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: 0392-4033

Abstract

top
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.

How to cite

top

Amendola, 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
  1. AMENDOLA, G., The minimum free energy for incompressible viscoelastic fluids, Math. Meth. Appl. Sci., 29 (2006), 2201-2223. Zbl1104.76032MR2273157DOI10.1002/mma.769
  2. 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
  3. BREUER, S. - ONAT, E.T., On recoverable work in linear viscoelasticity, Z. Angew. Math. Phys., 15 (1964), 12-21. Zbl0117.18801MR178644DOI10.1007/BF01602660
  4. CATTANEO, C., Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, 3 (1948), 83-101. MR32898
  5. COLEMAN, B. D., Thermodynamics of materials with memory, Arch. Rational Mech. Anal., 17 (1964), 1-46. MR171419DOI10.1007/BF00283864
  6. COLEMAN, B. D. - OWEN, R. D., A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal., 54 (1974), 1-104. Zbl0306.73004MR395502DOI10.1007/BF00251256
  7. 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
  8. 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
  9. 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
  10. 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
  11. FABRIZIO, M. - MORRO, A., Mathematical problems in linear viscoelasticity, SIAM, Philadelphia, 1992. Zbl0753.73003MR1153021DOI10.1137/1.9781611970807
  12. 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
  13. 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
  14. 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
  15. 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
  16. MCCARTY, M., Constitutive equations for thermomechanical materials with memory, Int. J. Engng. Sci., 8 (1970), 467-126. 
  17. MUSKHELISHVILI, N. I., Singular Integral Equations, Noordhoff, Groningen, 1953. MR355494
  18. NOLL, W., A new mathematical theory of simple materials, Arch. Rational Mech. Anal., 48 (1972), 1-50. Zbl0271.73006MR445985DOI10.1007/BF00253367
  19. NUNZIATO, J. W., On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204. Zbl0227.73011MR295683DOI10.1090/qam/295683

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.