Numerical solution of singular integral equations in stress concentration problems under longitudinal shear loading

The paper deals with numerical solutions of singular integral equations in stress concentration problems for longitudinal shear loading. The body forc e method is used to formulate the problem as a system of singular integral equations with Cauchy-type singularities, where unknown functions are densi ties of body forces distributed in the longitudinal direction of an infinit e body. First, four kinds of fundamental density functions are introduced t o satisfy completely the boundary conditions for an elliptical boundary in the range 0 less than or equal to phi(k) less than or equal to 2 pi. To exp lain the idea of the fundamental densities, four kinds of equivalent auxili ary body force densities are defined in the range 0 less than or equal to p hi(k) less than or equal to pi/2, and necessary conditions that the densiti es must satisfy are described. Then, four kinds of fundamental density func tions are explained as sample functions to satisfy the necessary conditions . Next, the unknown functions of the body force densities are approximated by a linear combination of the fundamental density functions and weight fun ctions, which are unknown. Calculations are carried out for several arrange ments of elliptical holes. It is found that the present method yields rapid ly converging numerical results. The body force densities and stress distri butions along the boundaries are shown in figures to demonstrate the accura cy of the present solutions.