An Optimal Condition of Compactness for Elasticity Problems Involving one Directional Reinforcement

Abstract

This paper deals with the homogenization of a homogeneous elastic medium reinforced by very stiff strips in dimension two. We give a general condition linked to the distribution and the stiffness of the strips, under which the nature of the elasticity problem is preserved in the homogenization process. This condition is sharper than the one used in Briane and Camar-Eddine (J. Math. Pures Appl. 88:483–505, 2007) and is shown to be optimal in the case where the strips are periodically arranged. Indeed, a fourth-order derivative term appears in the limit equation as soon as the condition is no more satisfied. In the periodic case the influence of oscillations in the medium surrounding the strips is also considered. The homogenization method is based both on a two-scale convergence for the strips and the use of suitable oscillating test functions. This allows us to obtain a distributional convergence of two of the three entries of the stress tensor contrary to the Γ-convergence approach of Briane and Camar-Eddine (J. Math. Pures Appl. 88:483–505, 2007).