# Uniform distribution of dislocations in Peierls-Nabarro models for semi-coherent interfaces

created by fanzon on 13 Aug 2019
modified on 30 Jul 2020

[BibTeX]

Published Paper

Inserted: 13 aug 2019
Last Updated: 30 jul 2020

Journal: Calculus of Variations and Partial Differential Equations
Year: 2019

ArXiv: 1908.04222 PDF

Abstract:

In this paper we introduce Peierls-Nabarro type models for edge dislocations at semi-coherent interfaces between two heterogeneous crystals, and prove the optimality of uniformly distributed edge dislocations. Specifically, we show that the elastic energy $\Gamma$-converges to a limit functional comprised of two contributions: one is given by a constant $c_\infty>0$ gauging the minimal energy induced by dislocations at the interface, and corresponding to a uniform distribution of edge dislocations; the other one accounts for the far field elastic energy induced by the presence of further, possibly not uniformly distributed, dislocations. After assuming periodic boundary conditions and formally considering the limit from semi-coherent to coherent interfaces, we show that $c_\infty$ is reached when dislocations are evenly-spaced on the one dimensional circle.