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

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 (covering creep or plasticity) are considered, using the additive Green-Naghdi's 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. The time-discrete suitably regularized scheme is devised. The numerical stability and, considering the multipolar 2nd-grade viscosity, also convergence towards weak solutions are proved, exploiting the convexity of the kinetic energy when written in terms of linear momentum instead of velocity and estimating the temperature gradient from the entropy-like inequality.