Effect of Crack Geometry on Dislocation Nucleation and Cleavage Thresholds

A continuum model based upon the Peierls-Nabarro description of a dislocation ahead of a crack is used to evaluate the critical mode I loading for dislocation nucleation at the tip of a finite, pre-blunted crack. A similar approach is used to evaluate the critical mode I loading for atomic decohesion. Results are presented for various crack tip root radii (a measure of bluntness), for several crack lengths. It is shown that increasing the crack length increases the critical energy release rate for both material behaviors. Increasing the bluntness of a crack tip always increases the required loading for atomic decohesion but nucleation thresholds are initially decreased by very small increases in crack tip bluntness. Nucleation thresholds are later increased after reaching significant crack tip blunting. Implications for ductile versus brittle competition are discussed by comparing the ongoing competition between these two different material behaviors. INTRODUCTION In 1974, Rice and Thomson (1) presented their well-known continuum model for quantifying the competition between dislocation nucleation and atomic decohesion. The basic approach was to evaluate the critical energy release rate of cleavage, given by the Griffith criterion, and to evaluate the critical energy release rate for the emission of a dislocation on a slip plane intersecting the crack tip (2). The event possessing the lowest critical energy release rate was predicted to be the dominant outcome. Emission of a dislocation implied that the material would continue to e mit dislocations thus shielding the stress singularity and preventing further brittle fracture. In contrast, if cleavage was predicted, brittle fracture would persist. The major grievances with the Rice-Thomson model are that it generally only provides a qualitative prediction of the material behavior, it does not address the actual nucleation event but instead calculates the energy to move a well developed dislocation, it uses a vaguely defined core cut-off parameter as the equilibrium formation position, and the model always assumes a perfectly sharp crack. Rice (3) addressed several of these problems when he presented a model that utilized a Peierls-Nabarro (4,5) dislocation description. Assuming a periodic relationship between the shear stress and slip displacement along a slip plane intersecting the crack tip, this model solves for a distribution of infinitesimal slip displacement that results from far field loading using a nonlinear integral equilibrium equation. Eventually, an instability is reached and the integral equation can no longer be solved for increased applied loading. This instability marks the nucleation of a dislocation from the crack tip. The efforts of many groups have continued to build upon and improve methods for evaluating the critical energy release rate of dislocation nucleation (6,7), but the sharp crack assumption is still universally adopted as the incipient dislocation forms. Results from various groups in the atomistic community suggest that crack tip geometry can have a substantial quantitative effect on the energies of nucleation and cleavage (8,9,10,11). Physically speaking, an atomically sharp crack is rare, and this is motivation enough to study the effects of crack geometry using continuum analyses. This paper examines the effects of crack tip blunting and crack length on the critical mode I loading for edge dislocation nucleation and atomic decohesion at the tip of a pre-blunted crack. THEORY To study the effect of crack geometry, an elliptically shaped crack is used as shown in Figure 1. Prescribing the major and minor axes of the ellipse defines a crack length and a crack tip root radius of curvature, ρ=d 2 /a, at the tip of the ellipse thereby approximating a blunt crack configuration.