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Fractional Heat Conduction and Related Theories of Thermoelasticity

  • Yuriy PovstenkoEmail author
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 219)

Abstract

This chapter is devoted to time- and space-nonlocal generalizations of the standard Fourier law, the corresponding generalizations of the classical heat conduction equation and formulation of associated theories of fractional thermoelasticity. Different kinds of boundary conditions for the time-fractional heat conduction equation are analyzed including the Dirichlet, mathematical and physical Neumann and Robin conditions, the conditions of perfect thermal contact and the moving interface boundary conditions at the solid-liquid interface. Representations of stresses in terms of the displacement potential, biharmonic Galerkin vector and biharmonic Love function are discussed.

Keywords

Heat Flux Heat Conduction Equation Caputo Fractional Derivative Telegraph Equation Classical Heat Conduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Arpaci, V.S.: Conduction Heat Transfer. Addison-Wesley, Reading (1966)zbMATHGoogle Scholar
  2. 2.
    Atanacković, T.M., Pilipović, S., Zorica, D.: Diffusion wave equation with two fractional derivatives of different order. J. Phys. A: Math. Theor. 40, 5319–5333 (2007)CrossRefzbMATHGoogle Scholar
  3. 3.
    Boley, B.A., Weiner, J.H.: Theory of Thermal Stresses. Wiley, New York (1960)zbMATHGoogle Scholar
  4. 4.
    Cattaneo, C.: Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena 3, 83–101 (1948)MathSciNetGoogle Scholar
  5. 5.
    Cattaneo, C.: Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée. C. R. Acad. Sci. 247, 431–433 (1958)Google Scholar
  6. 6.
    Chandrasekharaiah, D.S.: Thermoelasticity with second sound: a review. Appl. Mech. Rev. 39, 355–376 (1986)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chandrasekharaiah, D.S.: Hiperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–729 (1998)CrossRefGoogle Scholar
  8. 8.
    Compte, A., Metzler, R.: The generalized Cattaneo equation for the description of anomalous transport processes. J. Phys. A: Math. Gen. 30, 7277–7289 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Day, W.: The Thermodynamics of Simple Materials with Fading Memory. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
  10. 10.
    Demiray, H., Eringen, A.C.: On nonlocal diffusion of gases. Arch. Mech. 30, 65–77 (1978)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Duhamel, J.-M.-C.: Second mémoire sur les phénom\(\grave{\text{ e }}\)nes thermo-mécanique. J. Ecole Polytech. 15, 1–57 (1837)Google Scholar
  12. 12.
    Eringen, A.C.: Theory of nonlocal thermoelasticity. Int. J. Eng. Sci. 12, 1063–1077 (1974)CrossRefzbMATHGoogle Scholar
  13. 13.
    Eringen, A.C.: Vistas of nonlocal continuum physics. Int. J. Eng. Sci. 30, 1551–1565 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fourier, J.B.J.: Théorie analytique de la chaleur. Firmin Didot, Paris (1822)Google Scholar
  15. 15.
    Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinetti, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Wien (1997)CrossRefGoogle Scholar
  16. 16.
    Gorenflo, R., Mainardi, F.: Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal. 1, 167–191 (1998)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Gorenflo, R., Iskenderov, A., Luchko, Yu.: Mapping between solutions of fractional diffusion-wave equations. Fract. Calc. Appl. Anal. 3, 75–86 (2000)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31, 113–126 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hanyga, A.: Multidimensional solutions of space-fractional diffusion equations. Proc. R. Soc. Lond. A 457, 2993–3005 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Hanyga, A.: Multidimensional solutions of space-time-fractional diffusion equations. Proc. R. Soc. Lond. A 458, 429–450 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Hetnarski, R.B. (ed.): Encyclopedia of Thermal Stresses, in 11 vols. Springer, New York (2014)Google Scholar
  23. 23.
    Hetnarski, R.B., Eslami, M.R.: Thermal Stresses—Advanced Theory and Applications. Springer, New York (2009)zbMATHGoogle Scholar
  24. 24.
    Hetnarski, R.B., Ignaczak, J.: Generalized thermoelasticity. J. Therm. Stress. 22, 451–476 (1999)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Hetnarski, R.B., Ignaczak, J.: Nonclassical dynamical thermoelasticity. Int. J. Solids Struct. 37, 215–224 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Ignaczak, J., Ostoja-Starzewski, M.: Thermoelasticity with Finite Wave Speeds. Oxford University Press, Oxford (2009)CrossRefGoogle Scholar
  27. 27.
    Jiji, L.M.: Heat Conduction, 3rd edn. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  28. 28.
    Jordan, P.M., Puri, P.: Thermal stresses in a spherical shell under three thermoelastic models. J. Therm. Stress. 24, 47–70 (2001)CrossRefGoogle Scholar
  29. 29.
    Joseph, D.D., Preziosi, L.: Heat waves. Rev. Mod. Phys. 61, 41–73 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  31. 31.
    Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)CrossRefzbMATHGoogle Scholar
  32. 32.
    Mainardi, F., Luchko, Yu., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 153–192 (2001)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Moodi, T.B., Tait, R.J.: On thermal transients with finite wave speeds. Acta Mech. 50, 97–104 (1983)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Neumann, F.: Vorlesung über die Theorie des Elasticität des festen Körper und des Lichtäthers. Teubner, Leipzig (1885)Google Scholar
  35. 35.
    Nigmatullin, R.R.: To the theoretical explanation of the “universal response”. Phys. Status Solidi (B) 123, 739–745 (1984)CrossRefGoogle Scholar
  36. 36.
    Nigmatullin, R.R.: On the theory of relaxation for systems with “remnant” memory. Phys. Status Solidi (B) 124, 389–393 (1984)CrossRefGoogle Scholar
  37. 37.
    Noda, N., Hetnarski, R.B., Tanigawa, Y.: Thermal Stresses, 2nd edn. Taylor and Francis, New York (2003)Google Scholar
  38. 38.
    Norwood, F.R.: Transient thermal waves in the general theory of heat conduction with finite wave speeds. J. Appl. Mech. 39, 673–676 (1972)CrossRefGoogle Scholar
  39. 39.
    Nowacki, W.: Thermoelasticity, 2nd edn. PWN-Polish Scientific Publishers, Warsaw and Pergamon Press, Oxford (1986)zbMATHGoogle Scholar
  40. 40.
    Nunziato, J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–204 (1971)zbMATHMathSciNetGoogle Scholar
  41. 41.
    Parkus, H.: Instationäre Wärmespannungen. Springer, Wien (1959)CrossRefzbMATHGoogle Scholar
  42. 42.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  43. 43.
    Podstrigach, Ya.S., Kolyano, Yu.M.: Generalized Thermomechanics. Naukova Dumka, Kiev (1976) (in Russian)Google Scholar
  44. 44.
    Podstrigach, Ya.S., Povstenko, Y.Z.: Introduction to Mechanics of Surface Phenomena in Deformable Solids. Naukova Dumka, Kiev (1985) (in Russian)Google Scholar
  45. 45.
    Podstrigach, Ya.S., Shvets, R.N.: The quasi-static problem in coupled thermoelasticity. Int. Appl. Mech. 5, 33–39 (1969)Google Scholar
  46. 46.
    Povstenko, Y.: Fractional heat conduction equation and associated thermal stresses. J. Therm. Stress. 28, 83–102 (2005)CrossRefMathSciNetGoogle Scholar
  47. 47.
    Povstenko, Y.: Thermoelasticity based on fractional heat conduction equation. In: Ziegler, F., Heuer, R., Adam, C. (eds.) Proceedings of the 6th International Congress on Thermal Stresses, Vienna, Austria, 26–29 May 2005, vol. 2, pp. 501–504. Vienna University of Technology, Vienna (2005)Google Scholar
  48. 48.
    Povstenko, Y.: Thermoelasticity which uses fractional heat conduction equation. Math. Methods Phys.-Mech. Fields 51(2), 239–246 (2008)Google Scholar
  49. 49.
    Povstenko, Y.: Space-time-fractional heat conduction equation and the theory of thermoelasticity. In: 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey, 5–7 Nov 2008Google Scholar
  50. 50.
    Povstenko, Y.: Thermoelasticity which uses fractional heat conduction equation. J. Math. Sci. 162, 296–305 (2009)CrossRefMathSciNetGoogle Scholar
  51. 51.
    Povstenko, Y.: Theory of thermoelasticity based on the space-time-fractional heat conduction equation. Phys. Scr. T 136, 014017-1-6 (2009)Google Scholar
  52. 52.
    Povstenko, Y.: Theories of thermoelasticity based on space-time-fractional Cattaneo-type equation. In: Podlubny, I., Vinagre Jara, M.B., Chen, Y.Q., Felin Batlle, V., Tejado Balsera, I. (eds.) Proceedings of the 4th IFAC Workshop on Fractional Differentiation and Its Applications, Badajoz, Spain, 18–20 Oct 2010, Article No. FDA10-014Google Scholar
  53. 53.
    Povstenko, Y.: Different formulations of Neumann boundary-value problems for time-fractional diffusion-wave equation in a half-plane. In: Podlubny, I., Vinagre Jara, M.B., Chen, Y.Q., Felin Batlle, V., Tejado Balsera, I. (eds.) Proceedings of the 4th IFAC Workshop on Fractional Differentiation and Its Applications, Badajoz, Spain, 18–20 Oct 2010, Article No. FDA10-015Google Scholar
  54. 54.
    Povstenko, Y.: Fractional Cattaneo-type equations and generalized thermoelasticity. J. Therm. Stress. 34, 97–114 (2011)CrossRefGoogle Scholar
  55. 55.
    Povstenko, Y.: Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder. Fract. Calc. Appl. Anal. 14, 418–435 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
    Povstenko, Y.: Different formulations of Neumann boundary-value problem for time-fractional heat conduction equation in a half-space. In: Proceedings of the 9th International Congress on Thermal Stresses, Budapest, Hungary, 5–9 June 2011Google Scholar
  57. 57.
    Povstenko, Y.: Neumann boundary-value problems for a time-fractional diffusion-wave equation in a half-plane. Comput. Math. Appl. 64, 3183–3192 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  58. 58.
    Povstenko, Y.: Theories of thermal stresses based on space-time-fractional telegraph equations. Comput. Math. Appl. 64, 3321–3328 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  59. 59.
    Povstenko, Y.: Axisymmetric solutions to time fractional heat conduction equation in a half-space under Robin boundary conditions. Int. J. Differ. Equ. 2012, 154085-1-13 (2012)Google Scholar
  60. 60.
    Povstenko, Y.: Different kinds of boundary problems for fractional heat conduction equation. In: Petráš, I., Podlubny, I., Kostúr, K., Kačur, J., Mojžišová, A. (eds.) Proceedings of the 13th International Carpathian Control Conference, Podbanské, Hight Tatras, Slovak Republic, 28–31 May 2012, pp. 588–591. Institute of Electrical and Electronics Engineers, Košice (2012)Google Scholar
  61. 61.
    Povstenko, Y.: Fractional heat conduction in infinite one-dimensional composite medium. J. Therm. Stress. 36, 351–363 (2013)CrossRefGoogle Scholar
  62. 62.
    Povstenko, Y.: Fractional heat conduction in an infinite medium with a spherical inclusion. Entropy 15, 4122–4133 (2013)CrossRefMathSciNetGoogle Scholar
  63. 63.
    Povstenko, Y.: Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition. Eur. Phys. J. Spec. Top. 222, 1767–1777 (2013)CrossRefGoogle Scholar
  64. 64.
    Povstenko, Y.: Fundamental solutions to time-fractional heat conduction equations in two joint half-lines. Cent. Eur. J. Phys. 11, 1284–1294 (2013)CrossRefGoogle Scholar
  65. 65.
    Povstenko, Y.: Fractional thermoelasticity. In: Hetnarski, R.B. (ed.) Encyclopedia of Thermal Stresses, vol. 4, pp. 1778–1787. Springer, New York (2014)CrossRefGoogle Scholar
  66. 66.
    Roscani, S., Marcus, E.S.: Two equivalent Stefan’s problems for the time-fractional diffusion equation. Fract. Calc. Appl. Anal. 16, 802–815 (2013)CrossRefMathSciNetGoogle Scholar
  67. 67.
    Saichev, A.I., Zaslavsky, G.M.: Fractional kinetic equations: solutions and applications. Chaos 7, 753–764 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  68. 68.
    Sherief, H.H., El-Sayed, A.M.A., Abd El-Latief, A.M.: Fractional order theory of thermoelasticty. Int. J. Solids Struct. 47, 269–275 (2010)Google Scholar
  69. 69.
    Sneddon, I.N.: The Use of Integral Transforms. McGraw-Hill, New York (1972)zbMATHGoogle Scholar
  70. 70.
    Tamma, K.K., Zhou, X.: Macroscale and microscale thermal transport and thermo-mechanical interactions: some noteworthy perspectives. J. Therm. Stress. 21, 405–449 (1998)CrossRefGoogle Scholar
  71. 71.
    Voller, V.R., Falcini, F., Garra, R.: Fractional Stefan problem exhibiting lumped and distributed latent-heat memory effects. Phys. Rev. E 87, 042401-1-6 (2013)Google Scholar
  72. 72.
    Youssef, H.M.: Theory of fractional order generalized thermoelasticity. J. Heat Transf. 132, 061301-1-7 (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceJan Długosz UniversityCzęstochowaPoland

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