An Eulerian formulation for dissipative materials using Lie derivatives and GENERIC

TL;DR

This paper develops a systematic Eulerian framework for dissipative materials using Lie derivatives, integrating GENERIC to clearly separate reversible and irreversible effects, facilitating future extensions.

Contribution

It introduces a Lie derivative-based Eulerian formulation combined with GENERIC, providing a new perspective on modeling thermo-viscoelastic-viscoplastic materials.

Findings

Poisson structure naturally generated by Lie derivatives

Clear splitting of reversible and dissipative effects

Framework adaptable for adding new physical effects

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.