Homogenization of processes in nonlinear visco-elastic composites

The constitutive behaviour of a multiaxial visco-elastic material is here represented by the nonlinear relation epsilon - A(x): integral(t)(0) sigma(x, tau) d tau epsilon alpha (sigma, x), which generalizes the classical Maxwell model of visco-elasticity of fluid type. Here alpha(., x) is a (possibly multivalued) maximal monotone mapping, sigma is the stress tensor, epsilon 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 sigma = (f) over bar. 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 material, in which the functions alpha 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 problem can be represented in that way. The homogenized constitutive relation is represented by the minimization of a time-integrated functional, and is rather different from the above constitutive law. These results are also retrieved via De Giorgi's notion of Gamma-convergence. These conclusions are at variance with the outcome of so-called analogical models, that rest on an (apparently unjustified) mean-field-type hypothesis.