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Thermo-elastodynamics of nonlinearly viscous solids.

Stefano Almi1, Rufat Badal2, Manuel Friedrich3

  • 1Department of Mathematics and Applications "R. Caccioppoli", University of Naples Federico II, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy.

Calculus of Variations and Partial Differential Equations
|March 3, 2026
PubMed
Summary

This study establishes the existence of weak solutions for dynamic thermo-elastodynamics in nonlinear Kelvin-Voigt solids. Novel regularity properties of the deformation are derived, advancing the understanding of viscoelastic materials.

Keywords:
35A1535Q7435Q7974D1074F0574H20

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Area of Science:

  • Continuum Mechanics
  • Nonlinear Viscoelasticity
  • Thermo-elastodynamics

Background:

  • Kelvin-Voigt rheology models nonlinearly viscous solids.
  • Frame-indifference principle applies to both elastic and viscous stress tensors.
  • System includes force balance with inertia and Fourier heat transfer.

Purpose of the Study:

  • Establish the existence of weak solutions for dynamic thermo-elastodynamics.
  • Investigate the behavior of nonlinearly viscous solids under thermal and mechanical loads.
  • Explore novel regularity properties of deformation in nonlinear viscoelasticity.

Main Methods:

  • Combining staggered minimizing movement scheme with variational approach to hyperbolic PDEs.
  • Employing higher-order regularization for dissipation.
  • Utilizing regularity theory for the fourth-order p-Laplacian.

Main Results:

  • Existence of weak solutions demonstrated for the dynamic thermo-elastodynamic system.
  • Higher-order regularization is introduced and subsequently removed.
  • New regularity estimates for deformation are derived beyond standard energy bounds.

Conclusions:

  • The study provides a rigorous mathematical framework for dynamic thermo-elastodynamics of nonlinear Kelvin-Voigt solids.
  • Derived regularity properties offer new insights into material behavior and may have independent applications.
  • The findings advance the field of nonlinear viscoelasticity, including static and quasi-static cases.