All materials consist of some heterogeneity. In many cases heterogeneity can affect the properties of the whole sample, and this fact stimulates the desire to create heterogeneous materials with definite desired properties. Most of man-made materials such as composite materials, porous structures, powders, have periodical structure. In nature geological materials have structures, close to periodical. This is the reason to investigate the behavior of heterogeneous materials with periodical structure. In the majority of cases heterogeneous materials with a number of components, having different properties, cannot be described by direct considering each of the heterogeneities. The way to avoid intractable problems is to replace the heterogeneous medium by an equivalent homogeneous. Materials with periodic structures are investigated by means of multi-scale homogenization, which make it possible to derive macroscopic behavior of the material from its local structure.
Within the framework of multi-scale homogenization of differential equations with rapidly oscillating coefficients, the solution of the problem of heat conduction and filtration was built for heterogeneous media with a periodic structure. General equations to describe the processes of heat conduction with account for conductive and convective mechanisms of heat transfer, as well as for the possibility of phase transitions were obtained. Multi-scale homogenization was applied to the problem of deformation of heterogeneous material, composed of elastic components. As a result the effective stiffness tensor and effective technical deformation characteristics such as Young moduli, shear moduli and Poison ratios were defined.