Theoretical study of the dependences of the Young's and torsion moduli of thin single-layer carbon zigzag and armchair nanotubes on the geometric parameters

The Young's and torsion moduli of single-layer carbon (m, 0) and (m, m) nanotubes are studied. It is demonstrated that both moduli depend on the chirality, diameter, and length of the nanotube. It is found for the first time that the torsion modulus increases with the nanotube diameter and diminishes with an increase in its length. By considering nanotubes with various values of the diameter-to-length ratio, it is shown that the Young's and torsion moduli of the nanotubes saturate at a diameter-to-length ratio of similar to 0.3.The values of the torsion modulus as calculated from the Young's modulus we obtained and from the deformation energy do not coincide, which can be attributed to the effect of dangling bonds at the open ends of the nanotubes. Energy calculations are performed using the Goodwin modification of the semiempirical Harrison tight-binding method.