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

摘要

This paper deals with processes in nonlinear inelastic materials whose constitutivebehaviour is represented by the inclusion — B(x) :](x);here we denote by the stress tensor, by a the linearized strain tensor, by B(x) thecompliance tensor and by (.,x) the subdifferential of a convex function (, x).This relation accounts for elasto-viscoplasticity, including a nonlinear version of theclassical Maxwell model of viscoelasticity and the Prandtl- Reuss model ofelastoplasticity. The constitutive law is coupled with the equation of continuum dynamics, andwell-posedness is proved for an initial- and boundary-value problem. The function coand the tensor B are then assumed to oscillate periodically with respect to x and, asthis period vanishes, a two-scale model of the asymptotic behaviour is derived viaNguetseng's notion of two-scale convergence. A fully homogenized single-scale modelis also retrieved, and its equivalence with the two-scale problem is proved. Thisformulation is non-local in time and is at variance with that based on so-calledanalogical models that rest on a mean-field-type hypothesis.