Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates

Materials and Design 31 (2010) 3403–3411

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Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates Hui-Shen Shen a,b,*, Chen-Li Zhang a a b

School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 23 December 2009 Accepted 26 January 2010 Available online 1 February 2010 Keywords: A. Composites E. Thermal I. Buckling

a b s t r a c t Thermal buckling and postbuckling behavior is presented for functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs) subjected to in-plane temperature variation. The material properties of SWCNTs are assumed to be temperature-dependent and are obtained from molecular dynamics simulations. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. Based on the multi-scale approach, numerical illustrations are carried out for perfect and imperfect, geometrically mid-plane symmetric FG-CNTRC plates and uniformly distributed CNTRC plates under different values of the nanotube volume fractions. The results show that the buckling temperature as well as thermal postbuckling strength of the plate can be increased as a result of a functionally graded reinforcement. It is found that in some cases the CNTRC plate with intermediate nanotube volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Recently, a new member of advanced material family, carbon nanotube-reinforced composites, have been synthesized [1–3]. As the mechanical properties of composites depend directly upon the embedded fiber mechanical behavior, replacing conventional micro-sized fibers with CNTs can potentially improve composite properties, such as tensile strength and elastic modulus. Therefore, the introduction of carbon nanotubes into polymers may improve their applications in the fields of reinforcing composites, electronic devices and more. Most studies on carbon nanotube-reinforced composites (CNTRCs) have focused on their material properties [4–9]. It has been reported most CNTRCs have low volume fractions of nanotube [5,6]. Several investigations have shown that the addition of small amounts of carbon nanotube can considerably improve the mechanical, electrical and thermal properties of polymeric composites [4–9]. Instabilities in CNTRCs have also been of substantial interest and many experiments have observed buckling [10–13]. Even though these studies are quite useful in establishing the buckling properties of the nanocomposites, their use in actual structural applications is the ultimate purpose for the development * Corresponding author. Address: School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China. E-mail address: [email protected] (H.-S. Shen). 0261-3069/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2010.01.048

of this advanced class of materials. As a result, there is a need to observe the global response of CNTRCs in an actual structural element such as a beam as was done in Vodenitcharova and Zhang [14]. Motivated by these considerations, we aim to study the thermal buckling behavior of CNTRC plates subject to in-plane temperature variation. One of the problems is how to determine and increase the buckling temperature and thermal postbuckling strength of CNTRC plates under such a low nanotube volume fraction. The traditional approach to fabricating nanocomposites implies that the nanotube is distributed either uniformly or randomly such that the resulting mechanical, thermal, or physical properties do not vary spatially at the macroscopic level. Functionally graded materials (FGMs) are a new generation of composite materials in which the microstructural details are spatially varied through non-uniform distribution of the reinforcement phase. The concept of FGM can be utilized for the management of a material’s microstructure so that the buckling behavior of a plate/shell structure made of such material can be improved. Birman [15] made the first attempt to solve the compressive buckling problem of functionally graded hybrid composite plates. Feldman and Aboudi [16] studied the compressive buckling of composite plates with functionally graded distribution of reinforcement volume fraction. It is worthy to note that in [15,16] they only investigated the mid-plane symmetric composite plates. This is due to the fact that the bifurcation buckling does not exist due to the stretching–bending coupling

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effect for the simply supported FGM plates subjected to in-plane edge compressive loads or temperature variation. The notable recent contributions pertaining to the thermal buckling analysis of rectangular FGM plates are available in [17–20]. However, some previous analyses fell into a trap and yielded physically incorrect solutions, as mentioned in [21,22]. In these studies FGM means functionally graded ceramic–metal materials, and the material properties of which are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Motivated by these ideas, the concept of functionally graded materials has been applied to the nanocomposite plates and/or beams reinforced by SWCNTs [23,24]. The present paper extends the previous works [23] to the case of functionally graded CNTRC plates with low nanotube volume fractions. The temperature field considered is assumed to a parabolic distribution over the plate surface and uniform through the plate thickness. The material properties of SWCNTs are assumed to be size-dependent and temperature-dependent and are obtained from molecular dynamics (MD) simulations. The material properties of functionally graded CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model in which the CNT efficiency parameter is estimated by matching the elastic modulus of CNTRCs observed from the MD simulation results with the numerical results obtained from the extended rule of mixture. The governing equations are based on a higher order shear deformation plate theory with a von Kármán-type of kinematic nonlinearity and include thermal effects. The initial geometric imperfection of the plate is taken into account. All four edges of the plate are assumed to be simply supported with no in-plane displacement.

2. Material properties of functionally graded CNTRC plates Consider a rectangular CNTRC plate which consists of two layers made of functionally graded materials and is mid-plane symmetric, as shown in Fig. 1. The length, width and total thickness of the CNTRC plate are a, b and t. As usual, the coordinate system has  V  its origin at the corner of the plate on the middle plane. Let U,  and W be the plate displacements parallel to a right-hand set of axes (X, Y, Z), where X is longitudinal and Z is perpendicular to  y are the mid-plane rotations of the normals  x and W the plate. W about the Y and X axes, respectively. We assume that each CNTRC layer is made from a mixture of SWCNT, graded distribution in the thickness direction, and matrix which is assumed to be isotropic. At the nanoscale, the structure of the carbon nanotube strongly influences the overall properties of the composite. Several micromechanical models have been developed to predict the effective material properties of CNTRCs, e.g. Mori–Tanaka scheme [25,26] and the rule of mixture [2,5]. The Mori–Tanaka scheme is applicable to nanoparticles and the rule

of mixture is simple and convenient to apply for predicting the overall material properties and responses of the structures. The accuracy of the rule of mixture was discussed and a remarkable synergism between the Mori–Tanaka scheme and the rule of mixture for functionally graded ceramic–metal beams was reported in [27]. According to the extended rule of mixture, the effective Young’s modulus and shear modulus can be expressed as [23] m E11 ¼ g1 V CN ECN 11 þ V m E

g2 E22

g3 G12

¼

¼

V CN ECN 22 V CN GCN 12

þ

Vm Em

ð2Þ

þ

Vm Gm

ð3Þ

CN CN where ECN 11 , E22 and G12 are the Young’s moduli and shear modulus, respectively, of the carbon nanotube, and Em and Gm are corresponding properties for the matrix. 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.), we introduced gj (j = 1,2,3) into Eqs. (1)–(3) to consider the size-dependent material properties. gj is called the CNT efficiency parameter which will be determined later by matching the elastic modulus of CNTRCs observed from the MD simulation results with the numerical results obtained from the rule of mixture. VCN and Vm are the carbon nanotube and matrix volume fractions and are related by

V CN þ V m ¼ 1

ð4Þ

The material properties of functionally graded ceramic–metal materials vary continuously from one surface to the other and, therefore, the volume fractions of the constituents may follow a simple power law [17–22]. In contrast, for functionally graded fiber-reinforced composites, to avoid abrupt change of the material properties, only linear distribution can readily be achieved in practice. Consequently, we assume the volume fraction VCN for the top layer follows as

V CN ¼

  2t 1  2Z  V CN t1  t0

ð5Þ

and for the bottom layer follows as

V CN ¼

  2Z  2t1  V CN t2  t1

ð6Þ

in which

V CN ¼

wCN wCN þ ðqCN =qm Þ  ðqCN =qm ÞwCN

ð7Þ

where wCN is the mass fraction of nanotube, and qCN and qm are the densities of carbon nanotube and matrix, respectively. In such a way, the two cases of uniformly distributed (UD), i.e. V CN ¼ V CN , and functionally graded (FG) CNTRCs will have the same value of mass fraction of nanotube. Similarly, the thermal expansion coefficients in the longitudinal and transverse directions can be expressed as m a11 ¼ V CN aCN 11 þ V m a





f m m a22 ¼ 1 þ mCN 12 V CN a22 þ ð1 þ m ÞV m a  m12 a11

Fig. 1. Configuration of a functionally graded carbon nanotube-reinforced composite plate.

ð1Þ

ð8Þ ð9Þ

m CN where aCN are thermal expansion coefficients, and mCN 11 , a22 and a 12 and mm are Poisson’s ratios, respectively, of the carbon nanotube and matrix. Note that a11 and a22 are also graded in the Z direction. It is assumed that the material property of nanotube and matrix is a function of temperature, so that the effective material properties of

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CNTRCs, like Young’s modulus, shear modulus and thermal expansion coefficients, are functions of temperature and position. The Poisson’s ratio depends weakly on temperature change and position [28] and is expressed as m m12 ¼ V CN mCN 12 þ V m m

ð10Þ

3. Governing equations of functionally graded CNTRC plates The plate is assumed to be geometrically imperfect, and is subjected to in-plane temperature variation. The temperature field is assumed to be a parabolic distribution in the XY-plane of the plate, but uniform through the plate thickness [29]

" TðX; YÞ ¼ T 1 þ T 2 1 



2X  a a

2 #"

 2 # 2Y  b 1 b

ð11Þ

where T1 is the uniform temperature rise and T2 is the temperature gradient. Reddy [30,31] developed a simple higher order shear deformation plate theory. This theory assumes that the transverse shear strains are parabolically distributed across the plate thickness. The advantages of this theory over the first order shear deforma V,  tion theory are that the number of independent unknowns (U;  W  y ) is the same as in the first order shear deformation  x and W W, theory, and no shear correction factors are required. Denoting the  (X, Y) be the addi  (X, Y), let W initial geometric imperfection by W  tional deflection and F (X, Y) be the stress function for the stress  , N  y ¼ F;     x ¼ F; resultants defined by N YY XX and N xy ¼ F;XY , where a comma denotes partial differentiation with respect to the corresponding coordinates. Based on Reddy’s higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity and including thermal effects, the governing differential equations for an FG-CNTRC plate can be derived in terms of a stress function   two rotations W  y , and a transverse displacement W,  x and W F;   along with the initial geometric imperfection W . They are

~L11 ðWÞ   ~L12 ðW   ~L15 ðN  T Þ  ~L16 ðM  TÞ  x Þ  ~L13 ðW  y Þ þ ~L14 ðFÞ  þW   ; FÞ  ¼ ~LðW

ð12Þ

2

~L31 ðWÞ  þ ~L32 ðW   ~L35 ðN  T Þ  ~L36 ðST Þ ¼ 0  x Þ  ~L33 ðW  y Þ þ ~L34 ðFÞ

ð14Þ

  ~L42 ðW   ~L45 ðN  T Þ  ~L46 ðST Þ ¼ 0 ~L41 ðWÞ  x Þ þ ~L43 ðW  y Þ þ ~L44 ðFÞ

ð15Þ

Note that the geometric nonlinearity in the von Kármán sense is given in terms of ~L () in Eqs. (12) and (13), and the other linear operators ~ Lij () are defined as in Shen [32,33].  T, P  T , and  T, M ST are the forces, moIn the above equations, N ments and higher order moments caused by elevated temperature, and are defined by

2 T Nx 6N T 4 y T  N

xy

T M x T M y

T M xy

2 3 T 3 P Z tk Ax x X 6 7 3 T 7 P 4 Ay 5 ð1; Z; Z ÞDT dZ y 5 ¼ t k1 k¼1 T Axy P k

xy

2 T 3 2  T 3 2 T 3 Mx Px Sx 4 6 ST 7 6 M 7 6P T  T 7 ¼  4 y 5 4 y 5 24 y 5 3t ST T T M P xy

xy

ð16Þ

ð17Þ

xy

where DT = T  T0 is temperature rise from the reference temperature T0 at which there are no thermal strains, and

2

 11 Ax Q 6 7 6 4 Ay 5 ¼ 4 Q 12  16 Axy Q

 12 Q  22 Q  26 Q

32

3

 16 1 0  Q  6 7 a11 ðTÞ  26 7 54 0 1 5 Q a22 ðTÞ  66 0 0 Q

ð18Þ

where a11 and a22 are the thermal expansion coefficients measured in the longitudinal and transverse directions for kth ply, in particu ij are lar for an CNTRC layer they are given in detail in Eq. (4), and Q the transformed elastic constants with details being given in  ij ¼ Q in which [30,31]. Note that for an FG-CNTRC layer, Q ij

E11 ðTÞ ; 1  m12 m21 m21 E11 ðTÞ ¼ ; 1  m12 m21

E22 ðTÞ ; 1  m12 m21

Q 11 ¼

Q 22 ¼

Q 12

Q 16 ¼ Q 26 ¼ 0;

Q 66 ¼ G12 ðTÞ

ð19Þ

where E11, E22, G12, m12 and m21 have their usual meanings, in particular for an CNTRC layer they are given in detail in Eqs. (1), (2), and (10). All four edges of the plate are assumed to be simply supported with no in-plane displacements. The boundary conditions are X = 0, a:

 ¼W y ¼ 0 W

ð20Þ

 ¼0 U

ð21Þ

 xy ¼ 0; N

x ¼P x ¼ 0 M

ð22Þ

Y = 0, b:

 ¼W x ¼ 0 W

ð23Þ

 ¼0 V

ð24Þ

 xy ¼ 0; N

y ¼P y ¼ 0 M

ð25Þ

 x and M  y are the bending moments and P  x and P  y are the where M higher order moments as defined in [30,31].  ¼ 0 (on The condition expressing the immovability condition, U  = 0 (on Y = 0, b), is fulfilled on the average sense as X = 0, a) and V

Z 0

 þ 2W   ; WÞ  þ ~L22 ðW   ~L25 ðN  T Þ ¼  1 ~LðW  ~L21 ðFÞ  x Þ þ ~L23 ðW  y Þ  ~L24 ðWÞ 2 ð13Þ

3

b

Z 0

a

 @U dX dY ¼ 0; @X

Z

a 0

Z 0

b

@ V dY dX ¼ 0 @Y

ð26Þ

This condition in conjunction with Eqs. (27) and (28) below provides the compressive stresses acting on the edges X = 0, a and Y = 0, b. The average end-shortening relationships are defined as Z bZ a  Dx 1 @U ¼ dX dY a ab 0 0 @X    Z b Z a (" 1 @ 2 F @ 2 F 4 @ Wx A11 2 þ A12 2 þ B11  2 E11 ¼ @X ab 0 0 3t @Y @X !#      2 2  2  4 @ Wy 4 @ W 1 @W  @ W þ E   2 E11 þ B12  2 E12 12 @Y 2 @X 3t 3t @X 2 @Y 2 )    @W @W  T þ A N T  ð27Þ  ðA11 N 12 y Þ dX dY x @X @X

Z

Z

 @V dY dX 0 0 @Y " (    Z aZ b 1 @ 2 F @ 2 F 4 @ Wx A22 2 þ A12 2 þ B21  2 E21 ¼ @X ab 0 0 3t @X @Y !#      2 2  2  4 @ Wy 4 @ W @ W 1 @W þ B22  2 E22 þ E22   2 E21 2 2 @Y 2 @Y 3t 3t @X @Y )    @W @W  T þ A N T  ð28Þ  ðA12 N 22 y Þ dY dX x @Y @Y

Dy 1 ¼ b ab

a

b

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where Dx and Dy are plate end-shortening displacements in the X and Y directions, and for the case of immovable edges Dx and Dy must be zero-valued. In the above equations, the reduced stiffness matrices [Aij ], [Bij ], [Dij ], [Eij ], [F ij ] and [Hij ] are functions of T and Z, determined through relationship [32,33]

and the details of which can be found in Appendix A. The nonlinear Eqs. (12)–(15) may then be written in dimensionless form as

A ¼ A1 ;

L21 ðFÞ 

B ¼ A1 B;

¼ A1 E;

D ¼ D  BA1 B;

F ¼ F  EA1 B;

E

H ¼ H  EA1 E

ð29Þ

where Aij, Bij, etc., are the plate stiffnesses, defined by

ðAij ; Bij ; Dij ; Eij ; F ij ; Hij Þ X Z tk ¼ ðQ ij Þk ð1; Z; Z 2 ; Z 3 ; Z 4 ; Z 6 ÞdZ

ði; j ¼ 1; 2; 6Þ

ð30Þ

ði; j ¼ 4; 5Þ

ð31Þ

t k1

k¼1

ðAij ; Dij ; F ij Þ ¼

XZ k¼1

tk

t k1

ðQ ij Þk ð1; Z 2 ; Z 4 ÞdZ

It is evident that the above equations involve the stretchingbending coupling, as predicted by Bij and Eij. Eqs. (12)–(15) are identical in form to those of unsymmetric cross-ply laminated plates, and applying in-plane compressive loads to such plates will cause bending curvature to appear. Consequently, the bifurcation buckling did not exist due to the stretching-bending coupling effect, as previously proved by Leissa [34], and Qatu and Leissa [35]. For this reason, we assume the FG-CNTRC plate is geometrically mid-plane symmetric, as previous reported in Feldman and Aboudi [16]. In such a case, the stretching-bending coupling is L14 ¼ ~L15 ¼ ~L22 ¼ zero-valued, i.e. Bij = Eij = 0. As a result, ~ ~ ~ ~ ~ ~ ~L23 ¼ ~ L24 ¼ L25 ¼ L34 ¼ L35 ¼ L44 ¼ L45 ¼ 0. It is worthy to note that for fully clamped FG-CNTRC plates the buckling loads do exist and the assumption of mid-plane symmetric structure is unnecessary. The solutions must be obtained by solving Eqs. (12)–(15) combined with clamped boundary conditions.

L11 ðWÞ  L12 ðWx Þ  L13 ðWy Þ ¼ c14 b2 LðW þ W  ; FÞ 32

p2

1 kT C 1 ¼  c24 b2 LðW þ 2W  ; WÞ 2

 W  Þ X Y a ðW; x ¼ p ; y ¼ p ; b ¼ ; ðW;W  Þ ¼     1=4 ; a b b ½D11 D22 A11 A22 

  1=2  x; W  yÞ a ðW D22 ; ð W ; W Þ ¼ ; c ¼ ; x y p ½D11 D22 A11 A22 1=4 14 D11 ½D11 D22 1=2   1=2 T T A A12 a2 ðAx ;Ay Þ c24 ¼ 11 ; c ¼  ; ð c ; c Þ ¼ ; 5 T1 T2 A22 A22 p2 ½D11 D22 1=2   a2 1  x; 4 P x ; ðMx ; Px Þ ¼ 2      1=4 M 2 p D11 ½D11 D22 A11 A22  3t   2 Dx Dy b ; k T ¼ a 0 DT ð32Þ ; ðdx ;dy Þ ¼ a b 4p2 ½D11 D22 A11 A22 1=2 F

F¼

where DT = T2  T0 for the in-plane parabolic temperature variation and DT = T1  T0 for the uniform temperature rise, and a0 is an arbitrary reference value, and

a11 ¼ a11 a0 ; a22 ¼ a22 a0

ð33Þ

in Eq. (32) ATx and ATy are defined by

"

ATx ATy

#

DT ¼ 

XZ k¼1

tk tk1



ð37Þ

L41 ðWÞ  L42 ðWx Þ þ L43 ðWy Þ ¼ 0

ð38Þ

where all dimensionless linear operators Lij () and nonlinear operator L () are defined as in [32,33]. The boundary conditions expressed by Eqs. (20)–(25) become x = 0, p:

W ¼ Wy ¼ 0

ð39Þ

dx ¼ 0

ð40Þ

F;xy ¼ M x ¼ Px ¼ 0

ð41Þ

y = 0, p:

W ¼ Wx ¼ 0

ð42Þ

dy ¼ 0

ð43Þ

F;xy ¼ M y ¼ Py ¼ 0

ð44Þ

and the unit end-shortening relationships become

dx ¼ 

Z p Z p ("

1 4p2 b2 c24

0

Ax Ay

DT dZ k

0

Z p Z p (" 2 @ F

dy ¼ 

1 4p2 b2 c24

0

0 

c24 b2

 2 @2F @2F 1 @W  c  c 5 24 @y2 @x2 2 @x )

@x2

 c5 b2

ð45Þ

 2 @2F 1 2 @W  c b 24 @y2 2 @y )

@W @W þ ðcT2  c5 cT1 ÞkT C 3 dy dx @y @y

ð46Þ

Note that in Eqs. (36), (45), and (46), for the in-plane non-uniform parabolic temperature loading case, C 1 ¼ b2 ðc224 cT1  c5 cT2 Þ ðx=p  x2 =p2 ÞþðcT2  c5 cT1 Þðy=p  y2 =p2 Þ; C 3 ¼ T 1 =T 2 þ 16ðx=p x2 =p2 Þðy=p  y2 =p2 Þ; and for a uniform temperature rise, C1 = 0, C3 = 1.0. Applying Eqs. (35)–(46), the thermal postbuckling behavior of perfect and imperfect, FG-CNTRC plates under thermal loading is now determined by means of a two step perturbation technique, for which the small perturbation parameter has no physical meaning at the first step, and is then replaced by a dimensionless deflection at the second step. The essence of this procedure, in the present case, is to assume that

Wðx; y; eÞ ¼

X

ej wj ðx; yÞ; Fðx; y; eÞ ¼

j¼1

Wx ðx; y; eÞ ¼

X

X

ej fj ðx; yÞ

j¼0

ej wxj ðx; yÞ; Wy ðx; y; eÞ ¼

j¼1

X

ej wyj ðx; yÞ

ð47Þ

j¼1

where e is a small perturbation parameter and the first term of wj (x, y) is assumed to be the classical buckling mode of the plate, i.e. ð1Þ

ð34Þ

c224 b2

@W @W  c24 þ ðc224 cT1  c5 cT2 ÞkT C 3 dx dy @x @x

w1 ðx; yÞ ¼ A11 sin mx sin ny



ð36Þ

L31 ðWÞ þ L32 ðWx Þ  L33 ðWy Þ ¼ 0

4. Solution procedure Having developed the theory, we are in a position to solve Eqs. (12)–(15) with boundary conditions (20)–(25). Before carrying out the solution process, it is convenient first to define the following dimensionless quantities

ð35Þ

ð48Þ

and the initial geometric defection is assumed to have a similar form

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H.-S. Shen, C.-L. Zhang / Materials and Design 31 (2010) 3403–3411 ð1Þ

W  ðx; y; eÞ ¼ ea11 sin mx sin ny ¼ elA11 sin mx sin ny

ð49Þ

ð1Þ

where l ¼ a11 =A11 is the imperfection parameter. Substituting Eq. (47) into Eqs. (35)–(38) and collecting the terms of the same order of e, a set of perturbation equations is obtained. By using Eqs. (48) and (49) to solve these perturbation equations of each order, the amplitudes of the terms wj(x, y), fj(x, y), wxj(x, y) and wyj(x, y) are determined step by step [33]. As a result, up to 4th-order asymptotic solutions are obtained as ð1Þ

ð3Þ

ð3Þ

W ¼ e½A11 sin mx sin ny þ e3 ½A13 sin mx sin 3ny þ A31 5

 sin 3mx sin ny þ Oðe Þ

ð50Þ

 2   2  y5 y6 x5 x5 ð0Þ x ð0Þ y  b00  C5 þ C5  C6 þ C6 F ¼ B00 2 120 360p 2 120 360p   2   2  y5 y6 x5 x5 ð2Þ x ð2Þ y 2 þ e B00  b00  C5 þ C5  C6 þ C6 2 120 360p 2 120 360p   2  i 5 6 y y y ð2Þ ð2Þ ð4Þ þB20 cos2mx þ B02 cos2ny þ e4 B00  C5 þ C5 2 120 360p  2  x5 x5 ð4Þ x ð4Þ ð4Þ  b00 þ B20 cos2mx þ B02 cos2ny  C6 þ C6 2 120 360p ð4Þ

ð4Þ

ð4Þ

ð4Þ

þ B22 cos2mxcos2ny þ B40 cos4mx þ B04 cos4ny þ B24 cos2mxcos4ny i ð4Þ ð51Þ þB42 cos4mxcos2ny þ Oðe5 Þ ð3Þ 3 ð3Þ 5 Wx ¼ e½C ð1Þ 11 cosmxsinny þ e ½C 13 cosmxsin3ny þ C 31 cos3mxsinny þ Oðe Þ

ð52Þ ð3Þ ð3Þ 3 Wy ¼ e½Dð1Þ 11 sin mx cos ny þ e ½D13 sin mx cos 3ny þ D31

 sin 3mx cos ny þ Oðe5 Þ

ð53Þ

It is mentioned that for the case of uniform temperature rise C5 = C6 = 0 in Eq. (51), and all coefficients in Eqs. (50)–(53) are reð1Þ lated and can be expressed in terms of A11 but, for the sake of brevity, the detailed expressions are not shown. Next, upon substitution of Eqs. (50)–(53) into the boundary conditions dx = 0 and dy = 0, the thermal postbuckling equilibrium path can be written as

kT ¼ kð0Þ þ kð2Þ ðA11 eÞ2 þ kð4Þ ðA11 eÞ4 þ    ð1Þ

ð1Þ

ð54Þ

ð1Þ (A11

In Eq. (54), e) is taken as the second perturbation parameter relating to the dimensionless maximum deflection Wm. From Eq. (50), taking (x, y) = (p/2m, p /2n) yields ð1Þ

A11 e ¼ W m þ H1 W 2m þ   

ð55Þ

Eq. (54) may then be re-written as ð0Þ

ð2Þ

ð4Þ

kT ¼ kT þ kT W 2m þ kT W 4m þ    ðiÞ kT

ð56Þ

It is noted that (i = 0,2,4,. . .) are related to the material properties and are all functions of T and Z, and the details of which may be found in [21,33]. Eq. (56) can be employed to obtain numerical results for thermal postbuckling load–deflection curves of simply supported FGCNTRC plates subjected to in-plane temperature variation. The buckling temperature of a perfect plate can be obtained from the   =t = 0 (or present solution by imposing the conditions that W  l = 0) and W=t = 0 (or Wm = 0). In the present case, the minimum temperature (called buckling temperature) and corresponding buckling mode (m, n) can be determined by comparing temperatures [obtained from Eq. (56)] under various values of (m, n), which determine the number of half-waves in the X and Y directions. The major difference herein is that the CNTRC stiffnesses are determined based on a micromechanical model and the stiffness matrixes of FG-CNTRC plates are altered. Note that in the present

study the material properties for both matrix and SWCNTs are assumed to be temperature-dependent as shown in the next section. 5. Numerical results and discussions Numerical results are presented in this section for perfect and imperfect, geometrically mid-plane symmetric FG-CNTRC plates subjected to in-plane temperature variation. We first need to determine the effective material properties of CNTRCs. Poly (methyl methacrylate), referred to as PMMA, is selected for the matrix, and the material properties of which are assumed to be mm = 0.34, am = 45(1 + 0.0005DT)  106/K and Em = (3.52  0.0034T) GPa, in which T = T0 + DT and T0 = 300 K (room temperature). In such a way, am = 45.0  106/K and Em = 2.5 GPa at T = 300 K. (10, 10) SWCNTs are selected as reinforcements. It has been shown [36–40] the material properties of SWCNTs are anisotropic, chirality- and size-dependent and temperature-dependent. Also the buckling and postbuckling behavior of nanotubes is very sensitive to the material properties and effective wall thickness [41,42]. Therefore, all effective elastic properties of a SWCNT need to be carefully determined, otherwise the results may be incorrect. The molecular dynamics simulations are first carried out. We use the many-body reactive empirical bond order potential developed by Brenner et al. [43] to describe the interaction between carbon atoms. The long-range van der Waals interaction between CNTs are described by the Lennard-Jones potential ELJ given as ELJ = A/r12  C/r6 [44], where r is the distance between two nonbonding atoms, constants A = 24086 eV Å12 and C = 15 eV Å6 are fitted to reproduce the structural properties of graphite. To account for the thermal effect, we use the Nose-Hoover thermostat [45] to maintain the temperature of the system. This thermostat provides good conservation of energy and lead to less fluctuation in temperature. In the frame work of MD simulation, the nanotube can be considered as a congeries of individual atoms. The integration of Newtonian dynamics function is used to determine the variation of the instantaneous location and velocity of each atom. The perfect CNTs subjected to axial compression and torsion are simulated under temperature varying from 300 K to 1000 K. Fixed boundary condition is assumed to be at one end of the tube, and axial compressive force P or torque Ms is applied on the other end with the appropriate constraints [40]. From MD simulation results the size-dependent and temperature-dependent material properties for armchair (10, 10) SWCNT can be obtained numerically. Typical results are listed in Table 1. It is noted that the effective wall thickness obtained for (10,10)-tube is h = 0.067 nm, and the wide used value of 0.34 nm for tube wall thickness is thoroughly inappropriate to SWCNTs. The key issue for successful application of the extended rule of mixture to CNTRCs is to determine the CNT efficiency parameter gj (j = 1,2,3). For short fiber composites g1 is usually taken to be 0.2 [46]. However, there are no experiments conducted to determine the value of gj for CNTRCs. In present study, we give the estimation of CNT efficiency parameters g1 and g2 by matching the Young’s moduli E11 and E22 of CNTRCs obtained by the rule of mixture to those from the MD simulations given by Han and Elliott [6].

Table 1 Temperature-dependent material properties for (10, 10) SWCNT (L = 9.26 nm, R = 0.68 nm, h = 0.067 nm, mCN 12 =0.175). Temperature (K)

ECN 11 (TPa)

ECN 22 (TPa)

GCN 12 (TPa)

6 aCN 11 (10 /

6 aCN 22 (10 /

K)

K)

300 500 700 1000

5.6466 5.5308 5.4744 5.2814

7.0800 6.9348 6.8641 6.6220

1.9445 1.9643 1.9644 1.9451

3.4584 4.5361 4.6677 4.2800

5.1682 5.0189 4.8943 4.7532

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H.-S. Shen, C.-L. Zhang / Materials and Design 31 (2010) 3403–3411

plates is the same, i.e. t = 2.0 mm. It should be appreciated that in   =t denotes the dimensionless maximum initial geoall figures W metric imperfection of the plate. Tables 4 and 5 present, respectively, the buckling temperatures for perfect, CNTRC plates with three different values of the nanotube volume fraction V CN (=0.12, 0.17 and 0.28) subjected to a uniform temperature rise (T2 = 0) and non-uniform parabolic temperature variations with three thermal load ratio T1/T2 (=0.0, 0.5, and 1.0). Here, UD represents uniformly distributed CNTRC plate and FG represents functionally graded CNTRC plate. It is worthy to note that for these three cases of V CN the plate has the same nanotube mass fraction wCN = 0.142, 0.2 and 0.321, respectively, by taking the density of carbon nanotube qCN = 1.4 g/cm3 and the density of matrix qm = 1.15 g/cm3 in Eq. (7). T-D represents material properties are temperature-dependent, whereas T-ID represents material properties are temperature-independent, i.e. in a fixed temperature T = 300 K. Note that, for T-D case, since the material properties are temperature-dependent, an iterative numerical procedure is necessary, as previously reported in [21]. It can be seen that the buckling temperature decreases when the temperature dependency is put into consideration. The effect of temperature dependency on the buckling temperature of CNTRC plate with b/ t = 20 is rather small, but this effect becomes pronounced when b/t = 10. It can also be seen that the buckling temperature of FGCNTRC plate is larger than that of the UD-CNTRC plate. For example, the percentage increase is about 4.1–5.2% for the square plate under uniform temperature rise, and is about 5.6–6.7% under nonuniform parabolic temperature variation with T1/T2 = 0.0 under T-D case, as shown in the brackets. It can be found that the buckling temperatures are increased with increase in nanotube volume fraction V CN for the plate with b/t = 20. In contrast, for the case of b/ t = 10 the plate with V CN ¼ 0:17 will have highest buckling temperature among the three. In other words, the plate with intermediate nanotube volume fraction will not have intermediate buckling

Table 2 Comparisons of Young’s moduli for PMMA/CNT composites reinforced by (10, 10)tube under T = 300 K. V CN

0.12 0.17 0.28

MD [6]

Rule of mixture

E11 (GPa)

E22 (GPa)

E11 (GPa)

g1

E22 (GPa)

g2

94.6 138.9 224.2

2.9 4.9 5.5

94.78 138.68 224.50

0.137 0.142 0.141

2.9 4.9 5.5

1.022 1.626 1.585

Through comparison, we find that the Young’s moduli obtained from the rule of mixture and MD simulations can match very well if the CNT efficiency parameters g1 and g2 are properly chosen, as shown in Table 2. Note that there are no MD results for shear modulus G12 in [6]. In addition, a comparison study is carried out for polymer/CNT composites [4] and the results are listed in Table 3, from which g3:g2 = 0.7:1. These values will be used in all the following examples, in which we assume that G13 = G12 and G23 = 1.2 G12 [47]. The accuracy and effectiveness of the present method for the thermal buckling and postbuckling analyses of isotropic and orthotropic plates subjected to in-plane uniform or non-uniform temperature variation, were examined by many comparison studies given in Shen [21,48–50]. These comparisons show that the results from present method are in good agreement with existing results for the limiting case. Since there are no thermal buckling results for an CNTRC plate, no new additional comparison studies will be given in the present study. A parametric study has been carried out and typical results are shown in Tables 4 and 5, and Figs. 2–5. For these examples, the plate width-to-thickness ratio b/t = 10, 20 and 50, and the thickness of each FGM layer is taken to be 1.0 mm. A double-thickness uniformly distributed CNTRC plate is also considered as a comparator. In such a way, the total thickness of these two types of CNTRC

Table 3 Comparisons of elastic moduli for Polymer/CNT composites reinforced by (10, 10)-tube under T = 300 K (Em = 0.85 GPa, mm = 0.44). V CN

0.028

MD [4]

Rule of mixture

E11 (GPa)

E22 (GPa)

G12 (GPa)

E11 (GPa)

g1

E22 (GPa)

g2

G12 (GPa)

g3

1.74

0.81

0.19

1.74

0.0058

0.81

0.931

0.19

0.642

Table 4 Comparisons of buckling temperatures T1 (in K) for perfect, CNTRC plates under uniform temperature rise [t = 2 mm, (m, n) = (1, 1)]. b/t

T-ID 20

T-D 20

T-ID 10

T-D 10

a

a/b

V CN ¼ 0:12

V CN ¼ 0:17

V CN ¼ 0:28

UD

FG

UD

FG

UD

FG

1.0 1.5 2.0

340.74 323.47 317.52

355.85(+4.4%)a 332.67(+2.8%) 323.59(+1.9%)

344.54 325.59 319.34

362.79(+5.3%) 336.57(+3.4%) 326.77(+2.3%)

345.88 326.62 319.51

366.21(+5.9%) 340.41(+4.2%) 329.69(+3.2%)

1.0 1.5 2.0

339.42 323.04 317.25

353.20(+4.1%) 331.85(+2.7%) 323.13(+1.8%)

343.00 325.08 319.01

359.52(+4.8%) 335.54(+3.2%) 326.16(+2.2%)

344.08 326.05 319.17

362.06(+5.2%) 339.03(+4.0%) 328.91(+3.0%)

1.0 1.5 2.0

398.48 371.95 360.66

419.65(+5.3%) 391.31(+5.2%) 376.86(+4.5%)

412.36 380.81 368.21

439.88(+6.7%) 405.58(+6.5%) 389.16(+5.7%)

403.63 377.63 365.47

430.26(+6.6%) 405.27(+7.3%) 392.24(+7.3%)

1.0 1.5 2.0

388.19 366.89 357.02

403.91(+4.0%) 382.93(+4.4%) 371.09(+3.9%)

399.44 374.59 363.67

419.09(+4.9%) 394.66(+5.3%) 381.51(+4.9%)

391.62 371.43 361.10

410.58(+4.8%) 393.28(+5.9%) 383.35(+6.2%)

Difference = 100% [T1(FG)  T1(UD)]/T1(UD).

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H.-S. Shen, C.-L. Zhang / Materials and Design 31 (2010) 3403–3411 Table 5 Comparisons of buckling temperatures T2 (in K) for perfect, CNTRC plates under non-uniform parabolic temperature variations [a/b = 1.0, t = 2 mm, (m, n) = (1, 1)]. b/t

T-ID 20

T-D 20

T-ID 10

T-D 10

a

V CN ¼ 0:12

T1/T2

V CN ¼ 0:17

V CN ¼ 0:28

UD

FG

UD

FG

UD

FG

0.0 0.5 1.0

365.64 336.35 325.14

389.13(+6.4%)a 349.57(+3.9%) 334.34(+2.8%)

371.64 339.71 327.47

399.60(+7.5%) 355.55(+4.7%) 338.51(+3.4%)

373.13 340.70 328.19

403.12(+8.0%) 357.97(+5.1%) 340.32(+3.7%)

0.0 0.5 1.0

362.27 335.29 324.63

382.55(+5.6%) 347.47(+3.6%) 333.31(+2.7%)

367.75 338.48 326.87

391.63(+6.5%) 352.97(+4.3%) 337.25(+3.2%)

368.71 339.28 327.50

393.54(+6.7%) 354.79(+4.6%) 338.75(+3.4%)

0.0 0.5 1.0

458.68 387.88 360.77

490.93(+7.0%) 406.20(+4.7%) 373.55(+3.5%)

480.70 400.16 369.28

521.87(+8.6%) 423.74(+5.9%) 385.79(+4.5%)

465.18 391.92 363.68

502.89(+8.1%) 414.06(+5.6%) 379.33(+4.3%)

0.0 0.5 1.0

433.61 379.58 356.67

453.89(+4.7%) 393.60(+3.7%) 367.26(+3.0%)

449.53 389.76 364.13

473.95(+5.4%) 407.17(+4.5%) 377.46(+3.7%)

436.94 382.38 358.93

459.48(+5.2%) 398.73(+4.3%) 371.55(+3.4%)

Difference = 100% [T2(FG)  T2(UD)]/T2(UD).

temperature in some cases. It can also be found that the buckling temperatures are decreased with increase in thermal load ratio T1/T2 and the plate aspect ratio a/b for both T-ID and T-D cases. Fig. 2 shows the effect of the nanotube volume fraction V CN (=0.12, 0.17 and 0.28) on the thermal postbuckling behavior of FG- and UD-CNTRC plate with b/t = 10 subjected to a uniform temperature rise. It can be seen that the plate with V CN ¼ 0:17 has

highest buckling temperature and initial thermal postbuckling strength, whereas in the deep postbuckling region the plate with V CN ¼ 0:28 becomes stiffer when the deflection is sufficiently large. It can be found that the buckling temperature as well as thermal postbuckling strength of FG-CNTRC plate is higher than that of the UD-CNTRC plate. The thermal postbuckling equilibrium paths

900

900

*

*

UD: V CN =0.12 *

W /t = 0.0 *

W /t = 0.1

0.5

1.0

1.5

FG: β = 1.0 UD: β = 1.0

300

* FG: V CN =0.17 * UD: V CN =0.17 * FG: V CN =0.28 * UD: V CN =0.28

300

0.0

T1 (K)

*

FG: V CN =0.12

0

b/t = 10, VCN =0.17 (m, n)=(1, 1)

600

T1 (K)

600

uniform temperature CNTRC plate

uniform temperature CNTRC plate β = 1.0, b/t = 10 (m, n)=(1, 1)

FG: β = 1.5 UD: β = 1.5 FG: β = 2.0 UD: β = 2.0

*

W /t = 0.0 *

W /t = 0.1

0

2.0

0.0

0.5

1.0

1.5

W (mm) Fig. 2. Effect of nanotube volume fraction on the thermal postbuckling behavior of CNTRC plates subjected to a uniform temperature rise.

Fig. 4. Effect of plate aspect ratio on the thermal postbuckling behavior of CNTRC plates subjected to a uniform temperature rise.

900

900 uniform temperature CNTRC plate *

β = 1.0, V CN =0.17

parabolic temperature CNTRC plate

*

W /t = 0.0 *

*

W /t = 0.1

(m, n)=(1, 1)

T2 (K)

600

T1 (K)

600

VCN = 0.17 β = 1.0, b/t = 10 (m, n)=(1, 1)

FG: T1/T2 = 0.0 UD: T1/T2 = 0.0 FG: T1/T2 = 0.5 UD: T1/T2 = 0.5 FG: T1/T2 = 1.0 UD: T1/T2 = 1.0

300

300

*

FG: b/t = 10 UD: b/t = 10

0

2.0

W (mm)

0.0

0.5

1.0

W /t = 0.0

FG: b/t = 50 UD: b/t = 50

FG: b/t = 20 UD: b/t = 20

2.0

1.5

W (mm) Fig. 3. Effect of plate thickness ratio b/t on the thermal postbuckling behavior of CNTRC plates subjected to a uniform temperature rise.

*

W /t = 0.1

0

0.0

0.5

1.0

1.5

2.0

W (mm) Fig. 5. Effect of thermal load ratio T1/T2 on the thermal postbuckling behavior of CNTRC plates under non-uniform parabolic temperature variations.

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H.-S. Shen, C.-L. Zhang / Materials and Design 31 (2010) 3403–3411

for both FG- and UD-CNTRC plates are stable, and the plate structure is imperfection-insensitive. Fig. 3 shows the effect of plate width-to-thickness ratio b/t (=10, 20 and 50) on the thermal postbuckling behavior of FG- and UDCNTRC plates with V CN ¼ 0:17 subjected to a uniform temperature rise. It can be seen that the buckling temperature and thermal postbuckling strength of a shear deformable plate (b/t = 10) are higher and are considerably greater than those of the thin plate (b/t = 50). It can be found that the difference between two thermal postbuckling curves of FG- and UD-CNTRC plates with b/t = 10 is more pronounced. Fig. 4 shows the effect of plate aspect ratio b (=a/b = 1.0, 1.5 and 2.0) on the thermal postbuckling behavior of FG- and UD-CNTRC plates with V CN ¼ 0:17 subjected to a uniform temperature rise. As expected, these results show that the buckling temperature and thermal postbuckling strength are decreased by increasing plate aspect ratio b. Fig. 5 shows the effect of different values of the thermal load ratio T1/T2 (=0.0, 0.5, 1.0) on the thermal postbuckling behavior of FG- and UD-CNTRC plate subjected to non-uniform parabolic temperature variations. It can be seen that both buckling temperature and thermal postbuckling strength are decreased by increasing T1/ T2, when the nanotube volume fraction V CN ¼ 0:17. 6. Concluding remarks Thermal postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates subjected to in-plane temperature variation has been presented on the basis of a micromechanical model and multi-scale approach. The scale effects of nanocomposites are considered by two ways. The size-dependent and temperature-dependent material properties of SWCNTs are obtained from molecular dynamics simulations. The CNT efficiency parameter gj is estimated by matching the elastic modulus of CNTRCs observed from the MD simulation results with the numerical results obtained from the extended rule of mixture. The new finding is that in some cases the plate with intermediate nanotube volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength. The results show that the buckling temperature as well as thermal postbuckling strength of the plate can be increased as a result of a functionally graded reinforcement. The results reveal that the thermal postbuckling behaviors of CNTRC plates are significantly influenced by the thermal load ratio, the transverse shear deformation, the plate aspect ratio as well as the nanotube volume fraction. Acknowledgments This work is supported in part by the National Natural Science Foundation of China under Grant 10802050. The authors are grateful for this financial supports. Appendix A In Eq. (34)

t m m fð1 þ m21 ÞEm am þ ½ð1 þ m21 Þðg1 ECN 11  E Þa 1  m12 m21  CN m m CN m þ Em ðm21 ð1 þ mCN 12 Þða22  a Þ þ ð1  m21 m Þða11  a ÞÞðV CN Þ 4 m CN CN m m CN m þ ½ðg1 ECN 11  E Þðm21 ð1 þ m12 Þða22  a Þ þ ð1  m21 m Þða11  a ÞÞ 3  2 CN m m CN m  Em m21 ðmCN 12  m Þða11  a ÞðV CN Þ  2ðg1 E11  E Þm21

ATx ¼

 3 m CN m  ðmCN 12  m Þða11  a ÞðV CN Þ g

ð57Þ

T

Ay ¼

t 1  m12 m21

( m m m CN m ðg2 þ m21 ÞEm am þ m21 ½ðg1 ECN 11  E Þa þ E ða11  a Þ

"

 ðV CN Þ þ g2 Em am

m ECN 22  E

ECN 22

#

! CN m m CN m þ ð1 þ mCN 12 Þða22  a Þ  m ða11  a Þ

4 4 m  2 m CN m  ðV CN Þ þ ðg1 ECN 11  E Þm21 ða11  a ÞðV CN Þ þ g2 E 3 3 2 !2 ! m ECN  Em ECN CN m m CN m 22  E  4am 22 CN þ ðð1 þ mCN Þð a  a Þ  m ð a  a ÞÞ 12 22 11 E22 ECN 22 2 !3 CN m   2 m 4 m E22  E m CN m a þ ðð1 þ mCN ðmCN 12  m Þða11  a Þ ðV CN Þ þ 2g2 E 12 Þ ECN 22 !2 m ECN m m CN m m CN m 22  E  ðmCN  ðaCN 22  a Þ  m ða11  a ÞÞ 12  m Þða11  a Þ CN E22 2 !# !4 m CN m ECN 16  3 m 4 m E22  E CN m 22  E ðV CN Þ þ g2 E a þ ðð1 þ mCN  12 Þða22  a Þ 5 ECN ECN 22 22 !3 !2 3 m m ECN ECN m CN m CN m CN m 22  E 22  E 5 m ða11  a ÞÞ  ðm12  m Þða11  a Þ ECN ECN 22 22 2 !5 16 ECN  Em CN m  ðV CN Þ4 þ g2 Em 4am 22 CN þ ðð1 þ mCN 12 Þða22  a Þ 3 E22 !4 !3 3 ) m m ECN ECN  5 m CN m CN m CN m 22  E 22  E 5 m ða11  a ÞÞ  ðm12  m Þða11  a Þ ðV CN Þ ECN ECN 22 22 ð58Þ

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