Fractional Thermoelasticity of Thin Shells

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


The notion of a thin shell (a solid with one size being small with respect to two other sizes) allows reducing of a three-dimensional problem to a two-dimensional one for the median surface. In such an approach, investigation of thermomechanical state of a considered three-dimensional solid is reduced to investigation of thermomechanical state of the median surface endowed with equivalent properties characterizing deformation and heat conduction. Equations of fractional thermoelasticity of thin shells are formulated for theories based on the time-fractional heat conduction equation as well as on the time-fractional telegraph equation. The generalized boundary conditions of nonperfect thermal contact for the time-fractional heat conduction in composite medium are also obtained. These boundary conditions take into account the reduced heat capacity, reduced thermal conductivity and reduced thermal resistance of the median surface modeling the transition region between solids in contact.


  1. 1.
    Awrejcewicz, J., Krysko, V.A., Krysko, A.V.: Thermo-dynamics of Plates and Shells. Springer, Berlin (2007)zbMATHGoogle Scholar
  2. 2.
    Bolotin, V.V.: Equations of nonstationary temperature fields in thin shells under existence of heat sources. Appl. Math. Mech. 24, 361–363 (1960) (in Russian)Google Scholar
  3. 3.
    Danilovskaya, V.I.: Approximate solution of the problem on stationary temperature field in a thin shell of arbitrary form. Izvestia Acad. Sci. SSSR. Ser. Mech. Mech. Eng. 9, 157–158 (1957) (in Russian)Google Scholar
  4. 4.
    Goldenveiser, A.L.: Theory of Thin Shells. Pergamon Press, Oxford (1961)Google Scholar
  5. 5.
    Lurie, A.I.: Spatial Problems of the Theory of Elasticity. Gostekhizdat, Moscow (1955) (in Russian)Google Scholar
  6. 6.
    Marguerre, K.: Thermo-elastische Platten-Gleichungen. Z. Angew. Math. Mech. 15, 369–372 (1935)CrossRefzbMATHGoogle Scholar
  7. 7.
    Marguerre, K.: Temperaturverlauf und Temperaturspannungen in platten- und schalenförmigen Körpern. Ing. Arch. 8, 216–228 (1937)CrossRefzbMATHGoogle Scholar
  8. 8.
    Motovylovets, I.O.: On derivation of heat conduction equations for a plate. Prikl. Mekh. (Appl. Mech.) 6, 343–346 (1960)Google Scholar
  9. 9.
    Naghdi, P.M.: The theory of shells and plates. In: Truesdell, C. (ed.) Handbuch der Physik, vol. VI a/2, pp. 425–640. Springer, Berlin (1972)Google Scholar
  10. 10.
    Novozhilov, V.V.: Thin Shell Theory. Noordholf, Groningen (1964)CrossRefGoogle Scholar
  11. 11.
    Podstrigach, Ya.S.: Temperature field in thin shells. Dop. Acad. Sci. Ukrainian SSR (5), 505–507 (1958) (in Ukrainian)Google Scholar
  12. 12.
    Podstrigach, Ya.S.: Temperature field in a system of solids conjugated by a thin intermediate layer. Inzh.-Fiz. Zhurn. 6, 129–136 (1963) (in Russian)Google Scholar
  13. 13.
    Podstrigach, Ya.S., Shvets, R.N.: Thermoelasticity of Thin Shells. Naukova Dumka, Kiev (1978) (in Russian)Google Scholar
  14. 14.
    Povstenko, Y.: Fractional Cattaneo-type equations and generalized thermoelasticity. J. Therm. Stress. 34, 97–114 (2011)CrossRefGoogle Scholar
  15. 15.
    Povstenko, Y.: Different kinds of boundary problems for fractional heat conduction equation. In: Petrá\(\check{\text{ s }}\), I., Podlubny, I., Kostúr, K., Ka\(\check{\text{ c }}\)ur, J., Moj\(\check{\text{ z }}\)i\(\check{\text{ s }}\)ová, 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, Ko\(\check{\text{ s }}\)ice (2012)Google Scholar
  16. 16.
    Povstenko, Y.: Fractional heat conduction in infinite one-dimensional composite medium. J. Therm. Stress. 36, 351–363 (2013)CrossRefGoogle Scholar
  17. 17.
    Povstenko, Y.: Fractional heat conduction in an infinite medium with a spherical inclusion. Entropy 15, 4122–4133 (2013)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Povstenko, Y.: Thermoelasticity of thin shells based on the time-fractional heat conduction equation. Cent. Eur. J. Phys. 11, 685–690 (2013)CrossRefGoogle Scholar
  19. 19.
    Povstenko, Y.: Fractional thermoelasticity. In: Hetnarski, R.B. (ed.) Encyclopedia of Thermal Stresses, vol. 4, pp. 1778–1787. Springer, New York (2014)CrossRefGoogle Scholar
  20. 20.
    Povstenko, Y.: Fractional thermoelasticity of thin shells. In: Pietraszkiewicz, W., Górski, J. (eds.) Shell Structures, vol. 3, pp. 141–144. CRC Press, Boca Raton (2014)Google Scholar
  21. 21.
    Povstenko, Y.: Generalized boundary conditions for time-fractional heat conduction equation. In: International Conference on Fractional Differentiation and Its Applications, Catania, Italy, 23–25 June 2014Google Scholar
  22. 22.
    Vekua, I.N.: Some General Methods of Constructing Different Variants of Shell Theory. Nauka, Moscow (1982) (in Russian)Google Scholar
  23. 23.
    Ventsel, E., Krauthammer, T.: Thin Plates and Shells: Theory, Analysis, and Applications. Marcel Dekker, New York (2001)CrossRefGoogle Scholar
  24. 24.
    Vodi\(\check{\text{ c }}\)ka, V.: Stationary temperature fields in a two-layer plate. Arch. Mech. Stos. 9, 19–24 (1957)Google Scholar
  25. 25.
    Vodi\(\check{\text{ c }}\)ka, V.: Stationary temperature distribution in cylindrical tubes. Arch. Mech. Stos. 9, 25–33 (1957)Google Scholar
  26. 26.
    Wempner, G., Talaslidis, D.: Mechanics of Solids and Shells. CRC Press, Boca Raton (2003)zbMATHGoogle 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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