Thermoelastic theory with microtemperatures and dissipative thermodynamics

ABSTRACT

In this article we derive a nonlinear theory of thermoelastic materials with microtemperatures based on the entropy balance of type III postulated by Green and Naghdi. The work is motivated by an increasing use of materials which possess thermal variation at a microstructure level such that both thermal and microtemperatures waves can propagate with finite speeds and energy dissipation. The equations of the linear theory are also obtained. Then, we use a semigroup approach to derive an existence and uniqueness result for the solutions to the anisotropic problem and to study the asymptotic behavior. Finally, we investigate the impossibility of the localization in time of solutions.

KEYWORDS:

• Asymptotic behavior
• Green–Naghdi theory of type III
• impossibility of localization
• microtemperatures
• thermoelasticity
• well posedness

Acknowledgments

The authors would like to thank Prof. Vincenzo Tibullo for the valuable discussions and helpful suggestions. Part of this work was done when the first author visited the Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, in July 2016 and February and July 2017. He thanks them for their hospitality.

Notes

¹The inequality sign is a consequence of the Second Law of Thermodynamics, which requires the non-negativeness of the functional ξ [11] and of our choice (31)₅.