Elastic fields of double branched and Kalthoff–Winkler cracks in a half-plane

Abstract

We demonstrate in this paper a combination of the Schwarz–Christoffel mapping and Muskhelishvili’s approach with fractional function series in solving the elastic fields of a cracked half-plane, and zoom in on two typical problems, a double branched crack with two rays emanating from one point on the edge and two edge cracks spaced by a certain distance. Typical loading conditions are considered, including far-field uniform tensile stress and concentrated loads along either the tangential or the normal direction of the free surface. We supply a semi-analytic solution to those boundary-value problems in the cracked half-plane, and validate the theory by comparing the theoretical results in terms of stress fields, stress intensity factors (SIFs) and crack opening displacement (COD) with those from finite-element simulations. The theoretical approach shows how two edge cracks may shield the stress intensity factors of each other in a quantitative manner. For the typical Kalthoff–Winkler cracks of length a and being spaced by a distance d, their SIFs KI decay with decreasing d, and KI=KI0−KI1[1−exp(−a/d)]. It converges to KI0—the SIF of a single edge crack when d approaches to infinity. Those observations and the theory approach itself provide a general way to analyze the mechanical consequence of edge cracks in engineering practice.