Convergence from Atomistic Model to Peierls-Nabarro Model for Dislocations in Bilayer System with Complex Lattice

TL;DR

This paper proves the mathematical convergence of the Peierls-Nabarro model to the atomistic model for dislocations in complex bilayer lattices, establishing second-order accuracy and using polynomial approximations of atomic interactions.

Contribution

It provides a rigorous proof of convergence from the atomistic to the continuum PN model for complex bilayer systems, with a focus on accuracy and stability.

Findings

Dislocation solutions of PN model converge to atomistic solutions with second-order accuracy.

The approach uses polynomial approximations to handle complex lattice interactions.

The proof relies on the consistency of the PN model and the stability of the atomistic model.

Abstract

In this paper, we prove the convergence from the atomistic model to the Peierls--Nabarro (PN) model of two-dimensional bilayer system with complex lattice. We show that the displacement field of the dislocation solution of the PN model converges to the dislocation solution of the atomistic model with second-order accuracy. The consistency of PN model and the stability of atomistic model are essential in our proof. The main idea of our approach is to use several low-degree polynomials to approximate the energy due to atomistic interactions of different groups of atoms of the complex lattice.