Asymptotic analysis of crack interaction with free boundary

Abstract

This paper employs the beam and dipole asymptotic techniques for modelling interaction of a crack with parallel free boundaries. Two configurations are considered: (1) a crack in a half-plane and (2) a crack in the centre of an infinite strip. Both, the stress intensity factors and the areas of the crack opening are calculated. For the crack situated close to the boundary, the part of the material between the crack and the boundary is represented by a beam (plate in plane-strain). This allows calculating the area of the crack opening. The stress intensity factors are calculated by matching the beam approximation with Zlatin and Khrapkov's solution (Zlatin and Khrapkov, 1986) for a semi-infinite crack parallel to the boundary of a half-plane or with Entov and Salganik's solution (Entov and Salganik, 1965) for a central semi-infinite crack in a strip. It has been shown that this asymptotic method allows obtaining two leading terms for the SIFs and the crack opening area. When the distance between the crack and the free surface is large, the problem is treated in the far field approximation. This, dipole asymptotic method allows finding the leading asymptotic terms responsible for the crack–boundary interaction. For intermediate distances between the crack and the boundary, simple interpolating formulas are derived. Particular examples of cracks loaded by pair of concentrated forces and for uniform loading are considered. The obtained results are compared with available numerical solutions.