Time discretization in convected linearized thermo-visco-elastodynamics at large displacements
Tomáš Roubíček
TL;DR
This work develops a fully implicit backward-Euler time discretization for compressible thermo-viscoelastic solids modeled in Eulerian coordinates with large displacements. It formulates a rate-based, thermodynamically consistent model using Green-Naghdi decomposition and Zaremba-Jaumann derivatives, incorporating multipolar (second-grade) viscosity and Jeffreys rheology. The authors prove stability and convergence to weak solutions by leveraging entropy inequalities and the convexity of the kinetic energy in momentum, and they analyze a tractable special case with a partly linearized thermo-mechanical coupling. The results extend prior isothermal analyses to anisothermal, nonlinear, fully coupled thermo-viscoelastic systems with higher-gradient dissipation, ensuring energy-dissipation balance and providing a rigorous foundation for numerical simulations of large-deformation viscoelastic materials.
Abstract
The fully-implicit time discretization (i.e. the backward Euler formula) is applied to compressible nonlinear dynamical models of thermo-viscoelastic solids in the Eulerian description, i.e. in the actual deforming configuration, formulated in terms of rates. The Kelvin-Voigt rheology or also, in the deviatoric part, the Jeffreys rheology (which covers creep or plasticity) are considered, using the additive Green-Naghdi decomposition of total strain into the elastic and the inelastic strains formulated in terms of (objective) rates exploiting the Zaremba-Jaumann time derivative. A linearized convective model at large displacements is considered, focusing on the case where the internal energy additively splits the (convex) mechanical and the thermal parts.A fully implicit time-discrete scheme is devised. Considering the multipolar 2nd-grade viscosity, the numerical stability and convergence towards weak solutions are proven by exploiting, in particular, the convexity of the kinetic energy when written in terms of linear momentum instead of velocity and by estimating the temperature gradient from the entropy-like inequality.