A mathematical methodology for analysing pile-ups of large numbers of dislocations is described. As an example, the pile-up of n identical screw or edge dislocations in a single slip plane under the action of an external force in the direction of a locked dislocation in that plane is considered. As n ! 1 there is a well-known formula for the number density of the dislocations, but this density is singular at the lock and it cannot predict the stress field there or the force on the lock. This poses the interesting analytical and numerical problem of matching a local discrete model near the lock to the continuum model further away. r

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The methodology developed in the precursor to this paper is used to find the positions of n screw dislocations in a pile-up against an interface bonding two crystalline solids. The pile-up is caused by a constant applied stress and, as n ! 1, the dislocations are located with sufficient accuracy to predict the large but finite stress distribution at the interface. Such a prediction is impossible using a conventional continuum dislocation density.