Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity

TL;DR

This paper establishes the existence of weak solutions for a complex coupled PDE system modeling viscoplasticity and damage in thermoviscoelastic materials, addressing nonlinear challenges and providing conditions for uniqueness.

Contribution

It introduces two weak solution concepts and proves existence results for a highly nonlinear PDE system coupling viscoplasticity, damage, and heat transfer.

Findings

Existence of entropic and weak energy solutions for the model.

A continuous dependence and uniqueness result under simplified conditions.

Development of a time discretization scheme for the nonlinear PDE system.

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.