On Porous Matrices with Three Delay Times: A Study in Linear Thermoelasticity

Manuela Carini,Vittorio Zampoli

doi:10.3390/math8030371

Abstract

Through the present work, we want to lay the foundation of the well-posedness question for a linear model of thermoelasticity here proposed, in which the presence of voids into the elastic matrix is taken into account following the Cowin–Nunziato theory, and whose thermal response obeys a three-phase lag time-differential heat transfer law. By virtue of the linearity of the model investigated, the basic initial-boundary value problem is conveniently modified into an auxiliary one; attention is paid to the uniqueness question, which is addressed through two alternative paths, i.e., the Lagrange identity and the logarithmic convexity methods, as well as to the continuous dependence issue. The results are achieved under very weak assumptions involving constitutive coefficients and delay times, at most coincident with those able to guarantee the thermodynamic consistency of the model.

Highlights

The use of multiple relaxation times or phase lags in heat conduction constitutive equations is today more than ever a subject of ever-increasing interest, especially with regard to the related time-differential formulations: this is driven by the potential of such models to accurately predict an actual physical phenomenology in reference to extremely small spatial and temporal contexts.The literature on the subject, and more in general on topics involving multi-phase lag approaches, is absolutely extensive and in particular, in the last years, the number of research works investigating the so-called three-phase lag model of heat conduction is rapidly increasing
In the present case, assuming that for adequately short spatial scales the distortions due to thermal effects are small enough to allow to be described under linear hypotheses, we investigate a thermoelastic porous medium following the Cowin–Nunziato model, but this time adding a third relaxation time according to [2]
Uniqueness and continuous dependence of the solution are demonstrated proceeding through the following scheme: in Section 2, the basic equations of the time-differential three-phase lag model for a porous material are described along with the presentation of related mathematical manipulations; suitable initial-boundary value problems are defined

Summary

Introduction

The use of multiple relaxation times or phase lags in heat conduction constitutive equations is today more than ever a subject of ever-increasing interest, especially with regard to the related time-differential formulations: this is driven by the potential of such models to accurately predict an actual physical phenomenology in reference to extremely small spatial and temporal contexts. The present work has to be considered as the natural continuation of the path recently traced in [26,27], where porous material matrices were investigated when coupled with heat transfer phenomena modeled through time-differential constitutive laws with two relaxation times This was done in terms of well-posedness of the theory, and obtaining interesting information. Uniqueness and continuous dependence of the solution are demonstrated proceeding through the following scheme: in Section 2, the basic equations of the time-differential three-phase lag model for a porous material are described along with the presentation of related mathematical manipulations; suitable initial-boundary value problems are defined. We emphasize that in [26] something similar has been done taking into account only two relaxation times; differently, in this case, two different approaches aimed at verifying the uniqueness of the solution are proposed besides the proof of a continuous dependence theorem

Mathematical Formulation

First Result

Second Result

Third Result

Conclusions