Nonlinear and Linearized Models in Thermoviscoelasticity

Badal R, Friedrich M, Kružík M (2023)

Publication Type: Journal article

Publication year: 2023

Journal

Archive for Rational Mechanics and Analysis Springer Verlag (Germany)

Book Volume: 247

Article Number: 5

Journal Issue: 1

DOI: 10.1007/s00205-022-01834-9

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 and Roubíček (Arch Ration Mech Anal 238:1–45, 2020) 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.

Authors with CRIS profile

Rufat Badal Lehrstuhl für Angewandte Mathematik (Modellierung und Numerik) Manuel Friedrich

Involved external institutions

Westfälische Wilhelms-Universität (WWU) Münster DE Germany (DE) Institute of Information Theory and Automation (UTIA) CZ Czech Republic (CZ)

How to cite

APA:

Badal, R., Friedrich, M., & Kružík, M. (2023). Nonlinear and Linearized Models in Thermoviscoelasticity. Archive for Rational Mechanics and Analysis, 247(1). https://dx.doi.org/10.1007/s00205-022-01834-9

MLA:

Badal, Rufat, Manuel Friedrich, and Martin Kružík. "Nonlinear and Linearized Models in Thermoviscoelasticity." Archive for Rational Mechanics and Analysis 247.1 (2023).

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