Fractional thermoelasticity problem for an infinite solid with a penny-shaped crack under prescribed heat flux across its surfaces

Abstract

The time-nonlocal generalization of the Fourier law with the ‘long-tail’ power kernel can be interpreted in terms of fractional calculus and leads to the time-fractional heat conduction equation with the Caputo derivative. The theory of thermal stresses based on this equation was proposed by the first author ( J. Therm. Stresses 28 , 83–102, 2005 ( doi:10.1080/014957390523741 )). In the present paper, the fractional heat conduction equation is solved for an infinite solid with a penny-shaped crack in the case of axial symmetry under the prescribed heat flux loading at its surfaces. The Laplace, Hankel and cos-Fourier integral transforms are used. The solution for temperature is obtained in the form of integral with integrands being the generalized Mittag-Leffler function in two parameters. The associated thermoelasticity problem is solved using the displacement potential and Love’s biharmonic function. To calculate the additional stress field which allows satisfying the boundary conditions at the crack surfaces, the dual integral equation is solved. The thermal stress field is calculated, and the stress intensity factor is presented for different values of the order of the Caputo time-fractional derivative. A graphical representation of numerical results is given. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.