A fracture criterion for three-dimensional crack problems

A criterion for predicting the growth of three-dimensional cracks is developed on the basis of the strain energy density concept which has been used successfully for treating two-dimensional crack problems. Fracture is assumed to initiate from the nearest neighbor element located by a set of spherical coordinates (r, θ, φ) attached to the crack border. The new fracture surface is described by a locus of these elements whose locations correspond to the strain energy function, dW/dV, being a minimum. The function dW/dV is found to be singular of the type 1/r and is of quadratic form in the three stress intensity factors k₁, k₂ and k₃ expressed through the strain energy density factor S. It is postulated that unstable crack propagation initiates from a region where S reaches a critical value S_cr = r₀(dW/dV)_cr. The locations of failure lying on the fracture surface is determined by holding (dW/dV)_cr = S_min/r₀ constant. The quantity S_min stands for the value of S minimized with respect to θ and φ and r₀ is a radial distance measured from the crack border.

An example of failure prediction for an embedded elliptical crack subjected to both normal and shear loads is presented. According to the S-criterion, fracture initiation takes place at the ends of the minor axis. An unexpected result is that for a narrow elliptical crack and Poisson's ratio of 1/3 the lowest failure load occurs when the uniaxial tensile load makes an angle of approximately 60° with the crack surface and is in the plane of the major axis. This is in contrast to the expectation that the lowest critical load occurs when the uniaxial tension is perpendicular to the crack surface. In the limit as the elliptical crack becomes increasingly narrower, the result reduces to the two dimensional line crack case of Mode I and III loading. The S-criterion is also applied to the failure prediction of three dimensional cracks under compressive loads.