Elastic properties of single-walled carbon nanotubes

Analytical expressions for the velocities of the longitudinal and the torsi onal sound waves in single-walled carbon nanotubes are derived using Born's perturbation technique within a lattice-dynamical model. These expressions are compared to the formulas for the velocities of the sound waves in an e lastic hollow cylinder from the theory of elasticity to obtain analytical e xpressions for the Young's and shear moduli of nanotubes. The calculated el astic moduli for different chiral and achiral (armchair and zigzag) nanotub es using force constants of the valence force field type are compared to th e existing experimental and theoretical data.