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Entropy-energy solutions for Thermo-Visco-Elastic systems with Mróz-type inelastic behavior

Tomasz Cieślak, Sebastian Owczarek, Karolina Wielgos

TL;DR

The paper addresses global solvability for a thermo-visco-elastic system with temperature-dependent coefficients and Mróz-type inelasticity by framing a weak entropy-energy solution that respects thermodynamic consistency. It introduces a two-level semi-Galerkin approximation, reformulates the energy balance as an entropy equation, and leverages Young measures and Minty arguments to handle nonlinear dissipation. The main result proves the existence of a global-in-time weak entropy-energy solution with a defect measure, along with a total energy-dissipation inequality and preservation of positivity for the temperature. This work advances the mathematical analysis of temperature-dependent inelastic materials without relying on favorable structural assumptions like Kelvin-Voigt regularization, offering a robust framework for thermodynamically consistent existence results in multi-dimensional settings.

Abstract

In this article, we study a thermodynamically consistent thermo-visco-elastic model describing the balance of internal energy in a heat-conducting inelastic body. In the considered problem, the temperature dependence appears in both the elastic and inelastic constitutive relations. For such a system, we introduce the concept of a weak entropy-energy solution, which satisfies the entropy equality instead of the internal energy equation. Although the model does not possess any mathematically favorable structural properties, such as Kelvin-Voigt type effects or simplifications eliminating temperature from the constitutive relations, we prove the global-in-time existence of weak entropy-energy solutions for large initial data.

Entropy-energy solutions for Thermo-Visco-Elastic systems with Mróz-type inelastic behavior

TL;DR

The paper addresses global solvability for a thermo-visco-elastic system with temperature-dependent coefficients and Mróz-type inelasticity by framing a weak entropy-energy solution that respects thermodynamic consistency. It introduces a two-level semi-Galerkin approximation, reformulates the energy balance as an entropy equation, and leverages Young measures and Minty arguments to handle nonlinear dissipation. The main result proves the existence of a global-in-time weak entropy-energy solution with a defect measure, along with a total energy-dissipation inequality and preservation of positivity for the temperature. This work advances the mathematical analysis of temperature-dependent inelastic materials without relying on favorable structural assumptions like Kelvin-Voigt regularization, offering a robust framework for thermodynamically consistent existence results in multi-dimensional settings.

Abstract

In this article, we study a thermodynamically consistent thermo-visco-elastic model describing the balance of internal energy in a heat-conducting inelastic body. In the considered problem, the temperature dependence appears in both the elastic and inelastic constitutive relations. For such a system, we introduce the concept of a weak entropy-energy solution, which satisfies the entropy equality instead of the internal energy equation. Although the model does not possess any mathematically favorable structural properties, such as Kelvin-Voigt type effects or simplifications eliminating temperature from the constitutive relations, we prove the global-in-time existence of weak entropy-energy solutions for large initial data.

Paper Structure

This paper contains 14 sections, 12 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Entropy equation and total energy dissipation formula
  4. Main result
  5. Approximation of the problem
  6. Existence of approximate solutions
  7. Limit passage in $k$ for fixed $n$
  8. Convergences for fixed $n$
  9. Convergence Analysis and Proof of the Main Theorem
  10. Rewriting the Energy Equation as an Entropy Equation in the Approximation
  11. Estimates with respect to $n$
  12. Convergence Properties of the $n$-th Approximation
  13. Limit passage as $n\to\infty$
  14. Total Energy Dissipation Inequality and Proof of the Main Result

Key Result

Theorem 1.2

(Main result) Let the elasticity tensor $\mathbb{C}:\mathcal{S}^3 \to \mathcal{S}^3$ be symmetric and positive definite. Additionally, let us assume that given data $u_{0}$, $u_{1}$, $\mathbb{T}_{0}$, $\tau_{0}$, $e^{\tau_{0}}$ and $f$ have the regularities specified in conditions p_1 and p_6 and th

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Remark 3.2
  • ...and 23 more