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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on July 17, 2008
The Quarterly Journal of Mechanics and Applied Mathematics 2008 61(4):523-547; doi:10.1093/qjmam/hbn016
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© The author 2008. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Fractional heat conduction equation and associated thermal stresses in an infinite solid with spherical cavity

Y. Z. Povstenko{dagger}

( Institute of Mathematics and Computer Science, Jan Dlugosz University of Czestochowa, al. Armii Krajowej 13/15, 42–200 Czestochowa, Poland )

{dagger} (j.povstenko{at}ajd.czest.pl)

Received 25 October 2007. Revise 14 January 2008. Revise 21 May 2008. Accepted 10 June 2008.


  Abstract

In this work, the temperature distribution and thermal stresses in an infinite medium with a spherical cavity are studied in the framework of a quasi-static uncoupled theory of thermoelasticity based on heat conduction equation with a time fractional derivative of order 0 < {alpha} ≤ 2. The Caputo fractional derivative is used. As the fractional heat conduction equation in the case 1 ≤ {alpha} ≤ 2 interpolates the standard heat conduction equation ({alpha} = 1) and the wave equation ({alpha} = 2), the proposed theory interpolates the classical thermoelasticity and the thermoelasticity without energy dissipation introduced by Green and Naghdi. The solution is obtained using the integral transform technique. Numerical results are illustrated graphically.


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