Crack Path in a Homogeneous Crystalline Material.

In a separate paper[], we have shown that in an amorphous material, where the surface energy is completely isotropic in direction, a crack under mixed modes will deviate out of the initial crack path into the path where on the branching crack. In such cases, crack deviation is, of course, easy, and widely observed. But in a lattice, where has cusps on the high density planes, only certain special planes are allowed for cleavage. In this paper, we have shown that significant Mode II can be sustained by a crack, up to the limits governed by the stability diagram. Indeed, for any given lattice, branching may be quite difficult, because not only must the Griffith relation on the branching plane be satisfied (with any additional cornering resistance which may be necessary), but dislocation emission on one of the allowed planes might intervene before the relevant Griffith relation is satisfied.

We performed a simulation to explore the case of branching in a completely homogeneous lattice where all interfacial and chemical effects were taken out. For the UBER force law, we found that branching was possible only for a range parameter corresponding to the largest possible unstable stacking fault, for the reason quoted above-sufficiently large (negative) values were necessary, such that dislocation emission tended to occur on the cleavage plane. When branching was finally induced, however, it became possible only for a very narrow, essentially unique, load range, for which it was found that was not zero, but rather when branching occurred. We consider this finding to be a significant violation of the expectation of He and Hutchinson[] that at branching. Presumably the reason for the violation is that the stability diagram on the branching plane covers a range of values, and in the UBER law, only the extreme lower limit of the branching stability curve is reachable, and then only for a specific value of the range parameter.

Thus, in the lattice, in spite of the continuum prediction that is maximized when , the stability line on the branching plane encompasses a range of values. Therefore, a complete analysis of the stability diagram for the given force law, including competing dislocation emission on all possible planes is necessary, in order to predict when branching will occur.

Hutchinson Mear and Rice[] also apply the criterion for predicting the plane on which cracking will occur when the crack runs parallel to an interface. Presumably similar difficulties will arise in that case when the material is a crystal and not amorphous. Any plane parallel to the interface where the crack satisfies the stability criterion will be a possible plane of cracking. Since can reach values in the approximate range of 0.25 to 0.5 for reasonable force laws for stable cracks, and a crack, once established on a stable plane will not deviate from it, there should be a range of possible planes where a crack will be stable on a plane parallel to an interface.