Homogenization of nonlinear visco-elastic composites

Abstract

Quasi-static processes in nonlinear visco-elastic materials of solid-type are here represented by the system: (∗) σ − B ( x ) : ∂ ε ∂ t ∈ β ( ε , x ) , − div σ = f → , coupled with initial and boundary conditions. Here σ denotes the stress tensor, ε the linearized strain tensor, B ( x ) the viscosity tensor, β ( ⋅ , x ) a (possibly multi-valued) maximal monotone mapping, and f → an applied load. Existence and uniqueness of the weak solution are proved. A composite material in which the data β and B rapidly oscillate in space is then considered, and a two-scale model is derived via Nguetseng's notion of two-scale convergence. Although neither the stress nor the strain need be mesoscopically uniform, it is proved that their coarse-scale averages solve a global-in-time single-scale homogenized problem ( upscaling). From any solution of the latter a solution of the two-scale problem is then reconstructed ( downscaling). These results are at variance with the outcome of so-called analogical models, that assume a mean-field-type hypothesis. Finally, we represent the system (∗) as a minimum problem, and interpret the above results in terms of two- and single-scale Γ-convergence.