Thermodynamics of a non-simple heat conductor with memory
Authors:
Giovambattista Amendola, Mauro Fabrizio and Murrough Golden
Journal:
Quart. Appl. Math. 69 (2011), 787-806
MSC (2010):
Primary 80A20, 74F05
DOI:
https://doi.org/10.1090/S0033-569X-2011-01228-5
Published electronically:
July 1, 2011
MathSciNet review:
2894001
Full-text PDF Free Access
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Abstract: A generalization is given of the linearized constitutive equation, proposed by Gurtin and Pipkin for the heat flux in a rigid heat conductor, which includes the effects of both the histories of the temperature gradient $\mathbf {g}$ and of $\nabla \mathbf {g}$. This new contribution yields a non-simple material, for which the Second Law of Thermodynamics assumes a modified form, characterized by an extra flux, depending on the material. Some standard free energy functionals are adapted to these new materials, including an explicit formula for the minimum free energy.
References
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References
- G. Amendola, A. Manes and C. Vettori, Maximum recoverable work for a rigid heat conductor with memory, Acta Appl. Math. 110 (3) (2010), 1011–1036. MR 2639155 (2011e:80004)
- G. Amendola and S. Carillo, Thermal work and minimum free energy in a heat conductor with memory, Quart. J. Mech. Appl. Math. 57 (3) (2004), 429-446. MR 2088844 (2005f:80002)
- C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1948), 83-101. MR 0032898 (11:362d)
- V. A. Cimmelli and K. Frischmuth, Gradient generalization to the extended thermodynamic approach and diffusive-hyperbolic heat conduction, Physica B: Condensed Matter (1-2) 400 (2007), 257-265.
- B.D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1-46. MR 0171419 (30:1650)
- L. Deseri, M. Fabrizio and J.M. Golden, The concept of minimal state in viscoelasticity: new free energies and applications to PDEs, Arch. Rational Mech. Anal. 181(1) (2006), 43-96. MR 2221203 (2009a:74024)
- M. Fabrizio, Free energies in the materials with fading memory and applications to PDEs, Proceedings WASCOM 2003, World Sci., Singapore, 2004. MR 2089846
- M. Fabrizio, G. Gentili and D.W. Reynolds, On rigid heat conductors with memory, Int. J. Engrg. Sci. 36 (1998), 765-782. MR 1629806 (99i:80006)
- M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal. 125 (1994), 341-373. MR 1253168 (95j:73012)
- M. Fabrizio and J.M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math. LX (2) (2002), 341-381. MR 1900497 (2003b:74013)
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- G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quart. Appl. Math. LX (1) (2002), 153-182. MR 1878264 (2002m:74011)
- J.M. Golden, Free energy in the frequency domain: The scalar case, Quart. Appl. Math. LVIII (1) (2000), 127-150. MR 1739041 (2001e:74025)
- L. Deseri, G. Gentili, J. M. Golden, An explicit formula for the minimum free energy in linear viscoelasticity, J. Elasticity 54 (1999) 141-185. MR 1728444 (2001i:74012)
- D. Graffi, Sull’espressione analitica di alcune grandezze termodinamiche nei materiali con memoria, Rend. Sem. Mat. Univ. Padova, 68 (1982), 17-29.
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- R. A. Guyer and J. A. Krumhansl, Solution of the linearized Phonon Boltzmann equation, Phys. Rev. 148 (1966), 766-778.
- I. Müller, On the entropy inequality, Arch. Rational Mech. Anal. 26 (1967), 118-141. MR 0214336 (35:5187)
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- V. Volterra, Theory of functional and of integral and integro-differential equations, Blackie & Son Limited, London, 1930.
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Additional Information
Giovambattista Amendola
Affiliation:
Dipartimento di Matematica Applicata “U. Dini”, via F. Buonarroti 1c, 56127-Pisa, Italy
MR Author ID:
25450
Email:
amendola@dma.unipi.it
Mauro Fabrizio
Affiliation:
Dipartimento di Matematica, Piazza di Porta S. Donato 5, 40127-Bologna, Italy
MR Author ID:
64800
Email:
fabrizio@dm.unibo.it
Murrough Golden
Affiliation:
School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
MR Author ID:
213598
Email:
murrough.golden@dit.ie
Keywords:
Memory effects,
heat conduction
Received by editor(s):
May 11, 2010
Published electronically:
July 1, 2011
Additional Notes:
The work of the first two authors was performed with support from the Italian C.N.R. and M.I.U.R
The research of the third author was supported by the Dublin Institute of Technology
Article copyright:
© Copyright 2011
Brown University