Applicability and limitations of Donnell shell theory for vibration modelling of double-walled carbon nanotubes

In this paper, the comparison is conducted between two shell theories as applied to the vibrations of double-walled carbon nanotubes (DWCNTs); specifically, the evaluation of the natural frequencies is conducted via Donnell and Sanders shell theories. The discrete DWCNTs are modelled by means of couples of concentric continuous circular cylindrical shells, where van der Waals interaction forces between the two shells are modelled via He's formulation. In order to take into account the chirality of DWCNTs, an anisotropic elastic shell model is used. Simply supported, clamped and free boundary conditions are applied and Rayleigh–Ritz method is adopted to obtain approximate natural frequencies and mode shapes. At the beginning, comparisons with experimental and molecular dynamics data are made, from which it is confirmed that anisotropic elastic shell model is more accurate than isotropic one and it is obtained that Sanders shell theory is more accurate than Donnell one. Then, a parametric analysis considering different values of aspect ratio and numbers of waves along the longitudinal and circumferential directions is carried out in the framework of the anisotropic elastic shell model. From these simulations, it is found that Donnell shell theory yields unsatisfactory results for relatively low numbers of longitudinal and circumferential waves, and for relatively high values of aspect ratio, with respect to Sanders shell theory. Therefore Donnell shell theory cannot be used for vibration modelling of DWCNTs in a large range of longitudinal and circumferential wavenumbers and aspect ratios.