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arXiv · 2312.07196

Derivation of a von K\'arm\'an plate theory for thermoviscoelastic solids

Rufat Badal·Manuel Friedrich·Lennart Machill
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Abstract

We derive a von K\'arm\'an 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-Roub\'{i}\v{c}ek '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 K\'arm\'an plates, previously derived in [Friedrich-Kru\v{z}\'ik '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.

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Rufat Badal, Manuel Friedrich, Lennart Machill. 2023-12-12. Derivation of a von K\'arm\'an plate theory for thermoviscoelastic solids. https://doi.org/10.1142/s0218202524500581

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