Homogenization of Some Multiparametric Problems

Abstract

Inhomogeneous media are considered in this paper, e.g., composites and mixtures. Let L be the length scale of the problem and d be the typical length of inhomogeneities. The ratio ε = d/L is supposed to be small. Then the averaged equations can often be obtained that describe a certain homogeneous medium and have solutions close in some sense to solutions of original equations. For periodic media an asymptotic homogenization method to obtain the averaged equations is developed making use of the presence of a small parameter ε.

In many problems there are additional small parameters γi, besides ε. For example the following parameters can be small: the ratios of different phases moduli, the ratios of coefficients determining different properties of a phase, the ratios of inhomogeneity scales in different directions. The averaged equations essentially depend on the relations between small parameters ε and γi. In some cases the homogenized equations are of another type than equations describing the process in original medium. For example, instead of differential equations we obtain integro-differential equations.

Construction of averaged equations for periodic media includes solution of the so-called cell-problems. They are boundary-value problems for partial differential equations. As a rule they can be solved only numerically. In some cases analytical approximate solutions to cell-problems and explicit formulae for effective coefficients can be obtained due to presence of additional small parameters. The explicit formulae for effective moduli are very useful, especially in optimal design of materials and constructions.

The paper is a brief review of some author’s results concerning the effect of different small parameters.

The work was supported by Russian Foundation for Basic research (Projects No 02-01-00490, 02-01-00613)