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

Abstract

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 ∼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 ∼340 N/m and the total mass equal to that of the SLGS, the thickness of the SLGS is found to be ∼1 Å. Thus Young's modulus and the shear modulus of a SLGS equal ∼3.4 TPa and ∼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 freefree 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 Å 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.

Transverse vibration of single-layer graphene sheets

“…However, edge effects have been demonstrated to play a significant role in the static [29,34,35,48] and dynamic [33] mechanical properties of SLGS, leading to an equivalent orthotropic, rather than isotropic material model for the graphene. Although the dimensions of the SLGS considered in this work are dissimilar, we note a general agreement in terms of magnitude between the eigenvalues calculated with our MM approach, and the results using the MM3 potential in [21]. Figure 8 shows equivalent plots for the SLGS with cantilevered boundary condition.…”

“…The vibration of single and multiple layer graphene sheets has been investigated by several authors, using continuum mechanics approaches [19], equivalent lattice structures made by atomistic-continuum models representing the C-C bonds [20], and molecular dynamics approaches combined with continuum mechanics for thickness identification [21]. The out-of-plane deformation of SLGS has been considered using the continuum mechanics models [13,22], together with continuum and truss-like structural assemblies [23][24][25][26][27][28][29][30].…”

Mode-I stress intensity factor in single layer graphene sheets

“…We note that Gupta and Batra [31] studied vibrations of free single layer zigzag and armchair graphene sheets by using molecular mechanics simulations with the MM3 potential, equated frequencies so found with those of a continuous structure of the same size as the graphene sheet and found t varying between 0.82 and 1.0 Å. Using different techniques other authors had found t between 0.618 and 3.4 Å.…”

Linear free vibration of graphene sheets with nanopore via Aifantis theory and Ritz method

“…Gupta and Batra (2010) implemented MSM simulation to investigate the effect of pretention on the natural frequencies of SLGS. The comparison between the results obtained by the MSM method and those predicted by an equivalent linear elastic isotropic continuum model showed that there was a noticeable difference between the mode shapes corresponding to the several lowest frequencies of the SLGS and those of an equivalent linear elastic isotropic continuum model (Gupta and Batra, 2010). Sadeghi and Naghdabadi (2010) introduced a hybrid atomistic structural element to model the nonlinear behavior of SLGS.…”