An edge crack and a crack close to the vertex of a wedge

Two model problems of an elastic wedge with an internal and edge crack are analyzed. The problem of an internal crack reduces to an order-four vector Riemann–Hilbert problem whose matrix kernel entries are meromorphic functions and have exponential factors. When the internal crack is located along one of the wedge sides, an efficient method of solution is proposed. It requires a factorization of the order-two matrix coefficient associated with the corresponding problem of an edge crack and the solution of an infinite system of linear algebraic system with an exponential rate of convergence of an approximate solution to the exact one. The original order-two Khrapkov's factorization is modified by splitting the matrix kernel into a scalar dominant function and a “regular” matrix whose factorization is more convenient for numerical purposes. Expressions for the stress intensity factors and the potential energy released when the crack advances are derived. Asymptotic relations for the stress intensity factors and the potential energy when one of the crack tips is close to the wedge vertex are obtained.