Third-order elastic moduli of single-layer graphene

In the framework of the Keating model with allowance made for the anharmonic constant of the central interaction between the nearest neighbors mu, analytical expressions have been obtained for three third-order independent elastic constants c (ijk) (mu, zeta) of single-layer graphene, where zeta = (2 alpha - beta)/(4 alpha + beta) is the Kleinman internal displacement parameter and alpha and beta are the harmonic constants of the central interaction between the nearest neighbors and the noncentral interaction between the next-nearest neighbors, respectively. The dependences of the second-order elastic constants on the pressure p have been determined. It has been shown that the moduli c (11) and c (22) differently respond to the pressure. Therefore, graphene is isotropic in the harmonic approximation, whereas the inclusion of anharmonicity leads to the appearance of the anisotropy.