Elasticity of Interfacial Cracks.

Figure shows the coordinate system for a crack of length lying on an interface between materials 1 and 2, and emitting a dislocation at an angle, , to the crack and interface plane. Without the dislocation, the shear stress field of the finite loaded with a point load at its center can be written []

is the (complex) load stress intensity factor of the bulk material (i.e. without the interface) written as , and will sometimes be referred to as the ``lab'' stress intensity factor. is the phase angle of the load. The stress intensity factor of the interfacial crack is written as with the connection given above to the shear stress, . This definition for the stress intensity factor differs slightly from that in common use[], but is appropriate for the crack and load geometry in use in this work. In these equations, the additional phase angle at the crack tip generated by the elastic mismatch at the interface is given by . is seen to be a singular logarithmic function of the distance, , from the crack tip, which is the mode mixing anomaly characteristic of interfacial cracks commented on above. is a constant which depends on the elastic mismatch, where and are the standard isotropic elastic parameters for the two materials. The expressions for are given by[]

We will interpret our lattice results in terms of the ``effective G'' criterion introduced by Rice for the Mode I emission in a homogeneous bulk solid[]. That is, we convert the -field for the straight interface crack into the -field for a kinked crack, with the kink lying in the emission direction[]. We then consider the emission process to be equivalent to the growth of the crack in the emission direction as a pure shear kink. The Mode II component of the kinking configurational force on the crack is then set equal to some lattice resistance, . In Rice's formulation, , the unstable stacking fault energy, but ZCT found that is more complicated than that. The Mode II stress intensity factor for the kinking crack in the Cotterell/Rice approximation is given by

Here, we have used the idea of the ``local core phase shift'', , introduced by Zhou and Thomson[]. That is, the phase shift at the ``tip'' of the crack is undefined at because of the logarithmic singularity in the definition of . Thus, the appropriate phase from which to compute the kinked crack stresses is not defined. But the atomic crack has no such difficulties, because the phase in the core region of the crack, where the physics of the crack is determined, lies at some nonzero distance from the mathematical tip. Thus, we define a distance, as the core size, and use that in the expressions above. (We return later to an operational definition of what to choose for .) With this definition of , we note that the phase, , for the kinked crack does not depend on the length of the kink, because the kink lies in homogeneous material, and in this sense, the kink stress field is a ``standard'' crack field, not an interface field with a singular phase shift at its tip. We do have to determine what phase the kink has to start with, because it is starting out of the core of the main interface crack, but once started, the kink no longer possesses the interfacial phase anomaly as a function of its (kink) length.

According to Rice[], in terms of the kinking crack, the emission criterion, , for the Mode I configuration is

Here, is the appropriate elastic modulus for the plane stress crack extension force, and is a lattice resistance, which will be determined from the lattice computations. Since the dislocation is emitted into the ``ductile'' side of the couple, must correspond to that bulk elastic parameter. (In the Rice proposal, .) The plane stress elastic moduli are used because the 2D simulations correspond to that case. Thus, , where and are the usual plane strain elastic constants. Finally, with the previous equations, the critical load modulus, , for emission is given by , and

This last equation will be used to interpret the lattice simulations. In the bulk simulations of ZCT, it was found that , and in the current work, we will again explore the appropriate functionality for . Notice that in (), and are fixed by the elastic mismatch, and by the phase of the external load.

Cleavage on the interface is governed by the simple mechanics crack extension force law,

where is the applied crack extension force, which has been found to be independent of the interface phase effects. In our atomic simulations it is always found that the cleavage criterion is quite strictly given by the quadratic form given above. That is, the criterion is not determined primarily by the local value of [].

The crack extension force above is written in terms of the effective force on the kinking crack. It is more natural to express it in terms of the lab stress intensity factors and lab crack forces, as

In this equation, is expressed in terms of the modulus of the loading stress intensity factor, at the critical load when emission occurs. This form of the criterion contains, explicitly, the various factors for converting to the slip plane or kinked crack system.

A ductility parameter can be defined as the ratio of the square of the critical load stress intensity factors for cleavage and emission,

where is the effective surface energy defined by (). The factor, , contains the kinking and interface physics. When the critical load for emission is equal to that for cleavage, and , the material undergoes a crossover from ductile behavior () to brittle behavior (). For homogeneous material[], where the misfit goes to zero, and the load is pure Mode I, . Thus, the ductility parameter summarizes the crucial and complicated physics of the ductility question in a simple number. The ductility for an interface, of course, depends on the force laws to be used, as well as the load phase, elastic mismatch, and the geometry of the cleavage plane/slip plane.