Linear vibrations of triple-walled carbon nanotubes

Matteo Strozzi,Francesco Pellicano

doi:10.1177/1081286517727331

Abstract

In this paper, the linear vibrations of triple-walled carbon nanotubes (TWNTs) are investigated. A multiple elastic thin shell model is applied. The TWNT dynamics is studied in the framework of the Sanders–Koiter shell theory. The van der Waals interaction between any two layers of the TWNT is modelled by a radius-dependent function. The shell deformation is described in terms of longitudinal, tangential and radial displacements. Simply supported, clamped and free boundary conditions are applied. The three displacement fields are expanded by means of a double mixed series based on Chebyshev polynomials for the longitudinal variable and harmonic functions for the tangential variable. The Rayleigh–Ritz method is applied to obtain approximate natural frequencies and mode shapes. The present model is validated in the linear field by means of comparisons with data from the literature. This study is focused on determining the effect of geometry and boundary conditions on the natural frequencies of TWNTs.

Highlights

Carbon nanotubes (CNTs) were first discovered in 1991 by S
This paper presents an investigation on the dynamical properties of single-walled carbon nanotubes (SWCNTs), nonlinear modal interaction and energy exchange are analysed in detail
The Lagrange equations are used to get the system of nonlinear ordinary differential equations of motion, which is numerically solved by means of the implicit Runge-Kutta iterative method

Summary

Introduction

Carbon nanotubes (CNTs) were first discovered in 1991 by S. One of the most controversial topics in modelling the discrete CNTs as continuous cylindrical shells is denoted by the inclusion of the size effects, i.e. surface stresses, strain gradients and nonlocalities, into the elastic shell theory considered It was found in literature [42] that the size effects influence the vibration characteristics of the CNTs in the higher region of their frequency spectrum (vibration modes with relatively high number of longitudinal half-waves). Nonlinear resonance interaction between conjugate circumferential flexural modes in single-walled carbon nanotubes where the first term of the right-hand side of equation (9) is the membrane energy ( referred to stretching energy) and the second one is the bending energy Another relevant issue of the thin shell modelling of SWCNTs is given by the choice of an isotropic or anisotropic model; CNTs are discrete systems, i.e., they are intrinsically non-isotropic. The dimensionless mode shape (U,V ,W ) is expanded by considering a double mixed series in terms of m-th order dimensionless Chebyshev polynomials Tm*( ) along the longitudinal direction and harmonic functions (cos n ,sin n ) along the circumferential direction; following Ref. [38], the expansion reads: Mu N

Mw N

Aspect ratio

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