A self-consistent model for cleavage in the presence of plastic flow

A theory is proposed for cleavage in the presence of plastic flow, in circumstances which do not involve strong viscoplastic retardation of dislocation motion. We build upon recent notions that recognize the large disparity between relevant length scales involved in plastic flow processes around cracks in metals and on metal—ceramic interfaces. The lengths consist of (1) the Burgers vector, (2) the nominal dislocation spacing, (3) the elastic “cell” dimension, and (4) the overall plastic zone size. Of particular interest is the phenomenon of “brittle” crack growth in the presence of pre-existing, apparently mobile, dislocations, which has been observed in several material systems. A continuum elastic—plastic finite element model is utilized that assumes the presence of a dislocation-free strip of elastic material of height D surrounding a crack tip, from which dislocations are assumed not to emit. The parameter D is self-consistently chosen by identifying a maximum equivalent Mises stress in the plastic zone with that predicted by a phenomenological strength law of the type first used by Taylor and Orowan, in which strength varies inversely with nominal dislocation spacing or with cell size, either of which is identified as D in different interpretations of the model. For steady-state crack growth to occur, it is found that the applied energy release rate G must generally be several orders of magnitude larger than the ideal work necessary to separate the interface, at least when D is taken as dislocation spacing. Furthermore, this “shielding” ratio is found to be strongly sensitive to the ideal work of fracture itself, as well as other material properties.