Time discretization in visco-elastodynamics at large displacements and strains in the Eulerian frame

TL;DR

This paper develops and analyzes fully-implicit time discretization schemes for large displacement viscoelastic solid models in the Eulerian frame, ensuring stability and convergence for complex nonlinear rheologies like Kelvin-Voigt and Jeffreys.

Contribution

It introduces regularized, stable, and convergent time discretization schemes for large strain viscoelastic models in the Eulerian frame, including nonlinear and convex energy formulations.

Findings

Proved numerical stability of the schemes.

Established convergence towards weak solutions.

Demonstrated applicability to neo-Hookean and Mooney-Rivlin materials.

Abstract

The fully-implicit time discretization (i.e. the backward Euler formula) is applied to compressible nonlinear dynamical models of viscoelastic solids in the Eulerian description, i.e. in the actual deforming configuration. The Kelvin-Voigt rheology or also, in the deviatoric part, the Jeffreys rheology are considered. Both a linearized convective model at large displacements with a convex stored energy and the fully nonlinear large strain variant with a (possibly generalized) polyconvex stored energy are considered. The time-discrete suitably regularized schemes are devised for both cases. 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. In the fully nonlinear case, the examples of neo-Hookean and Mooney-Rivlin materials are presented. A comparison with models of viscoelastic barotropic fluids is also made.