Nonlinear and Linearized Models in Thermoviscoelasticity

0573354 - ÚTIA 2024 RIV DE eng J - Článek v odborném periodiku
Badal, R. - Friedrich, M. - Kružík, Martin
Nonlinear and Linearized Models in Thermoviscoelasticity.
Archive for Rational Mechanics and Analysis. Roč. 247, č. 1 (2023), č. článku 5. ISSN 0003-9527. E-ISSN 1432-0673
Grant CEP: GA ČR GF21-06569K
Institucionální podpora: RVO:67985556
Klíčová slova: quasistatic nonlinear model * thermoviscoelasticity * Kelvin-Voigt rheology
Obor OECD: Pure mathematics
Impakt faktor: 2.5, rok: 2022
Způsob publikování: Omezený přístup
http://library.utia.cas.cz/separaty/2023/MTR/kruzik-0573354.pdf https://link.springer.com/article/10.1007/s00205-022-01834-9

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.
Trvalý link: https://hdl.handle.net/11104/0343815