ИСТИНА

The well-known asymptotic method of homogenization is usually applied to materials with rapidly varying (at the micro-scale level) almost periodic (at the macro-scale level) structure. Nevertheless the method of asymptotic homogenization can be applied to much wider class of problems. Our work is concerned with the problem of dimension reduction: how can one derive the equations of two-dimensional elasticity (especially the equations of classical and not-so-classical plate bending theories) and which hypotheses are needed to perform such a derivation. We show that a specific form of asymptotic series typically used in homogenization methods, provides an almost mechanical way to obtain equations of plate bending of any order (in the sense of asymptotic approximation). Thus the (hard in some cases) problem of solving the equations of full three-dimensional elasticity is reduced to a (more tractable) sequence of two-dimensional problems. The first of these two-dimensional problems shares its partial differential equation (in the interior of the domain concerned) with the classical Kirchhoff plate theory. Taking into account the problems of higher orders, one gets the equations which can be reduced to a Reissner-Mindlin-type plate theory. One more interesting aspect of the described approach is that the formal asymptotic solution to the mentioned above infinite sequence of two-dimensional problems can be shown to solve the plate-bending equations obtained by more traditional approaches. So we present a (unfortunately, not so well known) way to obtain the classical plate theories by means of orthogonal projection relative to the energetic inner product and then explain in what sense the asymptotic solution happens to be a solution to corresponding variational problems.