scholarly journals Discussion: “Applicability and Limitations of Simplified Elastic Shell Equations for Carbon Nanotubes” (Wang, C. Y., Ru, C. Q., and Mioduchowski, A., 2004, ASME J. Appl. Mech., 71, pp. 622–631)

2005 ◽  
Vol 72 (6) ◽  
pp. 981-981 ◽  
Author(s):  
J. G. Simmonds
Keyword(s):  
Carbon Nanotubes ◽  
Elastic Shell ◽  
Shell Equations

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.

Variational Principles for the Stability Analysis of Multi-Walled Carbon Nanotubes Based on a Nonlocal Elastic Shell Model

2010 ◽  
Author(s):  
Mohsen Asghari ◽  
Jacob Rafati
Keyword(s):  
Carbon Nanotubes ◽  
Stability Analysis ◽  
Inverse Method ◽  
Variational Principles ◽  
Elastic Shell ◽  
Multi Walled Carbon Nanotubes ◽  
Natural Boundary Conditions ◽  
The Stability ◽  
Walled Carbon Nanotubes ◽  
Continuum Theories

The nonlocal continuum theories are capable to reflect the small length characteristic of nanostructures. In this work, variational principles are presented for the stability analysis of multi-walled carbon nanotubes under various mechanical loadings based on the nonlocal elastic Donnell’s shell by the semi-inverse method. In this manner, a set of proper essential and natural boundary conditions for each layer of the multi-walled nanotube is derived.

Scale Coefficient, Length, Mode and Radius Effect on Critical Axial Compressed Buckling Load of Carbon Nanotubes via Nonlocal Continuum Model

2012 ◽  
Vol 629 ◽  
pp. 296-301
Author(s):  
Hong Liang Tian
Keyword(s):  
Carbon Nanotubes ◽  
Continuum Model ◽  
Elastic Shell ◽  
Elastic Beam ◽  
Analytic Solutions ◽  
Buckling Load ◽  
Beam Model ◽  
Mechanical Behaviors ◽  
Nonlocal Beam ◽  
Scale Coefficient

Some exact concise analytic solutions of critical axial compressed buckling load for carbon nanotubes are derived via nonlocal beam. Scale coefficient, length, mode and radius effect on nonlocal critical axial compressed buckling load of CNTs is established and can be analyzed in terms of the general solutions. Radius effect on nonlocal critical axial compressed buckling load is only found through nonlocal elastic shell model but not derived via nonlocal elastic beam model. Numerical calculations of CNTs show that local critical axial compressed buckling load through local elastic theory is overestimated. Scale coefficient, length, mode and radius effect should be taken into account in predicting more accurate results for mechanical behaviors of CNTs via continuum model.

Thermal Buckling of Multi-Walled Carbon Nanotubes by Nonlocal Elasticity

2006 ◽  
Vol 74 (3) ◽  
pp. 399-405 ◽  
Author(s):  
Renfu Li ◽  
George A. Kardomateas
Keyword(s):  
Carbon Nanotubes ◽  
Thermal Loading ◽  
Scale Effects ◽  
Elastic Shell ◽  
Thermal Buckling ◽  
Small Scale ◽  
Internal Length ◽  
Internal Characteristic ◽  
Multi Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

The small internal length scales of nanomaterials/nano-devices may call the direct application of classical continuum models into question. In this research, a nonlocal elastic shell model, which takes the small scale effects into account, is developed to study the thermal buckling behavior of multi-walled carbon nanotubes. The multi-walled carbon nanotubes are considered as concentric thin shells coupled with the van der Waals forces between adjacent nanotubes. Closed form solutions are formulated for two types of thermal buckling of a double-walled carbon nanotube: Radial thermal buckling (as in a shell under external pressure) and axial thermal buckling. The effects of small scale effects are demonstrated, and a significant influence of internal characteristic parameters such as the length of the C‐C bond has been found on the thermal buckling critical temperature. The study interestingly shows that the axial buckling is not likely to happen, while the “radial” buckling may often take place when the carbon nano-tubes are subjected to thermal loading. Furthermore, a convenient method to determine the material constant, “e0” and the internal characteristic parameter, “a,” is suggested.

Analytical Treatment of the Free Vibration of Single-Walled Carbon Nanotubes Based on the Nonlocal Flugge Shell Theory

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
R. Ansari ◽  
H. Rouhi
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Scale Effects ◽  
Elastic Shell ◽  
Nonlocal Parameter ◽  
Single Walled Carbon Nanotubes ◽  
Small Scale ◽  
Resonant Frequencies ◽  
Single Walled Carbon ◽  
Walled Carbon Nanotubes

In the current work, the vibration characteristics of single-walled carbon nanotubes (SWCNTs) under different boundary conditions are investigated. A nonlocal elastic shell model is utilized, which accounts for the small scale effects and encompasses its classical continuum counterpart as a particular case. The variational form of the Flugge type equations is constructed to which the analytical Rayleigh–Ritz method is applied. Comprehensive results are attained for the resonant frequencies of vibrating SWCNTs. The significance of the small size effects on the resonant frequencies of SWCNTs is shown to be dependent on the geometric parameters of nanotubes. The effectiveness of the present analytical solution is assessed by the molecular dynamics simulations as a benchmark of good accuracy. It is found that, in contrast to the chirality, the boundary conditions have a significant effect on the appropriate values of nonlocal parameter.

Combined Torsional Buckling of Carbon Nanotubes Subjected to Thermo-Electro-Mechanical Loadings with Consideration of Scale Effect

2013 ◽  
Vol 562-565 ◽  
pp. 744-749 ◽  
Author(s):  
Xiao Hu Yao ◽  
Yu Gang Sun ◽  
Han Zhou Li
Keyword(s):  
Carbon Nanotubes ◽  
Scale Effect ◽  
Elastic Shell ◽  
Nonlocal Elasticity Theory ◽  
Torsional Buckling ◽  
Coupling Condition ◽  
Field Coupling ◽  
Buckling Behavior ◽  
Applied Torque ◽  
Effects Of Temperature

The present study has theoretically investigated the combined torsional buckling behavior of carbon nanotubes (CNTs) with consideration of scale effect in the multi-field coupling condition. The generalized governing equation of buckling for CNTs subjected to thermo-electro-mechanical loadings has been established based on an elastic shell model of continuum mechanics, in which scale effect is taken account of through the nonlocal elasticity theory. Except the applied torque and torsion-related axial load, the Van der Waals forces between adjacent nanotubes, as well as effects of temperature change and voltage load, is taken into consideration at the meantime. Numerical experiments are conducted to demonstrate the influences of different factors. The conclusions provided herein will be helpful and valuable for the dependent designs and related applications of CNT-based nano-structures serving in the complex thermal and electrical environment.

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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