Discrete-to-continuum limits of planar disclinations

TL;DR

This paper develops a mathematical framework for modeling wedge disclinations in materials, deriving a continuum limit from an atomistic model, and demonstrating the recovery of classical disclination results.

Contribution

It introduces a novel atomistic model with boundary conditions for disclinations and proves a discrete-to-continuum limit using relaxation and density theorems.

Findings

Discrete-to-continuum limit established for the disclination energy

Examples of disclinations constructed via numerical minimization

Classical wedge disclination results recovered from the model

Abstract

In materials science, wedge disclinations are defects caused by angular mismatches in the crystallographic lattice. To describe such disclinations, we introduce an atomistic model in planar domains. This model is given by a nearest-neighbor-type energy for the atomic bonds with an additional term to penalize change in volume. We enforce the appearance of disclinations by means of a special boundary condition. Our main result is the discrete-to-continuum limit of this energy as the lattice size tends to zero. Our proof method is relaxation of the energy. The main mathematical novelty of our proof is a density theorem for the special boundary condition. In addition to our limit theorem, we construct examples of planar disclinations as solutions to numerical minimization of the model and show that classical results for wedge disclinations are recovered by our analysis.