Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach

Accepted Manuscript Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach L.W. Zhang, Z.X. Lei, K.M. Liew PII:

S1359-8368(15)00050-5

DOI:

10.1016/j.compositesb.2015.01.033

Reference:

JCOMB 3382

To appear in:

Composites Part B

Received Date: 6 December 2014 Revised Date:

19 January 2015

Accepted Date: 21 January 2015

Please cite this article as: Zhang LW, Lei ZX, Liew KM, Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach, Composites Part B (2015), doi: 10.1016/ j.compositesb.2015.01.033. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach L. W. Zhang1,2, Z. X. Lei2,3, K. M. Liew2,4* 1

College of Information Technology, Shanghai Ocean University, Shanghai 201306, China Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR 3

School of Sciences, Nanjing University of Science and Technology, Nanjing 210094, China

City University of Hong Kong Shenzhen Research Institute Building, Shenzhen Hi-Tech Industrial Park, Nanshan District, Shenzhen, China

Abstract

SC

4

RI PT

2

The buckling behavior of functionally graded carbon nanotube (FG-CNT) reinforced

M AN U

composite thick skew plates is studied. The CNTs are reinforced uniaxially aligned in the axial direction. Material properties of the nanocomposites are assumed to be graded in the thickness direction. The element-free IMLS-Ritz method is employed for the numerical analysis. The theoretical formulation has incorporated the effects of transverse shear deformation and rotary inertia through employing the first-order shear deformation theory (FSDT). A few numerical examples are chosen to demonstrate the numerical stability and

TE D

accuracy of the IMLS-Ritz method. The validity of the IMLS-Ritz results is examined by comparing them with those of the known data in the literature. Parametric studies are conducted for various types of CNTs distributions, CNT ratios, skew plates, aspect ratios and

EP

thickness-to-height ratios under different boundary conditions. Some conclusions are drawn on the parametric studies with respect to the buckling characteristics.

AC C

Keywords: A. Plates; B. Buckling; C. Numerical Analysis

*

Corresponding author. E-mail address: [email protected] Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR

1

ACCEPTED MANUSCRIPT 1. Introduction The discovery of carbon nanotubes (CNTs) has motivated researchers to work on better and stronger structural materials mainly because CNTs have remarkable physical and chemical properties, such as high strength, high stiffness and high aspect ratio but low density. Guided by the concept of functionally graded (FG) materials, a class of new emerging composite

RI PT

materials, the FG-CNT reinforced composite, has been proposed making use of CNTs as the reinforcements in a functionally graded pattern. This new type of FG-CNT reinforced composite will need further research so as to find out its mechanical properties. In this paper, the buckling behavior of FG-CNT reinforced composite skew plates will be studied under

SC

various geometrics and boundary conditions.

The CNT-based FGMs were first studied by Shen [1]. He considered the composite with CNT distributions within an isotropic matrix designed specifically to grade them with certain

M AN U

rules along the desired directions to improve its mechanical property. A series of investigations about FG-CNT reinforced composite beam, plate and shell were then conducted to study their mechanical behaviors. The vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels based on the Eshelby-Mori-Tanaka approach was studied by Aragh et al. [2]. The static stress analysis of carbon nanotube

TE D

reinforced composite cylinder under non-axisymmetric thermo-mechanical loads and uniform electro-magnetic fields was carried out by Arani et al. [3]. The mechanical buckling of FGCNT reinforced composite plates subjected to uniaxial and biaxial in-plane loadings was studied by Mehrabadi et al. [4]. The first-order plate theory was employed to derive the

EP

equilibrium and stability equations. The buckling behavior of quadrilateral laminated thin-tomoderately thick plates consisting of perfectly bonded carbon CNT reinforced composite

AC C

layers was examined by Malekzadeh and Shojaee [5]. The Trefftz buckling criterion was used in obtaining the stability equations. A mapping-differential quadrature technique was used to solve the stability equations subjected to arbitrary boundary conditions. The buckling analysis of FG-CNT reinforced composite plates under various in-plane mechanical loads was conducted by Lei et al. [6]. The kp-Ritz method was employed in their study. The effective material properties of plates reinforced by CNTs were estimated through a micromechanical model based on either the Eshelby-Mori-Tanaka approach or the extended rule of mixture. Based on Reissner’s mixed variational theorem [7], a unified formulation of finite layer methods for three-dimensional buckling of FG-CNT reinforced composite plates was developed by Wu and Chang [8]. The FG-CNT reinforced composite plates with their surface-bonded piezoelectric actuator and sensor layers were studied under bi-axial 2

ACCEPTED MANUSCRIPT compressive loads. Before the instability occurred, a set of membrane stresses was assumed to exist, and determined using the predefined 3D deformations for the pre-buckling state. The thermal buckling and postbuckling behaviors of functionally graded carbon nanotubereinforced composite cylindrical shells was examined by Shen [9]. The same method [9] was employed by Shen and Xiang [10] to study the postbuckling of axially compressed nanotube-

RI PT

reinforced composite cylindrical panels resting on elastic foundations in thermal environments. Besides, there were also some other works that are relevant to the mechanical analysis of FG materials [11-13].

From a recent literature survey [14], one can find that there are numerous studies on the

SC

buckling of FG-CNT reinforced composite plate using different analytical and numerical approaches, however, only a limited work was reported on the FG-CNT reinforced composite plate of skew domain. In this study, the buckling solution of FG-CNT reinforced composite

M AN U

skew plate is obtained using an element/mesh-free method. Unlike any traditionally discretization techniques such as the finite element method and the finite difference method, the element/mesh-free method furnishes solution with only a minimum of meshing or no meshing at all, and a set of scattered nodes are used instead of meshing the problem domain [15]. The method has become a powerful numerical tool for determining approximate

TE D

solutions for boundary value problems. In the existing literature, numerous element/meshfree methods have been developed such as the element-free Galerkin method [15], the improved element-free Galerkin (IEFG) method [16, 17], the element-free kp-Ritz method [18-22], the element-free IMLS-Ritz method [23], the meshless local Petrov-Galerkin method

EP

[24], and the local Kriging meshless method [25] [26]. The buckling analysis of this study will be performed using the IMLS-Ritz method.

AC C

The problem considered in this paper is to obtain the buckling solution of FG-CNT reinforced composite thick skew plate using the IMLS-Ritz method, in which an orthogonal function system with a weight function is used as the basis for the displacement field [23]. It is known that the effects of transverse shear deformation and rotary inertia will need to be taken into account for analysis of thick plate. In the theoretical formulation, the transverse shear deformation and rotary inertia effects will be taken care through employing the firstorder shear deformation theory (FSDT). With CNTs assumed uniaxially aligned in axial direction and functionally graded in thickness direction of the plates, the effective material properties of FG-CNT reinforced composite plates will be estimated through a micromechanical model. Improvement of computational efficiency and eliminate shear and membrane locking are performed using a stabilized conforming nodal integration scheme to 3

ACCEPTED MANUSCRIPT evaluate the system bending stiffness. The cubic spline function of high continuity [27-30] is used as the weight function in the formulation. Several example problems will be considered in the buckling study. The effects of CNT volume fraction, plate thickness-to-width ratio, plate aspect ratio, height-to-width ratio, plate geometric angle and boundary conditions on the critical buckling loads of the plates will be examined. Carefully planned convergence studies

RI PT

will be carried out to ascertain the accuracy of the IMLS-Ritz solutions. The validity of the computed results for various examples will be verified with the known data in the literature. Finally, detailed parametric studies will be carried out to examine the influences of variations of the skew angles, thickness-to-height ratio, plate aspect ratio and material composition on

SC

the critical buckling loads of FG-CNT reinforced composite skew plates.

2. Material properties of FG-CNT reinforced composite plates

M AN U

Consider FG-CNT reinforced composite skew plates, having three types of distributions of CNTs, with length a, width b and thickness h. Material properties of the FG-CNT reinforced composites are assumed to be graded through thickness direction according to a linear distribution of the volume fraction of carbon nanotubes. Distributions of CNTs along the thickness direction of FG-CNT reinforced composite plates, as shown in Fig. 1, are assumed

TE D

to be

EP

 * VCNT    2z * VCNT ( z ) = 2 1 −  VCNT t     2z * 2( )VCNT  t

( UD

CNT reinforced composite )

( FG-O CNT reinforced composite ) ,

(1)

( FG-X CNT reinforced composite )

AC C

where UD represents a uniform distribution and the other two types of functionally graded distributions of CNTs are denoted by FG-O and FG-X, respectively. The CNT volume fraction of UD-CNT reinforced composite plate and the other two types of FG-CNT * reinforced composite plates are assumed to be VCNT = VCNT , that means all these three types of

FG-CNT reinforced composite plates having the same mass volume of CNTs. An embedded carbon nanotube in a polymer matrix is considered, therefore, there is no abrupt interface between the CNT and the polymer matrix. It is assumed the FG-CNT reinforced composite plates are made of a mixture of single-walled carbon nanotubes (SWCNTs) and an isotropic matrix. The rule of mixture is employed to estimate the effective material

properties

of

FG-CNT

reinforced

composite

plates.

The

poly{(m4

ACCEPTED MANUSCRIPT phenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene) vinylene]}, referred to as PmPV, is selected for the matrix. The (10, 10) SWCNTs are used as the reinforcements. The detailed material properties of SWCNTs used for the present analysis of FG-CNT reinforced composite plates are selected from the simulated results reported by Shen and Zhang [31]

RI PT

which are tabulated in Table 1.

3. Energy functional for FG-CNT reinforced composite skew plates 3.1. Governing differential Equations

Consider a flat FG-CNT reinforced composite skew plate of uniform thickness t, length a,

SC

oblique width b, with an arbitrary combination of boundary conditions along the four edges. The plate is subjected to normal in-plane loads, Nx and Ny, as shown in Fig. 2. The problem is to determine the elastic buckling load of the plates.

M AN U

The energy functional of the FG-CNT reinforced composite plate is given by

Π = Uε − Wg ,

(2)

in which the strain energy of the plate is given by Uε =

1 ε T SεdΩ , ∫ Ω 2

(3)

TE D

and the potential energy due to in-plane loads is given by

Wg = ∫ ε NTτε N dΩ Ω

(4)

where S is the material property matrix.

AC C

be expressed as

EP

Based on the FSDT, the displacement fields of the FG-CNT reinforced composite plate can

u( x, y, z) = u0 ( x, y) + zθx ( x, y) ,

(5)

v ( x , y , z ) = v0 ( x , y ) + zθ y ( x , y ) ,

(6)

w( x, y, z ) = w0 ( x, y) ,

(7)

in which u0, v0 and w0 are the translation displacements of a point at the mid-plane of the plate in x, y and z directions, respectively, and θ x and θ y are the rotations of a transverse normal about positive y and negative x axes, respectively. In view of equation (3) and using Green’s definition for strain

5

ACCEPTED MANUSCRIPT  ∂w   ε0   ∂x    ε =  κ  , εN =   ,  ∂w  γ   0  ∂y 

(8)

the linear strain-displacement relationships can be expressed as

where

M AN U

SC

 ∂u0     ∂x   ∂v0  ε0 =  ,  ∂y   ∂u0 ∂v0  +    ∂y ∂x 

RI PT

ε xx  γ yz    ε yy  = ε 0 + zκ ,   = γ 0 , γ xz  γ   xy 

 ∂θ x    ∂x    ∂θ y  κ= , ∂y    ∂θ ∂θ y   x+  ∂x   ∂y

(10)

(11)

TE D EP AC C

(9)

∂w0   θ y + ∂y  γ0 =  . θ + ∂w0   x ∂x 

(12)

The stress τ is given by

γ N 0 τ = 1 x  0

0  . γ 2 N y0 

(13)

The material property S is given by

A B 0    S = B D 0  ,  0 0 As   

( Aij , Bij , Dij ) = ∫

h /2

− h /2

(14)

Qij (1, z, z 2 )dz , Aijs = K ∫

h /2

− h /2

Qij dz ,

(15)

6

ACCEPTED MANUSCRIPT s in which Aij , Bij , Dij and Aij are the extensional, coupling, bending, and transverse shear

stiffness, respectively, and K is the transverse shear correction coefficient which is suggested as K = 5 / 6 for isotropic materials [32]. For FGMs, the shear correction coefficient is taken to be K =5 / (6 − (v1V1 + v2V2 )) [33]. It should be noted that the coupling stiffness in Eq. (14)

plane (see Fig.1).

RI PT

should equal to zero because the material property given by Eq. (1) is symmetric in the x-y

In order to satisfy the boundary conditions of the FG-CNT reinforced composite skew plate, it is expedient to use oblique co-ordinates ( ξ , η ) instead of the above orthogonal coordinates ( x, y ). From simple geometry, the oblique co-ordinates ξ , η are given by

η = y sec α ,

SC

ξ = x − y tan α ,

(16) (17)

M AN U

The rotations θ can be expressed in the oblique co-ordinates as

θ x ( x, y ) = θξ (ξ ,η ) cos α ,

θ y ( x, y ) = −θξ (ξ ,η )sin α + θη (ξ ,η ),

(18) (19)

and the partial derivatives are related by

TE D

∂( ) 1 ∂( ) , = ∂x a ∂ξ

∂( ) 1 ∂( ) 1 ∂( ) , = − tan α + ∂y a ∂ξ b cos α ∂η

(20)

(21)

EP

in which α is the skew angle (see Fig. 2).

AC C

In view of Eq. (7), the strain fields can be expressed as

  1 ∂u0   a ∂ξ     ∂v0 1 1 ∂v0 ε0 =  − tan α + , a ∂ξ b cos α ∂η    1 ∂u0 1 ∂u0 1 ∂v0  + +  − tan α  ∂ξ b cos α ∂η a ∂ξ   a

(22)

7

ACCEPTED MANUSCRIPT 1 ∂φξ a ∂ξ

   ∂φη 1 ∂φη  + , ∂ξ b cos α ∂η  1 ∂φξ 1 ∂φη  + +  b cos α ∂η a ∂ξ 

RI PT

   1  κ= − tan α a   1 ∂φξ − tan α ∂ξ  a

(24)

SC

∂w0 1 1 ∂w0   φη + − a tan α ∂ξ + b cos α ∂η    γ0 =  . ∂ w 1 0   φξ +  a ∂ξ 

(23)

In Eq. (3), the potential energy due to in-plane loads can be given in the oblique co-

M AN U

ordinates as

(25)

γ 1 Nξ0 0  τ = . γ 2 Nη0   0

(26)

TE D

1 ∂w0     a ∂ξ , εN =  ∂w0 1 ∂w0   1 − a tan α ∂ξ + b cos α ∂η   

3.2. Two-dimensional IMLS shape functions

EP

The IMLS method is employed to construct the shape functions in the element-free method [14] [34]. For a plate, which is discretized by a set of nodes xi , i =1, … , m, in a

AC C

computational domain Ω , the displacement field can be expressed as m

u h ( x ) = ∑ p i ( x ) a i ( x ) = p T ( x )a ( x ) ,

(27)

i =1

where pi ( x ), i = 1, 2, …, m are the basis functions, m is the number of terms in the basis, and the coefficients ai ( x ) are functions of the spatial coordinates x, which are obtained by performing a weighted least-squares fit. This procedure thus yields the quadratic form n

J = ∑ w( x − x I )[u h ( x , x I ) − u ( x I )]2 I =1

8

ACCEPTED MANUSCRIPT 2

m  = ∑ w( x − x I ) ∑ pi ( x I ) ⋅ ai ( x ) − u ( x I ) , I =1  i =1  n

(28)

where w( x − x I ) is a weight function with a domain of influence, and x I ( I = 1, 2,..., n ) are the nodes with domains of influence that cover the point x .

J = ( Pa − u)T W ( x )( Pa − u) .

∂J = 0, ∂a

M AN U

and we have

SC

The coefficients a ( x ) are obtained by

RI PT

Eq. (28) can be written as

(31)

A( x ) = P TW ( x ) P ,

(32)

B ( x ) = P TW ( x ) .

(33)

u = ( u1 , u2 ,L , un ) T ,

EP

 p1 ( x1 )  p (x ) P= 1 2  M   p1 ( x n )

AC C

(30)

A( x )a ( x ) = B ( x ) u ,

TE D

where

(29)

(34)

pm ( x1 )  p2 ( x 2 ) L pm ( x 2 )  , M O M   p 2 ( x n ) L pm ( x n )  p 2 ( x1 ) L

0  w( x − x1 )  w( x − x 2 ) 0 W ( x) =   M M  0 0 

  L 0  .  O M  L w( x − x n ) L

(35)

0

(36)

In this study, the cubic spline function is chosen as the weight function, w( x − x I ) , which is given by

9

ACCEPTED MANUSCRIPT

zI =

x − xI dI

,

1  2   1  for < zI ≤ 1  , 2  otherwise    for 0 ≤ z I ≤

RI PT

2 2 3  3 − 4 zI + 4 zI  4 4 w( x − x I ) = wI ( x ) =  − 4 z I + 4 z I2 − z I3 3 3 0  

(37)

(38)

where d I is the size of the support domain of node I, determined by

SC

d I = d max cI ,

(39)

M AN U

in which d max is a scaling factor ranging from 2.0 to 4.0, and cI is the average distance between nodes in the influenced domain of the point x.

On the Hilbert space span ( p ) , the following inner product n

( f , g ) = ∑ w( x − x I ) f ( x I ) g ( x I ) ,

(40)

I =1

TE D

is defined. ∀f ( x ) and g ( x ) ∈ span ( p ) .

The equation system A( x ) a ( x ) = B ( x ) u can then be expressed as

L ( p1 , pm )   a1 ( x )   ( p1 , uI )  L ( p2 , pm )   a2 ( x )   ( p2 , uI )   = . O M  M   M      L ( pm , pm )   am ( x )  ( pm , uI ) 

(41)

AC C

EP

 ( p1 , p1 ) ( p1 , p2 ) ( p , p ) ( p , p ) 2 2  2 1 M  M  ( pm , p1 ) ( pm , p2 )

This algebraic equation system may sometime be ill-conditioned or presence of singularity. Thus we assume that the basis function set pi ( x ) ∈ span( p ) , i = 1,2,..., m, is a weighted orthogonal function set about points { xi } , i.e. if ( pi , p j ) = 0,

(i ≠ j ) ,

(42)

then Eq. (41) becomes

10

ACCEPTED MANUSCRIPT 0 ( p1 , p1 )  0 ( p2 , p2 )  M  M  0  0

  a1 ( x )   ( p1 , uI )  0   a2 ( x )   ( p2 , uI )  L  = . O M  M   M      L ( pm , pm )   am ( x )  ( pm , uI )  L

0

(43)

ai ( x ) =

( pi , uI ) , ( pi , pi )

RI PT

From the above equation, coefficients ai ( x) can be determined accordingly: i = 1, 2,L , m ;

i.e.,

SC

a( x ) = A% ( x ) B( x )u ,

0

L

1 ( p2 , p2 )

L

M

O

0

L

TE D

1   (p , p )  1 1  0  A% ( x ) =   M    0  

M AN U

where

    0   .  M   1  ( pm , pm ) 

(44)

(45)

0

(46)

Obviously, coefficients ai ( x) in the IMLS approximation are obtained directly without

system.

EP

matrix inversion. It is, therefore, avoiding forming an ill-conditioned or a singular equation

AC C

From Eqs. (27) and (45), the approximation function u h ( x ) is given by n

% ( x )u = ∑ Φ % ( x )u , uh ( x) = Φ I I

(47)

I =1

% ( x ) is given by in which Φ

% ( x ) = (Φ % ( x ), Φ % ( x ),L , Φ % ( x )) = p T ( x ) A% ( x ) B( x ) , Φ 1 2 n

(48)

and m

% ( x ) = ∑ p ( x )[ A% ( x ) B( x )] , Φ I j jI

(49)

j =1

11

ACCEPTED MANUSCRIPT that represents the shape function of the IMLS approximation corresponding to node I . % ( x ) lead to Meanwhile, the partial derivatives of Φ I m

% ) + p ( A% B + AB % ) ]. % ( x ) = ∑ [ p ( AB Φ I ,i j,i jI j ,i ,i jI

(50)

j =1

RI PT

The weighted orthogonal basis function set p = ( pi ) is formed by using the Gram-Schmidt method as

p1 = 1, M ( r i −1 , pk ) pk , k =1 ( pk , pk ) i −1

pi = r i −1 − ∑

(51)

SC

i = 2, 3,L ,

M AN U

in which r = x12 + x22 or r = x1 + x2 for a two-dimensional problem.

Moreover, the weighted orthogonal basis function set p = ( pi ) can be formed in terms of the polynomial function as

p% = ( p% i ) = (1, x1 , x2 , x3 , x1 x2 , x1 x3 , x2 x3 , x12 , x22 , x32 ...) ,

(52)

TE D

and the weighted orthogonal basis function set can be generated by i −1

( p% i , pk ) pk , k =1 ( pk , pk )

pi = p% i − ∑

i = 1, 2,3,... .

(53)

EP

When the above weighted orthogonal basis functions are adopted, there will be fewer coefficients in the trial function. Thus the computational efficiency should be improved due

AC C

to fewer nodes are required to be included in the influenced domain.

3.3. Buckling of FG-CNT reinforced composite skew plates For a FG-CNT reinforced composite plate discretized by a set of nodes ξ I , I=1,…, NP, we can obtain the approximations of displacements based on Eq. (47) as

 u0h   uI   h     v0  NP  vI  NP h h %  wI  eiωt = ∑ Φ % (ξ )u eiωt . u 0 =  w0  = ∑ Φ I I I  h  I =1   I =1 θ x   θ xI  h θ  θ   yI   y 

(54)

Since the shape function does not have the Kronecker delta property, the essential boundary 12

ACCEPTED MANUSCRIPT conditions cannot be directly imposed. In this study, the transformation method [19-21] is used to enforce the essential boundary conditions even though some other useful methods, such as the penalty method, and Lagrange multipliers, could also be considered. Substituting Eq. (54) into Eq. (1) and performing the Ritz procedure to the total energy functional yields the buckling eigen-equation as follows cr

Kg )u = 0 ,

(55)

RI PT

(K + N

where N cr is the critical buckling load of FG-CNT reinforced composite plates; K and K g represent the linear stiffness matrix and geometric stiffness matrix, respectively, and are given by

SC

K = Kb + K m + K s ,

K bIJ = ∫ BbI DBbJ dΩ , T

M AN U

Ω

(57)

K mIJ = ∫ B mI AB mJ dΩ + ∫ B mI BB bJ dΩ + ∫ B bI BB mJ dΩ , T

T

Ω

T

Ω

(56)

Ω

(58)

K sIJ = ∫ B sI A s B sJ dΩ ,

(59)

K g = ∫ G TI NG J dΩ .

(61)

T

Ω

TE D

Ω

Stiffness matrices in Eqs. (56)-(61) are evaluated via the stabilized nodal integration and direct nodal integration [35]. Compared with the Gauss integration, the stabilized nodal integration and direct nodal integration may reduce the high computational cost and eliminate

EP

the errors caused by the mismatch between quadrature cells and shape function supports [36].

AC C

Based on nodal integration, Eqs. (56)-(61) can be approximated as

NP

NP

T K bIJ = ∑ B% bI (x L )DB% bJ (x L ) AL ,

(62)

L =1

K mIJ = ∑ B mI (x L ) AB mJ (x L ) + B mI (x L )BB bJ (x L ) + BbI (x L )BB mJ (x L )  AL   L =1 T

T

NP

K sIJ = ∑ B sI (x L ) A s B sJ (x L ) AL , T

T

(63)

(64)

L =1

NP

K wIJ = ∑ B wI (x L ) K w B wJ (x L ) AL , T

(65)

L =1 NP

K g = ∑ G TI (x L )NG J (x L ) AL ,

(66)

L =1

13

ACCEPTED MANUSCRIPT where x L and AL denote the node coordinate and nodal representative area, respectively. Matrices B% bI (x L ) , B mI ( x L ) , B bI (x L ) , B sI (x L ) , B wI (x L ) , G ( x L ) and N are calculated by  0 0 0 b%Ix ( x L ) 0    b b%Iy ( x L )  , B% I ( x L ) =  0 0 0 0   % %  0 0 0 bIy ( x L ) bIx ( x L ) 

1 b%Iy (x L ) = AL

∫

ΓL

ψ I ( x L ) n x ( x L ) dΓ ,

ΓL

ψ I ( x L ) n y ( x L ) dΓ ,

1 ∂ψ I a ∂ξ

∂ψ I 1 ∂ψ I + ∂ξ b cos α ∂η

∂ψ I 1 1 ∂ψ I − tan α + a ∂ξ b cos α ∂η 1 ∂ψ I a ∂ξ

AC C

EP

1 ∂ψ I  0 0 a ∂ξ B sI =  1 ∂ψ I 1 ∂ψ I  0 0 − a tan α ∂ξ + b cos α ∂η   0 0 G= 1  0 0 − a tan α 

ψI

1 ∂ψ I a ∂ξ ∂ψ I 1 ∂ψ I + ∂ξ b cos α ∂η

γ 1 N xx N=  0

(69)

   ∂ψ I 1 1 ∂ψ I  , (70) − tan α + ∂ξ b cos α ∂η  a  1 ∂ψ I  a ∂ξ  0

TE D

0

(68)

0

M AN U

 1 ∂ψ I 0 0 0 a ∂ξ   BbI = 0 0 0 0   ∂ψ I 1 1 ∂ψ I 0 0 0 − a tan α ∂ξ + b cos α ∂η      B mI =    1  − a tan α 

RI PT

∫

SC

1 b%Ix (x L ) = AL

(67)

0  . γ 2 N yy 

0

 0 0 0   0 0 0  , (71)   0 0 0 

 0 ,  ψI  

(72)

 0 0 ,  0 0 

(73)

(74)

The buckling load intensity factor k = N cr b 2 / Emt 3 is obtained by solving the generalized eigenvalue problem defined by Eq. (55).

14

ACCEPTED MANUSCRIPT 4. Numerical results and discussion Numerical computation has been carried out to explore the buckling behavior of FG-CNT nodal integration skew plates using the IMLS-Ritz method. The geometry and configuration of a FG-CNT reinforced composite skew plate is presented in Fig. 1. Various types of CNTs distributions, CNTs ratios, skew plates ( α ), aspect ratios (h/a) and thickness-to-height ratios

RI PT

(t/h) under different combinations of supporting conditions are considered in the numerical studies. For easy reference, the boundary conditions at four edges of the FG-CNT reinforced composite skew plates are denoted by four upper case latters, for example, CSFS denotes clamped at the edge AB, simply-supported at the edge BC, free at the edge CD and simply-

SC

supported at the edge DA, respectively. The effective material properties of SWCNTs are obtained from simulated results reported by Shen and Zhang [31] and listed in Table 1. The material

properties

of

the

PmPV

matrix

are

assumed

to

be: v m = 0.34,

M AN U

α m = 45(1 + 0.0005∆T ) ×10−6 / K and E m = (3.51 − 0.0047T ) GPa, and T = 300 K (at room temperature).

4.1. Comparison and Convergence studies

To validate the applicability and versatility of the IMLS-Ritz method, several numerical

TE D

experiments are performed on the buckling analysis of FG-CNT reinforced composite skew plates. A convergence study is carried out to establish the number of nodes required for an accurate solution. Since no results are available for FG-CNT reinforced composite skew plates, comparison studies can only be carried out on the isotropic plates. The IMLS-Ritz

EP

solutions are compared with the pb-2 Ritz results [37] in Table 2 where the buckling load intensity factors ( k = N cr b 2 / (π 2 D) ) are presented for isotropic skew Mindlin plates (with

AC C

a/b=1 and t/b=0.001 and 0.2) under uniaxial compressive pressure ( γ 1 = −1 , γ 2 = 0 ) for various numbers of nodes. Two types of boundary conditions are considered in this convergence and comparison studies, i.e. CCCC and S*S*S*S*. The symbol C denotes a clamped edge, while the S* condition requires only the transverse deflection along the support to be zero. According to the results presented in Table 2, it is evident that the critical buckling coefficients of both pb-2 Ritz method [37] and IMLS-Ritz method agree very well. Besides, for all cases it is observed that as N increases more than 26×26, the IMLS-Ritz solution tend to be closer to the results in [37]. It is also evident that relatively more nodes are needed for the highly slewed plate when compared to a rectangular plate. It can be evident from these studies that the IMLS-Ritz method has high accuracy for furnishing numerical 15

ACCEPTED MANUSCRIPT solution to this problem.

4.2. Buckling behavior of FG-CNTRC skew plates Detailed parametric studies are carried out to investigate the effects of thickness-to-height ratios, aspect ratios, CNT ratios and distributions of CNTs on the buckling load parameters

RI PT

k = N cr b 2 / Emt 3 of the FG-CNT reinforced composite skew plates. In these studies, different combinations of free, simply-supported and clamped edge conditions are also considered. A further convergence study is carried out before the following parametric studies. From Fig. 3, it is apparent that the IMLS-Ritz method converges well with increasing of nodes in the case

SC

of simply-supported UD CNT reinforced composite skew plates (CNT ratio=11%) under uniaxial compression. The effect of d max shrinks as N increases from 13×13 to 31×31.

M AN U

Therefore, d max =3 and N=31×31 are adopted for the following numerical studies. The effect of skew angle on the critical buckling load parameters is studied for FG-CNT reinforced composite skew plates under various loading conditions. In Table 3, the buckling load intensity factors, k = N cr b 2 / Emt 3 , are presented for FG-CNT reinforced composite skew plates subjected to uniaxial compression ( γ 1 = −1 , γ 2 = 0 ), biaxial compression

TE D

( γ 1 = −1 , γ 2 = −1 ) and biaxial compression and tension ( γ 1 = −1 , γ 2 = 1 ) having different CNT volume fractions (from 11% to 17%). The thickness-to-height ratio of the plates is set to t / h =0.01 and aspect ratio h/a=1 under the simply-supported boundary condition (SSSS). The

ratio h/a is kept constant in this analysis, i.e. plate area and volume remain unchanged. This

EP

enables us to investigate the influence of a single factor, i.e. the skew angle, on the buckling load by maintaining the area and volume constant. From Figs. 4-6, the variation of critical

AC C

buckling loads on the skew angle can be observed apparently. It is evident from the table and figures that the critical buckling loads of FG-CNT reinforced composite skew plates increase significantly as the skew angle increases. It is also noted that the change of skew angle has relatively smaller effect on the buckling load parameter for FG-CNT reinforced composite plates under the biaxial compression and tension ( γ 1 = −1 , γ 2 = 1 ), than those under the other two types of compression, i.e. uniaxial compression and biaxial compression. When we focus our study on the effect of volume fraction of CNT, i.e. from 11% to 17%, on the critical buckling load, it is observed that for all three types of FG-CNT reinforced composite skew plates (UD, FG-O, FG-X), the increase in CNTs ratio and critical buckling load k is directly related. This is likely due to the effect of increase in stiffness of the FG-CNT reinforced 16

ACCEPTED MANUSCRIPT composite plate as the CNT volume fraction enhanced. Besides, from the analysis, we found that FG-X plates having the highest buckling load values while FG-O plates having the lowest ones. This phenomenon agreed with the previous finding [38], i.e. CNT reinforcements distributed close to the top and bottom are more efficient than those distributed near the mid-plane for increasing the stiffness of FG-CNT reinforced composite

RI PT

plates. Moreover, when comparing the results across Figs. 4-6, it is evident that the critical buckling load is larger when the FG-CNT reinforced composite skew plate is under the uniaxial compression than that under the biaxial compression.

An investigation on the plate buckling under eight different boundary conditions, i.e.

SC

CCCC, CSCS, SCSC, CCCF, SSSS, SSSF, SFSF and FSFS, is carried out. Numerical results are given in Table 4. All these results are obtained using a/b=1, t/b=0.01 under the uniaxial compression ( γ 1 = −1 , γ 2 = 0 ). These results are useful to examine the effect of different

M AN U

boundary conditions on the buckling load parameters of CNT distributions. It is evident that as the boundary condition changes from the fully clamped to simply-supported and/or free for the corresponding support edges, for example from CCCC to SFSF, the buckling load parameter becomes lower. As the number of simply-supported edge or free edge increases, the buckling load parameter of the corresponding mode become lower. This is because a

buckling load.

TE D

higher constraint at the edge increases the flexural rigidity of the plate, resulting in a higher

A combined effect of thickness-to-height ratios (t/h) and skew angle is studied on the critical buckling load of a simply supported FG-CNT reinforced composite skew plate under

EP

the uniaxial compression. The results are given in Table 5, in which t/h is varied from 0.01 to * = 11% and the aspect ratio is 1. The 0.2, while the CNT volume fractions are fixed at VCNT

AC C

buckling load parameters of various FG-CNT reinforced composite skew plates are plotted in Figs. 7-9. As expected, the shear deformation effect becomes more pronounce as the plate thickness increases. As listed in Table 5, the critical buckling load ratio increases along with the increasing skew angle α . However, negative correlations are observed between the critical buckling load ratio and thickness-to-height ratio. It is noted that the effect of skew angle α on the critical buckling load k shrinks when the plate is getting thicker. Besides, the effect of thickness-to-height ratio seems to be more significant than that of the skew angle. From the study, it is also evident that the effect of distribution of CNTs becomes weaker for the moderately thick FG-CNTRC skew plates. Finally, an examination on the influence of another geometric factor, i.e. plate aspect ratio,

17

ACCEPTED MANUSCRIPT h/a, is considered in this numerical study. Table 6 and Figs. 10-12 illustrate the variation of buckling load parameters of SSSS FG-CNT reinforced composite skew plates with different distributions of CNTs versus plate width-to-thickness ratio. The CNT volume fractions and * = 11% and 1, respectively. According to these thickness-to-height ratio are fixed at VCNT

results, it is apparent that the buckling load parameter shows increasing tendency as the plate

RI PT

aspect ratio h/a varying from 1.0 to 3.0, for UD, FG-O and FG-X types of FG-CNT reinforced composite skew plates. This is likely due to an increase in area as the stiffness of the plate increases, leading to the enhancement of buckling load.

SC

5. Conclusions

The main purpose of the present work is to investigate the buckling behavior of FG-CNT reinforced composite thick skew plates under different compressions. The formulations are

M AN U

established based on FSDT and the governing equations are solved using the element-free IMLS-Ritz method. Through extending this robust and efficient method on the buckling analysis of FG-CNT reinforced composite skew plates, we have examined the composite effect of skew angle and carbon nanotube volume fraction, plate thickness-to-height ratio, plate aspect ratio on the plate’s critical buckling load. It is observed that the increase of skew

TE D

angle is likely to pronounce the influence of geometric and other parameters on the buckling load intensity factors. Besides, the geometric factor, t/h, diminishes the effect of skew angle on the critical buckling load. These studies enable us to better understand the buckling behavior of FG-CNT reinforced composite skew plates, based on which the nonlinear

EP

mechanical behaviors could be studied in the near future.

AC C

Acknowledgements

The work described in this paper was fully supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 9042047, CityU 11208914) and the National Natural Science Foundation of China (Grant No. 11402142 and Grant No. 51378448).

18

ACCEPTED MANUSCRIPT References [1] Shen H-S. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Composite Structures. 2009;91(1):9-19. [2] Aragh BS, Nasrollah Barati AH, Hedayati H. Eshelby–Mori–Tanaka approach for

Compososites Part B: Engineering 2012;43:1943–54.

RI PT

vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels.

[3] Arani GA, Haghparast E, Maraghi KE, Amir S. Static stress analysis of carbon nano-tube reinforced composite (CNTRC) cylinder under non-axisymmetric thermo-mechanical loads and uniform electro-magnetic fields. Composites Part B: Engineering 2015;68:136–45.

SC

[4] Jafari Mehrabadi S, Sobhani Aragh B, Khoshkhahesh V, Taherpour A. Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled

M AN U

carbon nanotubes. Composites Part B: Engineering. 2012;43(4):2031-40.

[5] Malekzadeh P, Shojaee M. Buckling analysis of quadrilateral laminated plates with carbon nanotubes reinforced composite layers. Thin-Walled Structures. 2013;71(0):108-18. [6] Lei ZX, Liew KM, Yu JL. Buckling analysis of functionally graded carbon nanotubereinforced composite plates using the element-free kp-Ritz method. Composite Structures. 2013;98(0):160-8.

TE D

[7] Wu C-P, Li H-Y. The RMVT- and PVD-based finite layer methods for the threedimensional analysis of multilayered composite and FGM plates. Composite Structures. 2010;92(10):2476-96.

[8] Wu C-P, Chang S-K. Stability of carbon nanotube-reinforced composite plates with

EP

surface-bonded piezoelectric layers and under bi-axial compression. Composite Structures. 2014;111(0):587-601.

AC C

[9] Shen H-S. Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells. Compos B-Eng 2012;43:1030–8. [10] Shen H-S, Xiang Y. Postbuckling of axially compressed nanotube-reinforced composite cylindrical panels resting on elastic foundations in thermal environments. Compos B-Eng 2014;67: 50–61.

[11] Shen H-S, Zhu ZH. Buckling and postbuckling behavior of functionally graded nanotube-reinforced composite plates in thermal environments. CMC-Computers, Materials & Continua 2010;18: 155-182. [12] Sheng GG, Wang X. Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells. Applied Mathematical Modelling 2010;34: 2630-2643.

19

ACCEPTED MANUSCRIPT [13] Meng F, Wang H, Wang X, Li Z. Elliptically delaminated buckling near the surface of piezoelectric laminated shells under electric and thermal loads. Composite Structures 2010; 92: 684-690. [14] Liew KM, Lei ZX, Zhang LW. Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review. Composite Structures. 2015;120(2):90-7.

Numerical Methods in Engineering. 1994;37(2):229-56.

RI PT

[15] Belytschko T, Lu YY, Gu L. Element-free Galerkin methods. International Journal for

[16] Zhang Z, Zhao P, Liew KM. Improved element-free Galerkin method for twodimensional

potential

problems.

Engineering

Analysis

Boundary

Elements.

SC

2009;33(4):547-54.

with

[17] Zhang LW, Deng YJ, Liew KM. An improved element-free Galerkin method for

Boundary Elements. 2014;40(0):181-8.

M AN U

numerical modeling of the biological population problems. Engineering Analysis with

[18] Lei ZX, Liew KM, Yu JL. Large deflection analysis of functionally graded carbon nanotube-reinforced composite plates by the element-free kp-Ritz method. Computer Methods in Applied Mechanics and Engineering. 2013;256(0):189-99. [19] Liew KM, Hu YG, Zhao X, Ng TY. Dynamic stability analysis of composite laminated

TE D

cylindrical shells via the mesh-free kp-Ritz method. Computer Methods in Applied Mechanics and Engineering. 2006;196(1–3):147-60. [20] Lei ZX, Liew KM, Yu JL. Free vibration analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method in thermal

EP

environment. Composite Structures. 2013;106(0):128-38. [21] Lei ZX, Zhang LW, Liew KM, Yu JL. Dynamic stability analysis of carbon nanotube-

AC C

reinforced functionally graded cylindrical panels using the element-free kp-Ritz method. Composite Structures. 2014;113(0):328-38. [22] Zhang LW, Lei ZX, Liew KM, Yu JL. Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels. Computer Methods in Applied Mechanics and Engineering. 2014;273(0):1-18. [23] Zhang LW, Lei ZX, Liew KM. Free vibration analysis of functionally graded carbon nanotube-reinforced composite triangular plates using the FSDT and element-free IMLS-Ritz method. Composite Structures. 2015;120(0):189-99. [24] Sator L, Sladek V, Sladek J. Coupling effects in elastic analysis of FGM composite plates by mesh-free methods. Composite Structures. 2014;115(0):100-10.

20

ACCEPTED MANUSCRIPT [25] Zhu P, Liew KM. Free vibration analysis of moderately thick functionally graded plates by local Kriging meshless method. Composite Structures. 2011;93(11):2925-44. [26] Zhang LW, Zhu P, Liew KM. Thermal buckling of functionally graded plates using a local Kriging meshless method. Composite Structures. 2014;108(0):472-92. [27] Li H, Wang QX, Lam KY. Development of a novel meshless Local Kriging (LoKriging)

RI PT

method for structural dynamic analysis. Comput Meth Appl Mech Eng. 2004;193(2326):2599-619.

[28] Lam KY, Wang QX, Li H. A novel meshless approach - Local Kriging (LoKriging) method with two-dimensional structural analysis. Comput Mech. 2004;33(3):235-44.

SC

[29] Li H, Wang QX, Lam KY. Development of a novel meshless Local Kriging (LoKriging) method for structural dynamic analysis. Computer Methods in Applied Mechanics and Engineering. 2004;193(23–26):2599-619.

M AN U

[30] Gu YT, Wang QX, Lam KY. A meshless local Kriging method for large deformation analyses. Computer Methods in Applied Mechanics and Engineering. 2007;196(9–12):167384.

[31] Shen H-S, Zhang C-L. Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates. Materials & Design. 2010;31(7):3403-

TE D

11.

[32] Liew KM. On the use of pb-2 Rayleigh-Ritz method for free-flexural vibration of triangular plates with curved internal supports. Journal of Sound and Vibration. 1993;165:329-40.

EP

[33] Efraim E, Eisenberger M. Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. Journal of Sound and Vibration. 2007;299(4–5):720-38.

AC C

[34] Ren H, Cheng Y. The interpolating element-free Galerkin (IEFG) method for twodimensional

potential

problems.

Engineering

Analysis

with

Boundary

Elements.

2012;36(5):873-80.

[35] Chen JS, Wu CT, Yoon S, You Y. A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng. 2001;50(2):435-66. [36] Beissel S, Belytschko T. Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng. 1996;139(1-4):49-74. [37] Kitipornchai S, Xiang Y, Wang CM, Liew KM. Buckling of thick skew plates. International Journal for Numerical Methods in Engineering. 1993;36(8):1299-310.

21

ACCEPTED MANUSCRIPT [38] Zhu P, Lei ZX, Liew KM. Static and free vibration analyses of carbon nanotubereinforced composite plates using finite element method with first order shear deformation

AC C

EP

TE D

M AN U

SC

RI PT

plate theory. Composite Structures. 2012;94(4):1450-60.

22

ACCEPTED MANUSCRIPT Table 1. Young’s moduli for PmPV/CNT composites reinforced by (10, 10) SWCNT under T = 300 K .

Rule of mixture [31] E11 (GPa) 94.57 120.09 145.08

0.11 0.14 0.17

Table

2.

Convergence

and

η1

η2

E22 (GPa) 0.149 2.2 0.150 2.3 0.149 3.5

comparison

studies

0.934 0.941 1.381

of

RI PT

* VCNT

buckling

load

intensity

and t/b=0.001 and 0.2. t/b=0.001 15 30

0

45

0

t/b=0.2 15 30

45

4.3854 4.3899 4.3904 4.3911 4.3940

5.8355 5.8712 5.9013 5.9014 5.9026

9.7716 10.0022 10.1321 10.1330 10.1428

2.8455 2.8512 2.8754 2.8755 2.8766

3.0999 3.1006 3.1038 3.1041 3.1111

4.0549 4.0481 4.0215 4.0136 3.9226

5.5795 5.5624 5.5547 5.5545 5.5479

9.9012 9.9028 9.9078 9.9099 10.0738

10.7800 10.8056 10.8293 10.8318 10.8345

13.2154 13.3993 13.4566 13.4568 13.5377

19.6267 19.8808 20.0955 20.0984 20.1115

5.1371 5.2710 5.2933 5.3043 5.3156

5.3553 5.4070 5.4370 5.4553 5.4913

5.8760 5.8927 5.9229 5.9264 6.0328

6.8770 6.8119 6.9549 6.9612 6.9712

EP

TE D

3.8691 3.8714 3,8715 3.9736 3.9998

AC C

S*S*S*S* 21×21 23×23 27×27 29×29 Pb-2 Ritz CCCC 21×21 23×23 27×27 29×29 Pb-2 Ritz

M AN U

Skew angle α °

N

SC

factors, k = N cr b 2 / (π 2 D) , by varying the number of nodes (N) for isotropic plates with a/b=1

23

ACCEPTED MANUSCRIPT Table 3. Buckling load parameter k = N cr b 2 / Emt 3 for various types of FG-CNT reinforced composite plates (h/a=1, t/h=0.01) under compression and tension with SSSS boundary conditions.

TE D

EP

AC C

45°

63.2226 75.4869 89.8620 42.5896 48.6334 58.4320 85.3655 100.8541 119.7000

122.3462 141.6644 168.6500 76.7106 90.4980 107.2656 158.7182 183.1880 218.8127

216.6947 260.7669 308.3364 130.3856 157.7246 185.2418 287.3778 343.5769 408.8017

15.7522 17.4593 25.5845 12.7576 12.9233 19.0046 18.5529 21.0794 30.7477 = 1) 233.8050 288.6736 336.5686 134.2894 164.3948 192.1009 327.4068 405.3863 473.4253

29.5266 32.9821 45.4556 22.4801 25.0019 34.8340 34.6072 38.5770 55.2056

62.9565 71.1082 98.1507 45.3420 52.1485 68.4423 75.8789 86.5262 120.4126

304.3710 371.7221 435.9453 176.0935 216.1693 252.4383 416.1738 507.2465 597.0713

657.7089 801.9131 945.7881 373.6782 460.7288 537.3876 909.5909 1113.9781 1308.9980

M AN U

0° uniaxial compression ( γ 1 = −1 , γ 2 = 0 ) UD 11% 37.8553 14% 49.1426 17% 57.5401 FG-O 11% 21.3235 14% 26.3994 17% 31.2210 FG-X 11% 56.7107 14% 71.7667 17% 83.7358 biaxial compression ( γ 1 = −1 , γ 2 = −1 ) 11% 11.4511 UD 14% 13.3746 17% 17.8289 11% 7.7310 FG-O 14% 8.8137 17% 12.0926 11% 14.3439 FG-X 14% 16.3778 17% 23.7981 biaxial compression and tension ( γ 1 = −1 , γ 2 11% 186.9243 UD 14% 235.1053 17% 272.2822 11% 99.2650 FG-O 14% 124.5118 17% 143.9923 11% 270.9198 FG-X 14% 340.2747 17% 395.0279

RI PT

Critical buckling load ratio 15° 30°

CNT ratio

SC

Types

24

ACCEPTED MANUSCRIPT Table 4. Buckling load parameter k = N cr b 2 / Emt 3 for various types of FG-CNT reinforced composite plates (b/a=1, t/b=0.01) under uniaxial compression ( γ 1 = −1 , γ 2 = 0 ) with different boundary conditions.

CCCF SSSS

SSSF SFSF

RI PT

30° 149.1832 82.8979 213.0034 149.1828 82.8976 213.0015 102.9716 67.5650 131.3016 139.045 73.6114 201.6874 92.5900 60.3045 122.6200 91.6234 57.0086 119.1749 90.0412 55.9202 117.3037 10.0429 9.2109 10.8710

45° 170.8123 102.7086 236.4000 170.8111 102.7073 236.3982 144.5785 95.6111 185.0553 141.7337 75.9204 204.8227 118.5677 74.0455 158.0032 111.027 65.6242 148.1237 108.6756 63.9564 145.1185 27.3313 24.0351 30.0913

AC C

FSFS

15° 144.0686 78.1404 207.502 144.0677 78.1402 207.5009 65.7618 44.6058 85.1663 137.9727 72.8127 200.1253 59.6296 40.4401 80.4751 59.8536 38.7915 78.9768 59.1491 38.1508 78.2144 5.8164 5.4128 6.2589

SC

SCSC

0° 142.9444 77.0899 206.2944 142.9436 77.0897 206.2932 42.7758 24.8459 60.6283 137.6183 72.5938 199.4954 39.1158 21.3509 56.7423 38.0432 20.2474 55.6336 37.3990 19.6595 54.9203 4.8893 4.5673 5.2535

M AN U

CSCS

UD FG-O FG-X UD FG-O FG-X UD FG-O FG-X UD FG-O FG-X UD FG-O FG-X UD FG-O FG-X UD FG-O FG-X UD FG-O FG-X

Critical buckling load ratio

TE D

CCCC

Types

EP

Boundary conditions

25

ACCEPTED MANUSCRIPT Table 5. Buckling load parameter k = N cr b 2 / Emt 3 for various types of FG-CNT reinforced composite plates (h/a=1) under uniaxial compression ( γ 1 = −1 , γ 2 = 0 ) in different thicknessto-width ratio (SSSS).

AC C

EP

TE D

FG-X

0° 37.8553 30.9218 19.0902 11.7900 7.8014 21.3235 18.7625 13.8123 9.7167 6.9590 56.7107 40.8409 22.1615 12.8645 8.0335

RI PT

FG-O

0.01 0.05 0.1 0.15 0.2 0.01 0.05 0.1 0.15 0.2 0.01 0.05 0.1 0.15 0.2

Critical buckling load ratio 15° 30° 45° 63.2226 122.3462 216.6947 34.2129 46.2466 77.4031 20.6021 26.4520 42.4321 12.6845 15.9547 25.4873 8.3769 10.2885 16.4615 42.5896 76.7106 130.3856 21.1928 29.9404 52.6220 15.0412 19.6184 32.3622 10.4772 13.3187 21.6393 7.4571 9.2737 15.0663 85.3655 158.718 287.3778 44.8860 59.4846 97.2334 23.9930 30.6185 48.5578 13.8406 17.3021 27.4866 8.7864 10.7439 16.6154

SC

UD

t/ h

M AN U

Types

26

ACCEPTED MANUSCRIPT Table 6. Buckling load parameter k = N cr b 2 / Emt 3 for various types of FG-CNT reinforced composite plates (t/a=0.01) under uniaxial compression ( γ 1 = −1 , γ 2 = 0 ) in different aspect ratio (SSSS).

1.5

2

2.5

AC C

EP

TE D

3

0° 37.8553 21.3235 56.7107 54.4549 28.8634 75.5829 83.4605 43.5372 122.5596 146.8624 75.9185 215.6541 228.6242 118.1844 337.8747 328.3944 169.6473 480.8008

RI PT

1.2

UD FG-O FG-X UD FG-O FG-X UD FG-O FG-X UD FG-O FG-X UD FG-O FG-X UD FG-O FG-X

Critical buckling load ratio 15° 30° 45° 63.2226 122.3462 216.6947 42.5896 76.7106 130.3856 85.3655 158.718 287.3778 89.1181 170.0083 309.9035 59.4616 108.7348 185.9580 120.8473 226.3031 419.4576 137.5283 269.9084 449.1103 91.1964 168.1614 262.3633 186.9340 350.9974 611.4877 242.5961 404.8584 606.3358 160.181 235.2299 357.6215 330.033 549.7768 825.2163 377.1930 547.1262 795.8951 241.9311 314.7471 462.6182 501.9328 765.1999 1122.274 539.8882 708.0200 1067.891 325.7737 395.2985 587.4352 715.9362 1015.133 1530.234

SC

1

Types

M AN U

h/ a

27

ACCEPTED MANUSCRIPT

Figures z

RI PT

x

b y

M AN U

a

SC

t

UD

FG-X

FG-O

y

EP

TE D

Figure 1. Geometry and Configurations of FG-CNT reinforced composite skew plates

η

Ny

AC C

D

Nx

h

α

Nη C

D

C

b

Nξ

b

ξ

x A

B a

B

A a

Figure 2. Transformation of geometry and coordinate system of the FG-CNT reinforced composite skew plate: (a) actual skew plate, and (b) square computational domain

28

38.4

dmax=3

38.3

dmax=2.8 dmax=2.6

RI PT

38.2 38.1 38 37.9 37.8

15×15

19×19

23×23 N

SC

Buckling load intensity factor (k = Ncrb2/Emt3)

ACCEPTED MANUSCRIPT

27×27

31×31

M AN U

Figure 3. Convergence of buckling load intensity factors, k = N cr b 2 / Emt 3 , for simply supported UD CNT reinforced composite skew plates with h/a=1, t/a=0.01, α = 0°

150 100 50

0 0

15

30

45

400 350

UD FG-O FG-X

300 250 200 150 100

50 0 0

Buckling load intensity factor (k = Ncrb2/Emt3)

EP

300

200

450

TE D

350

250

CNT ratio=14%

Buckling load intensity factor (k = Ncrb2/Emt3)

UD FG-O FG-X

400

AC C

Buckling load intensity factor (k = Ncrb2/Emt3)

CNT ratio=11% 450

15 30 45 Skew angle, α°

CNT ratio=17% 450 UD FG-O FG-X

400 350 300 250 200 150 100 50 0 0

15

30

45

Figure 4. Effects of skew angle on the critical buckling load ( k = N cr b 2 / Emt 3 ) of FG-CNT reinforced composite skew plates under uniaxial compression

29

ACCEPTED MANUSCRIPT

40 20 0 0

15

30

60 40 20 0 0

45

80

RI PT

60

80

100

60 40

SC

80

100

UD FG-O FG-X

120

15 30 45 Skew angle, α°

M AN U

100

UD FG-O FG-X

120

CNT ratio=17% Buckling load intensity factor (k = Ncrb2/Emt3)

UD FG-O FG-X

120

CNT ratio=14% Buckling load intensity factor (k = Ncrb2/Emt3)

Buckling load intensity factor (k = Ncrb2/Emt3)

CNT ratio=11%

20

0 0

15

30

45

Figure 5. Effects of skew angle on the critical buckling load ( k = N cr b 2 / Emt 3 ) of FG-CNT reinforced composite skew plates under biaxial compression and tension

600 400 200

15

30

45

1200

UD FG-O FG-X

1000

800 600 400 200 0 0 15 30 45 Skew angle, α°

Buckling load intensity factor (k = Ncrb2/Emt3)

EP

1000

0 0

1400

TE D

1200

800

CNT ratio=14%

Buckling load intensity factor (k = Ncrb2/Emt3)

UD FG-O FG-X

AC C

Buckling load intensity factor (k = Ncrb2/Emt3)

CNT ratio=11% 1400

CNT ratio=17% 1400 UD FG-O FG-X

1200 1000 800 600 400 200 0 0

15

30

45

Figure 6. Effects of skew angle on the critical buckling load ( k = N cr b 2 / Emt 3 ) of FG-CNT reinforced composite skew plates under biaxial compression 30

ACCEPTED MANUSCRIPT

t/h=0.01 t/h=0.05 t/h=0.1 t/h=0.15 t/h=0.2

200

RI PT

150

100

50

SC

Buckling load intensity factor ( k = Ncrb2/Emt3)

UD

0 0

45

M AN U

15 30 Skew angle, α °

Figure 7. Effect of thickness-to-height ratio t/h on the critical buckling load ( k = N cr b 2 / Emt 3 ) of UD CNT reinforced composite skew plates

TE D

100

t/h=0.01 t/h=0.05 t/h=0.1 t/h=0.15 t/h=0.2

EP

Buckling load intensity factor (k = Ncrb2/Emt3)

FG-O

150

AC C

50

0 0

15 30 Skew angle, α °

45

Figure 8. Effect of thickness-to-height ratio t/h on the critical buckling load ( k = N cr b 2 / Emt 3 ) of FG-O CNT reinforced composite skew plates

31

ACCEPTED MANUSCRIPT

t/h=0.01 t/h=0.05 t/h=0.1 t/h=0.15 t/h=0.2

250

RI PT

200 150 100

SC

Buckling load intensity factor (k = Ncrb2/Emt3)

FG-X 300

50

M AN U

0 0

15 30 Skew angle, α °

45

Figure 9. Effect of thickness-to-height ratio t/h on the critical buckling load ( k = N cr b 2 / Emt 3 )

TE D

UD

1200 1000

EP

Buckling load intensity factor ( k = Ncrb2/Emt3)

of FG-X CNT reinforced composite skew plates

AC C

800

α=0° α=15° α=30° α=45°

600 400 200 0 1

1.2

1.5

2

2.5

6

h/a

Figure 10. Effect of plate aspect ratio h/a on the critical buckling load ( k = N cr b 2 / Emt 3 ) of UD CNT reinforced composite skew plates 32

600

400

RI PT

300 200 100 0 1

1.2

1.5

2 h/a

SC

500

α =0° α =15° α =30° α =45°

2.5

3

M AN U

Buckling load intensity factor ( k = Ncrb2/Emt3)

ACCEPTED MANUSCRIPT

Figure 11. Effect of plate aspect ratio h/a on the critical buckling load ( k = N cr b 2 / Emt 3 ) of

TE D

1500

EP

Buckling load intensity factor ( k = Ncrb2/Emt3)

FG-O CNT reinforced composite skew plates

AC C

1000

α =0° α =15° α =30° α =45°

500

0 1

1.2

1.5

2

2.5

3

h/a

Figure 12. Effect of plate aspect ratio h/a on the critical buckling load ( k = N cr b 2 / Emt 3 ) of FG-X CNT reinforced composite skew plates 33