The Lattice Resistance.

The lattice modeling uses the same lattice Green's function techniques used in previous work[], and the general methodology is given in Thomson, etal.[] Also we use the same 2D hexagonal lattice with the same set of nearest neighbor forces. Figure 3 shows the lattice with a crack on an interface between atoms of one kind below and a second kind above. In the lattice case, we are at liberty to make the bonding between the layers different from that of either bulk, as in the real physical situation. The elastic constant of material 2 is arbitrarily normalized to unity, since all the physical results scale with the elastic constant. Also, we allow the crack to emit a dislocation into material 2, only, and assume that material 1 is strong, incapable of deforming or cleavage. We also make the assumption that dislocation emission takes place on the ``forward'' slip plane, , because the shear stress is largest on this plane. Subsequent emission could conceivably take place on the second slip plane, , but multiple emission of dislocations is not explored in this work. The crack is loaded at the center of the crack with a concentrated load, see Fig. 3. This method of loading means that the crack system is stable, because the at the crack tip decreases as the crack length increases by the elastic equation

where is the point load on upper and lower crack planes, and is the half length of the parent crack.

As in previous work, our simulations are for a bimaterial slab. The slab is atoms thick, with the interface running down the center. The slab has periodic boundary conditions in the lateral direction, again with repeat distance of atom spacings. The crack itself is 201 atom spacings in total length, the cohesive zone is 12 atom spacings long on the cleavage plane to the right, and the inclined slip plane is 16 atom spacings long. Thus we have no worries about short crack effects, or interactions with neighboring cracks in the repeating cells, or with the free surfaces.

In contrast with our previous work, we have restricted the force law in this work to be the UBER of Rose, etal.[] derived from the energy expression,

is the lattice spring constant, is the displacement from the equilibrium distance between two atoms, and is the range parameter. The lattice parameter will be normalized to unity. This equation has been modified from the standard UBER expression so it has zero energy and force at the second neighbor position, . It will be convenient to normalize the lattice spring constant in sublattice 2 to unity, .

We have performed a set of simulations for a variety of elastic mismatches, Figs. 4,5. In all these simulations, the range parmeters are held fixed (), and . Thus, for all cases, the interface energy and unstable stacking fault in material 2 are held constant, and , in the natural units of the simulation (lattice parameter and spring constant, , normalized to unity). Note that with these choices, the bonds in the interface are stronger than in the matrix. This choice amounts to an assumption about the kind of chemical interaction taking place between the two interfaces to the right of the crack tip, and the dislocation emission criterion depends on this chemistry. Nevertheless, the choice we make allows us to explore the form of the stability diagram, which is our purpose. We will return at a later point to discuss the implications of the interfacial chemistry problem.

In these simulations, we have arbitrarily cut the bonds to the left of the base of the spur, so that the crack does not have to grow through ``good material'' to reach the branching plane. This corresponds to a chemical knife which has acted over the cleavage plane up to the crack tip, in the time honored manner of the continuum mechanics community. That is, a chemical adsorbate is assumed to have formed over the cleavage plane which has zero interaction with the atoms on the opposite cleavage plane-highly unlikely physically, but perfectly possible, mathematically. It has the advantage of allowing the branching crack to grow out of the tip, without regard for the problem of how it got there. Specifically, the first active bond at the tip is the one at 120 from the base of the right hand spur line where it meets the lower cleavage plane, and is depicted as the first connected bond shown in Fig. 3.

In Figs. 4,5, as the solid line, we plot the appropriate to a simple Griffith law, where for material 2. The individual points are the simulation results, and they span the complete range of possible mixed loads, , for which branching cleavage is possible. The upper limits correspond to lattice shear breakdown and dislocation emission on the inclined plane, while the lower values correspond to the lowest loads where the crack is stable on the inclined plane.

The results show that the expected Griffith condition is not satisfied for the branching crack, but that an extra ``cornering'' energy is required to turn the crack out of its initial cleavage plane. This cornering energy is increased by increasing the lattice mismatch. The line of points, however, is parallel to the expected -curve, showing that the driving force for cleavage is correctly given by the sum of the squares of the local 's in (6). The emission point obtained is consistent with the analytic emission criterion given in the previous paper[]. We note again, that the lower limit in the line of simulation points simply corresponds to the load where the first bond of the branching crack is not broken. For lower loads, the crack opens up to the artificially created crack tip, but cannot go further.

In the case of the zero misfit, the numerical results for are known, and can be compared with the actual stability diagram in Fig. 4. As noted in §II, the analytic expression is larger than the numerical value, so the cornering energy is slightly larger than the 30%shown in Fig. 4. Unfortunately, the numerical results of He and Hutchinson[] are given for misfit parameters which differ completely from those of the present work, so we cannot comment on what portion of the cornering energy in Fig. 5 might be due to an error in the analytic Cotterell/Rice approximation for the interface. However, we see no physical reason why the error should be significantly larger than in the zero misfit case, and believe that increasing the misfit increases the cornering energy, as shown in Fig. 5.

There is another possibly important effect for the interface, however, associated with the dependence of on the length of the kink when the elastic mismatch is nonzero. (See the discussion following Eqn. (5).) Because of the length dependence of the local for the kinking crack, the crack may require a larger to nucleate than to propagate. This effect could masquerade for the increased cornering energy we observe for the high elastic mismatch case. We would not be able to tell the difference between the oscillation in at the growing kink and an increase in cornering resisitance. In He and Hutchinson[], the oscillations in are noted, but not quantified, because they have no way to infer the actual phase at the nucleating kink. (They simply suggest that the phase be ignored, which is not a good approximation in our results. In our work, we calculate it in terms of the force law range parameter.) In our simulations, the kinking crack always jumps forward, once it has been nucleated, which would be consistent with an oscillatory similar to that shown in Fig. 2 of their paper, but it is also consistent with the existence of a local cornering energy for nucleating the kink. So, it is possible that the additional cornering energy we observe for large elastic mismatch case could be an effect due to oscillations in . That there is a physical cornering resistance, however, is clear, because it occurs for zero elastic mismatch, where the phase oscillation in would disappear.

Figs. 4,5 represent the competition between cleavage on the spur plane and emission on that plane. But we noted in the Introduction that there is also competition with events on the cleavage plane to the right of the spur. Depending on circumstances, it is possible for the crack to either cleave or emit on that plane, as well, as shown in Fig. 1. The total picture is obtained by overlaying events on the interface plane with events on the spur, as shown scematically in Fig. 6. The radius of the Griffith circle for cleavage on the interface is proportional to , and by changing the chemical bonding between the two interfaces, this circle can be expanded or contracted, relative to branching, as shown. If the events on the spur lie below the Griffith circle for cleavage (or for nonblunting emission) on the interface plane, then events on the spur plane will dominate over those on the interface plane. Alternatively, if the interface bonding is weak, then events on the spur plane can dominate. (The reader will now appreciate why it was necessary to choose a strong force law on the interface to explore events on the spur.)

In the Fig. 6, the branching plane stability line is shown crossing the smaller Griffith circle. In this case, events on the upper portion of the Griffith circle for the interface will be stable relative to branching, while for loads on the lower portion, branching will be stable relative to events on the interface plane. Such a degenerate situation can easily be set up in the computer with precisely the results predicted by the diagram, but it requires a careful choice of relative bonding between matrix and interface.

Note that ``crossover'' from the interface plane to the spur plane takes place over a range of loads, but in the particular situation depicted in Fig. 6, there is no combination of loads where the crack will emit a dislocation on the spur plane for the small Griffith circle case. For this particular choice of force laws, then, the crack may cleave or emit on the interface plane, or it may branch onto the spur plane as a cleavage crack. But it will not ever be possible to emit a blunting dislocation, and in this sense, the material is intrinsically brittle. However, there exists a combination of force laws such that the situation depicted in Fig. 7 is realized, and for a particular choice of loads, the crack stability is degenerate-it can either emit or cleave, corresponding to a crossover from brittle to ductile behavior. In previous papers[][], we have worked with pure Mode I loads, and this is seen as a special case of the more general criterion for crossover, where the loading mode is not specified. In Fig. 7, zero elastic mismatch is assumed for clarity, and the upper limit of the branching curve where emission occurs lies precisely on the axis. In general, of course, the emission point does not lie on the axis. But it is true that negative Mode II enhances cleavage on the branching plane, so the upper limit of the branching stability line will tend to be in the vicinity of the axis, if not precisely on it. (This statement, of course, does not apply to the interface case.)

Crossover from cleavage to emission can also be described in terms of the ductility parameter, , defined as

When the material is ductile, brittle otherwise. Because both and are quantities depending on the mixing of the modes, the crossover condition defined in this way is not unique, but depends somewhat on the mode mixity, , as described graphically, above.