Continuum analysis of dislocation pile-ups: influence of sources

One explanation of the Hall-Fetch relationship is that dislocation pile-ups serve to enhance the stress felt at grain boundaries. This dislocation pile-up model results in a d(-1/2) dependence of the yield strength, with d the grain size. In fact, this dependence is observed often. However, the traditional pile-up picture neglects two important aspects of pile-up formation: firstly the existence of a threshold stress for dislocation production, and secondly the necessity of a finite-sized dislocation-fi-ee region in which a source may operate. In this paper, both of these aspects are addressed within a continuum model of the dislocation pile-ups. A closed-form expression is obtained for the dependence of yield stress on grain size and source characteristics. The continuum model agrees closely with the corresponding discrete dislocation model even when the pile-up contains as few as ten dislocations.