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.

NONLOCAL ANALYTICAL FLUGGE SHELL MODEL FOR AXIAL BUCKLING OF DOUBLE-WALLED CARBON NANOTUBES WITH DIFFERENT END CONDITIONS

NANO ◽  
2012 ◽  
Vol 07 (03) ◽  
pp. 1250018 ◽  
Author(s):  
HESSAM ROUHI ◽  
REZA ANSARI
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Shell Model ◽  
Ritz Method ◽  
Nonlocal Parameter ◽  
Geometrical Parameters ◽  
Field Equations ◽  
Axial Buckling ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

In this paper, a nonlocal Flugge shell model is utilized to investigate the axial buckling behavior of double-walled carbon nanotubes (DWCNTs) under various boundary conditions. According to the nonlocal elasticity theory, the displacement field equations coupled by the van der Waals interaction are derived. The set of governing equations of motion is then solved by the Rayleigh–Ritz method. The present analysis can treat boundary conditions in a layer-wise manner. The effects of nonlocal parameter, layer-wise boundary conditions and geometrical parameters on the mechanical behavior of DWCNTs are examined. Furthermore, molecular dynamics simulations are performed to assess the validity of the results and also to predict the appropriate values of nonlocal parameter. It is found that the type of boundary conditions affects the proper value of nonlocal parameter.

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 Linear vibrations of triple-walled carbon nanotubes

2017 ◽  
Vol 23 (11) ◽  
pp. 1456-1481 ◽  
Author(s):  
Matteo Strozzi ◽  
Francesco Pellicano
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Mode Shapes ◽  
Displacement Fields ◽  
Linear Vibrations ◽  
Longitudinal Variable ◽  
Walled Carbon Nanotubes ◽  
Linear Field

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.

Applicability and Limitations of Simplified Elastic Shell Equations for Carbon Nanotubes

2004 ◽  
Vol 71 (5) ◽  
pp. 622-631 ◽  
Author(s):  
C. Y. Wang ◽  
C. Q. Ru ◽  
A. Mioduchowski
Keyword(s):  
Carbon Nanotubes ◽  
Cylindrical Shells ◽  
Elastic Shell ◽  
Multiwall Carbon Nanotubes ◽  
Detailed Comparison ◽  
Single Equation ◽  
Simplified Models ◽  
Shell Models ◽  
Shell Equations ◽  
Relative Errors

This paper examines applicability and limitations of simplified models of elastic cylindrical shells for carbon nanotubes. The simplified models examined here include Donnell equations and simplified Flugge equations characterized by an uncoupled single equation for radial deflection. These simplified elastic shell equations are used to study static buckling and free vibration of carbon nanotubes, with detailed comparison to exact Flugge equations of cylindrical shells. It is shown that all three elastic shell models are in excellent agreement (with relative errors less than 5%) with recent molecular dynamics simulations for radial breathing vibration modes of carbon nanotubes, while reasonable agreements for various buckling problems have been reported previously for Donnell equations. For general cases of buckling and vibration, the results show that the simplified Flugge model, which retains mathematical simplicity of Donnell model, is consistently in better agreement with exact Flugge equations than Donnell model, and has a significantly enlarged range of applicability for carbon nanotubes. In particular, the simplified Flugge model is applicable for carbon nanotubes (with relative errors around 10% or less) in almost all cases of physical interest, including some important cases in which Donnell model results in much larger errors. These results are significant for further application of elastic shell models to carbon nanotubes because simplified shell models, characterized by a single uncoupled equation for radial deflection, are particularly useful for multiwall carbon nanotubes of large number of layers.

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