Effective elastic properties of solids with cavities of various shapes are derived in two approximations: the approximation of non-interacting cavities and the approximation of the average stress field (Mori-Tanaka’s scheme); the latter appears to be appropriate when mutual positions of defects are random. We construct the elastic potential of a solid with cavities. Such an approach covers, in a unified way, cavities of various shapes and any mixture of them. No degeneracies (or a need in a special limiting procedure) arise when cavities shrink to cracks. It also provides a unified description of both isotropic and anisotropic effective properties and recovers results available in the literature for special cases. Elastic potentials dictate the choice of proper parameters of cavity density. These parameters depend on defect shapes. Even in the case of random orientations, the isotropic overall properties cannot be characterized in terms of porosity alone; for elliptical holes, for example, a second parameter - “eccentricity” - is needed.

The theory of elasticity is a basic discipline in solid mechanics. The classical theory of elasticity was developed during the nineteenth century and is linked up to the names of Cauchy, Navier, Poisson, St. Venant et al. The classical theory is limited to geometrical and material linearity and is a matured field of mathematics. So a great number of solutions of theoretical and applied problems are available, but many of the solutions cannot be found in textbooks but they are available in scientific articles.This handbook of elasticity is an extensive collection of closed solutions relevant to general solid mechanics and applications in material sciences, but not to structural mechanics. Solutions for beams, plates, shells etc. are not included and the content is strong limited to the linear elasticity.The handbook is subdivided into 11 basic chapters, 2 appendices and the references. Chap. 1 describes the fundamental equations of the linear elasticity in Cartesian, cylindrical, and spherical coordinates (strains, compatibility equations, equations of equilibrium, and the Hooke's law of anisotropic materials). Fundamental solutions of point forces and systems of point forces in the infinite isotropic solid and at the boundary of the isotropic half-space (forces, force couples, rotation centres, dipoles, dilatation centres) are given in Chap. 2. Chap. 3 is a collection of selected two-dimensional problems (plane stress and strain, infinite isotropic and orthotropic 2Dsolids, 2D-half-plane, stress concentration near holes and inclusions, and equilibrium of elastic wedges). Solutions for crack problems are given in the Chaps. 4, 5, and 6. There are three dimensional crack problems for the isotropic and the transversal isotropic infinite solid, cracks in an infinite two-dimensional isotropic solid, and cracks in an infinite twodimensional anisotropic solid.Chap. 7 demonstrates solutions of thermo-elasticity. There are given the basic equations, stationary and nonstationary 3-D and 2-D problems, and thermal stresses in heated infinite solids containing inhomogeneities or cavities. Contact problems are discussed in Chap. 8. One can find solutions for 2-D and 3-D problems of rigid punches with rectilinear or circular base and different force loadings. Also solutions for 2-D and 3-D Hertzian contact and dynamic contact of two colliding elastic spheres are given.Chap. 9 concentrates to Eshelby's problem and related results, Chap. 10 to elastic spaces containing rigid ellipsoidal inclusions subjected to translation and rotation and Chap. 11 contains selection of basic stress intensity factors.All solutions which are given in the Chaps. 1 till 11 are formulated in the most used coordinate systems, i.e. in Cartesian, cylindrical, and spherical coordinates. Appendix A provides the mathematical tool for a formulation of the elasticity problems in an arbitrary curvilinear orthogonal coordinate system and specifies the basic equations of elasticity for elliptic cylindrical, bipolar-cylindrical, toroidal, and ellipsoidal coordina...

Effective, generally anisotropic, conductivity of a material with ellipsoidal inclusions is analyzed. The results are given in closed form. They cover, in a unified way, mixtures of inclusions of diverse eccentricities ͑including cracks͒ and arbitrary nonrandom orientational distributions. Proper parameter of inclusions concentration, in whose terms the effective conductivities are to be expressed, is identified. This parameter correctly represents contributions of the individual inclusions to the overall conductivity. It reflects inclusion shapes and is tensorial; generally, it cannot be replaced by the volume fraction parameter.

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