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Effects of nanotube helical angle on mechanical properties of carbon nanotube reinforced polymer composites Z. Matin Ghahfarokhi a, H. Golestanian b,c,⇑ a

Mechanical Engineering Department, University of Shahrekord, Shahrekord, Iran Faculty of Engineering, University of Shahrekord, Shahrekord 8818634141, Iran c Nanotechnology Research Center, University of Shahrekord, Shahrekord 8818634141, Iran b

a r t i c l e

i n f o

Article history: Received 12 April 2011 Received in revised form 25 May 2011 Accepted 29 May 2011 Available online 25 June 2011 Keywords: Helical nanotube Straight nanotube Mechanical properties Helical angle

a b s t r a c t Carbon nanotubes (CNTs) possess exceptional mechanical properties and are therefore suitable candidates for use as reinforcements in composite materials. The CNTs, however, form complicated shapes and do not usually appear as straight reinforcements when introduced in polymer matrices. This results in a decrease in nanotube effectiveness in enhancing the matrix mechanical properties. In this paper, theory of elasticity of anisotropic materials and ﬁnite element method (FEM) are used to investigate the effects of CNT helical angle on effective mechanical properties of nanocomposites. Helical nanotubes with different helical angles are modeled to investigate the effects of nanotube helical angle on nanocomposite effective mechanical properties. In addition, the results of models consisting of helical nanotubes are compared with the effective mechanical properties of nanocomposites reinforced with straight nanotubes. Ultimately, the effects of helical CNT volume fraction on nanocomposite longitudinal modulus are investigated. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Since their discovery in the early 1990s, carbon nanotubes have excited scientists and engineers with their unusual physical and mechanical properties. Besides, their extraordinary small size, it has been suggested that carbon nanotubes are half as dense as aluminum, 1.33–1.4 g/cm3 [1]. CNTs have Young’s modulus of about 1 TPa and tensile strength 20 times that of steel alloys. To take advantage of their unique combination of size and properties, a wide variety of applications have been proposed for CNTs. These applications include reinforcement in nanocomposites, ultraﬁne sensors, bullet-proof vests, chemical and genetic probes, hydrogen and ion storage, structural materials, and particle transport [2–4]. Carbon nanotubes can be classiﬁed into two categories: singlewalled nanotubes (SWNT), multi-walled nanotubes (MWNT). SWNTs consist of a single layer of carbon atoms wrapped into a cylindrical shape. Multi-walled nanotubes consist of several concentric layers of individual carbon nanotubes that are attracted to each other through van der Waals forces. Nanotubes usually form bundles or ropes consisting of several nanotubes arranged in a closepacked cluster inside the matrix [3]. In general, embedded nanotubes seldom appear as straight inclusions but are often characterized by a certain degree of waviness along their axial direction. ⇑ Corresponding author at: Faculty of Engineering, University of Shahrekord, Shahrekord 8818634141, Iran. Tel./fax: +98 381 4424438. E-mail address: [email protected] (H. Golestanian). 0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2011.05.046

The efﬁciency of the nanotube in reinforcing the matrix is strongly dependent on its geometry, diameter, and ﬁber/matrix interface characteristics [2,3,5,6]. Experimental data show that nanotube waviness can limit modulus enhancement in comparison to the straight nanotubes [3]. One of the ﬁrst investigations in this ﬁeld was performed by Fisher et al. [2,3]. They used ﬁnite element method and micromechanics approach to evaluate mechanical properties of polymers reinforced with curved nanotubes of the form y ¼ acos 2pl z . Their results showed that even slight nanotube curvature signiﬁcantly reduces the reinforcement capacity of the CNT compared to straight nanotubes. To calculate the mechanical properties of nanocomposites with a variety of nanotubes, both molecular dynamics and continuum mechanics have been implemented. Ren et al. [7] used molecular mechanics simulation and tensile loading to investigate fatigue failure mechanisms of SWCNT bundles embedded in epoxy matrix. They found that SWCNTs within the rope may not break at the same time or at the same location. Walters et al. [8] calculated yield strength of 45 ± 7 GPa for the nanotube ropes. Odegard et al. evaluated effective elastic properties of CNT-based composites using a molecular dynamics (MD)-based multi-scale approach [9]. They calculated the properties of an effective ﬁber that contains a CNT surrounded by the polymer matrix of cylindrical shape. They computed effective Young’s and shear moduli using this approach for different CNT lengths and volume fractions. Their results showed good agreement with available experimental data. Griebel and Hamaekers [10] evaluated the effective properties of

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CNT-based polymer composites using the MD-approach. Their models predicted that the effective elastic modulus of the polymer reinforced with short CNTs to be twice as much as that of pure polymer. In case of long CNTs, nanocomposite modulus was predicted to be about 30 times polymer modulus. Liu et al. [11] developed a new continuum model and fast multipole boundary element method for large-scale analysis of CNT-based composites. In this approach, they proposed that CNTs are treated as rigid ﬁbers in an elastic matrix. Their results demonstrate that the developed fast multipole boundary element method could be used as a fast and ﬁrst-order numerical tool for large-scale characterization of carbon nanotube reinforced composites. Li and Chou performed a multi-scale analysis of the compressive behavior of carbon nanotube polymer composites [12]. They modeled the nanotube at the atomistic scale and analyzed matrix deformation using ﬁnite element method. Ajayan et al. [13] indicated weak interfacial bonding between nanotubes and resin matrix. The measurement of Schadler et al. [14] indicated better load sharing efﬁciency when the composites are under compression than tension. Wise and Hinkley used molecular dynamics simulation to investigate the interface between a SWNT and matrix [15]. Lordi and Yao [16] used force-ﬁeld-based molecular mechanics to model the interactions between nanotube and several different kinds of polymers. Li and Chou [17] used the structural mechanics approach to model the deformation of carbon nanotubes. Their computations showed that elastic moduli of nanotubes vary with nanotube diameter and are affected by CNT helical angle. Liu and Chen applied ﬁnite element and boundary element methods to determine CNT-based nanocomposite modulus containing single and multiple CNTs. They modeled the nanotubes as thin elastic layers in the shape of a capsule (closed ends) or an open cylinder [18–20]. Bonora and Ruggiero developed a unit cell model for Metal Matrix Composites (MMCs) and used their model to predict the macroscopic response of SiC/Ti unidirectional composite laminates [21]. These investigators used ﬁnite element method (FEM) and indicated that unit cell approach is successful in predicting overall behavior of composite laminates. Xu and Sengupta [22] used ﬁnite element method to investigate the interfacial stress transfer and possible stress singularities in a nanocomposite. These investigators performed micromechanical analysis on a representative volume element. They investigated the reasons for nanocomposite low failure strains. These investigators maintain that the higher Young’s modulus of the reinforcement leads to higher interfacial stresses and severe stress singularity at the ends of the nanoﬁber. In this paper, the effective mechanical properties of CNT-based nanocomposites are determined using ﬁnite element method. Analysis is performed on an RVE consisting of straight and helical nanotubes to determine the effects of nanotube characteristics on nanocomposite mechanical properties. To achieve our goals, generalized Hooke’s law is used to determine mechanical properties of helical CNT/polymer nanocomposites. Next, the effects of volume fraction and helical angle of helical nanotubes on reinforcement of polymer matrix are investigated. Ultimately, the results of the cases of helical and straight nanotube reinforced polymer composites are compared at a volume fraction of 2.4%. The analysis and results of this investigation are presented in the following sections. 2. Generalized Hooke’s law for anisotropic body Generalized Hooke’s law is valid for every continuous medium, regardless of its physical properties. Hooke’s law explains the relations between strain and stress components in the body. The generalized Hooke’s law is represented in Eq. (1). In general the number of elastic constants is 36.

8 > > > > > > > > > <

x y z

9 > > > > > > > > > =

2

1 6 Evxx 6 xy 6 Eyy

1 Eyy

v zx Exx v zy Eyy

v yz Ezz

gyz;x

gzx;x

Exx

Exx

gyz;y

gzx;y

Eyy

Eyy

1 Ezz

gyz;z

gzx;z

Ezz

Ezz

gy;yz

gz;yz Gyz

1 Gyz

lzx;yz

Gyz

gy;zx

gz;zx

lyz;zx

Gxz

Gxz

Gxz

1 Gxz

gx;xy

gy;xy

gz;xy

lyz;xy

lyz;xy

Gxy

Gxy

Gxy

Gxy

Gxy

6 6 v xz 6 Ezz ¼6 6 gx;yz > > c > yz > 6 Gyz > > > > > > 6 > 6 gx;zx cxz > > > > > 6 > :c > ; 4 Gxz xy

v yx Exx

Gyz

3 8 rx 9 > gxy;y 7 7> > > > > 7 > Eyy > ry > 7> > > > > v xy;z 7> > < = 7 r Ezz z 7 lxy;yz 7 > syz > > Gyz 7> > > > 7> > > lxy;zx 7> s xz > > > > > 7 ; Gxz 5: s gxy;x Exx

1 Gxy

ð1Þ

xy

In this equation, Exx, Eyy, and Ezz are Young’s moduli in directions x, y, and z. Gxy, Gxz, and Gyz are the shear moduli for planes parallel to the coordinate planes. In addition tij are Poisson’s ratios characterizing the strain in jth direction due to the applied stress in ith direction. The coefﬁcients lzx,yz , . . . , lzx,xy characterize shears in planes parallel to the coordinate planes produced by shearing stresses acting in other planes parallel to the coordinate planes. These coefﬁcients are called Choentsov’s coefﬁcients. The constants gyz,x, . . . , gxy,z characterize extension in the directions of the coordinate axes produced by shearing stresses acting in the coordinate planes. They are termed the mutual inﬂuence coefﬁcients of the ﬁrst kind. Finally, gx,yz . . . gz,xy, are called the mutual inﬂuence coefﬁcients of the second kind. These coefﬁcients express shears in the coordinate planes due to normal stresses acting in the directions of the coordinate axes [23]. These constants are usually non-zero for an anisotropic elastic body and are zero for an isotropic body. 3. Analysis models In this section, the models used to investigate the effects of nanotube helical angle on the effective mechanical properties of nanocomposite are presented. Two RVEs, one reinforced with a helical nanotube, and the other reinforced with a straight nanotube are considered. Perfect bonding is established between the nanotube and the matrix in both cases. CNT volume fraction in both RVEs is equal to 2.4%. Elasticity solutions can be obtained under certain load cases. The RVEs have several independent material constants that need to be determined. We used generalized Hooke’s law Eq. (1) to determine these material constants. Next, the effects of change in helical angle and CNT volume fraction on nanocomposite mechanical properties are investigated. To achieve this goal, the CNT helical angle is changed from 5° to 20°. Also, CNT volume fraction is changed from 2.4% to 5.194%. The analysis approach and the results are represented in the next sections for each case. 3.1. Analysis of the RVE reinforced with straight nanotube The square RVE used to model the nanocomposite reinforced with straight nanotube is shown in Fig. 1. The directions of the coordinate axes (1)–(3) are also shown in this ﬁgure. In our formulations and in presenting the results, however, we use x, y, and z to denote these coordinate axes, respectively. This RVE is symmetric, thus it has ﬁve independent material constants, namely; Ex = Ey, Ez, txy, txz, and Gxz. Four of these properties are determined in this paper and the shear modulus, Gxz, is not considered in this investigation. Since the RVE reinforced with the straight nanotube is transversely isotropic, the shearing and normal stress coupling coefﬁcients in Eq. (1) will be zero, that is: gij;k¼0 ; gi;jk ¼ 0; lij;km ¼ 0. Elasticity solutions for this RVE can be obtained under certain load cases [1]. Thus, the ﬁrst three relations in Eq. (1) reduce to:

8 9 > < x > = > :

y > ¼ z ;

2

1 E 6 vxxxy 6 4 Exx v zx Ezz

v xy Exx 1 Exx v zx Ezz

v zx Ezz v zx Ezz 1 Ezz

38 9 > rx > 7< = 7 ry 5> > : ;

rz

ð2Þ

Z. Matin Ghahfarokhi, H. Golestanian / Computational Materials Science 50 (2011) 3171–3177

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CNT

Matrix

Fig. 1. Cut-through view of the RVE used to model the straight CNT-based nanocomposite.

To calculate the four constants (Ez, Ex, txy, txz) we need four independent equations. To derive these equations, two different loading cases have been devised as illustrated in Figs. 2 and 3 [20]. Formulation for each of these loading cases is presented in the following sections. 3.1.1. Square RVE under an axial elongation DL In this case, shown in Fig. 2, let the length of simulated RVE in the z-direction be denoted by L and the cross sectional dimension by letter a. An elongation DL in the z-direction is imposed on the RVE in the z-direction. A change of dimension equal to Da is created in the RVE cross sectional dimensions due to Poisson’s ratio effect. Stress and strain components on the lateral surface are as follows:

rx ¼ ry ¼ 0; z ¼

y

DL Dx ; x ¼ along x ¼ a and; L a

Dy along y ¼ a ¼ a

Integrating and averaging the third equation in (2) on the plane z ¼ 2L, we obtain:

Ez ¼

rav e L ¼ r z DL a v e

ð3Þ

Fig. 3. The FEA model of the RVE under a transverse distributed load in the ydirection.

In Eq. (4), A is the RVE cross sectional area. The value of rave is evaluated by averaging stress in the z-direction on the plane z ¼ 2L in the model. Using the ﬁrst relation in Eq. (2) and the relation between stress and strain, along x = ±a, we have:

x ¼

1 A

Z

rz ðx; y; L=2Þdx dy

ð4Þ

A

Ez

rz ¼ mzx

DL Da ¼ L a

ð5Þ

Thus, we can obtain an expression for the Poisson’s ratio:

Da DL = a L

mzx ¼

ð6Þ

Eqs. (3) and (6) can be applied to estimate the effective Young’s modulus Ez and Poisson’s ratio tzx (=tzy), once the contraction Da and the average stress, rave, are obtained from FEA results. 3.1.2. Square RVE under a lateral uniform load In this load case, Fig. 3, the RVE is placed under a lateral uniform load, p, in the y-direction. The RVE is constrained in the z-direction so that the plane strain condition is maintained in order to simulate the interactions between the nanotube and matrix materials existing in the z-direction. Thus, since ez = 0 for plane strain case, Eq. (2) reduces to:

where rave is the averaged value of stress in the z-direction, given by:

rav e ¼

mzx

x y

2 ¼4

1 Ex

mxy Ex

m2zx Ez

mxy Ex

m2zx Ez

1 Ex

m2zx Ez

m2zx Ez

3 rx 5

ry

ð7Þ

In this load case, the values of stress and strain components at a point on the lateral surfaces are given by:

rx ¼ 0; ry ¼ p; x ¼

y ¼

Dx along x ¼ a and; a

Dy along y ¼ a a

where Dx and Dy are changes of dimensions in the x and y directions, respectively. Applying the ﬁrst and second equations in (7) for points along x = ±a and y = ±a, together with the above conditions, we obtain:

Dx p¼ a Ex Ez 1 m2zx Dy p¼ ¼ þ Ex Ez a

x ¼ y Fig. 2. The model of the RVE under an axial elongation DL in the z-direction. In helical CNT case an axial distributed load is applied in the z-direction.

mxy

þ

m2zx

ð8Þ

Solving these two equations gives the effective Young’s modulus and Poisson’s ratio in the transverse direction, 1–2-plane in Fig. 1, to be:

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Ex ¼ Ey ¼

Dx pa

mxy ¼ Dy pa

þ þ

1 Dy pa

þ

ð9Þ

m2zx Ez

m2zx Ez

ð10Þ

m2zx Ez

The results of axial elongation loading case are used in Eqs. (9) and (10) for Ez and tzx. Once the changes in dimensions, Dx and Dy, are determined for the square RVE from a ﬁnite element analysis, Ex = (Ey) and txy can be computed from Eqs. (9) and (10), respectively. 3.2. Analysis of RVE reinforced with helical nanotube The RVE reinforced with helical nanotube is not symmetric and is considered as an anisotropic body. In this case, to extract material constants of helical nanotube-based nanocomposite the ﬁrst three relations in Eq. (1) are used. Thus, we have:

8 9 > < x > =

y ¼ > : > z ;

2

1 Exx 6 mxy 6 4 Eyy mxz Ezz

myx Exx 1 Eyy

mzx Exx mzy Eyy

myz Ezz

1 Ezz

gyz;x

gzx;x

Exx

Exx

gyz;y

gzx;y

Eyy

Eyy

gyz;z

gzx;z

Ezz

Ezz

8 rx 9 > > > > > > > gxy;x 3> ry > > > > > > Exx > > < = 7 r gxy;y z 7 Eyy 5 > > > syz > > gxy;z > > > > > > > Ezz > > sxz > > : ;

4. Results and discussion

ð11Þ

sxy

In this paper we calculated the nine independent material properties namely (three moduli of elasticity and six Poisson’s ratios) for the anisotropic nanocomposite. The rest of the properties (shear moduli) can be calculated using torsional loadings applied to the RVE in different directions. Shear moduli are not treated in this paper. To determine the nine unknown material constants, nine independent equations are needed. Three different loading cases have been devised to provide these equations. In these loading cases, illustrated in Figs. 2–4 through, coefﬁcients gij,k are set equal to zero since sij = 0. Then, Eq. (11) is reduced to:

8 9 > < x > = > :

y ¼ > z ;

2

1 Exx 6 mxy 6 E 4 yy mxz Ezz

myx Exx 1 Eyy myz Ezz

mzx Exx mzy Eyy 1 Ezz

38 9 > rx > 7< = 7 ry 5> > : ;

ð12Þ

rz

Initially, the RVE shown in Fig. 4 is loaded by a tensile stress along the x axis. Then, using the ﬁrst relation in Eq. (12) Young’s modulus Exx can be calculated. In case of the helical CNT nanocomposite, three distributed loading cases were applied. Each of these three loading cases consisted of a distributed load applied in one of the coordinate directions, Figs. 2–4. That is, three different models were created and in each one the distributed load was applied in one coordinate direction. Using a similar method, for loadings and using the second and the third relations in Eq. (12) Young’s moduli Eyy and Ezz, are determined. In Eq. (12), strain components in different directions are calculated using;

e¼

u L

Fig. 4. The RVE model of the helical nanotube-polymer nanocomposite under a transverse distributed load in the x-direction.

ð13Þ

where u is the average value of displacement of nodes on the considered plane and L is the initial length of RVE in the corresponding direction. Finally, using three loading cases and strains calculated in directions x, y, and z along with the applied arbitrary stresses, Poisson’s ratios can be calculated. For example, to calculate ex, the RVE is cut at an arbitrary location perpendicular to the CNT in the composite region. Then, the displacements in the x-direction of nodes located on the cross section are averaged. Finally, the calculated average value is divided by the initial width of the RVE in the x-direction.

Models are developed based on the above discussions to determine nanocomposite effective mechanical properties. The material properties of nanocomposite constituents are listed in Table 1. The dimensions of the RVEs are as follows: length = 266 nm, width = 29.3 nm. The nanotube dimensions are: length = 236 nm, radius = 2.72 nm. In case of the helical nanotube, the RVE consists of a CNT with a helical angle of 16°. The outer diameter of the helix is 16.32 nm, and the diameter of helical nanotube in the helix is 5.44 nm. The helical nanotube has a pitch of 118 nm. 4.1. Straight CNT-based nanocomposite results The RVE reinforced with straight nanotube was modeled on the basis of the previous discussions. Von Misses stress contour plot under the axial elongation of the RVE is shown in Fig. 5. Note that maximum stress occurs in the nanotube, indicating the role of nanotube in carrying most of the load. The calculated mechanical properties for straight CNT/polymer nanocomposite are listed in Table 2. The results indicate that addition of only 2.4% nanotube to the matrix results in an increase in matrix modulus by a factor of 3.48 in the longitudinal direction. The ratio of nanocomposite transverse modulus to matrix modulus is 1.01. This result suggests that the straight nanotube does not contribute to nanocomposite strength in the transverse direction. 4.2. Helical CNT-based nanocomposite results In this section, the results of nanocomposite reinforced with helical nanotube are presented. As mentioned above, helical angle of helical nanotube is 16° and CNT volume fraction is 2.4%. Von Misses stress contour plot for helical nanotube case is shown in Figs. 6 and 7. Only portions of the helical CNT can be seen in Fig. 6. Fig. 7 shows the helical CNT without the surrounding matrix for a better observation of the stresses in the CNT. Note that the stresses vary along the helical nanotube and are not uniform as they were in the straight nanotube shown in Fig. 5. The results of helical CNT-based nanocomposite are listed in Tables 3 and 4.

Table 1 Mechanical properties of nanocomposite constituents.

t

E (GPa)

Material

0.3 0.3

3.2 1000

Matrix CNT

Z. Matin Ghahfarokhi, H. Golestanian / Computational Materials Science 50 (2011) 3171–3177

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Fig. 5. Von Misses stress contour plot for straight nanotube-polymer nanocomposite under an axial elongation DL.

Table 2 Mechanical properties of straight CNT nanocomposite.

txy

tzx

Ex Em

0.42

0.3

1.01

E

¼ Emy

Ez Em

3.48

The ratio of nanocomposite modulus to matrix modulus in the longitudinal direction, Ez / Em, is equal to 2.21 in this case. This is about 36% lower than the straight CNT reinforced nanocomposite longitudinal modulus. This comparison indicates that the strengthening efﬁciency of helical nanotube decreases in the longitudinal direction due to the CNT helical angle. The ratios of transverse moduli to matrix modulus, however, are equal to 1.15. These results suggest that the helical CNTs reinforce the matrix in all three directions. In fact, in case of transverse moduli, the portion of the CNT which is oriented along the x- and y-directions contributes to the nanocomposite moduli in those directions.

Next, to investigate the effects of change in CNT helical angle on nanocomposite mechanical properties, the CNT helical angle is changed from 5° to 20°. Due to the importance of axial modulus, the effect on this parameter is investigated. The variation of nanocomposite longitudinal modulus to matrix modulus, Ez/Em, with helical angle is shown in Fig. 8. The results suggest that Ez/Em decreases with increasing the CNT helical angle, as expected. This is due to the fact that less and less nanotube length is oriented along the RVE axis as the CNT helical angle increases. A helical angle of only 5° results in an almost 30% decrease in nanocomposite longitudinal modulus.

4.3. Effects of CNT volume fraction To investigate the effects of CNT volume fraction on nanocomposite longitudinal modulus, models were created with helical CNT volume fractions ranging from 2.4% to 5.19%. The models

Fig. 6. Von Misses stress contour plot for helical nanotube-polymer nanocomposite under a distributed load in the z-direction.

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Fig. 7. Von Misses stress contour plot in the helical nanotube inside the RVE, with matrix removed, under a distributed load in the z-direction.

Table 3 Young’s moduli for helical CNT nanocomposite with helical angle of 16°. Ex Em

E

¼ Emy

1.15

Ez Em

Em (GPa)

2.21

3.2

Table 4 Poisson’s ratio of helical CNT nanocomposite with helical angle of 16°.

tyx = txy

tyz

tzy

txz

tzx

Em (GPa)

0.34

0.66

0.35

0.66

0.337

3.2

Fig. 9. Variation of Ez/Em with volume fraction for helical nanotube polymer nanocomposite.

Fig. 8. Variation of Ez/Em with angle for helical nanotube polymer nanocomposite.

consisted of a helical CNT with helical angle of 16°. The variation of nanocomposite longitudinal modulus to matrix modulus, Ez/Em, with CNT volume fraction is shown in Fig. 9. These results indicate a linear increase in the nanocomposite longitudinal modulus with CNT volume fraction. Note that addition of 5.19% CNT to the matrix increases its longitudinal modulus by a factor of 3.24. 5. Conclusions In this article the effects of CNT geometry and volume fraction on nanocomposite mechanical properties are investigated. FEA

results suggest that the maximum stress occurs in the CNT indicating that the nanotube takes most of the applied load in the nanocomposite. Nanocomposite reinforced with straight nanotubes has a higher longitudinal modulus compared to the nanocomposite reinforced with helical nanotubes. This indicates a higher efﬁciency of straight nanotubes in reinforcing the matrix. As the nanotube helical angle increases, nanocomposite longitudinal modulus decreases. A helical angle of only 5° results in an almost 30% decrease in nanocomposite longitudinal modulus. Finally, the nanocomposite longitudinal modulus increases with increasing CNT volume fraction.

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