Multiple dislocation pile-ups in small grains at small strains: implications for the Hall-Petch relationship and backstress screening

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A classic explanation for the Hall-Petch relationship is given by the stress field of a single dislocation pile-up perpendicular to the grain boundary. Similarly, the gradual compensation of the stress fields of pile-ups on both sides of the boundary has been invoked to explain the transitory effects observed in the stress- strain curves of ultrafine grained polycrystals. This paper studies the effects of introducing deviations of the highly simplified geometry mentioned above, using the proper mathematical approximations of linear elastic dislocation theory. Multiple pile-ups invalidate the conclusions drawn from the single pile-up model. Pile-ups in multiple grains are assessed by a highly idealised model of an infinite array of periodical pile-ups. In the latter case, screening is always perfect. By considering the Peach-Köhler force between dislocations mutually disoriented grains, the magnitude of the fluctuations around such ideal case can be estimated. However, using sound probabilistic arguments to calculate the free path for dislocation slip in fine-grained polycrystals, it is found that the amount of dislocations that can be stored in the pile- ups is generally too small to explain the strong grain size effects observed experimentally.