Concentration and homogenization in composites with total flux interface conditions

Abstract 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 $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> . 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 $$\varepsilon \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , 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 $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> . The thin layer presents an inner interface with imperfect contact conditions of non-local type and we deal with the concentration, as $$\eta \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , 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.