There were two distinct approaches to the description of dislocations in solids. Often discrete dislocation modelling was too time-consuming and computationally intensive, whereas the solution of the equivalent problem in a continuum approximation could be obtained relatively easily. Although such solutions could provide information about the macroscopic stress it cannot be applied at the scale of the separation between neighbouring dislocations. Robust proof had already been provided for the interconnection between continuum and discrete approaches to dislocation description and a methodology was suggested for analyzing pile-ups of screw and edge dislocations in a uniform material and a pile-up of screw dislocations near to an interface in a bimetallic solid. This was developed further here in order 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 were located with sufficient accuracy that the stress distribution could be evaluated by a simple computational procedure. The stress was determined from a lumped discretisation of super-dislocations away from the interface. An example was presented in which a hundred dislocations could be replaced by just four super-dislocations with only a 1% error consequent in the computation.

Matched Asymptotic Expansion in Modelling of Edge Dislocation Pile-Ups. R.E.Voskoboinikov, S.J.Chapman, J.R.Ockendon: IOP Conference Series - Materials Science and Engineering, 2009, 3[1], 012017