Heat conduction in a functionally graded medium with an arbitrarily oriented crack

To characterize the influence of cracking defect on the heat conduction in a functional graded structure, the problem of an infinite nonhomogeneous plane containing an arbitrarily oriented crack under uniform remote heat flux is considered. In the mathematical formulation the crack is approximated as a cut with temperature discontinuity. By using Fourier transformation, the mixed boundary value problem is reduced to a Cauchy-type singular integral equation for an unknown density function. The singular integral equation is then regularized by approximating the density function with a Chebyshev polynomial-based series, and the resulting linear equation is solved by using a collocation technique. Temperature distributions along the crack surface planes and the heat flux intensity factors at crack tips are calculated for quantifying the singular temperature gradient and heat flux in the neighborhood of the crack tip, and to evaluate the effects of grading inhomogeneity, crack orientation, and crack thermal permeability on the post-damage performance of the thermal system. It is shown that the inhomogeneity in thermal conductivity around crack tip results in a higher temperature gradient, and that the heat flux intensity factor is strongly influenced by crack orientation as well as material inhomogeneity.