Boundary-layer analysis of a pile-up of walls of edge dislocations at a lock

Abstract

In this paper we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum non-local energy Eγ modelling the interactions-at a typical length-scale of 1/γ-of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy Eγ in powers of 1/γ by Γ-convergence, in the limit γ → ∞. While the zero-order term in the expansion, the Γ-limit of Eγ , captures the 'bulk' profile of the density of dislocation walls in the pile-up domain, the first-order term in the expansion is a 'boundary-layer' energy that captures the profile of the density in the proximity of the lock.This study is a first step towards a rigorous understanding of the behaviour of dislocations at obstacles, defects, and grain boundaries.

Many-particle limits and non-convergence of dislocation wall pile-ups

“…The limiting energy of the intermediate regime 1 α n n has the porous medium equation as its gradient flow (see [Oel90] for V regular enough). The minimisers exhibits intricate boundary layers [HCO10,GvMPS16], and again the proof for the many-particle limit differs substantially. A more detailed discussion on all scaling regimes of α n and their connection is given in [GPPS13,SPPG14].…”

Asymptotic Analysis of Boundary Layers in a Repulsive Particle System

“…Our study takes place in the context of a variety of recent mathematical results aiming to better understand surface effects in similar particle systems, notably [HCO10,GvMPS15] in the setting of dislocation pile-ups. While [HCO10] focuses on the case in which the interaction potential is homogeneous, and [GvMPS15,Hal11] studies boundary layers in a continuum model for the x(0) = 0…”

“…particle density, our contribution is the derivation of a discrete description of the boundary layer for a general class of interaction potentials. Two particular examples that we have in mind are pile-ups of dislocation walls [GvMPS15] and pile-ups of dislocation dipoles [HCO10].…”

“…The numerical scheme approximates (7) by assuming that χ(i) = χ(i−1)+1 for all i larger than a fixed index. We compare its solution to the minimiser x n of (2) for various numbers of n particles and for two physically relevant choices for V : the case of dislocation dipoles ([HCO10], V (x) = x −2 ) and dislocation walls ( [GvMPS15], V has a logarithmic singularity at 0 and tails which vanish exponentially fast). We observe that the convergence rate of nx n (i) to χ(i) as n → ∞ is close to O(n −1 ), and independent of i.…”

Regularity of the minimiser of one-dimensional interaction energies

“…In Section 7 we demonstrate the applicability of Theorem 1.4 and Theorem 1.5 by lifting the limitations in (A1)-(A3) and extending the class of potentials U in the result of [DKM98]. In a future publication, we use Theorems 1.4 and 1.5 to pursue (A2) and to tackle the open problem on the discrete part of the boundary layer result in [GvMPS16].…”