Thermo-elastodynamics of finitely-strained multipolar viscous solids with an energy-controlled stress

TL;DR

This paper develops a thermodynamically consistent Eulerian model for finitely-strained multipolar viscous solids with energy-controlled stress, enabling existence proofs and modeling of neo-Hookean materials.

Contribution

It introduces a novel Eulerian formulation for viscoelastic solids with higher-order viscosity and energy-controlled stress, ensuring thermodynamic consistency and mathematical rigor.

Findings

Proved existence and regularity of weak solutions.

Demonstrated model applicability to neo-Hookean-type materials.

Ensured non-negative entropy and compliance with the third law of thermodynamics.

Abstract

The thermodynamical model of viscoelastic deformable solids at finite strains with Kelvin-Voigt rheology with a higher-order viscosity (using the concept of multipolar materials) is formulated in a fully Eulerian way in rates. Assumptions used in this paper allow for a physically justified free energy leading to non-negative entropy that satisfies the 3rd law of thermodynamics, i.e. entropy vanishes at zero temperature, and energy-controlled stress. This last attribute is used advantageously to prove the existence and a certain regularity of weak solutions by a simplified Faedo-Galerkin semi-discretization, based on estimates obtained from the total-energy and the mechanical-energy balances. Some examples that model neo-Hookean-type materials are presented, too.