A crack on an interface in a material can take any of four different event paths: It can cleave straight ahead on the interface plane, emit a dislocation on this plane, or it can branch in cleavage on an inclined cleavage plane, or it can emit a dislocation on an inclined slip plane. Depending on the symmetry of the lattice and the geometry of the crack line and cleavage plane, a number of slip or branching planes may be available.
The subject of this paper is to give a
unified presentation of the criteria for these various events, and
introduce a graphical representation from which it is possible to pick
off what path a crack under a particular (mixed) load will take, given
that the force laws and crystal type are known. Since it is
impossible to give criteria for the general crystal including all
possible bonding types, we will illustrate the process with a generic
2D hexagonal lattice with a coherent interface, using UBER and
associated pair forces. The criterion for branching is presumably
associated with a Griffith condition with an appropriate
crack driving force (energy release rate) for the particular plane.
The emission condition is more complicated, and will be analyzed in
terms of Rice's unstable stacking fault energy[],
and its extensions[][][].
Both emission and cleavage events must be a balance between a driving
force for the event, and a lattice resistance to the event. Since
branching is a cleavage process, the driving force must be the
standard energy release rate on the branching plane, written in terms
of the local stress intensity factors, , for the branching crack
in the limit of zero length for the kink.
(We will alternatively refer to the branching event as a kinking
of the original crack, and the emerging crack on the branching plane
as a kink.) The limiting local stress intensity factors for the kink
are not obtainable from the stress intensity factors of the main crack
in an analytic form, but must be obtained
numerically[][][][]. The more complicated interfacial case
has been analyzed by He and Hutchinson[]. Even though analytic
results are not possible, Cotterell and Rice[] have shown that
the kink stress intensity factors are closely related to the stress
fields surrounding the main crack, and have written an analytic
approximation for the local
's which is remarkably accurate. Since
the continuum approximation introduces inaccuacies of its own, the
Cotterell/Rice expressions may be as close as one can come to these
quantities in the continuum approximation, and will be used as
the analytic basis for our lattice comparisons.
The correct driving force for cleavage has not been clearly identified until lattice studies[] showed it to be given by the standard mechanics expression
where is an appropriate elastic shear modulus. It has
sometimes been claimed[] that the cleavage condition should be
predominantly a
matter of the Mode I loading, because the opening mode is the only one
to actually force the cleavage planes apart; otherwise the crack should
simply reclose itself.
But the lower limit for
is actually determined
by the fact that the crack will break down in shear and emit a
dislocation when the shear load is sufficiently great on the Griffith
circle, and for shear loads below this
limit, the criterion is given by the
formula above. Thus,
contrary to He and Hutchinson[], we will always take the local
on the kinking crack as the appropriate driving force. When this
quadratic expression is transformed back to the remote loading stress
intensity factors, of course, a more complex quadratic form in
and
results.
The appropriate driving force for emission has been proposed by Rice[]
to be the Mode II component of , or
Again, when this equation is transformed back to the remote or lab system of stress intensity factors, a more complex form results, to be discussed in §III.
The lattice resistance for an event will be computed in the simulations to be reported. One would expect that the lattice resistance for cleavage should simply be the intrinsic surface energy of the lattice, but we will show that a special resistance due to turning the corner of the kink apparently comes into play, so that the standard Griffith relation is not satisfied exactly in this case.
For emission, the lattice resistance has been the subject of intense
study, and it has been shown[]
that when the crack is not blunted by the
emission, the lattice resistance is simply given by the unstable
stacking fault, , as proposed by Rice[]. When the
crack is blunted by the emission, however, the lattice resistance is
more complex, and involves a product of
, as shown
by Zhou, Carlsson and Thomson[] and by Thomson and
Carlsson[]. Blunting emission for the interface was studied by
Thomson[].
The lattice modeling will be done with the methodology described elsewhere[][] for the 2D hexagonal lattice with nearest neighbor central force laws, and the reader is referred to the references for the details of the simulations. The reason for choosing the hexagonal lattice is that it is elastically isotropic in the continuum limit, so anisotropic corrections need not be considered for the analysis to be carried out. Also, our purpose is to illucidate the generic physics of cracks in lattices, and a method which gives quick results for a variety of different force laws in a simulation where the load mix and lattice mismatch can be varied easily is desired.