It was explained in §II that was a core parameter to be
determined from the atomic simulations. There are several ways to
address this question. Rice, Suo and Wang[] simply
assume a cut-off at the lattice parameter in the elastic
equations. However, with the results of the simulations in hand,
other options exist, because one can actually try to measure the phase
shift in the core, and hence calculate the correct cut-off from
(
). This is the approach taken in the earlier Mode II
paper on interfaces[]. In the present case, we can do
this either on the inclined plane or on the cleavage plane.
Unfortunately, we found that there was no clear way to infer the
elastic core phase from the actual configurations because of various
nonlinear interactions in the actual core, which go beyond the elastic
picture.
Our best explanation for this failure is to note that the phase shift is an
elastic concept. It will be affected by the actual nonlinearities
which the crack configuration generates. If the crack opening behind
the crack is sufficiently large, then the core on the inclined plane
will reflect this opening, in a way not anticipated by the elastic solution.
For this reason, it would be wrong to measure the core phase in the
inclined plane, and infer in (
) from it. In addition,
when the core phase is measured on the cleavage plane, there will be
elastic dislocation shielding effects from nonlinearities occurring in
the inclined plane, which will alter the observed shear in the
cleavage plane core. So there is little in the actual core
structures which accurately reflects the elastic prediction for
the phase. Nonetheless, the actual cores do exhibit the predicted
phase shifts in a qualitative way, in the sense that shear loadings
are certainly induced by the elastic mismatch, especially when the
lattice mismatch is large enough not to be masked by core
nonlinearities.
In particular, when , the
phase in the cleavage plane core is not swamped by the shielding in
the inclined plane if the shear in that plane is not allowed to grow
too large by controlling the loads on the crack. In this
case, we could get quantitative measures of the core shift, which will
be used in the discussion of crack length scaling effects. But for
moderate values of elastic mismatch, any attempt to measure the core
phase is doomed to failure, because of the interactions between the
different contributions to the nonlinear core configuration.
What can one do? We remember that the purpose of this work is to
see to what extent one can understand the actual crack in terms of
elastic concepts, and the various lattice breakdown energies.
We have argued that the phase shift which appears in the elastic
analysis for the emission criterion should be measured at the
core radius, as a cutoff for the elastic theory, suggesting that some
version of the Rice, Suo and Wang[] approach might be
appropriate. In our
simulations, we find that the actual phase shifts in the cores, when
they can be measured, are force law dependent. So we have adopted
an empirical rule to set the core size in () equal to the range
parameter, , in the Uber force law.
In the Gaussian force law, we set
, because
has the dimensions of length.
The efficacy of this rule will be demonstrated entirely on the basis
of its utility in comparing elastic predictions with simulation results.