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

Tomáš Roubíček

TL;DR

This work develops a rigorous staggered (semi-implicit) time discretization for finitely-strained, heterogeneous visco-elastodynamics formulated in the Eulerian frame, incorporating Kelvin-Voigt and Jeffreys rheologies. By truncating the stored energy and exploiting convexity of the kinetic energy in the linear momentum, the authors establish stability and convergence to weak solutions in 3D, without requiring polyconvex stored energy, and derive a comprehensive energy-dissipation balance. The scheme is designed to decouple the main mass/ momentum update from the evolution of internal variables, enabling extensions to multiphysics couplings. The paper further extends the framework to damage and diffusion models, detailing how internal variables modify stresses and energetics while preserving the staggered structure and convergence properties, thereby providing a robust, flexible tool for heterogeneous, large-strain materials.

Abstract

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.

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

TL;DR

This work develops a rigorous staggered (semi-implicit) time discretization for finitely-strained, heterogeneous visco-elastodynamics formulated in the Eulerian frame, incorporating Kelvin-Voigt and Jeffreys rheologies. By truncating the stored energy and exploiting convexity of the kinetic energy in the linear momentum, the authors establish stability and convergence to weak solutions in 3D, without requiring polyconvex stored energy, and derive a comprehensive energy-dissipation balance. The scheme is designed to decouple the main mass/ momentum update from the evolution of internal variables, enabling extensions to multiphysics couplings. The paper further extends the framework to damage and diffusion models, detailing how internal variables modify stresses and energetics while preserving the staggered structure and convergence properties, thereby providing a robust, flexible tool for heterogeneous, large-strain materials.

Abstract

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.

Paper Structure

This paper contains 8 sections, 1 theorem, 83 equations.

Table of Contents

  1. Introduction
  2. The visco-elastodynamic system and its energetics
  3. The kinematics in brief
  4. The visco-elastic rheology and the dynamics
  5. The staggered time discretization
  6. Extension towards some multiphysics models
  7. Damage
  8. Diffusion by the chemical potential gradient

Key Result

Proposition 3.2

$_{}$ Let (Euler-large-ass) holds with $r>3$ and let also $p>3$. Then: (i)For any $\lambda$ and for all sufficiently small time steps $\tau>0$, the staggered scheme (Euler-large-disc) has a solution $(\varrho_{\tau}^k,{\bm v}_{\tau}^k,{\bm F}_{\space\mathrm e,\tau}^k,\bm\xi_{\tau}^k)\in W^{1,r}(\var

Theorems & Definitions (8)

  • Definition 3.1: Weak formulation of (\ref{['Euler-hypoplast-p-heter']})
  • Proposition 3.2: Stability and convergence of (\ref{['Euler-large-visco-ED+discr']}) and existence of solutions to (\ref{['Euler-hypoplast-p-heter']})
  • proof : Sketch of the proof
  • Remark 3.3: Higher regularity of $\bm\xi$
  • Remark 3.4: More general $\bm g$
  • Remark 3.5: A gradient on inelastic distortion rate
  • Remark 3.6: Discrete Gronwall inequality
  • Remark 4.1: Diffusant content $\alpha$ as an extensive variable