3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Accepted Manuscript 3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates Bo Yang, Jing Mei, Ding Chen, Feng Yu, Jie Yang PII: DOI: Reference:

S0263-8223(17)32827-1 https://doi.org/10.1016/j.compstruct.2017.09.086 COST 8949

To appear in:

Composite Structures

Received Date: Accepted Date:

1 September 2017 26 September 2017

Please cite this article as: Yang, B., Mei, J., Chen, D., Yu, F., Yang, J., 3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates, Composite Structures (2017), doi: https:// doi.org/10.1016/j.compstruct.2017.09.086

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Bo Yang* Department of Civil Engineering, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China

Jing Mei Department of Civil Engineering, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China

Ding Chen Department of Civil Engineering, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China

Feng Yu Department of Civil Engineering, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China Jie Yang* School of Engineering, RMIT University, Bundoora, VIC 3083, Australia

_____________________________________________________________________________ Corresponding authors: B Yang, Department of Civil Engineering, Zhejiang Sci-Tech University, Hangzhou, 310018, P. R. China Email: [email protected] J Yang, School of Engineering, RMIT University, PO Box 71, Bundoora, VIC 3083 Australia Email: [email protected]

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

ABSTRACT This paper presents an analytical investigation on the thermo-elastic response of a clamped elliptical plate made of either transversely isotropic functionally graded materials (FGMs) or a novel functionally graded nanocomposite reinforced with graphene nanoplatelets (GPLs) whose weight fraction varies continuously and smoothly along the thickness direction according to three uniform and non-uniform distribution patterns. Based on three-dimensional elasticity theory and the generalized Mian and Spencer’s method, 3D elasticity solutions are obtained for the elliptical plate under thermo-mechanical loading whose mid-plane displacements are constructed to satisfy the clamped boundary conditions in which the unknown constants are determined from the governing equations of the plate. The present analytical solutions are validated through comparisons with those available in open literature. A parametric study is then conducted, with a particular focus on the effects of GPL weight fraction, distribution pattern, geometry and size as well as the major to minor axis ratio of the plate on the thermo-mechanically induced stress and deformation fields.

Keywords: Elliptical plate; Functionally graded material; Graphene nanoplatelets; Thermo-elasticity; Analytical solution

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

1． Introduction Functionally graded materials (FGMs) are novel microscopically inhomogeneous composites in which the material composition and mechanical properties are graded in one or more directions. Numerous studies have been conducted on the study of elastic responses of FGM plates with different shape under various loading conditions [1-3]. The bending of elliptical plates that are widely used in many engineering fields is one of the most important classical topics in the theory of elasticity [4]. Compared with the analyses of FGM rectangular and circular plates, investigations on FGM elliptical plates are limited. Cheng and Batra [5] used an asymptotic expansion method to obtain a three dimensional (3D) thermo-elastic solution for an isotropic linear functionally graded elliptic plate rigidly clamped at the edges. Based on the classical plate theory, Hsieh and Lee [6] studied the inverse problem of a functionally graded elliptical plate with large deflection and disturbed boundary under uniform load. Kumar et al. [7] carried out parametric studies on the vibro-acoustic response of an elliptic disc of functionally graded material. Çeribaşı [8] investigated static and dynamic linear problems of clamped thin super elliptical FGM plates subject to a uniform load. By employing the classical plate theory (CPT), Jazi and Farhatnia [9] studied the elastic buckling of functionally graded super elliptical plate using Pb-2 Ritz method. To the authors’ knowledge, however, no closed-form solution has been reported for the bending of transversely isotropic FGM elliptical plates based on 3D theory of elasticity in the literature. Since the discovery of graphene, huge attentions from both research and engineering communities have been focused on its extraordinary properties and great potential in various engineering applications. Attributed to its two-dimensional geometry, graphene platelets (GPLs) which is composed of multiple graphene sheets that are stacked together remarkably outperform carbon nanotubes in flexibility, thermal expansion, and electrical and thermal conductivities [10, 11] hence are considered to be better reinforcement than carbon nanotubes for the development of nanocomposites. In order to make the best use of graphene’s exceptionally high Young’s modulus to develop advanced lightweight structures, Yang and his co-workers [12-22] introduced the concept of functionally graded (FG) graphene reinforced multilayer structures and investigated the buckling, postbuckling, free and forced vibration, nonlinear bending, and dynamic stability of functionally graded graphene reinforced polymer nanocomposite beams, plates and shells.

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Their results showed that dispersing GPL nanofillers with a lower content into polymer matrix according to an appropriate non-uniform distribution pattern can achieve much better reinforcing effect than the conventional uniform pattern. Most recently, Shen et al. [23-25] studied the buckling, postbuckling, nonlinear bending and vibration behaviors of piece-wise graded graphene-reinforced laminated plates in thermal environments. However, the thermo-mechanically loaded elliptical plates have not been considered in the above studies [12-25]. Within the framework of 3D theory of elasticity, this paper aims to investigate the thermo-elastic bending behaviour of clamped elliptical plates made of either transversely isotropic functionally graded materials (FGMs) or functionally graded polymer nanocomposite reinforced with GPLs that are non-uniformly dispersed within the matrix with weight fraction varying continuously and smoothly along the thickness direction. The analytical solution procedure developed by Mian and Spencer [26] for isotropic FGM plates with tractions free on both upper and bottom surfaces, which was later successfully extended to uniformly loaded transversely isotropic FGM plates in our previous work [27], will be further utilized to obtain the 3D elasticity solution of the plate. It is assumed that the plate is subjected to a uniform transverse load in combination with a steady temperature field along the thickness direction due to heat conduction. A parametric study is conducted to examine the effects of weight fraction, distribution pattern, geometry and size of GPLs as well as the major to minor axis ratio of the plate on the stress and deformation distributions of the plate. 2． The generalized Mian and Spencer method The schematic diagram of an elliptical plate of thickness h , half-length of major axis a and half-length of minor axis b is shown in Fig. 1. The rectangular Cartesian coordinates system Oxyz in which the xy plane coincides with the mid-plane of the plate is used. The plate is

under the action of a uniformly distributed pressure q1 and temperature T1 on the bottom surface and a uniformly distributed pressure q2 and temperature T2 on the top surface of the plate. Let u , v and w be the displacement components, and  ij (i, j  x, y, z ) be the stress components, respectively.

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Fig. 1

In the absence of body forces, the equations of equilibrium are

 ij , j  0 ,

(1)

where a comma denotes partial differentiation with respect to the indicated variable. The linear thermo-elastic stress-strain relationship for transversely isotropic material can be expressed as

 xx  c11  u, x  1T   c12  v, y  1T   c13  w, z  3T  ,  yy  c12  u, x  1T   c11  v, y  1T   c13  w, z  3T  ,

 zz  c13  u, x  1T   c13  v, y  1T   c33  w, z  3T  ,  yz  c44  w, y  v, z  ,

 zx  c44  u, z  w, x  ,

 xy  c66  u, y  v, x  ,

(2)

where cij with 2c66  c11  c12 are elastic constants and  i the linear thermal expansion coefficients of the plate while T is the temperature change from the initial stress-free state. Here, the z-axis coincides with the material symmetry axis. For the functionally graded plate, its material parameters are position dependent, i.e., cij  cij ( z ) , i  i ( z ) and T  T ( z ) . According to Mian and Spencer [26], the solutions of Eqs (1) and (2) take the following forms

u  x, y, z   u  x, y   F , x  Aw, x  B2 w, x , v  x, y, z   v  x, y   F , y  Aw, y  B2 w, y , w  x, y, z   w  x, y   G  C2 w  D ,

(3)

where A , B , C , D , F and G are the functions of z which can be determined from the traction conditions on the top and bottom surfaces of the plate, and   ( x, y)  u, x  v, y , 2   2 x2   2 y 2 .

(4)

The displacement field (u, v, w) in Eq. (3) can be obtained after mid-plane displacements (u , v , w) have been determined for a given boundary value problem. To this end, the governing

equations for (u , v , w) will be derived in the first place through the following process. Substituting Eq. (3) into Eq. (2) gives

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

 xx   c12  c13G     c12 A  c13C    2 w  c12  F  2   B 4 w   c13 D  2c66  u, x  F , xx  Aw, xx  B 2 w, xx    c11  c12  1  c13 3  T ,

 yy   c12  c13G     c12 A  c13C    2 w  c12  F  2   B 4 w   c13 D

 2c66  v, y  F , yy  Aw, yy  B 2 w, yy    c11  c12  1  c13 3  T ,

 zz   c13  c33G     c13 A  c33C    2 w  c13  F  2   B 4 w   c33 D   2c131  c33 3  T ,

 yz  c44  A  1 w, y   F   G  , y   B  C  2 w, y  ,

 zx  c44  A  1 w, x   F   G  , x   B  C  2 w, x  ,  xy  c66  u, y  v, x  2F , xy  2 Aw, xy  2B2 w, xy  ,

(5)

where the prime denotes partial derivative with respect to z . Substituting Eq. (5) into Eq. (1), one has  2   c11  c13G  c44  F   G    , x  c66 , y  c11 F   , x  c44  A  1  w, x  





(6)

 c11 A  c13C   c44 ( B  C )  2 w, x  c11B 4 w, x  0,

 2   c11  c13G  c44  F   G    , y  c66, x  c11F   , y  c44  A  1 w, y      c11 A  c13C   c44  B  C    2 w, y  c11B 4 w, y  0,  

(7)

 c13  c33G   c44  F   G    c13 F    2   c44  A  1   c13 A  c33C    2 w 





 c44  B  C    c13 B    4 w  c33 D   2c131  c33 3 T   0.  



(8)

On setting

c44  A  1   0 ,

(9)

 c13  c33G  0 ,

(10)

c11  c13G  c44 ( F   G)  c661 ,

(11)

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

c11 A  c13C   c44  B  C   c66 2 .

(12)

c44  A  1   c13 A  c33C    0 ,

(13)

where 1 and  2 are arbitrary constants. Eqs (6)-(8) become

c66 1, x  , y   22 w, x   c11  F 2   B4 w, x  0 ,

(14)

c66 1, y  , x   22 w, y   c11  F 2   B4 w, y  0 ,

(15)

H 2   I 4 w  c33 D   2c131  c333  T   0 ,

(16)

in which   v, x  u, y ,

H  c44  F   G    c13 F  ,

I  c44  B  C    c13 B  .

(17)

If H and I are not zero simultaneously, it follows from Eq. (16) that 4 w   4 .

2    3 ,

(18)

where  3 and  4 are arbitrary constants. Eqs (14)-(16) then become

     w 

 , y  0 ,

(19)

     w 

 , x  0 ,

(20)

2

1

2

,x

2

1

2

,y

H  3  I  4  c33 D   2c131  c333  T   0 .

(21)

With Eq. (18) in mind, it can be obtained from Eqs (19) and (20) that

13   2 4  0 .

(22)

Eqs (18)2, (19), and (20) constitute the governing equations for unknown displacement functions u ( x, y) , v ( x, y) , and w( x, y) , defined as u ( x, y)  u( x, y,0) ,

v ( x, y)  v( x, y,0) ,

w( x, y)  w( x, y,0) .

(23)

By virtue of Eq. (23) and the traction stress conditions at the top and bottom surfaces of the plate, i.e.,  xz   yz  0 at z   h 2 ,  z  q1 at z   h 2 , and  z  q2 at z  h 2 , the unknown functions A( z ) , B( z ) , C ( z ) , D( z ) , F ( z ) and G( z ) can be determined as

A z   z .

(24)

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

B  z    2 f6  z   f 4  z   g11  z    z  h 2 C0  B0 .

(25)

C  z   g10  z   C0 .

(26)

D  z   3 f7  z    4 f8  z   q1 f3  z   f9  z   D0

(27)

F  z   1 f 6  z   f5  z   g01  z    z  h 2  G0  F0 .

(28)

G  z    g00  z   G0 .

(29)

where the related functions and constants can be found in Yang et al. [17]. It is obvious that the displacement and stress fields of the plate can be determined only when the unknown displacement functions u ( x, y) , v ( x, y) , and w( x, y) have been found, the solution process of which will be discussed in Section 5. 3． Temperature field T ( z ) This paper considers the one-dimensional steady state temperature field T ( z ) along the thickness direction due to heat conduction induced at the top and bottom surfaces of the elliptical plate whose cylindrical boundaries are assumed to be adiabatic. According to the Fourier law of heat conduction [28], the thermal flux q z in the z-direction is qz  k  z 

dT . dz

(30)

where k  k ( z ) is the thermal conductivity coefficient. Based on this equation, one has 

d  dT  k  z 0.  dz  dz 

(31)

The temperature field can be obtained from Eq. (31) as follows

T  z   C1 

z

h 2

d  C2 . k  

(32)

where C1 and C2 are integral constants to be determined by the thermal conditions on the top and bottom surfaces of the plate. The following three typical thermal boundary conditions are described (1) The Dirichlet conditions used at z   h 2 , namely T

z  h 2

 T1 ,

T

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z h 2

 T2 .

(33)

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

where T1 and T2 are known constants. It follows from Eqs (32) and (33) that

C1  T2  T1 

d , h 2 k   



h2

C2  T1 .

(34)

(2) The Dirichlet condition used at z  h 2 and the Neumann condition employed at

z   h 2 , namely qz

z  h 2

 1 ,

T

z h 2

 T2 .

(35)

where 1 is a known constant. The two integral constants can be found from Eqs (30), (32) and (35) as

C1  1 ,

C2  T2  1 

h2

h 2

d . k  

(36)

(3) The Dirichlet condition at z  h 2 and the Newton’s law of cooling condition at

z   h 2 , namely

T   qz  z h 2  1 ,

T

z h 2

 T2 .

(37)

where  and 1 are known constants. Solution of Eqs (30), (32) and (37) gives h2  d  , C1   1  T2      h 2 k     

h 2 d   C2    T2  1   h 2 k    

h 2 d      h 2 . k    

(38)

After the integral constants C1 and C2 have been determined as above, the temperature field in Eq. (32) can be completely determined. 4． Functionally graded polymer nanocomposite reinforced with GPLs The isotropic functionally graded GPL reinforced nanocomposite material under current consideration can be considered as a special case of transversely isotropic functionally graded

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

materials by setting c11  c33 , c12  c13 , c44  c66 and 1   3 , in which the elastic constants are related to the effective Young’s modulus Ec and Poisson’s ratio  c by

c1 2  c 1 3 where

Ec

Ec 1  c  Ec c , c11  c33  , 1  c  1 2c  1  c 1  2 c 

and  c

c44  c66 

Ec . 2 1  c 

are the effective Young’s modulus and Poisson’s ratio of the

nanocomposites, respectively. According to the modified Halpin-Tsai micromechanics model, the effective Young’s modulus Ec for the isotropic homogeneous nanocomposite reinforced with uniformly distributed GPLs can be predicted by [12-21]

 E  E E 1  E 1  1   L  GPL M 1  W  GPL M  VGPL  VGPL EGPL EM   L  EGPL EM  W  3 5   Ec  EM  EM . 8 8  EGPL EM  1   EGPL EM  1  1  1   VGPL  VGPL  EGPL EM   L   EGPL EM  W 

(39)

where L , W and t are the average length, width and thickness of the GPLs; EGPL and EM represent the moduli of the GPLs and the polymer matrix, respectively;  L and W are the two geometry parameters describing the effects of GPLs’ shape and size in the longitudinal and transverse directions which are expressed as

L  2  L t  ,

 W  2 W t  .

(40)

The volume fraction VGPL of GPL nanofillers can be deduced from its weight fraction

WGPL and mass density GPL by VGPL 

WGPL

 WGPL  GPL 1  WGPL  M

.

(41)

in  M is the mass density of the matrix. The effective Poisson’s ratio  c and thermal expansion coefficient  c of the GPL reinforced nanocomposite can be calculated the rule of mixture [29], by

 c  VGPL GPL  Vm m .

(42)

c  VGPLGPL  Vm m .

(43)

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

where  GPL ,  m ,  GPL and  m are Poisson’s ratio and thermal expansion coefficient of GPLs and the polymer matrix, respectively. The volume fraction of the matrix Vm and that of GPLs VGPL are related by VGPL  Vm  1 .

(44)

The effective thermal conductivity kc of the GPL reinforced nanocomposite can be derived from [30]

kc V  1  GPL km 3

  2 1   .  H  1  k x km  1 1  H  2  1  k z km  1 

(45)

where km is the thermal conductivity of the matrix and





ln  p  p 2  1 p  1   . H   2 3 2 p  1  p  1

kx 

kg 2 Rk k g L  1

kz 

,

kg 2 Rk k g t  1

(46)

.

(47)

in which Rk is an average interfacial thermal resistance between the GPLs and matrix, k g is GPL’s intrinsic thermal conductivity along the in-plane direction, and p  L t . As for the functionally graded polymer nanocomposites where GPLs are non-uniformly dispersed within the polymer matrix, the elastic and thermal constants in Eqs (39), (42), (43) and (45) are position-dependent as the functions of thickness coordinate of the plate. In the present work, three typical GPL distribution models [18] as shown in Fig. 2 are considered, that is 1) Linear distribution with GPL weight fraction changing from the highest on the top surface to zero at the bottom surface, i.e. 0 WGPL ( z )  1WGPL (1 2  z h) .

(48)

2) Parabolic distribution where the GPL weight fraction is the maximum at the top and bottom surfaces and zero at the mid-plane, i.e. WGPL ( z ) 

4 0 2WGPL z2 . 2 h

- 11 -

(49)

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

3) Uniform GPL distribution with constant weight fraction along the plate thickness, i.e. 0 . WGPL ( z )  3WGPL

(50)

0 where WGPL is a certain GPL weight fraction, 1 , 2 and 3 are weight fraction indices.

Fig. 2

5． Mid-plane displacement solution for a clamped elliptical plate For the clamped elliptical plate considered in the present study, it is assumed that its mid-plane displacements that satisfy the boundary conditions are of the form as 2

 x2 y 2  u  A 1  2  2  x ,  a b 

 x2 y 2  w  C 1  2  2  , b   a

 x2 y 2  v  B 1  2  2  y ,  a b 

(51)

where the transverse displacement function w was used by Timoshenko and WoinowskyKrieger [4] for a clamped isotropic homogeneous elliptical thin plate, and constants A, B and C are to be determined. Substituting w in Eq. (51) into the second equation in Eq. (18) leads to C

4 24 16 24  2 2 4 4 a ab b

.

(52)

Substituting Eq. (51) into Eqs (19) and (20) gives  3A B  2 2 a a

3C  B A   C   4 2  2 2  4   2  2  0 . a  a b  a b

(53)

3C  B A  A 3B   C + 2   4 2  2 2  4   2  2  0 . 2 b  b  a b b a b

(54)

1 

1 

from which A and B can be obtained A

 2 4  6a 4 6  1  4  2  2  b a   b 2

B

,

The expression of the constant  2 is [31]

- 12 -

 2 4  6b 4 6 1  4  2  2  a b   a 2

. (55)

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

2  

h10  h 2  , g6  h 2 

(56)

where

C132   h  z    h  [C11    ]d .  C33   2

g6  z    h C66  d , z



0 1

2

z

(57)

It can be found from Eq. (56) that  2 =0 for a homogeneous material. Consequently, constants A, B and u , v are all equal to zero, which are consistent with the results in simplified plate theories for homogeneous materials. With unknown displacements u ( x, y) , v ( x, y) and w( x, y) having been determined as described above, the analytical solutions of all 3D displacement and stress components of the plate can now be obtained from Eqs (3) and (5). 6． Numerical results and discussions 6.1 Comparison study In order to validate the present analytical solutions, a numerical example is given for an isotropic homogeneous thin elliptical plate with clamped edge subjected to uniform loads.

q1 =q0 and q2 =0 . The dimensionless deflection w0 = w(0,0,0) Eh3 12q0 is listed in Table 1 for plates with b=1 m and various aspect ratios a b . Our results are in excellent agreement with the solutions by Çeribaşı et al. [32] based on classical plate theory and those by Zhang [33] based on Reddy’s high order shear deformation theory.

Table 1

Our analytical solutions is further validated by considering the thermally induced bending of an isotropic FGM elliptical plate. According to the Mori-Tanaka homogenization method, the effective bulk modulus K and effective shear modulus  of the FGM are as follows [5]:  K  K1 K 2  K1   V2 1  1  V2  , K  4 3  K 2  K1   1 1  

- 13 -

   1   1   V2 1  1  V2  2 , 1  f1  2  1 

(58)

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

where n

1 z V2     , 2 h

f1 

1  9K1  81  . 6  K1  21 

(59)

The local effective heat conductivity coefficient k and coefficient of linear thermal expansion  are determined from

  1 V2 ,   2  1 1  1  V2    2  1  31 

  1  1 1   1 1       .  2  1  K K1   K 2 K1 

(60)

Based on Eq. (58), the effective Young’s modulus E and Poisson’s ratio  can be computed from

E

9K  , 3K  



3K  2  . 2  3K   

(61)

Monel and Zironia are selected as the constituent material 1 and material 2 of the FGM plate in this example, with the following material properties: K1 = 227.24GPa, μ1 = 65.55GPa, α1 = 15×10-6/K, κ1 = 25W/mK; K2 = 125.83GPa, μ2 = 58.077GPa, α2 = 10×10-6/K, κ2 = 2.09W/mK. Figs 3 and 4 plot, respectively, the through-thickness distribution of the dimensionless deflection w1  w (5h *T2 ) and dimensionless normal stress  x* =  x ( E* *T2 ) of the thermally loaded FGM elliptical plate. Here, E* = 1GPa, α* = 10-6 /K, a = 0.1m, h = 0.02m. Again, excellent agreement with existing results [5] is observed.

Fig. 3

Fig. 4

6.2 Thermo-mechanical behaviour of transversely isotropic FGM elliptical plates A transversely isotropic FGM elliptical plate under a combined thermal and mechanical loading is studied in this example with a = 0.1m, b = 0.05m, h = 0.02m, q1  106 N m2 ,

q2  0 , T1 = 10K, and T2 = 150K. Unless stated otherwise, numerical results are provided for the deflection and stress at the centre ( x  0 , y  0 ) of the plate. The material properties P( z ) such

- 14 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

as cij ,  i , and k are graded along the thickness direction according to an exponential function below P  z   P0 e  0.5 z h ,

(58)

where P0 denotes the material property value at z   h 2 given in Table 2, and  is the gradient index characterizing the material inhomogeneity. As a special case,   0 corresponds to homogeneous material.

Table 2

Fig. 5

Fig. 5 depicts the distribution of the dimensionless elastic constant c11 c110 along the thickness direction of an elliptical plate, reflecting the stiffness change in the thickness direction of the plate. It can be found that the dimensionless elastic constant decreases slowly along the thickness direction when   0 but increases drastically with an increasing positive  .

Table 3 Table 3 lists the dimensionless deflection w/h and normal stresses  x q1 and  y q1 of the clamped elliptical plate subjected to a uniform load. The deflection significantly decreases while the normal stresses show a considerable increase as  increases from a negative value to a positive value. This is owing to the fact that the plate becomes much stiffer with an increasing  . For a given  , the normal stress  x q1 is nearly half the normal stress  y q1 .

Fig. 6 Fig. 6 displays the distribution of the dimensionless deflection w h105 and normal stress  x q1 along the thickness direction of the elliptical plate under a mechanical load. It is shown that the deflection varies almost linearly along the thickness direction for  =0,1 but

- 15 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

becomes nonlinear for  =-1 which corresponds to a plate with a relatively small stiffness. It should be noted that the real through-thickness deflection distribution can be captured by 3D elasticity theory only because in all classical, first-order and higher-order plate theories, the deflections are assumed to be constant through the thickness direction. As for the normal stress, the distribution is linear for a homogeneous plate and nonlinear for an FGM plate and the stress value of the homogeneous plate is in the middle of those of FGM plates, except for two special points at which they are almost the same.

Fig.7

Fig. 7 plots the distribution of the dimensionless deflection w h103 and normal stress

 x c110 104 along the thickness direction of the elliptical plate under a temperature change that varies in the thickness direction only but remains uniform in the x-y plane of the plate. Hence, only the through-thickness distributions of the deflection and stress induced by this temperature filed will be discussed. It is observed that regardless of the value of  , both the deflection and normal stress change nonlinearly along the thickness direction and the results of the homogeneous plate lie in between those of two FGM plates with   1 and   1 . 6.3 Thermo-mechanical behaviour of functionally graded GPL reinforced elliptical plates In this section, a functionally graded GPL reinforced elliptical plate under a combined thermal and mechanical loading is considered to investigate the influences of the weight fraction, distribution pattern, geometry and size of GPL nanofillers as well as the major to minor axis ratio of the plate on its stress and displacement fields. For this purpose, epoxy resin is selected as the polymer matrix and the related parameters are [18]: L  2.5 µm, W  1.5 µm,

t  1.5 nm, EGPL  1.01 TPa, EM  2.85 GPa, M  1.2 103 kg/m3, GPL  1.06  103 kg/m3; kM  0.2 W mK , kg  2000 W mK , Rk  108 m2 K W ;  M  8.2 10 5 K ; GPL  0.006 ,

GPL  2.35 10 5 K ;  M  0.34 ,

0 WGPL  1% ,

q1  103 N/m2, q2  0 , T1  280 K , T2  370K .

Table 4

- 16 -

a  0.1 m,

b  0.05 m,

h  0.02 m,

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Table 4 lists the relationship between GPL gradient index i and the total GPL content * for three GPL distribution models in which 1 , 2 and 3 correspond to linear, WGPL

parabolic, and uniform GPL distributions, respectively. Fig. 8 displays the effects of GPL dimension and geometry on the dimensionless deflection

w h 106 and normal stress  x Em 106 of the elliptical plate under a mechanical load. The GPL length L is kept constant. W = L and W = L/2 correspond to square and rectangular shaped GPLs, respectively. It is noted that the surface area of the squared GPL is twice that of the rectangular one and a bigger t/L ratio means more graphene monolayers stacked together in GPLs. Fig. 8a shows that the deflection is effectively reduced by using square-shaped GPLs and this effect tends to be more pronounced when GPLs with more graphene monolayers are used. This is because square GPL nanofillers have a higher stiffness and better load transfer capability due to its bigger surface area which can most significantly strengthen the plate stiffness. Fig. 8b shows that with more graphene monolayers involved, the normal stress increases for nonlinear GPL distribution and decreases for linear GPL distribution. A reduced W/L ratio leads to a slightly increased (reduced) normal stress for nonlinear (linear) GPL distribution pattern. When GPLs are uniformly distributed, the effects of both GPL size and geometry on the normal bending stress are insignificant. Among the three GPL distribution models considered, the parabolic pattern (λ2 = 3) is the most effective way to reduce both bending deflection and normal stress while the linear pattern (λ1 = 2) is the worst as it leads to a much bigger deflection and normal stress.

Fig.8

Fig.9

Fig. 9 illustrate the effect of major to minor axis ratio a/b on the dimensionless deflection

w h106 and normal stress  x Em 106 at the top surface of the elliptical plate under a mechanical load. Here, the half-length of the major axis a is fixed to be 0.1m while that of the minor axis b is varied. As can be seen, both the deflection and normal stress decrease at a bigger

- 17 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

a/b due to the increased bending rigidity of the plate. Among the three GPL distributions, the parabolic pattern yields the lowest deflection at all a/b ratios and the smallest normal stress when a/b ≥1.5. Once again, it is confirmed that the linear pattern gives the poorest performance.

Fig.10

Fig. 10 displays the effect of major to minor axis ratio on the dimensionless deflection

w h102 and normal stress  x Em 102 at the top surface of the plate under a thermal load. It is found that the deflection and normal stress almost keep constant with a/b which means that the results duo to the thermal load are the same for a plate of arbitrary shape. Among the three GPL distributions, the parabolic pattern ones all show the biggest values.

Fig.11

Fig. 11 plots the effect of a/b ratio on the distribution of dimensionless deflection

w h106 and normal stress  x Em 106 of the plate under a mechanical load. Only the parabolic distribution pattern with 2 =3 is considered herein for brevity. Numerical results demonstrate that as a/b is varied, the deflection is basically unchanged along the thickness direction of the plate whereas the through-thickness normal stress distribution of a circular plate (a/b = 1) is greatly affected but that of an elliptical plate is much less sensitive to a/b. It is also noted that at a/b = 10, the normal stress is tensile at the bottom surface and compressive at the top surface of the plate, which is different from the results at a/b = 1 and 2. Fig. 12 shows the dimensionless deflection at point (0, 0, h/2) versus load parameter P  q1a5 ( Em h5 ) of the elliptical plate under a combined thermo-mechanical loading. Here

symbols © and ® refer to circular (b=a) and elliptical (b=a/2) plates, respectively. It is observed that the deflection grows linearly as P increases. The deflection is positive because q1 is positive and the thermal expansion at the top surface is also higher due to a higher temperature at this location, both resulting in an upward deflection. Compared to the circular plate, the deflection of the elliptical plate is smaller due to its bigger bending stiffness. In addition, it is seen again the parabolic distribution pattern gives the smallest deflection whereas

- 18 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

the linear distribution pattern produces the biggest deflection among the three GPL distribution patterns.

Fig.12

3

Conclusions Thermo-mechanical bending of a transversely isotropic functionally graded and

functionally graded GPL reinforced elliptical plate has been investigated based on the 3D theory of elasticity and the generalized Mian and Spencer’s method. 3D elasticity solutions are obtained for clamped plates under a combined thermo-mechanical loading. Unlike simplified plate theory based solutions in which the deflection of the plate is assumed to be constant along the plate thickness, our 3D elasticity solution is capable of providing real deflection and stress distributions. It is found from numerical results for functionally graded GPL reinforced elliptical plates that GPLs with a larger surface area and fewer graphene monolayers offer better reinforcing effect. Among the three GPL distribution patterns considered, the parabolic pattern yields the lowest deflection at all a/b ratios and the smallest normal stress when a/b ≥1.5. Compared with circular plate, the through-thickness normal stress distribution of an elliptical plate is much less sensitive to a/b. Furthermore, the deflection and normal stress of the thermally loaded functionally graded GPL reinforced elliptical plate remain almost the same regardless of the change in a/b. It should be pointed out that the present closed-form solutions exactly satisfy the 3D equilibrium equations and the traction boundary conditions on the top and bottom surfaces of the plate. The cylindrical boundary conditions in the plate are satisfied in the Saint-Venant sense. Thus, the proposed 3D elasticity solutions can serve as a benchmark for the thermo-mechanical bending solutions based on various simplified plate theories or numerical methods.

Acknowledgement The work was supported by the Natural Science Foundation of Zhejiang Province, China (No. LY18A020009), the Science Foundation of Zhejiang Sci-Tech University (Grant No.

- 19 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

16052188-Y) and two research grants from the Australian Research Council under Discovery Project scheme (DP140102132, DP160101978). Prof. Feng Yu is also grateful for the support from the National Nature Science Foundation of China (No. 41472284) and Natural Science Foundation of Zhejiang Province, China (No. LZ17E080002).

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[2]

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[3]

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[4]

Timoshenko SP, Goodier JN. Theory of Elasticity. 3rd. New York: McGraw-Hill; 1970.

[5]

Cheng ZQ, Batra RC. Three-dimensional thermo-elastic deformations of a functionally graded elliptic plates. Compos Part B-Eng 2000; 31(2): 97-106.

[6]

Hsieh JJ, Lee LT. An inverse problem for a functionally graded elliptical plate with large deflection and slightly disturbed boundary. Int J Solids Struct 2006; 43 (20): 5981-5993.

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Kumar B R, Ganesan N, Sethuraman R. Vibro-acoustic analysis of functionally graded elliptic disc under thermal environment. Mech Adv Mater Struct 2009; 16 (2): 160-172.

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Çeribaşı S. Static and dynamic analyses of thin uniformly loaded super elliptical FGM plates. Mech

[9]

Adv Mater Struct 2012; 19 (5): 323–335.

Jazi SR, Farhatnia F. Buckling analysis of functionally graded super elliptical plate using Pb-2 Ritz method. Adv Mater Res 2012; 383–390: 5387-5391.

[10] Rafiee MA, Rafiee J, Wang Z, Song H, Yu ZZ, Koratkar N. Enhanced mechanical properties of

nanocomposites at low grapheme content. Acs Nano 2009; 3(12): 3884-3890. [11] Xie SH, Liu YY, Li JY. Comparison of the effective conductivity between composites reinforced

by graphene nanosheets and carbon nanotubes. Appl Phys Lett 2008; 92(24): 243121. [12] Yang J, Wu HL, Kitipornchai S. Buckling and postbuckling of functionally graded multilayer

graphene platelet-reinforced composite beams. Compos Struct 2017; 161: 111-118. [13] Wu HL, Yang J, Kitipornchai S. Dynamic instability of functionally graded multilayer graphene

nano-composite beams in thermal environment. Compos Struct 2017; 162: 244-254. [14] Kitipornchai S, Chen D, Yang J. Free vibration and elastic buckling of functionally graded porous

beams reinforced by graphene platelets. Mater Des 2017;116: 656-665.

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

[15] Feng C, Kitipornchai S, Yang J. Nonlinear bending of polymer nanocomposite beams reinforced

with non-uniformly distributed graphene platelets (GPLs). Compos Part B-Eng 2017; 110: 132-140. [16] Song MT, Kitipornchai S, Yang J. Free and forced vibrations of functionally graded polymer

composite plates reinforced with graphene nanoplatelets. Compos Struct 2017; 159: 579-588. [17] Yang B, Yang J, Kitipornchai S. Thermo-elastic analysis of functionally graded graphene

reinforced rectangular plates based on 3D elasticity. Meccanica 2017; 52(10): 2275-2292. [18] Yang B, Kitipornchai S, Yang YF, Yang J. 3D thermo-mechanical bending solution of functionally

graded graphene reinforced circular and annular plates. Appl Math Model. 2017; 49: 69-86. [19] Wu HL, Kitipornchai S, Yang J. Thermal buckling and postbuckling of functionally graded

graphene nanocomposite plates. Mater Des 2017; 132: 430-441. [20] Chen D, Yang J, Kitipornchai S. Nonlinear vibration and postbuckling of functionally graded

graphene reinforced porous nanocomposite beams. Compos Sci Technol 2017; 142: 235-245. [21] Feng C, Kitipornchai S, Yang J. Nonlinear free vibration of functionally graded polymer

composite beams reinforced with graphene nanoplatelets (GPLs). Eng Struct 2017; 140: 110-119. [22] Wang Y, Feng C, Yang J, Zhao Z. Buckling of graphene platelet (GPL) reinforced composite

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graphene-reinforced composite laminated plates in thermal environments. Compos Part B-Eng 2017; 119: 67-78. [24] Shen H S, Xiang Y, Lin F. Nonlinear bending of functionally graded graphene-reinforced

composite laminated plates resting on elastic foundations in thermal environments. Compos Struct 2017; 170: 80-90. [25] Shen H S, Xiang Y, Lin F. Nonlinear vibration of functionally graded graphene-reinforced

composite laminated plates in thermal environments. Comput Method Appl Mech Engrg 2017; 319: 175-193. [26] Mian MA, Spencer AJM. Exact solutions for functionally graded and laminated elastic materials. J

Mech Phys Solids 1998; 46(12): 2283-2295. [27] Yang B, Ding HJ, Chen WQ. Elasticity solutions for a uniformly loaded rectangular plate of

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

[29] Ke LL, Yang J, Kitipornchai S. Nonlinear free vibration of functionally graded carbon

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3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Captions of Figures

Fig. 1 Schematic diagram of FGM elliptical plate and its coordinates: (a) 3D model, (b) front view, and (c) mid-plane Fig. 2 Schematic GPL distribution patterns along the plate thickness: (a) linear, (b) parabolic; and (c) uniform Fig. 3 Through-thickness deflection distribution of the FGM elliptical plate under a thermal load Fig. 4 Through-the-thickness distribution of the dimensionless normal stress  x* of the FGM elliptical plate under a thermal load Fig. 5 Through-thickness distribution of elastic constant c11 c110 of the transversely isotropic functionally graded elliptical plate Fig. 6 Through-thickness deflection and normal stress distributions of the transversely isotropic functionally graded elliptical plate under a mechanical load: (a) deflection; and (b) normal stress Fig. 7 Through-thickness deflection and normal stress distributions of the transversely isotropic functionally graded elliptical plate under a thermal load: (a) deflection; and (b) normal stress Fig. 8 Effects of GPL dimension and geometry on the bending bahavior of the functionally graded GPL reinforced elliptical plate under a mechanical load: (a) deflection; and (b) normal stress

- 23 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Fig. 9 Effect of major to minor axis ratio on the deflection and normal stress on the top surface of the functionally graded GPL reinforced elliptical plate under a mechanical load: (a) deflection; and (b) normal stress Fig. 10 Effect of major to minor axis ratio on the deflection and normal stress on the top surface of the functionally graded GPL reinforced elliptical plate under a thermal load: (a) deflection; and (b) normal stress Fig. 11 Effect of major to minor axis ratio on the bending behavior of the functionally graded GPL reinforced elliptical plate under a mechanical load: (a) deflection; and (b) normal stress Fig. 12 Dimensionless deflection versus load parameter P of the functionally graded GPL reinforced elliptical plate under a combined thermo-mechanical load

- 24 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

z

q2

T2 o

x

q1

T1

(b) Front view y b o

a

(a) 3D model

x

(c) Mid-plane Fig. 1 Schematic diagram of FGM elliptical plate and its coordinates: (a) 3D model, (b) front view, and (c) mid-plane

- 25 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

0 2WGPL

0 1WGPL

0 3WGPL

0 2WGPL

(a)

(b)

(c)

Fig. 2 Schematic GPL distribution patterns along the plate thickness: (a) linear, (b) parabolic; and (c) uniform

- 26 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

2.4 [5](ceramic) [5](n=1) [5](metal) Present(ceramic) Present(n=1) Present(metal)

1.8

w1

1.2 0.6

0 -0.6 -1.2 -0.5

-0.375

-0.25

-0.125

0

0.125

0.25

0.375

0.5

z/h

Fig. 3 Through-thickness deflection distribution of the FGM elliptical plate under a thermal load

- 27 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

0

[5](ceramic) [5](n=1) [5](metal) Present(ceramic) Present(n=1) Present(metal)

-500

σx

*

-1000

-1500

-2000

-2500 -0.5

-0.375

-0.25

-0.125

0

0.125

0.25

0.375

0.5

z/h Fig. 4 Through-the-thickness distribution of the dimensionless normal stress  x* of the FGM elliptical plate under a thermal load

- 28 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

8 λ=-1 λ=0 λ=1 λ=-2 λ=2

7 6

0

c11/c11

5 4 3 2 1

0 -1 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

z/h Fig. 5 Through-thickness distribution of elastic constant c11 c110 of the transversely isotropic functionally graded

elliptical plate

- 29 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

20 18 16

w/h×105

λ=-1

λ=0

λ=1

14 12 10 8 6 4 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

z h

(a)

3 2

σx/q1

1 0 -1

-2 λ=-1

-3

λ=0

λ=1

-4 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

z h

(b)

Fig. 6 Through-thickness deflection and stress distributions of the - 30 normal transversely isotropic functionally graded elliptical plate under a mechanical load: (a) deflection; and (b) normal stress

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

3.5

2.5

w/h×103

λ=-1

λ=0

λ=1

1.5

0.5

-0.5

-1.5 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

z h

(a)

10 0

0

σx/c11×104

-10 -20 -30 -40 -50 λ=-1

-60

λ=0

λ=1

-70 -80 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

z h (b)

Fig. 7 Through-thickness deflection and normal stress distributions of the transversely isotropic functionally graded elliptical plate under a thermal load: (a) deflection; and (b) normal stress - 31 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

2.2

λ₁=2(W=L) λ₂=3(W=L) λ₃=1(W=L) λ₁=2(W=0.5L) λ₂=3(W=0.5L) λ₃=1(W=0.5L)

2 1.8

w/h×106

1.6 1.4 1.2

1 0.8 0.6 0.4 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t/L (×10-3) (a)

1.4

1.2

σx/Em×106

1

λ₁=2(W=L)

λ₂=3(W=L)

λ₃=1(W=L)

λ₁=2(W=0.5L)

λ₂=3(W=0.5L)

λ₃=1(W=0.5L)

0.8 0.6 0.4 0.2 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

-3

t/L (×10 ) (b)

Fig. 8 Effects of GPL dimension and geometry on the bending bahavior of the functionally graded GPL reinforced elliptical plate under a mechanical load: (a) deflection; and (b) normal stress - 32 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

30

w/h×106

25 λ₁=2

20

λ₂=3

λ₃=1

15 10 5 0 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

ab (a)

16 14

σx/Em×106

12 λ₁=2

10

λ₂=3

λ₃=1

8 6 4 2 0 -2 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

ab (b)

Fig. 9 Effect of major to minor axis ratio on the deflection and normal stress on the top surface of the functionally graded GPL reinforced elliptical plate under a mechanical load: (a) deflection; and (b) normal stress - 33 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

2.95

w/h×102

2.925

2.9 λ₁=2

λ₂=3

λ₃=1

2.875

2.85

2.825 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

ab (a)

-10

σx/Em×102

-20

-30

-40 λ₁=2

λ₂=3

λ₃=1

-50

-60 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

ab

(b)

Fig. 10 Effect of major to minor axis ratio on the deflection and normal stress on the top surface of the functionally graded GPL reinforced elliptical plate under a - 34and - (b) normal stress thermal load: (a) deflection;

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

6 5

w/h×106

4 3

a/b=1

a/b=2

a/b=10

2 1 0 -1 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

z h (a)

6

σx/Em×106

4

a/b=1

a/b=2

a/b=10

2 0 -2 -4

-6 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

z h (b)

Fig. 11 Effect of major to minor axis ratio on the bending behavior of the functionally graded GPL reinforced elliptical plate under a mechanical load: (a) deflection; and (b) normal stress

- 35 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

1.4 λ₁=2© λ₂=3© λ₃=1© λ₁=2® λ₂=3® λ₃=1®

w/h×10

1.2

1 0.8 0.6 0.4 0.2 0

1

2

3

4

5

6

7

8

9

10

Load parameter P

Fig. 12 Dimensionless deflection versus load parameter P of the functionally graded GPL reinforced elliptical plate under a combined thermo-mechanical load

- 36 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Captions of Tables

Table 1. Dimensionless deflection w0 of the clamped elliptical plate under a uniform load. Table 2. Material properties of hexagonal Zinc Table 3. Dimensionless deflection and normal stresses of the clamped transversely isotropic functionally

graded elliptical plate under a uniform load. Table 4. Relationship between gradient index and total weight fraction of GPLs

- 37 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Table 1. Dimensionless deflection w0 of the clamped elliptical plate under a uniform load

a/b

Çeribaşı et al. [32]

Zhang [33]

Present

1

0.01563

0.01563

0.01562

1.2

0.02142

0.02143

0.02142

1.4

0.02603

0.02604

0.02603

1.6

0.02949

0.02950

0.02949

1.8

0.03203

0.03204

0.03203

2

0.03390

0.03391

0.03390

3

0.03835

0.03836

0.03835

4

0.03985

0.03986

0.03985

5

0.04052

0.04054

0.04052

7

0.04109

0.04110

0.04109

9

0.04132

0.04133

0.04132

10

0.04139

0.04140

0.04139

15

0.04154

0.04156

0.04154

20

0.04160

0.04161

0.04160

- 38 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Table 2. Material properties of hexagonal Zinc

Elastic constants (GPa)

c110

c120

c130

0 c33

0 c44

162.8

50.8

36.2

62.7

38.5

Thermal moduli ( 105 /K)

Thermal conductivity (W/mK)

10

 30

k0

0.5818

1.535

124

- 39 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Table 3. Dimensionless deflection and normal stresses of the clamped transversely isotropic functionally graded elliptical plate subject to a uniform load.



w(0,0, h 2) h (105 )

 x (0,0, h 2) q1

 y (0,0, h 2) q1

-3

47.5685

0.6295

1.3283

-2

28.7039

0.9076

1.8968

-1

17.3691

1.2687

2.6228

0

10.6435

1.7378

3.5515

1

6.6219

2.3576

4.7651

2

4.1587

3.1954

6.3995

3

2.6054

4.3457

8.6531

- 40 -

3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates

Table 4. Relationship between gradient index and total weight fraction of GPLs 1 (Linear

2 (Paraboli

3 (Unifor

* (% WGPL

)

c)

m)

)

0

0

0

0

2/3

1

1/3

0.33

2

3

1

1

- 41 -