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

Marc Geers, Ron Peerlings, Mark Peletier, Lucia Scardia

Research output: Contribution to journalArticle

37 Citations (Scopus)

### Abstract

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 Γ-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter β n , corresponding to $${\beta_n \ll 1/n, 1/n \ll \beta_n \ll 1}$$, and $${\beta_n \gg 1}$$, and the two critical regimes β n ~ 1/n and β 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 β n .
Original language English 495-539 45 Archive for Rational Mechanics and Analysis 209 2 15 May 2013 https://doi.org/10.1007/s00205-013-0635-7 Published - Aug 2013

### Fingerprint

Edge dislocations
Dislocation
Piles
Asymptotic Behavior
Shear stress
Dimensionless
Horizontal
Energy
Upscaling
Gamma Convergence
Limiting Behavior
Stationary States
Shear Stress
Straight
Forcing
Pairwise
Open Problems
Vertical
Infinity
Tend

### Cite this

Asymptotic behaviour of a pile-up of infinite walls of edge dislocations. / Geers, Marc; Peerlings, Ron; Peletier, Mark; Scardia, Lucia.

In: Archive for Rational Mechanics and Analysis, Vol. 209, No. 2, 08.2013, p. 495-539.

Research output: Contribution to journalArticle

Geers, Marc ; Peerlings, Ron ; Peletier, Mark ; Scardia, Lucia. / Asymptotic behaviour of a pile-up of infinite walls of edge dislocations. In: Archive for Rational Mechanics and Analysis. 2013 ; Vol. 209, No. 2. pp. 495-539.
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abstract = "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 Γ-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter β n , corresponding to $${\beta_n \ll 1/n, 1/n \ll \beta_n \ll 1}$$, and $${\beta_n \gg 1}$$, and the two critical regimes β n ~ 1/n and β 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 β n .",
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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 Γ-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter β n , corresponding to $${\beta_n \ll 1/n, 1/n \ll \beta_n \ll 1}$$, and $${\beta_n \gg 1}$$, and the two critical regimes β n ~ 1/n and β 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 β n .

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