Generalized solutions of boundary problems for layered composites with notches or cracks

A method is presented for solutions of a class of boundary value problems corresponding to problems of a layered composite with a notch, or in particular, a crack. In this paper the method is applied to problems reducible to Poisson's partial differential equation, namely heat conduction, mass diffusion in solid bodies, consolidation, and antiplane fracture mechanics. The examples which we discuss in this paper refer to problems of heat conduction in solids. Such problems have a direct physical explanation. It is a matter of replacing the coefficients of thermal conductivity λiby the shear moduli μito obtain antiplane problems of fracture mechanics. We apply the Fourier and Mellin transforms technique for generalized functions and reduce the problem to solving a singular integral equation with fixed singularities on the semi-axis. The method is a generalization of the classical approach on the cases when we deal with distributions.