Local and global effects of small holes periodically distributed on a surface embedded in an axisymmetrical elastic medium

We study an axisymmetric structure made of a linearly isotropic and homogeneous elastic material containing small holes whose centers are periodically distributed on a surface. The global structure then consists of two homogeneous media connected by a thin pierced layer which contains the holes. We use a multiple scaling method involving both periodic homogenization and matched asymptotic expansions. We aim at obtaining on the one hand the stresses close to the holes, and on the other hand the global behavior of the body. The stress field in the neighborhood of the hole is obtained through the solution of three elastic problems posed in an infinite strip containing a hole. It also allows to calculate the coefficients entering into the effective behaviour of the body. These coefficients depend on the shape of the holes. The second order problem involves jumps of the displacement and the stress fields which depend on the first order terms, and which exhibit singularities. (authors)