Elastic constants and phonon dispersion relation analysis of graphene sheet with varied Poisson's ratio

Through the achievement of elastic parameter equivalence by the structural mechanics method, the beam–truss space frame model for graphene sheet has been simplified; moreover a representative unit cell of the periodic graphene structure has been extracted. On this basis, the Bloch theorem is combined with the finite element method to determine the phonon dispersion relation of graphene. For the widely used thickness values of the carbon–carbon (C–C) bond, the aspect ratios (thickness/length) are so large that the shear deformation should be considered such that as a result, the negative Poisson's ratio (NPR) characteristic of the C–C bond is observed. Analytical expressions for the elastic constants of graphene sheet have been obtained with varied C–C bond Poisson's ratios. The reasonable coherence with previous results reveals the effectiveness of the structural mechanics method for analysing the mechanical properties of nanostructures. The effect of deformation modes on the phonon dispersion relation of graphene sheet is further discussed; it is concluded that the acoustic phonon modes are sensitive to the deformation modes of the C–C bond, whereas these deformation modes negligibly influence the optical phonon modes. The numerical results also reveal that the shear deformation and the resulting NPR effect play an important role in the mechanical property analysis of graphene sheet. To summarise, the interrelation between NPR materials and nano materials has stimulated a wide field of research; moreover, numerous novel phenomena have been observed.