It was recalled that one explanation of the Hall-Petch relationship was that dislocation pile-ups served to enhance the stress which was experienced by grain boundaries. This dislocation pile-up model predicted a d-½ dependence of the yield strength; where d was the grain size. On the other hand, this familiar picture neglected the existence of a threshold stress for dislocation production, as well as the need for a finite-sized dislocation-free region in which a source could operate. Both of these aspects were addressed here within a continuum model for dislocation pile-ups. A closed-form expression was obtained for the dependence of the yield stress upon grain size and source characteristics. This continuum model agreed closely with the corresponding discrete-dislocation model; even when the pile-up contained as few as 10 dislocations.

Continuum Analysis of Dislocation Pile-Ups: Influence of Sources L.H.Friedman, D.C.Chrzan: Philosophical Magazine A, 1998, 77[5], 1185-204