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

Matteo Strozzi,Oleg V Gendelman,Isaac E Elishakoff,Francesco Pellicano

doi:10.3390/c7030061

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.

Highlights

IntroductionThin-walled mechanical components, such as beams, plates and shells, are adopted as structural elements in several engineering fields, in particular mechanical, aeronautical and aerospace, due to their specific mechanical properties, such as high strength and stiffness, low weight and light inertia
Thin-walled mechanical components, such as beams, plates and shells, are adopted as structural elements in several engineering fields, in particular mechanical, aeronautical and aerospace, due to their specific mechanical properties, such as high strength and stiffness, low weight and light inertia.As an example, monolithic rectangular tanks are used as engineering structures in sewage and water treatment plants, and recreational facilities
The aim of the present study is to propose a new simplified elastic shell model applicable with satisfactory accuracy for linear vibrations of double-walled carbon nanotubes (DWCNTs) in a large range of geometries and wavenumbers, including cases in which membrane and moment shell models result in large relative errors

Summary

Introduction

Thin-walled mechanical components, such as beams, plates and shells, are adopted as structural elements in several engineering fields, in particular mechanical, aeronautical and aerospace, due to their specific mechanical properties, such as high strength and stiffness, low weight and light inertia. Monolithic rectangular tanks are used as engineering structures in sewage and water treatment plants, and recreational facilities. In most cases they are made with constant thickness walls, but adoption of variable thickness (e.g., trapezoidal crosssection) walls is justified in structures where hydrostatic pressure acts on the walls [1,2]. The wide application of thin shell structures in engineering field is due to their very high load-carrying capability, strength-to-weight ratio and structural stiffness. Thin shell structures are able to efficiently support applied external forces by virtue of their characteristic spatial curvature, and, as a result, thin shells are much stronger and stiffer than all other structural forms [4]

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