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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
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.
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.
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