A variational model for dislocations at semi-coherent interfaces

TL;DR

This paper introduces a simple variational model for dislocations at semi-coherent interfaces, balancing surface energy and elastic energy, with rigorous derivation and analysis of scaling behaviors.

Contribution

It presents a new continuum variational model derived from semi-discrete dislocation theory, analyzing the energy scaling at semi-coherent interfaces.

Findings

Surface energy scales with interface area.

Elastic energy scales with interface diameter.

Linearized elasticity emerges in large interface limit.

Abstract

We propose and analyze a simple variational model for dislocations at semi-coherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations that compensate the lattice misfit at the interface, and a far field elastic energy, spent to decrease the amount of needed dislocations. We prove that the former scales like the surface area of the interface, the latter like its diameter. The proposed continuum model is deduced from some rigorous derivation from the semi-discrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the far field strain vanishes as the interface size increases.