Advertisement

Axisymmetric Problems in Cylindrical Coordinates

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

Abstract

Problems of fractional thermoelasticty based on the time-fractional heat conduction equation are considered in cylindrical coordinates. Heat conduction in a long cylinder, in an infinite solid with a long cylindrical cavity and in a half-space is investigated. The solutions to the fractional heat conduction equation under the Dirichlet boundary condition with the prescribed boundary value of temperature and the physical Neumann boundary condition with the prescribed boundary value of the heat flux are obtained. Different kinds of integral transforms (Laplace, Fourier and Hankel) are used. Associated thermal stresses are investigated. The representations of stresses in terms of the displacement potential and the biharmonic Love function allows us to satisfy the prescribed boundary condition for the stress tensor.

References

  1. 1.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Tables of Integral Transforms, vol. 1. McGraw-Hill, New York (1954)Google Scholar
  2. 2.
    Galitsyn, A.S., Zhukovsky, A.N.: Integral Transforms and Special Functions in Heat Conduction Problems. Naukova Dumka, Kiev (1976) (in Russian)Google Scholar
  3. 3.
    Nowacki, W.: Thermoelasticity, 2nd edn. PWN-Polish Scientific Publishers, Warsaw and Pergamon Press, Oxford (1986)zbMATHGoogle Scholar
  4. 4.
    Parkus, H.: Instationäre Wärmespannungen. Springer, Wien (1959)CrossRefzbMATHGoogle Scholar
  5. 5.
    Podstrigach, Ya.S., Povstenko, Y.Z.: Introduction to Mechanics of Surface Phenomena in Deformable Solids. Naukova Dumka, Kiev (1985) (in Russian)Google Scholar
  6. 6.
    Povstenko, Y.: Fractional heat conduction equation and associated thermal stresses. J. Therm. Stress. 28, 83–102 (2005)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Povstenko, Y.: Thermoelasticity which uses fractional heat conduction equation. J. Math. Sci. 162, 296–305 (2009)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Povstenko, Y.: Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry. Nonlinear Dyn. 59, 593–605 (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    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
  10. 10.
    Povstenko, Y.Z.: Solutions to time-fractional diffusion-wave equation in cylindrical coordinates. Adv. Differ. Equ. 2011, 930297-1-14 (2011)Google Scholar
  11. 11.
    Povstenko, Y.Z.: Non-axisymmetric solutions to time-fractional heat conduction equation in a half-space in cylindrical coordinates. Math. Methods Phys.-Mech. Fields 54(1), 212–219 (2011)Google Scholar
  12. 12.
    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
  13. 13.
    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
  14. 14.
    Povstenko, Y.: Fractional thermoelasticity. In: Hetnarski, R.B. (ed.) Encyclopedia of Thermal Stresses, vol. 4, pp. 1778–1787. Springer, New York (2014)CrossRefGoogle Scholar
  15. 15.
    Povstenko, Y.: Axisymmetric thermal stresses in a half-space in the framework of fractional thermoelasticity. Sci. Issues Jan Dlugosz Univ. Czestochowa, Math. 19, 207–216 (2014)Google Scholar
  16. 16.
    Povstenko, Y.: Axisymmetric solution to time-fractional heat conduction equation in an infinite cylinder under local heating and associated thermal stresses. Int. J. Mech. 8, 383–390 (2014)Google Scholar
  17. 17.
    Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series. Volume 1: Elementary Functions. Gordon and Breach, Amsterdam (1986)Google Scholar
  18. 18.
    Prudnikov, A.P., Brychkov, YuA, Marichev, O.I.: Integrals and Series. Volume 2: Special Functions. Gordon and Breach, Amsterdam (1986)Google Scholar
  19. 19.
    Sneddon, I.N.: The Use of Integral Transforms. McGraw-Hill, New York (1972)zbMATHGoogle Scholar
  20. 20.
    Titchmarsh, E.C.: Eigenfunction Expansion Associated with Second-Order Differential Equations. Clarendon Press, Oxford (1946)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

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

Personalised recommendations