Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Minimum free energy in linear thermoelectromagnetism

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.

References [Enhancements On Off] (What's this?)

  • 1. W.A. Day, Reversibility, recoverable work and free energy in linear viscoelasticity, Quart. J. Mech. Appl. Math. 23, 1-15 (1970)
  • 2. S. Breuer and E.T. Onat, On recoverable work in viscoelasticity, Z. Angew. Math. Phys. 15, 13-21 (1981) MR 0178644 (31:2901)
  • 3. J.M. Golden, Free energy in the frequency domain: the scalar case, Quart. Appl. Math. LVIII (1), 127-150 (2000) MR 1739041 (2001e:74025)
  • 4. G. Gentili, Thermodynamic potentials for electromagnetic fields in the ionosphere, Int. J. Engng. Sci. (33) 11, 1561-1575 (1995) MR 1343117 (96h:78001)
  • 5. G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quart. Appl. Math. LX (1), 153-182 (2002) MR 1878264 (2002m:74011)
  • 6. G. Gentili and V. Berti, The minimum free energy for isothermal dielectrics with memory, J. Non-equil. Thermodyn. 24, 154-176 (1999)
  • 7. M. Fabrizio, Teoremi di approssimazione e restrizioni termodinamiche per le equazioni costitutive del campo elettromagnetico, Atti Accad. Sci. Ist. Bologna XII (10), 97(1973) MR 0347238 (49:11958)
  • 8. M. Fabrizio and J.M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math. LX (2), 341-381 (2002) MR 1900497 (2003b:74013)
  • 9. M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, Philadelphia, 1992 MR 1153021 (93a:73034)
  • 10. M. Fabrizio and A. Morro, Thermodynamics of electromagnetic isothermal systems with memory, J. Non-equil. Thermodyn. 22, 110-128 (1997)
  • 11. M. Fabrizio and A. Morro, Electromagnetism of continuous media, Oxford University Press, 2003 MR 1996323 (2004j:78001)
  • 12. M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal. 125, 341-373 (1994) MR 1253168 (95j:73012)
  • 13. G. Amendola, Asymptotic behaviour for electromagnetic fields in the ionosphere, Ann. Univ. Ferrara Sez. VII - Sci. Mat. XLVIII, 165-187 (2002) MR 1980831 (2004f:86006)
  • 14. G. Amendola, The minimum free energy for an electromagnetic conductor, Appl. Anal. (1) 84, 67-87 (2005) MR 2113656
  • 15. G. Amendola, Linear stability for a thermoelectromagnetic material with memory, Quart. Appl. Math. LIX (1), 67-84 (2002) MR 1811095 (2001j:35261)
  • 16. G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal. 138, 1-35 (1997) MR 1463802 (98i:73023)
  • 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)
  • 18. C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3, 83-101 (1948) MR 0032898 (11:362d)
  • 19. B.D. Coleman and E.H. Dill, On the thermodynamics of electromagnetic fields in materials with memory, Arch. Rational Mech. Anal. 41, 132-162 (1971) MR 0347245 (49:11965)
  • 20. B.D. Coleman and E.H. Dill, Thermodynamic restrictions on the constitutive equations of electromagnetic theory, ZAMP 22, 691-702 (1971)
  • 21. B.D. Coleman and D.R. Owen, A mathematical foundation of Thermodynamics, Arch. Rational Mech. Anal. 54, 1-104 (1974) MR 0395502 (52:16299)
  • 22. W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal. 48, 1-50 (1972) MR 0445985 (56:4318)
  • 23. N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen, 1953 MR 0058845 (15:434e)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 78A25, 74A15

Retrieve articles in all journals with MSC (2000): 78A25, 74A15

Additional Information

Giovambattista Amendola
Affiliation: Dipartimento di Matematica Applicata “U.Dini”, Facoltà di Ingegneria, via Diotisalvi 2, 56126-Pisa, Italy

Adele Manes
Affiliation: Dipartimento di Matematica “L.Tonelli”, via F.Buonarroti 2, 56127-Pisa, Italy

DOI: https://doi.org/10.1090/S0033-569X-05-00983-2
Keywords: Thermoelectromagnetism, fading memory, free energy
Received by editor(s): December 1, 2004
Published electronically: September 22, 2005
Additional Notes: This work was performed under the support of C.N.R. and M.I.U.R
Article copyright: © Copyright 2005 Brown University

American Mathematical Society