From atomistic model to the Peierls--Nabarro model with \(\gamma\)-surface

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 connection between a full atomistic model and a PN model with \(\gamma\)-surface for the dislocation in a bilayer system (e.g. bilayer graphene). Under some stability condition, we prove that the displacement field of the atomistic model is asymptotically close to that of the dislocation solution of the PN 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.