Incipient fracture angle,fracture loci and critical stress for mixed mode loading

A prediction of the direction of incipient crack growth in brittle-like materials and the associated fracture loci under mixed mode loading is proposed. It is postulated that the direction of unstable crack propagation is determined by the “weakest” near-tip element defined as the one which would relax maximum potential energy upon prospective crack extension. Starting from the energy rate principle of crack extension (Eshelby energy-momentum tensor and Rice $\text{J-}\text{internal}$ vector) it is deduced that a crack will extent in the direction along which the following stress criterion is satisfied, $\text{(δ}{}_{\mathrm{\theta \theta }}{}^2\text{? δ}{}_{\mathrm{rr}}{}^2\text{) →}\text{maximum (for}\text{δ}{}_{\mathrm{\theta \theta \; >\; 0)}}$ The fracture angle in pure Mode II (70.4° away from the original straight path) is shown to be unstable in the sense that any slight tension along the crack (non-singular at the crack tip) affects considerably (up to 22%) the directionality of crack extension. It appears to be sensitive to the extent of the near-tip zone ($\text{r}{}_0$) in which linear elasticity does not hold and the non-singular stress term (squared).The fracture loci in mixed mode loading (generated by projecting the $\text{J-}\text{integral}$ vector along the prospective fracture path and letting this scalar function attain a critical value) is quadratic in $\text{K}{}_1$ and $\text{K}{}_2$ with an interactive cross product term $\text{K}{}_1\text{× K}{}_2$.The suggested criterion with its implication in predicting critical fracture load, exhibits behavior which is consistent with experimental observations collected from several sources. The common and uncommon features with respect to other known criteria are compared and discussed.