Derivation of a von Kármán plate theory for thermoviscoelastic solids

Abstract

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.

57 pages

References in corpus (6)

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• One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams
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