Stress intensity factors for curvilinear cracks loaded under anti-plane strain (mode III) conditions

Abstract

Fracture of a linearly elastic solid containing a slightly curved crack and being loaded under antiplane strain conditions is investigated. The internal shear stresses and the normal displacement are represented by complex holomorphic functions and calculated by using the technique of Hilbert problems and Cauchy integrals. The crack is assumed to have slight curvature and a linearization with respect to the crack shape function is employed. The perturbation solution is thus correct up to first order in the deviation of the crack shape from a straight line. The mode III stress intensity factor is expressed in the shear stresses exerted on the crack flanks, the uniform stresses applied at remote positions, and the shape of the curved crack. Examples for some specific loading configurations and crack geometries are given. Results for the circular-arc crack show good agreement with known results from the literature over a wide range of arc angles.

In addition, similarities with the stress intensity factors for curvilinear cracks in planar deformation and plate bending are indicated. The local crack-tip stress and displacement fields for anti-plane strain or mode III fracture are compared with those for the fracture of thin flat plates loaded by perpendicular forces (mode 3). A simple equation is derived relating the respective stress intensity factors, namely K ₃=3/2K _III, which satisfies the physical condition that the eergy release rates per unit area are equivalent. These results suggest that only five independent stress intensity factors exist for the analysis of mixed-mode fracture under general loading conditions.