Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method

Accepted Manuscript Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method Jie Yang, Da Chen, Sritawat Kitipornchai PII: DOI: Reference:

S0263-8223(18)30786-4 https://doi.org/10.1016/j.compstruct.2018.03.090 COST 9537

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

25 February 2018 22 March 2018 26 March 2018

Please cite this article as: Yang, J., Chen, D., Kitipornchai, S., Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct.2018.03.090

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Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method Jie Yang a,*, Da Chen b, Sritawat Kitipornchai b a b

School of Engineering, RMIT University, PO Box 71, Bundoora, VIC 3083 Australia

School of Civil Engineering, the University of Queensland, Brisbane, St Lucia 4072, Australia

__________________________________________________________________________ Abstract This paper is concerned with the buckling and free vibration behaviors of functionally graded (FG) porous nanocomposite plates reinforced with graphene platelets (GPLs). The porous plates are constructed based on a multiplayer model with GPLs uniformly or non-uniformly distributed in the metal matrix containing open-cell internal pores. The modified Halpin-Tsai micromechanics model, the extended rule of mixture, and the typical mechanical properties of open-cell metal foams are used to determine the effective properties of the porous nanocomposite. By using the first-order shear deformation plate theory (FSDT) to account for the transverse shear strain and Chebyshev-Ritz method to discretize the displacement fields, the governing equations are derived and then solved to calculate the critical uniaxial, biaxial and shear buckling loads and natural frequencies of the plates with different porosity distributions and GPL dispersion patterns. After a convergence and validation study to verify the present analysis, a comprehensive parametric investigation on the influences of the weight fraction and geometric parameters of GPL nanofiller and the porosity coefficient is conducted to identify the most effective way to achieve improved buckling and vibration resistances of the porous nanocomposite plate. Keywords: Porous nanocomposite plate; graphene platelet; elastic buckling; free vibration; ChebyshevRitz method. 1. Introduction Since the discovery in 1991, carbon nanotubes (CNTs) are widely used as the reinforcing nanofillers to develop high-strength nanocomposites owing to their exceptional mechanical properties and chemical stability [1-11]. However, a newly-developed carbon material – graphene nanoplatelets (GPLs) which are the two-dimensional counterparts of CNTs can ___________________________________________ *Corresponding author E-mail address: [email protected] (J. Yang).

1

provide even better reinforcement effects when dispersed at a low concentration, as the higher surface-volume ratio of GPLs boosts the load-carrying capacity with stronger bonding between the matrix and nanofillers [12-16]. Recent studies by Yang and his co-workers [17-19] on a novel class of functionally graded nanocomposites with non-uniform dispersions of GPLs have led to an explosion of active research into their advanced structural properties. Song et al. [20-22] presented a systematic study regarding the bending, buckling and vibration analyses of FG graphene reinforced polymer plates based on FSDT and Navier solution. Wu et al. [23] examined the stability of FG nanocomposite plates under combined thermal and mechanical loadings by using the differential quadrature method. Shen et al. [24] investigated the nonlinear vibration response of nanocomposite laminated beams on elastic foundations subjected to thermal variations with uniform and FG piece-wise distributions of graphene reinforcements. Kiani and Mirzaei [25] conducted a thermal postbuckling analysis of graphene reinforced composite beams using von Kármán type nonlinearity and Ritz method. Gholami and Ansari [26] employed the third-order shear deformable theory to study the geometrically nonlinear forced vibration of nanocomposite rectangular plates under harmonic excitations. García-Macías et al. [14] compared the bending and free vibration behaviors of FG plates reinforced by GPLs and CNTs with the aid of the Mori-Tanaka micromechanics model and finite element formulation. Their results confirmed that the stiffening effect of GPLs is more superior than that of CNTs. Yang et al. [27] calculated the thermo-mechanical axisymmetric bending solutions of circular and annular plates in the framework of the three-dimensional elasticity theory. Along with these works reporting on the fascinating capacities of graphene reinforced nanocomposites, much research effort has also been devoted to the theoretical and experimental investigations of FG porous structures, which process novel mechanical properties as an advanced multifunctional ultralight structural form. Chen et al. [28] computed the elastic buckling loads and static bending deflections of FG open-cell steel beams according to Timoshenko beam theory. Later they [29, 30] extended their study to the linear and nonlinear free vibration, and forced vibration analyses of the similar beams. Grygorowicz and Magnucka-Blandzi [31] proposed the mathematical modelling of static and dynamic stability of an axially compressed sandwich beam with an FG metal foam core on the basis of the classical broken line hypothesis and Bubnov-Galerkin method. Shafiei et al. [32] investigated the size-dependent nonlinear vibration behavior of FG porous microbeams with the modified couple stress and Euler-Bernoulli theories, and evaluated the effects of uniform and non-uniform porosities. Jabbari and Tayebi [33] analytically calculated the timedependent electro-magneto-thermoelastic stresses of a fluid-saturated porous hollow sphere. Hangai et al. [34] used the X-ray computed tomography to present a non-destructive observation of the deformation of closed-cell FG aluminium foams during the compression tests. Koohbor and Kidane [35] manufactured discretely-layered graded foams and measured the influence of density gradation on the load-carrying and energy absorption capacities. Chen et al. [36] numerically examined the crushing progress of FG porous structures, and pointed out that a certain type of non-uniformly asymmetric porosities is able to evidently promote the energy absorption of metal foams under the high-velocity impacting. 2

Introducing non-uniformly or uniformly dispersed GPLs into the matrix of porous structures results in the FG porous nanocomposites, making promise for developing ultralight high-strength structures with supreme static stiffnesses and dynamic responses, as reported by recent works [37-39]. Despite the presented exciting results, the number of studies focused on this new structural form is still rather scarce. A large amount of corresponding research works need to be carried out to fully reveal the structural properties of FG porous nanocomposites before these novel structures can be actually employed in engineering applications. This paper conducts the uniaxial, biaxial, shear buckling and free vibration analyses on the rectangular nanocomposite plates, comprising non-uniform porosity distributions and graded GPL dispersion patterns simultaneously. The dimensionless buckling loads and natural frequencies are obtained within the framework of FSDT and ChebyshevRitz method. The effects of varying parameters of fully clamped (CCCC) and simply supported (SSSS) plates, concerning the porosity and nanofiller fractions as well as the plate and GPL geometries, are presented in both tabular and graphical forms to discuss the structural performance of FG porous nanocomposite plates under different loading conditions. 2. Porous nanocomposite plates Three distributions of internal pores inside of the proposed porous plates are illustrated in Fig. 1 with two types of non-uniformly symmetric FG porosities and a uniform porosity distribution. The largest-size pores are located on the mid-plane for non-uniform distribution 1 while on the top and bottom surfaces for distribution 2, leading to the corresponding variations of material properties given in Eqs. (1) and (2) for distributions 1 and 2, respectively. The constant material properties for the uniform porosity are described by using Eq. (3). Meanwhile, three GPL dispersion patterns regarding the varying nanofiller volume contents VGPL across the thickness are depicted in Fig. 2 and quantified in Eq. (15).

Max Material properties Min

Min

h/2

Material properties

-h / 2

-h / 2

x Max

h/2

z

(A) Porosity distribution 1 (symmetric I)

z

Max

x

Min

(B) Porosity distribution 2 (symmetric II)

3

Material properties Uniform

-h / 2

x

h/2

z

(C) Uniform porosity distribution Fig. 1. Different porosity distributions.

si1

-h / 2 0

-h / 2

z

si 3

-h / 2 si 2

VGPL si1

h/2

0

h/2 0 z

VGPL

VGPL

h/2 z

(A) GPL pattern A (B) GPL pattern B (C) GPL pattern C (symmetric I) (symmetric II) (uniform) Fig. 2. Different GPL dispersion patterns.

Non-uniform porosity distribution 1 (symmetric I):  E  z   E * 1  e0 cos  z / h     * G  z   G 1  e0 cos  z / h    *    z    1  em cos  z / h   

(1)

Non-uniform porosity distribution 2 (symmetric II):  E  z   E * 1  e0* 1  cos  z / h        * * G  z   G 1  e0 1  cos  z / h     * *       z    1  em 1  cos  z / h   

(2)

Uniform porosity distribution:  E  z   E *  * G  z   G   *    z     '

(3)

where E  z  , G  z  and   z  are Young’s modules, shear modules and mass density of porous plates, E * , G * and  * are the corresponding properties of graphene reinforced 4

nanocomposites without internal pores, e0 and e0* ( 0  e0  e0*   1 ) are the porosity coefficients for distributions 1 and 2, respectively, em and em* are the corresponding coefficients of mass densities,  and  ' are the parameters for the uniform porosity. It should be noted that the proposed porosity coefficient is introduced to reflect the effects of varying porosities. The growing size and density of internal pores increase the porosity coefficient, leading to the decreasing material properties, suggesting a weakening effect from the porosities. The varying value of porosity coefficient represents the general changing trend of internal pore size and density in metal foams. Young’s modulus E * of the non-porous nanocomposite is determined based on Halpin-Tsai micromechanics model [12, 40-42] as

3  1   LGPLLGPLVGPL  5  1  WGPLWGPLVGPL  E*   E   m   Em 8  1   LGPLVGPL  8  1  WGPLVGPL 

(4)

where

 LGPL 

2lGPL tGPL

(5)

WGPL 

2wGPL tGPL

(6)

 LGPL 

EGPL  Em EGPL   LGPL Em

(7)

WGPL 

EGPL  Em EGPL  WGPL Em

(8)

of which EGPL and Em are Young’s moduli of GPLs and the nanocomposite matrix which is assumed to be metal in this paper, lGPL , wGPL and tGPL are the average dimensions of nanofiller platelets, i.e., length, width and thickness, and VGPL is the GPL volume content. The extended rule of mixture [24, 43] is adopted to obtain the mass density and Poisson’s ratio of the nanocomposite:

 *  GPLVGPL  m 1  VGPL 

(9)

v*  vGPLVGPL  vm 1  VGPL 

(10)

where GPL and vGPL are the mass density and Poisson’s ratio of GPLs, while  m and vm are the corresponding parameters of the metal matrix. It should be noted that the value of Poisson’s ratio is fixed for open-cell metal foams [44-47]. The shear modulus G * of the nanocomposite is calculated by

5

G* 

E* 2 1  v* 

(11)

The typical mechanical property of open-cell metal foams [44, 45, 48, 49], shown in Eq. (12), is employed to establish the relationships in Eq. (13) between the mass density coefficients and porosity coefficients for different porosity distributions. E  z   z   *  E*   

2

(12)

1  em cos  z / h   1  e0 cos  z / h    * * 1  em 1  cos  z / h    1  e0 1  cos  z / h     '   

Porosity distribution 1 Porosity distribution 2

(13)

Uniform porosity distribtuion

The masses of all porous nanocomposite plates with varying porosities and GPL dispersions are set to be equivalent, resulting in h /2  h /2 *  1  e0 1  cos  z / h  dz = 1  e0 cos  z / h dz  0 0  h /2 h /2   dz = 1  e0 cos  z / h dz   0 0

 



(14)



which can be used to determine e0* and  with a given value of e0 , as tabulated in Table 1. It is seen that e0* rises dramatically with the increasing of e0 . When e0 reaches 0.6, e0* (  0.9612 ) is close to the upper limit. Therefore, the selected range of e0 ( e0  0, 0.6 ) is applied in the following numerical calculations. Table 1 Porosity coefficients for different distributions. e0

e0*



0.1 0.2 0.3 0.4 0.5 0.6

0.1738 0.3442 0.5103 0.6708 0.8231 0.9612

0.9361 0.8716 0.8064 0.7404 0.6733 0.6047

Due to the limitation of the current manufacture technology, to the best of our knowledge, the continuous variations of porosities and nanofiller contents along certain directions cannot be achieved simultaneously. Hence, a multilayer plate model (length a, width b, thickness h) 6

is adopted in this paper to compute the material properties of the proposed FG porous nanocomposite plates, and is defined in a Cartesian coordinate system (x, y, z) as depicted in Fig. 3. The origin point of this coordinate system is located in the centre of the plate with x-, y- and z-axes parallel to the length, width and thickness directions, respectively. Each layer is of the same thickness with the uniform distributions of both internal pores and GPLs. The optimal total layer number n is discussed and determined based on a convergence analysis in section 4.1. a b

h

Layer number n n-1 … 2 1 y

x

z Fig. 3. Configuration of a multilayer plate model.

The GPL volume content VGPL varies along the thickness direction according to the following equations for different dispersion patterns.

 si1 1  cos  z / h    VGPL  z    si 2 cos  z / h  s  i 3

GPL Pattern A GPL Pattern B

(15)

GPL Pattern C

where si1 , si 2 and si 3 are the maximum values of the volume content, and i  1, 2, 3 correspond to the porosity distributions 1, 2 and the uniform distribution, respectively. The T total GPL volume content VGPL is calculated from the nanofiller weight fraction GPL in Eq.

(16), and then is used to determine si1 , si 2 and si 3 by Eq. (17) with the aid of the multilayer plate model. T VGPL 

GPL m GPL m  GPL  GPL GPL

(16)

7

 n

T GPL

V

j 1

  si1   (z j )    si 2 *   s  i3 

 n

j 1

(z j )    1  cos  z j / h    *   

 cos  z / h  n

j

j 1

 n

j 1

(z j )   * 

(17)

(z j ) *

in which 1 1 j zj      h , j  1, 2,3,..., n  2 2n n 

(18)

3. Formulations 3.1 Total energy functionals This paper employs FSDT with the following displacement field for the multilayer model illustrated in Fig. 3.

u  x, y, z , t   u0  x, y, t   zx  x, y, t   v  x, y, z, t   v0  x, y, t   z y  x, y, t    w  x, y, z, t   w0  x, y, t 

(19)

where u, v, w represent the displacements along x-, y-, z- axes, u0 , v0 , w0 are the corresponding mid-plane displacement components,  x ,  y are the slope rotations in the x-z and y-z planes, and t denotes the time. The linear strain-displacement relationship for the inplane and transverse strains can be expressed as   xx    yy    xy    yz    xz 

u0  z x x x  v  0 z y y y



    u0 v0   z x  y  y x x   y w  y  0 y w  x  0 x 

(20)

8

where  xx ,  yy ,  xy  are the in-plane strains and  yz ,  xz  are the average transverse strains, which are used to determine the stress distributions given by  xx  Q11 xx  Q12 yy   yy  Q21 xx  Q22 yy   xy  Q66 xy   yz  Q44 yz   Q  55 xz  xz

(21)

in which  xx ,  yy ,  xy  and  yz ,  xz  are the in-plane and transverse stresses, respectively, and Q11  Q22  E  z  / 1  v*2  , Q12  Q21  v* E  z  / 1  v*2  , Q66  Q44  Q55  G  z  . Thus, the strain energy of the porous nanocomposite plate is taken as

             u v  1   A A B B   2    x y x

U 

1 2

0.5b

0.5 a

0.5 h

0.5b

0.5 a

0.5 h

0.5b

0.5 a

xx xx

yy

0

0.5b

0.5 a

yy

xy

0

11

xy

yz

x

21

11

21

 yz   xz xz dzdxdy  y  u0  u v    A12 0  A22 0  B12 x  y  x  x y x

 y  v0   u0 v0    y x    u0 v0   u0 v   A66      B21 0 (22)      B66     B11 y  y   y x  y    y x   x y  x          u  u v   D11 x  D21 y  x   B12 0  B22 0  D12 x  D22 y  y   B66  0 x y  x  x y x y  y   y  B22

   v   0   D66  y  x x  y  x

    y x    y    x

2

 w0  w0  s  s    A44   y    A55  x   y  x    

2

  dxdy 

where the plate stiffness elements are calculated by n  h Q11  z j  , Q12  z j  , Q22  z j  , Q66  z j   A11 , A12 , A22 , A66   n j 1   n h  Q11  z j  , Q12  z j  , Q22  z j  , Q66  z j  z j  B11 , B12 , B22 , B66   n  j 1  n  D , D , D , D   h Q11  z j  , Q12  z j  , Q22  z j  , Q66  z j  z 2j  11 12 22 66 n j 1 

(23)

 A21, B21, D21   A12 , B12 , D12 

(24)

 A , A   hn s 44

s 55

 k Q n

44













 z  , Q  z  j

55

(25)

j

j 1

9

of which k   5 / 6  denotes the shear correction factor and z j is given in Eq. (18). The potential energy due to the in-plane normal ( N x and N y ) and shear ( N xy ) loadings is shown as follows

1 V  2

2  w 2   w0  w0 w0 0 N  2 N  N  xy   y dxdy  x  x  y  y 0.5 a  x     

  0.5b

0.5b

0.5 a

(26)

The corresponding kinetic energy based on FSDT can be obtained as

1 K 2 1  2

 u 2  v 2  w  2    z          dzdxdy 0.5 h  t   t   t  

   0.5b

0.5 a

0.5b

0.5 a

2 2   u0 2 u0 x v0  y  x   v0   I2   I0    2 I1   I0    2 I1 t t t t 0.5 a   t   t    t 

  0.5b

0.5b

0.5 h

0.5 a

(27)

2   y   w0    I2    I0    dxdy  t    t  2

where the inertias are given by

h I 0 , I1 , I 2   n

   z 1, z , z  n

j

j

2 j

(28)

j 1

In terms of the following dimensionless quantities

 ,    

x , a

A110

a a b y  u0 v0 w0   , u1 , v1 , w1   , ,  , x , y   x ,  y  , 1  , 2  , 3  , b h h b h h h 

 A11 A12 A22 A66   A44s A55s  Em h s s , a , a , a , a    , , , ,   , a , a    , 1  vm2  11 12 22 66  A110 A110 A110 A110  44 55  A110 A110  N N   B11 N B  B B , 12 , 22 , 66  ,  px , p y , pxy    x , y , xy  ,  A110 h A110 h A110 h A110h   A110 A110 A110 

b11 , b12 , b22 , b66   

 D11 D66  D D22 , , 12 2 , , 2 2 2  A110 h A110 h A110 h A110 h 

d11 , d12 , d22 , d66   

I I I  I10  m h , 0 , 1 , 2    0 , 1 , 2 2  ,   A110 h2  I10 I10 h I10 h 

(29)

the strain energy and potential energy of static plates can be rewritten in dimensionless forms as

10

U 1 U     2 *

2 2  a  u  2  v1   u1  u1 v1 u1 v1 11 1  1a22      2a12   1a66    2a66     0.5  1          

  0.5

0.5

0.5

a  v  b u  u  v  v  u   66  1   2 11 1 x  2b12 1 y  2b12 1 x  21b22 1 y  2b66 1 y 1    1           2

   b v    y u  v  d    21b66 1 x  2 66 1 y  2b66 1 x  11  x   2d12 x  1d 22  y  (30)   1     1         2

2

 x d       w s s s  w1   66  y   2d 66 y  1d 66  x   23a44  y2  22 a44  y 1  1a44   1             2 s  w1   w1 a55 s 2 s 23 a55x  23 a55x     d d  1     2

V 1 V    2 *

2

2  p  w 2  w1   w1 w1 x 1  1 p y      2 pxy  d d   0.5  1        

  0.5

0.5

2

0.5

(31)

The dimensionless displacement and rotation quantities for the harmonic vibration response can be further considered as

  u0 v0 w0  i u1 , v1 , w1 e   h , h , h      ,   ei   ,   x y  x y

(32)

of which

i  1 ,   a

I10 t , A110 a

A110 I10

(33)

Therefore, the corresponding dimensionless strain and kinetic energies for plates undergoing free harmonic vibrations are U*  e2i U*

K* 

K 2  e2i  1

(34)

1 2 1 2 1 2 1 2 1 2  0u1  1u1x  2x  0v1  1v1y  2y  0 w1  d d (35) 2 2 2 2 0.5  2 

  0.5

0.5

0.5

The total energy functional  is computed with the sum of strain and potential energies for the elastic buckling analysis, while the sum of strain and kinetic energies for the free vibration analysis. In this way, we can obtain the following dimensionless forms of total energy functionals for both buckling and vibration analyses. * 

* *   U   V  * *   U   K

For the elastic buckling analysis For the free vibration analysis

11

(36)

3.2 Chebyshev-Ritz method The dimensionless displacement and rotation quantities are discretised with the Chebyshev polynomials and multiplied by auxiliary functions, which can be expressed as Tx  u u1  R  ,    i 1  Tx  v v1  R  ,   i 1   Tx  w  w1  R  ,    i 1  Tx  x x  R  ,   i 1  Tx    R y  ,    y i 1 

Ty

        u ij

i

j

j 1

Ty

        v ij

i

j

j 1

Ty



w ij

 i    j   

(37)

j 1

Ty

        x ij

i

j

j 1 Ty



y ij

 i    j   

j 1

where R  ,   ,   u , v, w, x, y, are the auxiliary functions, Tx and Ty are the total numbers of Chebyshev polynomial items in length and width directions, respectively,  iju ,  ijv ,

 ijw ,  ijx and  ijy are the unknown coefficients corresponding to different quantities ( 1  i  Tx , 1  j  Ty ), i   and  j    are the one-dimensional Chebyshev polynomials given by

 i    cos   i  1 arccos  2      j     cos   j  1 arccos  2  

(38)

The auxiliary function is written in the following form with exponents determined based on different boundary conditions and loading patterns for the buckling and vibration analyses, as tabulated in table 2. Two typical types of boundary conditions with CCCC and SSSS edges are examined in this paper. R  ,    1  2 

a

1  2  1  2  1  2  b

c

d

12

(39)

Table 2 Exponents in the auxiliary function for the buckling and vibration analyses of CCCC and SSSS plates. au, bu cu, du av, bv cv, dv aw, bw cw, dw ax, bx cx, dx ay, by cy, dy 0

0

0

1

0

0

1

1

0

1

Uniaxial, biaxial and shear buckling analyses (CCCC) 0 0 1 1 1 1 Uniaxial and biaxial buckling analyses (SSSS) 1 0 1 1 0 1 Shear buckling analysis (SSSS) 0 0 1 1 0 0 Free vibration analysis (CCCC) 1 1 1 1 1 1 Free vibration analysis (SSSS) 1 0 1 1 0 1

1

1

1

0

0

0

1

1

1

0

It should be mentioned that in Table 2, ‘0’ or ‘1’ denotes that the corresponding edge displacement is unrestrained or restrained, respectively. For instance, au  bu  0 and cu  du  1 indicate that the in-plane dimensionless displacement u1 is restrained on the edges of   0.5 while unstrained on the edges of   0.5 . The governing equations for the buckling and vibration analyses can be derived by substituting Eq. (37) into Eq. (36) then minimizing the total energy functionals with respect to  iju ,  ijv ,  ijw ,  ijx and  ijy

 *  *  *  *  *  0 , , , ,  0  0  0 0  iju  ijv  ijw  ijx  ijy

(40)

The following equations governing the uniaxial, biaxial and shear buckling responses of the proposed plates can be obtained according to the foregoing formulations.

KL  p V d = 0 ,

p y  pxy  0

KL  p V d = 0 ,

pcr  px  p y , pxy  0

KL  p V d = 0 ,

px  p y  0

x

cr

xy

(uniaxial buckling)

(41)

(biaxial buckling)

(42)

(shear buckling)

(43)

where  KL  and  V  are the stiffness and geometric matrices, and d is the unknown



coefficient vector specified as d   iju   ijv   ijw   ijx   ijy  T

T

T

T



T T

. The governing

equation for the free vibration analysis is expressed as

KL   M d = 0 2

(44)

in which  M  denotes the mass matrix. Eqs. (41)-(44) are the typical eigenvalue problems with the dimensionless critical buckling loads and fundamental natural frequency as the lowest eigenvalues. 13

4. Results and discussions 4.1 Convergence and validation analyses The computational efficiency and accuracy are ensured by adopting the suitable values of the total numbers ( Tx and Ty ) of Chebyshev polynomial items in the trial functions, of which the effects are examined in Table 3 regarding the biaxial buckling factors of simply supported FG rectangular plates with homogeneous porosities and the following material properties. 1   z 1 P( z )   PT  PB      PB   PT  PB  1 2 h 2

k

(45)

where P( z ) denotes the varying parameters along the thickness direction, including Young’s modulus E ( z ) , shear modulus G( z ) and Poisson’s ratio v( z ) , ET (= 151 GPa) and EB (= 210 GPa) correspond to Young’s moduli of Zirconia and steel as the top-surface and bottomsurface materials, vT  vB  0.3 , k1 and 1 represent the material gradation parameter and the porosity volume, respectively. The plate dimensions are set as a  b  3 m and h  0.2 m. The biaxial buckling factor is

N cr  N x

a2 a2  N y EB h3 EB h3

(46)

As can be observed, the convergent biaxial buckling factors for different material gradations and porosity volumes can be obtained with the increasing total numbers of Chebyshev polynomial items. Our results agree well with those calculated by Akbaş [50] using the classical plate theory and generalized differential quadrature method when

Tx  Ty  8 which is adopted in this study. Table 3

Biaxial buckling factors of simply supported FG plates with homogeneous porosities. 1

0

0.05

0.1

k1

0.5 2 10 0.5 2 10 0.5 2 10

Tx  Ty

4 1.4430 1.5540 1.6657 1.3485 1.4579 1.5686 1.2567 1.3645 1.4743

5 1.4430 1.5540 1.6657 1.3485 1.4579 1.5686 1.2567 1.3645 1.4743

6 1.4426 1.5535 1.6652 1.3482 1.4575 1.5682 1.2564 1.3641 1.4739

14

7 1.4426 1.5535 1.6652 1.3482 1.4574 1.5682 1.2564 1.3641 1.4739

8 1.4426 1.5535 1.6652 1.3482 1.4574 1.5682 1.2564 1.3641 1.4739

Ref. [50] Fig. 3 1.4419 1.5508 1.6648 1.3652 1.4762 1.5882 1.2885 1.4005 1.5125

The normal buckling and free vibration analyses of CNT-reinforced polymer-matrix nanocomposite plates are conducted to verify the correctness of the presented formulations. The extended rule of mixture is used to calculate the mechanical properties as

 E11  1*VCNT E11CNT  Vm E m  * CNT m 2 / E22  VCNT / E22  Vm / E  * CNT m 3 / G12  VCNT / G12  Vm / G

(47)

CNT of which E11CNT , E22 and G12CNT are Young’s moduli and shear modulus of single-walled

CNTs, E m and G m are the corresponding moduli of the polymer, 1* ,  2* and 3* are the CNT efficiency parameters, VCNT and Vm (  1  VCNT ) are the volume contents of CNTs and matrix materials, respectively. Uniform and FG dispersions of CNTs are employed here with the following variations of nanofiller volume contents. * VCNT  VCNT  * VCNT  z   1  2 z / h  VCNT  * VCNT  z   2 1  2 z / h VCNT  * VCNT  z    4 z / h VCNT

UD  uniform dispersion  FG-V  FG dispersion  FG-O  FG dispersion 

(48)

FG-X  FG dispersion 

* where VCNT is the total volume content of CNTs. Poisson’s ratio and the mass density of the

nanocomposite plate are expressed as * v12  VCNT v12CNT  Vm v m   CNT  Vm  m     VCNT 

(49)

of which v12CNT and  CNT are Poisson’s ratio and the mass density of CNTs, respectively, and v m and  m are those of the polymer. The revised transverse shear correction factor is equal

to k

6  V

5

* CNT CNT 12

v

(50)

 Vm v m 

Tables 4-6 present the dimensionless uniaxial and biaxial buckling loads, and the fundamental and higher-order natural frequencies of CNT-reinforced nanocomposite plates with different boundary conditions, based on the following material constants: E11CNT  5.6466 CNT TPa, E22  7.0800 TPa, G12CNT  1.9445 TPa, v12CNT  0.175 ,  CNT  1400 kg/m3, E m  2.1

* GPa,  m  1150 kg/m3, v m  0.34 , 1*  0.149 and 2*  3*  0.934 when VCNT  0.11 , * 1*  0.150 and 2*  3*  0.941 when VCNT  0.14 , 1*  0.149 and 2*  3*  1.381 when

15

* VCNT  0.17 , and G12  G23  G13 . It should be noted that v12 is fixed in the vibration analysis.

The dimensionless buckling load and vibration frequency in this example can be written as

pcr  N x

b2 b2  N y E m h3 E m h3

(51)

 m  a2  = m     h

(52)

Our results are compared to the normal buckling loads calculated by Zhang et al. [51] with an element-free based improved moving least squares-Ritz method, as well as the fundamental natural frequencies given by Zhu et al. [49] using finite element method and the higher-order frequencies presented by Lei et al. [50] employing the element-free kp-Ritz method. It can be seen that excellent agreements are achieved in both normal buckling and free vibration analyses.

Table 4 Dimensionless uniaxial and biaxial buckling loads of CNT-reinforced nanocomposite plates (SSSS plate, a / b  1 , b / h  100 ). * VCNT  0.11

CNT pattern

Present

Ref. [51]

UD FG-O FG-X

39.3633 21.4830 57.1048

39.1158 21.3316 56.7373

UD FG-O FG-X

11.5585 7.8264 14.6022

11.4103 7.7153 14.2345

* VCNT  0.14

Present Ref. [51] Uniaxial buckling 49.4162 49.0816 26.5753 26.3572 72.0431 71.5516 Biaxial buckling 13.6454 13.7293 8.9275 8.7948 16.6659 16.2428

16

* VCNT  0.17

Present

Ref. [51]

60.4855 32.7296 88.1271

57.4776 31.2163 83.6142

17.9249 11.9614 23.5661

17.7656 12.0667 23.6906

Table 5 Dimensionless fundamental natural frequencies of square CNT-reinforced nanocomposite plates. UD FG-V FG-O FG-X * VCNT

0.11

0.14

0.17

0.11

0.14

0.17

b/h

Present

[52]

Present

10 20 50 10 20 50 10 20 50

13.693 17.404 19.175 14.507 19.020 21.344 17.000 21.505 23.631

13.532 17.355 19.223 14.306 18.921 21.354 16.815 21.456 23.697

12.563 15.125 16.206 13.403 16.564 17.978 15.583 18.646 19.919

10 20 50 10 20 50 10 20 50

17.967 28.744 39.663 18.495 30.341 43.623 22.417 35.711 48.965

17.625 28.400 39.730 18.127 29.911 43.583 22.011 35.316 49.074

17.510 26.531 34.046 18.121 28.235 37.526 21.893 32.932 41.909

[52]

Present

SSSS plate 12.452 11.629 15.110 13.522 16.252 14.259 13.526 12.451 16.510 14.815 17.995 15.782 15.461 14.368 18.638 16.620 19.982 17.486 CCCC plate 17.211 16.971 26.304 24.637 34.165 30.167 17.791 17.608 27.926 26.354 37.568 33.293 21.544 21.139 32.686 30.482 42.078 37.063

[52]

Present

[52]

11.550 13.523 14.302 12.338 14.784 15.801 14.282 16.628 17.544

14.832 20.039 22.939 15.623 21.806 25.561 18.527 24.870 28.348

14.616 19.939 22.984 15.368 21.642 25.555 18.278 24.764 28.413

16.707 24.486 30.303 17.311 26.127 33.369 20.833 30.325 37.247

18.456 30.884 46.195 18.987 32.401 50.572 23.189 38.596 57.240

18.083 30.421 46.166 18.593 31.857 50.403 22.748 38.062 57.245

Table 6 Dimensionless higher-order natural frequencies of square CNT-reinforced nanocomposite plates * ( VCNT  0.11 , b / h  10 ).

Mode

UD Present

[53]

1 2 3 4 5 6

13.693 17.794 19.427 19.427 27.336 33.127

13.495 17.629 19.399 19.404 27.307 32.466

1 2 3 4 5 6

17.967 23.232 33.353 34.372 37.274 37.625

17.587 22.933 33.170 33.612 36.905 37.238

FG-V FG-O Present [53] Present [53] SSSS plate 12.563 12.416 11.629 11.514 17.107 16.984 16.282 16.187 19.476 19.448 19.477 19.449 19.477 19.452 19.477 19.454 27.067 27.069 26.217 26.240 31.879 31.309 30.662 30.163 CCCC plate 17.510 17.171 16.971 16.667 22.971 22.704 22.377 22.138 33.282 32.939 32.560 32.237 33.650 33.121 32.908 32.424 37.080 36.405 36.313 35.674 37.394 37.357 37.404 37.367

17

FG-X Present [53] 14.832 18.789 19.477 19.477 28.323 34.240

14.578 18.579 19.449 19.454 28.261 33.510

18.456 23.822 34.122 35.156 37.404 38.444

18.045 23.498 33.915 34.361 37.367 37.693

In another effort, the shear buckling behavior of CNT-reinforced nanocomposite plates was investigated by Kiani [54] with another set of material parameters: 1*  0.137 and * *  0.12 , 1*  0.142 and 2*  1.626 when VCNT  0.17 , 1*  0.141 and 2*  1.022 when VCNT

* 2*  1.585 when VCNT  0.28 , 3*  0.72* , E m  2.5 GPa, and G23  1.2G12  1.2G13 . The

dimensionless shear buckling load takes the form as

pxy  N xy

b2 D m 2

(53)

where Dm 

E m h3

(54)

2 12 1   v m    

According to the comparisons given in Table 7, it is observed that the shear buckling loads obtained based on our method show a good agreement with those listed by Kiani [54]. Table 8 examines the effect of the total layer number n on the buckling and vibration responses of porous nanocomposite plates to determine its optimal value, which is important to ensure the excellent simulation accuracy and economical manufacturing efficiency simultaneously. It is found that the results converge monotonically with the increasing of the total layer number, and the difference between the results of n  12 and n  10000 is less than 2.0 %. Therefore, n  12 is applied in the following calculations. Table 7 Dimensionless shear buckling loads of CNT-reinforced nanocomposite plates * ( VCNT  0.28 , SSSS plate, FG-X).

b/h 100 50 20 10 b/h 100 50 20 10 b/h 100 50 20 10

a / b  1.0 Present Ref. [54] 168.3787 167.7654 146.5772 145.2256 82.7024 80.7348 37.8722 36.5894 a / b  1.5 Present Ref. [54] 86.2573 86.2611 81.0320 80.7696 57.5610 56.6194 30.5247 29.6973 a / b  3.0 Present Ref. [54] 31.3900 31.4335 30.9409 30.9547 28.3232 28.2052 22.5143 22.2321

18

a / b  1.2 Present Ref. [54] 120.0230 119.8693 108.8934 108.2733 70.0288 68.6953 35.1230 34.0093 a / b  2.0 Present Ref. [54] 50.7663 50.8047 48.8678 48.8065 39.5024 39.1084 25.6054 25.0887

Table 8 Dimensionless buckling loads and natural frequencies of GPL-reinforced porous nanocomposite plates: effect of total layer number (porosity distribution 1, GPL pattern A, e0  0.5 , a / b  1 , a / h  20 ).

 GPL

1.0 wt.%

0.0 wt.%

n 4 6 8 10 12 14 10000 12 10000

Uniaxial buckling CCCC SSSS 0.02548 0.01063 0.02767 0.01159 0.02844 0.01192 0.02879 0.01208 0.02899 0.01217 0.02910 0.01222 0.02943 0.01236 0.01571 0.00654 0.01588 0.00662

Biaxial buckling CCCC SSSS 0.01360 0.00531 0.01479 0.00579 0.01520 0.00596 0.01539 0.00604 0.01550 0.00608 0.01556 0.00611 0.01574 0.00618 0.00838 0.00327 0.00847 0.00331

Shear buckling CCCC SSSS 0.03522 0.02309 0.03812 0.02506 0.03913 0.02575 0.03960 0.02607 0.03985 0.02625 0.04001 0.02635 0.04043 0.02664 0.02174 0.01426 0.02198 0.01442

Free vibration CCCC SSSS 0.65977 0.37080 0.68664 0.38661 0.69577 0.39200 0.69994 0.39447 0.70220 0.39580 0.70355 0.39660 0.70728 0.39881 0.49853 0.27997 0.50118 0.28152

4.2 Uniaxial, biaxial and shear buckling analyses In what follows, a detailed parameter study is conducted to reveal the buckling and vibration performance of GPL-reinforced porous metal-matrix nanocomposite plates with the material properties and GPL dimensions as Em  130 GPa, m  8960 kg/m3, vm  0.34 for copper [42, 55, 56], and EGPL  1.01 TPa, GPL  1062.5 kg/m3, vGPL  0.186 , wGPL  1.5 μm,

lGPL  2.5 μm, tGPL  1.5 nm for GPLs [12, 57]. And the plate thickness h is set as 0.1 m. The effect of porosity coefficient e0 is illustrated in Fig. 4 concerning the buckling behavior of porous nanocomposite plates with different porosity distributions and GPL patterns. Results show that an increase in the porosity coefficient leads to the decreasing of uniaxial, biaxial and shear buckling loads of porous plates, as the larger size and higher density of internal pores decrease the plate stiffness. It can also be seen that the highest critical buckling loads, i.e., the stiffest plates, are obtained with the combination of the nonuniformly symmetric porosity distribution 1 and GPL pattern A. Thus, dispersing nanofillers with larger volume contents around the surface areas, as well as reducing the size and density of internal pores on the surfaces, results in the higher structural stiffness of porous plates. Non-uniformly symmetric porosity distribution 2 and GPL pattern B introduce less pores and more nanofillers on the mid-plane, leading to even lower plate stiffness compared to that of the nanocomposite plate with uniformly distributed internal pores and GPLs.

19

0.020

Porosity 1 (symmetric I) Porosity 2 (symmetric II) Uniform porosity

GPL A (symmetric I)

Dimensionless Biaxial buckling load

Dimensionless uniaxial buckling load

0.04

0.03

GPL C (uniform)

0.02

GPL B (symmetric II) a / b = 1 a / h = 20GPL = 1.0 wt.%

0.01 0.0

0.1

0.2

0.3

CCCC plate 0.4

0.5

0.6

GPL A (symmetric I)

Porosity 1 (symmetric I) Porosity 2 (symmetric II) Uniform porosity

0.015

GPL C (uniform)

0.010

GPL B (symmetric II) a / b = 1 a / h = 20GPL = 1.0 wt.% 0.005 0.0 0.1 0.2 0.3

CCCC plate 0.4

0.5

0.6

Porosity coefficient e0

Porosity coefficient e0

(A) Uniaxial buckling

(B) Biaxial buckling

Dimensionless shear buckling load

0.06 Porosity 1 (symmetric I) Porosity 2 (symmetric II) Uniform porosity

GPL A (symmetric I) 0.05

0.04

0.03

0.02

GPL C (uniform) GPL B (symmetric II)

a / b = 1 a / h = 20GPL = 1.0 wt.% 0.01 0.0 0.1 0.2 0.3

CCCC plate 0.4

0.5

0.6

Porosity coefficient e0

(C) Shear buckling Fig. 4. Dimensionless buckling loads of GPL-reinforced porous nanocomposite plates: effect of porosity coefficient ( a / b  1 , a / h  20 , GPL  1.0 wt.%, CCCC plate).

Fig. 5 and Fig. 6 examine the effect of GPL weight fraction GPL on the porous nanocomposite plates with non-uniform porosity distributions 1 and 2, respectively. It is observed that GPLs present remarkable reinforcement effects, as evidenced by the monotonous increase of buckling loads with the addition of the nanofiller weight fraction. For CCCC plates with porosity distribution 1 and GPL pattern A, the increments of uniaxial, biaxial and shear buckling loads are up to 84.7%, 84.5% and 83.9% with only 1.0 wt.% of GPLs dispersed in the metal matrix. The reinforcement effect varies according to the employed dispersion patterns under the same weight fraction of nanofillers. Non-uniformly symmetric GPL pattern A is the most preferred to achieve the best enhancement, followed by the uniform pattern C. While the non-uniformly symmetric pattern B provides the least pronounced improvement of the buckling resistance. In addition to the observations from these two figures, it is discerned that the value of the shear buckling load is the maximum compared to those of normal buckling loads for the same porous plate, of which the uniaxial buckling load is larger than the biaxial one. This is due to the fact that rectangular plates subjected to the biaxial compression are easier to buckle than those under the uniaxial 20

compression, whereas for plates with shear loads, it is the most difficult to induce the structural instability.

0.04

0.03

0.03

GPL A (symmetric I) GPL B (symmetric II) GPL C (uniform)

Porosity 1 (symmetric I) a / b = 1 a / h = 20 e0 = 0.5 CCCC plate

GPL A (symmetric I) Porosity 1 (symmetric I) GPL B (symmetric II) a / b = 1 a / h = 20 e0 = 0.5 SSSS plate GPL C (uniform)

Dimensionless buckling load

Dimensionless buckling load

0.05

Shear buckling

0.02 Uniaxial buckling 0.01

0.02

Shear buckling

0.01

Uniaxial buckling

Biaxial buckling 0.00 0.0

0.2

Biaxial buckling

0.4

0.6

0.8

0.00 0.0

1.0

0.2

GPL weight fraction (wt.%)

0.4

0.6

0.8

1.0

GPL weight fraction (wt.%)

(A) CCCC plate (B) SSSS plate Fig. 5. Dimensionless buckling loads of GPL-reinforced porous nanocomposite plates: effect of GPL weight fraction (porosity distribution 1, a / b  1 , a / h  20 , e0  0.5 ).

0.020 GPL A (symmetric I) GPL B (symmetric II) GPL C (uniform)

Porosity 2 (symmetric II) a / b = 1 a / h = 20 e0 = 0.5 CCCC plate

Dimensionless buckling load

Dimensionless buckling load

0.03

0.02 Shear buckling

0.01 Uniaxial buckling

GPL A (symmetric I) GPL B (symmetric II) GPL C (uniform) 0.015 Shear buckling 0.010

Uniaxial buckling 0.005

Biaxial buckling 0.00 0.0

0.2

Porosity 2 (symmetric II) a / b = 1 a / h = 20 e0 = 0.5 SSSS plate

Biaxial buckling 0.4

0.6

0.8

1.0

0.000 0.0

0.2

0.4

0.6

0.8

1.0

GPL weight fraction (wt.%)

GPL weight fraction (wt.%)

(A) CCCC plate (B) SSSS plate Fig. 6. Dimensionless buckling loads of GPL-reinforced porous nanocomposite plates: effect of GPL weight fraction (porosity distribution 2, a / b  1 , a / h  20 , e0  0.5 ).

Table 9 tabulates the dimensionless buckling loads of porous nanocomposite plates and highlights the effect of thickness ratio a / h . As expected, an dramatic decrease of normal and shear buckling loads can be found with an increase in the thickness ratio, indicating a sharp drop of the plate stiffness. It is also noted that compared to the simply-supported porous plates, those with full-clamped edges are stiffer and possess higher flexural rigidities.

21

Table 9 Dimensionless buckling loads of GPL-reinforced porous nanocomposite plates: effect of thickness

ratio ( a / b  1 , GPL  1.0 wt.%, e0  0.5 ). Porosity distribution 1 GPL A GPL B GPL C a/h 20 30 40 50

0.02899 0.01343 0.00767 0.00494

0.02037 0.00933 0.00531 0.00341

0.02384 0.01098 0.00625 0.00403

0.01217 0.00547 0.00309 0.00198

0.00842 0.00377 0.00213 0.00136

0.00992 0.00445 0.00251 0.00161

a/h 20 30 40 50

0.01550 0.00712 0.00405 0.00261

0.01084 0.00494 0.00280 0.00180

0.01271 0.00581 0.00330 0.00212

0.00608 0.00273 0.00154 0.00099

0.00421 0.00189 0.00106 0.00068

0.00496 0.00223 0.00126 0.00081

0.02452 0.01130 0.00644 0.00415

0.01718 0.00783 0.00445 0.00286

0.01990 0.00911 0.00518 0.00333

0.00734 0.00329 0.00185 0.00119

0.00526 0.00235 0.00132 0.00085

0.00590 0.00264 0.00149 0.00095

0.01022 0.00459 0.00259 0.00166

0.00706 0.00316 0.00178 0.00114

0.00822 0.00368 0.00208 0.00133

0.00948 0.00431 0.00244 0.00157

0.00686 0.00309 0.00175 0.00112

0.00767 0.00347 0.00196 0.00126

0.01308 0.00599 0.00340 0.00219

0.00913 0.00414 0.00235 0.00151

0.01059 0.00482 0.00273 0.00176

0.00367 0.00164 0.00093 0.00059

0.00263 0.00118 0.00066 0.00042

0.00295 0.00132 0.00074 0.00048

0.00511 0.00229 0.00129 0.00083

0.00353 0.00158 0.00089 0.00057

0.00411 0.00184 0.00104 0.00067

0.03391 0.01604 0.00924 0.00598

0.02410 0.01120 0.00640 0.00413

0.02776 0.01299 0.00744 0.00481

0.02222 0.01030 0.00589 0.00380

0.01562 0.00716 0.00407 0.00262

0.01807 0.00831 0.00474 0.00305

Shear buckling, CCCC plate 0.03985 0.01900 0.01097 0.00711

0.02838 0.01330 0.00763 0.00493

0.03303 0.01559 0.00897 0.00580

0.02625 0.01223 0.00700 0.00452

0.01850 0.00852 0.00486 0.00313

0.02162 0.01001 0.00572 0.00369

a/h 20 30 40 50

0.01446 0.00656 0.00372 0.00239

Biaxial buckling, SSSS plate

a/h 20 30 40 50

0.01295 0.00586 0.00332 0.00213

Biaxial buckling, CCCC plate

a/h 20 30 40 50

0.01783 0.00814 0.00463 0.00297

Uniform porosity GPL A GPL B GPL C

Uniaxial buckling, SSSS plate

a/h 20 30 40 50

Porosity distribution 2 GPL A GPL B GPL C Uniaxial buckling, CCCC plate

0.02497 0.01163 0.00666 0.00429

0.01834 0.00842 0.00479 0.00308

0.02042 0.00941 0.00536 0.00345

Shear buckling, SSSS plate 0.01620 0.00743 0.00423 0.00272

0.01180 0.00536 0.00304 0.00195

0.01317 0.00600 0.00340 0.00219

Fig. 7(A) and Fig. 7(B) are focused on the effect of GPL shape on the uniaxial buckling loads of porous plates with CCCC and SSSS boundary conditions, respectively. The influence from different shape ratios is investigated, including lGPL / tGPL and lGPL / wGPL . It is obvious from these two figures that decreasing lGPL / wGPL with a fixed lGPL / tGPL or raising

lGPL / tGPL under a constant lGPL / wGPL grows the uniaxial buckling loads of both CCCC and SSSS plates. Therefore, smaller and thinner GPLs provide the better stiffening effect, which is not evidently observed when lGPL / tGPL  103 , suggesting the reinforcement limitation with a certain value of nanofiller weight fraction. Similar conclusions can be obtained from the biaxial and shear buckling analyses, which are not presented here for brevity. 22

0.013

Dimensionless uniaxial buckling load

Dimensionless uniaxial buckling load

0.030

0.028

0.026 lGPL / wGPL = 1

Porosity 1 (symmetric I) GPL A (symmetric I) a / b = 1 a / h = 20 e0 = 0.5 GPL = 1.0 wt.%

lGPL / wGPL = 3

0.024

lGPL / wGPL = 5 lGPL / wGPL = 8 lGPL / wGPL = 10

0.022 1 10

10

2

10

CCCC plate

3

10

4

10

0.012

0.011 lGPL / wGPL = 1 0.010

lGPL / wGPL = 5 lGPL / wGPL = 8 lGPL / wGPL = 10

0.009 1 10

5

Porosity 1 (symmetric I) GPL A (symmetric I) a / b = 1 a / h = 20 e0 = 0.5 GPL = 1.0 wt.%

lGPL / wGPL = 3

10

2

10

SSSS plate

3

10

4

10

5

lGPL / tGPL

lGPL / tGPL

(A) CCCC plate (B) SSSS plate Fig. 7. Dimensionless uniaxial buckling loads of GPL-reinforced porous nanocomposite plates: effect of GPL shape (porosity distribution 1, GPL pattern A, a / b  1 , a / h  20 , e0  0.5 , GPL  1.0 wt.%).

4.3 Free vibration analysis The dimensionless fundamental natural frequency versus porosity coefficient curves for CCCC and SSSS porous nanocomposite plates are plotted in Fig. 8(A) and Fig. 8(B), respectively. It is interesting to see that the effect of porosity coefficient is not pronounced on the vibration behavior of porous plates with non-uniform symmetric porosity distribution 1, as under the increasing porosity coefficient, only a minor decrease can be observed in the corresponding fundamental natural frequency which however drops evidently for plates with distribution 2 and uniform porosity. This finding is further confirmed by the higher-order frequencies given in Table 10, where the first six dimensionless natural frequencies are presented for CCCC plates. It can be explained by the similar reduction rates of the plate stiffness and cross-section rotary inertia for porosity distribution 1, whereas for distribution 2 and uniform porosity, the stiffness reduction rate turns out to be much higher than that of the inertia with a larger porosity coefficient. Please also note that 2 and 3 are basically the same, and 5 and 6 are fairly close in all cases. Moreover, the highest frequency is based on the non-uniformly symmetric porosity distribution 1. This is to be expected, since the porosity distribution 1 provides the best rigidity, resulting in the highest natural frequency, as well as the largest buckling load.

23

0.45 Porosity 1 (symmetric I) Porosity 2 (symmetric II) Uniform porosity

GPL A (symmetric I)

Dimensionless fundamental frequency

Dimensionless fundamental frequency

0.8

0.7

0.6 GPL C (uniform)

0.5

GPL B (symmetric II) a/b=1

0.4 0.0

0.1

a / h = 20GPL = 1.0 wt.% 0.2

0.3

CCCC plate 0.4

0.5

0.6

Porosity 1 (symmetric I) Porosity 2 (symmetric II) Uniform porosity

GPL A (symmetric I) 0.40

0.35

0.30

0.25

GPL C (uniform) GPL B (symmetric II) a/b=1

0.20 0.0

0.1

a / h = 20GPL = 1.0 wt.% 0.2

Porosity coefficient e0

0.3

SSSS plate 0.4

0.5

0.6

Porosity coefficient e0

(A) CCCC plate (B) SSSS plate Fig. 8. Dimensionless fundamental natural frequencies of GPL-reinforced porous nanocomposite plates: effect of porosity coefficient ( a / b  1 , a / h  20 , GPL  1.0 wt.%). Table 10 Dimensionless higher-order natural frequencies of GPL-reinforced porous nanocomposite plates: effect of porosity coefficient ( a / b  1 , a / h  20 , GPL  1.0 wt.%, CCCC plate). GPL pattern

GPL A

GPL B

GPL C

GPL A

GPL B

GPL C

GPL A

GPL B

GPL C

e0

1

0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6

0.7075 0.7035 0.7017 0.5918 0.5888 0.5888 0.6383 0.6364 0.6379

0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6

0.6558 0.5885 0.5099 0.5501 0.4973 0.4394 0.5894 0.5288 0.4610

0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6

0.6887 0.6612 0.6285 0.5762 0.5532 0.5259 0.6202 0.5954 0.5660

3

2

Porosity distribution 1 1.4057 1.4057 1.3958 1.3958 1.3897 1.3897 1.1832 1.1832 1.1755 1.1755 1.1732 1.1732 1.2729 1.2729 1.2674 1.2674 1.2677 1.2677 Porosity distribution 2 1.3068 1.3068 1.1756 1.1756 1.0223 1.0223 1.1027 1.1027 0.9991 0.9991 0.8851 0.8851 1.1791 1.1791 1.0605 1.0605 0.9273 0.9273 Uniform porosity distribution 1.3696 1.3696 1.3149 1.3149 1.2500 1.2500 1.1532 1.1532 1.1071 1.1071 1.0525 1.0525 1.2383 1.2383 1.1888 1.1888 1.1301 1.1301

24

4

5

6

2.0262 2.0099 1.9985 1.7139 1.7009 1.6949 1.8403 1.8301 1.8276

2.4337 2.4125 2.3968 2.0650 2.0479 2.0387 2.2147 2.2008 2.1955

2.4501 2.4290 2.4134 2.0779 2.0609 2.0520 2.2289 2.2152 2.2102

1.8879 1.7020 1.4845 1.6009 1.4531 1.2901 1.7088 1.5401 1.3502

2.2709 2.0499 1.7912 1.9315 1.7550 1.5602 2.0595 1.8585 1.6319

2.2857 2.0628 1.8020 1.9432 1.7653 1.5690 2.0723 1.8696 1.6413

1.9758 1.8969 1.8032 1.6718 1.6050 1.5258 1.7918 1.7202 1.6353

2.3744 2.2795 2.1670 2.0153 1.9348 1.8393 2.1574 2.0712 1.9690

2.3902 2.2947 2.1814 2.0278 1.9467 1.8506 2.1711 2.0844 1.9815

The reinforcement effect of GPLs on the vibration response of different porous nanocomposite plates is thoroughly revealed in Fig. 9. A higher fundamental natural frequency can be obtained with an increase in the GPL weight fraction. The reinforcement effect of GPLs with symmetric pattern A is the most obvious, compared to those of GPLs with patterns B and C. Therefore, applying the suitable dispersion pattern can make the best use of the nanofillers, which should be distributed symmetrically around surfaces to provide the highest improvement of the vibration resistance, and consequently achieve the most stiffened nanocomposite plate.

0.7

GPL A (symmetric I) GPL B (symmetric II) GPL C (uniform)

a/b=1 e0 = 0.5

0.40

a / h = 20 CCCC plate

Dimensionless fundamental frequency

Dimensionless fundamental frequency

0.8

Porosity 1 (symmetric I) 0.6

Uniform porosity

0.5

0.4 Porosity 2 (symmetric II) 0.3 0.0

0.2

0.4

0.6

0.8

1.0

GPL A (symmetric I) GPL B (symmetric II) GPL C (uniform) 0.35

Porosity 1 (symmetric I) Uniform porosity

0.30

0.25 a/b=1 e0 = 0.5

Porosity 2 (symmetric II) 0.20 0.0

GPL weight fraction (wt.%)

0.2

0.4

0.6

a / h = 20 SSSS plate 0.8

1.0

GPL weight fraction (wt.%)

(A) CCCC plate (B) SSSS plate Fig. 9. Dimensionless fundamental natural frequencies of GPL-reinforced porous nanocomposite plates: effect of GPL weight fraction ( a / b  1 , a / h  20 , e0  0.5 ).

Table 11 lists the fundamental natural frequencies of porous nanocomposite plates under varying thickness ratios. It is found that a notable drop in the vibration resistance, indicated by the decreasing fundamental frequency, is induced by the increasing thickness ratio. Among all the porous nanocomposite plates examined, the CCCC plate with porosity distribution 1 and GPL pattern A presents the highest flexural rigidity, which is consistent with the finds in the elastic buckling analysis given in Table 9.

25

Table 11 Dimensionless fundamental natural frequencies of GPL-reinforced porous nanocomposite plates: effect of thickness ratio ( a / b  1 , GPL  1.0 wt.%, e0  0.5 ). Porosity distribution 1 GPL A GPL B GPL C a/h 20 30 40 50 a/h 20 30 40 50

0.7022 0.4783 0.3616 0.2904

0.5883 0.3987 0.3008 0.2413

0.6366 0.4324 0.3265 0.2620

0.3958 0.2657 0.1997 0.1600

0.3293 0.2207 0.1658 0.1328

0.3574 0.2397 0.1801 0.1442

Porosity distribution 2 GPL A GPL B GPL C CCCC plate 0.5502 0.4686 0.4953 0.3720 0.3155 0.3339 0.2805 0.2375 0.2515 0.2249 0.1903 0.2016 SSSS plate 0.3072 0.2601 0.2754 0.2057 0.1740 0.1843 0.1545 0.1306 0.1384 0.1237 0.1046 0.1108

Uniform porosity GPL A GPL B GPL C 0.6456 0.4387 0.3313 0.2659

0.5402 0.3652 0.2753 0.2207

0.5814 0.3938 0.2971 0.2383

0.3627 0.2433 0.1828 0.1464

0.3014 0.2018 0.1516 0.1214

0.3252 0.2179 0.1637 0.1311

Fig. 10 and Fig. 11 depict the variations of dimensionless fundamental natural frequencies of porous nanocomposite plates with the changing GPL shape ratios lGPL / tGPL and lGPL / wGPL , respectively. We can discover that higher lGPL / tGPL and lower lGPL / wGPL lead to a larger value of the fundamental natural frequency.

GPL A (symmetric I) GPL B (symmetric II) GPL C (uniform)

GPL = 1.0 wt.%

a/b=1 e0 = 0.5

0.5

lGPL / wGPL = 1

Dimensionless fundamental frequency

Dimensionless fundamental frequency

0.8

a / h = 20 CCCC plate

0.7 Porosity 1 (symmetric I) Uniform porosity

0.6

0.5

Porosity 2 (symmetric II) 0.4 1 10

10

2

10

3

10

4

10

5

GPL A (symmetric I) GPL B (symmetric II) GPL C (uniform)

GPL = 1.0 wt.%

a/b=1 e0 = 0.5

lGPL / wGPL = 1

a / h = 20 SSSS plate

Porosity 1 (symmetric I) 0.4

Uniform porosity 0.3

Porosity 2 (symmetric II) 0.2 1 10

lGPL / tGPL

10

2

10

3

10

4

10

5

lGPL / tGPL

(A) CCCC plate (B) SSSS plate Fig. 10. Dimensionless fundamental natural frequencies of GPL-reinforced porous nanocomposite plates: effect of lGPL / tGPL ( a / b  1 , a / h  20 , lGPL / wGPL  1 , e0  0.5 , GPL  1.0 wt.%).

26

GPL A (symmetric I) GPL B (symmetric II) GPL C (uniform)

GPL = 1.0 wt.%

0.5

lGPL / tGPL = 100

Dimensionless fundamental frequency

Dimensionless fundamental frequency

0.8

a / b = 1 a / h = 20 e0 = 0.5 CCCC plate

0.7 Porosity 1 (symmetric I) Uniform porosity

0.6

0.5

Porosity 2 (symmetric II) 0.4

1

2

4

6

8

GPL A (symmetric I) GPL B (symmetric II) GPL C (uniform)

10

a/b=1 e0 = 0.5

lGPL / tGPL = 100

a / h = 20 SSSS plate

Porosity 1 (symmetric I)

0.4 Uniform porosity

0.3

Porosity 2 (symmetric II) 0.2

lGPL / wGPL

GPL = 1.0 wt.%

1

2

4

6

8

10

lGPL / wGPL

(A) CCCC plate (B) SSSS plate Fig. 11. Dimensionless fundamental natural frequencies of GPL-reinforced porous nanocomposite plates: effect of lGPL / wGPL ( a / b  1 , a / h  20 , lGPL / tGPL  100 , e0  0.5 , GPL  1.0 wt.%).

5. Conclusions The combination of the FG porosity design and graphene reinforcement is able to produce an advanced ultralight high-strength structure for civil, automotive and aerospace industries to build construction, protection and load-bearing elements. Their unique thermal and electrical properties also make promise for the development of superior capacitors, batteries and heat exchangers. In this paper, the elastic buckling and free vibration responses of porous nanocomposite plates are revealed with a detailed examination on the effects of the porosity coefficient, GPL weight fraction, plate thickness ratio, and GPL shape ratios. Our results show that (1) The uniaxial, biaxial and shear buckling loads, as well as the fundamental natural frequencies of the proposed plates, decrease with the increasing of porosity coefficient, while grow evidently with the addition of GPL weight fraction, highlighting the weakening effect of internal pores and the impressive reinforcement effect of GPLs. (2) An increase in the thickness ratio leads to a dramatic drop in the plate flexural rigidity, as evidenced by the decreasing of the elastic buckling loads and fundamental frequencies. (3) Changing the GPL shape ratios by increasing lGPL / tGPL and reducing lGPL / wGPL results in the nanofiller platelets with thinner thicknesses and smaller surface areas, as well as a better reinforcement effect on the plate stiffness. (4) The best buckling and vibration resistance can be achieved with the non-uniformly symmetric porosity distribution 1 and GPL pattern A, indicating that centralizing internal pores on the mid-plane and dispersing nanofillers around the surfaces can obtain the highest flexural rigidity of porous nanocomposite plates under the identical consumptions of the matrix materials and nanofillers.

27

Acknowledgements The work described in this paper was fully funded by an Australian Government Research Training Program Scholarship and four research grants from the Australian Research Council under Discovery Project scheme (DP160101978, DP140102132) and Linkage Project scheme (LP150100103, LP150101033). The authors are grateful for their financial supports. References [1]

[2]

[3]

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