Solution concepts for a model of visco-elasto-plasticity with slight compressibility

TL;DR

This paper develops a mathematical framework for modeling nearly incompressible visco-elasto-plastic materials, incorporating pressure wave propagation and non-isochoric deformation, with solutions constructed via a time-discrete saddle-point scheme.

Contribution

It introduces a weak formulation with variational inequalities for a complex visco-elasto-plastic model and explores regularization and relaxed solution concepts for well-posedness.

Findings

Existence of solutions via time-discrete saddle-point scheme

Regularization with stress diffusion improves solution properties

Energy-variational solutions enable analysis of non-smooth plasticity

Abstract

We study a model for the deformation of a visco-elasto-plastic material that is nearly incompressible. It originates from geophysics, is given in the Eulerian description and combines a Kelvin-Voigt rheology in the spherical part with a Jeffreys-type rheology in the deviatoric part. Despite a constant density, the model allows for non-isochoric deformation and the propagation of pressure waves. An additive decomposition of the strain rate into elastic and inelastic parts leads to an evolution equation for the small elastic strain, which is coupled with an adapted momentum equation. As plasticity is modeled through a non-smooth dissipation potential, we introduce a weak formulation in terms of a variational inequality. Since the well-posedness in such a weak setting is out of reach, we study two possible modifications: the regularization in terms of stress diffusion, and the relaxation of the solvability concept by transition to energy-variational solutions. In both cases, solutions are constructed by the same time-discrete scheme, consisting of solving a saddle-point problem in each time step.