# Thermo-elastodynamics of nonlinearly viscous solids

## TL;DR

This paper investigates the dynamic behavior of nonlinearly viscous solids under thermal and mechanical effects, proving the existence of weak solutions using advanced mathematical techniques.

## Contribution

The paper introduces a novel variational approach combining regularization and energy balance to prove existence of solutions in nonlinear viscoelasticity.

## Key findings

- Weak solutions exist for dynamic thermo-elastodynamics in nonlinearly viscous solids using a staggered minimizing movement scheme.
- Higher-order regularization and a weaker heat-transfer formulation enable the removal of regularization while preserving energy balance.
- New regularity properties for the p-Laplacian in nonlinear viscoelasticity may have broader theoretical significance.

## 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 [2, 37] with a variational approach to hyperbolic PDEs developed in [5], 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.