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TECHNICAL PAPERS

# Coupling of Extension and Twist in Single-Walled Carbon Nanotubes

[+] Author and Article Information
Karthick Chandraseker

Department of Theoretical and Applied Mechanics, Kimball Hall, Cornell University, Ithaca, NY 14853

Subrata Mukherjee1

Department of Theoretical and Applied Mechanics, Kimball Hall, Cornell University, Ithaca, NY 14853sm85@cornell.edu

It is noted that the term $(r∕R)erieRi$ is missing in Eq. (38) of (32) and in Eqs. (25), (29) of (33). This is because 2 D deformation is considered in these papers while a cylindrical reference configuration is used for deformation analysis of a SWNT in the present work.

Equation 32$(TZiϴi=TϴiZi=0)$ in (32) is expected to be true for armchair and zig-zag nanotubes but not for a chiral SWNT which tends to twist when extended. It is noted that for this problem, using $T=JF−1∙σ∙F−T$ [where $σ$ is the Cauchy stress and $J=det(F)$], one can show that $TϴiZi=TZiϴi=(r∕R)σθizi−[kr2∕(R(1+ϵ))]σzizi$. Now, if twist is not allowed, one has $k=0,σθizi≠0$; while if twist is allowed, $σθizi=0,k≠0.$ In either case, in general, $TϴiZi≠0$. It is noted that numerical results are presented in (32) for only armchair and zig-zag nanotubes, and not for chiral ones.

1

To whom correspondence should be addressed.

J. Appl. Mech 73(2), 315-326 (Aug 22, 2005) (12 pages) doi:10.1115/1.2125987 History: Received April 10, 2005; Revised August 22, 2005

## Abstract

This paper presents a study of the deformation behavior of single-walled carbon nanotubes (SWNTs) subjected to extension and twist. The interatomic force description is provided by the Tersoff-Brenner potential for carbon. The rolling of a flat graphene sheet into a SWNT is first simulated by minimizing the energy per atom, the end result being the configuration of an undeformed SWNT. The Cauchy-Born rule is then used to connect the atomistic and continuum descriptions of the deformation of SWNTs, and leads to a multilength scale mechanics framework for simulating deformation of SWNTs under applied loads. Coupled extension and twist of SWNTs is considered next. As an alternative to the Cauchy-Born rule for coupled extension-twist problems, a direct map is formulated. Analytic expressions are derived for the deformed bond lengths using the Cauchy-Born rule and the direct map for this class of deformations. Numerical results are presented for kinematic coupling, for imposed extension and imposed twist problems, using the Cauchy-Born rule as well as the direct map, for representative chiral, armchair and zig-zag SWNTs. Results from both these approaches are carefully compared.

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## Figures

Figure 1

(a) Rolling of a graphene sheet into a SWNT. Single walled carbon nanotubes drawn on the same scale: (b) Chiral (9,6); (c) Armchair (5,5) (d) Zig-zag (10,0). The figures (bd) have been drawn by using the software in (1)

Figure 2

A representative atom A and its nearest neighbors B,C,D and bonds AB,AC,AD. a1=BC⃗ and a2=DC⃗

Figure 3

Atomic structure for (a) armchair r(0)(A,B)=r(0)(A,D),r(0)(C,B)=r(0)(C,D) (b) zig-zag r(0)(A,B)=r(0)(A,C),r(0)(D,B)=r(0)(D,C) The figures show nearest neighbors B,C,D of atom A, and the nearest neighbors of B,C,D

Figure 4

The shaded parallelogram shown above is the unit cell in a graphene sheet

Figure 5

Undeformed and deformed cross sections of a SWNT

Figure 6

The direct map F(F(P)=p)

Figure 7

Kinematic coupling plots for parameter set 1 (k for (5,5) and (10,0) coincide in (b))

Figure 8

Kinematic coupling plots for parameter set 2 (k for (5,5) and (10,0) coincide in (b))

Figure 9

Comparison of results from the Cauchy-Born rule and the direct map for parameter set 1. Very similar results are obtained for parameter set 2.

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