Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl—Reuss model of elastoplasticity

Abstract

This paper deals with processes in nonlinear inelastic materials whose constitutive behaviour is represented by the inclusionhere we denote by σ the stress tensor, by ε the linearized strain tensor, by B(x) the compliance tensor and by ∂ϕ(·, x) the subdifferential of a convex function ϕ(·, x). This relation accounts for elasto-viscoplasticity, including a nonlinear version of the classical Maxwell model of viscoelasticity and the Prandtl—Reuss model of elastoplasticity.The constitutive law is coupled with the equation of continuum dynamics, and well-posedness is proved for an initial- and boundary-value problem. The function ϕ and the tensor B are then assumed to oscillate periodically with respect to x and, as this period vanishes, a two-scale model of the asymptotic behaviour is derived via Nguetseng's notion of two-scale convergence. A fully homogenized single-scale model is also retrieved, and its equivalence with the two-scale problem is proved. This formulation is non-local in time and is at variance with that based on so-called analogical models that rest on a mean-field-type hypothesis.