Thermodynamics of a non-simple heat conductor with memory

Amendola, Giovambattista; Fabrizio, Mauro; Golden, Murrough

doi:10.1090/S0033-569X-2011-01228-5

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
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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

G. Amendola, A. Manes, and C. Vettori, Maximum recoverable work for a rigid heat conductor with memory, Acta Appl. Math. 110 (2010), no. 3, 1011–1036. MR 2639155, DOI https://doi.org/10.1007/s10440-009-9491-8
G. Amendola and S. Carillo, Thermal work and minimum free energy in a heat conductor with memory, Quart. J. Mech. Appl. Math. 57 (2004), no. 3, 429–446. MR 2088844, DOI https://doi.org/10.1093/qjmam/57.3.429
Carlo Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1949), 83–101 (Italian). MR 0032898

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.

Bernard D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1–46. MR 171419, DOI https://doi.org/10.1007/BF00283864

Luca Deseri, Mauro Fabrizio, and Murrough Golden, The concept of minimal state in viscoelasticity: new free energies and applications to PDEs, Arch. Ration. Mech. Anal. 181 (2006), no. 1, 43–96. MR 2221203, DOI https://doi.org/10.1007/s00205-005-0406-1

Mauro Fabrizio, Free energies in the materials with fading memory and applications to PDEs, “WASCOM 2003”—12th Conference on Waves and Stability in Continuous Media, World Sci. Publ., River Edge, NJ, 2004, pp. 172–184. MR 2089846, DOI https://doi.org/10.1142/9789812702937_0022

M. Fabrizio, G. Gentili, and D. W. Reynolds, On rigid linear heat conductors with memory, Internat. J. Engrg. Sci. 36 (1998), no. 7-8, 765–782. MR 1629806, DOI https://doi.org/10.1016/S0020-7225%2897%2900123-7

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

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 Barbara Lazzari, On asymptotic stability for linear viscoelastic fluids, Differential Integral Equations 6 (1993), no. 3, 491–505. MR 1202554

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

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

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

Luca Deseri, Giorgio Gentili, and Murrough Golden, An explicit formula for the minimum free energy in linear viscoelasticity, J. Elasticity 54 (1999), no. 2, 141–185. MR 1728444, DOI https://doi.org/10.1023/A%3A1007646017347

D. Graffi, Sull’espressione analitica di alcune grandezze termodinamiche nei materiali con memoria, Rend. Sem. Mat. Univ. Padova, 68 (1982), 17-29.

Dario Graffi, More on the analytic expression of free energy in materials with memory, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 120 (1986), no. suppl., 111–124 (1987) (Italian). Writings in mathematical physics in honor of the ninetieth birthday of Cataldo Agostinelli (Italian). MR 958166

Morton E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), no. 2, 113–126. MR 1553521, DOI https://doi.org/10.1007/BF00281373

R. A. Guyer and J. A. Krumhansl, Solution of the linearized Phonon Boltzmann equation, Phys. Rev. 148 (1966), 766-778.

Ingo Müller, On the entropy inequality, Arch. Rational Mech. Anal. 26 (1967), 118–141. MR 214336, DOI https://doi.org/10.1007/BF00285677

N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 0355494

V. Volterra, Theory of functional and of integral and integro-differential equations, Blackie & Son Limited, London, 1930.