Asymptotic behavior of thin elastic interphases

The asymptotic behavior of a linearly elastic composite material that contains a thin interphase is described and analyzed by means of two complementary methods: the asymptotic expansions method and the study of the weak form using variational methods on Sobolev spaces. We recover the solution of the system of linearized elasticity in the two dimensional vectorial case and we find limit transmission conditions. The same steps are followed for harmonic oscillations of the elasticity system, and different solutions are found for concentrated mass densities. The cases in which the elastic coefficients depend on the thickness of the small parameter, for soft as well as stiff materials are considered. An approximated solution is found for harmonic oscillations of the elasticity system and limit transmission conditions are derived. Considering a bounded rectangular composite domain, with a thin interphase, we describe the weak formulation of the linearized system of elasticity. In the case of constant elastic coefficients, we estimate the bounds of the strain tensor and so, the energetic functional in the rescaled domain. We perform a variational formulation of the system of linearized elasticity and find estimates for the energetic functional of the system.