| We studied the problem of crack propagation in the field of fracture mechanics. Our interest was the prediction of the path taken by a growing crack under the assumption that dynamic effects can be neglected. We used the criterion of local symmetry to determine the crack path. That is, we solved the following problem: Given a two-dimensional domain with a crack and some loading and constraints for the domain, find a curve which is an extension of the existing crack, such that if the crack propagates along this curve, then the Mode II stress intensity factor is zero at the leading edge of the crack at all times.; The major portion of the analysis for this problem is performed for the model problem, Laplace's equation on a two-dimensional domain with a crack. Using expressions for the stress intensity factors at the tip of a crack, we derived an integral equation for the difference between a given possible crack extension and the real extension.; We developed a numerical method for determining the path taken by a propagating crack. We applied this method to the real problem, the linear elasticity problem in a two-dimensional domain with a crack, and compared it to a typical method used by engineers to determine crack paths. Our method performs very favorably in terms of accuracy and work required. The integral equation that we derived is used to generate an a-posteriori error estimate for the error in the crack path generated by the method. |
