Homogenization of a Highly Heterogeneous Elastic-Viscoelastic Composite Materials

Abstract

We consider the homogenization of a highly heterogeneous periodic medium made of two connected components filled with two linear elastic materials having elasticity tensors of order O(1), separated by a third viscoelastic layer, having a thickness of the same order \({\varepsilon}\) as the basic periodicity cell, and a stiffness (resp. viscosity) tensor of order \({\varepsilon^r ({\rm resp.}, \varepsilon^s) (r, s) \in \mathbb{R}_{d}x+^{2}}\). Using the method of two-scale convergence, a unified approach yielding rigorous proofs is given covering different scalings. In particular, it is shown that the medium behaves asymptotically as an elastic homogeneous one, except in the case m : = min(r, s) = 0 where it becomes viscoelastic with long memory.