Slightly curved or kinked cracks in anisotropic elastic solids

Abstract

Slightly curved or kinked cracks in anisotropic elastic solids are studied via a perturbation analysis valid for the second order accuracy in the deviation of the crack surfaces from a straight line. The analysis is based on complex variable representations in the Stroh formalism and known solutions for a perfectly straight reference crack. First and second order perturbation solutions are given for the stress intensity factors at the tip of a finite curved crack under remote stresses in a solid with arbitrary anisotropy, and are used to establish an approximate relationship between the apparent and local stress intensities at a crack tip, in view of the possible shielding effects of the crack surface morphology near the tip. The role of anisotropy is examined for a number of problems including circular arc cracks, slightly jogged or kinked cracks, cosine wavy cracks, etc. In several cases, further insight into the anisotropic effect on crack curving or kinking is provided by specializing the general perturbation formulae to cracks in materials with orthotropic symmetry. In the limiting case of a crack with an infinitesimal branch length, comparison with numerical results reported in the literature indicates that our perturbation solutions are accurate over the full range of practically important branch angles, up to nearly 150.For anisotropic elastic solids, various fracture criteria such as those based on energy release rate, G. and stress intensity factors, K1 and K11. cease to give consistent predictions on crack behaviour. By investigating the effects of mode mixity. material anisotropy and non-singular T-stress on the behaviour of a nearly symmetric crack path in an orthotropic solid, we show that the K-based fracture criteria lead to very peculiar, perhaps even physically unreasonable, predictions such as: (i) branch angle at a mixed mode crack tip becomes infinite as the compliances perpendicular and parallel to the crack. Svv/Svv reach a four-fold difference: (ii) the sign of the branch angle is reversed when Svv/Svv > 4. with the consequence of reversing the role of the non-singular T stress on the stability of a symmetric fracture path, namely, the path becomes stable for T > 0 and unstable for T < 0. Further, the mode 1 stress intensity factor, Kt, becomes a local minimum with respect to crack angle once Svv/Svv > 4. In contrast, the G-based criterion gives reasonable predictions such as: (i) cracks under mixed mode loading always tend to branch toward symmetric orientations; (ii) material and loading asymmetries play an equivalent role in affecting crack branching near a symmetric orientation; (iii) a compressive T stress always tends to stabilize a symmetric fracture path while a tensile T stress destabilizes such a path. For general crack path stability, the role of anisotropy can be manifested through (i) the value of T-stress at the crack tip and (ii) the variation of fracture resistance with respect to crack orientation.