Superposition method for calculating singular stress fields at kinks, branches and tips in multiple crack arrays

Abstract

A method is developed for calculating stresses and displacements around arrays of kinked and branched cracks having straight segments in a linearly elastic solid loaded in plane stress or plain strain. The key idea is to decompose the cracks into straight material cuts we call `cracklets', and to model the overall opening displacements of the cracks using a weighted superposition of special basis functions, describing cracklet opening displacement profiles. These basis functions are specifically tailored to induce the proper singular stresses and local deformation in wedges at crack kinks and branches, an aspect that has been neglected in the literature. The basis functions are expressed in terms of dislocation density distributions that are treatable analytically in the Cauchy singular integrals, yielding classical functions for their induced stress fields; that is, no numerical integration is involved. After superposition, nonphysical singularities cancel out leaving net tractions along the crack faces that are very smooth, yet retaining the appropriate singular stresses in the material at crack tips, kinks and branches. The weighting coefficients are calculated from a least squares fit of the net tractions to those prescribed from the applied loading, allowing accuracy assessment in terms of the root-mean-square error. Convergence is very rapid in the number of basis terms used. The method yields the full stress and displacement fields expressed as weighted sums of the basis fields. Stress intensity factors for the crack tips and generalized stress intensity factors for the wedges at kinks and branches are easily retrieved from the weighting coefficients. As examples we treat cracks with one and two kinks and a star-shaped crack with equal arms. The method can be extended to problems of finite domain such as polygon-shaped plates with prescribed tractions around the boundary.