Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes

doi:10.1016/j.compositesb.2013.08.020

Composites Part B: Engineering

Volume 57, February 2014, Pages 21-24

https://doi.org/10.1016/j.compositesb.2013.08.020 Get rights and content

Abstract

In this paper, the mechanical buckling properties of a zigzag double-walled carbon nanotube (DWCNT) with both chirality and small scale effects are studied. Based on the nonlocal continuum theory and the Timoshenko beam model, the governing equations are derived and the critical buckling loads under axial compression are obtained. The DWCNTs are considered as two nanotube shells coupled through the van der Waals interaction between them. The equivalent Young’s modulus and shear modulus for zigzag DWCNT are derived using an energy-equivalent model. The results show that the critical buckling load can be overestimated by the local beam model if the small-scale effect is overlooked for long nanotubes. In addition, significant dependence of the critical buckling loads on the chirality of zigzag carbon nanotube is confirmed. These findings are important in mechanical design considerations of devices that use carbon nanotubes.

Introduction

Carbon nanotubes (CNTs) have excellent mechanical, chemical, electronic, and thermal properties which make them attractive for a wide range of applications, for example, energy storage media, drug delivery, novel probes and sensors, ultrafine nanocomponents, etc. [1], [2], [3], [4]. CNTs subjected to compression or bending are especially prone to buckling because of their high aspect ratios [5]. The occurrence of buckling of CNTs can lead to potential applications as nanometersized tunnel barriers for electron transport [6] and fluid-flow control nanovalves [7], [8].

It is still a challenge for experimental study of the buckling behavior of CNTs due to the difficulties encountered on the nanoscale at the current stage. Therefore, the theoretical methods, including atomistic simulations and continuum mechanics, are often applied for studying the buckling behavior of CNTs. For example, Ranjbartoreh et al. [9] investigated the buckling behavior and critical axial pressure of double-walled carbon nanotubes with surrounding elastic medium using the numerical method based on the classical theory of plates and shells and the Galerkin method. Cao and Chen [10] carried out atomistic simulations to study the effect of the displacement increment on the critical buckling strain of single-walled carbon nanotubes (SWCNTs) under axial compression.

However, in the classical elastic model, the small scale effects are not considered. The nonlocal continuum theory initiated by Eringen [11], [12], assumes the stress at a reference point is considered as a function of the strain at every point in the body. Here, it should mention some pioneer work on the mechanical behaviors of carbon nanotube with the nonlocal continuum theory. Sudak firstly developed the nonlocal multi-beam model to discuss the buckling properties [13]. Zhang et al. firstly presented the nonlocal multi-shell model [14] and estimated a value of the scale effect parameter e₀ for nanotubes [15]. As a result, the nonlocal continuum theory can present the more reliable analysis and show accurate results [16], [17], [18], [19], [20], [21], [22], [23].

To the best of authors’ knowledge, the nonlocal buckling analysis of zigzag DWCNTs using nonlocal Timoshenko beam theory has not been studied yet.

In this paper, the buckling behavior of zigzag DWCNT under axial compression is investigated based on nonlocal Timoshenko beam model. The equivalent Young’s modulus and shear modulus for zigzag DWCNT are derived using an energy-equivalent model developed by Wu et al. [24]. The obtained results in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the mechanical buckling properties of zigzag double-walled carbon nanotubes.

Section snippets

Atomic structure of carbon nanotube

Carbon nanotubes are considered to be tubes formed by rolling a graphene sheet about the $\vec{T}$ vector. A vector perpendicular to the $\vec{T}$ is the chiral vector denoted by ${\vec{C}}_{h}$ .

The chiral vector and the corresponding chiral angle define the type of CNT, i.e. zigzag, armchair, chiral ${\vec{C}}_{h}$ can be expressed with respect to two base vectors ${\vec{a}}_{1}$ and ${\vec{a}}_{2}$ as under: ${\vec{C}}_{h} = n {\vec{a}}_{1} + m {\vec{a}}_{2}$ where n and m are the indices of translation, which decide the structure around the circumference. Fig. 1 depicts the lattice indices

The nonlocal Timoshenko beam model

Response of materials at the nanoscale is different from those of their bulk counterparts. Nonlocal elasticity is first considered by Eringen [11], [12]. He assumed that the stress at a reference point is a functional of the strain field at every point of the continuum. Nonlocal stress tensor t at point x′ is defined by: $σ = \int_{V} K (| x - x^{'} |, τ) S (x^{'}) d x$ where S(x′) is the classical, macroscopic stress tensor at point x′, K(|x − x′|, τ) is the kernel function and τ is a material constant that depends on

Results and discussions

In this section, numerical calculations for the mechanical buckling properties of zigzag double-walled nanotubes are carried out. The parameters used in calculations for the zigzag DWCNTs are given as follows: the effective thickness of CNTs taken to be 0.258 nm [24], the force constants K/2 = 46,900 kcal/mol/nm² and C/2 = 63 kcal/mol/rad² [34], the mass density ρ = 2.3 g cm⁻³ [27]. The shear coefficient β = 9/10. It should be noted that according to the previous discussions about the values of e₀ and a in

Conclusions

The nonlocal Timoshenko beam model was applied to analyze the chirality and small scale effects on the critical buckling load of zigzag DWCNT subjected to axial compression. According to the analysis, the following results were obtained:

•
The effect of chirality on the critical buckling load of DWCNT increased with increasing the index of translation, especially at higher-order modes.
•
Increasing the value of scale coefficient decreased the critical buckling load, especially at higher order modes.
•

Acknowledgments

This research was supported by the Algerian national agency for development of university research (ANDRU) and university of Sidi Bel Abbes (UDL SBA) in Algeria.

References (35)

A.R. Ranjbartoreh et al.
Physica E
(2007)
A.C. Eringen
Int J Eng Sci
(1972)
M. Şimşek
Physica E
(2010)
M. Şimşek
Comput Mater Sci
(2012)
Y. Wu et al.
Thin-Walled Struct
(2006)
H. Heireche et al.
Physica E
(2008)
C.Y. Wang et al.
Int J Solids Struct
(2003)
R.C. Batra et al.
Int J Solids Stuct
(2007)
P.M. Ajayan et al.
Appl Phys
(2001)

Y. Zhou

The Open Nanosci J

(2008)

S. Iijima et al.

J Chem Phys

(1996)

H.W.C. Postma et al.

Science

(2001)

S.D. Solares et al.

Nanotechnology

(2004)

M. Grujicic et al.

Mater Sci Eng B

(2005)

G. Cao et al.

Nanotechnology

(2006)

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Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes

Abstract

Introduction

Section snippets

Atomic structure of carbon nanotube

The nonlocal Timoshenko beam model

Results and discussions

Conclusions

Acknowledgments

Physica E

Int J Eng Sci

Physica E

Comput Mater Sci

Thin-Walled Struct

Physica E

Int J Solids Struct

Int J Solids Stuct

Appl Phys

The Open Nanosci J

J Chem Phys

Science

Nanotechnology

Mater Sci Eng B

Nanotechnology