# Minimum Free Energy for a Rigid Heat Conductor and Application to a Discrete Spectrum Model

Giovambattista Amendola; Adele Manes

Bollettino dell'Unione Matematica Italiana (2007)

- Volume: 10-B, Issue: 3, page 969-987
- ISSN: 0392-4041

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topAmendola, Giovambattista, and Manes, Adele. "Minimum Free Energy for a Rigid Heat Conductor and Application to a Discrete Spectrum Model." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 969-987. <http://eudml.org/doc/290377>.

@article{Amendola2007,

abstract = {A general closed expression is given for the minimum free energy for a rigid heat conductor with memory effects. This formula, derived in the frequency domain, is related to the maximum recoverable work we can obtain from the material at a given state, which is characterized by the temperature and the past history of its gradient. Another explicit formula of the minimum free energy is also derived and used to obtain the results related to the particular case of a discrete spectrum model material response.},

author = {Amendola, Giovambattista, Manes, Adele},

journal = {Bollettino dell'Unione Matematica Italiana},

language = {eng},

month = {10},

number = {3},

pages = {969-987},

publisher = {Unione Matematica Italiana},

title = {Minimum Free Energy for a Rigid Heat Conductor and Application to a Discrete Spectrum Model},

url = {http://eudml.org/doc/290377},

volume = {10-B},

year = {2007},

}

TY - JOUR

AU - Amendola, Giovambattista

AU - Manes, Adele

TI - Minimum Free Energy for a Rigid Heat Conductor and Application to a Discrete Spectrum Model

JO - Bollettino dell'Unione Matematica Italiana

DA - 2007/10//

PB - Unione Matematica Italiana

VL - 10-B

IS - 3

SP - 969

EP - 987

AB - A general closed expression is given for the minimum free energy for a rigid heat conductor with memory effects. This formula, derived in the frequency domain, is related to the maximum recoverable work we can obtain from the material at a given state, which is characterized by the temperature and the past history of its gradient. Another explicit formula of the minimum free energy is also derived and used to obtain the results related to the particular case of a discrete spectrum model material response.

LA - eng

UR - http://eudml.org/doc/290377

ER -

## References

top- AMENDOLA, G., The minimum free energy for incompressible viscoelastic fluids, Math. Meth. Appl. Sci., 29 (2006), 2201-2223. Zbl1104.76032MR2273157DOI10.1002/mma.769
- AMENDOLA, G. - CARILLO, S., Thermal work and minimum free energy in a heat conductor with memory, Quart. Jl. Mech. Appl. Math., 57 (3) (2004), 429-446. Zbl1151.80301MR2088844DOI10.1093/qjmam/57.3.429
- BREUER, S. - ONAT, E.T., On recoverable work in linear viscoelasticity, Z. Angew. Math. Phys., 15 (1964), 12-21. Zbl0117.18801MR178644DOI10.1007/BF01602660
- CATTANEO, C., Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3 (1948), 83-101. MR32898
- COLEMAN, B. D., Thermodynamics of materials with memory, Arch. Rational Mech. Anal., 17 (1964), 1-46. MR171419DOI10.1007/BF00283864
- COLEMAN, B. D. - OWEN, R. D., A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal., 54 (1974), 1-104. Zbl0306.73004MR395502DOI10.1007/BF00251256
- DAY, W. A., Reversibility, recoverable work and free energy in linear viscoelasticity, Quart. J. Mech. Appl. Math., 23 (1970), 1-15. Zbl0219.73040MR273881DOI10.1093/qjmam/23.4.469
- FABRIZIO, M. - GENTILI, G. - REYNOLDS, D. W., On rigid heat conductors with memory, Int. J. Engng. Sci., 36 (1998), 765-782. Zbl1210.80007MR1629806DOI10.1016/S0020-7225(97)00123-7
- FABRIZIO, M. - GIORGI, C. - MORRO, A., Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal., 125 (1994), 341-373. Zbl0806.73006MR1253168DOI10.1007/BF00375062
- FABRIZIO, M. - GOLDEN, J. M., Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math., LX (2) (2002), 341-381. Zbl1069.74008MR1900497DOI10.1090/qam/1900497
- FABRIZIO, M. - MORRO, A., Mathematical problems in linear viscoelasticity, SIAM, Philadelphia, 1992. Zbl0753.73003MR1153021DOI10.1137/1.9781611970807
- GENTILI, G., Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quart. Appl. Math., LX (1) (2002), 153-182. Zbl1069.74009MR1878264DOI10.1090/qam/1878264
- GIORGI, C. - GENTILI, G., Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math., LVIII (51) 2 (1993), 343-362. Zbl0780.45011MR1218373DOI10.1090/qam/1218373
- GOLDEN, J. M., Free energy in the frequency domain: the scalar case, Quart. Appl. Math., LVIII (1) (2000), 127-150. Zbl1032.74017MR1739041DOI10.1090/qam/1739041
- GURTIN, M. E. - PIPKIN, A. C., A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126. Zbl0164.12901MR1553521DOI10.1007/BF00281373
- MCCARTY, M., Constitutive equations for thermomechanical materials with memory, Int. J. Engng. Sci., 8 (1970), 467-126.
- MUSKHELISHVILI, N. I., Singular Integral Equations, Noordhoff, Groningen, 1953. MR355494
- NOLL, W., A new mathematical theory of simple materials, Arch. Rational Mech. Anal., 48 (1972), 1-50. Zbl0271.73006MR445985DOI10.1007/BF00253367
- NUNZIATO, J. W., On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204. Zbl0227.73011MR295683DOI10.1090/qam/295683

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