Positive temperature in nonlinear thermoviscoelasticity and the derivation of linearized models

TL;DR

This paper demonstrates that solutions to a nonlinear thermoviscoelastic model with positive temperature remain bounded away from zero and converge to linearized models near a critical temperature, extending previous linearization results.

Contribution

It extends the linearization of nonlinear thermoviscoelasticity models to include positive critical temperatures, showing convergence of solutions near identity deformations.

Findings

Solutions maintain positive temperature bounds over time.

Weak solutions converge to linearized models near critical temperature.

The result generalizes previous linearization work to positive temperature regimes.

Abstract

According to the Nernst theorem or, equivalently, the third law of thermodynamics, the absolute zero temperature is not attainable. Starting with an initial positive temperature, we show that there exist solutions to a Kelvin-Voigt model for quasi-static nonlinear thermoviscoelasticity at a finite-strain setting [Mielke-Roub\'i\v{c}ek '20], obeying an exponential-in-time lower bound on the temperature. Afterwards, we focus on the case of deformations near the identity and temperatures near a critical positive temperature, and we show that weak solutions of the nonlinear system converge in a suitable sense to solutions of a system in linearized thermoviscoelasticity. Our result extends the recent linearization result in [Badal-Friedrich-Kru\v{z}\'ik '23], as it allows the critical temperature to be positive.