Matched asymptotic expansion in modelling of edge dislocation pile-ups

Abstract

There are two distinct approaches to the description of dislocations in solids. Often discrete dislocation modelling is too time-consuming and computationally intensive, whereas the solution of the equivalent problem in a continuum approximation can be obtained relatively easily. Although such solutions can provide information about the macroscopic stress it cannot be applied at the scale of the separation between neighbouring dislocations. We have already provided robust proof of the interconnection between continuum and discrete approaches to dislocation description and suggested a methodology for analyzing pile-ups of screw and edge dislocations in a uniform material and a pile-up of screw dislocations near an interface in a bimetallic solid. Here it is developed further to derive the equilibrium distribution of n edge dislocations in a linear pile-up stressed against an interface in a bimetallic solid. As n → ∞ the dislocation positions are located with sufficient accuracy that the stress distribution can be evaluated by a simple computational procedure. The stress is determined from a lumped discretisation of superdislocations away from the interface. We present an example in which a hundred dislocations can be replaced by just four superdislocations with only 1 % error in the computation.