  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 the atomically thin membrane. In this paper, we perform
modal analysis to estimate the effective macroscopic 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 at high frequencies in other
two-dimensional nano-structures at finite temperatures.
