Concentration and homogenization in composites with total flux interface conditions

The model analyzed in this paper describes a composite made by a hosting medium containing a periodic array of inclusions, of size $\eps$, coated by a thin layer of a different material with thickness of the order $\eps\eta$. The thin layer presents an inner interface with imperfect contact conditions of non-local type. We deal with the concentration of the coating layer and with the homogenization, via the periodic unfolding method, of the resulting model. The construction of the homogenized matrix and of the limit source term deserves a deep investigation, despite the fact that the limit problem looks like a standard Dirichlet problem for an elliptic equation.The goal of this paper is to obtain, via the periodic unfolding method, the homogenized limit of a stationary diffusion model describing a composite made by a hosting medium containing a periodic array of inclusions of size $\eps$. The thermal potentials of the two phases are connected through suitable imperfect contact conditions imposed on the interface separating the two materials. Despite the fact that the limit model, obtained as $\eps\to 0$, is governed by a standard Dirichlet problem for an elliptic equation, the construction of the homogenized matrix and of the limit source term deserves a deep investigation. We propose this microscopic model inspired by the result of a concentration procedure performed in a simplified flat geometry, where we have two different bulk materials separated by a thin layer of another material with thickness of the order $\eta$. The thin layer presents an inner interface with imperfect contact conditions of non-local type and we deal with the concentration, as $\eta\to 0$, of such a layer. The final interface conditions thus obtained are exactly the interface conditions we impose in the above mentioned microscopic model set in a more general geometry.