Complete asymptotic expansion M. Williams near the crack tips of collinear cracks of equal lengths in an infinite plane medium

The paper is aimed at analytical determination of the coefficients in crack tip expansions for two collinear finite cracks of equal length in an infinite plane medium under mixed mode loading conditions. The study is based on the solutions of the complex variable theory in plane elasticity theory and the complete asymptotic Williams expansion of the stress field in the vicinity of the crack tip. From the practical point of view, it is very important to know: 1) analytical dependence of coefficients on geometrical parameters of specimens and applied loads; 2) the number of higher-order terms in the asymptotic Williams expansions that we need to keep to accurately describe the stress and displacement fields in the neighborhood of the crack tip. In the paper the authors have provided a multiparametric presentation of the stress filed near the crack tips in the infinite plate with two collinear cracks of finite length. We have presented the analytical determination method of coefficients related to the complete asymptotic expansion. The method is based on the complex variable theory and the classical Kolosoff-Muskhelishvili approach. The expansion of the Kolosoff-Muskhelishvili's potential in the vicinity of the crack tip allows to find the analytical presentation for coefficients of the complete Williams asymptotic expansion. The analytical solution gives the dependence of the amplitude coefficients on the lengths of the crack and the applied loads for pure mode I, pure mode II and mixed mode I/II conditions. It is shown that together with the main term including the stress intensity factor and the term called T-stress, it is necessary to hold higher-order terms in the asymptotic solution. The obtained solution permits to construct the asymptotic stress field expansion containing an arbitrary preassigned number of higher-order terms.