Elastic Properties and Frequencies of Free Vibrations of Single-Layer Graphene Sheets

We determine the basal plane stiffness and Poisson's ratio of single layer graphene sheets (SLGSs) in armchair and zigzag directions by using molecular mechanics simulations of their uniaxial tensile deformations with the MM3 potential, and of their axial and bending vibrations. Both approaches give the basal plane stiffness equal to similar to 340 N/m which agrees well with that reported in the literature and derived from results of indentation experiments on SLGSs and from the first principle calculations. The computed value of Poisson's ratio equals 0.21 in both armchair and zigzag directions. Assuming that the response of a SLGS is the same as that of a plate made of a linear elastic, homogeneous, and isotropic material having Poisson's ratio = 0.21, the in-plane stiffness of similar to 340 N/m and the total mass equal to that of the SLGS, the thickness of the SLGS is found to be similar to 1 angstrom. Thus Young's modulus and the shear modulus of a SLGS equal similar to 3.4 TPa and similar to 1.4 TPa, respectively. It is shown that mode shapes corresponding to the several lowest frequencies of the SLGS differ noticeably from those of an equivalent thin layer made of a linear elastic isotropic material with Young's modulus = 3.4 TPa and the shear modulus = 1.4 TPa. Furthermore, a free- free SLGS vibrates about a plane bisecting its width rather than its thickness as predicted by the Euler-Bernoulli beam theory. We also investigate the effect of pretension on the natural frequencies of SLGSs using MM simulations and correlate it to that of 1 angstrom thick linear elastic plate found by analyzing its three-dimensional deformations. These results will help design SLGS nanomechanical resonators having frequencies in the THz range.