The lattice modeling uses the same lattice Green's function techniques we have used in the previous interface work[], and the general methodology is given in Thomson, etal[]. Also, we use the same 2D hexagonal lattice used previously[][], with the same set of nearest neighbor pair forces. It might be useful to note here that the use of the 2D lattice is not a simplification of something more general, but the 2D case is exactly the one pertaining to the physics of the problem. The reason is that we are exploring the lattice stability of the straight crack. It is true that at finite temperatures, a dislocation can be nucleated at a stable sharp crack by the generation of a dislocation loop, locally in 3D. But in the present case, we simply wish to explore the temperature independent mechanical stability of the straight crack in the lattice, and not the ease of generation of fluctuations from that state. Thus, this is one of those rare cases in physics where the simple 2D problem is precisely the one of interest. As in previous work, we also justify our use of the hexagonal lattice because it is the lattice which is isotropic in the continuum limit. Since the previous work by Rice[] and the bulk of the interfacial studies are performed in the isotropic limit, we remain in that regime for the work here.
Figure 
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 than that of either bulk, as in the real physical situation.
We will arbitrarily set the elastic constant of material 2 to unity,
since all the physical results scale with the elastic constant. (Of
course, the ratio of the different elastic constants is a physically
important quantity.) Also, we will allow the crack to emit a
dislocation into only material 2, and assume that material 1 is
brittle, incapable of deforming. 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. 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 crack half length.
The present paper differs in one important respect from our earlier simulations. In this work, it is assumed that the atoms on either side of the cleavage plane and those on either side of the slip plane are in the nonlinear zone. That is, these atoms interact with all their neighbors with nonlinear bonds. In previous work, only the bonds crossing the cleavage or slip plane were considered to be nonlinear, but we recently discovered that there were significant nonlinear forces acting through bonds which were assumed to be linear, so we have extended the nonlinear zone to include more bonds. Also, the atoms on the open surface are subject to a particularly subtle and important force when the surface atoms are rotated, as they are near the crack tip. Thus, the crack is always allowed to move well into the nonlinear zone to make these contributions to the crack physics negligible. When the criteria for cleavage and for emission are compared between the current method and the older one, we find that physical trends are preserved, but that small changes in critical parameters of the order of 10%are observed. These errors are not considered damaging to previous conclusions, but they are sufficiently bothersome to make the effort to reduce them further, as we have in the present paper. See the paper by Canel[], etal for further discussion of this point.
Finally, we have adopted another convention in this paper not used in the previous work. In the nonlinear zone on the crack surface, we assume that the atoms are defined to be broken up to some point about half way into the cohesive zone, and begin the cracking at this point. The physical idea is that in standard crack analysis, it is assumed that the crack is made by slitting bonds on the cleavage plane, so that a crack-like singularity can be formed at the crack tip. One needs to do that here as well, because the investigation is focussed on growing cracks, not their nucleation from the perfect lattice. But in the case of the ductile lattice, there is a paradox, because no brittle crack is presumably possible! We get around this both physically and mathematically, by supposing that the brittle crack has been created by some chemical agent which acts at the growing crack tip till it reaches the size from which the computer studies begin. After the brittle crack has thus been ``prepared'', then the assumed bonds of the material come into play, and the crack may either cleave, or emit a dislocation. The only concern is how the core of the crack interacts with the plane on which it can emit a dislocation. In our simulations, emission on only one plane is allowed. This plane is assumed to intersect the crack tip at the exact point where the nonlinear bonds begin to act. We have also explored the effect of allowing the core to build up at the crack tip before it encounters the allowed slip plane, but this introduces only minor effects. Thus the results are quoted for the physical case shown in the figure.
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.
Once the Green's function for the cracked lattice with interface is found[], then the displacement field for the cracked lattice in the linear approximation is given by the ``master'' equation for the Green's function,
If now one recognizes that there are given external forces, , as
well as nonlinear forces acting at bonds stretched into their
nonlinear regimes, then these nonlinear forces can be treated,
mathematically, as external forces, so long as the forces at these
atoms are consistent with the bond stretch (or in general, with the
configuration of the atom and its neighborhood). Thus one can write
where is the force on the atom at position
considered as a functional of that position. This is a set
of nonlinear equations to be solved for the set of atoms having
nonlinear bonds-what we term the cohesive zone of the crack, to be
solved self consistently with the force laws assumed to be operating
in the solid. Use will be made here of the simplest form of these
force laws, nearest neighbor central forces, but the formalism is
quite general. These equations may be solved either by a simple
relaxation program using (
) directly, or by use of a more
efficient energy technique[]. The methodology is accurate,
provided the linear part of the lattice outside the cohesive zone is
only subjected to small scale shear or to small
rotations. (See Canel, etal[] for further discussion of these
restrictions.)
Two kinds of force law are used. The first is 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 second bonding law is a Gaussian generalization of the
UBER,
where is the range parameter. (In actual practice, this law
is too soft in repulsion, and a Uber form of law is welded smoothly
onto it for repulsion displacements.)