Homogenization of processes in nonlinear visco-elastic composites

Abstract

The constitutive behaviour of a multiaxial visco-elastic material is here represented by the nonlinear relation ε − A(x) : t 0 σ (x, τ) dτ ∈ α(σ, x), which generalizes the classical Maxwell model of visco-elasticity of fluid-type. Here α(·, x) is a (possibly multivalued) maximal monotone mapping, σ is the stress tensor, ε is the linearized strain tensor, and A(x) is a positive-definite fourth-order tensor. The above inclusion is here coupled with the quasi-static force-balance law, − div σ = f . Existence and uniqueness of the weak solution are proved for a boundary-value problem. Convergence to a two-scale problem is then derived for a composite ma-terial, in which the functions α and A periodically oscillate in space on a short length-scale. It is proved that the coarse-scale averages of stress and strain solve a single-scale homogenized problem, and that conversely any solution of this prob-lem can be represented in that way. The homogenized constitutive relation is represented by the minimization of a time-integrated functional, and is rather dif-ferent from the above constitutive law. These results are also retrieved via De Giorgi's notion of -convergence. These conclusions are at variance with the outcome of so-called analogical models, that rest on an (apparently unjustified) mean-field-type hypothesis. Mathematics Subject Classification (2010): 35B27 (primary); 49J40, 73E50, 74Q (secondary).