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

TL;DR

This paper investigates the nonlinear vibrational characteristics of graphene-based structures, incorporating contact and material nonlinearities, using a calibrated hyperelastic shell model validated through analytical and numerical methods.

Contribution

It introduces a calibrated nonlinear hyperelastic shell model for graphene structures and provides analytical solutions for natural frequencies under various conditions.

Findings

Analytical frequencies underestimate real frequencies, important for device design.

Model accurately predicts frequencies for different boundary conditions and loadings.

Performance validated through multiple numerical examples.

Abstract

The nonlinear frequencies of pre-stressed graphene-based structures, such as flat graphene sheets and carbon nanotubes, 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 of various plates are obtained for the Canham bending model by assuming infinitesimal strains. These solutions are used for the verification of the numerical results. The performance of the model is illustrated by means of several examples. Modal analysis is performed for square plates under pure dilatation or uniaxial stretch, circular plates 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.