Analytical prediction of Young's modulus of carbon nanotubes using a variational method

Molecular mechanics and solid mechanics are linked to establish, a nanoscale analytical continuum theory for determination of stiffness and Young's modulus of carbon nanotubes. A space-frame structure consisted of representative unit cells has been introduced to describe the mechanical response of carbon nanotubes to the applied loading. According to this assumption a novel unit cell, given the name mechanical unit cell here is introduced to construct a graphene sheet or the wall of the carbon nanotubes. Incorporating the Morse potential function with the strain energy of the mechanical unit cells in a carbon nanotube is the key point of this study. The structural model of the carbon nanotube is solved to obtain its Young's modulus by using the principle of minimum total potential energy. It was found that the Young's modulus of the zigzag and armchair single-walled carbon nanotubes are 1.42 and 1.30 TPa, respectively. The results indicate sensitivity of the stiffness and Young's modulus of carbon nanotubes to chirality but show no dependence on its diameter. The presented analytical investigation provides a very simple approach to predict the Young's modulus of carbon nanotubes and the obtained results are in good agreement with the existing experimental and theoretical data.