Solution concepts for a model of visco-elasto-plasticity with slight compressibility
Authors
- Eiter, Thomas
ORCID: 0000-0002-7807-1349
2020 Mathematics Subject Classification
- 35A15 35Q35 35Q74 35Q86 76A10 74B20
Keywords
- Visco-elasto-plasticity, slight compressibility, non-smooth potential, energy-variational solutions, stress diffusion, saddle-point structure
DOI
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.
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