# From atomistic model to the Peierls-Nabarro model with $\gamma$-surface for dislocations

## TL;DR

This paper rigorously demonstrates the convergence of the atomistic model to the Peierls-Nabarro model with a $ ext{γ}$-surface for dislocations in bilayer systems, bridging atomistic and continuum descriptions.

## Contribution

It provides a mathematical proof of the asymptotic closeness between atomistic and Peierls-Nabarro models for dislocations, extending previous convergence analyses.

## Key findings

- Displacement fields of the models are asymptotically close.
- Total energies of dislocation solutions converge.
- Generalizes atomistic to continuum convergence analysis.

## Abstract

The Peierls-Nabarro (PN) model for dislocations is a hybrid model that incorporates the atomistic information of the dislocation core structure into the continuum theory. In this paper, we study the convergence from a full atomistic model to the PN model with $\gamma$-surface for the dislocation in a bilayer system (e.g. bilayer graphene). We prove that the displacement field of and the total energy of the dislocation solution of the PN model are asymptotically close to those of the full atomistic model. Our work can be considered as a generalization of the analysis of the convergence from atomistic model to Cauchy-Born rule for crystals without defects in the literature.