An integral equation method for dynamic crack growth problems

Within the assumptions of linear elastic fracture mechanics, dynamic stresses generated by a crack growth event are examined for the case of an infinite body in the state of plane strain subjected to mode I loading.The method of analysis developed in this paper is based on an integral equation in one spatial coordinate and in time. The kernel of this equation, i.e., the influence or Green's function, is the response of an elastic half-space to a concentrated unit impulse acting on its edge. The unknown function is the normal stress distribution in the plane of the crack, while the free term represents the effect of external loading.The solution for the stresses is obtained with the assumption that its spatial distribution contains a square root singularity near the tip of the crack, while its intensity is an unknown function of time. Thus, the orginal integral equation in space and time reduces to Volterra's integral equation of the first kind in time. The equation is singular, with the singularity of the kernel being a combined effect of the singularity of the influence function and the singularity of the dynamic stresses at the tip of the crack. Its solution is obtained numerically with the aid of a combination of quadrature and product integration methods. The case of a semi-infinite crack moving with a prescribed velocity is examined in detail.The method can be readily extended to problems involving mode II and mixed mode crack propagation as well as to problems of dynamic external loadings.