The 2D crystals mechanics: transition from atomic lattice description to the elastic theory equations

A two-dimensional (2D) crystal formed by the system of identical atoms with a pair centrosymmetrical interaction between them is considered. It is suggested that in the initial equilibrium state the atoms occupy the sites of the plane translation-symmetrical lattice and the deformation state appears due to their displacement within the crystal plane (longitudinal deformation) and in the normal-to-it-direction (flexural strain). It is shown that in the continual description to this crystal corresponds an infinitely thin anisotropic film with a finite mass density that can undergo elastic longitudinal and flexural deformations. In the context of classical mechanics are derived the base relations and equations for atomic displacements and the relevant equations of the theory of elasticity that describe the both modes of deformation for the 2D crystal in the linear approximation as well as with the account of anharmonicity. Explicit expressions are obtained which relate the linear and nonlinear elastic moduli of the crystal to the potential of interatomic interaction and the geometrical characteristics of the plane crystal lattice.