Homogenization of elastic media with gaseous inclusions

Abstract

We study the asymptotic behavior of a system modeling a composite material made of an elastic periodically perforated support, with period epsilon > 0, and a perfect gas placed in each of these perforations, as epsilon goes to zero. The model we use is linear, corresponding to deformations around a reference configuration. We apply both two-scale asymptotic expansion and two-scale convergence methods in order to identify the limit behaviors as epsilon goes to 0. We state that in the limit, we get a two-scale linear elasticity-like boundary value problem. From this problem, we identify the corresponding homogenized and periodic cell equations which allow us to find the first corrector term. The analysis is performed both in the case of an incompressible and a compressible material. We derive some mechanical properties of the limit materials by studying the homogenized coefficients. Finally, we calculate numerically the homogenized coefficients in the incompressible case for different types of elastic materials.