Thermo-elastodynamics of nonlinearly viscous solids

TL;DR

This paper establishes the existence of weak solutions for the thermo-elastodynamics of nonlinearly viscous solids using a variational approach, regularization techniques, and advanced regularity theory, advancing understanding of nonlinear viscoelastic behavior.

Contribution

It introduces a novel combination of regularization and variational methods to prove existence of solutions in nonlinear thermo-viscoelasticity, with new regularity results for the deformation.

Findings

Existence of weak solutions for the dynamic thermo-elastodynamics system.

Development of a regularization approach that can be removed for a weaker formulation.

New regularity estimates for the deformation involving the fourth order p-Laplacian.

Abstract

In this paper, we study the thermo-elastodynamics of nonlinearly viscous solids in the Kelvin-Voigt rheology where both the elastic and the viscous stress tensors comply with the frame-indifference principle. The system features a force balance including inertia in the frame of nonsimple materials and a heat-transfer equation which is governed by the Fourier law in the deformed configuration. Combining a staggered minimizing movement scheme for quasi-static thermoviscoelasticity with a variational approach to hyperbolic PDEs, our main result consists in establishing the existence of weak solutions in the dynamic case. This is first achieved by including an additional higher-order regularization for the dissipation. Afterwards, this regularization can be removed by passing to a weaker formulation of the heat-transfer equation which complies with a total energy balance. The latter description hinges on regularity theory for the fourth order $p$-Laplacian which induces regularity estimates of the deformation beyond the standard estimates available from energy bounds. Besides being crucial for the proof, these extra regularity properties might be of independent interest and seem to be new in the setting of nonlinear viscoelasticity, also in the static or quasi-static case.