A simple technique for constructing the full stress and displacement fields in elastic plates with multiple cracks

Abstract

Full displacement and stress fields in a two-dimensional elastic solid with multiple interacting cracks is constructed in an approximate way, in elementary functions. It is done by approximating the actual displacement discontinuity (COD) on each crack, in each of modes I and II, by a product of the ellipse corresponding to an isolated crack and a quadratic polynomial. The polynomial’s coefficients, that reflect crack interactions, are chosen to match the SIFs and the average tractions on cracks (quantities found, for example, by a simple method of analysis of crack interactions; Kachanov, M. Elastic solids with many cracks: a simple method of analysis. International Journal of Solids and Structures, 23, 1987. 23–43). We then use this approximated CODs in the representation of the displacement field in the solid in terms of integrals over crack lines, with CODs as kernels. The constructed field differs from the actual one by the field generated by such traction distributions on cracks that have zero averages and produce zero stress intensity factors. The accuracy of the suggested method depends on the configuration, but, generally, remains good at spacings between cracks that are substantially smaller than the crack lengths.