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Nonlinear and Linearized Models in Thermoviscoelasticity

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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.

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Acknowledgements

This work was funded by the DFG Project FR 4083/5-1 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 -390685587, Mathematics Münster: Dynamics–Geometry–Structure. The work was further supported by the DAAD Project 57600633, and by the Project DAAD-22-03. Martin Kružík acknowledges support by GAČR-FWF Project 21-06569K and by the Erwin Schrödinger International Institute for Mathematics and Physics during his stay in Vienna in 2022.

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Authors and Affiliations

  1. Department of Mathematics, Friedrich-Alexander Universität Erlangen-Nürnberg, Cauerstr. 11, 91058, Erlangen, Germany

    Rufat Badal

  2. Mathematics Münster, University of Münster, Einsteinstr. 62, 48149, Münster, Germany

    Manuel Friedrich

  3. Institute of Information Theory and Automation, Czech Academy of Sciences, Pod vodárenskou věží 4, 182 00, Prague 8, Czechia

    Martin Kružík

  4. Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166 29, Prague 6, Czechia

    Martin Kružík

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  1. Rufat Badal
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  2. Manuel Friedrich
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  3. Martin Kružík
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Correspondence to Martin Kružík.

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Communicated by M. Ortiz.

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Badal, R., Friedrich, M. & Kružík, M. Nonlinear and Linearized Models in Thermoviscoelasticity. Arch Rational Mech Anal 247, 5 (2023). https://doi.org/10.1007/s00205-022-01834-9

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