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Research Papers

# A Finite-Deformation Shell Theory for Carbon Nanotubes Based on the Interatomic Potential—Part II: Instability Analysis

[+] Author and Article Information
J. Wu, K. C. Hwang

FML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P.R. China

J. Song

Department of Mechanical Science and Engineering, University of Illinois, 1206 W. Green Street, Urbana, IL 61801

Y. Huang1

Department of Civil and Environmental Engineering, and Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208y-huang@northwestern.edu

The complete expansion of the velocity $vθ$ includes the terms $sin nθ sin mπZ/L$, $sin nθ cos mπZ/L$, $cos nθ sin mπZ/L$, and $cos nθ cos mπZ/L$ ($n=0,1,2,...$,$m=1,2,3,...$), which are equivalent to $cos(mπZ/L−nθ)$, $cos(mπZ/L+nθ)$, $sin(mπZ/L−nθ)$, and $sin(mπZ/L+nθ)$. It can be shown that these four terms lead to the same bifurcation condition as the first one $cos(mπZ/L−nθ)$ in $vθ$ in Eq. 67.

1

Corresponding author.

J. Appl. Mech 75(6), 061007 (Aug 20, 2008) (7 pages) doi:10.1115/1.2965367 History: Received June 16, 2007; Revised May 10, 2008; Published August 20, 2008

## Abstract

Based on the finite-deformation shell theory for carbon nanotubes established from the interatomic potential in Part I of this paper, we have studied the instability of carbon nanotubes subjected to different loadings (tension, compression, internal and external pressures, and torsion). Similar to the conventional shells, carbon nanotubes may undergo bifurcation under compression/torsion/external pressure. Our analysis, however, shows that carbon nanotubes may also undergo bifurcation in tension and internal pressure, though the bifurcation modes for tension and compression are very different, and so are the modes for the internal and external pressures. The critical load for instability and bifurcation depends on the interatomic potential used.

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## Figures

Figure 1

The critical strain for bifurcation in tension based on the Brenner potential (7) versus mR/L for the (8,8) armchair carbon nanotube, where m(=1,2,3,...) is the bifurcation mode number, R and L are the radius and length of the carbon nanotube, respectively, n=0 represents the axisymmetric bifurcation, and n=1,2,3,... represents the nonaxisymmetric bifurcation

Figure 2

The critical strain for bifurcation in tension based on the second-generation interatomic potential (8) versus mR/L for the (8,8) armchair carbon nanotube, where m(=1,2,3,...) is the bifurcation mode number, R and L are the radius and length of the carbon nanotube, respectively, n=0 represents the axisymmetric bifurcation, and n=1,2,3,... represents the nonaxisymmetric bifurcation

Figure 3

The critical strain for bifurcation in tension based on the Brenner potential (7) versus mR/L for several armchair and zigzag carbon nanotubes, where m(=1,2,3,...) is the bifurcation mode number, and R and L are the radius and length of the carbon nanotube, respectively

Figure 4

The critical strain for bifurcation in compression based on the Brenner potential (7) versus L/(mR) for the (8,8) armchair carbon nanotube, where m(=1,2,3,...) is the bifurcation mode number, R and L are the radius and length of the carbon nanotube, respectively, n=0 represents the axisymmetric bifurcation, and n=1,2,3,... represents the nonaxisymmetric bifurcation

Figure 5

The critical strain for bifurcation in compression based on the second-generation interatomic potential (8) versus L/(mR) for the (8,8) armchair carbon nanotube, where m(=1,2,3,...) is the bifurcation mode number, R and L are the radius and length of the carbon nanotube, respectively, n=0 represents the axisymmetric bifurcation, and n=1,2,3,... represents the nonaxisymmetric bifurcation

Figure 6

The critical strain for bifurcation in compression based on the Brenner potential (7) versus L/(mR) for the (8,8), (12,12), and (16,16) armchair carbon nanotubes, where m(=1,2,3,...) is the bifurcation mode number, R and L are the radius and length of the carbon nanotube, respectively, and n=1,2,3,... represents the nonaxisymmetric bifurcation

Figure 7

The critical internal pressure for bifurcation based on the Brenner potential (7) versus L/(mR) for the (8,8), (12,12), and (16,16) armchair carbon nanotubes, where m(=1,2,3,...) is the bifurcation mode number, and R and L are the radius and length of the carbon nanotube, respectively. The bifurcation corresponds to n=1 (nonaxisymmetric bifurcation).

Figure 8

The critical internal pressure for bifurcation based on the second-generation interatomic potential (8) versus L/(mR) for the (8,8), (12,12), and (16,16) armchair carbon nanotubes, where m(=1,2,3,...) is the bifurcation mode number, and R and L are the radius and length of the carbon nanotube, respectively. The bifurcation corresponds to n=1 (nonaxisymmetric bifurcation).

Figure 9

The critical external pressure for bifurcation based on the Brenner potential (7) versus L/(mR) for the (8,8) armchair carbon nanotube, where m(=1,2,3,...) is the bifurcation mode number, R and L are the radius and length of the carbon nanotube, respectively, n=0 represents the axisymmetric bifurcation, and n=1,2,3,... represents the nonaxisymmetric bifurcation

Figure 10

The critical external pressure for bifurcation based on the second-generation interatomic potential (8) versus L/(mR) for the (8,8) armchair carbon nanotube, where m(=1,2,3,...) is the bifurcation mode number, R and L are the radius and length of the carbon nanotube, respectively, n=0 represents the axisymmetric bifurcation, and n=1,2,3,... represents the nonaxisymmetric bifurcation

Figure 11

The critical external pressure for bifurcation based on the Brenner potential (7) versus L/(mR) for the (8,8), (12,12), and (16,16) armchair carbon nanotubes, where m(=1,2,3,...) is the bifurcation mode number, R and L are the radius and length of the carbon nanotube, respectively, and n=1,2,3,... represents the nonaxisymmetric bifurcation

Figure 12

The critical twist kcr for bifurcation based on the Brenner potential (7) versus L/(mR) for the (8,8), (12,12), and (16,16) armchair carbon nanotubes, where m(=1,2,3,...) is the bifurcation mode number, R and L are the radius and length of the carbon nanotube, respectively, and n=1,2,3,... represents the nonaxisymmetric bifurcation

Figure 13

The critical twist kcr for bifurcation based on the second-generation interatomic potential (8) versus L/(mR) for the (8,8), (12,12), and (16,16) armchair carbon nanotubes, where m(=1,2,3,...) is the bifurcation mode number, R and L are the radius and length of the carbon nanotube, respectively, and n=1,2,3,... represents the nonaxisymmetric bifurcation

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