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

Published online by Cambridge University Press:  12 November 2008

Augusto Visintin
Affiliation:
Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, 38050 Povo (Trento), Italy (visintin@science.unitn.it)
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Abstract

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This paper deals with processes in nonlinear inelastic materials whose constitutive behaviour is represented by the inclusion

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 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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2008