Journal of Mechanical Science and Technology 26 (8) (2012) 2565~2572

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-012-0639-5

The effect of CNT volume fraction on the magneto-thermo-electro-mechanical

behavior of smart nanocomposite cylinder† A. Ghorbanpour Arani1,2,*, M. Rahnama Mobarakeh1, Sh. Shams3 and M. Mohammadimehr1

1Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran 2Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Iran

3Department of mechanical Engineering, Faculty of Engineering, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Isfahan, Iran

(Manuscript Received June 9, 2011; Revised February 17, 2012; Accepted March 26, 2012)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract In this article, using analytical approach, the stress analysis of a long piezoelectric polymeric hollow cylinder reinforced with carbon

nanotube (CNT) under combined magneto-thermo-electro-mechanical loading is investigated. Considering three combined loading con-ditions such as pressure-electric, pressure-electric magnetic and pressure-electric thermal, the governing equation of the problem is ob-tained. The rule of mixture and modified multiscale bridging model are used to predict effective properties of nanocomposite. The mag-neto-thermo-electro-mechanical stresses in hollow cylinder are discussed in detail. It can be concluded that increasing CNT volume frac-tion enhances strength of the nanocomposite cylinder. The results of this work could be useful in view of optimum design of the smart nanocomposite cylinder under magneto-thermo-electro-mechanical loadings and could also be as a reference for future related works.

Keywords: Magneto-thermo-electro-mechanical stress; Smart nanocomposite; Hollow cylinder; CNTs ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Numerous structural components of piezoelectric compos-ites have cylindrical like configuration. Piezoelectric materials have been used extensively in engineering applications. Pie-zoelectric materials produce an electric field during deforma-tion and deformed when subjected to an electric field. Ad-vancement of smart structure has attracted interests of many researchers in the precise prediction of response characteristics of composite hollow cylinder. Recently, magneto-electro-thermo-mechanical coupling behavior in piezoelectric com-posites has been studied widely in the literature. According to superior material properties of CNTs, they have also been employed as reinforcing factor in improving composites. Mitchell and Reddy [1] presented a power series solution for the static analysis of an axisymmetric composite cylinder with surface bonded or embedded piezoelectric laminate. Wang and Lu [2] investigated a theoretical method to analyze mag-neto–thermo–elastic waves and perturbation of the magnetic field vector produced by thermal shock in a solid conducting cylinder. By virtue of the separation of variables technique, analytical solution of a special non-homogeneous pyroelectric

hollow cylinder for piezothermoelastic axisymmetric plane strain dynamic problem was investigated by Ding et al. [3]. Dai and Wang [4] presented an analytical solution for mag-neto–thermo–electro–elastic problems of a piezoelectric hol-low cylinder placed in an axial magnetic field subjected to arbitrary thermal shock, mechanical load and transient electric excitation. Hou et al. [5] illustrated transient responses of magneto-electro-elastic hollow cylinders by virtue of the sepa-ration of variables and orthogonal expansion technique. Ghorbanpour et al. [6] investigated a stress and electric poten-tial field in rotating hollow circular shaft made from function-ally graded piezoelectric material. Their results show that the material in-homogeneity has a significant influence on the electro-thermo-mechanical behavior of the FPGM rotating shaft.

The buckling analysis of laminated composite plates rein-forced by single-walled carbon nanotubes (SWCNTs) is car-ried out using an analytical approach as well as the finite ele-ment method was studied by Ghorbanpour et al. [7]. Their results showed that the critical buckling load obtained from FEM for different layup and boundary conditions is in good agreement with those obtained by the analytical solution. They also concluded that the agglomeration of CNTs has significant influence on the buckling load and properties of CNT rein-forced composite.

In this study, a CNT reinforced hollow cylinder subjected to

*Corresponding author. Tel.: +98 3615912447, Fax.: +98 3615912424 E-mail address: aghorban@kashanu.ac.ir; a_ghorbanpour@yahoo.com

† Recommended by Editor Maenghyo Cho © KSME & Springer 2012

2566 A. G. Arani et al. / Journal of Mechanical Science and Technology 26 (8) (2012) 2565~2572

magnetic, thermal, and mechanical loads together with a po-tential difference induced by electrodes attached to the inner and outer surface of the cylinder is investigated. Matrix mate-rial is poly vinylidene fluoride (PVDF). Due to high length to diameter ratio, the plane strain state is considered. The re-sponse of nanocomposite hollow cylinder to the combined loadings has general and particular solutions. General solution is obtained from Cauchy-Euler method and the particular re-spond specify from indefinite coefficients approach. The mix-ture law method is utilized to model and specify effective properties of nanocomposite. Bonding between nanotube and matrix is perfect and interface debonding is neglected. Finally the effects of CNTs reinforcement on the magneto-thermo-electro-mechanical stresses are discussed in detail.

2. Effective properties of nanocomposite

Several studies on carbon nanotubes-reinforced composites (CNTRCs) have focused on their material properties [8-14]. Most of these investigations have shown that the addition of small amount of CNT can considerably improve the mechani-cal, electrical and thermal properties of polymeric composites [8-12]. When the size of CNT diameter is within nano scale, then both CNT material property and interphase properties vary significantly according to the size of CNT [8-12]. Me-chanical and thermal properties of CNTRC are obtained using the rule of mixture and modified multiscale bridging models [12]. However, magnetic and electric coefficients are obtained using the rule of mixture. CNTs are assumed isotropic, elastic and uniformly distributed in matrix. For simplicity, the mix-ture law is developed for magneto-thermo-electro-mechanical coefficients of nanocomposite are defined as follows [13]:

,cnt m

ij cnt ij m ijC c C c C= + (1a)

,cnt mij cnt ij m ije c e c e= + (1b)

,cnt mij cnt ij m ijc cν ν ν= + (1c)

1 1 1 ,cnt mcnt mc cα α α= + (1d)

2 2 2 1(1 ) (1 ) ,cnt cnt m mcnt mc cα ν α ν α να= + + + − (1e)

,cntij m ijcμ μ= (1f)

,mij m ijc∈ = ∈ (1g)

( ) ( )cnt mij cnt ij m ijc cβ β β= + (1h)

where ijC , ije , ijν , iα , ijμ , ij∈ , ijβ are the elastic stiffness constants, piezoelectric constants, Poisson’s ratio, thermal expansion coefficients, magnetic permeability, dielectric con-stant and pyroelectric coefficient, respectively. cntc and

mc indicate fiber volume fraction and matrix volume fraction, respectively. Superscripts cnt and m refer to fiber and matrix, respectively.

According to the modified multiscale bridging model, the effective stiffness of the nanocomposite is presented as [12]

( )

1inf

11inf inf

. ( ). . . .( . ) ,

.

cnt p

M

C C I S I c R C I S

C C C S

−

−−

⎡ ⎤= + − Φ + + Φ⎣ ⎦

⎡ ⎤Φ = − −⎢ ⎥⎣ ⎦

(2)

where C, S, R, I are nanocomposite stiffness tensor, Eshelby tensor, compliance tensor, identity tensor, respectively. SM is the modified Eshelby tensor. pC is the modified stiffness tensor. Φ is the fourth-order tensors. The subscript inf indi-cates interface of the CNT and matrix [12]. It should be men-tioned that for small volume fractions, the elastic stiffness of the matrix phase is enough to be used as the elastic stiffness of the infinite matrix [12]. It is noted that the electric properties of CNTs and magnetic properties of PVDF are neglected in this study.

3. Governing equations

As presented in Fig. 1, a long piezoelectric fiber reinforced composite hollow cylinder (PFRCHC) is considered in an axial magnetic field (0, 0, )zH H=

r and radial temperature

gradient T(r). a and b denote the inner and outer radii of cylinder, respectively.

In the cylindrical coordinates ( , , )r zθ system, assuming axi-symmetric plane strain state, the components of displacement and electric potential are 0,zu uθ = = ( ),ru u r= and

( ),rϕ ϕ= respectively. The constitutive equations for piezo-electric composites are written as [14]

11 12 11 1 ( ),rr rr rC C e E T rθθσ ε ε λ= + − − (3a)

12 22 12 2 ( ),rr rC C e E T rθθ θθσ ε ε λ= + − − (3b)

11 12 11 ( )r rr rD e e E T rθθε ε β= + −∈ + , (3c)

1 11 1 12 2 2 12 1 22 2, ,C C C Cλ α α λ α α= + = + (3d)

where ijσ , ijε , rD , rE are the component of stresses, component of strains, radial electric displacement and radial electric potential, respectively. In order to develop the analyti-cal solution, the dimensionless parameters are introduced as

11 12 22 11 12

1 2 3 1 211 11

, , , , ,C C C e eC C C e eY Y Y Y Y

= = = = =∈ ∈

Fig. 1. A schematic of a long polymeric piezoelectric hollow cylinder reinforced with CNTs under internal pressure and applied voltage.

A. G. Arani et al. / Journal of Mechanical Science and Technology 26 (8) (2012) 2565~2572 2567

11

11

1 21 2

11

. , , , ,.

, , , ,

, , ,

rr rrr r

r zzz

DDY b Y Y Y

u a r fU s f bb b b Y

Y YY

θθθ

φ σ σ

ρ

β λ λβ

∈Φ = Σ = Σ = =

∈

= = = =

= Γ = Γ =∈

(4)

where Y is the yield stress of the matrix.

The strain-displacement equations and the relation between the electric field and electric potential are respectively ex-pressed as [6]

, ,r rrr

du udr rθθε ε= = (5)

.dEdrφ

= − (6)

Substituting Eqs. (4)-(6) into Eq. (3) can be represented as

1 2 1 1 ,r rdU U dC C e Td dρ ρ ρ

ΦΣ = + + −Γ (7a)

2 3 2 2 ,rdU U dC C e Td dθ ρ ρ ρ

ΦΣ = + + −Γ (7b)

1 2 .r rdU U dD e e Td d

βρ ρ ρ

Φ= + − + (7c)

Considering that the magnetic permeability ( )μ of the

PFRCHC equals the magnetic permeability of the medium around it, the governing electrodynamics Maxwell equations are written as follows [4]:

( ,0,0), (0,0, ) , (0, ,0),

(0, ,0), ( )

rr z z

z r rz z

uU u h h e Ht

h u uJ h Hr r r

μ ∂= = = −

∂∂ ∂

= − = − +∂ ∂

rr r

r (8)

where J

r, er , h

r, Ur

and (0, 0, )zH H=r

are electric cur-rent density vector, perturbation of electric field vector, per-turbation of magnetic field vector, displacement vector and magnetic intensity vector, respectively. The electromagnetic equilibrium equations of PFRCHC can be expressed as [15, 16]

0,rr rrzz

d fdr r

θθσ σ σ−+ + = (9a)

0rr rrdD Ddr r

+ = (9b)

where zzf is Lorentz’s force, which defined as [4]

2( ) . ( ) .r r

zz zu uf J H H

r r rμ μ ∂ ∂

= × = +∂ ∂

rv (9c)

Substituting Eq. (7) into Eq. (9), the dimensionless forms of the equilibrium equation is rewritten as follows:

2 2

1 22 2 2

34 5

1

0,

d U dU U dI Idd d

I d dT TI Id d

ρ ρρ ρ ρ

ρ ρ ρ ρ

Φ+ − +

Φ+ − − =

(10a)

2 21 2

1 2 2

1 0

d U e e dU dedd d

d dT Td d

ρ ρρ ρ

β βρ ρ ρ ρ

+ Φ+ −

Φ− + + =

(10b)

where I1, I2, I3, I4 and I5 are defined as

2

31

1 22 2

1 1

1 2 13 42 2

1 1

1 25 2

1

, ,

, ,

.

HC eYI IH HC CY Y

e eI IH HC CY Y

IHCY

μ

μ μ

μ μ

μ

+= =

+ +

− Γ= =

+ +

Γ −Γ=

+

(11)

Considering teρ = , Eqs. (10a)-(10b) are reformulated as

2 21 2 3 2

4 5

( )

0,t t

D U I U I D I I DdTI e I e Tdt

− + Φ + − Φ

− − = (12a)

2 21 2

0t t

e D U e DU DdTD e e Tdt

β β

+ − Φ

− Φ + + = (12b)

where /D t= ∂ ∂ and 2 2 2/D t= ∂ ∂ . The governing differen-tial equation system has general and particular solutions. The general solution is obtained using Cauchy-Euler method ac-cording to procedure are presented as follows extracting U from Eq. (12a) yields

2

2 3 22

1

( ( ) ) .I D I I DUD I

+ − Φ=

− + (13)

Substituting Eq. (13) into Eq. (12b), we have

4 31 2 1 3 1 2 2 2

22 3 2 2 1 1

( 1) ( 1)0.

( )

e I D e I e I e I D

e I e I I D I D

⎡ ⎤+ + − + +⎢ ⎥Φ =⎢ ⎥+ − − −⎣ ⎦

(14)

The corresponding general solution for above equation is

expressed in the following form:

2568 A. G. Arani et al. / Journal of Mechanical Science and Technology 26 (8) (2012) 2565~2572

1 2w t w tg A Bt Ce DeΦ = + + + (15)

in which A ,B ,C and D are unknown constants obtained from boundary conditions. Moreover, 1w and 2w are two constants and t is an arbitrary variable.

Substituting Eq. (15) into Eq. (13), the general solution are obtained as

1 1

1

2

23 2 1 23 2

21 1

23 2 2 2 2

21 2

( )

( ) .

w tg

w t

I I w I wI IU B CeI I w

I I w I w DeI w

− +−= +

−

− ++

−

(16)

To obtain the particular solution, the thermal analysis is

needed. Heat equation for one-dimensional conduction in cylindrical coordination without heat generation can be writ-ten as [17]

1 ( ( )) 0.d rT rr dr

= (17)

The solution of Eq. (17) is obtained as

1 2( ) ( )T r K Ln r K= + (18)

where 1K and 2K are obtained from boundary conditions. The dimensionless form of Eq. (18) is expressed as

1 2( ) ( ) .oT K Ln r Kρ ρ= + (19)

Substituting Eq. (19) into Eqs. (12a) and (12b) and using

indefinite coefficients approach, the particular solutions is obtained as follows:

1 2 ( ),p x x Lnρ ρ ρΦ = + (20a)

6 7 ( )pU x x Lnρ ρ ρ= + (20b)

where x1, x2, x3 and x4 are related to K , e , β , Γ and .C Total solution for Φ and U can be expressed as

,g pU U U= + (21)

1

2

22 3 2 1 3 2 1

21 1 1

22 2 3 2 2

21 2

6 7

( )

( )

( ),

w

w

I I I w I I wU B CI I w

I w I I w DI w

x x Ln

ρ

ρ

ρ ρ ρ

− + + −= +

−

+ −+

−+ +

(22)

,g pΦ = Φ +Φ (23) 1 2

1 2( ) ( ).w wA BLn C D x x Lnρ ρ ρ ρ ρ ρΦ = + + + + + (24) Substituting Eqs. (22) and (24) into Eq. (7), the radial and

circumferential stresses for nanocomposite hollow cylinder under magneto-thermo-electro-mechanical loads are obtained as

1

2

12 32 1

12

2 1 3 2 11 1 2

1 12

12 1 3 2 12 1 12

1 12

2 2 3 2 21 2 2

1 22

12 2 3 2 22 1 22

1 2

1 6 7 1 2 7

2 6 1 1 2

( )

( )( ( )

( )( ) )

( )( ( )

( )( ) )

[ ( ) ( ) ( )( )

r

w

w

I IC e BI

I w I I wC wI w

I w I I wC e w CI w

I w I I wC wI w

I w I I wC e w DI w

C x x C C x LnC x e x x e

ρ

ρ

ρ

ρ

−

−

−

− +Σ = − +

+ −+

−

+ −+ +

−

+ −+

−

+ −+ +

−+ + + ++ + + + 1 2 1( ) ( )],x Ln Tρ ρ−Γ

(25a)

1

2

12 33 2

12

2 1 3 2 12 1 2

1 12

2 1 3 2 13 2

1 12

1 2 2 3 2 22 1 2 2 2

1 22

12 2 3 2 23 2 22

1 2

2 6 7 2 3 7

3 6 2 1 2

( )

( )( ( )

( )( )

( )) ( ( )

( )( ) )

[ ( ) ( ) ( )( )

w

w

I IC e BI

I w I I wC wI w

I w I I wCI w

I w I I we w C C wI w

I w I I wC e w DI w

C x x C C x LnC x e x x e

θ ρ

ρ

ρ

ρ

−

−

−

− +Σ = − +

+ −+

−

+ −+

−

+ −+ +

−

+ −+ +

−+ + + ++ + + + 2 2 2( ) ( )].x Ln Tρ ρ−Γ

(25b)

4. Numerical results and discussions

Stress analysis of a hollow composite cylinder reinforced by CNTs is considered. The variation of radial, circumferential and Von Mises stresses are shown in Figs. 2 to 10. The tem-perature gradient is considered from ( ) 300T a K= to

( ) 350T b K= for all boundary conditions. All quantities are plotted respect to dimensionless radius. In numerical calcula-tions, the CNTs constants are presented in Table 1 [18].

Table 1. Material properties of CNT.

Elastic stiffness constants C11 (TPa) 1.o554

Elastic stiffness constants C12 (TPa) 0.1517

Elastic stiffness constants C22 (TPa) 0.4113

Poisson’s ratio 0.278

Thermal expansion coefficients α11 (1/K) 5.1×10-6

Thermal expansion coefficients α12 (1/K) 3.45×10-6

Magnetic permeability 4π×10-7

Magnetic intensity vector 2.23×109

A. G. Arani et al. / Journal of Mechanical Science and Technology 26 (8) (2012) 2565~2572 2569

The material, thermal, and electrical constants for PVDF are given in Table 2 [19].

Two cases for combined magneto-thermo-electro-mechanical loading are considered in this study. The first case is combination of electric potential and magnetic fields in presence of internal pressure. In the second case the electric potential field, thermal gradient and internal pressure are ap-plied. The uniform electric potential field is considered across the annulus. Moreover, the magnetic field is uniformly applied in longitudinal direction. These boundary conditions can be mathematically expressed as

( ) 0.5, (1) 0, ( ) 0, (1) 1.rr rrρ ρΣ = − Σ = Φ = Φ = (26)

Figs. 2 to 4 show the effects of thermal and magnetic fields

on stresses of the nanocomposite cylinder. Fig. 2 depicts the distribution of radial stress along the dimensionless radius. As can be seen the radial stresses at internal and external radii of nanocomposite cylinder satisfy boundary conditions. It can also be observed that employing magnetic field in longitudinal direction of the cylinder increases the radial stress. The pre-scribed temperature gradient reduces the radial stress of the cylinder.

As can be seen in Fig. 3, magnetic and thermal fields have both the same decreasing influence on tangential stress. This

figure illustrates that the tangential stresses in internal and external radius of the cylinder conform to the boundary condi-tions. The maximum value of tangential stress occurs in inter-nal radius where there is the possibility of crack propagation and growth. As can be also seen from this figure that the max-imum tangential stress is related to pressure-electric loading and the minimum is associated with pressure-electric-thermal loading condition.

Fig. 4 shows the Von Mises stress for three prescribed load-ing conditions. It can be seen that this stress decreases with increasing in dimensionless radius. Moreover, applying tem-perature gradient and magnetic field reduced Von Mises stress.

In Figs. 5-10, the influence of CNT volume fraction on cor-responding stresses of the cylinder is investigated. In Figs. 5-7, the nanocomposite cylinder is under combined magneto-electro-mechanical loadings. The influence of magnetic field appears as Lorentz’s force in radial direction. Although CNT existence improved mechanical properties of nanocomposite cylinder, variation in volume fraction of CNTs doesn’t have influence on properties of nanocomposite cylinder. This hap-pens due to the fact that the magnetic coefficients of CNTs are negligible. In Figs. 8-10, the nanocomposite cylinder is under electro-thermo-mechanical loadings. Applying temperature gradient decreases radial, tangential and Von Mises stresses. By in-creasing volume fraction of CNT, the tangential stress change from tension stress to compression stress as shown in Fig. 9.

Table 2. Material properties of PVDF.

Elastic stiffness constants C11 (GPa) 3.8

Elastic stiffness constants C12 (TPa) 1.9

Elastic stiffness constants C22 (TPa) 3.2

Poisson’s ratio 0.3

Thermal expansion coefficients α11 (1/K) 1.45×10-4

Thermal expansion coefficients α12 (1/K) 0.13×10-4

Yield Strength (GPa) 54

Piezoelectric constants e11 (C/m2) -0.027

Piezoelectric constants e12 (C/m2) 0.024

Dielectric constant ∈ 0.65×10-10

Pyroelectric coefficient β 3.8×10-6

Fig. 2. Dimensionless radial stress versus dimensionless radius under three cases of combined loading conditions.

Fig. 3. Dimensionless tangential stress versus dimensionless radius under three cases of combined loading conditions.

Fig. 4. Dimensionless Von Mises stress versus dimensionless radius under three cases of combined loading conditions.

2570 A. G. Arani et al. / Journal of Mechanical Science and Technology 26 (8) (2012) 2565~2572

Therefore, this phenomenon improved mechanical proper-ties of nanocomposite cylinder in fatigue life problems. Fig. 10 shows that Von Mises stress as a function of dimensionless radius for electro-thermo-mechanical loading condition. As can be observed from this figure for each CNT volume frac-tion Von Mises stress is decreased with increasing radius. This means maximum Von Mises stress is occurred at the inner surface of the cylinder where the possibility of fatigue crack growth exists. Also, as can be seen from Fig. 10, with increas-ing CNT volume fraction, maximum Von Mises stress at the inner surface of the cylinder is decreasing. As a result, the increase of CNT volume fraction enhances the strength of the nanocomposite cylinder.

Using Eqs. (1) and (2), the dimensionless stresses of the cyl-inder were estimated according the rule of mixture and the modified multiscale bridging model, respectively (Figs. 11-

Fig. 5. Dimensionless radial stress versus dimensionless radius under magneto-electro-mechanical loading.

Fig. 6. Dimensionless tangential stress versus dimensionless radiusunder magneto-electro-mechanical loading.

Fig. 7. Dimensionless Von Mises stress versus dimensionless radius under magneto-electro-mechanical loading.

Fig. 8. Dimensionless radial stress versus dimensionless radius underelectro-thermo-mechanical loading.

Fig. 9. Dimensionless tangential stress versus dimensionless radius under electro-thermo-mechanical loading.

Fig. 10. Dimensionless Von Mises stress versus dimensionless radiusunder electro-thermo-mechanical loading.

Fig. 11. Dimensionless radial stress versus dimensionless radius under electro-thermo-mechanical loading using two models of evaluating effective elastic stiffness.

A. G. Arani et al. / Journal of Mechanical Science and Technology 26 (8) (2012) 2565~2572 2571

13). As can be seen the estimated stresses according to the modified multiscale bridging model is larger in magnitude than those obtained using the rule of mixture. This happens because of the fact that in the former the effect of weakened interface is considered.

5. Conclusions

In this article, the magneto-thermo-electro-mechanical be-havior of the smart polymeric piezoelectric hollow cylinder reinforced with CNT was investigated. An analytical solution technique was developed for this study, where stresses were produced by the combined effects of temperature gradient, magnetic field, internal pressure and electric potential. Dimen-sionless stress distributions, for different CNT volume fraction were investigated. The results showed that the influence of internal pressure on the radial stress is larger than thermal, magnetic and electric fields. The presence of magnetic field decreases the tangential and Von Mises stresses. Moreover, in presence of magnetic field, although adding CNTs to the ma-trix improve mechanical properties of the nanocomposite, variation in CNT volume fraction has not considerable effect on the corresponding stresses. Also, the existence of CNTs in the matrix have the same influence on the mechanical proper-ties of nanocomposite for two combined loading conditions, but the effect of increasing volume fiber fraction in the former is smaller than later in magnitude. In addition, elastic proper-ties of nanocomposite using the modified multiscale bridging

instead of the rule of mixture are evaluated precisely. It is seen that the dimensionless Von Mises, circumferential and radial stresses of the modified multiscale bridging model is larger than those obtained using the rule of mixture. This happens because of the fact that in the former the effect of weakened interface is considered. It is to say, the nanocomposite struc-ture investigated here are useful from a design point of view in that they can be tailored for specific applications to control the distributions of magneto-thermo-electro-mechani-cal stresses.

Acknowledgement

The authors would like to thank the referees for their valu-able comments. The authors are grateful to University of Ka-shan for supporting this work by Grant No. 65475/4. They would also like to thank the Iranian Nanotechnology Devel-opment Committee for their financial support.

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Ali Ghorbanpour Arani received his B.Sc. degree from Sharif University of Technology, Tehran, Iran, in 1988. He then received his M.Sc. degree from Amirkabir University of Technology, Tehran, Iran, in 1991 and his Ph.D degree from the Esfahan University of Technol-ogy, Esfahan, Iran, in 2001. Dr. Ali

Ghorbanpour Arani is a Professor in the Mechanical Engineer-ing Department of University of Kashan, Kashan, Iran. His current research interests are stress analyses, stability and vibra-tion of nanotubes and functionally graded materials (FGMs).

Maedeh Rahnama Mobarakeh re-ceived her B.Sc. degree from Sharif University of Technology in Tehran, Iran, in 2006. She then received her M.Sc. degree from University of Kashan in Kashan, Iran, in 2010. Her research interests are stress analysis, nanome-chanics and composite materials.

Shahrooz Shams received his B.Sc. degree from Isfahan University of Tech-nology in Isfahan, Iran, in 2007. He then received his M.Sc. degree from University of Kashan in Kashan, Iran, in 2010. He is currently a Ph.D student at University of Kashan in Kashan, Iran. His research interests are continuum

mechanics, nanomechanics, composites and functionally graded materials (FGMs); finite element and mesh free meth-ods.

Mehdi Mohammadimehr received his B.Sc. degree from the University of Kashan in Kashan, Iran, in 2002. He then received his M.Sc and Ph.D de-grees from Shahid Bahonar University of Kerman in Kerman, Iran, in 2004 and 22 May 2010. Dr. Mehdi Mohammadi-mehr is currently an Assistant Professor in Mechanical Engineering Department

of University of Kashan in Kashan, Iran. His research interests include elasticity, plasticity, continuum mechanics, nanome-chanics, composite materials, functionally graded materials (FGMs), beams, plates and shells theories, buckling, post-buckling and vibration analyses of carbon nanotubes (CNTs), and finite element method (FEM).