Applicability and Limitations of Simplified Elastic Shell Theories for Vibration Modelling of Double-Walled Carbon Nanotubes

Abstract

The applicability and limitations of simplified models of thin elastic circular cylindrical shells for linear vibrations of double-walled carbon nanotubes (DWCNTs) are considered. The simplified models, which are based on the assumptions of membrane and moment approximate thin-shell theories, are compared with the extended Sanders–Koiter shell theory. Actual discrete DWCNTs are modelled by means of couples of concentric equivalent continuous thin, circular cylindrical shells. Van der Waals interaction forces between the layers are taken into account by adopting He’s model. Simply supported and free–free boundary conditions are applied. The Rayleigh–Ritz method is considered to obtain approximate natural frequencies and mode shapes. Different aspect and thickness ratios, and numbers of waves along longitudinal and circumferential directions, are analysed. In the cases of axisymmetric and beam-like modes, it is proven that membrane shell theory, differently from moment shell theory, provides results with excellent agreement with the extended Sanders–Koiter shell theory. On the other hand, in the case of shell-like modes, it is found that both membrane and moment shell theories provide results reporting acceptable agreement with the extended Sanders–Koiter shell theory only for very limited ranges of geometries and wavenumbers. Conversely, for shell-like modes it is found that a newly developed, simplified shell model, based on the combination of membrane and semi-moment theories, provides results in satisfactory agreement with the extended Sanders–Koiter shell theory in all ranges.

Nonlocal anisotropic elastic shell model for vibrations of double-walled carbon nanotubes under nonlinear van der Waals interaction forces

“…, 𝛾 / "%,! ) at a generic point of the 𝑖-th cylindrical shell are related to the dimensionless middle surface strains and changes in curvature and torsion of the 𝑖-th shell by means of the following relationships [20]:…”

“…The dimensionless elastic strain energy 𝑈 B ! of the 𝑖-th cylindrical shell, which models the single-walled carbon nanotube, under plane stress hypothesis (𝜎 & = 0), can be expressed as follows [20]:…”

“…The dimensionless kinetic energy of the 𝑖-th cylindrical shell, which models the single-walled carbon nanotube, by neglecting the rotary inertia effect, is given by [20]:…”

“…The dimensionless van der Waals interaction energy of the i-th cylindrical shell, which models the single-walled carbon nanotube, is expressed as follows [20]:…”

“…In these models, actual discrete CNTs are replaced by equivalent continuous homogeneous structures, ignoring their intrinsic atomic nature and therefore reducing the number of degrees of freedom. Both beam-like [11][12][13][14][15] and shell-like [16][17][18][19][20] continuous models have been proposed.…”