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

2013 ◽  
pp. 97-107
Keyword(s):  
Carbon Nanotubes ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

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.

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.

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 of Cantilevered Double-Walled Carbon Nanotubes Predicted by Timoshenko Beam Model and Molecular Dynamics

2015 ◽  
Vol 12 (04) ◽  
pp. 1540017 ◽  
Author(s):  
Rumeng Liu ◽  
Lifeng Wang
Keyword(s):  
Carbon Nanotubes ◽  
Molecular Dynamics ◽  
Natural Frequencies ◽  
Md Simulations ◽  
Beam Model ◽  
Timoshenko Beams ◽  
Fundamental Frequencies ◽  
The Difference ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

Vibration of double-walled carbon nanotubes (DWCNTs) with one end fixed and the other end free is studied by using different beam models of continuum mechanics and the molecular dynamics (MD) simulations. The models of the double-Euler beams (DEB) and the double-Timoshenko beams (DTB) of cantilevered case, with the van der Waals interaction between layers taken into consideration, are applied to predict the natural frequencies of DWCNTs. An analytical solution is first obtained for the DTB model with cantilevered boundary condition. The fundamental frequencies obtained by the DEB model and the DTB model are very close, for the relatively long DWCNTs. The MD simulations show that these two models can predict the natural frequencies well. However, the difference between the DEB model and the DTB model becomes obvious, for the vibration of the relatively short DWCNTs. The DTB model can offer a much better prediction than the DEB model when the DWCNT is very short especially for high-order frequencies.

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