Derivation of a von Kármán plate theory for thermoviscoelastic solids

Badal R, Friedrich M, Machill L (2024)

Publication Type: Journal article

Publication year: 2024

Journal

Mathematical Models & Methods in Applied Sciences World Scientific Publishing

Book Volume: 34

Pages Range: 2749-2824

Journal Issue: 14

DOI: 10.1142/S0218202524500581

Abstract

In this paper, 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 [A. Mielke and T. Roubíček, Thermoviscoelasticity in Kelvin-Voigt rheology at large strains, Arch. Ration. Mech. Anal. 238 (2020) 1-45]. 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 and Kružík [Derivation of von Kármán plate theory in the framework of three-dimensional viscoelasticity, Arch. Ration. Mech. Anal. 238 (2020) 489-540], 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 L1-data to thin domains.

Authors with CRIS profile

Rufat Badal Lehrstuhl für Angewandte Mathematik (Modellierung und Numerik) Manuel Friedrich Professur für Angewandte Mathematik (Angewandte Analysis)

Involved external institutions

Westfälische Wilhelms-Universität (WWU) Münster DE Germany (DE)

How to cite

APA:

Badal, R., Friedrich, M., & Machill, L. (2024). Derivation of a von Kármán plate theory for thermoviscoelastic solids. Mathematical Models & Methods in Applied Sciences, 34(14), 2749-2824. https://doi.org/10.1142/S0218202524500581

MLA:

Badal, Rufat, Manuel Friedrich, and Lennart Machill. "Derivation of a von Kármán plate theory for thermoviscoelastic solids." Mathematical Models & Methods in Applied Sciences 34.14 (2024): 2749-2824.

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