Minimum free energy in linear thermoelectromagnetism

Amendola, Giovambattista; Manes, Adele

doi:10.1090/S0033-569X-05-00983-2

Authors: Giovambattista Amendola and Adele Manes
Journal: Quart. Appl. Math. 63 (2005), 645-672
MSC (2000): Primary 78A25, 74A15
DOI: https://doi.org/10.1090/S0033-569X-05-00983-2
Published electronically: September 22, 2005
MathSciNet review: 2187924
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A closed expression is given for the minimum free energy of a linear thermoelectromagnetic conductor, whose constitutive equations relative to the electric current density and to the heat flux have memory effects. This expression, derived in the frequency domain, is related to the maximum recoverable work, which can be obtained from a given state of the material. Another equivalent expression of the minimum free energy is deduced; it allows us to give explicit formulae for the case of a discrete spectrum model.

1 W.A. Day, Reversibility, recoverable work and free energy in linear viscoelasticity, Quart. J. Mech. Appl. Math. 23, 1-15 (1970)

Shlomo Breuer and E. Turan Onat, On recoverable work in linear viscoelasticity, Z. Angew. Math. Phys. 15 (1964), 12–21 (English, with German summary). MR 178644, DOI https://doi.org/10.1007/BF01602660
J. M. Golden, Free energies in the frequency domain: the scalar case, Quart. Appl. Math. 58 (2000), no. 1, 127–150. MR 1739041, DOI https://doi.org/10.1090/qam/1739041
Giorgio Gentili, Thermodynamic potentials for electromagnetic fields in the ionosphere, Internat. J. Engrg. Sci. 33 (1995), no. 11, 1561–1575. MR 1343117, DOI https://doi.org/10.1016/0020-7225%2895%2900019-T
Giorgio Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quart. Appl. Math. 60 (2002), no. 1, 153–182. MR 1878264, DOI https://doi.org/10.1090/qam/1878264

6 G. Gentili and V. Berti, The minimum free energy for isothermal dielectrics with memory, J. Non-equil. Thermodyn. 24, 154-176 (1999)

Mauro Fabrizio, Teoremi di approssimazione e restrizioni termodinamiche per le equazioni costitutive del campo elettromagnetico, Atti Accad. Sci. Ist. Bologna Cl. Sci. Fis. Rend. (12) 10 (1972/73), no. 1, 97–110 (Italian). MR 347238
M. Fabrizio and J. M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math. 60 (2002), no. 2, 341–381. MR 1900497, DOI https://doi.org/10.1090/qam/1900497
Mauro Fabrizio and Angelo Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1153021

10 M. Fabrizio and A. Morro, Thermodynamics of electromagnetic isothermal systems with memory, J. Non-equil. Thermodyn. 22, 110-128 (1997)

Mauro Fabrizio and Angelo Morro, Electromagnetism of continuous media, Oxford Science Publications, Oxford University Press, Oxford, 2003. Mathematical modelling and applications. MR 1996323
Mauro Fabrizio, Claudio Giorgi, and Angelo Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal. 125 (1994), no. 4, 341–373. MR 1253168, DOI https://doi.org/10.1007/BF00375062
Giovambattista Amendola, Asymptotic behaviour for electromagnetic fields in the ionosphere, Ann. Univ. Ferrara Sez. VII (N.S.) 48 (2002), 165–187 (English, with English and Italian summaries). MR 1980831
Giovambattista Amendola, The minimum free energy for an electromagnetic conductor, Appl. Anal. 84 (2005), no. 1, 67–87. MR 2113656, DOI https://doi.org/10.1080/00036810410001712835
Giovambattista Amendola, Linear stability for a thermoelectromagnetic material with memory, Quart. Appl. Math. 59 (2001), no. 1, 67–84. MR 1811095, DOI https://doi.org/10.1090/qam/1811095
Gianpietro Del Piero and Luca Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal. 138 (1997), no. 1, 1–35. MR 1463802, DOI https://doi.org/10.1007/s002050050035

17 M.E. Gurtin and A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31, 113-126 (1968)

Carlo Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1949), 83–101 (Italian). MR 0032898
Bernard D. Coleman and Ellis H. Dill, On the thermodynamics of electromagnetic fields in materials with memory, Arch. Rational Mech. Anal. 41 (1971), 132–162. MR 347245, DOI https://doi.org/10.1007/BF00281371

20 B.D. Coleman and E.H. Dill, Thermodynamic restrictions on the constitutive equations of electromagnetic theory, ZAMP 22, 691-702 (1971)

Bernard D. Coleman and David R. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. 54 (1974), 1–104. MR 395502, DOI https://doi.org/10.1007/BF00251256
Walter Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal. 48 (1972), 1–50. MR 445985, DOI https://doi.org/10.1007/BF00253367
N. I. Muskhelishvili, Singular integral equations. Boundary problems of function theory and their application to mathematical physics, P. Noordhoff N. V., Groningen, 1953. Translation by J. R. M. Radok. MR 0058845