We believe the model of Thomson and Carlsson[]
provides a reasonable physical basis for the cleavage/emission
competition, when the model is extended to incorporate the observed
``lower shelf''
of Fig. 
. Of course, the model is only an approximation
(e.g. the hysteresis of Fig () is not explained), but we
will show that it can form the basis for a discussion of trends, and as a
kind of ``rule of thumb'' even for
more realistic materials than are covered here.
Specifically, it is possible to incorporate the entire set of
simulation results from Figs. ()-() into a single
emission criterion, which is an extension of the form proposed by
Thomson and Carlsson. This equation, expressed for the two
separate low and high regimes, is
where is the generic lattice resistance defined in (
).
The first equation represents the main series of results shown in
Fig. () where the emission is linear in , while the
second represents the lower shelf regime. Note that the second
equation can be written as a pure Rice type of criterion with no
empirical factors, because all the lower plateaus level off at the
observed
. The first of these equations has precisely the form
proposed by Thomson and Carlsson, except that the intercept is not at
the
value. Instead,
appears as the
asymptotic position for the shelf.
These equations can be expressed in terms of the crossover parameter,
, of Eqn (
). For high values of , the
ductility parameter has the form
This equation predicts the anomalous result, first observed by
ZCT[], that the ductility is
independent of , and only depends on
. That is,
the crossover,
, occurs at a critical value of
. Thus, high
dislocation mobility and crack ductility go hand in hand. This result
is not due to any mechanistic connection between dislocation mobility and
the intrinsic ductility of the material, but rather
to the fact that the same
parameters of the force law control both the ductility crossover and
the dislocation mobility.
In the shelf regime of the criterion, where is low, the
ductility parameter takes the more expected form,
Although these criteria are constructed directly from the simulation
results, a separate and partially independent check can be performed
by comparing the observed left hand limiting points of the curves of
Fig. (), where cleavage takes over from emission, with the
computed values of cleavage from the Griffith relation. This check is
displayed in the Table, where the results are deemed to be satisfactory.
Parenthetically, we note that the crossover in the simulations also
exhibit the shifts which
the elastic mismatch ratio in () and ()
predict. That is, increasing the elastic mismatch at the interface
drives the material into the brittle direction.
The most important implication of Eqn. () is that
the ductile/brittle crossover is
independent of in the high
regime.
This result has already been observed in
the earlier work of ZCT, for the completely homogeneous case, and it
is instructive to compare their results with the current case. As
noted earlier, the difference between the present work and ZCT is that
when
, and the elastic mismatch disappears in Fig (
),
there is still a bonding discontinuity at the ``interface'', because the
bonds at the interface are stronger than those of the matrix on either
side. When there is zero elastic mismatch, , and in pure
Mode I loading with
, then for
,
. Setting
to find the condition for the
brittle/ductile crossover,
from (
), whereas ZCT obtained 0.012 for the
crossover critical value.
This result shows that having a bonding inhomogeneity in the lattice does change the ductility crossover criterion significantly from the truly homogeneous case. The criterion is shifted in the ductile direction by the presence of the chemical inhomogeneity. That is, the chemically inhomogeneous material is more ductile than predicted by ZCT in the homogeneous lattice. For the opposite case where the bonds in the interface are weaker, we would expect the shift to be opposite of that above so that the material would be more brittle than the homogeneous results predict.
The major consequence of the deviation from ZCT
is that chemical embrittlement is a subtle effect which goes well
beyond the ability of simple rules such as Eqns. () and
() to describe completely.
The physical reason is that the emission criterion and
ductility parameter must be sensitive to the range (and form) of
the bonding force laws in the vicinity of the interface. This follows
from Thomson and Carlsson's model of the ledge contribution when the
crack is blunted. In their model, the lattice resistance is composed
of a conventional unstable stacking fault term of the Rice form, and a
second term arising from a correction due to the ledge energy.
The ledge correction term contains the ledge suface energy, ,
averaged over a suitable number of bonds as the ledge is created,
multiplied by a
factor which arises from the localization of the emerging dislocation
density in the core. In their paper, Thomson and Carlsson calculate
the localization
from a Peierls dislocation model, which brings in the shear bonding
between lattice planes in the form of
. Both of these
factors in the ledge correction term obviously depend on the range
and form of the force laws. In
the nearest neighbor forces used in the current work,
as well as by Thomson and
Carlsson and ZCT, the ledge energy is averaged
over two bonds crossing the cleavage plane. Thus, when a localized
layer of strong bonds exists, as in the current work, both the
averaging over the ledge energy, and the localization of the emerging
dislocation core will depend on the type, form, and range of the force
law used, and, in general, will lead to results different from the
homogeneous solid.
Indeed, this argument shows that the criterion for ductility involves
a considerable subltety when it comes to judging how important the
ledge term is in comparison to the standard Rice term in the
emission criterion. When the force law has the pair form and is
cut off at the second neighbor distance, as in the current work and
that of ZCT, the ledge correction term is probably maximized. For
example, in metals where the opening of the ledge during cleavage
involves a geometry where the many body character of the force law
would lead to a weak debonding during the formation of a single step
from a pure cleavage crack, the ledge term should be smaller than we
have estimated. Thus, the issue of how important the ledge term is,
will be one which should only be answered in the context of a full
simulation of the physical geometry of the cracking lattice, with
appropriate force laws.
Nevertheless, keeping in mind these provisos and warnings, the general
picture we have developed on the basis of the Thomson/Carlsson
physical model should provide one with considerable insight into the
factors which control the intrinsic ductile/brittle criterion in a
material. There should always be a term associated with the
theoretical shear strength of the solid (), and a term
associated with the ledge formation. Although Rice and his coworkers
have devoted considerable attention to the possible importance of the
tension/shear coupling in the
part of the criterion,
we have found
that the simpler unvarnished (but relaxed)
describes the
results for the set of force laws and lattices we have used. On the
basis of the Thomson/Carlsson model, and on the basis of the
simulations done here and by ZCT, the ledge term appears to be
composed of a product of both
and
. But we
believe that even if this functionality is preserved for more
realistic bonding and lattices, the relative magnitudes of the
pure unstable stacking fault term and the ledge correction term will
depend on the type of bonding. Thus, trends may be discernable from
our results, which are probably applicable to real materials, but
quantitative predictions are probably not justified, at least for
chemical embrittlement situations.
In spite of these provisos and warnings, it will always be true that
cleavage on the interface is governed by the Griffith condition
with the interface , so that weakening the bonding at the
interface will always enhance failure on the interface. But it need
not be a brittle failure, unless
!
Likewise, a strong matrix will always tend to
suppress dislocation emission through the
parameter.
To summarize our general conclusions with a ``rule of thumb''
statement which we believe will survive more extensive investigation:
Equations () and ()
predict two regimes of behavior, and only in the case of low
interfacial bonding, will embrittlement be induced by a simple
lowering of . In general, the behavior will be more
complex, (and interesting!) than that. Exactly what ``low'' and
``high'' mean quantitatively, must be left for more realistic
modeling.
We have learned that the purely interfacial effects from the lattice mismatch can be incorporated into the standard elastic descriptions of interfacial cracks, provided a core phase angle and core stress intensity factor are defined using the range of the force law as the core size.
To return to the question asked at the beginning of this paper, ``Can ductility in a material be determined in terms of bulk material parameters?'', the answer is only provisionally ``yes''-if trends and qualitative features are desired. But localized chemical embrittlement, such as occurs during segregation at an interface, appears to require a full crack simulation for definitive answers.