Alexander Mielke

WIAS Berlin

Global existence for finite-strain viscoelasticity with temperature coupling

We describe recent work with Tomas Roubíček concerning the coupling of finite-strain visco-elasticity with temperature effects. Major nonlinearities arise through the static and time-dependent frame indifference, the thermomechanical coupling, and the viscous heating. To tackle these difficulties, we consider a regularized model (a so-called second-grade material) such that the deformation tensor is continuous. Using a uniform version of the Healey–Krömer estimate we are then able to show that the determinant of the deformation gradient is bounded away from 0. To control the time derivatives we rely on a generalized version of Korn’s inequality developed by Neff and Pompe.

A. Mielke, T. Roubíček. Thermoviscoelasticity in Kelvin–Voigt rheology at large strains. Arch. Rational Mech. Analysis 238 (2020) 1–45.