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

Geers, M.G.D., Peerlings, R.H.J., Peletier, M.A. and Scardia, L. (2013) Asymptotic behaviour of a pile-up of infinite walls of edge dislocations. Archive for Rational Mechanics and Analysis, 209(2), pp. 495-539. (doi: 10.1007/s00205-013-0635-7)

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 $n$ of walls tends to infinity, and characterise this behaviour in terms of $\Gamma$-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter $\beta_n$, corresponding to $\beta_n\ll 1/n$, $1/n\ll\beta_n \ll 1$, and $\beta_n\gg1$, and the two critical regimes $\beta_n\sim 1/n$ and $\beta_n\sim 1$. 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 $\beta_n$.

• Item Type: Articles
• Additional Information: The final publication is available at link.springer.com
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Scardia, Dr Lucia
• Authors: Geers, M.G.D., Peerlings, R.H.J., Peletier, M.A., and Scardia, L.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Archive for Rational Mechanics and Analysis
• Publisher: Springer Verlag
• ISSN: 0003-9527
• ISSN (Online): 1432-0673
• Copyright Holders: Copyright © 2013 Springer Verlag
• First Published: First published in Archive for Rational Mechanics and Analysis 209(2):495-539
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher