Communications in Mathematical Sciences

Volume 20 (2022)

Number 4

Convergence from atomistic model to Peierls–Nabarro model for dislocations in bilayer system with complex lattice

Pages: 947 – 986

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n4.a2

Authors

Yahong Yang (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong)

Tao Luo (School of Mathematical Sciences, Institute of Natural Sciences, and Qing Yuan Research Institute, Shanghai Jiao Tong University, Shanghai, China)

Yang Xiang (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong)

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 and the total energy of the solution of the PN model converge to those of the full atomistic model with second-order accuracy $O(\varepsilon^2)$, where $\varepsilon$ is a small dimensionless parameter characterizing a wide dislocation core with respect to the lattice constant. 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.

Keywords

dislocations, complex lattice, interpolation polynomial, Peierls–Nabarro model

2010 Mathematics Subject Classification

35Q70, 35Q74, 74A50, 74G10

Received 16 March 2021

Received revised 19 September 2021

Accepted 13 October 2021

Published 11 April 2022