Modal analysis of graphene-based structures for large deformations, contact and material nonlinearities.

The nonlinear frequencies of pre-stressed graphene based structures are calculated. These structures are modeled with a nonlinear hyperelastic shell model. The model is calibrated with quantum mechanics data and is valid for high strains. Analytical solutions of the natural frequencies are obtained for the Canham bending model by assuming infinitesimal strains. These solutions are used for verification of the numerical simulation. The performance of the model is illustrated by means of several examples. The modal analysis is performed for a square plate under pure dilatation or uniaxial stretch, a circular plate under pure dilatation or under the effects of an adhesive substrate, and carbon nanotubes under uniaxial compression or stretch. The adhesive substrate is modeled with van der Waals interaction (based on the Lennard-Jones potential) and a coarse grained contact model. It is shown that the analytical natural frequencies underestimate the real ones, and this should be considered in the design of devices based on graphene structures.