Multiple interacting curvilinear crack problems: A method of solution and numerical results

Abstract

The importance of accounting for multiple cracks and the blocky structure of rock is emphasized and a method to solve plane problems for such structures is developed. The method makes use of (i) hypersingular equations to deal with discontinuities combined with (ii) complex variables to facilitate evaluation of singular and hypersingular integrals. New approximations for the accurate representation of displacement discontinuities are suggested; they include linear combination of common and conjugated polynomials for ordinary elements together with a special form of asymptotic function for crack tip elements. This results in simple analytical recurrence formulae for the “crucial” (singular and hypersingular) integrals over arbitrary curvilinear elements (both ordinary and tip). The remaining non-singular integrals over elements are efficiently evaluated by common methods. Moreover, in the important particular cases of a straight element and a circular are element they are also evaluated in a simple analytical form. All the significant details of the method are presented. A computer program, worked out on this basis, provides an opportunity do deal with blocky systems, defined by multiple straight and curvilinear cracks as well as a basis for the simulation of propagating curvilinear cracks. An example illustrating the efficiency of the method is given.