Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 1, pp. 235-245, Warsaw 2014

ELASTIC MODULI OF CARBON NANOTUBES WITH NEW GEOMETRY
BASED ON FEM

Abdolhossein Fereidoon

Semnan University, Faculty of Mechanical Engineering, Semnan, Iran

Morteza Rajabpour

Young Researchers Club, Semnan branch, Islamic Azad University, Semnan, Iran

Hossein Hemmatian

Semnan University, Faculty of Mechanical Engineering, Semnan, Iran

e-mail: hoseinhemmatian@gmail.com

In this paper, the elastic moduli of elliptic single walled carbon nanotubes (ESWCNTs)
are described. A three-dimensional finite element (FE) model for such carbon nanotubes is
proposed. The covalent bonds are simulated by beam elements in the FEmodel. The elastic
moduli of beam elements are ascertained from a linkage betweenmolecular and continuum
mechanics. The deformations of the FE model are subsequently used to predict the elastic
moduli of ESWCNTs. In order to demonstrate the FE performance, the influence of length,
chirality, diameter and cross sectional aspect ratios on the elasticmoduli (Young’s modulus
and shearmodulus) of ESWCNTs is investigated. It is found that the cross sectional aspect
ratio of ESWCNTs significantly affects the elastic moduli. With increasing cross sectional
aspect ratio, the Young’smodulus and shearmodulus decrease.As a result, every change in
geometry operates as a defect and decreases the elastic moduli.With increasing the length,
Young’s modulus increases and the shear modulus decreases.

Key words: elastic moduli, elliptic single walled carbon nanotubes, length, chirality, cross
sectional aspect ratio, finite element model

1. Introduction

Since discovery of carbonnanotubes (CNTs) they have attracted considerable attention in scien-
tific communities. This is due to their remarkablemechanical, electrical and thermal properties.
In particular, composite materials such as carbon nanotube, nanoparticle-reinforced polymers
andmetals have shown potentially wide applications (Meo and Rossi, 2007).

The novelty in the field of mechanics includes their greatest Young’s modulus and tensiles
trengthamongall knownmaterials (Thostensonet al., 2001).Asaconsequence, theymayprovide
the ideal reinforcing material for a new class of nanocomposites (Lau and Hui, 2002a,b). Their
practical application, however, poses a great challenge to nanotechnology due to their nanoscale
size (Zaeri et al., 2010). Recent investigations (Fereidoon et al., 2012b, 2013; Hemmatian et al.,
2011; Rajabpour et al., 2012) have shown that CNTs when aligned perpendicular to cracks are
able to slow down the crack growth by bridging up the crack path. Although CNTs can be
used as conventional carbon fibers to reinforce polymermatrix in order to form advanced nano-
composites, they may be also used to improve the out-of-plane and interlaminar properties of
current advanced composite structures (Lau and Hui, 2002a). These composites have attracted
a lot of attention due to their high strength-to-weight ratio (Thostenson et al., 2001).

Qian et al. (2002) reported on one of the early investigations on nanotube-based composites;
it was observed that when nanotubes of about 1% weight are dispersed homogenously into



236 A. Fereidoon et al.

polystyrenematrices, the elastic modulus increases by 36-42% and the tensile strength improves
by about 25%. Parl et al. (2002) and Smith et al. (2002) developed and characterized nanotube
polymer composite films for spacecraft applications.

Twomain classes of theoretical methods are the atomistic based methods (Yakobson et al.,
1996; Lu, 1997; Sanchez-Portal et al., 1999) and the continuum mechanics based ones (Li and
Chou, 2003a; Chang and Gao, 2003; Odegard et al., 2002a). A few recent publications have
proposed the development of structure-property relationships for nanotubes and nanostructured
materials through the substitution of the discretemolecular structurewith equivalent continuum
models (Odegard et al., 2001, 2002b). Li and Chou (2003a, 2004) suggested a linkage betwe-
en molecular mechanics and structural mechanics in terms of geometric parameters of frame
structures.

Some researchers have attempted to characterize CNTs based on the effective bending mo-
duli. Wang et al. (2004) studied the relation between bending moment and bending curvature
of CNTs to describe the variation of effective bending modulus to Young’s modulus ratio with
ripples in nanotubes. Another investigation by Wang et al. (2004) studied the non-linear ben-
dingmoment-curvature relationship of CNTswith rippling deformation.Various theoretical and
experimental studies have been carried out to predict and characterize themechanical properties
of CNTs and related materials, all of which generally confirm their superiority. As an example,
although the results cover awide range of 270-5500GPa forYoung’smodulus and 290-2300GPa
for shear modulus of different kinds of CNT, which are several times greater than of steel (Lau
et al., 2004; Meo and Rossi, 2006; Qian et al., 2002; Thostenson et al., 2005; Yu, 2004).

Fundamental to Li and Chou (2003a,b) approach was the notion that CNTs are geometrical
space-frame structures and therefore, can be analyzed by classical structural mechanics. In this
research, based on the concept of Li et al. (2003a), three-dimensional (3D) FE models are
proposed for investigating ESWCNTs.

2. FE modeling

CNTs atoms are bonded together with covalent bonds forming a hexagonal lattice. An elliptic
singlewalled carbonnanotube(ESWCNT) is akindofnanotubethathasoval cross sectional area
(Fereidoon et al., 2012a). This kind of CNT may be produced in special states and conditions.
TheESWCNTwith a/b=1 is knownasSWCNT,where aand baremajor andminordiameters
of ellipse, respectively.

As shown in Fig. 1, these bonds have a characteristic bond length αC−C and bond angle
in the 3D space. The displacement of individual atoms under an external force is constrained
by the bonds. Therefore, the total deformation of the nanotube is a result of the interactions
between the bonds. By considering the bonds as connecting load-carrying elements, and the
atoms as joints of the connecting elements, CNTs may be simulated as space-frame structures.
In this work, a 3D FE model of ESWCNTs is proposed to assess the mechanical properties.
The 3D FEmodel is developed using the ANSYS commercial FE code. For themodeling of the
bonds, the 3D elastic BEAM4 element is used. The properties of these elements are obtained by
linking the potential energy of bonds (from a chemical point of view) and the strain energy of
mechanical elements (fromamechanical point of view). Considering a circular beamof length l,
diameter d, Young’smodulus E, and shearmodulus G, themodel represents the covalent bond
between carbon atoms.Theproperties of this element are given inTable 1 (Li andChou, 2003a).

The simulation relates the bond length αC−C with the element length L as well as the wall
thickness twith the element diameter d, as shown in Fig. 1.



Elastic moduli of carbon nanotubes with new geometry based on FEM 237

Fig. 1. Simulation of a SWCNT as a space-frame structure

Table 1.The properties of beam elements for real carbon nanotube (Li and Chou, 2003a)

Parameters Value

Nanotube diameter d 1.466Å

The length of carbon-carbon bondL 1.421Å

Cross-sectional area A 1.68794Å2

Polar inertia momentum Ixx 0.453456Å
4

Inertia momentum Izz = Iyy = I 0.22682Å
4

YoungmodulusE 5.188 ·10−8N/Å2

Shear modulusG 8.711 ·10−9N/Å2

3. Calculation of elastic moduli

3.1. Young’s modulus

Young’smodulus of amaterial is the ratio of normal stress to normal strain as obtained from
a uni-axial tension test (σ=Eε), and the strain energy of ESWCNTs is

U =
1

2
σεAL U =

1

2
Eε2A0L (3.1)

Therefore, Young’s modulus of ESWCNT is calculated using the following equation

E =
2U

A0Lε2
(3.2)

where U is total strain energy of ESWCNT obtained from summing individual strain energy of
all elements of the nanotube, ε – total applied strain, A0 – cross sectional area and L – initial
length. A0 is

A0 =π
[(

a+
t

2

)(

b+
t

2

)

−

(

a−
t

2

)(

b−
t

2

)]

(3.3)

In order to apply the conditions of tension, the nodes of the bottom end of the ESWCNT
havebeen fullybuilt-in (zero displacement and rotation conditions),while thenodes of theupper
end are subjected to tension (Jalalahmadi and Naghdabadi, 2007) (Fig. 2a). In this paper, it
has been assumed that the wall thickness of the ESWCNT is 0.34nm.

3.2. Shear modulus

For calculating the shear modulus of ESWCNTs, the following relation is used

G=
TL

Jθ
(3.4)



238 A. Fereidoon et al.

Fig. 2. Isometric view of FE meshes of the (10,10) elliptic CNT along with the applied boundary
conditions: (a) tension, (b) torsion

where T stands for the torque acting at one end of the ESWCNT, θ for the torsional angle of
the tube and J for the polar moment of inertia of the cross-sectional area. For calculating J,
the ESWCNT is considered as a hollow tube with elliptic cross section. In this case J is

J =
π

4

{(

a+
t

2

)(

b+
t

2

)[(

a+
t

2

)2

+
(

b+
t

2

)2]

−

(

a−
t

2

)(

b−
t

2

)[(

a−
t

2

)2

+
(

b−
t

2

)2]}

(3.5)

The torsional angle θ is calculated by the FE model. In order to apply the conditions of
torsion, the nodes of the bottom end of the ESWCNT have been fully built-in, while the nodes
of the upper end, were constrained frommoving in the radial direction (UR =0) and subjected
to tangential forces (Jalalahmadi and Naghdabadi, 2007) (Fig. 2b).
The values of Young’s and shear modulus calculated for SWCNTs (a/b = 1) have been

comparedwith the corresponding values from the literature. Table 2 summarizes all calculations
and the comparison. As may be seen in Table 2, elastic moduli computed by the present FE
model agrees very well with each elastic moduli obtained by the corresponding methods, and
the present method is verified.

Table 2.Comparison of elastic moduli by different researchers

Researcher Element E [TPa] G [TPa]

Li and Chou (2003a) beam 1.010 0.475

Tserpes and Papanikos (2005) beam 1.028 0.410

Kalamkarov et al. (2006) beam 0.980 0.350

Giannopoulos et al. (2008) spring 1.307 0.383

Shokrieh and Rafiee (2010) beam 1.033-1.042 0.417

Present work beam 1.033-1.035 0.46-0.49

4. Results and discussion

The mechanical properties of ESWCNTs depend on their length, chirality, diameter and cross
sectional aspect ratio (a/b). Several works investigating the dependence of elastic moduli of
SWCNTs on their diameter and chirality have been reported. In themajority of them, armchair
and zig-zag SWCNTs have been only included. In the present work, the FEmodel is applied to
investigate the effect of length, diameter, chirality and cross sectional aspect ratio (a/b) on the
elastic moduli of ESWCNTs. All three types of SWCNTs, namely, armchair, zig-zag and chiral
are included in the investigation.
The boundary conditions for tension and torsion are shown in Figs. 2a and 2b, respectively.

Cross sections of ESWCNTs for different a/b from 1 to 2 are represented in Fig. 3.
Young’s and shear modulus of ESWCNTs for some kind of armchair and zig-zag nanotubes

withdifferent lengths andcross sectional aspect ratios are calculated.Young’s and shearmodulus



Elastic moduli of carbon nanotubes with new geometry based on FEM 239

of ESWCNTs with different lengths of 25, 50, 75, 100Å for armchair nanotubes with chiralites
of (5,5), (10,10), (15,15), (20,20) are plotted against the cross sectional aspect ratio (a/b) in
Figs. 4-7.

Fig. 3. Cross sections of ESWCNTs for different a/b

Fig. 4. Variation of Young’s modulus (a) and shear modulus (b) of (5,5) with the cross sectional aspect
ratio (a/b) for different lengths

Fig. 5. Variation of Young’s modulus (a) and shear modulus (b) of (10,10) with the cross sectional
aspect ratio (a/b) for different lengths

Similarly, Figs. 8-11 showthevariation ofYoung’s and shearmoduluswith the cross sectional
aspect ratio (a/b) for different lengths of 25, 50, 75, 100 Å for zig-zag nanotubes with chirality
of (5,0), (10,0), (15,0), (20,0).

4.1. Effects of length and cross sectional aspect ratio (a/b) on the elastic moduli of
ESWCNTs

Figures 4-11 show the variation of elastic moduli of armchair, zig-zag ESWCNTs with dif-
ferent length and cross sectional aspect ratio (a/b). It can be observed that there is an evident
effect of these parameters onYoung’s and shearmodulusof the armchair and zig-zagESWCNTs.



240 A. Fereidoon et al.

Fig. 6. Variation of Young’s modulus (a) and shear modulus (b) of (15,15) with the cross sectional
aspect ratio (a/b) for different lengths

Fig. 7. Variation of Young’s modulus (a) and shear modulus (b) of (20,20) with cross sectional aspect
ratio (a/b) for different lengths

Fig. 8. Variation of Young’s modulus (a) and shear modulus (b) of (5,0) with the cross sectional aspect
ratio (a/b) for different lengths

Asmay be seen for all kinds of armchair and zig-zag nanotubes, with increasing the length,
Young’s modulus increases and the shear modulus decreases. Furthermore, for each length with
the increasing cross sectional aspect ratio (a/b), elasticmoduli decrease.Thefigures suggest that
the elastic moduli values are inversely proportional to the cross sectional aspect ratio. On the
other hand, elastic moduli of ESWCNTs are smaller than SWCNTs (a/b=1). So, by changing
geometry of SWCNTs, their elastic moduli decrease. As a result, every change in geometry
operates as a defect and decreases the elastic moduli.



Elastic moduli of carbon nanotubes with new geometry based on FEM 241

Fig. 9. Variation of Young’s modulus (a) and shear modulus (b) of (10,0) with the cross sectional
aspect ratio (a/b) for different lengths

Fig. 10. Variation of Young’s modulus (a) and shear modulus (b) of (15,0) with the cross sectional
aspect ratio (a/b) for different lengths

Fig. 11. Variation of Young’s modulus (a) and shear modulus (b) of (20,0) with the cross sectional
aspect ratio (a/b) for different lengths

4.2. Effects of chirality and cross sectional aspect ratio (a/b) on the elastic moduli of
ESWCNTs

Figures 12a and 12b indicate the variation of Young’s and shear modulus of armchair and
zig-zag ESWCNTs as a function of the cross sectional aspect ratio (a/b) with length 50Å. For
armchair and zig-zag nanotubeswith increasing chirality, the shearmodulus increases, andwith
increasing cross sectional aspect ratio (a/b), the shear modulus decreases.



242 A. Fereidoon et al.

Fig. 12. Variation of Young’s modulus (a) and shear modulus (b) with the cross sectional aspect
ratio (a/b) and length of 50Å

Young’s modulus of ESWCNTswith armchair and zig-zag chirality increases with the incre-
asing chirality for a/b=1. In themost of other cross sectional aspect ratios (a/b)with increasing
chirality, Young’s modulus decreases. With the increasing cross sectional aspect ratio (a/b) for
each chirality, Young’s modulus decreases.

4.3. Effects of chirality on the elastic moduli of ESWCNTs

In this section, elastic moduli of eight kinds of armchair, zig-zag and chiral nanotubes with
similar lengths and diameters and cross sectional aspect ratio (a/b) equal to 1.5 are calculated.
The diameters and lengths of these ESWCNTs are given in Table 3.

Table 3. Geometry and mechanical properties of different nanotubes with the same diameter
and a/b=1.5

Chirality Radius [Å] L [Å] E [TPa] G [TPa]

(10,10) 6.785 49.225 0.9415 0.3705

(11,9) 6.796 49.38 0.7837 0.3087

(12,8) 6.830 49.55 0.9017 0.3148

(13,7) 6.886 49.50 0.7844 0.3753

(14,5) 6.682 50.23 0.7096 0.3613

(15,4) 6.796 49.75 0.8788 0.3569

(16,2) 6.694 49.39 0.8314 0.3869

(17,0) 6.659 49.489 0.9204 0.3968

As shown in Fig. 13, the armchair nanotube (10,10) has the maximum Young’s modulus,
and the zig-zag nanotube (17,0) has the maximum shear modulus. The results indicate that
the chirality has evident influence on the elasticmoduli. In other words, with similar length and
diameter and different chirality, the elastic moduli are not equal. Therefore, the elastic moduli
of ESWCNTs are sensitive to their atomic arrangement.

5. Conclusions

Three-dimensional finite element models for ESWCNTs have been proposed. To create FEmo-
dels, nodes are placed at the locations of carbon atoms and the bondsbetween themaremodeled
using three-dimensional elastic beam elements. As the FE model comprises a small number of
elements, it performs under minimal computational time by requiring minimal computational



Elastic moduli of carbon nanotubes with new geometry based on FEM 243

Fig. 13. The elastic moduli versus chirality for ESWCNTwith the same diameter and a/b=1.5

power. This advantage, in combination with the modeling abilities of the FE method, extends
the model applicability to all kinds of nanotubes with a very large number of atoms. Themain
conclusions drawn from this study are as follows:

• With increasing the length of ESWCNTs, Young’s modulus increases and the shear mo-
dulus.

• For each length with the increasing cross sectional aspect ratio (a/b), elastic moduli de-
crease.

• With increasing chirality, the shear modulus of ESWCNTs increases.

• The elastic moduli of ESWCNTs are sensitive to their atomic arrangement.

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Manuscript received October 31, 2012; accepted for print September 6, 2013