Fracture of solids containing arrays of cracks

Abstract

This survey is largely a collection of the author's recent results on the fracture characteristics of elastic solids containing inhomogeneities in the form of slitlike cracks. The crack tip opening displacement is used as a measure of the fracture behaviour of the solid under three different deformation situations: (a) Opening Mode I; (b) Sliding Mode II; and (c) Tearing Mode III. Various crack geometries have been considered for which closed-form analytical solutions do not seem feasible. To this category belong (a) A stack of cracks (cracks with constant distance of vertical separation) and (b) A doubly-periodic array of cracks forming a rectangular or diamond-shaped pattern. Besides summarizing various results which have been published elsewhere in Scientific Literature, some new results, and necessary amplification of the previous results, are included herein to make the survey as self-contained as possible. It is hoped that it will be found useful by many research workers engaged in crack interaction and propagation problems. The survey is divided into five sections. Section 1 gives a general introduction to the problem at hand and brings out the importance of studying the subject. Sufficient reference is made to the available literature on the subject, without being unduly overemphatic. It is likely that many otherwise good papers have been omitted either through oversight or because they were thought to be peripheral to the subject matter of the survey. Interested readers should, however, be able to find sufficient cross references in the literature cited herein. This section also touches upon the necessity of using the dislocation formalism in solving the problems at hand. For obvious reasons, no attempt has been made to dwell upon this equivalence of slitlike cracks and straight dislocations, the interested reader being again referred to relevant literature. Section 2 formulates the problem in mathematical terms as one consisting in the solution of a singular integral equation. The latter results from the traction-free conditions on the crack faces. The equation is suitably non-dimensionalized and its kernel decomposed into a singular and a non-singular part as dictated by the method of solution. Section 3 presents a perturbation solution for widely spaced cracks. The solution is restricted to a stack of cracks under plane strain conditions. Section 4 deals with an approximate method used to solve the singular integral equation. It is based on an expansion of the non-singular part of the kernel in a series of orthogonal polynomials. The solution of the singular integral equation allows us to calculate the crack tip opening displacement as a function of the externally applied stress and the crack geometry. The results for various loading modes and crack configurations are presented graphically in Section 5 and discussed from the point of fracture initiation from multiple cracks. Where possible, the results are compared with that for an isolated relaxed crack for a better understanding of the change brought about by an array of interacting cracks. In order not to interrupt the text all complicated mathematical expressions have been grouped together and listed at the end of the relevant section.