Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and inextensional modes of vibration (original) (raw)
Related papers
Mechanics of Advanced Materials and Structures, 2018
In this article, vibration analysis of single-walled carbon nanotubes (SWCNTs) based on Love's thin shell theory has been investigated along with five sort of boundary conditions (S-S), (C-C), (C-F), (C-Sl), and (F-S). Three different shapes such as Armchair, Zigzag, and Chiral are taken into account under the influence of Winkler and Pasternak foundations. The wave propagation approach is employed to formulate the eigenvalue problem. MATLAB software package in used to obtain the vibrational natural frequencies of SWCNTs. The axial modal dependence is measured by the complex exponential functions implicating the axial modal numbers. Nomenclature E = Young's modulus h = shell thickness L = shell length v = Poisson's ratio ω = natural angular frequency E, v, ρ = effective material quantities L/R = length-to-radius ratio n = circumferential wave number Eh = in-plane rigidity ρh = mass density per unit lateral area θ = circumferential coordinate u(x, θ , t) = displacement functions in x direction v(x, θ , t) = displacement functions in θ direction w(x, θ , t) = displacement functions in z direction
Applied Mathematical Modelling, 2012
Single-walled carbon nanotubes (SWCNTs) exhibit remarkable chirality-dependent mechanical phenomena. In present paper, an anisotropic elastic shell model is developed to study the vibration characteristics of chiral SWCNTs. Analytical solution is presented by using the Flügge shell theory and complex method. The suggested model is justified by a good agreement between the present results and some experimental and numerical available data in literature. Furthermore, the model is used to elucidate the effect of tube chirality on the frequencies of SWCNTs. Finally, the influences of the externally applied mid-face axial force and torque on longitudinal, radial and torsional frequencies of SWCNTs are investigated.
2021
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, p...
Applied Mathematics and Computation, 2014
In this paper, a gradient elasticity shell formulation is presented for free vibration analysis of single-walled carbon nanotube placed on Winkler/Pasternak foundation. The proposed formulation is based on the combined strain-inertia gradient elasticity. The combined strain-inertia gradient elasticity provides an extension to the classical equations of elasticity with additional higher-order spatial derivatives of strains and two material length scale parameters related to the inertia and strain gradients, which enable formulation to investigate the size effect on the dynamic behavior of nanotubes. The effects of the length scale parameters, aspect ratio of single-walled carbon nanotube and foundation parameters on the fundamental frequencies for different values half-axial wave number and circumferential wave number are investigated. The natural frequencies obtained from the proposed shell formulation show the effects of size-dependent properties. It can be concluded that a continuum model enriched with higher-order inertia terms has been proposed as alternative to the continuum description obtained with classical elasticity theory.
In the present paper, the comparison is conducted between three classical shell theories as applied to the linear vibrations of single-walled carbon nanotubes (SWCNTs); specifically, the evaluation of the natural frequencies is conducted via Donnell, Sanders and Flügge shell theories. The actual discrete SWCNT is modelled by means of a continuous homogeneous cylindrical shell considering equivalent thickness and surface density. In order to take into account the intrinsic chirality of carbon nanotubes (CNTs), a molecular based anisotropic elastic shell model is considered. Simply supported boundary conditions are imposed and complex method is applied to solve the equations of motion and to obtain the natural frequencies. Comparisons with the results of molecular dynamics simulations available in literature are performed to check the accuracy of the three different shell theories, where Flügge shell theory is found to be the most accurate. Then, a parametric analysis evaluating the e...
Breakdown of structural models for vibrations of single-wall zigzag carbon nanotubes
Journal of Applied Physics, 2009
Free vibrations of zigzag single-wall carbon nanotubes ͑SWCNTs͒ of aspect ratio ͑length/diameter͒ ϳ6 and with ends traction-free have been studied using molecular mechanics ͑MM͒ simulations with the MM3 potential. It is found that the frequencies of inextensional ͑the Love and the Rayleigh͒ modes of an ͑n, 0͒ SWCNT saturate at the circumferential wave number of either ͑n −1͒ / 2 or n / 2 where n is odd or even. This is explained in terms of its molecular structure. Since the frequencies of the inextensional modes of vibration of a thin cylinder made of an isotropic linear elastic material do not saturate with an increase in the circumferential wave number, a continuum structure cannot represent all modes of vibration of a zigzag SWCNT. This result is independent of the value assigned to the wall thickness of the SWCNT. We have also found values of material and geometric parameters of a shell and a hollow cylinder by equating their frequencies of the inextensional, the radial breathing, the axial and the torsional modes of vibrations to the corresponding ones of a zigzag SWCNT, and by taking their mean diameter and length equal to those of the SWCNT. The frequencies of the extensional modes of oscillations of the two continuum structures for various axial half wave numbers and circumferential wave numbers are found to match well with those of the SWCNT obtained from the MM simulations. However, the frequencies of the inextensional modes of the continuum structures deviate noticeably from those of the SWCNT, and this deviation increases with an increase in the circumferential wave number.
Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes
2008
Axial, torsional and radial breathing mode (RBM) vibrations of free–free unstressed (ie, 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.
Natural Frequencies of Triple-Walled Carbon Nanotubes
2016
In this paper, the linear vibrations of Triple-Walled Carbon Nanotubes (TWNTs) are analysed. A multiple elastic shell model is considered. The shell dynamics is studied in the framework of the Sanders-Koiter shell theory. The van der Waals (vdW) interaction between two layers of the TWNT is modelled by a radius-dependent function. The circular cylindrical shell deformation is described in terms of longitudinal, circumferential and radial displacements. Simply supported, clamped and free boundary conditions are considered. The three displacement fields are expanded by means of a double mixed series based on Chebyshev orthogonal polynomials for the longitudinal variable and harmonic functions for the circumferential variable. The Rayleigh-Ritz method is applied to obtain approximate natural frequencies and mode shapes. The present model is validated in linear field by means of data derived from the literature. This study is focused on determining the effect of the geometry and boundary conditions on the natural frequencies of TWNTs.
Torsional Vibration and Static Analysis of the Cylindrical Shell Based on Strain Gradient Theory
In this paper, the free vibration and torsional static analysis of cylindrical shell are developed using modified strain gradient theory. In doing this, the governing equations and classical and non-classical boundary conditions are derived using Hamilton's principle. After obtaining equations governing the problem, the differential quadrature method is used to discretize the equations of motion of the vibration problem and to examine the single-walled carbon nanotube (SWCNT) with two clamped-free and clamped– clamped supports as a special application of this formulation. Also, torsional static analysis is carried out for the clamped– clamped SWCNT. Results reveal that SWCNT rigidity in strain gradient theory is higher than that in couple stress theory or the classical theory, which leads to increase in torsional frequencies and decrease in torsion of SWCNT. Results also demonstrate that the effect of size parameter and SWCNT torsion in different lengths and diameters is considerable.
Physica E: Low-dimensional Systems and Nanostructures, 2019
This piece of work describes a free vibration analysis of double walled carbon nanotubes (DWCNTs) based on non-local elastic shell model. The classical Donnell's shell theory is engrossed to account the effects of small scale by merging Eringen's non-local elasticity equations into it. The wave propagation approach is implemented to analyse the influence of various edge conditions and also van der Waals interaction forces between inner and outer tubes given due consideration. An eigen value problem is formed to investigate the frequency spectra of DWCNTs in relevance of aspect ratio and distinct values of non-local parameter. The results in current study show the ability to perform free frequency characteristics of DWCNTs by developed non-local elastic shell model with appropriate choice of material properties has been certified by experimental comparisons with molecular dynamic data and also verified by earlier computations.