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

here 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 initialand 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.