Prediction of the direction of crack propagation

The near-crack-tip stress field for a homogeneous, isotropic linear elastic material is given by

where r and $\theta$ are polar coordinates centered at the crack tip in a plane orthogonal to the crack front.

The direction of crack propagation can be obtained using either the condition $\partial {\sigma }_{\theta \theta }/\partial \theta =0$ or ${\tau }_{r\theta }=0$; i.e.,

where the crack propagation angle $\widehat{\theta }$ is measured with respect to the crack plane. $\widehat{\theta }=0$ represents the crack propagation in the “straight-ahead” direction. $\widehat{\theta }<0$ if $K_{II}>0$ while $\widehat{\theta }>0$ if $K_{II}<0$

Consider a crack segment of length a kinking out the plane of the crack at an angle $\widehat{\theta }$, as shown in Figure 1.

Figure 1. Contour for evaluation of the J-integral.

When a is infinitesimally small compared with all other geometric lengths (including the length of the parent crack), the stress intensity factors $K_I^k$ and $K_{II}^k$ at the tip of the putative crack can be expressed as linear combinations of $K_I$ and $K_{II}$, the stress intensity factors existing prior to kinking for the parent crack:

The $\widehat{\theta }$-dependences of the coefficients $c_{ij}$ are given by Hayashi and Nemat-Nasser (1981) and by He and Hutchinson (1989).

For the crack segment we also have the relation

The maximum energy release rate criterion postulates that the parent crack initially propagates in the direction that maximizes $G^k$.

This criterion simply postulates that a crack will initially propagate in the direction that makes $K_{II}^k=0$.

It can be seen from Figure 1 that the maximum energy release rate criterion and the $K_{II}=0$ criterion predict nearly coincident crack propagation angles. By comparison, the maximum tangential stress criterion predicts smaller crack propagation angles.