Discussion.

In this paper, we have presented a simple graphical construction in the remote K-space of the crack, with which one can describe the competition between emission and cleavage on two nonequivalent slip/cleavage planes. In general, there may be more planes where the lattice can either cleave or emit dislocations. These more complex situations can be described by simply overlaying the additional diagrams on the ones described here.

For events restricted to the initial cleavage plane, the stability diagram is a sector of a Griffith circle, where the circle limits represent lattice breakdown in shear and dislocation emission. For those force laws where lattice trapping is significant, then the Griffith circle fuzzes and develops a finite width, within which the lattice is stable under the relevant loads.

When events are possible on an inclined slip plane, then a similar (noncircular) contour line for the critical can be constructed in the same space. Dislocation emission corresponds to one limit on this curve, and the other limit corresponds to a critical local Mode I value below which the crack is no longer able to break the relevant bonds in the branching crack core. By overlaying these two plots, one can determine which events are favored, and which are not in a given load range. Moreover, the crossover between simple cleavage on the initial plane and blunting emission on an inclined plane can be determined by varying the interfacial bond strength. This crossover corresponds to the crossover between materials which are intrinsically brittle, and those which are intrinsically ductile. In this paper, we have used a simple lattice and simple force laws to illustrate the physical principles involved. In actual cases, one must carry out calculations for realistic force laws and lattices.

For the case of branching, the criterion is quite simple. Branching is expected to occur whenever the extended Griffith relation (including the cornering resistance) with the bulk in the stability diagram drops below the Griffith relation for simple straight crack extension with the interfacial . Since negative Mode II translates into a strong in the branching plane, mixed modes on the interface strongly influence branching. The magnitude of the cornering resistance can range from about 30%of the normal Griffith to a factor of about 2 for the highest elastic mismatch interface (where branching would be quite difficult). However, in a crystal lattice, where large Mode II may be necessary in order to induce branching, the competing dislocation emission may intervene. Thus, depending on the details of the force law, increasing Mode II in a lattice may branch the crack, or it may break it down in shear with dislocation emission.

It also follows that a crack which emits a dislocation on an inclined glide plane will develop a large mode mixity from the shielding by the emitted dislocation, and this could cause the crack to branch, as is sometimes seen in the computer simulations of Holian, etal.[].

We explored the prospects that branching will take place on the plane where the local . While this criterion does hold for an amorphous material[], we found that significant deviations from this popular criterion take place for crack branching in a crystal lattice.