Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and inextensional modes of vibration

Abstract

Abstract Free vibrations of thirty-three armchair, zigzag and chiral single-walled carbon nanotubes (SWCNTs) of aspect ratios (length/diameter) between ∼3 and ∼15 and having free ends have been studied using the MM3 potential. It is found that these tubes exhibit Rayleigh and Love inextensional modes of vibration. The lowest natural frequency of a mode of vibration corresponds to a circumferential wave number greater than one. Recall that a cylindrical shell of small aspect ratio and comprised of a linear elastic and isotropic material also exhibits the Rayleigh and the Love inextensional modes of vibration. In order to quantitatively compare frequencies of a shell with those of a SWCNT, we find geometric and material parameters for the shell in two ways. In the first approach, we require that the lowest Rayleigh and the radial breathing mode frequencies and the lowest frequencies of the axial and the torsional modes of vibration of a SWCNT match with the corresponding ones of the shell having length and mean diameter equal to those of the SWCNT. In the second technique, we account for the transverse inertia effects, and equate frequencies of the lowest Love, axial and torsional modes of vibration of a SWCNT to that of a shell. Each one of these two methods determines Young’s modulus and Poisson’s ratio of the material of the shell and its thickness, and enables us to explore similarities and differences between vibrations of a shell and of a SWCNT. It is found that the two techniques give very close results for the material and the geometric parameters of the shell, and hence of the SWCNT. The SWCNT thickness increases from ∼0.88 A to 1.37 A when the tube radius is increased from ∼3.6 A to 15 A and stays at 1.37 A for further increases in the tube radius. The wall thickness is essentially independent of the tube chirality. We use these results to provide an expression for the wall thickness in terms of the tube radius and the bond length in the initial relaxed configuration of a SWCNT. We also compare higher vibrational modes of shells and hollow cylinders with those of the corresponding SWCNTs. For a shell we use a first-order shear deformable shell theory (FSDST). Frequencies of a shell using the FSDST and of the hollow cylinder using the three-dimensional linear elasticity theory are computed with the finite element method. It is found that for low to moderate circumferential and axial wave numbers frequencies and mode shapes of the shell and of the hollow cylinder agree well with those of the corresponding SWCNT computed with the MM simulations.