Atomic lattice structure and continuum plate theories for the vibrational characteristics of graphenes

This paper is concerned with the equivalent extensional and flexural rigidities of a single layer graphene sheet by treating it as a plane lattice structure made of tightly packed carbon atoms into an array of honeycomb-shaped cells. Each carbon atom is modeled as a node with concentrated atomic mass and prescribed six degrees of freedom. The covalent bond between adjacent carbon atoms provides axial, bending, and torsional stiffness. Using the Poisson’s ratio of 0.16 and thickness of 3.4 Å, the equivalent Young’s moduli are found to be approximately 0.112 TPa for bending and in the range of 1.03–1.04 TPa for in-plane modes. Subsequently, the graphene structure is simulated by a classical plate with prescribed geometric and mechanical properties. The in-plane and out-of-plane free vibration analyses of the rectangular plate provide the natural frequencies and associated mode shapes. Results are compared with eigen analyses of the lattice structure model for different sizes of graphene. Examples are considered to show close agreement in the results from these two methods. Mode shapes reveal that the lattice structure model shows symmetry about the horizontal and vertical axes and also about the diagonals.