Homogenization of the elasticity problem with periodically located cracks

Abstract

In this paper we study the nonlinear problem of elasticity with periodically located cracks. On the edges of these cracks non-penetration conditions are given, which leads to a variational inequality. The period of distribution of cracks, as well as their sizes, depends on the small parameter.In this work the corresponding varational problem and its homogenized problems are presented. We derive the necessary estimates for the theorem about convergence of the solutions of the variational inequality to the solution of the homogenised problem. For the first corrector of the classical asymptotic expansion we construct a penalty equation and a linear iterative equation in integral form. The convergence theorems for corresponding problems are formulated.