Complex potentials and integral equations for curved crack and curved rigid line problems in plane elasticity

Abstract

In this paper, a simple layer potential and a double layer potential are suggested to solve the curved crack problem. The complex potentials in the simple layer case are formulated on the distributed dislocation along the curve. Meantime, the complex potentials in the double layer case are formulated on the crack displacement opening distribution. Behaviors of the complex potentials, for example the behaviors of increments of some physical quantities around a large circle, are analyzed in detail. Continuity and discontinuity of some physical quantities in the normal direction of the curve are analyzed, which are key points for formulating the integral equations of the problems. One weaker singular, two singular and one hypersingular integral equations are suggested to solve the problems. The relations between the kernels in different integral equations are addressed. Similarly, a simple layer potential and a double layer potential are suggested to solve the curved rigid line problem. The complex potentials in the simple layer case are formulated on the distributed forces along the curve. Meantime, the complex potentials in the double layer case are formulated on the resultant force function. One weaker singular, two singular and one hypersingular integral equations are suggested to solve the problems. When the resultant forces and moment are applied on the deformable line, the constraint equations are suggested. For more general cases, for example, in the case that the tractions applied on the two crack faces are not same in magnitude and opposite in direction, a singular integral equation is suggested. The equation is obtained by a superposition of two kinds of single layer potentials.