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.
In order to solve a differential equation you
look at it till a solution occurs to you. George Pólya
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Tables of Integral Transforms, vol. 1. McGraw-Hill, New York (1954)
Galitsyn, A.S., Zhukovsky, A.N.: Integral Transforms and Special Functions in Heat Conduction Problems. Naukova Dumka, Kiev (1976) (in Russian)
Nowacki, W.: Thermoelasticity, 2nd edn. PWN-Polish Scientific Publishers, Warsaw and Pergamon Press, Oxford (1986)
Parkus, H.: Instationäre Wärmespannungen. Springer, Wien (1959)
Podstrigach, Ya.S., Povstenko, Y.Z.: Introduction to Mechanics of Surface Phenomena in Deformable Solids. Naukova Dumka, Kiev (1985) (in Russian)
Povstenko, Y.: Fractional heat conduction equation and associated thermal stresses. J. Therm. Stress. 28, 83–102 (2005)
Povstenko, Y.: Thermoelasticity which uses fractional heat conduction equation. J. Math. Sci. 162, 296–305 (2009)
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)
Povstenko, Y.: Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder. Fract. Calc. Appl. Anal. 14, 418–435 (2011)
Povstenko, Y.Z.: Solutions to time-fractional diffusion-wave equation in cylindrical coordinates. Adv. Differ. Equ. 2011, 930297-1-14 (2011)
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)
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 2011
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)
Povstenko, Y.: Fractional thermoelasticity. In: Hetnarski, R.B. (ed.) Encyclopedia of Thermal Stresses, vol. 4, pp. 1778–1787. Springer, New York (2014)
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)
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)
Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series. Volume 1: Elementary Functions. Gordon and Breach, Amsterdam (1986)
Prudnikov, A.P., Brychkov, YuA, Marichev, O.I.: Integrals and Series. Volume 2: Special Functions. Gordon and Breach, Amsterdam (1986)
Sneddon, I.N.: The Use of Integral Transforms. McGraw-Hill, New York (1972)
Titchmarsh, E.C.: Eigenfunction Expansion Associated with Second-Order Differential Equations. Clarendon Press, Oxford (1946)
Author information
Authors and Affiliations
Corresponding author
Appendix: Integrals
Appendix: Integrals
Integrals containing elementary functions are taken from [17], integrals (4.216) and (4.217) containing Bessel function are taken from [18], integrals (4.212) and (4.214) are presented in [1]; integrals (4.202), (4.213) and (4.215) have been evaluated by the author.
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Povstenko, Y. (2015). Axisymmetric Problems in Cylindrical Coordinates. In: Fractional Thermoelasticity. Solid Mechanics and Its Applications, vol 219. Springer, Cham. https://doi.org/10.1007/978-3-319-15335-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-15335-3_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15334-6
Online ISBN: 978-3-319-15335-3
eBook Packages: EngineeringEngineering (R0)