Derivation of a Thermo-Visco-Elastic Plate Model at Small Strains

Abstract

We investigate a three-dimensional thermo-visco-elastic model with Kelvin–Voigt rheology under small strains confined to a thin domain. The model comprises a quasistatic linear momentum equation, with viscous stresses adhering to a Kelvin–Voigt viscosity law, coupled with a nonlinear heat equation governing temperature. The heat equation incorporates source terms arising from viscous dissipation and adiabatic heat sources due to thermal expansion. The model ensures thermodynamic consistency, maintaining energy conservation, positive temperature, and entropy production. We analyze the asymptotic behavior of solutions as the domain thickness approaches zero, deriving an effective two-dimensional model. This derivation involves rescaling the domain to a fixed thickness and establishing uniform a priori estimates relative to the plate’s thickness. In the limit, the temperature becomes vertically constant, and displacement are of Kirchhoff–Love type, enabling meaningful interpretation of the limiting objects within the plate’s two-dimensional cross-section. The mechanical equations consist of two parabolic equations, one for the membrane part and one for the bending part. Notably, the viscosity law in the limiting model departs from the Kelvin–Voigt form, reflecting nontrivial kinematic constraints on the rescaled out-of-plane strains. The bending of the plate does not depend on the temperature in the limit.