The mechanics of 2D crystals: A change from the atomic-lattice description to equations of the elasticity theory

Abstract

A two-dimensional (2D) crystal formed by a system of identical atoms with a pair centrosymmetric interaction between them is considered. It is assumed that in the initial state of equilibrium atoms occupy sites of a flat translation-symmetrical mesh, and the deformed state appears as a result of their displacements in the crystal plane (longitudinal deformations) and in the direction perpendicular to it (flexural deformations). It is shown that in the continuum description an infinitely thin anisotropic film with a finite mass density, which is capable of elastic longitudinal and flexural deformations, corresponds to this crystal. In the framework of classical mechanics we derive the basic relations and equations for atomic displacements and corresponding to them equations of the elasticity theory, describing both modes of deformation of a 2D crystal in the linear approximation as well as with taking into account anharmonicities. The explicit expressions which relates moduli of linear and nonlinear elasticity of the crystal with the potential of interatomic interaction and geometrical characteristics of the flat crystal lattice are obtained.