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

Thomas Eiter

TL;DR

The paper analyzes a geophysically motivated visco-elasto-plastic model in near-incompressible conditions by embedding a slight compressibility through an internal momentum term, enabling pressure waves in an Eulerian setting. It derives a coupled system with a spherical elastic stress and a deviatoric Maxwell/Jeffreys-type rheology governed by a non-smooth dissipation potential, and it recasts the evolution via a variational-inequality weak formulation. To overcome non-well-posedness in the non-smooth regime, two solution concepts are developed: weak solutions for a stress-diffusion regularization (γ>0) and energy-variational solutions for the original system (γ=0), both constructed through a unified time-discrete saddle-point scheme. The work provides existence results, consistency with strong solutions under additional regularity, and a robust framework for passing to the time-continuous limit, contributing to the mathematical foundation of geophysically relevant rock rheologies and their energetic structure.

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.

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

TL;DR

The paper analyzes a geophysically motivated visco-elasto-plastic model in near-incompressible conditions by embedding a slight compressibility through an internal momentum term, enabling pressure waves in an Eulerian setting. It derives a coupled system with a spherical elastic stress and a deviatoric Maxwell/Jeffreys-type rheology governed by a non-smooth dissipation potential, and it recasts the evolution via a variational-inequality weak formulation. To overcome non-well-posedness in the non-smooth regime, two solution concepts are developed: weak solutions for a stress-diffusion regularization (γ>0) and energy-variational solutions for the original system (γ=0), both constructed through a unified time-discrete saddle-point scheme. The work provides existence results, consistency with strong solutions under additional regularity, and a robust framework for passing to the time-continuous limit, contributing to the mathematical foundation of geophysically relevant rock rheologies and their energetic structure.

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.

Paper Structure

This paper contains 22 sections, 12 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Derivation of the model and weak formulation
  5. Visco-elasto-plastic rheology
  6. Slightly compressible approximation
  7. Energy dissipation
  8. Towards a weak formulation
  9. The generalized solution concepts
  10. General assumptions
  11. Weak solutions to the system with stress diffusion
  12. Energy-variational solutions to the original system
  13. Functional framework
  14. Consistency
  15. Time-discrete approximation
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $f\in L^1(0,T)$, $g\in L^\infty(0,T)$ and $g_0\in\mathbb{R}$. Then the inequality holds for all $\phi \in C^1_c ([0,T))$ with $\phi \geq 0$ if and only if the inequality holds for a.e. $s,\, t\in[0,T)$ with $s<t$, including $s=0$ if we replace $g(0)$ with $g_0$. In these cases, $g$ coincides a.e. with an element of $\mathrm{BV}([0,T])$ with $\lim_{s\searrow0}g(s)\leq g_0$ and $\lim_{s\searro

Theorems & Definitions (33)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • Definition 3.4
  • Theorem 3.5
  • Remark 3.6
  • Proposition 3.7
  • ...and 23 more