions 9.4 Dielectric properties of composites with high volume concentration of inclusions 9.4.1 The EFM in application to two-phase composites (the quasicrystalline approximation) 9.4.2 The EFM beyond the quasicrystalline approximation 9.4.3 Effective dielectric permittivity in 3D-case 9.4.4 Interaction between two inclusions in the 2D-case 9.4.5 Dielectric properties of the composites in 2D-case 9.4.6 Discussion and conclusion 9.5 Cross-properties relations 9.6 Notes A. Special tensor bases of four rank tensors A.1 E-basis A.2 P-basis A.3 9-basis A.4 R-basis <A.5 Averaging the elements of the E, P, 0, and R-bases A.6 Tensor bases of rank four tensors in 2D-space B. Generalized functions connected with the Green function of static elasticity B.1 The Green functions of static elasticity in the k-representation B.2 The Green functions of static elasticity in the x-representation B.3 The Green functions of static elasticity in 2D-case B.4 Special presentation of the K-operator C. Properties of some potentials of static elasticity concentrate on surfaces C.1 Gauss' and Stokes' integral theorems C.2 Derivatives of the double-layer potential of static elasticity C.3 Potentials with densities that are tensors of a surface D. Transition through the layers in the problems of thermoelasticity for multi-layered inclusions D.1 Elastic and thermoelastic problems for a spherical multi-layered inclusion D.2 Elastic and thermoelastic problems for a cylindrical multi-layered inclusion E. Correlation functions of random sets of spherical inclusions E.1 The Percus-Yevick correlation function of non-penetrating sets of spheres in the 3D-case E.2 The Percus-Yevick correlation function of non-penetrating sets of spheres in the 2D-case E.3 Correlation functions of the Boolean random sets of spheres and cylinders E.3. 1 Random models of two populations of inclusions References