Analysis.

The Cotterell/Rice[] prescription for the local stress intensity factor of a kinking crack when applied to an interface is given by[]

is the (complex) remote load stress intensity factor of the bulk material (i.e. without the interface) written as . is the phase angle of the remote load. The stress intensity factor of the unkinked interfacial crack is written as with the connection given above to the local stress intensity, , and 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 a singular logarithmic function of the distance, , from the crack tip-the mode mixing anomaly characteristic of interfacial cracks in the continuum limit. is a constant which depends on the elastic mismatch, where and are the standard isotropic elastic parameters for the two materials. The second form for in the last equation in (3) refers to the 2D hexagonal lattice, with the two spring constants, and .

The approximation by Cotterell and Rice[] is to recognize that the shear stress when normalized by the square root of the radius has the dimensions of the stress intensity factor, and when written in terms of the angular variables, and , has the actual character of the appropriate tensile or shear stress intensity factor for the branching crack. This approximation is rigorous, of course, in the limit of small .

Explicit expressions for the 's are given by[]

It is more useful to write these complicated expressions in terms of a set of angular phase shifts in the form

These last equations are written for a critical value of the local stress intensity factor (with a prime) corresponding to emission, cleavage, etc. The total local phase shift for the crack is composed of the separate contributions from the phase shift for the remote load, ; the geometrical contribution due to the kink angle, ; and the interfacial phase shift, , written in terms of the core size, . is a function of the interface mismatch, as well as the kink angle, . Earlier[], we have found that the core size, , can simply be set equal to the range parameter in the force law. The reader should note that the stress field of the kinked crack is not phase shifted by the length of the kink, because the crack is now off the interface in homogeneous material. However, the stress intensity factor in (3) does contain the phase shift[]. The physical reason why the local stress intensity of the kinked crack is phase shifted by the interfacial mismatch is that the kinked crack is ``loaded'' by the phase shifted stress field of the parent interface crack.

The criteria for critical events in terms of the local stress intensity factors are written with the subscripts corresponding to cleavage or emission. Thus

for cleavage, and

for emission. In these equations, an appropriate shear modulus must be taken. If the crack is kinked off the interface into material 2, then , while if the crack is on the interface, then , and the prime on the represents taking the appropriate plane strain or plane stress elastic modulus. In plane stress, which must be used in the 2D simulations, , where and have their standard elasticity definitions. The appropriate ``right hand sides'' for the lattice resistance in these equations are to be determined by computer simulations.

The accuracy of these Cotterell/Rice expressions are easily checked in the zero mismatch case from their paper, where comparisons are graphed as a function of the branching angle[]. The main deviation appears in the value for , where the analytic expression is about 10%higher than the correct numerical value at . All other 's are accurate to within a percent or so.