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A structural mechanics approach for the analysis of carbon nanotubes

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A structural mechanics approach for the analysis of carbonnanotubes
Chunyu Li, Tsu-Wei Chou
*
Department of Mechanical Engineering, Center for Composite Materials, University of Delaware,126 Spencer Laboratory, Newark, DE 19716-3140, USA
Received 14 March 2002; received in revised form 3 January 2003
Abstract
This paper presents a structural mechanics approach to modeling the deformation of carbon nanotubes. Funda-mental to the proposed concept is the notion that a carbon nanotube is a geometrical frame-like structure and theprimary bonds between two nearest-neighboring atoms act like load-bearing beam members, whereas an individualatom acts as the joint of the related load-bearing beam members. By establishing a linkage between structural me-chanics and molecular mechanics, the sectional property parameters of these beam members are obtained. The accuracyand stability of the present method is veriﬁed by its application to graphite. Computations of the elastic deformation of single-walled carbon nanotubes reveal that the Young
s moduli of carbon nanotubes vary with the tube diameter andare aﬀected by their helicity. With increasing tube diameter, the Young
s moduli of both armchair and zigzag carbonnanotubes increase monotonically and approach the Young
s modulus of graphite. These ﬁndings are in good agree-ment with the existing theoretical and experimental results.
2003 Elsevier Science Ltd. All rights reserved.
Keywords:
Carbon nanotube; Nanomechanics; Molecular mechanics; Force ﬁelds; Atomistic modeling; Structural mechanics
1. Introduction
The advancement of science and technology has evolved into the era of nanotechnology. The mostdistinct characteristic of nanotechnology is that the properties of nanomaterials are size-dependent. Due tothe extremely small size of nanomaterials, the evaluation of their mechanical properties, such as elasticmodulus, tensile/compressive strength and buckling resistance, presents signiﬁcant challenges to researchersin nanomechanics. While the experimental works has brought about striking progress in the research of nanomaterials, many researchers have also resorted to the computational nanomechanics. Because com-puter simulations based on reasonable physical models cannot only highlight the molecular features of nanomaterials for theoreticians but also provide guidance and interpretations for experimentalists. It is still
International Journal of Solids and Structures 40 (2003) 2487–2499www.elsevier.com/locate/ijsolstr
*
Corresponding author. Tel.: +1-302-831-2423/2421; fax: +1-302-831-3619.
E-mail address:
chou@me.udel.edu (T.-W. Chou).0020-7683/03/$ - see front matter
2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0020-7683(03)00056-8
an ongoing and challenging process to identify eﬀective and eﬃcient computational methods with respect tospeciﬁc nanomaterials.Among the many nanostructured materials, carbon nanotubes have attracted considerable attention.This kind of long and slender fullerene was ﬁrst discovered by Iijima (1991). They can be produced by anarray of techniques, such as arc discharge, laser ablation and chemical vapor deposition. A recent review of the processing and properties of carbon nanotubes and their composites is given by Thostenson et al.(2001). From the viewpoint of atomic arrangement, carbon nanotubes can be visualized as cylinders thatrolled from sheets of graphite. They assume either single-walled or multi-walled structures and their helicitymay also be diﬀerent (Iijima and Ichlhashi, 1993; Bethune et al., 1993). Since the discovery of carbon
nanotubes, much attention has been given to the investigation of their exceptional physical properties(Thostenson et al., 2001; Harris, 1999). It has been revealed that the conducting properties of carbon
nanotubes depend dramatically on their helicity and diameter (Terrones et al., 1999), and the stiﬀness,
ﬂexibility and strength of carbon nanotubes are much higher than those of conventional carbon ﬁbers(Treacy et al., 1996; Salvetat et al., 1999; Iijima et al., 1996). The extraordinary properties of carbon nano-
tubes have motivated researchers worldwide to study the fundamentals of this novel material as well as toexplore their applications in diﬀerent ﬁelds (Ajayan and Zhou, 2001).
Besides the great deal of experimental works on carbon nanotubes, many researchers have pursued theanalysis of carbon nanotubes by theoretical modeling (Harris, 1999; Saito et al., 1998). These modelingapproaches can be generally classiﬁed into two categories. One is the atomistic modeling and the majortechniques include classical molecular dynamics (MD) (Iijima et al., 1996; Yakobson et al., 1997), tight-
binding molecular dynamics (TBMD) (Hernandez et al., 1998) and density functional theory (DFT)(Sanchez-Portal et al., 1999). In principle, any problem associated with molecular or atomic motions can besimulated by these modeling techniques. However, due to their huge computational tasks, practical ap-plications of these atomistic modeling techniques are limited to systems containing a small number of molecules or atoms and are usually conﬁned to studies of relatively short-lived phenomena, from pico-seconds to nanoseconds.The other approach is the continuum mechanics modeling. Some researchers have resorted to classicalcontinuum mechanics for modeling carbon nanotubes. For examples, Tersoﬀ (1992) conducted simplecalculations of the energies of fullerenes based on the deformation of a planar graphite sheet, treated as anelastic continuum, and concluded that the elastic properties of the graphite sheet can be used to predict theelastic strain energy of fullerenes and nanotubes. Yakobson et al. (1996) noticed the unique features of fullerenes and developed a continuum shell model. Ru (2000a,b) followed this continuum shell model toinvestigate buckling of carbon nanotubes subjected to axial compression. This kind of continuum shellmodels can be used to analyze the static or dynamic mechanical properties of nanotubes. However, thesemodels neglect the detailed characteristics of nanotube chirality, and are unable to account for forces actingon the individual atoms.Therefore, there is a demand of developing a modeling technique that analyzes the mechanical res-ponse of nanotubes at the atomistic scale but is not perplexed in time scales. Such a modeling approachwould beneﬁt us in novel nanodevices design and multi-scale simulations of nanosystems (Nakano et al.,2001). In this paper, we extend the theory of classical structural mechanics into the modeling of carbonnanotubes. Our idea stems from that carbon nanotubes are elongated fullerenes, which were named afterthe architect known for designing geodesic domes, R. Buckmister Fuller. In fact, it is obvious that there aresome similarities between the molecular model of a nanotube and the structure of a frame building. In acarbon nanotube, carbon atoms are bonded together by covalent bonds. These bonds have their charac-teristic bond lengths and bond angles in a three-dimensional space. Thus, it is logical to simulate the de-formation of a nanotube based on the method of classical structural mechanics. In following sections, weﬁrst establish the bases of this concept and then demonstrate the approach by a few computational ex-amples.
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C. Li, T.-W. Chou / International Journal of Solids and Structures 40 (2003) 2487–2499
2. Brief review of structural mechanics for space frames
Structural mechanics analysis enables the determination of the displacements, strains and stresses of astructure under given loading conditions. Of the various modern structural analysis techniques, the stiﬀnessmatrix method has been by far the most generally used. The method can be readily applied to analyzestructures of any geometry and can be used to solve linear elastic static problems as well as problems in-volving buckling, plasticity and dynamics. In the following, we brieﬂy review the stiﬀness matrix method forlinearly elastic space frame problems, which is relevant to the present studies.For an element in a space frame as shown in Fig. 1, the elemental equilibrium equation can be written asfollowing (Weaver and Gere, 1990):
Ku
¼
f
;
ð
1
Þ
where
u
¼½
u
xi
;
u
yi
;
u
zi
;
h
xi
;
h
yi
;
h
zi
;
u
xj
;
u
yj
;
u
zj
;
h
xj
;
h
yj
;
h
zj
T
;
ð
2
Þ
f
¼½
f
xi
;
f
yi
;
f
zi
;
m
xi
;
m
yi
;
m
zi
;
f
xj
;
f
yj
;
f
zj
;
m
xj
;
m
yj
;
m
zj
T
ð
3
Þ
are the nodal displacement vector and nodal force vector of the element, respectively and
K
is the elementalstiﬀness matrix. The matrix
K
consists of following submatrices:
K
¼
K
ii
K
ij
K
T
ij
K
jj
;
ð
4
Þ
where
K
ii
¼
EA
=
L
0 0 0 0 00 12
EI
x
=
L
3
0 0 0 6
EI
x
=
L
2
0 0 12
EI
y
=
L
3
0
6
EI
y
=
L
2
00 0 0
GJ
=
L
0 00 0
6
EI
y
=
L
2
0 4
EI
y
=
L
00 6
EI
x
=
L
2
0 0 0 4
EI
x
=
L
2666666437777775
;
ð
5
Þ
K
ij
¼
EA
=
L
0 0 0 0 00
12
EI
x
=
L
3
0 0 0 6
EI
x
=
L
2
0 0
12
EI
y
=
L
3
0
6
EI
y
=
L
2
00 0 0
GJ
=
L
0 00 0 6
EI
y
=
L
2
0 2
EI
y
=
L
00
6
EI
x
=
L
2
0 0 0 2
EI
x
=
L
2666666437777775
;
ð
6
Þ
Fig. 1. Illustration of a beam element in a space frame.
C. Li, T.-W. Chou / International Journal of Solids and Structures 40 (2003) 2487–2499
2489
K
jj
¼
EA
=
L
0 0 0 0 00 12
EI
x
=
L
3
0 0 0
6
EI
x
=
L
2
0 0 12
EI
y
=
L
3
0 6
EI
y
=
L
2
00 0 0
GJ
=
L
0 00 0 6
EI
y
=
L
2
0 4
EI
y
=
L
00
6
EI
x
=
L
2
0 0 0 4
EI
x
=
L
2666666437777775
:
ð
7
Þ
It is observed from the above elemental stiﬀness matrices that when the length,
L
, of the element is known,there are still four stiﬀness parameters need to be determined. They are the tensile resistance
EA
, the ﬂexuralrigidity
EI
x
and
EI
y
and the torsional stiﬀness
GJ
. In order to obtain the deformation of a space frame, theabove elemental stiﬀness equations should be established for every element in the space frame and then allthese equations should be transformed from the local coordinates to a common global reference system.Finally, a system of simultaneous linear equations can be assembled according to the requirements of nodalequilibrium. By solving the system of equations and taking into account the boundary restraint conditions,the nodal displacements can be obtained.
3. Structural characteristics of carbon nanotubes
A single-walled carbon nanotube (SWNT) can be viewed as a graphene sheet that has been rolled into atube. A multi-walled carbon nanotube (MWNT) is composed of concentric graphitic cylinders with closedcaps at both ends and the graphitic layer spacing is about 0.34 nm. Unlike diamond, which assumes a 3-Dcrystal structure with each carbon atom having four nearest neighbors arranged in a tetrahedron, graphiteassumes the form of a 2-D sheet of carbon atoms arranged in a hexagonal array. In this case, each carbonatom has three nearest neighbors.The atomic structure of nanotubes can be described in terms of the tube chirality, or helicity, which isdeﬁned by the chiral vector
~
C C
h
and the chiral angle
h
. In Fig. 2, we can visualize cutting the graphite sheetalong the dotted lines and rolling the tube so that the tip of the chiral vector touches its tail. The chiralvector, also known as the roll-up vector, can be described by the following equation:
~
C C
h
¼
n
~
aa
1
þ
m
~
aa
2
;
ð
8
Þ
where the integers
ð
n
;
m
Þ
are the number of steps along the zigzag carbon bonds of the hexagonal lattice and
~
aa
1
and
~
aa
2
are unit vectors. The chiral angle determines the amount of
twist
in the tube. The chiral anglesare 0
and 30
for the two limiting cases which are referred to as zigzag and armchair, respectively (Fig. 3).
Fig. 2. Schematic diagram of a hexagonal graphene sheet (Thostenson et al., 2001).2490
C. Li, T.-W. Chou / International Journal of Solids and Structures 40 (2003) 2487–2499
In terms of the roll-up vector, the zigzag nanotube is denoted by
ð
n
;
0
Þ
and the armchair nanotube
ð
n
;
n
Þ
.The roll-up vector of the nanotube also deﬁnes the nanotube diameter.The physical properties of carbon nanotubes are sensitive to their diameter, length and chirality.In particular, tube chirality is known to have a strong inﬂuence on the electronic properties of carbonnanotubes. Graphite is considered to be a semi-metal, but it has been shown that nanotubes can be eithermetallic or semi-conducting, depending on tube chirality (Dresselhaus et al., 1996). The inﬂuence of chi-rality on the mechanical properties of carbon nanotubes has also been reported (Popov et al., 2000;Hernandez et al., 1998).
4. Structural mechanics approach to carbon nanotubes
From the structural characteristics of carbon nanotubes, it is logical to anticipate that there are potentialrelations between the deformations of carbon nanotubes and frame-like structures. For macroscopic spaceframe structures made of practical engineering materials, the material properties and element sectionalparameters can be easily obtained from material data handbooks and calculations based on the elementsectional dimensions. For nanoscopic carbon nanotubes, there is no information about the elastic andsectional properties of the carbon–carbon bonds and the material properties. Therefore, It is imperative toestablish a linkage between the microscopic computational chemistry and the macroscopic structural me-chanics.
4.1. Potential functions of molecular mechanics
From the viewpoint of molecular mechanics, a carbon nanotube can be regarded as a large moleculeconsisting of carbon atoms. The atomic nuclei can be regarded as material points. Their motions areregulated by a force ﬁeld, which is generated by electron–nucleus interactions and nucleus–nucleus inter-actions (Machida, 1999). Usually, the force ﬁeld is expressed in the form of steric potential energy. Itdepends solely on the relative positions of the nuclei constituting the molecule. The general expression of
Fig. 3. Schematic diagram of (a) an armchair and (b) a zigzag nanotube (Thostenson et al., 2001).
C. Li, T.-W. Chou / International Journal of Solids and Structures 40 (2003) 2487–2499
2491

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