Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity

Riccarda Rossi

doi:10.3934/dcdss.2017075

Article Contents

2017, Volume 10, Issue 6: 1413-1466. Doi: 10.3934/dcdss.2017075

This issue Previous Article On the modeling of transport phenomena in continuum and statistical mechanics Next Article Optimal control of a rate-independent evolution equation via viscous regularization

Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity

Riccarda Rossi

DIMI -Universitá degli studi di Brescia, V. Branze, 38, Brescia, I-25133, Italy

Received: September 2016

Revised: December 2016

Published: December 2017

The author has been partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Abstract

In this paper we address a model coupling viscoplasticity with damage in thermoviscoelasticity. The associated PDE system consists of the momentum balance with viscosity and inertia for the displacement variable, at small strains, of the plastic and damage flow rules, and of the heat equation. It has a strongly nonlinear character and in particular features quadratic terms on the right-hand side of the heat equation and of the damage flow rule, which have to be handled carefully. We propose two weak solution concepts for the related initial-boundary value problem, namely 'entropic' and 'weak energy' solutions. Accordingly, we prove two existence results by passing to the limit in a carefully devised time discretization scheme. Finally, in the case of a prescribed temperature profile, and under a strongly simplifying condition, we provide a continuous dependence result, yielding uniqueness of weak energy solutions.

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Mathematics Subject Classification: Primary: 35Q74, 74C10; Secondary: 74F05.

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