Using a combination of continuum modeling, atomistic simulations, and numerical optimization, we estimate the flexural rigidity of a graphene sheet. We consider a rectangular sheet that is initially parallel to a rigid substrate. The sheet interacts with the substrate by van der Waals forces and deflects in response to loading on a pair of opposite edges. To estimate the flexural rigidity, we model the graphene sheet as a continuum and numerically solve an appropriate differential equation for the transverse deflection. This solution depends on the flexural rigidity. We then use an optimization procedure to find the value of the flexural rigidity that minimizes the difference between the numerical solutions and the deflections predicted by atomistic simulations. This procedure predicts a flexural rigidity of 0.26？nN ？eV. 1. Introduction A graphene sheet is a one-atom thick lattice of carbon atoms arranged in hexagonal rings; see Figure 1. Graphite, a naturally occurring form of carbon, has the structure of stacks of graphene sheets, and consequently the properties of graphene have long been of interest [1]. In recent years, interest in graphene has intensified, driven by the discovery of novel forms of carbon like buckyballs and single-walled and multiwalled carbon nanotubes. Graphene is the fundamental structure for each of these new forms of carbon. For example, each wall of a nanotube is a graphene sheet wrapped into a cylinder. Figure 1: Schematic of the structure of a graphene sheet. Although graphene is a fundamental structure for other forms of carbon, for many years attempts to synthesize isolated graphene sheets failed [2, 3]. However, recently both mechanical and chemical methods have been developed for isolating individual graphene sheets [4–7]. Isolated graphene is especially exciting for its novel electronic transport properties [8–11], and significant experimental work is currently underway to develop nanoelectromechanical (NEM) systems, such as resonators, switches, and valves, based on isolated graphene [12, 13]. Electronic transport in graphene is coupled to its state of deformation. Hence to design NEM devices, it is essential to understand the basic mechanics of graphene. [14–17]. In this paper, we estimate the flexural rigidity of graphene by comparing atomistic and continuum models of a rectangular graphene sheet initially parallel to and supported by a rigid substrate. The geometry of our problem seems fundamental for the study of the mechanics of graphene because mechanical exfoliation, a technique for isolating individual
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