Nonlinear vibration of functionally graded graphene-reinforced composite laminated plates in thermal environments

Accepted Manuscript Nonlinear vibration of functionally graded graphene-reinforced composite laminated plates in thermal environments Hui-Shen Shen, Y. Xiang, F. Lin

PII: DOI: Reference:

S0045-7825(16)31743-1 http://dx.doi.org/10.1016/j.cma.2017.02.029 CMA 11356

To appear in:

Comput. Methods Appl. Mech. Engrg.

Received date : 6 December 2016 Revised date : 10 January 2017 Accepted date : 23 February 2017 Please cite this article as: H. Shen, et al., Nonlinear vibration of functionally graded graphene-reinforced composite laminated plates in thermal environments, Comput. Methods Appl. Mech. Engrg. (2017), http://dx.doi.org/10.1016/j.cma.2017.02.029 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.

Nonlinear vibration of functionally graded graphene-reinforced composite laminated plates in thermal environments 2,3

Hui-Shen Shen 1,*, Y. Xiang , F. Lin 1

2

School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China 2

School of Computing, Engineering and Mathematics and 3Centre for Infrastructure

Engineering, Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia

Abstract This

paper

deals

with

the

large

amplitude

vibration

of

functionally

graded

graphene-reinforced composite laminated plates resting on an elastic foundation and in thermal environments. The temperature-dependent material properties of piece-wise functionally graded graphene-reinforced composites (FG-GRCs) are assumed to be graded in the thickness direction of a plate, and are estimated through a micromechanical model. Based on a higher-order shear deformation plate theory, the motion equations are developed with geometric nonlinearity taking the form of von Kármán strains. The plate-foundation interaction and thermal effects are also included. The motion equations are then solved by a two-step perturbation technique to determine the nonlinear frequencies of the FG-GRC laminated plates. The numerical illustrations concern the nonlinear vibration characteristics of FG-GRC laminated plates under different sets of thermal environmental conditions, from which results for uniformly distributed GRC laminated plates are obtained as comparators. The effects of distribution type of reinforcements, temperature variation, foundation stiffness and different in-plane boundary conditions are also investigated.

Keywords:

Functionally

graded

materials;

Nanocomposites;

Nonlinear

Temperature-dependent properties; Elastic foundation

*

Corresponding author. E-mail address: [email protected] (H.-S. Shen) 1

vibration;

1. Introduction Since the discovery of graphene in 2004 [1], extensive researches have been carried out on the properties and behaviors of graphene sheets. The research findings revealed that graphene sheets possess remarkable mechanical, electrical and thermal properties [2-7] that can be utilized in various science and engineering applications. Like carbon nanotubes (CNTs), graphene sheets can be used to reinforce polymer resins to obtain high performing advanced nanocomposites [8-10]. Unlike CNTs, the sheet-like structure and high surface area of graphene sheets help to develop better interactions between the polymer chains and graphene sheets. Studies on graphene reinforced composites (GRCs) with superior mechanical, electrical and thermal properties have emerged as a hot research area in materials science and engineering. GRCs are now developed for general use as structural components in different fields such as electronics devices [11], energy storage [12] and sensors [13]. Since this area is relatively new, only a few investigations on GRC beam and plate structures subjected to mechanical loading are available in the literature. Among those, Chandra et al. [14] presented the natural frequencies and mode shapes of graphene/epoxy composite plates by using the finite element method (FEM) based on a multiscale approach. They found that the natural frequency of the graphene/epoxy composite plates decreases with increasing plate length and aspect ratio. Rafiee et al. [15] studied the buckling of graphene/epoxy composite beams experimentally. They found that the buckling load is increased by up to 52% compared to the pure epoxy beam when a small percentage of graphene (only 0.1% weight fraction) is added into the epoxy matrix. Parashar and Mertiny [16] studied the buckling of graphene/epoxy composite plates under uniaxial compression using the FEM approach. They found that the buckling strength of graphene/polymer composite is improved by 26% with only 6% filler volume fraction. In the aforementioned studies, however, the graphene fillers are distributed either uniformly or randomly in the matrix such that the resulting mechanical, thermal, or physical properties of the composites do not vary spatially at the macroscopic level. The major difference between the conventional carbon fiber-reinforced composites (FRCs) and the graphene-reinforced composites lies in that the former can contain a very high percentage of carbon fibers (usually over 60% by volume), while the latter only has a 2

low percentage of graphene fillers (about 0.5~50% by weight) [17,18]. This is due to the fact that larger graphene volume fractions in GRCs can actually lead to the deterioration of the mechanical properties of the composites [19]. One of the problems is how to determine and improve the vibration characteristics of GRC laminated plates under such a low graphene volume fraction. Functionally graded materials (FGMs) are a new generation of composite materials in which the microstructural details are spatially varied through nonuniform distribution of the reinforcement phase. Functionally graded carbon nanotube reinforced composites (FG-CNTRCs) were first proposed by Shen [20], and the graded distributions of CNT within an isotropic matrix were designed specifically under certain rules along the desired directions for the purposes of improved structural mechanical properties. Subsequently, the pioneering work of nonlinear vibrations of FG-CNTRC plates was performed by Wang and Shen [21]. Liew and his co-authors [22,23] studied the linear free vibration of FG-CNTRC plates without and resting on elastic foundations by using the element-free kp-Ritz method based on the first order shear deformation plate theory (FSDPT). The linear free vibration of FG-CNTRC plates with elastically restrained edges was performed by Zhang et al. [24], and the linear free vibration of FG-CNTRC plates of Levy-type was performed by Zhang et al. [25] based on the higher order shear deformation plate theory. Three-dimensional free vibration analysis of FG-CNTRC plates with various boundary conditions was also presented by Shahrbabaki and Alibeigloo [26], Nami and Janghorban [27] and Wu and Li [28]. Fan and Wang [29] studied the effect of matrix cracks on the nonlinear free and forced vibration of hybrid laminated plates containing CNTRC layers resting on elastic foundations. Recently, Yang and his co-authors presented the nonlinear bending, compressed postbuckling and dynamic instability analyses of functionally graded polymer nanocomposite beams reinforced with graphene platelets based on the Timoshenko beam theory [30-32], and the linear free and forced vibrations of functionally graded polymer composite plates reinforced with graphene platelets [33]. In their analysis the graphene platelets are assumed uniformly dispersed and randomly oriented in the matrix and the weight fraction of graphene platelets is assumed to vary linearly in the thickness direction of the beam or plate. The equivalent isotropic Young’s modulus of the nanocomposite is obtained by using a modified Halpin–Tsai model, and the 3

material properties are assumed to be independent of temperature. The concept of functionally graded materials can also be utilized for the FRC by non-homogeneous distribution of reinforcing fiber into the matrix with a piece-wise laminated type instead of a continuous type [34,35]. In the present work, we focus our attention on the nonlinear vibration of nanocomposite plates reinforced by graphene sheets with low graphene volume fractions. Two kinds of GRC laminated plates, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The temperature-dependent material properties of FG-GRC laminated plates are assumed to be graded in the thickness direction, and are estimated through a micromechanical model in which the graphene efficiency parameters are obtained by matching the elastic modulus of GRCs from the molecular dynamics (MD) simulation results with the numerical results from the extended Halpin–Tsai model. The motion equations are based on a higher-order shear deformation plate theory and general von Kármán-type equations that include plate-foundation interaction and thermal effects. Initial stresses caused by thermal loads or in-plane edge loads are also introduced. A two-step perturbation approach is employed to determine the nonlinear to linear frequency ratios of the GRC laminated plate. The numerical illustrations of the nonlinear vibration characteristics of GRC laminated plates resting on an elastic foundation under different sets of temperature conditions are then presented and discussed.

2. Multi-scale model for FG-GRC laminated plates A rectangular plate of length a, width b, and thickness h resting on an elastic foundation and in thermal environment is considered in the current study. The plate is made of N plies with each ply being a mixture of graphene reinforcement and polymer matrix. The graphene reinforcement is either zigzag (refer to as 0-ply) or armchair (refer to as 90-ply). The origin of the rectangular coordinate system is at the corner of the plate on the middle plane. The displacement fields of the plate along the X, Y and Z axes are assumed to be U , V and W , where Z is perpendicular to the plate. x and  y are the mid-plane rotations of the normals about the Y and X axes, respectively. The compliant Pasternak-type foundation, which means that no part of the plate lifts off the foundation during deflection, is employed in

4

the study. The foundation model can be defined by p 0  K 1W  K 2  2W , where p0 is the force per unit area, K1 is the Winkler foundation stiffness and K 2 is the shearing layer stiffness of the foundation, and  2 is the Laplace operator in X and Y. The plate is subjected to a transverse dynamic load q(X, Y, t) combined with initial in-plane edge loads. The graphene reinforcement is aligned in the X direction and is either uniformly distributed (UD) or functionally graded (FG) in the thickness direction with a piece-wise type. The Reddy higher order shear deformation plate theory [36] with a von Kármán-type of kinematic nonlinearity is employed to study the nonlinear vibration behavior of FG-GRC plates. The governing differential equations for the problem can be expressed as follows in which the plate-foundation interaction and thermal effects are taken into consideration ~ ~ ~ ~ ~ ~ L11 (W )  L12 ( x )  L13 (  y )  L14 ( F )  L15 ( N T )  L16 ( M T )  K 1W  K 2  2W

     ~ ~  y x  L (W , F )  L17 (W  )  I8   X Y 

 q  

(1)

1~ ~ ~ ~ ~ ~ L21 ( F )  L22 ( x )  L23 (  y )  L24 (W )  L25 ( N T )   L (W , W ) 2

(2)

 W ~ ~ ~ ~ ~ ~  L31 (W )  L32 ( x )  L33 (  y )  L34 ( F )  L35 ( N T )  L36 ( S T )  I 9  I 10  x X

(3)

 W ~ ~ ~ ~ ~ ~  L41 (W )  L42 ( x )  L43 (  y )  L44 ( F )  L45 ( N T )  L46 ( S T )  I 9  I 10  y Y

(4)

where F is the stress function defined by N x  F , YY , N y  F , XX and N xy   F , XY with a comma denoting partial differentiation with respect to the corresponding coordinates. The superposed dots are differentiation with respect to time and the inertias I8, I9 and I10 are ~ ~ given in detail in Eq. (19) below. Lij ( ) and L ( ) are linear and nonlinear operators as ~ defined in Shen [37], and L ( ) contains the geometric nonlinearity terms in the von Kármán sense, and can be expressed by 2 2 2 2 2 2 ~   L( )  2 XY XY Y 2 X 2 X 2 Y 2

(5)

The thermal forces N T , moments M T and S T , and higher order moments P T caused 5

by elevated temperature in Eqs. (1) to (4) are defined by

 N xT M xT PxT   T T T  N y M y Py   N T M T P T   xy xy xy 

 Ax    3  A y  (1, Z , Z )T dZ k 1 t k 1    Axy  N

tk



(6a)

k

 S xT   M xT   T  T 4 S y   M y   2  S T   M T  3h  xy   xy 

 PxT   T  Py  P T   xy 

(6b)

where T =T-T0 is the temperature rise from the reference temperature T0 at which there are free of thermal strains in the plate, and  Ax  Q11     A y    Q12   Q16   Axy 

Q12 Q22 Q26

Q16  1 0  11   Q26  0 1   22    Q66  0 0

(7)

in which  11 and  22 are the thermal expansion coefficients measured in the longitudinal and transverse directions, respectively, and Qij are the transformed elastic constants, details of which can be found in [36]. For a GRC layer reinforced by zigzag or armchair graphene sheet, Qij  Qij in which Q11 

E11 1   12 21

, Q22 

E 22 1   12 21

, Q12 

 21 E11 , 1   12 21

Q16=Q26= 0, Q66  G12 , Q44  G23 , Q55  G13 where E11, E22, G12, 

12

and 

21 have

(8)

their usual meanings.

A number of micromechanics models that have been proposed for the determination of effective material properties of GRCs based on the Halpin–Tsai model [38] and the Voigt model (rule of mixture) [39]. It has been reported that the Halpin–Tsai model as well as the Voigt model can not predict the effective material properties of GRCs accurately [40, 41]. Hence, in nanoscale both Halpin–Tsai model and Voigt model should be modified. According to the extended Halpin–Tsai model, the effective material properties of the GRCs can be expressed by E11  1

1  2(aG / hG ) 11VG m E 1   11VG

(9a)

6

E22   2

G12   3

1  ( 2bG / hG ) 22VG m E 1   22VG 1 1   12VG

(9b)

(9c)

Gm

in which aG , bG and hG are the length, width and effective thickness of the graphene sheet, and

 11 

G E11 / E m 1 G E11 / E m  2aG / hG

(10a)

 22 

G E22 / Em 1 G E22 / E m  2bG / hG

(10b)

 12 

G G12 / Gm 1 G G12 / Gm

(10c)

G G G where E11 , E22 and G12 are the Young’s moduli and shear modulus, respectively, of the

graphene sheet, and Em and Gm are corresponding properties for the matrix. Similar to the cases of CNTRCs, the load transfer between the graphene and polymeric phases is less than perfect (e.g. the surface effects, strain gradients effects, intermolecular coupled stress effects, etc.). The graphene efficiency parameters  j (j=1,2,3) are introduced into Eq. (9) to consider the small scale effect and other effects on the material properties of GRCs. The values of  j will be determined later by matching the elastic modulus of GRCs obtained from the MD simulation with the numerical results predicted by the extended Halpin–Tsai model. In Eq. (9), VG is the graphene volume fraction and the matrix volume fraction can be expressed by Vm = 1- VG, and the mass density of the GRC is defined by

  VG  G  Vm  m

(11)

where  G and  m are the densities of graphene and matrix, respectively. In Eq. (7), the thermal expansion coefficients in the longitudinal and transverse directions can be expressed as

7

11 

G G 11  Vm E m m VG E11 G VG E11  Vm E m

(12a)

G  22  (1   12G )VG 22  (1   m )Vm m  1211

(12b)

G G G where 11 ,  22 and  m are thermal expansion coefficients, and  12 and  m are

Poisson’s ratios, respectively, of the graphene sheet and matrix. It was reported that the material properties of both graphene sheet [42] and matrix [43] are functions of temperature T, therefore, the Young’s modulus, shear modulus and thermal expansion coefficients of GRCs are also functions of temperature T. The Poisson’s ratio depends weakly on temperature change and is expressed as

 12  VG 12G  Vm m

(13)

Simply supported boundary conditions of the GRC plates are considered in this study with two different in-plane displacement behaviors. Case 1: The edges are simply supported and freely movable in both the X and Y directions, and the uniaxial or biaxial edge loads are acting in the X and Y directions, respectively. Case 2: Four edges are simply supported with no in-plane displacements. The boundary conditions of the above two cases can be expressed as X = 0, a:

W  y  0

(14a)

M x  Px  0

(14b)

b

N

dY   x bh  0

x

(movable)

(14c)

0

U 0

(immovable)

(14d)

Y = 0, b: W  x  0

(14e)

M y  Py  0

(14f)

a

N

y

dX   y ah  0

(movable)

(14g)

0

8

V 0

(immovable)

(14h)

where  x and  y are average compressive stresses in the X and Y directions, M x and

M y are the bending moments and Px and Py are the higher order moments as defined in [36]. The immovability conditions (13d) and (13h) can be achieved on the average sense as b

U dXdY  0 X b V dYdX  0 0 Y

 0

a

(15a)

0

a

 0

(15b)

where 2 2 4 *   x  * 4 *   y 4 U  * *  F *  F A  2   B12  2 E12  A11    B11  2 E11   12 2 2 X 3h 3h 3h Y X  Y  X  



* E12

 2W Y 2

 1  W     2 X  

2

 * *   ( A11 N xT  A12 N yT ) 

(16a)

2 2 4 *   x  * 4 *   y 4 V  * *  F *  F A  2   B22  2 E 22  A22    B21  2 E 21   12 2 2 Y 3h 3h 3h X Y  Y   X 



* E 22

 2W Y 2

 1  W     2 Y  

 *  2W  E11  X 2 

 *  2W  E 21  X 2 

2

 * *   ( A12 N xT  A22 N yT ) 

(16b)

In Eq. (15) and, in what follows, [ Aij ], [ Bij ], [ Dij ], [ Eij* ], [ Fij* ] and [ H ij* ] (i,j = 1, 2, 6) are reduced stiffness matrices, determined through relationships [44] A* = A-1, B* = - A-1B, D* = D – BA-1B, E* = -A-1E, F* = F – EA-1B, H* = H – EA-1E

(17)

where Aij , Bij, Dij, etc., are the plate stiffnesses, defined by ( Aij , Bij , Dij , E ij , Fij , H ij ) 

N

hk

  (Q

ij ) k

(1, Z , Z 2 , Z 3 , Z 4 , Z 6 )dZ

(i,j = 1,2,6)

(18a)

k 1 hk 1

( Aij , Dij , Fij ) 

N

hk

  (Q

ij ) k

(1, Z 2 , Z 4 )dZ

(i,j = 4,5)

(18b)

k 1 hk 1

and the inertias I i ( i  1,2,3,4,5,7 ) are defined by (I1 , I 2 , I 3 , I 4 , I 5 , I 7 ) 

N

hk

k 1

hk 1



 k (1, Z , Z 2 , Z 3 , Z 4 , Z 6 )dZ

9

(19a)

in which  k is the mass density of the kth ply, and I 2  I2  I8 

8 16 4 4 I , I5  I5  2 I7 , I3  I3  2 I5  4 I7 , 2 4 3h 3h 3h 9h

I I I I I2I2 4 4  I 3  2 I 5 , I 9  2 ( I 5  2 4 ) , I 10  2 2  I 3 I1 I1 I1 3h 3h

(19b)

3. Solution procedure

For simplicity and convenience, the following dimensionless quantities are introduced x 

W F X Y a , y   ,   , W  * * * * 1/ 4 , F  * * 1/ 2 , b b a [ D11 D22 ] [ D11 D22 A11 A22 ]

(x ,  y ) 

* * * 1/ 4  [ D11* D22 ] A11 A22

( AxT , ATy )

a2

( T 1 ,  T 2 ) 

( x ,  y )

a

* 1/ 2  2 [ D11* D22 ]

( M x , Px ) 



1/ 2

,  24

( T 3 ,  T 4 ,  T 6 ,  T 7 ) 

,

a2 2

,  14

 D*    22 *   D11 

1 * * * * * 1/ 4 D11 D22 A11 A22 ] [ D11

 A*    11 *   A22 

1/ 2

, 5  

* A12 * A22

,

a2 4 4 ( D xT , D Ty , 2 FxT , 2 F yT ) , 2 *  hD11 3h 3h

4    M x , 2 Px  , 3h  

 a4  a2 b 4  b 2   ,  ( K 1 , k1 )  K 1  4 * , ( , ) , K k K 2 2 2 3    2 D* E h3  , 0 11   D11 E 0 h   



t

E0

a

0

,L   L

a



0 E0

( 80 ,  90 ,  10 )  ( I 8 , I 9 , I 10 )

q 

,  170  

E0

 0 D11*

I1 E0 a 2

  2

* 0 D11

, ( x ,  y ) 

,  171 

4E0 (I 5 I1  I 4 I 2 ) * 3 0 h 2 I 1 D11

( x b 2 ,  y a 2 )h * * 1/ 2 4 2 [ D11 D22 ]

qa 4 * * * 1/ 4  4 D11* [ D11* D22 A11 A22 ]

(20) in which the alternative dimensionless forms of foundation stiffnesses k1 and k2 are not needed until the numerical examples are considered,  0 and E0 are the reference values of  m and Em at the room temperature (T0=300 K), and AxT , D xT , FxT , etc., are defined by  AxT  T  Ay

DxT DTy

N FxT     FyT  K 1

 Ax  3   (1, Z , Z )dZ hk 1 A  y k



hk

10

(21)

where Ax and Ay are given in Eq. (7). The Eqs. (1)-(4) may then be rewritten in the following dimensionless form L11 (W )  L12 ( x )  L13 (  y )  14 L14 ( F )  L16 ( M T )  K 1W  K 2  2W

      y  x   14  2 L(W , F )  L17 (W )   80    q  x y  

(22)

1 L 21 ( F )   24 L22 ( x )   24 L23 (  y )   24 L24 (W )    24  2 L(W , W ) 2  W  L31 (W )  L32 (x )  L33 ( y )   14 L34 ( F )  L36 ( S T )   90   10  x x

L41 (W )  L42 ( x )  L43 ( y )   14 L44 ( F )  L46 ( S T )   90 

(23) (24)

 W    10  y y

(25)

where the dimensionless operators Lij( ) and L( ) are defined as in [37]. The boundary conditions of Eq. (14) become x = 0, π; W  y  0 1

 



0

0



2  0

(26a)

2F dy  4 x  2  0 (movable) y 2

(26b)

2  2 2 2F  y     x 2F  2W 2  W    511    24   611             24  5 24 233 244 2 2       x y x 2  x y y 2      



2



1  W  2   24    ( 24  T 1   5  T 2 )T dxdy  0 2  x 

(immovable)

(26c)

y = 0, π;

W  x  0 1

 



0

0





 0

(26d)

2F dx  4 y  0 (movable) x 2

(26e)

2 2F  y   x 2  F   220       2   5 24 522  x y y 2  x 



 W 1  24  2  2  y

2    2W 2  W    24   240    622   y 2 x 2  

   

2

   ( T 2   5  T 1 )T dydx  0 

where  ijk are defined in Shen [37]. 11

(immovable)

(26f)

We assume that the solutions of Eqs. (22)-(25) can be expressed as ~ W ( x, y ,  )  W * ( x , y )  W ( x , y ,  )

~ x ( x, y ,  )  x* ( x, y )  x ( x, y , ) ~  y ( x, y,  )   y* ( x, y )   y ( x, y , )

~ F ( x , y ,  )  F * ( x, y )  F ( x, y ,  ) (27) where W * ( x, y ) is an initial deflection due to initial thermal bending moment, and ~ W ( x, y , ) is an additional deflection. x* ( x, y ) ,  y* ( x, y ) and F * ( x, y ) are the mid-plane ~ ~ rotations and stress function corresponding to W * ( x, y ) . x ( x, y ,  ) ,  y ( x, y,  ) and ~ F ( x, y , ) are defined analogously to x* ( x, y ) ,  y* ( x, y ) and F * ( x, y ) , but is for ~ W ( x, y , ) . Note that for initially compressed GRC plate W * ( x, y ) = x* ( x, y ) =  y* ( x, y ) = F * ( x, y ) =0.

As mentioned previously, the bending-stretching coupling effect exists in the non-symmetric FG-GRC plates and a temperature rise will cause deflections and bending curvatures in the plates and in turn will change the vibration behavior of the plates. Therefore, we need to first obtain the pre-vibration solutions W * ( x, y ) , x* ( x, y ) ,  y* ( x, y ) using the following nonlinear equations L11 (W * )  L12 ( x* )  L13 (  y* )  14L14 ( F * )  L16 ( M T )  K 1W *  K 2  2W *

  14  2 L(W * , F * )

(28)

1 L 21 ( F * )   24 L22 (x* )   24 L23 ( y* )   24 L24 (W * )    24  2 L(W * ,W * ) 2

(29)

L31 (W * )  L32 ( x* )  L33 ( y* )   14 L34 ( F * )  L36 ( S T )  0

(30)

L41 (W * )  L42 ( x* )  L43 ( y* )   14 L44 ( F * )  L46 ( S T )  0

(31)

The solutions of Eqs. (28)-(31) can be obtained in the similar form as in [45]. Then an ~ ~ initially stressed FG-GRC plate is under consideration and W ( x, y,  ) , x ( x, y,  ) ,

12

~ ~  y ( x, y,  ) and F ( x, y , ) satisfy the nonlinear equations ~ ~ ~ ~ ~ ~ L11 (W )  L12 (x )  L13 ( y )   14 L14 ( F )  K 1W  K 2  2W

~  ~     y  ~ ~   x * ~  q  14 L(W  W , F )  L17 (W )   80   y   x   2

(32)

1 ~ ~ ~ ~ ~ ~ L 21 ( F )   24 L22 ( x )   24 L23 ( y )   24 L24 (W )    24  2 L(W  2W * , W ) 2

(33)

~  W ~ ~ ~ ~ ~  L31 (W )  L32 ( x )  L33 ( y )   14 L34 ( F )   90   10  x x

(34)

~  W ~ ~ ~ ~ ~  L41 (W )  L42 ( x )  L43 ( y )   14 L44 ( F )   90    10  y y

(35)

The initial conditions are assumed to be ~ ~  x W ~ ~ | 0  0 ,  x | 0  | 0  0 , W | 0   

~  y ~  y | 0  | 0  0 

(36)

A two step perturbation technique [46] is now used to solve Eqs. (32)-(35). The essence of this procedure, in the present case, is to assume that ~ W ( x, y, ~,  )   j w j ( x, y ,~ )

 j 1

~  x ( x, y ,~,  ) 

 

xj ( x,

y, ~ )

 

yj ( x,

y ,~ )

j

j 1

~  y ( x, y, ~,  ) 

j

j 1

~ F ( x, y ,~,  ) 



j

f j ( x, y, ~ )

j 1

 q ( x, y,~,  )    j  j ( x, y,~ ) j 1

(37) in which  is the small perturbation parameter, and    is introduced to improve perturbation procedure for solving nonlinear vibration problem. Substituting Eq. (37) into Eqs. (32)-(35), and collecting terms of the same order of  , a set of perturbation equations is obtained. Solving these equations step by step, we obtain asymptotic solutions as ~ ( 3) W ( x, y , )   [ A11(1) ( ) sin mx sin ny ]   3[ A31 ( ) sin 3mx sin ny

 A13(3) ( ) sin mx sin 3ny ]  O( 4 )

(38) 13

~ ( 2) x ( x, y, )   [C11(1) ( )  C11(3) ( )] cos mx sin ny   2C20 ( ) sin 2mx ( 3) ( ) cos 3mx sin ny  C13(3) ( ) cos mx sin 3ny ]  O( 4 )   3[C31

(39)

~  (3) ( )] sin mx cos ny   2 D ( 2) ( ) sin 2ny y ( x, y, )   [ D11(1) ( )  D 11 02 ( 3)   3[ D31 ( ) sin 3mx cos ny  D13(3) ( ) sin mx cos 3ny ]  O ( 4 )

(40)

~ (0) 2 ( 0) 2  (3) ( )] sin mx sin ny   2 (  B ( 2) y 2 / 2 F ( x, y , )   B00 y / 2  b00 x / 2   [ B11(1) ( )  B 11 00 ( 2) 2 ( 2) ( 2) ( 3) x / 2  B02 ( ) cos 2ny  B20 ( ) cos 2mx)   3[ B31 ( ) sin 3mx sin ny  b00

 B13(3) ( ) sin mx sin 3ny ]  O( 4 )

(41)

 (1) ( )] sin mx sin ny  (A (1) ( )) 2 ( g cos 2mx  g cos 2ny )  q ( x, y, )   [ g1 A11(1) ( )  g 4 A 20 02 11 11

 (A11(1) ( )) 3 g 3 sin mx sin ny  O( 4 )

(42)

In Eqs. (38)-(42)  is replaced by t and ( A11 ) is taken as the second perturbation parameter (1)

relating to the dimensionless vibration amplitude. For free vibration  q  0 . Applying Galerkin procedure to Eq. (42), we can obtain

g 40

d 2 (A11(1) )  g 41 (A11(1) )  g 42 (A11(1) ) 2  g 43 (A11(1) )3  0 d 2

(43)

where the terms g40, g41, g42 and g43 are described in detail in Appendix A, and the solution of which may be expressed by 1/ 2

 NL in

2  9 g g  10 g 42    L 1  41 43 2 A2  12 g 41  

which  L  [ g 41 / g 40 ]1 / 2

is

the

(44) dimensionless

linear

frequency,

and

A

* * * * 1/ 4 Wmax /[ D11 D22 A11 A22 ] is the dimensionless amplitude of the plate. According to Eq. (20),

the corresponding linear frequency can be expressed as  L   L ( / a )( E 0 /  0 )1 / 2 , where E 0 and  0 are defined as in Eq. (20).

4. Numerical examples and discussion

Numerical results are presented in this section for simply supported GRC laminated plates resting on an elastic foundation and in thermal environments. We first need to 14

determine the effective material properties of GRCs. Poly (methyl methacrylate), referred to as PMMA, is selected for the matrix, and the material properties of which are assumed to be

 m =1150 kg/m3,  m =0.34,  m = 45(1+0.0005  T )×10-6/K and E m = (3.52-0.0034T) GPa, in which T = T0 +  T and T0 = 300 K (room temperature). In such a way,  m = 45.0×10-6/K and E m =2.5 GPa at T=300 K. The zigzag (refer to as 0-ply) graphene sheets with effective thickness hG= 0.188 nm and G =4118 kg/m3 are selected as reinforcements. The material properties of which are temperature-dependent and listed in Table 1 [41]. The key issue for the successful application of the Halpin–Tsai model to GRCs is to determine the graphene efficiency parameters  j (j=1,2,3). However, there are no experiments conducted to determine the values of  j for GRCs. In the present study, we give the estimation of graphene efficiency parameters 1 ,  2 and  3 by matching the Young’s moduli E11 and E22 and shear modulus G12 of GRCs predicted from the Halpin–Tsai model to those from the MD simulations, as previously reported in Lin et al. [41]. These parameters are listed in Table 2, and we assume that G13= G23=0.5 G12. As part of the validation of the present solution method, the dimensionless fundamental frequencies ~  (a 2 / h)  E22

for square (0/90) and (0/90)S laminated plates with

different aspect ratios are calculated and compared in Table 3 with the HSDPT results of Reddy and Phan [47], the discrete layer HSDPT results of Cho et al. [48] and the 3D FEM solutions of Desai et al. [49]. In Table 3, the material properties are taken to be E11=275.6 GPa, E22=6.89 GPa, G12=4.134 GPa, G13=G23=3.445 GPa,  12 =0.25,  =1. As a second example, the dimensionless linear frequencies ~  (a 2 / h)  0 E0 for UD and FG CNT/PmPV plates with a/b=1.0, b/h =50, h=2 mm at T=300 K are calculated and compared in Table 4 with the FSDPT results of Zhu et al. [50], the Fourier series results of Alibeigloo and Emtehani [51], and the 3D elasticity solutions of Wu and Li [20]. In Table 4, the extended rule of mixture is adopted and the CNT efficiency parameters are taken to be 1 =0.149, *  2 =  3 =0.934 for the case of VCN =0.11, and 1 =0.150,  2 =  3 =0.941 for the case of

15

* * VCN =0.14, and 1 =0.149,  2 =  3 =1.381 for the case of VCN =0.17. As a last example, the

nonlinear to linear frequency ratios  NL /  L for CNT/PMMA plates with a/b=1.0, b/h =100, h=2 mm are plotted in Fig. 1 and are compared with the FSDPT results of Zhu et al. [50]. The * =0.17 and the CNT efficiency parameters are taken to be CNT volume fraction VCN

1 =0.142,  2 =1.626 and  3 =1.138. These comparisons show that the results from the present method are in good agreement with the existing results, thus verifying the reliability and accuracy of the present method. A parametric study has been carried out and typical results are shown in Tables 5-8 and

~ Figs. 2-6. The dimensionless natural frequency is defined by   (a 2 / h)  0 E 0 , where

 0 and E0 are the reference values of  m and Em at T=300 K. The total thickness of the laminated plate is h =2 mm which comprised of ten plies of identical thickness of 0.2mm. The plates may have different width-to-thickness ratios of b/h = 10, 20 and 100. Four types of FG-GRC laminated plates, i.e. FG-V, FG-  , FG-X and FG-O, are considered in this study. For Type FG-V, the graphene volume fractions are assumed to have graded distribution [(0.11)2/(0.09)2/(0.07)2/(0.05)2/(0.03)2] for ten plies. For Type FG-  , the distribution of graphene reinforcements is inversed, i.e. [(0.03)2/(0.05)2/(0.07)2/(0.09)2/(0.11)2]. For Type FG-X, a mid-plane symmetric graded distribution of graphene reinforcements is achieved, i.e. [0.11/0.09/0.07/0.05/0.03]S, and for type FG-O, the graphene volume fractions are assumed to be [0.04/0.05/0.07/0.09/0.11]S. A UD-GRC laminated plate with the same thickness is also considered as a comparator for which the graphene volume fraction VG of each ply is identical and VG = 0.07. In such a way, the two cases of UD and FG GRC laminated plates will have the same value of total fraction of graphene. The in-plane boundary condition is assumed to be immovable (Case 2) except for the results in Table 8 and Fig. 6, where the initial compressive edge load is considered when the plate edges are movable. Table 5 presents the natural frequencies of (0)10, (0/90/0/90/0)S and (0/90)5T GRC laminated square plates with b/h = 10 under thermal environments T= 300, 400 and 500K. It can be seen that the natural frequency of the GRC plate decreases when temperature rises due to the increased initial thermal deflection of the plates and the decreased GRC material 16

strength as temperature increases. It is found that the FG-X GRC laminated plate has the largest, while the FG-O GRC laminated plate has the lowest natural frequencies among the five plates. As expected, the natural frequencies of FG-V and FG-  are the same for both (0)10 and (0/90/0/90/0)S GRC laminated plates. Table 6 presents the natural frequencies of (0)10, (0/90/0/90/0)S and (0/90)5T GRC laminated square plates with b/h = 20 resting on elastic foundations and in room temperature (T = 300 K). Two foundation models are considered. The stiffnesses are (k1, k2)=(100, 10) for the Pasternak elastic foundation, (k1, k2)=(100, 0) for the Winkler elastic foundation and (k1, k2)=(0, 0) for the plate without any elastic foundation. As expected, the natural frequencies are ~ ~ increased with the increase in foundation stiffnesses. It can be seen that 12 =  21 and ~ ~ 13 = 31 for FG-X and FG-O (0/90)5T laminated plates. Table 7 shows the effect of temperature rise (T=300, 400 and 500 K) on the nonlinear to linear frequency ratios  NL /  L of (0/90)5T GRC laminated square plates with b/h = 10. Then Table 8 shows the effect of initial compressed stress on the nonlinear to linear frequency ratios  NL /  L of (0/90)5T GRC laminated square plates with b/h = 20 resting on elastic foundations. In these two examples the in-plane boundary condition is assumed to be movable (case 1). It can be seen that the fundamental frequencies of the GRC plates are increased, but the nonlinear to linear frequency ratios are decreased as the foundation stiffnesses are increased. It can also be seen that the initial compressed stress leads to the decrease of the fundamental frequency, but increase of the nonlinear to linear frequency ratio. As shown in Tables 7 and 8, the FG-X GRC plate has the lowest nonlinear to linear frequency ratios amongst the five GRC plates in the tables. Fig. 2 presents the frequency-amplitude curves of (0/90/0/90/0)S GRC laminated square plates with different types of graphene reinforcements. The results show that the FG-X (0/90/0/90/0)S plate has the highest fundamental frequency, but the lowest nonlinear to linear frequency ratios among the five cases. In contrast, the FG-O (0/90/0/90/0)S plate has the lowest fundamental frequency, but the highest nonlinear to linear frequency ratios among the same five plates. The results confirm that the nonlinear to linear frequency ratios of FG-V and

17

FG-  (0/90/0/90/0)S plates are the same. Hence, in the following examples only UD and FG-X plates are considered. Fig. 3 shows the effect of temperature rise on the nonlinear vibration behavior of UD and FG-X (0/90/0/90/0)S laminated square plates with b/h = 10 under thermal environments T= 300, 400 and 500K. The results from Table 7 and Fig. 3 confirm that the fundamental frequency of the GRC plate is decreased, but the nonlinear to linear frequency ratio is increased as temperature rises. Fig. 4 depicts the effect of plate width-to-thickness ratio b/h (=10, 20 and 100) on the nonlinear vibration behavior of UD and FG-X (0/90/0/90/0)S laminated square plates at T=300 K. It can be seen that the frequency-amplitude curves are decreased with increase in plate width-to-thickness ratio b/h for both UD and FG-X laminated plates. Fig. 5 presents the effect of foundation stiffness on the nonlinear vibration behavior of UD and FG-X (0/90/0/90/0)S laminated square plates resting on elastic foundations at T=300 K. Two foundation models are considered. In this example, (k1, k2)=(1000, 100) is for the Pasternak elastic foundation, and (k1, k2)=(1000, 0) is for the Winkler elastic foundation. The results confirm that the frequency-amplitude curves are decreased with increase in foundation stiffness for both UD and FG-X laminated plates. Fig. 6 shows the effect of the in-plane boundary conditions on the nonlinear vibration behavior of UD and FG-X (0/90/0/90/0)S laminated square plates. Two in-plane boundary condition, i.e., ’movable’ and ’immovable’, are considered. It can be seen that the nonlinear to linear frequency ratios for movable end conditions are much smaller than those for immovable end conditions when no initial compressed edge load is applied.

5.

Concluding remarks

Modeling and nonlinear vibration analysis of GRC laminated plates resting on elastic foundations and in thermal environments have been carried out by a multi-scale approach. Micromechanical model is used to estimate the temperature dependent material properties of GRCs. A parametric study for UD and FG GRC laminated plates with low graphene volume fractions has been conducted. Four types of FG-GRC laminated plates are considered. The numerical results reveal that the nonlinear vibration characteristics of the plates are 18

significantly influenced by a functionally graded reinforcement. The results also show that the natural frequencies of the plates are reduced but the nonlinear to linear frequency ratios are increased when temperature rises or foundation stiffness decreases.

Acknowledgments

This study was supported by the National Natural Science Foundation of China under Grant 51279103, and the Australian Research Council grant DP140104156. The authors are very grateful for these financial supports.

Appendix A

In Eq. (43) g 42  

g 43 

  g  2  14 24 m 2 n 2  2  8  9  4 05 (1  cos m )(1  cos n ) , 2 g 06  3 mn 6 7

 m4 n4 4  1  14 24    C33  16 6  7 

* g 40   g 08   14 24

* g 05 g 07  [ 170   171 (m 2  n 2  2 )] g 06

 m 2 g 04  n 2  2 g 03 m 2 g 02  n 2  2 g 01 g 05    80    14  24 g 00 g 00 g 06 

 ,  

(A.1)

for initially thermal stressed plates (immovable edge condition) g 41  Q11   14 ( T 1m 2   T 2 n 2  2 )T  2 g 42 (T )  2 g 43 2 (T ) , C33  2

2 4 4 m 4   24 n   2 5 m 2 n 2  2 , 2  24   52

(A.2)

and for initially compressed stressed plates (movable edge condition)  P ( m 2  n 2  2 )  g 41  Q11 1   , C33  0 , m2   Pcr

(A.3)

in which Pcr is the buckling load for the UD-GRC plate under uniaxial compression, and  is the load proportion ratio, defined by  y   x . In the above equations (with other symbols being defined as in [37])

19

Q11  g 08   14  24

2  6  1   14  24  220

 8   140

g 05 g 07  [ K 1  K 2 ( m 2  n 2  2 )] , g 06

 31

4m 2 4n 2  2 2 2 , ,        7 24 14 24 233   320 4m 2  42   432 4n 2  2

4m 2 4n 2  2   ,   120 220 ,     9 144 133 233  42   432 4n 2  2  31   320 4m 2 (A.4)

In Eq. (A2)  (T )     2 ( ) 2   3 ( ) 3  

(A.5)

where (with m=n=1) ( T 3   T 6 )m 2 g102  ( T 4   T 7 )n 2  2 g 101  16  2 2 2 T    2 (    ) m n T3 T4  g 00  G08    2 

h * * * * 1/ 4 [ D11 D22 A11 A22 ]

,

 g  8  14  24 m 2 n 2  2  8  9  4 05 g 06 3 G08 6 7 2

3  2 22 

  , 

  m4 n4 4 1  14 24    C33  , 16G08 6   7

 3  2 22 

2 4 4  m4 n4 4 m 4   24 n   2 5 m 2 n 2  2 1  14  24   2 2 6 16G 08  24   52  7

 ,  

g101  ( 31   320 m 2   322 n 2  2 )( 131 m 2   133 n 2  2 )   331 m 2 ( 120 m 2   122 n 2  2 ) , g102  ( 42   430 m 2   432 n 2  2 )( 120 m 2   122 n 2  2 )   331 n 2  2 ( 131 m 2   133 n 2  2 ) ,

G08  Q11   14 ( T 1 m 2   T 2 n 2  2 )T (A.6) References

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plates using a layerwise mixed finite element model, Compos. Struct. 59 (2003) 237–249. [50] P. Zhu, Z.X. Lei, K.M. Liew, Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory, Compos. Struct. 94 (2012) 1450-1460. [51] A. Alibeigloo, A. Emtehani, Static and free vibration analyses of carbon nanotube-reinforced composite plate using differential quadrature method, Meccanica 50 (2015) 61-76.

25

Table 1

Temperature-dependent material properties for monolayer graphene (aG=14.76 nm, bG=14.77 G = 0.177, G =4118 kg/m3) [41]. nm, thickness hG= 0.188 nm,  12

Temperature(K) 300 400 500

G E11 (GPa)

1812 1769 1748

G E22 (GPa)

1807 1763 1735

26

G G12 (GPa)

11G (x10-6/K)

G  22 (x10-6/K)

683 691 700

-0.90 -0.35 -0.08

-0.95 -0.40 -0.08

Table 2

Comparisons of Young’s and shear moduli for graphene/PMMA nanocomposites. MD [41] E11 (GPa) T=300 K 0.03 36.538 0.05 59.544 0.07 80.096 0.09 90.023 0.11 96.388 T=400 K 0.03 32.639 0.05 53.462 0.07 71.698 0.09 81.035 0.11 88.557 T=500 K 0.03 31.926 0.05 52.171 0.07 69.960 0.09 79.218 0.11 87.039

E22 (GPa)

G12 (GPa)

Halpin–Tsai model E11 E22 1 (GPa) (GPa)

35.613 57.479 78.843 88.750 94.265

11.388 15.655 23.644 33.635 34.713

12.47 19.41 26.58 34.01 41.71

2.929 3.068 3.013 2.647 2.311

31.750 51.661 70.928 78.091 86.063

11.572 12.919 19.574 25.566 26.735

10.96 17.09 23.43 30.01 36.82

31.854 50.225 67.453 76.019 84.743

11.700 11.450 17.106 22.202 23.478

9.424 14.72 20.21 25.90 31.82

VG

27

2

G12 (GPa)

3

12.47 19.41 26.58 34.01 41.71

2.855 2.962 2.966 2.609 2.260

0.962 0.982 1.003 1.025 1.048

11.842 15.944 23.575 32.816 33.125

2.977 3.128 3.060 2.701 2.405

10.96 17.09 23.43 30.00 36.82

2.896 3.023 3.027 2.603 2.337

0.831 0.848 0.867 0.886 0.906

13.928 15.229 22.588 28.869 29.527

3.388 3.544 3.462 3.058 2.736

9.420 14.71 20.20 25.89 31.80

3.382 3.414 3.339 2.936 2.665

0.700 0.715 0.730 0.746 0.763

16.712 16.018 23.428 29.754 30.773

Table 3

Comparison of fundamental frequency ~  (a 2 / h)  E22 for square laminated plates with different aspect ratios Lay-up (0/90)

(0/90)S

Source

Reddy & Phan [47] Cho et al. [48] Desai et al. [49] Present Reddy & Phan [47] Cho et al. [48] Desai et al. [49] Present

a/h 5

10

20

50

100

9.010 8.388 8.527 9.049 10.989 10.673 10.682 10.603

10.449 10.270 10.338 10.565 15.270 15.066 15.069 14.985

10.968 11.016 11.040 11.106 17.668 17.535 17.636 17.598

11.132 11.260 11.267 11.275 18.606 18.670 18.670 18.663

11.156 11.296 11.298 11.300 18.755 18.835 18.835 18.833

28

Table 4

Comparisons of natural frequency ~  (a 2 / h)  0 E0 for square CNT/PmPV plates (a/b=1, b/h=50, h=2 mm, T=300 K) Source

UD (1,1)

FG-V (1,2)

(1,3)

(1,1) * VCN

Zhu et al. [50] Alibeigloo [51] Wu & Li [28]

Present

19.223 19.168 19.155 19.154

23.408 23.284 23.270 23.273

34.669 34.054 34.038 34.055

FG-X (1,2)

(1,3)

(1,1)

(1,2)

(1,3)

21.142 21.046 20.998 20.825

33.350 32.785 32.702 32.403

22.984 22.898 22.902 23.144

26.784 26.617 26.630 27.263

37.591 36.919 36.938 37.812

22.643 22.566 22.519 22.359

34.660 34.111 34.032 33.745

25.555 25.491 25.499 25.833

29.192 29.042 29.061 29.880

39.833 39.171 39.200 40.281

28.987 28.825 28.810 28.813

43.165 42.386 42.367 42.388

28.413 28.264 28.275 28.627

33.434 33.163 33.192 34.035

47.547 46.605 46.650 47.752

=0.11

16.252 16.208 16.176 16.140 * VCN =0.14

Zhu et al. [50] Alibeigloo [51] Wu & Li [28]

Present

21.354 21.328 21.317 21.316

25.295 25.199 25.188 25.190

36.267 35.679 35.667 35.684

17.995 17.968 17.938 17.911 * VCN =0.17

Zhu et al. [50] Alibeigloo [51] Wu & Li [28]

Present

23.697 23.622 23.607 23.607

28.987 28.825 28.810 28.813

43.165 42.386 42.367 42.388

19.982 19.932 19.890 19.833

29

Table 5 ~ Natural frequency   (a 2 / h)  0 E 0 of GRC laminated plates in thermal environments (a/b=1, b/h=10, h=2 mm)

Lay-up

~ 11

(0)10

28.0982 25.3899 29.5212 23.5769 28.0982 25.3900 29.5212 23.5769 28.0982 25.3899 25.3899 29.5212 23.5769

(0)10

21.9591 19.7324 23.8137 17.5580 21.9591 19.7323 23.8138 17.5581 21.9591 19.7323 19.7323 23.8138 17.5580

(0)10

15.0773 14.8437 18.2243 9.8081 15.0776 14.8439 18.2245 9.8082

UD FG-V&  FG-X FG-O (0/90/0/90/0)S UD FG-V&  FG-X FG-O (0/90)5T UD FG-V FG-  FG-X FG-O UD FG-V&  FG-X FG-O (0/90/0/90/0)S UD FG-V&  FG-X FG-O (0/90)5T UD FG-V FG-  FG-X FG-O UD FG-V&  FG-X FG-O (0/90/0/90/0)S UD FG-V&  FG-X FG-O

~ 12

~  21

T=300K 64.8785 65.1157 59.1670 59.5288 65.6449 65.9070 56.0228 56.3974 64.9584 65.0364 59.2889 59.4084 65.7134 65.8394 56.2111 56.2103 64.9974 64.9974 59.3551 59.3422 59.3422 59.3551 65.7764 65.7764 56.2107 56.2107 T=400K 54.8037 57.6844 50.0996 53.1479 56.1356 58.8911 46.4214 49.6313 55.9631 56.5605 51.2779 52.0123 57.2699 57.7903 47.8107 48.2951 56.2626 56.2626 51.6404 51.6521 51.6521 51.6404 57.5307 57.5307 48.0534 48.0534 T=500K 45.2749 52.0752 45.6953 51.7534 48.1752 54.0462 37.5828 44.6734 48.0719 49.5070 48.1197 49.5077 50.6451 51.7409 40.7210 41.8332 30

~  22

~ 13

~ 31

95.6901 88.6176 95.6293 84.5665 95.6904 88.6182 95.6296 84.5669 95.6903 88.6181 88.6181 95.6296 84.5669

116.0301 107.3566 113.4456 103.7875 116.1739 107.5928 113.5548 104.1813 116.2548 107.7307 107.7107 113.6785 104.1862

116.4778 108.0807 113.9093 104.5809 116.3356 107.8485 113.8022 104.1911 116.2548 107.7107 107.7307 113.6785 104.1862

84.0640 78.3921 84.4575 73.9941 84.0641 78.3923 84.4584 73.9946 84.0641 78.3921 78.3921 84.4584 73.9945

101.1662 94.2289 99.7253 89.5692 102.8508 95.9517 101.4343 91.5963 103.2981 96.5026 96.5318 101.8561 91.9321

105.3867 98.7516 103.9394 94.2338 103.7435 97.0796 102.2760 92.2668 103.2981 96.5318 96.5026 101.8561 91.9321

74.6301 75.4082 76.4753 66.0903 74.6317 75.4086 76.4767 66.0909

88.3236 92.2884 89.1503 78.7036 92.2023 95.5734 92.7234 82.8460

97.9272 100.6377 97.7478 88.1220 94.2890 97.5242 94.3692 84.2405

(0/90)5T

UD FG-V FG-  FG-X FG-O

15.0774 48.7945 14.8437 48.8047 14.8437 48.8322 18.2245 51.1959 9.8081 41.2808

48.7945 48.8322 48.8047 51.1959 41.2808

74.6315 75.4084 75.4084 76.4767 66.0908

93.2513 96.5224 96.5845 93.5499 83.5461

93.2513 96.5845 96.5224 93.5499 83.5461

Table 6 ~ Natural frequency   (a 2 / h)  0 E 0 of GRC laminated plates resting on elastic foundations (a/b=1, b/h=20, h=2 mm, T=300K)

Lay-up

~ 11

(0)10

29.5854 26.4859 31.7471 24.3546 29.5854 26.4860 31.7471 24.3547 29.5854 26.4859 26.4859 31.7470 24.3547

(0)10

30.9790 28.0340 33.0496 26.0300 30.9790 28.0340 33.0496 26.0300 30.9790 28.0340 28.0340 33.0496 26.0300

(0)10

33.5605 30.8627 35.4806 29.0548 33.5605 30.8627 35.4806

UD FG-V&  FG-X FG-O (0/90/0/90/0)S UD FG-V&  FG-X FG-O (0/90)5T UD FG-V FG-  FG-X FG-O UD FG-V&  FG-X FG-O (0/90/0/90/0)S UD FG-V&  FG-X FG-O (0/90)5T UD FG-V FG-  FG-X FG-O UD FG-V&  FG-X FG-O (0/90/0/90/0)S UD FG-V&  FG-X

~ 12

~  21

(k1, k2) = (0, 0) 72.7627 73.0926 65.0535 65.5262 76.4747 76.8785 60.3844 60.8429 72.8777 72.9781 65.2165 65.3644 76.5884 76.7657 60.6172 60.6112 72.9279 72.9279 65.3005 65.2803 65.2803 65.3005 76.6769 76.6769 60.6142 60.6142 (k1, k2) = (100, 0) 73.3379 73.6652 65.6960 66.1641 77.0223 77.4233 61.0762 61.5295 73.4520 73.5516 65.8575 66.0039 77.1352 77.3112 61.3063 61.3004 73.5018 73.5018 65.9406 65.9207 65.9207 65.9406 77.2231 77.2231 61.3034 61.3034 (k1, k2) = (100, 10) 76.1127 76.4282 68.7791 69.2264 79.6695 80.0572 64.3811 64.8113 76.2227 76.3187 68.9333 69.0732 79.7785 79.9488 31

~  22

~ 13

~ 31

112.3927 101.5595 118.0847 94.3076 112.3928 101.5599 118.0849 94.3078 112.3927 101.5597 101.5597 118.0847 94.3078

140.7286 126.3209 144.7172 118.5110 140.9946 126.7107 144.9680 119.0813 141.1151 126.9157 126.8685 145.1749 119.0760

141.4999 127.4593 145.6302 119.6372 141.2357 127.0735 145.3820 119.0707 141.1151 126.8685 126.9157 145.1749 119.0760

112.7646 101.9707 118.4390 94.7502 112.7647 101.9710 118.4392 94.7503 112.7646 101.9709 101.9709 118.4390 94.7503

141.0252 126.6510 145.0060 118.8626 141.2906 127.0398 145.2563 119.4312 141.4109 127.2442 127.1971 145.4628 119.4259

141.7949 127.7864 145.9172 119.9855 141.5313 127.4016 145.6694 119.4207 141.4109 127.1971 127.2442 145.4628 119.4259

115.6587 105.1606 121.1999 98.1746 115.658 105.1609 121.2001

143.9198 129.8634 147.8257 122.2788 144.1799 130.2427 148.0713

144.6742 130.9713 148.7198 123.3709 144.4158 130.5958 148.4767

(0/90)5T

FG-O UD FG-V FG-  FG-X FG-O

29.0548 64.5995 33.5605 76.2706 30.8627 69.0128 30.8627 68.9937 35.4806 79.8636 29.0548 64.5967

64.5939 76.2706 68.9937 69.0128 79.8636 64.5967

98.1747 115.6588 105.1608 105.1608 121.1999 98.1747

122.8318 144.2978 130.4422 130.3963 148.2739 112.8266

122.8215 144.2978 130.3963 130.4422 148.2739 112.8266

Table 7

Nonlinear to linear frequency ratios  NL /  L for (0/90)5T GRC laminated plates in thermal environments (a/b=1, b/h=10, h=2 mm) ~ 

T (K) 300

400

500

UD FG-V&  FG-X FG-O UD FG-V&  FG-X FG-O UD FG-V&  FG-X FG-O

28.0982 25.3899 29.5212 23.5769 21.9591 19.7323 23.8138 17.5580 15.0774 14.8437 18.2245 9.8081

Wmax / h

0.2 1.0241 1.0265 1.0198 1.0308 1.0353 1.0393 1.0272 1.0494 1.0706 1.0669 1.0449 1.1475

32

0.4 1.0934 1.1023 1.0768 1.1180 1.1347 1.1492 1.1048 1.1854 1.2589 1.2463 1.1693 1.5056

0.6 1.1999 1.2183 1.1658 1.2498 1.2832 1.3121 1.2232 1.3826 1.5218 1.4982 1.3515 1.9622

0.8 1.3348 1.3641 1.2800 1.4140 1.4662 1.5110 1.3719 1.6188 1.8273 1.7924 1.5715 2.4632

1.0 1.4904 1.5313 1.4133 1.6004 1.6723 1.7334 1.5421 1.8794 2.1575 2.1113 1.8155 2.9863

Table 8

Nonlinear to linear frequency ratios  NL /  L for initially compressed (0/90)5T GRC laminated plates resting on elastic foundations (a/b=1, b/h=20, h=2 mm, T=300K)

(k1, k2)

P/Pcr

(0, 0)

0.0

0.5

(100, 0)

0.0

0.5

(100, 10)

0.0

0.5

~  UD FG-V&  FG-X FG-O UD FG-V&  FG-X FG-O UD FG-V&  FG-X FG-O UD FG-V&  FG-X FG-O UD FG-V&  FG-X FG-O UD FG-V&  FG-X FG-O

29.5854 26.4859 31.7470 24.3547 20.9200 18.7284 22.4485 17.2213 30.9790 28.0340 33.0496 26.0300 21.9055 19.8230 23.3696 18.4060 33.5605 30.8627 35.4806 29.0548 23.7309 21.8232 25.0886 20.5448

Wmax / h 0.2 1.0056 1.0063 1.0044 1.0074 1.0111 1.0124 1.0087 1.0148 1.0051 1.0056 1.0040 1.0065 1.0102 1.0111 1.0081 1.0130 1.0043 1.0046 1.0035 1.0052 1.0087 1.0092 1.0070 1.0104

33

0.4 1.0221 1.0248 1.0174 1.0294 1.0438 1.0487 1.0345 1.0580 1.0202 1.0222 1.0161 1.0258 1.0400 1.0437 1.0319 1.0509 1.0173 1.0183 1.0140 1.0208 1.0342 1.0362 1.0277 1.0411

0.6 1.0494 1.0549 1.0388 1.0650 1.0961 1.1066 1.0761 1.1263 1.0449 1.0492 1.0358 1.0571 1.0880 1.0958 1.0704 1.1114 1.0384 1.0408 1.0311 1.0461 1.0755 1.0798 1.0614 1.0903

0.8 1.0859 1.0958 1.0679 1.1130 1.1655 1.1829 1.1318 1.2155 1.0786 1.0859 1.0628 1.0995 1.1519 1.1648 1.1222 1.1908 1.0673 1.0715 1.0547 1.0806 1.1307 1.1379 1.1068 1.1556

1.0 1.1313 1.1461 1.1043 1.1718 1.2489 1.2743 1.1996 1.3214 1.1204 1.1314 1.0966 1.1518 1.2291 1.2480 1.1854 1.2857 1.1034 1.1097 1.0843 1.1234 1.1979 1.2086 1.1625 1.2346

2.0 1.8

NL/L

1.6 1.4

CNTRC plate a/b=1, b/h=100 * T=300 K, V CN=0.17

2

1: UD 2: FG-V

1

1.2

Present Zhu et al. [50]

1.0 0.8 0.0

0.2

0.4

0.6

0.8

1.0

W/h

Fig. 1. Comparisons of frequency-amplitude curves of UD and FG-V CNT/PMMA plates.

34

1.8 (0/90/0/90/0)S

NL/L

1.6

1.4

1.2

a/b=1, b/h=20 T=300 K 5 2&3 1

1: UD 2: FG-V 3: FG- 4: FG-X 5: FG-O

4

1.0

0.8 0.0

0.2

0.4

0.6

0.8

1.0

W/h

Fig. 2. Nonlinear vibration behavior of (0/90/0/90/0)S GRC laminated plates with different

types of graphene reinforcements.

35

2.2 2.0

NL/L

1.8 1.6 1.4

(0/90/0/90/0)S a/b=1, b/h=10

1: T=300 K 2: T=400 K 3: T=500 K

1: UD 1: FG-X 2: UD 2: FG-X 3: UD 3: FG-X

1.2 1.0 0.8 0.0

0.2

0.4

0.6

0.8

1.0

W/h

Fig. 3. The effect of temperature rise on the frequency-amplitude curves of (0/90/0/90/0)S

GRC laminated plates.

36

1.6 (0/90/0/90/0)S a/b=1, T=300 K

NL/L

1.4

1.2

1: b/h=10 2: b/h=20 3: b/h=100 1: UD 1: FG-X 2: UD 2: FG-X 3: UD 3: FG-X

1.0

0.8 0.0

0.2

0.4

0.6

0.8

1.0

W/h

Fig. 4. Effect of plate thickness ratio b/h on the frequency-amplitude curves of (0/90/0/90/0)S GRC laminated plates.

37

1.6 (0/90/0/90/0)S

NL/L

1.4

1.2

a/b=1, b/h=20 T=300 K 1: UD 1: FG-X 2: UD 2: FG-X 3: UD 3: FG-X

1: (k1, k2)=(0, 0) 2: (k1, k2)=(1000, 0) 3: (k1, k2)=(1000, 100)

1.0

0.8 0.0

0.2

0.4

0.6

0.8

1.0

W/h

Fig. 5. The effect of foundation stiffness on the frequency-amplitude curves of (0/90/0/90/0)S

GRC laminated plates resting on elastic foundations.

38

1.6 (0/90/0/90/0)S

NL/L

1.4

a/b=1, b/h=20 T=300 K

1.2

1: immovable 2: movable, P/Pcr = 0.0 3: movable, P/Pcr = 0.5

1: UD 1: FG-X 2: UD 2: FG-X 3: UD 3: FG-X

1.0

0.8 0.0

0.2

0.4

0.6

0.8

1.0

W/h

Fig. 6. Nonlinear vibration behavior of (0/90/0/90/0)S GRC laminated plates under two cases

of in-plane boundary conditions.

39

Figure legends Fig. 1. Comparisons of frequency-amplitude curves of UD and FG-V CNT/PMMA plates. Fig. 2. Nonlinear vibration behavior of (0/90/0/90/0)S GRC laminated plates with different

types of graphene reinforcements. Fig. 3. The effect of temperature rise on the frequency-amplitude curves of (0/90/0/90/0)S

GRC laminated plates. Fig. 4. Effect of plate thickness ratio b/h on the frequency-amplitude curves of (0/90/0/90/0)S

GRC laminated plates. Fig. 5. The effect of foundation stiffness on the frequency-amplitude curves of (0/90/0/90/0)S

GRC laminated plates resting on elastic foundations. Fig. 6. Nonlinear vibration behavior of (0/90/0/90/0)S GRC laminated plates under two cases

of in-plane boundary conditions.

40

   

The concept of functionally graded materials is extended to the GRC laminated plates. A multi-scale approach for nonlinear vibration analysis of FG GRC laminated plates is proposed. The plate-foundation interaction and temperature-dependent material properties are both taken into account. FG reinforcement has a significant effect on the vibration characteristics of GRC laminated plates.

41