Time discretization in convected linearized thermo-visco-elastodynamics at large displacements

The fully-implicit time discretization (ie. 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.