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

TL;DR

This work advances a fully Eulerian thermo-elastodynamics framework for finitely-strained multipolar viscous solids with Kelvin-Voigt dissipation, where the stress is energy-controlled via $E(F,\theta)$ and a higher-order dissipative term $H$ stabilizes the dynamics. By deriving the first and second laws in this setting and employing a simplified Faedo-Galerkin semi-discretization, the authors establish global existence of weak solutions with precise regularity and demonstrate that the total-energy and entropy balances hold, enabling rigorous thermodynamic consistency. The analysis leverages the energy-control bound on the Cauchy stress, a physically justified free energy, and ambient boundary conditions; the approach yields clear a priori estimates and compactness to pass to the limit, even under data allowing negative temperatures during the Galerkin approximation. The paper also provides neo-Hookean-type examples and discusses extensions to boundary heat flux and to multi-well free energies for shape-memory alloys, illustrating the practical applicability of the energy-controlled stress framework in modeling complex thermo-mechanical behavior.

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.

Figures (2)

• Figure 1: A schematic illustration of continuous extensions for negative temperatures of the thermal part of the internal energy $u=U({\boldsymbol F},\theta)$, the heat capacity $c=c({\boldsymbol F},\theta)$, the nonlinearity $\widetilde{E}$ from (\ref{['primitive-theta-c']}), and the modified continuous inverse $[U({\boldsymbol F},\cdot)]_{\text{\sc m}}^{-1}$ of $U({\boldsymbol F},\cdot)$ as defined by (\ref{['inverse-ext']}); here ${\boldsymbol F}$ is considered fixed.
• Figure 2: A comparison of Examples \ref{['exa-neo']} and \ref{['exa-c-bounded']} with $c_{\text{\sc v}}=1$ ignoring the mechanical part, i.e. $\upphi=0$. The former example is depicted for $\alpha=0.05$ (dashed lines) and for $\alpha=0.2$ (dotted lines) while the latter example is depicted for $\theta_{\text{\sc{r}}}=0.2$ (solid lines). Both examples yield a similar internal energy $E$ and a similar heat capacity $c$ for $\alpha\to0$ and $\theta_{\text{\sc{r}}}\to0$.

Theorems & Definitions (16)

• Remark 2.1: Selecting the mere stored temperature-independent energy out
• Remark 2.2: Thermoelasticity in an "engineering" formulation
• Definition 3.1: Weak solutions to (\ref{['Euler-thermo-finite']})
• Theorem 3.2: Existence and regularity of weak solutions
• proof
• Remark 3.3: Fourier boundary conditions
• Remark 3.4: Non-negativity of the temperature
• Remark 3.5: Smoothness of $\psi$
• Remark 3.6: Exploiting the entropy balance -- an open problem
• Example 4.1: A free energy of the neo-Hookean type