Effective Approximations for Natural Frequencies of Double-Walled Carbon Nanotubes Based on Donnell Shell Theory

2011 ◽  
Vol 2 (2) ◽  
Author(s):  
Demetris Pentaras ◽  
Isaac Elishakoff
Keyword(s):  
Carbon Nanotubes ◽  
Carbon Nanotube ◽  
Natural Frequency ◽  
Characteristic Equation ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Analytical Formulas ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes ◽  
First Time

The vibration behavior of double-walled carbon nanotube (DWCNT) is investigated based on Donnell shell theory with van der Waals interaction taken into consideration. In addition, new results are obtained for the natural frequencies of a DWCNT based on a simplified version of Donnell shell theory by neglecting the tangential inertia terms. The resulting reduced characteristic equation for the natural frequency represents the radial mode of vibration. The factor of neglecting tangential inertia relative to the values of frequencies obtained by full Donnell shell theory is also obtained with attendant interesting results. Further possible simplifications of Donnell shell theory are introduced. For the first time in the literature, the effect of neglect of tangential inertia terms in DWCNTs is investigated. Accurate approximate analytical formulas are uncovered for the fundamental natural frequencies and compared with the exact values.

Buckling Analyses of Double-Wall Carbon Nanotubes: A Shell Theory Based on the Interatomic Potential

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
W. B. Lu ◽  
J. Wu ◽  
X. Feng ◽  
K. C. Hwang ◽  
Y. Huang
Keyword(s):  
Carbon Nanotubes ◽  
Shell Theory ◽  
Interatomic Potential ◽  
Outer Wall ◽  
Critical Buckling Strain ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes ◽  
Double Wall ◽  
The Continuum ◽  
Vdw Interaction

Based on the finite-deformation shell theory for carbon nanotubes established from the interatomic potential and the continuum model for van der Waals (vdW) interactions, we have studied the buckling of double-walled carbon nanotubes subjected to compression or torsion. Prior to buckling, the vdW interactions have essentially no effect on the deformation of the double-walled carbon nanotube. The critical buckling strain of the double-wall carbon nanotubes is always between those for the inner wall and for the outer wall, which means that the vdW interaction decelerates buckling of one wall at the expenses of accelerating the buckle of the other wall.

Vibration Simulations of Double-Walled Carbon Nanotubes by Finite Element Method

2013 ◽  
Vol 774-776 ◽  
pp. 21-24
Author(s):  
Yu Jia ◽  
Li Jie Chen ◽  
Qi Zhao
Keyword(s):  
Carbon Nanotubes ◽  
Finite Element ◽  
Aspect Ratio ◽  
Natural Frequency ◽  
Van Der Waals ◽  
Van Der Waals Force ◽  
Vacancy Defects ◽  
Vibration Behavior ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

Finite element (FE) method is used to study the vibration behavior of armchair and zigzag double-walled carbon nanotubes (DWCNTs). In the analysis, nonlinear spring elements and the Lennard-Jones potential function are used to simulate the Van der Waals' force between non-bond atoms of different tube layers. We systematically analyze the effects of aspect ratio, double-atom vacancy defects and Van der Waals' force on the vibration behavior of DWCNTs. The simulation results show that the first order natural frequency decreases with the increase of length-to-diameter ratio (aspect ratio), the existence of Van der Waals' force causes the increase of natural frequency, and double-atom vacancy defects results in the decrease of each order natural frequency.

Buckling of a Clamped-Free Double-Walled Carbon Nanotube by the Bubnov-Galerkin Method

2012 ◽  
Vol 80 (1) ◽  
Author(s):  
Isaac Elishakoff ◽  
Simon Bucas
Keyword(s):  
Carbon Nanotubes ◽  
Carbon Nanotube ◽  
Galerkin Method ◽  
Compressive Load ◽  
Comparison Functions ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

This study deals with the Bubnov-Galerkin method applied to the buckling of clamped-free double-walled carbon nanotubes (DWCNTs) subjected to a concentrated compressive load at the free end. It was found that at least four comparison functions are needed in order to obtain accurate results.

Effects of Initial Axial Stress on the Vibration of Double-Walled Carbon Nanotubes under Temperature Field

2012 ◽  
Vol 443-444 ◽  
pp. 617-621
Author(s):  
Yan Yan ◽  
Wen Quan Wang ◽  
Li Xiang Zhang
Keyword(s):  
Carbon Nanotubes ◽  
Temperature Field ◽  
Natural Frequency ◽  
Temperature Change ◽  
Axial Stress ◽  
Mechanical Behaviors ◽  
Amplitude Ratios ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes ◽  
Vdw Interaction

The paper studies the linear vibrating mechanical behaviors of double-walled carbon nanotubes (DWCNTs) with the initial axial stress under the temperature field. The DWCNTs are modeled as double elastic shells coupled together through vdW interaction between inner and outer nanotubes. Based on the model, the relation between the amplitudes and the frequencies of the tubes is achieved. The results show that the initial axial stress could crucially affect the existence of the natural and intertube frequencies. Meanwhile, it is also concluded that the temperature change is significant for natural frequency, but weakly affects intertube frequency and amplitude ratios.

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

2021 ◽  
Vol 7 (3) ◽  
pp. 61
Author(s):  
Matteo Strozzi ◽  
Oleg V. Gendelman ◽  
Isaac E. Elishakoff ◽  
Francesco Pellicano
Keyword(s):  
Carbon Nanotubes ◽  
Shell Theory ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Elastic Shell ◽  
Mode Shapes ◽  
Simplified Models ◽  
Circular Cylindrical Shells ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

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.

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