Rate-independent damage in thermo-viscoelastic materials with inertia

TL;DR

This paper develops a comprehensive mathematical model for rate-independent damage in thermo-viscoelastic materials considering inertia and thermal effects, and proves existence and asymptotic properties of solutions.

Contribution

It introduces a novel coupled model incorporating damage, thermal effects, and inertia, with a new existence proof and asymptotic analysis for the rate-independent limit.

Findings

Existence of weak solutions for the coupled damage-thermal-inertia model.

Asymptotic limit yields a rate-independent model independent of temperature.

The model includes the Ambrosio-Tortorelli phase-field as a special case.

Abstract

We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is independent of temperature.