Strain energy and Young’s modulus of single-wall carbon nanotubes calculated from electronic energy-band theory

The strain energies in straight and bent single-walled carbon nanotubes (SWNT's) are calculated by taking account of the total energy of all the occupied band electrons. The obtained results are in good agreement with previous theoretical studies and experimental observations. Young's modulus and the effective wall thickness of SWNT's are obtained from the bending strain energies of SWNT's with various cross-sectional radii. The repulsion potential between ions contributes the main part of Young's modulus of SWNT's. The wall thickness of the SWNT comes completely from the overlap of electronic orbits and is approximately of the extension of the $\ensuremath{\pi}$ orbit of carbon atom. Both Young's modulus and the wall thickness are independent of the radius and the helicity of SWNT, and insensitive to the fitting parameters. The results show that continuum elasticity theory can serve well to describe the mechanical properties of SWNT's.