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

A method is developed for calculating stresses and displacements around arr ays of kinked and branched cracks having straight segments in a linearly el astic solid loaded in plane stress or plain strain. The key idea is to deco mpose the cracks into straight material cuts we call 'cracklets', and to mo del the overall opening displacements of the cracks using a weighted superp osition of special basis functions, describing cracklet opening displacemen t profiles. These basis functions are specifically tailored to induce the p roper singular stresses and local deformation in wedges at crack kinks and branches, an aspect that has been neglected in the literature. The basis fu nctions are expressed in terms of dislocation density distributions that ar e treatable analytically in the Cauchy singular integrals, yielding classic al functions for their induced stress fields; that is, no numerical integra tion is involved. After superposition, nonphysical singularities cancel out leaving net tractions along the crack faces that are very smooth, yet reta ining the appropriate singular stresses in the material at crack tips, kink s and branches. The weighting coefficients are calculated from a least squa res fit of the net tractions to those prescribed from the applied loading, allowing accuracy assessment in terms of the root-mean-square error: Conver gence is very rapid in the number of basis terms used. The method yields th e full stress and displacement fields expressed as weighted sums of the bas is fields. Stress intensity factors for the crack tips and generalized stre ss intensity factors for the wedges at kinks and branches are easily retrie ved from the weighting coefficients. As examples we treat cracks with one a nd two kinks and a star-shaped crack with equal arms. The method can be ext ended to problems of finite domain such as polygon-shaped plates with presc ribed tractions around the boundary.