Despite numerous theoretical investigations on the mechanical properties of graphene, an accurate identification of its material behavior is still unattained. One hypothesis for this uncertainty is that modeling graphene as a static membrane cannot describe the strong coupling between mechanics and thermodynamics of this structure. Therefore, characterization methods built upon static models could not capture these effects. In this paper, we propose a new method for building a reduced order model for the dynamics of thermalized graphene membranes. We apply the proper orthogonal decomposition algorithm on time responses obtained from molecular dynamics simulations. As a result, a set of orthogonal modes is obtained which are then employed to build a reduced order model. The proposed model can describe the motion of the suspended graphene membrane over the whole spatial domain accurately. Moreover, due to its computational efficiency, it is more versatile for exploring the nonlinear dynamics of the system. This model is then employed for studying the nonlinear dynamics of graphene membranes at large amplitudes to extract Young's modulus. The obtained Young's modulus incorporates the effects of nano-scaled thermally induced dynamic ripples and hence, is temperature and size dependent. Our proposed atomistic modal order reduction method provides a framework for studying the dynamics and extracting the mechanical properties of other nano-structures at the molecular level.

The bending rigidity of two-dimensional (2D) materials is a key parameter for understanding the mechanics of 2D NEMS devices. The apparent bending rigidity of graphene membranes at macroscopic scale differs from theoretical predictions at micro-scale. This difference is believed to originate from thermally induced dynamic ripples in these atomically thin membranes. In this paper, we perform modal analysis to estimate the effective bending rigidity of graphene membranes from the frequency spectrum of their Brownian motion. Our method is based on fitting the resonance frequencies obtained from the Brownian motion in molecular dynamics simulations, to those obtained from a continuum mechanics model, with bending rigidity and pretension as the fit parameters. In this way, the effective bending rigidity of the membrane and its temperature and size dependence, are extracted, while including the effects of dynamic ripples and thermal fluctuations. The proposed method provides a framework for estimating the macroscopic mechanical properties in other 2D nanostructures at finite temperatures.