Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T13:30:42.966Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermoelastic Interaction in an Infinite Long Hollow Cylinder with Fractional Heat Conduction Equation

Part of: Thin bodies, structures Equilibrium (steady-state) problems

Published online by Cambridge University Press:  09 January 2017

Ahmed. E. Abouelregal
Ahmed. E. Abouelregal*
Affiliation:
Department of Mathematics, Faculty of Science, Mansoura University, P.O. Box 35516, Egypt Department of Mathematics, College of Science and Arts, Aljouf University, Al-Qurayat, Saudi Arabia
*
*Corresponding author. Email:ahabogal@gmail.com (A. E. Abouelregal)
Get access

Abstract

In this work, we introduce a mathematical model for the theory of generalized thermoelasticity with fractional heat conduction equation. The presented model will be applied to an infinitely long hollow cylinder whose inner surface is traction free and subjected to a thermal and mechanical shocks, while the external surface is traction free and subjected to a constant heat flux. Some theories of thermoelasticity can extracted as limited cases from our model. Laplace transform methods are utilized to solve the problem and the inverse of the Laplace transform is done numerically using the Fourier expansion techniques. The results for the temperature, the thermal stresses and the displacement components are illustrated graphically for various values of fractional order parameter. Moreover, some particular cases of interest have also been discussed.

Keywords

Thermoelasticityfractional calculushollow cylinderLaplace transformtraction freeheat flux

MSC classification

Secondary: 74K25: Shells 74G60: Bifurcation and buckling
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 2 , April 2017 , pp. 378 - 392
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Biot, M., Thermoelasticity and irreversible thermodynamics, J. Appl. Phys., 27 (1956), pp. 240–253.CrossRefGoogle Scholar
[2] Lord, H. and Shulman, Y., A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solid, 15 (1967), pp. 299–309.CrossRefGoogle Scholar
[3] Dhaliwal, R. and Sherief, H., Generalized thermoelasticity for anisotropic media, Quart. Appl. Math., 33 (1980), pp. 1–8.Google Scholar
[4] Li, Changpin, Zhao, Zhengang and Chen, Yangquan, Numerical approximation and error estimation of a time fractional order diffusion equation, Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, August 30–September 2, 2009, San Diego, California, USA.Google Scholar
[5] Caputo, M. and Mainardi, F., A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91 (1971), pp. 134–147.Google Scholar
[6] Caputo, M. and Mainardi, F., Linear model of dissipation in anelastic solids, Rivis ta del Nuovo cimento, 1 (1971), pp. 161–198.Google Scholar
[7] Caputo, M., Vibrations on an infinite viscoelastic layer with a dissipative memory, J. Acoust. Society America, 56 (1974), pp. 897–904.Google Scholar
[8] Miller, K. S. and Ross, B., An Introduction to the Fractional Integrals and Derivatives—Theory and Applications, JohnWiley & Sons Inc, NewYork, 1993.Google Scholar
[9] Samko, S. G., Kilbas, A. A. and Marichev, O. I., Fractional Integrals and Derivatives—Theory and Applications, Gordon and Breach, Longhorne, PA, 1993.Google Scholar
[10] Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, 1974.Google Scholar
[11] Gorenflo, R. and Mainardi, F., Fractional Calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien, 1997.Google Scholar
[12] Podlubny, I., Fractional Differential Equations, Academic Press, NewYork, 1999.Google Scholar
[13] Hilfer, R., Applications of Fraction Calculus in Physics, World Scientific, Singapore, 2000.CrossRefGoogle Scholar
[14] Sherief, H., El-Sayad, A. M. A. and Abd El-Latief, A. M., Fractional order theory of thermoelasticity, Int. J. Solids Structures, 47 (2010), pp. 269–275.CrossRefGoogle Scholar
[15] Zenkour, M. and Abouelregal, A. E., State-space approach for an infinite medium with a spherical cavity based upon two-temperature generalized thermoelasticity theory and fractional heat conduction, Zeitschrift für angewandte Mathematik und Physik, 65(1) (2014), pp. 149–164.Google Scholar
[16] Abouelregal, A. E. and Zenkour, A. M., The effect of fractional thermoelasticity on a two-dimensional problem of a mode I crack in a rotating fiber-reinforced thermoelastic medium, Chinese Phys. B, 22(10) (2013), 108102.Google Scholar
[17] Mashat, D. S., Zenkour, A. M. and Abouelregal, A. E., fractional order thermoelasticity theory for a half-space subjected to an axisymmetric heat distribution, Mech. Adv. Mater. Structures, 22(11) (2015), pp. 925–932.Google Scholar
[18] Sur, A. and Kanoria, M., Fractional order generalized thermo-visco-elastic problem of a spherical shell with three-phase-lag effect, Latin American Journal of Solid and Structures, 11 (2014), pp. 1132–1162.Google Scholar
[19] Youssef Hamdy, M. and Al-Lehaibi Eman, A., Fractional order generalized thermoelastic half-space subjected to ramp-type heating, Mech. Res. Commun., 37 (2010), pp. 448–452.CrossRefGoogle Scholar
[20] Honig, G. and Hirdes, U., A method for the numerical inversion of the Laplace transform, J. Comput. Appl. Math., 10 (1984), pp. 113–132.CrossRefGoogle Scholar