Asymptotic analysis for vanishing acceleration in a thermoviscoelastic system

Abstract

We have investigated a dynamic thermoviscoelastic system (2003), establishing existence and uniqueness results for a related initial and boundary values problem. The aim of the present paper is to study the asymptotic behavior of the solution to the above problem as the power of the acceleration forces goes to zero. In particular, well-posedness and regularity results for the limit (quasistatic) problem are recovered.

Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials

“…Among others, we would like to cite the papers [3][4][5][6]8,16,23]. The analysis of a thermoviscoelastic system not subject to a phase transition has been tackled in [3,4], in which a linear viscoelastic equation for the displacement u and an internal energy balance equation for ϑ are considered. The latter parabolic equation has a quadratic contribution in ε(u t ) on the right-hand side.…”

“…The latter parabolic equation has a quadratic contribution in ε(u t ) on the right-hand side. Due to the highly nonlinear character of the system, only a local well-posedness result (proved in [3]) is available in the three-dimensional case, while in [4] an asymptotic analysis is performed. However, in this framework no degeneracy of the elliptic operator in the equation for u is allowed.…”

Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

“…Instead, in the papers [4], [5] the focus is on the analysis of a model for thermoviscoelastic systems not subject to phase transitions: the related (highly nonlinear) PDE system couples a linear viscoelastic equation for the displacement u and an internal energy balance equation for ϑ, while the equation of microscopic movements is not considered. The model is analyzed in [20] and in [21] pertains to nonlinear thermoviscoplasticity: in the one-dimensional (in space) case, the authors prove the global well-posedness of a PDE system, incorporating both hysteresis effects and modelling phase change, which however does not display a degenerating character.…”

On mathematical problems for viscoelastic multi‐mechanism models in the isothermal case

“…e.g. [15]) yields the heat-conduction equation (in the case of Fourier's law (9)), using the energy balance (1b), the well-known relation between free energy, internal energy and entropy ψ = e − θη as well as (4) and (6), and (8).…”

“…. Bonetti and Bonfanti, and , present some mathematical results regarding existence and uniqueness of the corresponding mathematical model for thermoviscoelastic behavior based on a Kelvin‐Voigt model. Continuing the works of Bonetti and Bonfanti, Fernández, and Kuttler deal with the variational analysis for a modified nonlinear Kelvin‐Voigt model for thermoviscoelasticity in .…”