Staggered time discretization in finitely-strained heterogeneous visco-elastodynamics with damage or diffusion in the Eulerian frame

The semi-implicit (partly decoupled, also called staggered or fraction-step) time discretization is applied to compressible nonlinear dynamical models of viscoelastic solids in the Eulerian description, i.e.\ in the actual deforming configuration, formulated fully in terms of rates. The Kelvin-Voigt rheology and also, in the deviatoric part, the Jeffreys rheology are considered. The numerical stability and, considering the Stokes-type viscosity multipolar of the 2nd-grade, also convergence towards weak solutions are proved in three-dimensional situations, exploiting the convexity of the kinetic energy when written in terms of linear momentum. No (poly)convexity of the stored energy is required and some enhancements (specifically towards damage and diffusion models) are briefly outlined, too.