TY - JOUR

T1 - Asymptotic behaviour of a pile-up of infinite walls of edge dislocations

AU - Geers, M.G.D.

AU - Peerlings, R.H.J.

AU - Peletier, M.A.

AU - Scardia, L.

PY - 2013

Y1 - 2013

N2 - We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional horizontal positions x_i > 0 of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x = 0 that prevents the walls from leaving through the left boundary. We study the behaviour of the energy as the number of walls, n, tends to infinity, and characterise this behaviour in terms of G-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter ß n , corresponding to ß_n>1 , and the two critical regimes ß_n ~ 1/n and ß_n ~ 1. As a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes. The results shed new light on the open problem of upscaling large numbers of dislocations. We show how various existing upscaled models arise as special cases of the theorems of this paper. The wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities. This additional information is encoded in the limit of the dimensionless parameter ß_n .

AB - We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional horizontal positions x_i > 0 of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x = 0 that prevents the walls from leaving through the left boundary. We study the behaviour of the energy as the number of walls, n, tends to infinity, and characterise this behaviour in terms of G-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter ß n , corresponding to ß_n>1 , and the two critical regimes ß_n ~ 1/n and ß_n ~ 1. As a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes. The results shed new light on the open problem of upscaling large numbers of dislocations. We show how various existing upscaled models arise as special cases of the theorems of this paper. The wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities. This additional information is encoded in the limit of the dimensionless parameter ß_n .

U2 - 10.1007/s00205-013-0635-7

DO - 10.1007/s00205-013-0635-7

M3 - Article

VL - 209

SP - 495

EP - 539

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -