Advertisement

Acta Mechanica

, Volume 230, Issue 6, pp 2137–2144 | Cite as

Fractional thermoelasticity problem for an infinite solid with a cylindrical hole under harmonic heat flux boundary condition

  • Yuriy PovstenkoEmail author
Open Access
Original Paper
  • 104 Downloads

Abstract

The time-fractional heat conduction equation with the Caputo derivative results from the law of conservation of energy and time-nonlocal generalization of the Fourier law with the “long-tail” power kernel. In this paper, we consider an infinite solid with a cylindrical cavity under harmonic heat flux boundary condition. The Laplace transform with respect to time and the Weber transform with respect to the spatial coordinate are used. The solutions are obtained in terms of integrals with integrands being the Mittag-Leffler functions. The numerical results are illustrated graphically.

Notes

References

  1. 1.
    Povstenko, Y.: Linear Fractional Diffusion-Wave Equation for Scientists and Engineers. Birkhäuser, New York (2015)CrossRefzbMATHGoogle Scholar
  2. 2.
    Povstenko, Y.: Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder. Fract. Calc. Appl. Anal. 14, 418–435 (2011).  https://doi.org/10.2478/s13540-011-0026-4 CrossRefzbMATHGoogle Scholar
  3. 3.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Amsterdam (1993)zbMATHGoogle Scholar
  4. 4.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  5. 5.
    Nigmatullin, R.R.: To the theoretical explanation of the “universal response”. Phys. Stat. Sol. (B) 123, 739–745 (1984)CrossRefGoogle Scholar
  6. 6.
    Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)CrossRefzbMATHGoogle Scholar
  7. 7.
    Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)CrossRefzbMATHGoogle Scholar
  8. 8.
    Povstenko, Y.: Fractional heat conduction equation and associated thermal stress. J. Therm. Stresses 28, 83–102 (2005)CrossRefGoogle Scholar
  9. 9.
    Sherief, H.H., El-Sayed, A.M.A., Abd El-Latief, A.M.: Fractional order theory of thermoelasticity. Int. J. Solids Struct. 47, 269–275 (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Youssef, H.M.: Theory of fractional order generalized thermoelasticity. J. Heat Transf. 132, 061301-1-7 (2010)CrossRefGoogle Scholar
  11. 11.
    Povstenko, Y.: Fractional Cattaneo-type equations and generalized thermoelasticity. J. Therm. Stresses 34, 97–114 (2011)CrossRefGoogle Scholar
  12. 12.
    Povstenko, Y.: Theories of thermal stresses based on space-time-fractional telegraph equations. Comput. Math. Appl. 64, 3321–3328 (2012)CrossRefzbMATHGoogle Scholar
  13. 13.
    Shaw, S.: Theory of generalized thermoelasticity with memory-dependent derivatives. J. Eng. Mech. (2019).  https://doi.org/10.1061/(ASCE)EM.1943-7889.0001569 Google Scholar
  14. 14.
    Povstenko, Y.: Fractional Thermoelasticity. Springer, New York (2015)CrossRefzbMATHGoogle Scholar
  15. 15.
    Ångström, A.J.: Neue Methode, das Wärmeleitungsvermögen der Körper zu bestimmen. Ann. Phys. Chem. 144, 513–530 (1861)Google Scholar
  16. 16.
    Ångström, A.J.: New method of determining the thermal conductivity of bodies. Philos. Mag. 25, 130–142 (1863)CrossRefGoogle Scholar
  17. 17.
    Mandelis, A.: Diffusion waves and their uses. Phys. Today 53, 29–33 (2000)CrossRefGoogle Scholar
  18. 18.
    Mandelis, A.: Diffusion-Wave Fields: Mathematical Methods and Green Functions. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  19. 19.
    Vrentas, J.S., Vrentas, C.M.: Diffusion and Mass Transfer. CRC Press, Boca Raton (2013)zbMATHGoogle Scholar
  20. 20.
    Nowacki, W.: State of stress in an elastic space due to a source of heat varying harmonically as function of time. Bull. Acad. Polon. Sci. Sér. Sci. Technol. 5, 145–154 (1957)Google Scholar
  21. 21.
    Nowacki, W.: Thermoelasticity, 2nd edn. PWN-Polish Scientific Publishers, Warsaw and Pergamon Press, Oxford (1986)zbMATHGoogle Scholar
  22. 22.
    Povstenko, Y.: Fractional heat conduction in a space with a source varying harmonically in time and associated thermal stresses. J. Therm. Stresses 39, 1442–1450 (2016)CrossRefGoogle Scholar
  23. 23.
    Povstenko, Y.: Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses. Mech. Res. Commun. 37, 436–440 (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    Povstenko, Y.: The Neumann boundary problem for axisymmetric fractional heat conduction equation in a solid with a cylindrical hole and associated thermal stress. Meccanica 47, 23–29 (2012)CrossRefzbMATHGoogle Scholar
  25. 25.
    Raslan, W.E.: Application of fractional order theory of thermoelasticity to a 1D problem for a cylindrical cavity. Arch. Mech. 66, 257–267 (2014)zbMATHGoogle Scholar
  26. 26.
    Povstenko, Y., Avci, D., Iskender Eroglu, B.B., Ozdemir, N.: Control of thermal stresses in axissymmetric problems of fractional thermoelasticity for an infinite cylindrical domain. Therm. Sci. 21(1A), 19–28 (2017)CrossRefGoogle Scholar
  27. 27.
    Parkus, H.: Instationäre Wärmespannungen. Springer, Wien (1959)CrossRefzbMATHGoogle Scholar
  28. 28.
    Titchmarsh, E.C.: Eigenfunction Expansion Associated with Second-Order Differential Equations. Clarendon Press, Oxford (1946)zbMATHGoogle Scholar
  29. 29.
    Galitsyn, A.S., Zhukovsky, A.N.: Integral Transforms and Special Functions in Heat Conduction Problems. Naukova Dumka, Kiev (1976). (in Russian)Google Scholar
  30. 30.
    Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  31. 31.
    Gorenflo, R., Loutchko, J., Luchko, Y.: Computation of the Mittag-Leffler function and its derivatives. Fract. Calc. Appl. Anal. 5, 491–518 (2002)zbMATHGoogle Scholar
  32. 32.
    Podlubny, I.: Matlab File Exchange 2005, Matlab-Code that calculates the Mittag-Leffler function with desired accuracy. www.mathworks.com/matlabcentral/fileexchange/8738-Mittag-Leffler-function

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceJan Dlugosz University in CzestochowaCzestochowaPoland

Personalised recommendations