Second-order approximation for heat conduction: dissipation principle and free energies.
In the context of new models of heat conduction, the second-order approximation of Tzou's theory, derived by Quintanilla and Racke, has been studied recently by two of the present authors, where it was proved equivalent to a fading memory material. The importance of determining free energy functionals for such materials, and indeed for any material with memory, is emphasized. Because the kernel does not satisfy certain convexity restrictions that allow us to obtain various traditional free energies for materials with fading memory, it is necessary to restrict the study to the minimum and related free energies, which do not require these restrictions. Thus, the major part of this work is devoted to deriving an explicit expression for the minimum free energy. Simple modifications of this expression also give an intermediate free energy and the maximum free energy for the material. These derivations differ in certain important respects from earlier work on such free energies.
Department of Mathematics , University of Pisa , Largo Bruno Pontecorvo 5, 56127 Pisa, Italy.
Giovambattista
G
Amendola
Department of Mathematics , University of Bologna , Piazza di Porta S. Donato 5, 40127 Bologna, Italy.
Mauro
0000-0003-0920-2963
M
Fabrizio
School of Mathematical Sciences , Dublin Institute of Technology , Kevin Street, Dublin 8, Ireland.
Murrough
M
Golden
Department of Mathematics , University of Bologna , Piazza di Porta S. Donato 5, 40127 Bologna, Italy.
Barbara
0000-0002-4212-0233
B
Lazzari
dual phase lag; heat conduction; minimum free energy