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

@article{Garroni2015BoundarylayerAO,
title={Boundary-layer analysis of a pile-up of walls of edge dislocations at a lock},
author={Adriana Garroni and Patrick van Meurs and Mark A. Peletier and Lucia Scardia},
journal={arXiv: Analysis of PDEs},
year={2015}
}
• Published 20 February 2015
• Materials Science
• arXiv: Analysis of PDEs
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 $E_\gamma$ modelling the interactions$-$at a typical length-scale of $1/\gamma$$-$of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy $E_\gamma$ in powers of $1/\gamma$ by $\Gamma$-convergence, in the limit $\gamma\to\infty$. While the zero…
13 Citations

## Figures and Tables from this paper

The continuum limit of interacting dislocations on multiple slip systems
In this paper we derive the continuum limit of a multiple-species, interacting particle system by proving a $\Gamma$-convergence result on the interaction energy as the number of particles tends to
Many-particle limits and non-convergence of dislocation wall pile-ups
The starting point of our analysis is a class of one-dimensional interacting particle systems with two species. The particles are confined to an interval and exert a nonlocal, repulsive force on each
Many-particle limits and non-convergence of dislocation wall pile-ups
The starting point of our analysis is a class of one-dimensional interacting particle systems with two species. The particles are confined to an interval and exert a nonlocal, repulsive force on each
Collective dynamics of dislocations
We derive the grand-canonical partition function of straight and parallel dislocation lines without making a priori assumptions on the temperature regime. Such a systematic derivation for
Discrete-to-continuum limits of interacting dislocations
This thesis studies discrete-to-continuum limits of models for interacting dislocations. This analysis contributes to the ultimate goal of obtaining a system of equations which accurately describes
Asymptotic Analysis of Boundary Layers in a Repulsive Particle System
• Mathematics
• 2016
This paper studies the boundary behaviour at mechanical equilibrium at the ends of a finite interval of a class of systems of interacting particles with monotone decreasing repulsive force. This
Elasto-plastic evolution of single crystals driven by dislocation flow
• Materials Science
Mathematical Models and Methods in Applied Sciences
• 2022
This work introduces a model for large-strain, geometrically nonlinear elasto-plastic dynamics in single crystals. The key feature of our model is that the plastic dynamics are entirely driven by the
Boundary-Layer Analysis of Repelling Particles Pushed to an Impenetrable Barrier
• P. Meurs
• Mathematics
SIAM J. Math. Anal.
• 2022
This paper considers the equilibrium positions of n particles in one dimension. Two forces act on the particles; a nonlocal repulsive particle-interaction force and an external force which pushes
Upscaling of the dynamics of dislocation walls
• Physics
• 2014
We perform the discrete-to-continuum limit passage for a microscopic model describing the time evolution of dislocations in a one dimensional setting. This answers the related open question raised ...
Convergence Rates for Discrete-to-Continuum Limits in 1D Particle Systems
We contribute to a recent series of papers on discrete-to-continuum limits of one-dimensional particle systems governed by nonlocal and unbounded interactions. While convergence of the equilibrium

## References

SHOWING 1-10 OF 37 REFERENCES
Asymptotic Behaviour of a Pile-Up of Infinite Walls of Edge Dislocations
• Mathematics
• 2013
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
Upscaling of dislocation walls in finite domains
• Mathematics
European Journal of Applied Mathematics
• 2014
We wish to understand the macroscopic plastic behaviour of metals by upscaling the micro-mechanics of dislocations. We consider a highly simplified dislocation network, which allows our discrete
Asymptotics of Edge Dislocation Pile-Up against a Bimetallic Interface
• Materials Science
• 2009
The approach developed in preceding papers is extended to derive the equilibrium positions of n edge dislocations in a linear pile-up stressed by a constant applied loading against an interface in a
Γ-Limit of a Phase-Field Model of Dislocations
• Mathematics
SIAM J. Math. Anal.
• 2005
A regime corresponding to a diluted distribution of obstacles is considered, by means of Γ-convergence, the asymptotic behavior of a variational problem modeling a dislocation ensemble moving on a slip plane through a discrete array of obstacles.
A 1D Macroscopic Phase Field Model for Dislocations and a Second Order Γ-Limit
• Mathematics
Multiscale Model. Simul.
• 2008
The study under consideration is motivated by the analysis of a variational model for a very important class of defects in crystals, the dislocations, and the derivation of macroscopic models for plasticity.