Edge crack problem in a semi-infinite FGM plate with a bi-directional coefficient of thermal expansion under two-dimensional thermal leading

The temperature distribution in structural elements in practical cases usually changes in two or three directions. Based on such facts, aiming at more effectiveness, a Functionally graded material (FGM), whose properties change in two or three directions, is introduced, that investigated here called bi-directional FGM. The current study aims at the formulation: solution and investigation of a semiinfinite edge cracked FGM plate problem with a bi-directional coefficient of thermal expansion under two-dimensional thermal loading. The solution of the boundary value problem that one obtains from the mathematical formulation of the current crack problem under thermal loading reduces to an integral equation with a generalized Cauchy kernel. This integral equation contains many two-dimensional double strongly singular integrals, which can be solved numerically. In order to separate the singular terms and overcome the divergence of the integrals an asymptotic analysis for the singular parts in the obtained integral equation was carried out. Also, the exact solution for many singular integrals is obtained. The obtained numerical results are used in the representation of the thermal stress intensity factor versus the thermal/mechanical nonhomogeneous parameters. The numerical results show that it is possible to reduce and control the thermal stress intensity Factor.