Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes

Axial, torsional and radial breathing mode (RBM) vibrations of free–free unstressed (i.e., relaxed) single-walled carbon nanotubes (SWCNTs) of different helicities having aspect ratio (length/diameter) of about 15 have been studied using the MM3 potential. It is found that for axial and torsional vibrations, frequencies of the second and the third modes of SWCNTs equal, respectively, twice and three times that of the corresponding first mode. A similar relation also holds for axial and torsional vibrations of a homogeneous linear elastic prismatic body. The RBM frequencies are also used to validate computed frequencies of SWCNTs.Equivalent continuum structures (ECSs) whose frequencies in axial, torsional and radial breathing modes are equal to those of the SWCNTs are identified. The consideration of free ends eliminates the effect of boundary conditions and avoids resolving equivalence between boundary conditions in the analysis of SWCNTs and their ECSs. It is found that the ECS made of a linear elastic homogeneous material is a cylindrical tube of mean radius and length equal to those of the SWCNT. Poisson’s ratio of the ECS (and hence of the SWCNT) computed without assuming any value for the wall thickness converges with an increase in the diameter of the SWCNT to 0.20 for armchair, 0.23 for zigzag, and 0.21 for chiral tubes. For a wall thickness of the ECS equal to 3.4 Å, Young’s modulus of the material of the ECS (and hence of the SWCNT) equals ∼1 TPa and is independent of the helicity and the diameter of the SWCNT. However, the shear modulus varies with the diameter and the helicity of the underlying SWCNT. In the more common terminology, normal modes of vibrations of a SWCNT give Young’s modulus of ∼1 TPa in the axial and the circumferential directions, and the shear modulus of ∼0.4 TPa. Whereas Young’s modulus of a SWCNT is found to be independent of its diameter and helicity, the shear modulus depends weakly upon them.