Continuous theory and problems of static bending, natural vibrations and stability of a graphene sheet

Abstract

A beam model of graphene deformations is constructed, in which a continuous-moment model of an elastic thin beam is chosen as a model of an individual beam, the deformation of which follows the “shear plus rotation” concept. By passing to the limit from the beam model, a transition has been made to the continuous model of graphene deformations, which consists of two parts: a) a model of graphene deformations in its plane, b) a model of graphene bending deformations from its plane. It is proved that both deformation models of the graphene are identical, respectively, to the models of plane stress state and bending, the so-called moment-membrane theory of elastic thin plates, constructed on the basis of the moment theory of elasticity with independent fields of displacements and rotations. By comparing the corresponding deformation models of graphene and an elastic thin plate, all six elastic constants of the moment theory of elasticity are determined through the physical parameters of the graphene atomic structure. Further, based on the continuous model of bending deformation of graphene from its plane, the problems of static bending, natural vibration, and stability of a rectangular graphene sheet are studied.