Positive temperature in nonlinear thermoviscoelasticity and the derivation of linearized models
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913413966331904 |
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| author | Badal, Rufat Friedrich, Manuel Kružík, Martin Machill, Lennart |
| author_facet | Badal, Rufat Friedrich, Manuel Kružík, Martin Machill, Lennart |
| contents | 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-Roubíček '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-Kružík '23], as it allows the critical temperature to be positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02035 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positive temperature in nonlinear thermoviscoelasticity and the derivation of linearized models Badal, Rufat Friedrich, Manuel Kružík, Martin Machill, Lennart Analysis of PDEs 35A15, 35Q74, 74A15, 74D05, 74D10 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-Roubíček '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-Kružík '23], as it allows the critical temperature to be positive. |
| title | Positive temperature in nonlinear thermoviscoelasticity and the derivation of linearized models |
| topic | Analysis of PDEs 35A15, 35Q74, 74A15, 74D05, 74D10 |
| url | https://arxiv.org/abs/2407.02035 |