Theoretical consideration of a microcontinuum model of graphene

A microcontinuum model of graphene is proposed based on micromorphic theory, in which the planar Bravais cell of graphene crystal is taken as the basal element of finite size. Governing equations including the macro-displacements and the micro-deformations of the basal element are modified and derived in global coordinates. Since independent freedom degrees of the basal element are closely related to the modes of phonon dispersions, the secular equations in micromorphic form are obtained by substituting the assumed harmonic wave equations into the governing equations, and simplified further according to the properties of phonon dispersion relations of two-dimensional (2D) crystals. Thus, the constitutive equations of the microcontinuum model are confirmed, in which the constitutive constants are determined by fitting the data of experimental and theoretical phonon dispersion relations in literature respectively. By employing the 2D microcontinuum model, we obtained sound velocities, Rayleigh velocity and elastic moduli of graphene, which show good agreements with available experimental or theoretical values, indicating that the current model would be another efficient and reliable methodology to study the mechanical behaviors of graphene.