Solution of the problem of stress concentration around intersecting defects by using the riemann problem with an infinite index

Abstract

An analytic method is proposed for solving discontinuous boundary value problems for harmonic and biharmonic operators based on relying on apparatus developed to solve the Riemann boundary value problem with an infinite index. Boundary value problems for differential equations are first reduced to a system of two singular integral equations (SIE) with a fixed singularity by the generalized method of integral transforms, and then to a certain Riemann problem with zero index on a contour parallel to the imaginary axis. Subsequent transformations reduce this problem to two successively solvable scalar Riemann problems (the first with a plus-infinity index and the second with a minus-infinity index). The first problem is solved to entire-function accuracy, found from the condition for the second problem to be solvable (a convolution type Fredholm integral equation on a segment). This method is applied to the solution of the antiplane problem for a plane with a Γ-shaped slit, and also to the plane problem for a plane containing a cruciform slit (the slit edges are free of tangential loads), where the slit branches are of different length in both problems. The singularity in the solution of the SIE system at zero (the intersection of the slit branches) and also the stress intensity coefficients are found. The final formulas are reduced to a form convenient for numerical realization. Earlier /1/ SIE systems analogous to that under consideration were solved by approximate methods without taking account of the presence of the fixed singularity in the kernel.