Asymptotic expressions for the nearest and furthest dislocations in a pile-up against a grain boundary

In 1965, Armstrong and Head (Acta Metall. 13(7):759–764, 1965) explored the problem of a pile-up of screw dislocations against a grain boundary. They used numerical methods to determine the positions of the dislocations in the pile-up and they were able to fit approximate formulae for the locations of the first and last dislocations. These formulae were used to gain insights into the Hall-Petch relationship. More recently, Voskoboinikov et al. (Phil. Mag. Lett. 87(9):669-676, 2007) used asymptotic techniques to study the equivalent problem of a pile-up of a large number of screw dislocations against a bimetallic interface. In this paper, we extend the work of Voskoboinikov et al. to construct systematic asymptotic expressions for the formulae proposed by Armstrong and Head. The further extension of these techniques to more general pile-ups is also outlined. As a result of this work, we show that a pile-up against a grain boundary can become equivalent to a pile-up against a locked dislocation in the case where the mismatch across the boundary is small