Subsonic propagation of a crack parallel to the boundary of a half-plane

Abstract

A two-dimensional steady-state model problem on a half-plane with a semi-infinite crack propagating at constant speed parallel to the boundary of the half-plane is considered. The crack faces are subjected to normal and tangential loads, while the boundary of the half-plane is free of traction. The problem is formulated as an order-2 vector Riemann–Hilbert problem and then reduced to a system of singular integral equations in a semi-infinite segment with respect to the derivatives of the displacement jumps. The solution to the system of integral equations is represented in a series form in terms of an orthonormal basis of the associated Hilbert space, the orthonormal Jacobi polynomials. The coefficients of the expansions solve an infinite system of linear algebraic equations of the second kind. The stress intensity factors and the weight functions are determined and computed. The Griffith energy criterion is applied to derive a crack growth criterion in terms of the K I - and K II -stress intensity factors, the crack speed, the shear and dilatational waves speeds and the Griffith material constant.