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The bending rigidity of two-dimensional (2D) materials is a key parameter forunderstanding the mechanics of 2D NEMS devices. The apparent bending rigidityof graphene membranes at macroscopic scale differs from theoretical predictionsat micro-scale. This difference is believed to originate from thermally induceddynamic ripples in the atomically thin membrane. In this paper, we performmodal analysis to estimate the effective macroscopic bending rigidity ofgraphene membranes from the frequency spectrum of their Brownian motion. Ourmethod is based on fitting the resonance frequencies obtained from the Brownianmotion in molecular dynamics simulations, to those obtained from a continuummechanics model, with bending rigidity and pretension as the fit parameters. Inthis way, the effective bending rigidity of the membrane and its temperatureand size dependence, are extracted, while including the effects of dynamicripples and thermal fluctuations. The proposed method provides a framework forestimating the macroscopic mechanical properties at high frequencies in othertwo-dimensional nano-structures at finite temperatures.

Modal analysis for determining the size-and temperature-dependent bending rigidity of graphene