Elastic properties of single-walled carbon nanotubes

Analytical expressions for the velocities of the longitudinal and the torsional 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 elastic hollow cylinder from the theory of elasticity to obtain analytical expressions for the Young's and shear moduli of nanotubes. The calculated elastic moduli for different chiral and achiral (armchair and zigzag) nanotubes using force constants of the valence force field type are compared to the existing experimental and theoretical data.