Communications in Mathematical Sciences
Volume 20 (2022)
Convergence from atomistic model to Peierls–Nabarro model for dislocations in bilayer system with complex lattice
Pages: 947 – 986
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.
dislocations, complex lattice, interpolation polynomial, Peierls–Nabarro model
2010 Mathematics Subject Classification
35Q70, 35Q74, 74A50, 74G10
This work was supported by the Hong Kong Research Grants Council General Research Fund 16313316.
Received 16 March 2021
Received revised 19 September 2021
Accepted 13 October 2021
Published 11 April 2022