Somewhat circular tensile cracks

Abstract

In this paper we apply the method developed by Rice [1], of solving for the elastic field of a crack with a front perturbed from some reference shape, to solve the elasticity problems of somewhat circular planar tensile cracks under arbitrary load distributions. The method is based on a known solution for the stress intensity factor along a circular crack due to a pair of wedge-opening point forces on its surfaces. A full solution, accurate to first order in the deviation from a circular shape, is derived for the stress intensity factor and the crack opening displacement distributions. The results of a perturbation in a harmonic wave form suggest that a circular crack, under axially symmetric loading, can be configurationally unstable (not grow as a circle) for loadings that increase in intensity with distance from the center. Circular cracks with harmonic shape perturbations are found to have the same form of variation of the stress intensity factor with arc length along the crack edge (to first order accuracy) as found in previous work for a half plane crack. As a test case for the perturbation solution, an elliptical planar tensile crack under uniform tension is viewed as being perturbed from a circular crack. Results derived from the perturbation formulae through numerical evaluation are compared with the exact solutions existing in the literature. The perturbation results show a very good match with the exact solutions even when the semi-axis lengths of the elliptical crack differ by a factor of two (and by as much as a factor of three when special choices of the reference circular crack location are made). This suggests that the perturbation procedure presented here, while theoretically exact only to first order, can be used to produce acceptable results for some planar cracks whose shapes deviate appreciably from a circle.