Continuum and discrete models of dislocation pile-ups. II. Pile-up of screw dislocations at the interface in a bimetallic solid

Abstract

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.

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

“…The former situation is identical to the problem investigated by Voskoboinikov et al [8] and we analyse it first.…”

“…This implies the situation of a pile-up against a lock and a grain boundary, which we discuss in Section 3. As described by Voskoboinikov et al [8], there are three spatial scalings to consider for (1). In the region where x is sufficiently far from the boundary layers near x = 0 and x = x n , the spatial variable can be rescaled by introducing x = n ξ to recover a singular integral equation for the leading-order dislocation density, ρ 0 (ξ).…”

“…Since this dislocation will lie in a boundary layer region near x = 0, it requires a different spatial scaling from the one described above. Following Voskoboinikov et al [8], we introduce an inner scaling of the form x = n −a/(1−a) η, where a is defined as above. As shown in [8], the leading-order inner equation takes the form subject to the matching condition…”

“…The strength of the image dislocation is an explicitly known quantity that depends on the degree of elastic anisotropy in the crystal and on the misorientation across the boundary. Analogous pile-ups of screw dislocations against a bimetallic interface have recently been investigated by Voskoboinikiov et al [8] in the asymptotic limit as the number of dislocations tends to infinity. In [1], numerical methods were used to solve the equations of equilibrium for pile-ups of up to 64 dislocations.…”

“…In all cases described, it was found that the dimensionless position of the first dislocation could be simply expressed in terms of the number of real dislocations in the pile-up and the comparative strength of an image dislocation relative to a real dislocation. In this paper, the results of Voskoboinikov et al [8] are used to develop asymptotic expressions for the positions of the first and last dislocations in a pile-up of screw dislocations against a grain boundary. These expressions give a good justification for the formulae obtained in [1], but we also find some important differences.…”