Derivation of a von Kármán plate theory for thermoviscoelastic solids
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| Format: | Preprint |
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2023
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| _version_ | 1866915025571020800 |
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| author | Badal, Rufat Friedrich, Manuel Machill, Lennart |
| author_facet | Badal, Rufat Friedrich, Manuel Machill, Lennart |
| contents | We derive a von Kármán plate theory from a three-dimensional quasistatic nonlinear model for nonsimple thermoviscoelastic materials in the Kelvin-Voigt rheology, in which the elastic and the viscous stress tensor comply with a frame indifference principle [Mielke-Roubíček '20]. In a dimension-reduction limit, we show that weak solutions to the nonlinear system of equations converge to weak solutions of an effective two-dimensional system featuring mechanical equations for viscoelastic von Kármán plates, previously derived in [Friedrich-Kružík '20], coupled with a linear heat-transfer equation. The main challenge lies in deriving a priori estimates for rescaled displacement fields and temperatures, which requires the adaptation of generalized Korn's inequalities and bounds for heat equations with $L^1$-data to thin domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_07196 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Derivation of a von Kármán plate theory for thermoviscoelastic solids Badal, Rufat Friedrich, Manuel Machill, Lennart Analysis of PDEs 35A15, 35A23, 35Q74, 35Q79, 74A15, 74D10 We derive a von Kármán plate theory from a three-dimensional quasistatic nonlinear model for nonsimple thermoviscoelastic materials in the Kelvin-Voigt rheology, in which the elastic and the viscous stress tensor comply with a frame indifference principle [Mielke-Roubíček '20]. In a dimension-reduction limit, we show that weak solutions to the nonlinear system of equations converge to weak solutions of an effective two-dimensional system featuring mechanical equations for viscoelastic von Kármán plates, previously derived in [Friedrich-Kružík '20], coupled with a linear heat-transfer equation. The main challenge lies in deriving a priori estimates for rescaled displacement fields and temperatures, which requires the adaptation of generalized Korn's inequalities and bounds for heat equations with $L^1$-data to thin domains. |
| title | Derivation of a von Kármán plate theory for thermoviscoelastic solids |
| topic | Analysis of PDEs 35A15, 35A23, 35Q74, 35Q79, 74A15, 74D10 |
| url | https://arxiv.org/abs/2312.07196 |