Nonlinear and Linearized Models in Thermoviscoelasticity

TL;DR

This paper develops a quasistatic nonlinear thermoviscoelastic model with frame indifference, proves existence of weak solutions without viscosity regularization, and demonstrates convergence to linearized solutions near the identity and small temperatures.

Contribution

It refines previous models by removing viscosity regularization and establishes convergence of nonlinear solutions to linearized models in thermoviscoelasticity.

Findings

Weak solutions obtained without viscosity regularization.

Convergence of nonlinear solutions to linearized models near identity.

Time-discrete approximations also converge to linearized solutions.

Abstract

We consider a quasistatic nonlinear model in thermoviscoelasticity at a finite-strain setting in the Kelvin-Voigt rheology where both the elastic and viscous stress tensors comply with the principle of frame indifference under rotations. The force balance is formulated in the reference configuration by resorting to the concept of nonsimple materials whereas the heat transfer equation is governed by the Fourier law in the deformed configurations. Weak solutions are obtained by means of a staggered in-time discretization where the deformation and the temperature are updated alternatingly. Our result refines a recent work by Mielke & Roub\'{\i}\v{c}ek [arXiv:1903.11094] since our approximation does not require any regularization of the viscosity term. Afterwards, we focus on the case of deformations near the identity and small temperatures, and we show by a rigorous linearization procedure that weak solutions of the nonlinear system converge in a suitable sense to solutions of a system in linearized thermoviscoelasticity. The same property holds for time-discrete approximations and we provide a corresponding commutativity result.