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An Eulerian formulation for dissipative materials using Lie derivatives and GENERIC

Alexander Mielke

TL;DR

This work develops an Eulerian continuum framework for dissipative solids by unifying Lie derivatives with the GENERIC structure. It proves that transport along the velocity field naturally induces a Poisson (Jacobi-satisfying) structure, enabling a clean separation of reversible Hamiltonian and irreversible dissipative dynamics. The authors formulate a thermo-visco-elastoplastic model with a multiplicative F- split in Eulerian variables $q=({\boldsymbol\pi},{\mathbf F},{\mathbf F}_{\mathrm p},\tau)$, deriving explicit expressions for the Hamiltonian and dissipative contributions and expressing driving forces through transformed energy/entropy gradients. A key result is Theorem JacobiLie, plus a constructive approach to assemble the full model via a simple Poisson operator and a block-structured dissipation potential, ensuring energy conservation and entropy production in line with the first and second laws. The framework promises robust, thermodynamically consistent extension to additional effects (e.g., reactive species) and applications to geophysical and materials engineering problems where Eulerian descriptions and complex couplings are essential.

Abstract

We recall the systematic formulation of Eulerian mechanics in terms of Lie derivatives along the vector field of the material points. Using the abstract properties of Lie derivatives we show that the transport via Lie derivatives generates in a natural way a Poisson structure on the chosen phase space. The evolution equations for thermo-viscoelastic-viscoplastic materials in the Eulerian setting is formulated in the abstract framework of GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling). The equations may not be new, but the systematic splitting between reversible Hamiltonian and dissipative effects allows us to see the equations in a new light that is especially useful for future generalizing of the system, e.g., for adding new effects like reactive species.

An Eulerian formulation for dissipative materials using Lie derivatives and GENERIC

TL;DR

This work develops an Eulerian continuum framework for dissipative solids by unifying Lie derivatives with the GENERIC structure. It proves that transport along the velocity field naturally induces a Poisson (Jacobi-satisfying) structure, enabling a clean separation of reversible Hamiltonian and irreversible dissipative dynamics. The authors formulate a thermo-visco-elastoplastic model with a multiplicative F- split in Eulerian variables , deriving explicit expressions for the Hamiltonian and dissipative contributions and expressing driving forces through transformed energy/entropy gradients. A key result is Theorem JacobiLie, plus a constructive approach to assemble the full model via a simple Poisson operator and a block-structured dissipation potential, ensuring energy conservation and entropy production in line with the first and second laws. The framework promises robust, thermodynamically consistent extension to additional effects (e.g., reactive species) and applications to geophysical and materials engineering problems where Eulerian descriptions and complex couplings are essential.

Abstract

We recall the systematic formulation of Eulerian mechanics in terms of Lie derivatives along the vector field of the material points. Using the abstract properties of Lie derivatives we show that the transport via Lie derivatives generates in a natural way a Poisson structure on the chosen phase space. The evolution equations for thermo-viscoelastic-viscoplastic materials in the Eulerian setting is formulated in the abstract framework of GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling). The equations may not be new, but the systematic splitting between reversible Hamiltonian and dissipative effects allows us to see the equations in a new light that is especially useful for future generalizing of the system, e.g., for adding new effects like reactive species.

Paper Structure

This paper contains 17 sections, 3 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Lie derivatives
  3. Differential geometric approach
  4. Differential forms
  5. Tensors with values in a linear space
  6. Lie derivatives of vectors, operators, and volume forms
  7. GENERIC structures and Eulerian mechanics
  8. Setup of GENERIC
  9. Eulerian thermo-elastoplasticity and its kinematics
  10. Energy, entropy and driving forces
  11. The Poisson structure for the Hamiltonian part
  12. The Eulerian form of thermo-visco-elastoplasticity
  13. The Hamiltonian part of the model
  14. The dissipative part of Eulerian thermo-elastoplasticity
  15. The Eulerian model for thermo-visco-elastoplasticity
  16. ...and 2 more sections

Key Result

Lemma 3.1

Considering ${\bm v}$, ${\bm F}$, and $\rho$ as above we have the relations (see eq:LieD.0.d.form and eq:LieD.vec.covec)

Theorems & Definitions (3)

  • Lemma 3.1: Kinematic relation
  • Proposition 3.2
  • Theorem 3.3: Jacobi identity via Lie derivatives