Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes

Composites: Part B 43 (2012) 2031–2040

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Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes S. Jafari Mehrabadi ⇑, B. Sobhani Aragh, V. Khoshkhahesh, A. Taherpour Department of Mechanical Engineering, Arak Branch, Islamic Azad University, Arak, Iran

a r t i c l e

i n f o

Article history: Received 15 April 2011 Received in revised form 15 January 2012 Accepted 24 January 2012 Available online 2 February 2012 Keywords: A. Plates B. Buckling B. Mechanical properties C. Numerical analysis Carbon nanotube

a b s t r a c t In this paper, the mechanical buckling of a functionally graded nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes (SWCNTs) subjected to uniaxial and biaxial in-plane loadings is investigated. The material properties of the nanocomposite plate are assumed to be graded in the thickness direction and vary continuously and smoothly according to two types of the symmetric carbon nanotubes volume fraction profiles. The material properties of SWCNT are determined according to molecular dynamics (MDs), and then the effective material properties at a point are estimated by either the Eshelby–Mori–Tanaka approach or the extended rule of mixture. The equilibrium and stability equations are derived using the Mindlin plate theory considering the first-order shear deformation (FSDT) effect and variational approach. The results for nanocomposite plate with uniformly distributed CNTs, which is a special case in the present study, are compared with those of the symmetric profiles of the CNTs volume fraction. A numerical study is performed to investigate the influences of the different types of compressive in-plane loadings, CNTs volume fractions, various types of CNTs volume fraction profiles, geometrical parameters and different types of estimation of effective material properties on the critical mechanical buckling load of functionally graded nanocomposite plates. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Many researches regarding composite structures reinforcement by nanotubes have been performed in the last few years. Most of these researches relate to carbon nanotubes and composite structures properties. In this introduction we introduce a few of them. Fukuda and Kawata [1] studied the Young modulus of composite structures reinforced by short fibers. The mentioned paper is one of the first studies concerning the properties of reinforced structures. Also Jin and Yuong [2] determined the elasticity properties of single-walled carbon nanotubes (CNTs) applying the molecular dynamic method in. In fact this macroscopic behavior was analyzed by studying the interaction of the atomic force and dynamic response in nanostructures affected by insignificant strain. Chang and Gao [3] studied the dependence of SWCNTs elastic properties on its dimensions according to the molecular mechanics model. In fact this model is one of the first studies for developing the analytical method application of molecular mechanic for modeling of nanostructures. Griebel and Hamaekers [4] analyzed the elastic modulus of composite structures under CNTs reinforcement by simulating the molecular dynamic. The research carried out on

⇑ Corresponding author. Tel.: +98 9364978879; fax: +98 8613288444. E-mail address: [email protected] (S. Jafari Mehrabadi). 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2012.01.067

the composite structures made of polyethylene reinforced by CNTs. One of the interesting results of the mentioned research was that using long nanotubes can lead to a better reinforcement. For constant tensional forces, nanotubes should be arranged in parallel with the exerted force and for general loadings, many long nanotubes should be randomly arranged in order to achieve the same results. Vodenitcharova and Zhang [5] studied the shear bending and shear with local buckling of nanocomposite beams reinforced by SWCNTs. For the analysis of problem they use the Airy stress function. Special attention has been given on the radial flexibility of SWCN in the stress–strain and buckling. Seidel and Lagoud [6] carried out a research on micromechanical elastic properties of reinforced structures by CNTs for determination of elastic properties of nanotubes. In above-mentioned paper, the graphite plate properties were used to determine the nanotubes elastic features. Zhang and Shen [7] investigated CNTs elastic properties depended on temperature by molecular dynamic simulation. In the mentioned paper, they studied the mechanical properties of SWCN by molecular dynamic. The obtained interesting result was that the Young modulus decreases by an increase in temperature meanwhile the shearing modulus increases for temperatures less than 700 K and remains almost constant for temperatures exceeding 700 K. Han and Elliott [8] conducted a research on the elastic properties of composites–polymer utilizing the molecular dynamic method.

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In the mention paper, the molecular dynamic simulation was studied in order to analyze and produce composites with arrangement of CNTs (10, 10) in two kind of matrix polymer, poly methyl phenylene-vinylene (PMPV) and polyethylene methacrylate (PMMA) with two different of volume fraction for CNTs. They use the minimum potential energy method for calculating the axial and shear elastic modulus. Zhu et al. [9] studied the stress–strain behavior of Epon 862 composite strengthened by carbon nanotubes in 2006. The subject that studied in this paper was the stress–strain behavior of Epon 862 composite under the reinforcement of CNTs (10, 10). Song and Youn [10] investigated a model to determine the elastic properties of composite structures reinforced by carbon nanotubes. The asymptotic expansion homogenization (AEH) method was used to calculate the elastic properties. They assumed the nanocomposite epoxy have repetitive geometric and is in accordance with the microscopic dimension. The elastic properties calculated using the homogenization method and for achieving higher accuracy they used the tensorial calculation. The obtained results indicated that the calculated modulus was greatly compatible with Halpin– Tsai micromechanical model. Xiaodong et al. [11] analyzed the reinforcement mechanism of polymer composites strengthened by SWCNTs. The approaches of composite reinforcement by SWCNTs were studied based on the micromechanical model. The obtained results from Helpin-Tesi and Mori–Tanaka model were very compatible with the experimental results. These studies showed that, the random or irregular distribution of CNTs in reinforcement of composite polymer are very coinciding to each other. Ke et al. [12] analyzed the nonlinear vibration of nanocomposite beam reinforced by SWCNTs according to Timoshenko beam theory and Von-Karman nonlinear geometry. The material properties vary in the thickness direction, i.e. functionally graded and obey from the rule of mixture. Rio et al. [13] investigated the effect of SWCN on the elastic constants of poly (etheylen-terephthatat). The obtained results indicated that by adding the SWCN in the matrix, the Young’s modulus increases and the Poission ratio decreases. Also, they showed that the micromechanical model for nanocomposite with random distribution of CNTs is not suitable because the effect of curving the CNTs is ignored. In another paper, Shen [14] studied the nonlinear bending of reinforced composite plate by SWCN under the sinusoidal loading in thermal environment. The results obtained indicated that the effect of FG distribution of CNTs causes increase in the moment and stress. Also Shen and Zhang [15] investigated the thermal buckling of composite plate reinforced by CNT (FG distribution). The obtained results of the paper shown that the FG distribution of CNTs causes critical buckling temperature to be higher than the UD distribution of CNTs. Shen [16] investigated the buckling and postbuckling of nanocomposite plates with functionally graded nanotube reinforcements subjected to uniaxial compression in thermal environments. The effective material properties of the nanocomposite plates were derived by the use of extended rule of mixture. They found that the linear functionally graded nano-reinforcement has a quantitative effect on the uniaxial buckling load as well as postbuckling strength of the plates. The main objective of the present work is to investigate the buckling behavior of the functionally graded nanocomposite rectangular plates reinforced by straight, single-walled CNTs subjected to uniaxial and biaxial in-plane loadings. The material properties of the nanocomposite plate are assumed to be graded in the thickness direction and vary continuously and smoothly according to two types of the symmetric carbon nanotubes volume fraction profiles. The material properties of SWCNT are determined according to molecular dynamics (MDs) and then the effective material properties at a point are estimated by either the Eshelby–Mori–Tanaka approach or the extended rule of mixture. The equilibrium and

stability equations are derived using the Mindlin plate theory considering the first-order shear deformation (FSDT) effect and variational approach. Resulting equations are employed to obtain the closed-form solution for the critical buckling load for each loading case. A detailed parametric study is carried out to investigate the influences of the different types of compressive in-plane loadings, CNTs volume fractions, various types of CNTs volume fraction profiles, geometrical parameters and different types of estimation of effective material properties on the critical mechanical buckling load of functionally graded nanocomposite plates. 2. Modeling of functionally graded CNT-reinforced composites As it is shown in Fig. 1, a polymer rectangular plate with the length of a, width of b and the thickness of t reinforced by SWCNTs, graded distribution in the thickness direction, is assumed. It is also assumed that the mentioned nanocomposite plate is being influenced by plane forces Nx and Ny which are in x and y direction respectively. In this paper, for the first time, two types of symmetric profiles for CNTs volume fractions are configurated. As can be seen from Fig. 2, for the first type, a mid-plane symmetric graded distribution of CNT reinforcements is achieved and both top and bottom surfaces are CNT-rich referred to as Type I FG. For the second type, the distribution of CNT reinforcements is inversed and both top and bottom surfaces are CNT-poor, whereas the mid-plane surface is CNT-rich, referred to as Type II FG. We assume the CNTs volume fraction for Type I FG follows as:

V CN ¼

4jzj  V h CN

ð1Þ

In which:

V CN ¼

wCN wCN þ ðqCN =qm Þ  ðqCN =qm ÞwCN

ð2Þ

where wCN is the mass fraction of nanotube [17,18], and qCN and qm are the densities of CNT and matrix, respectively. The CNTs volume fraction for Type II FG follows as:

Fig. 1. Schematic of nanocomposite plate reinforced by CNTs.

Fig. 2. Configurations of FG nanocomposite plate (a): Type I FG and (b): Type II FG).

S. Jafari Mehrabadi et al. / Composites: Part B 43 (2012) 2031–2040

V CN

  jzj  ¼ 4 0:5  V CN h

ð3Þ

Note that V CN ¼ V CN corresponds to the uniformly distributed CNTR plate, referred to as UD. It should be mentioned that these two FG plates and the UD plate have the same CNT mass fraction. 2.1. Extended rule of mixture According to the extended rule of mixture, the effective Young’s modulus and shear modulus of FG nanocomposite plate are expressed by the following relations [14–17]: m E11 ¼ g1 V CN ECN 11 þ V m E g2 V CN V m ¼ þ m E22 ECN E 22

g3 G12

¼

V CN GCN 12

þ

CN CN where ECN 11 , E22 and G12 are the Young’s and shear moduli of the m m CNTs, E and G are the corresponding properties for the matrix, and the gi (i = 1, 2, 3) are the CNT efficiency parameters, respectively. With the knowledge that load transfer between the nanotube and polymeric phases is less than perfect (e.g. the surface effects, strain gradients effects, intermolecular coupled stress effects, etc.), Shen [14] introduced the CNT efficiency parameters to account load transfer between the nanotube and polymeric phases and other effects on the material properties of CNTRCs. They determined CNT efficiency parameters by matching the elastic modulus of CNTRCs observed from the MD simulation results with the numerical results obtained from the extended rule of mixture. VCN and Vm; are the CNT and matrix volume fractions and are related by:

ð5Þ

Note that the Poisson’s ratio coefficient with regard to its independence to temperature is written as: m t12 ¼ V CN tCN 12 þ V m t

characteristics of continuously graded carbon nanotube-reinforced cylindrical panels. In the present paper, the proposed model is framed within the Eshelby theory for elastic inclusions. The original theory of Eshelby [24–26] is restricted to one single inclusion in a semi-infinite elastic, homogeneous and isotropic medium. The theory, generalized by Mori–Tanaka [27], allows extending the original approach to the practical case of multiple inhomogeneities embedded into a finite domain. The Eshelby–Mori–Tanaka approach, based on the equivalent elastic inclusion idea of Eshelby and the concept of average stress in the matrix due to Mori–Tanaka, is also known as the equivalent inclusion-average stress method [28,29]. According to Benveniste’s revision [19], the following expression of the effective elastic tensor is obtained:

C ¼ Cm þ V CN hðCCN  Cm Þ  Ai  ½V m I þ V CN hAi1 ð4Þ

Vm Gm

V CN þ V m ¼ 1

2033

ð6Þ

2.2. Eshelby–Mori–Tanaka approach Previous studies have examined the validity of the Eshelby– Mori–Tanaka approach in determining the effective properties of composites reinforced with misaligned, carbon fibers, and with carbon nanotubes [19–22]. For instance, Shady and Gowayed [21] modified the effective fiber model used to calculate the elastic properties of nanocomposites in order to include the effect of the curvature of nanotubes. They used the effective fiber model to calculate the elastic properties of nanocomposites using the Eshelby– Mori–Tanaka. As another example, Odegard et al. [22] developed constitutive models for SWNT-reinforced polymer composite materials based on the equivalent continuum modeling technique for nano-structured materials. It was proposed that the nanotube, the local polymer near the nanotube, and the nanotube/polymer interface can be modeled as an effective continuum fiber by using an equivalent-continuum modeling method. They employed the Eshelby–Mori–Tanaka approach to determine the bulk constitutive properties of the SWNT/polymer composite with aligned and random nanotube orientations and with various nanotube lengths and volume fractions. In addition, predicted values of modulus were compared with experimental data obtained from mechanical testing. Very recently, Sobhani Aragh et al. [23] exploited Eshelby– Mori–Tanaka approach to compute the elastic stiffness properties of nanocomposite materials reinforced by graded oriented, straight CNTs. They investigated the impacts of the volume fraction of oriented CNTs and different CNTs distributions on the vibrational

ð7Þ

where VCN and Vm are the fiber and matrix volume fractions, respectively, I represents the fourth-order unit tensor, Cm is the stiffness tensor of the matrix material, CCN is the stiffness tensor of the CNT, the brackets denote an average overall possible orientations of the inclusions. It should be noted that (.) denotes the dot product. A is the fourth-order tensor referred to as concentration factor: 1 A ¼ ½I þ S  C1 m  ðCCN  Cm Þ

ð8Þ

where S is the fourth-order Eshelby tensor [24] and Mura [30] that is specialized to the case of cylindrical inclusions representative of the CNTs and depends on their orientation. The terms enclosed with angle brackets in Eq. (7) represent the average value of the term over all orientations defined by transformation from the local fiber coordinates ðo  x01 x02 x03 Þ to the global coordinates (o  x1x2x3) (Fig. 3). We first consider a polymer composite reinforced with straight CNTs aligned in the x2-axis direction. The matrix is assumed to be elastic and isotropic, with Young’s modulus Em and Poisson’s ratio mm. Each straight CNT is modeled as a fiber with transversely isotropic elastic properties. Therefore, the composite is also transversely isotropic. The substitution of non-vanishing components of the Eshelby tensor S for a straight, long fiber along the x2-direction [29] in Eq. (8) gives the dilute mechanical strain concentration tensor. Then the substitution of A Eq. (8) into Eq. (7) gives the tensor of effective elastic moduli of the composite reinforced by aligned and straight CNTs. In particular, the Hill’s elastic moduli are found as:

k¼

Em fEm V m þ 2kr ð1 þ mm Þ½1 þ V CN ð1  2mm Þg    2ð1 þ mm Þ Em ð1 þ V CN  2mm Þ þ 2V m kr 1  mm  2m2m

l¼

   Em nm V CN ½Em þ 2kr ð1 þ nm Þ þ 2V CN kr 1  n2m    ð1 þ nm Þ Em ð1 þ V CN  2nm Þ þ 2V CN kr 1  nm  2n2m

ð9Þ

ð10Þ

 2 E2m V m ð1 þ V CN  V m mm Þ þ 2V m V CN kr nr  lr ð1 þ mm Þ2 ð1  2mm Þ    n¼ ð1 þ mm Þ Em ð1 þ V CN  2mm Þ þ 2V m kr 1  mm  2m2m h i Em 2V 2m kr ð1  mm Þ þ V CN nr ð1 þ V CN  2mm Þ  4V m lr mm   þ Em ð1 þ V CN  2mm Þ þ 2V m kr 1  mm  2m2m ð11Þ p¼

Em ½Em V m þ 2pr ð1 þ mm Þð1 þ V CN Þ 2ð1 þ mm Þ½Em ð1 þ V CN Þ þ 2V m pr ð1 þ mm Þ

k¼

E ½E V þ 2mr ð1 þ mm Þð3 þ V CN  4mm Þ m m m   2ð1 þ mm Þ Em ½V m þ 4V CN ð1  mm Þ þ 2V m mr 3  mm  4m2m

ð12Þ

ð13Þ where k, l, m, n, and p are Hill’s elastic moduli of the composite; k is the plane-strain bulk modulus normal to the fiber direction, n is the

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Fig. 3. Representative volume element (RVE) including straight CNTs.

uniaxial tension modulus in the fiber direction, l is the associated cross modulus, m and p are the shear moduli in planes normal and parallel to the fiber direction, respectively. kr, lr, mr, nr, and pr are the Hill’s elastic moduli for the reinforcing phase (CNTs) [31]. The elastic moduli parallel and normal to CNTs are related to Hill’s elastic moduli by: 2

Ejj ¼ n 

l ; k

4mðkn  l Þ 2

kn  l þ mn

ð14Þ

Q 11

Q 11

0

0

Q 11

0

0

0

Q 11

0

0

0

Q 11

0

0

0

9 38 exx > > > > > > 7> e > 0 7> > yy > > = 7< c 0 7 xy 7> > 7> > cxz > > 0 5> > > > > > : cyz ; Q 11 0

According to the based on the FSDT of Mindlin [32–34], the displacement field of the rectangular plate is considered as:

Q 22

uðx; y; zÞ ¼ zwx ðx; yÞ

E11 1  t12 t21 t12 E22 t21 E11 ¼ ¼ 1  t12 t21 1  t12 t21 E22 ¼ 1  t12 t21 ¼ G12

Q 66

ð15Þ

wðx; y; zÞ ¼ wðx; yÞ In which u, v and w are the displacement components in x, y and z directions respectively. øx and øy are also the neutral plate rotation around y and x axes respectively. In accordance with the above displacement field, the strain components are [35,36]

Q 55 ¼ G13 The coefficients G23, G13, E22, E11, m21 and m12 are calculated by using either the modified rule of mixtures or Eshelby–Mori–Tanaka approach. Now in order to achieve the equilibrium equations, the relations of plane forces, moments and shear forces are written as follows:

R t=2 ðNi ; M i Þ ¼ t=2 ri ð1; zÞdz i; j ¼ x; y; z R t=2 Q i ¼ k t=2 riz dz i; j ¼ x; y; z

ex ¼ u;x ey ¼ v ;y ez ¼ w;z cxy ¼ u;y þ v ;x cxz ¼ w;x þ u;z cyz ¼ v ;z þ w;y

ð16Þ

By substituting the Eq. (15) into Eq. (16), the strain–displacement relations are derived:

feij g ¼ fe0ij g þ zfe1ij g i; j ¼ x; y; z

ð17Þ

In which

8 9 0 > > > > > > > > > > 0 > > > > > > <0 = zz ¼ > > c0xy > > >0 > > > > > > > > > > > > > > 0 > > > > þ w w ;x > > cxz > > x > > > > > > : > ; : 0 ; w þw ;y

ð20Þ

Q 44 ¼ G23

v ðx; y; zÞ ¼ zwy ðx; yÞ

8 0 9 exx > > > > > > e0 > > > > > > yy > > > > 0 > > < e =

ð19Þ

Q 11 ¼ Q 12

y

2

6 6 Q 11 6 sxy ¼ 6 6 0 > > > > 6 > > s > > yz > > 4 0 > > : ; 0 sxz

3. Equilibrium equations

cyz

9 > > > > > > =

In which the Qij components are expressed as:

2

E? ¼

8 rx > > > > > > < ry

8 1 9 exx > > > > > > e1 > > > > > > yy > > > > 1 > > < e =

8 9 wx;x > > > > > > > > > > w > > y;y > > > > < = 0 zz ¼ > c1xy > > wx;y þ wy;x > > > > > > > > > > > > > >0 > > 1 > > > > > > cxz > > > > > > > : > > ; : 1 ; 0

ð18Þ

cyz

On the other hand the stress–strain relations for orthotropic plates are written as follows:

ð21Þ

In which Nij and Mij are the forces and the resultant moments. Qij is the sheer force and k is the shear correction factor coefficient considered as k = 5/6. In addition rij is the stress components obtained based on Hooke’s law.By calculating the stresses and substituting them into Eq. (21), the forces and subsequent moments are obtained as follows:

h



i 1 0 0 1 1 Nx ¼ 1m12 ðE þ E Þ e þ m e þ E Þ e þ m e þ ðE 1a 21 1e 21 1d 1b xx yy xx yy m21 h



i 1 0 0 1 1 Ny ¼ 1m12 m21 ðE2a þ E2d Þ eyy þ m12 exx þ ðE2b þ E2e Þ eyy þ m12 exx Nxy ¼ ðG12a þ G12d Þc0xy þ ðG12b þ G12e Þc1xy h



i 1 0 0 1 1 Mxx ¼ 1m12 m21 ðE1b þ E1e Þ exx þ m21 eyy þ ðE1c þ E1f Þ exx þ m21 eyy h



i 1 0 0 1 1 Myy ¼ 1m12 m21 ðE2b þ E2e Þ eyy þ m12 exx þ ðE2c þ E2f Þ eyy þ m12 exx Mxy ¼ ðG12b þ G12e Þc0xy þ ðG12c þ G12f Þc1xy Q x ¼ kðG12a þ G12d Þc0xz Q y ¼ kðG12a þ G12d Þc0yz ð22Þ

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wx ! wx0 þ wx1 wy ! wy0 þ wy1

In which the coefficients are as follows:

ðE1a ; E1b ; E1c Þ ¼

R0

ðE1d ; E1e ; E1f Þ ¼

R t=2

ðE2a ; E2b ; E2c Þ ¼

R0

2

t=2 0

E11 ð1; z; z Þdz

E11 ð1; z; z2 Þdz

E ð1; z; z2 Þdz t=2 22

R t=2

ð23Þ

E22 ð1; z; z2 Þdz R0 ðG12a ; G12b ; G12c Þ ¼ t=2 G12 ð1; z; z2 Þdz R t=2 ðG12d ; G12e ; G12f Þ ¼ 0 G12 ð1; z; z2 Þdz ðE2d ; E2e ; E2f Þ ¼

0

ð24Þ

In which V is the overall potential energy, U is the elastic potential energy and O is work done by the external forces the relations of which are written as follows:

1 U¼ 2

ZZZ

t=2 t=2

ðrx ex þ ry ey þ rz ez þ sxy cxy þ syz cyz þ szx czx Þdz dx dy ð25Þ

 ZZ  1 1 px u;x þ py v ;y  pn w dx dy X¼ b a   1 @wx 1 @wy px þ py  pn w dx dy ¼ b a @x @y

F¼





e00 þ e01 þ z e10 þ e11



ð30Þ

Subsequently the relations of the forces and moments are written as follow:

Ni ¼ Ni0 þ Ni1

i ¼ x; y; xy

Mi ¼ M i0 þ Mi1

i ¼ x; y; xy

Q i ¼ Q i0 þ Q i1

i ¼ x; y

ð31Þ

Now, by applying the Eqs. (21), (29), and (30) the relations of the forces and moments are written in the form below: 1 Nx0 ¼ 1m12 m21 ðE1b þ E1e Þðwx0;x þ t21 wy0;y Þ 1 Nx1 ¼ 1m12 m21 ðE1b þ E1e Þðwx1;x þ t21 wy1;y Þ 1 Ny0 ¼ 1m12 m21 ðE2b þ E2e Þðwy0;y þ t12 wx0;x Þ

1 Myy0 ¼ 1m12 m21 ðE2c þ E2f Þðwy0;y þ t12 wx0;x Þ

ð26Þ

h

 0 0 ðE1a þ E1d Þ e02 xx þ m21 exx eyy

2ð1  m12 m21 Þ

 þðE1b þ E1e Þ 2e0xx e1xx þ m21 e0xx e1yy þ m21 e1xx e0xx



 þðE1c þ E1f Þ e1xx 2 þ m21 e1xx e1yy þ ðE2a þ E2d Þ e0yy 2 þ m12 e0xx e0yy

 þðE2b þ E2e Þ  2e0yy e1yy þ m12 e0xx e1yy þ m12 e1xx e0yy

i þðE2c þ E2f Þ e1yy 2 þ m12 e1yy e1xx þ ðG12b þ G12e Þc0xy c1xy

 1 1 þ ðG12c þ G12f Þc1xy 2 þ ðG12a þ G12d Þ c0xy 2 þ c0yz 2 þ c0zx 2 2 2 1 1 ð27Þ þ px zwx;x þ py zwy;y  pn w b a

Now by applying the Euler–Lagrange equations to the above equation, the equilibrium equations of the nanocomposite plate are obtained based on Midlin plate theory in the form below:

M x;x þ Mxy;y  Q x ¼ 0 M y;y þ Mxy;x  Q y ¼ 0



 @ @ Nx @w þ Nxy @w N xy @w þ N y @w ðQ x;x þ Q y;y Þ þ @x þ @y ¼0 @x @y @x @y



1 Ny1 ¼ 1m12 m21 ðE2b þ E2e Þðwy1;y þ t12 wx1;x Þ

by substituting the stress components of Eq. (19), into Eq. (25), and calculating the total potential energy, (V), the functional of energy is obtained as:

1

In which the subscript 0 indicates the equilibrium state and the subscript 1 expresses a minute change in the plate equilibrium condition. Regarding the incremented variation to the rotation and the displacement components, the overall strain is expressed as follows:

feg ¼

In order to obtain the equilibrium relations, the energy method will be used. Therefore, the plate overall potential energy is written as follows:

V ¼UþX

ð29Þ

w ! w0 þ w1

ð28Þ

1 Myy1 ¼ 1m12 m21 ðE2c þ E2f Þðwy1;y þ t21 wx1;x Þ 1 Mxx0 ¼ 1m12 m21 ðE1c þ E1f Þðwx0;x þ t21 wy0;y Þ 1 Mxx1 ¼ 1m12 m21 ðE1c þ E1f Þðwx1;x þ t21 wy1;y Þ

ð32Þ

Nxy0 ¼ ðG12b þ G12e Þðwx0;y þ wy0;y Þ Nxy1 ¼ ðG12b þ G12e Þðwx1;y þ wy1;y Þ Mxy0 ¼ ðG12c þ G12f Þðwx0;y þ wy0;x Þ Mxy1 ¼ ðG12c þ G12f Þðwx1;y þ wy1;x Þ Q x0 ¼ kðG12a þ G12d Þðwx0 þ w0;x Þ Q x1 ¼ kðG12a þ G12d Þðwx1 þ w1;x Þ Q y0 ¼ kðG12a þ G12d Þðwy0 þ w0;y Þ Q y1 ¼ kðG12a þ G12d Þðwy1 þ w1;y Þ By substituting the aforementioned relations into the equilibrium equations and applying the following assumptions; the linear stability equations will be obtained in the form of the Eq. (33). (a) w0 and all its derivatives are zero. (b) The expressions conclude the Ni0, Mi0 and Qi0 indicate the initial equilibrium condition and should be eliminated. (c) The expressions consists of multiplying the (Qi0, Mi0 and Ni0) in (wi0, wx0 and wy0) are negligible and should be eliminated.

Mx1;x þ M xy1;y  Q x1 ¼ 0 My1;y þ M xy1;x  Q y1 ¼ 0



 @ @ 1 1 1 1 Nx0 @w þ Nxy0 @w N xy0 @w þ Ny0 @w þ @y ¼0 ðQ x1;x þ Q y1;y Þ þ @x @x @y @x @y ð33Þ 5. Buckling analysis of the polymer plate reinforced by CNTs

4. Stability equations In this section, the stability equations of the FG nanocomposite rectangular plate are derived by using the adjacent equilibrium criterion [35]. We give small increments to the displacement and rotation variables and examine the two adjacent configurations represented by the displacements and rotations before and after the increment, as follows:

In this section, the determination of the critical buckling force of the functionally graded composite plate under the effect of plane forces on the basis of Mindlin plate theory has been discussed. It has been assumed that the forces are Nx and Ny in x and y directions respectively and their relation is Ny = cNx (see Fig. 4). By substituting the Eq. (32) into Eq. (33) the stability equations will be written in terms of the displacement components:

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After substituting the Eq. (37) into Eqs. (36) we get a systems of three equations for finding the Wmn, Xmn and Ymn. By equaling the determinant of coefficient to zero we have:

2 6 4

s11  Ny ðb2 þ ca2 Þ s12 s21

s22

s31

s32

9 38 s13 > < W mn > = 7 ¼0 s23 5 X mn > > : ; Y mn s33

ð39Þ

s11 þ Ny b2 þ Nx a2 ¼ kðG12a þ G12d Þða2 þ b2 Þ þ N y ðb2 þ ca2 Þ s12 ¼ kðG12a þ G12d Þa s13 ¼ kðG12a þ G12d Þb s21 ¼ s12 s22 ¼ 1t1 ðE1c þ E1f Þa2  ðG12c þ G12f Þb2  kðG12a þ G12d Þ 12 t21 s23 ¼ 1tt1221t21 ðE1c þ E1f Þab  ðG12c þ G12f Þab Fig. 4. Nanocomposite rectangular plate reinforced by SWCNTs under the plane forces in x and y directions (Nx = N, Ny = cNx).

1 ð1  m12 m21 Þ

ð40Þ

 ðwx1;yy þ m21 wy1;xy Þ  kðG12a þ G12d Þðwx1 þ w1;x Þ ð34-1Þ

1 ð1  m12 m21 Þ

 ðwx1;yx þ m21 wy1;xx Þ  kðG12a þ G12d Þðwy1 þ w1;y Þ

On the other hand, by applying plate boundary condition (34-3) and regarding that Nx and Ny are the plane forces per unit length applied on the plate edges one would get:

Ny ¼ 

py b

kðG12a þ G12d Þðwx1;x þ w1;xx þ wy1;y þ w1;yy Þ 

ð35Þ py p w1;yy  x w1;xx ¼ 0 a b ð36Þ

In which Px and Py are the total imposed forces on the plate in the direction of x and y respectively.According to the simply supported boundary conditions for the mentioned plate edges, the following functions have been assumed for quantities of w1, wx1 and wx1

w1 ðx; yÞ ¼ wx1 ðx; yÞ ¼ wy1 ðx; yÞ ¼

1 P 1 P

W mn sin ax sin by

n¼1 m¼1 1 P 1 P n¼1 m¼1 1 P 1 P

X mn cos ax sin by

ð37Þ

Y mn sin ax cos by

n¼1 m¼1

In which Wmn, Xmn and Ymn are the amplitude of the above-mentioned functions. m and n indicate also number of half wave in x and y directions respectively and the following relations express the relationship between m and n with a and b

a¼

mp a

b¼

np b

 2 þ nb2 sc

ð41Þ

sd ¼ 2s13 s12 s23  s22 s213 þ s11 s22 s33  s11 s223  s33 s212

ð42Þ

6. Results and discussion

ð34-3Þ

px a

sd

sc ¼ s22 s33  s223

kðG12a þ G12d Þðwx1;x þ w1;xx þ wy1;y þ w1;yy Þ     @ @w1 @w1 @ @w1 @w1 þ Nx0 Nxy0 þ Nxy0 þ þ N y0 @x @y @x @y @x @y

Nx ¼ 

p



2 2 m c a2

In which sd and sc is given as follows:

ð34-2Þ

¼0

In Eq. (40) Nx or Ny (Nx = cNy) indicated the critical mechanical buckling load as:

Ncr ¼ 

ðE2c þ E2f Þðwy1;yy þ m12 wx1;yx Þ þ ðG12c þ G12cf Þ

¼0

s32 ¼ s23 s33 ¼ 1t1 ðE2c þ E2f Þb2  ðG12c þ G12f Þa2  kðG12a þ G12d Þ 12 t21

ðE1c þ E1f Þðwx1;xx þ m21 wy1;yx Þ þ ðG12c þ G12cf Þ

¼0

s31 ¼ s13

ð38Þ

In this section of paper for giving the new results, by solving a numerical example we calculate the critical buckling load of a rectangular nanocomposite plates reinforced by aligned and straight CNTs by changing the problem parameters. For more explanation we choose the loading of plate as follows: (6.1) Plane loading in the x direction. (6.2) Plane loading in the y direction. (6.3) Plane loading in the x and y directions. For uniaxially compressed nanocomposite plate (Sections 6.1 and 6.2), the extended rule of mixture is used for predicting the overall material properties and responses of the plate, while for the case of biaxial in-plane loadings (Section 6.3), effective elastic moduli are computed by using Eshelby–Mori–Tanaka approach. In this paper, Poly (methyl methacrylate), referred to as PMMA, is selected for the matrix, and the material properties of which are assumed to be vm = 0.34, Em ¼ 2:5 GPa at room temperature (300 K) [15,16]. The (10, 10) SWCNTs are selected as reinforce22 ments. Han and Elliott [8] chose E11 CN ¼ 600 GPa, ECN ¼ 10 GPa, 12 GCN ¼ 17:2 GPa for (10, 10) SWCNTs. Such a low value of Young’s modulus, as those of Odegard et al. [22], is due to the fact that the effective thickness of CNTs is assumed to be 0.34 nm or more. However, as reported recently the effective thickness of SWCNTs should be smaller than 0.142 nm [37]. Therefore, all material properties and effective thickness of SWCNTs used for the present analysis properly are selected according to the MD simulation results of Shen [15,16]. It is noted that the effective wall thickness obtained for (10, 10)-tube is 0.067 nm, which satisfies the Vodenitcharova– Zhang criterion [37], and the wide used value of 0.34 nm for tube wall thickness is thoroughly inappropriate for SWCNTs.

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S. Jafari Mehrabadi et al. / Composites: Part B 43 (2012) 2031–2040 Table 1 Comparison between critical buckling load N cr x (KN) versus a/b for various types of CNTs volume fraction profiles (t/b = 0.1). a/b

Type I FG V CN

1 2 3 4 5 6

¼ 0:12

0.8514 5.1456 11.054 18.789 30.055 42.035

UD V CN

¼ 0:17

1.1959 8.6542 18.562 32.325 50.054 71.519

V CN

¼ 0:28

1.4426 10.549 22.853 39.564 61.593 88.324

V CN

Type II FG ¼ 0:12

0.6219 4.0212 9.0954 14.029 23.548 33.521

The key issue for successful application of the extended rule of mixture to CNTRCs is to determine the CNT efficiency parameter gi. There are no experiments conducted to determine the value of gi for CNTRCs. Shen [15,16] determined the CNT efficiency parameters g1, g2 and g3 by matching the Young’s moduli E11 and E22 and shear modulus G12 of CNTRCs predicted from the extended rule of mixture to those from the MD simulations given by Han and Elliott [8] and Griebel and Hamaekers [4]. For example, g1 = 0.137, g2 = 1.022 and g3 = 0.715 for the case of V CN ¼ 0:12 and g1 = 0.142, g2 = 1.626 and g3 = 1.138 for the case of V CN ¼ 0:17 and g1 = 0.141, g2 = 1.585 and g3 = 1.109 for the case of V CN ¼ 0:28. These values will be used in all the following examples, in which taking g3:g2 = 0.7:1 and G13 = G12 and G23 = 1.2G12 [16].

V CN

¼ 0:17

1.1721 5.5487 15.156 27.298 38.129 52.529

V CN

¼ 0:28

1.3265 9.2018 18.201 33.235 50.129 70.219

V CN ¼ 0:12

V CN ¼ 0:17

V CN ¼ 0:28

0.5587 3.3214 8.2487 12.897 20.784 29.587

1.1665 5.2135 12.598 23.520 33.351 46.651

1.2256 8.7521 16.258 30.298 45.098 64.032

high aspect ratio of a/b. To assess the effects of various types of volume fraction profiles and aspect ratio of t/b, critical load for different values of CNTs volume fraction are presented in Table 2. It is observed from Tables 1 and 2 that nanocomposite plates with Type I FG and Type II FG have highest and lowest critical load, respectively. This means that the functionally graded nanocomposite plates with symmetric profiles of the CNTs volume fraction can likely be designed according to the actual requirement and it is a potential alternative to the CNTRc plates with uniformly distributed CNTs.

6.1. Loading of plate only in x direction In this section, nanocomposite rectangular plate is subjected to a uniform compressive load on edges x = 0 and x = a. A parametric study has been carried out and typical results for plate with 5 mm thickness are shown in Table 1 and Figs. 5–7. Various types of CNTs volume fraction profiles, i.e. Type I FG, UD and Type II FG, are considered. Figs. 5–7 show the critical load versus the aspect ratio of a/ b with different CNTs volume fractions and various types of CNTs volume fraction profiles. Because the drawn curves in Figs. 5–7 are close to each other, the values of critical load for UD and FG distribution of CNTs are listed in Table 1. From Figs. 5–7, it is concluded that the critical buckling load of various types of CNTs volume fraction profiles become larger when the CNTs volume fraction increases. Another important result of Figs. 5–7 is that the influence of the CNTs volume fraction on the critical buckling load for different types of CNTs profiles is generally significant at

Fig. 6. Critical load (Nx)cr in the case of UD distributed SWCNTs for various values of a/b (t/b = 0.1, Ny = 0).

Fig. 5. Critical load (Nx)cr in the case of Type I FG distributed SWCNTs for various values of a/b (t/b = 0.1, Ny = 0).

Fig. 7. Critical buckling load (Nx)cr in the case of Type II FG distributed SWCNTs for various values of a/b (t/b = 0.1, Ny = 0).

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S. Jafari Mehrabadi et al. / Composites: Part B 43 (2012) 2031–2040

Table 2 Comparison between critical load N cr y (KN) versus t/b for various types of CNTs volume fraction profiles (a/b = 1). t/b

Type I FG V CN

0.1 0.2 0.3 0.4 0.5 0.6

¼ 0:12

0.8514 6.0523 8.4126 9.8521 11.265 13.026

UD V CN

¼ 0:17

1.1959 10.561 13.265 16.249 19.025 22.165

V CN

¼ 0:28

1.4426 11.235 16.265 20.152 23.475 27.016

V CN

Type II FG ¼ 0:12

0.6219 5.0215 7.5129 8.8541 10.025 11.526

V CN

¼ 0:17

1.1721 9.2351 11.851 14.238 16.851 19.321

V CN

¼ 0:28

1.3265 10.001 14.766 18.324 21.164 24.513

V CN ¼ 0:12

V CN ¼ 0:17

V CN ¼ 0:28

0.5587 4.7512 7.2315 8.4523 9.5136 11.032

1.1665 9.0126 11.266 13.483 16.010 18.016

1.2256 9.7464 14.236 17.686 20.253 23.256

Table 3 Critical buckling load N cr y (KN) for various types of CNTs volume fraction profiles versus a/b (t = 5 mm, Nx = 0). a/b

1 2 3 4 5 6

Type I FG

UD

Type II FG

V CN ¼ 0:12

V CN ¼ 0:17

V CN ¼ 0:28

V CN ¼ 0:12

V CN ¼ 0:17

V CN ¼ 0:28

V CN ¼ 0:12

V CN ¼ 0:17

V CN ¼ 0:28

1.1034 1.2112 1.4396 1.7649 2.1574 2.5649

1.4863 1.5258 1.8253 2.1002 2.5000 2.9742

1.7067 1.8264 2.0875 2.3209 2.6999 3.2370

0.9977 1.1143 1.2667 1.4909 1.8387 2.2989

1.1573 1.2887 1.5014 1.8319 2.3190 2.6823

1.5010 1.6999 1.9053 2.1876 2.4372 2.9074

0.9011 1.0021 1.1743 1.5196 1.7784 2.1635

1.0052 1.1362 1.3999 1.6243 2.1842 2.4987

1.3893 1.6012 1.8111 1.9903 2.3732 2.8222

6.2. . Loading of plate only in the y direction

6.3. . Loading of plate in x and y directions

In this section for more attention to the effect of CNTs, we assume that loading of plate is only in the y direction. Critical buckling loads of the nanocomposite plates subjected to uniaxial in-plane loading (Nx = 0) are listed in Table 3 for various aspect ratio of a/b. It can be concluded from Table 3 that the increase of the CNTs volume fractions yields an increase in the critical buckling load. Variation of the critical buckling in-plane load N cr y against the aspect ratio t/b for Type I FG nanocomposite plate is plotted in Fig. 8. It can be inferred from Table 3 and Fig. 8 that with increasing the CNTs volume fractions, influence of the aspect ratio a/b on the critical buckling loads is more significant. The variation of the critical buckling load versus the aspect ratio of t/b for various types of CNTs volume fraction profiles is shown in Fig. 9 for V CN ¼ 0:12. The results in Fig. 9 indicate that discrepancy among the various types of CNTs profiles increases with the increasing values of the aspect ratio of t/b.

In this section we consider two axial loading of plate (Nx and Ny) and their relation is Nx = cNy. Also the material properties of the FGM nanocomposite plate are assumed to be graded in the thickness direction and estimated though the Eshelby–Mori–Tanaka approach. Influence of different types of CNTs volume fraction profiles on critical biaxial buckling load against aspect ratio a/b and t/b is presented in Figs. 10 and 11 for V CN ¼ 0:12. It is worthy of mention that critical biaxial buckling load of the Type II FG nanocomposite plate is lower than that of one with symmetric profile (Type I FG) and close to that of the uniformly distributed CNTs. Moreover, it can be concluded that for various types of estimation of effective material properties (extended rule of mixture and Mori–Tanaka method) and different loading of plate, nanocomposite plates with Type I FG and Type II FG have highest and lowest critical load, respectively. It is seen that as the aspect ratio a/b increases, the critical biaxial buckling load mainly decreases, while

Fig. 8. Variation of the critical buckling in-plane load (Ny)cr against the aspect ratio a/b for Type I FG nanocomposite plate (t/b = 0.1, Nx = 0).

Fig. 9. Critical load (Ny)cr against the aspect ratio t/b for various types of CNTs volume fraction profiles (a/b = 1, Nx = 0, V CN ¼ 0:12).

S. Jafari Mehrabadi et al. / Composites: Part B 43 (2012) 2031–2040

Fig. 10. Influence of different types of CNTs volume fraction profiles on critical buckling load against aspect ratio a/b (t/b = 0.1, c = 1, V CN ¼ 0:12).

2039

 It results that the functionally graded nanocomposite plates with symmetric profiles of the CNTs volume fraction can likely be designed according to the actual requirement and it is a potential alternative to the CNTR plates with uniformly distributed CNTs.  In the case of uniaxial loading, the discrepancy among the various types of CNTs profiles increases with the increasing values of the aspect ratio of t/b and a/b.  The critical buckling load of the Type II FG nanocomposite plate is lower than that of one with symmetric profile (Type I FG) and close to that of the uniformly distributed CNTs.  The achieved results show that for various types of estimation of effective material properties (extended rule of mixture and Eshelby–Mori–Tanaka) and different in-plane loadings of plate, nanocomposite plates with Type I FG and Type II FG have highest and lowest critical load, respectively.  The methods of the estimation of effective material properties presented in this paper can be used in mechanical and thermal stress analysis, vibration and buckling problems of the functionally graded nanocomposite structures.  It is observed that as the aspect ratio a/b increases, the critical biaxial buckling load mainly decreases, while an inverse behavior is experienced for critical uniaxial buckling load. Moreover, it is found that the critical biaxial buckling load increases rapidly with increasing the aspect ratio t/b and then remains almost unaltered for t/b > 0.5.

References

Fig. 11. Influence of different types of CNTs volume fraction profiles on critical buckling load against aspect ratio t/b (a/b = 1, c = 1, V CN ¼ 0:12).

an inverse behavior is experienced for critical uniaxial buckling load. In addition, it is found that the critical biaxial buckling load increases rapidly with increasing the aspect ratio t/b and then remains almost unaltered for t/b > 0.5. 7. Conclusion remarks In this research work, buckling analysis of functionally graded nanocomposite rectangular plates reinforced by aligned and straight single-walled carbon subjected to uniaxial and biaxial in-plane loadings was investigated. In order to equilibrium and stability equations of the rectangular plate under in-plane load, the analysis procedure was based on the Mindlin plate theory considering the first-order shear deformation effect. Resulting equations were employed to obtain the closed-form solution for the critical buckling load for each loading case. The effective material properties of the nanocomposite plate were assumed to be graded in the thickness direction and estimated by either the Eshelby–Mori–Tanaka approach or the extended rule of mixture. Two types of the symmetric carbon nanotubes volume fraction profiles were presented in this paper. New and interesting numerical results show that:  In the case of uniaxial loading, the influence of the CNTs volume fraction on the critical buckling load for different types of CNTs profiles is generally significant at high aspect ratio of a/b.

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