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In the computer simulations, we search for stable configurations of
the crack just before it cleaves on the interface, or just before it emits
a dislocation on the inclined slip plane. The loads are some
combination of tension (Mode I) and shear forces (Mode II)
exerted to the crack surfaces
at the center of the actually open crack plane. The effective
load stress
intensity factor, 
, with its associated phase angle, 
, can be
measured by using the magnitudes of the applied forces at the critical
equilibrium, and the length of the crack from (
). Figure
shows a typical result for the atoms of the cohesive zone of the crack
plus its slip plane.
There are four different paths the crack process can take. The crack
can cleave on the original cleavage plane. It can emit a dislocation
on either the interface plane or the inclined plane. And the crack
can kink or branch onto the inclined plane as a cleavage crack.
Under certain
conditions of loading, primarily for relatively large amounts of
shear, emission will take place ahead of the crack on the cleavage
plane. This case is not explored here, and if the shear load is modest,
then such emission does not occur. (See the previous paper by
Zhou and Thomson[].) For certain force
laws and loading combinations, the crack will form a kink,
and cleave on the
inclined plane. Again, the branching case is not studied here, but
will be the subject of a later paper. The two cases studied here are
emission on the inclined plane and cleavage on the initial plane,
which address the issue of cleavage vs. shear break down of the tip
of the crack with blunting of the tip.
Simulations for three different values of the elastic
mismatch, 
and 
, are carried out.
The first choice
represents quite an extreme value of mismatch, the second moderate
mismatch, and the third represents zero mismatch. In each case,
different range parameters for region 1, region 2, and for
the interfacial connecting bonds between crystal blocks 1 and 2 are assumed.
The range and
spring constant parameters in region 2 determine the unstable stacking
fault, 
in that region and the parameters for the connecting
bonds in the interface determine the interface energy, 
.
The range parameter for material 1 is less important, and was chosen
to be a nominal value. The spring constant in region 2 is normalized
to be unity throughout. The reader will note that for the case
where , the material is not necessarily homogeneous,
because the range parameters of the atoms at the interface and those
on either side may be quite different. That is, such a case corresponds
to a kind of degenerate grain boundary with a different
chemical species segregated there. Although the spring constant of
the interface connecting bonds is at our disposal, we set this
spring constant arbitrarily equal to unity so that 
.
Physically, there is no reason to do this, as 
will in general
be different from either bulk material, but we found that varying this
parameter did not have a strong effect on the results, independent of
.
It is clear that in the interface modeling, a wide range of
parameter space is available for exploration.
There is the elastic mismatch, and
the three bond range parameters which can all be varied at will.
In the context of the earlier work, our focus is on the two
parameters, and 
. But clearly, in the
modeling, we have 5 independent material parameters at our disposal,
as well as the phase, , of the load, so the
problem is over determined in this sense. Reversing the argument,
the question is whether the physical behavior is determined by
only two parameters, and 
, and that is the
question to which we now turn.
First, however, we note that the natural units are to take both the
lattice parameter and a spring constant to unity. As noted above,
. In these units, and this lattice,
and 
.
Next: Core Phase Shift
Up: The Intrinsic Ductility Criterion
Previous: Lattice Model