Mechanical and thermal post-buckling analysis of FGM rectangular plates with various supported boundaries resting on nonlinear elastic foundations

Thin-Walled Structures 89 (2015) 142–151

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Mechanical and thermal post-buckling analysis of FGM rectangular plates with various supported boundaries resting on nonlinear elastic foundations Da-Guang Zhang n, Hao-Miao Zhou College of Information Engineering, China Jiliang University, Hangzhou 310018, China

art ic l e i nf o

a b s t r a c t

Article history: Received 21 September 2014 Received in revised form 18 November 2014 Accepted 29 December 2014

Mechanical and thermal post-buckling analysis is presented for FGM rectangular plates resting on nonlinear elastic foundations using the concept of physical neutral surface and high-order shear deformation theory, and investigations on post-buckling behavior of FGM rectangular plates with two opposite simply supported edges and other two opposite clamped edges are also new. Approximate solutions of FGM rectangular plates are given out using multi-term Ritz method, and influences played by different supported boundaries, foundation stiffnesses, thermal environmental conditions and volume fraction index are discussed in detail. It is worth noting that the effect of nonlinear elastic foundation is small at the pre-buckling and initial post-buckling state and is significant with increasing deflection at the deep post-buckling state. Especially, comparisons of post-buckling for FGM rectangular plates resting on nonlinear elastic foundations with movable simply supported edge subjected to compression acting on the geometric middle surface and the physical neutral surface are innovative, and may be helpful to clarify typical mistakes in literature. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Functionally graded materials Nonlinear elastic foundations Physical neutral surface High order shear deformation theory Post-buckling

1. Introduction Functionally graded materials (FGMs) are usually made from a mixture of metals and ceramics phases with a continuously variable composition. FGMs possess a number of advantages that make them attractive in potential applications, including a potential reduction of in-plane and transverse through-the-thickness stresses, an improved residual stress distribution, enhanced thermal properties, higher fracture toughness, and reduced stress intensity factors. Comprehensive works on the bucking and post-bucking of FGM plates have been reported in the literature, for example, Javaheri and Eslami [1–3] presented buckling of FGM rectangular plates under inplane mechanical or thermal loads based on the classical and higherorder shear deformation plate theories, respectively. Najafizadeh and Eslami [4,5] considered axisymmetric buckling of simply supported and clamped circular FGM plates under uniform radial or thermal loads. Ma and Wang [6–8] investigated the post-buckling and bending behavior of FGM circular plates under uniformly distributed radial compression and thermal loading on the basis of classical

n

Corresponding author. E-mail address: [email protected] (D.-G. Zhang).

http://dx.doi.org/10.1016/j.tws.2014.12.021 0263-8231/& 2015 Elsevier Ltd. All rights reserved.

nonlinear plate theory. Wu [9] studied thermal post-buckling behavior of simply supported FGM rectangular plates under uniform temperature rise and gradient through the thickness based on the first-order shear deformation plate theory. Liew et al. [10,11] presented compressive post-buckling and thermal post-buckling behavior of FGM plates with two opposite edges clamped and with or without surface-bonded piezoelectric actuators. Na and Kim [12–14] used solid finite elements to calculate buckling temperature of FGM plates with fully clamped edges. Woo et al. [15] studied the postbucking behavior of FGM plates and shallow shells under edge compressive loads and a temperature field based on the higher order shear deformation theory. Park and Kim [16] presented thermal postbucking and vibration of simply supported FGM plates with temperature dependent materials properties by using finite element method. Li et al. [17] analyzed nonlinear thermo-mechanical postbuckling of circular FGM plate with geometric imperfection. More works can be available in Refs. [18–31]. It has been pointed out by Shen [32], the governing differential equations for FGM plates are identical in form to those of asymmetric cross-ply laminated plates, and the bifurcation buckling does not exist due to the stretching/bending coupling effect for FGM plates with simply supported edges, as previously proved by Leissa [33], and Qatu and Leissa [34]. Therefore, Shen [35] indicated that the buckling solutions obtained by Javaheri and Eslami [1–3] and Wu [9]

D.-G. Zhang, H.-M. Zhou / Thin-Walled Structures 89 (2015) 142–151

for simply supported FGM plates subjected to uniaxial compression and/or thermal loads may be incorrect. Recently, Naderi and Saidi [36] pointed out that those reported results in Refs. [1,4,19,37–41] may be correct, if the in-plane compressive loads act on the physical neutral surface. This physical neutral surface is defined by that there are no strains and stresses on this surface when the plate is subjected to pure bending. Zhang and Zhou [42] derived the governing equations of a FGM thin rectangular plate based on concept of physical neutral surface and classical laminated plate theory, the explicit and simple form solutions for vibration, bucking, linear and nonlinear bending were obtained. Moreover, Ma and Lee [43,44] derived governing equations for both the static behavior and the dynamic response of FGM beams subjected to uniform inplane thermal loading, based on the physical neutral surface and the first order shear deformation beam theory. Prakash et al. [45] investigated the influence of the physical neutral surface position on the nonlinear stability behavior of FGM plates. Singha et al. [46] discussed the bending behavior of FGM plates and concluded that physical neutral surface theories can make a noticeable difference in results with geometric middle surface theories. Fekrar et al. [47] put forward a new five-unknown refined theory based on physical neutral surface position for bending analysis of exponential graded plates. As pointed out by Abrate [48], the stretching–bending coupling can easily be eliminated in the governing equations based on the classical plate theory and/or the first order shear deformation theory by an appropriate choice of the reference surface, and it is difficult to achieve based on a high order shear deformation theory, while models of beam, plate and shell [49–53] are successfully put forward by physical neutral surface and high order shear deformation theory. Especially, it is worth noting that comparisons of postbuckling load–deflection curves for FGM rectangular plates with movable simply supported edge subjected to biaxial compression acting on the geometric middle surface and the physical neutral surface are presented in Ref. [52]. Investigations in nonlinear behaviors analysis of FGM plates resting on an elastic foundation are limited in number. Kiani and Eslami [31] presented thermal buckling and post-buckling response of imperfect temperature-dependent sandwich FGM plates resting on elastic foundation. Yang and Shen [54] studied a large deflection analysis of thin FGM plates resting on a Pasternak elastic foundation subjected to combined transverse and in-plane loads. Shen and Wang [55] presented nonlinear bending of FGM plates subjected to combined loading and resting on an elastic foundation of Pasternaktype, then Wang and Shen [56] investigated nonlinear analysis of sandwich plates with FGM face sheets resting on an elastic foundation. Shen and Zhu [57] examined post-buckling of sandwich plates with nanotube-reinforced composite face sheets resting on elastic foundations. Duc and Tung [58] calculated mechanical and thermal post-buckling of higher order shear deformable functionally graded plates on elastic foundations. Sepahi et al. [59] presented large deflection analysis of thermo-mechanical loaded annular FGM plates on nonlinear elastic foundation via differential quadrature method. Zhang [60] first investigated nonlinear bending analysis of functionally graded elliptical plates resting on two-parameter elastic foundations, then Zhang [61] examined nonlinear bending analysis of FGM rectangular plates with various supported boundaries resting on two-parameter elastic foundations based on the concept of physical neutral surface and high-order shear deformation theory. The present paper extends the previous works [52,61] to the case of post-bucking analysis for FGM rectangular plates resting on nonlinear elastic foundations. The material properties of FGMs are assumed to be graded in thickness direction according to a volume fraction power law distribution and are expressed as a nonlinear function of temperature. Nonlinear approximate solutions of FGM rectangular plates with various boundary conditions are given out using multi-term Ritz method. Influences played by different

143

supported boundaries, thermal environmental conditions, foundation stiffness and volume fraction index are discussed in detail.

2. Modeling of FGM plates resting on nonlinear elastic foundations based on physical neutral surface and high order shear deformation theory Consider a FGM rectangular plate of length Lx , width Ly and thickness h, which is made from a mixture of ceramics and metals. The coordinate system is illustrated in Fig. 1. The material properties (like Young’s modulus E, thermal conductivity κ and thermal expansion coefficient α) are temperature-dependent, and vary along the thickness, while Poisson’s ratio ν depends weakly on temperature change and position and is assumed to be a constant. As is commonly done, the foundation is assumed to be an attached foundation, meaning no part of the plate lifts off the foundation in the deformed region. The load–displacement relationship of the foundation is assumed to be p ¼ K 1 w  K 2 ∇2 w þK 3 w3 , where p is the force per unit area, K 1 is the Winkler foundation stiffness and K 2 is a constant showing the effect of the shear interactions of the vertical elements, K 3 is nonlinear elastic foundation coefficients, and ∇2 is the Laplace operator in x and y. According to the model of the FGM rectangular plates [52] based on physical neutral surface (z ¼ z0 ) and high order shear deformation theory, the displacements, the strains and the stresses have the same form as the previous works [52,61]. For the sake of simplicity, the deducing process of the formulae is omitted, and the governing equations can be derived according to energy variational principle. !   2 2 2 ~ 11 ∂ ψ x þ 1  ν ∂ ψ x þ 1 þ ν ∂ ψ y  4F 11 ∂ ∇2 w  A~ 44 ψ þ ∂w  ∂M T ¼ 0 D x 2 ∂x ∂x 2 ∂y2 2 ∂x∂y ∂x ∂x2 3h

ð1aÞ !   2 2 2 ~ 11 ∂ ψ y þ 1  ν ∂ ψ y þ 1 þ ν ∂ ψ x  4F 11 ∂ ∇2 w  A~ 44 ψ þ ∂w  ∂M T ¼ 0 D y 2 2 2 2 ∂x 2 ∂x∂y ∂y ∂y ∂y 3h ∂y

ð1bÞ

A~ 44



þ

∂ψ y ∂ψ x 2 ∂x þ ∂y þ∇ w



þ 4F 112 ∇2 3h



∂ψ y ∂ψ x ∂x þ ∂y



 16H411 ∇4 w  9h

4 ∇2 P T 3h2

 2 2  ∂ φ∂ w ∂2 φ ∂2 w ∂2 φ ∂2 w þ  K 1 w þ K 2 ∇2 w  K 3 w3 þ q ¼ 0  2 ∂x∂y ∂x∂y ∂x2 ∂y2 ∂y2 ∂x2

ð1cÞ " 4

2

2

∇ φ þ ð1  νÞ∇ N T ¼ ð1 ν ÞA11

# 2 ∂2 w ∂2 w ∂2 w  2 ∂x∂y ∂x ∂y2

ð1dÞ

All symbols used in Eqs. (1a–d) are defined in Refs. [52,61]. Depending upon the in-plane behavior at the edges, eight cases will be considered.

Fig. 1. Geometry and coordinates of a FGM rectangular plate resting on nonlinear elastic foundation.

144

D.-G. Zhang, H.-M. Zhou / Thin-Walled Structures 89 (2015) 142–151

Case 1: The edges are simply supported and freely movable in both the x and y directions, and the biaxial edge loads are acting in the x- and y-directions on the geometric middle surface, referred to as SS1. Case 2: The edges are simply supported and freely movable in both the x and y directions, and the biaxial edge loads are acting in the x- and y-directions on the physical neutral surface, referred to as SS2. Case 3: All four edges are simply supported with no in-plane displacements, i.e. prevented from moving in the x- and y-directions, referred to as SS3. Case 4: The edges are clamped and freely movable in both the x and y directions, and the biaxial edge loads are acting in the x- and y-directions on the geometric middle surface, referred to as CC1. Case 5: All four edges clamped supported with no in-plane displacements, referred to as CC2. Case 6: Two opposite edges are simply supported and freely movable in the x direction and two opposite edges are clamped and freely movable in the y direction, and the biaxial edge loads are acting in the x- and y-directions on the geometric middle surface, referred to as SC1. Case 7: Two opposite edges are simply supported and freely movable in the x direction and two opposite edges are clamped and freely movable in the y direction, and the x-axial edge loads are acting in the x-directions on the physical neutral surface and the y-axial edge loads are acting in the y-directions on the geometric middle surface, referred to as SC2. Case 8: Two opposite edges are simply supported with no inplane displacements in the x direction and two opposite edges are clamped with no in-plane displacements in the y direction, referred to as SC3.

in which Δn is the plate displacement in the n-direction, and defined by h2 i R L n R Ln n 2 ∂ φ 1   ν∂∂nφ2 þ ð1  νÞN T Δn ¼ 0 n ∂u ∂n z ¼ 0 dn ¼ 0 A11 ð1  ν2 Þ ∂s2    ) 1 ∂w 2 ∂ψ 4c0 ∂2 w ∂ψ n dn ð3Þ  z0 n þ 2 þ  2 ∂n ∂n 3h ∂n2 ∂n

3. Multi-term Ritz method for approximate solutions of nonlinear problems of FGM rectangle plates A multi-term Ritz method is adopted in this section to obtain approximate solutions of FGM rectangular plates. The key issue is first to assume the deflection of the plate w¼

M X

ð4Þ

Wm

m¼1

where M is total number of series. For symmetry problems about the plate with four edges simply supported, it can be assumed that 2m 1 X

Wm ¼

aij sin

i ¼ 1;3⋯

iπx jπy sin Lx Ly

ðj ¼ 2m  iÞ

ð5aÞ

where aij are undetermined coefficients. It can be assumed that the temperature variation is uniform or occurs in the thickness direction only, i.e. NT , M T and P T are constants for FGM plates. Substituting Eq. (5a) into Eqs. (1a), (1b) and (1d), ψ x , ψ y and φ can be determined as ψx ¼

XX i

cij cos

j

XX iπx jπy iπx jπy sin ; ψy ¼ dij sin cos Lx Ly L Ly x i j ð5b; cÞ

For these eight cases the associated boundary conditions are φ ¼ ð1  ν4 ÞA11 2

^ n M n ¼  z0 N ^ n ; Pn w ¼ ψ s ¼ 0; N ns ¼ 0; Nn ¼ N ^ n ; ðfor case SS1Þ ¼  c0 N

ð2aÞ

^ n ; M n ¼ P n ¼ 0; w ¼ ψ s ¼ 0; N ns ¼ 0; Nn ¼ N

ð2bÞ

w ¼ ψ s ¼ 0; N ns ¼ 0; Δn ¼ 0; M n ¼  z0 ∂2 φ ¼  c0 2 ; ∂s

ðfor case SS2Þ

2

∂ φ ; Pn ∂s2

ðfor case SS3Þ

ð2cÞ

w¼

∂w ^ n; ¼ ψ n ¼ ψ s ¼ 0; N ns ¼ 0; Nn ¼ N ∂n

w¼

∂w ¼ ψ n ¼ ψ s ¼ 0; N ns ¼ 0; Δn ¼ 0; ∂n

ðfor case CC1Þ ðfor case CC2Þ

ð2dÞ ð2eÞ

^ x ; P x ¼  c0 N ^ x ; ðat x ¼ 0; Lx Þ ^ x ; M x ¼ z0 N w ¼ ψ y ¼ N xy ¼ 0; Nx ¼ N ∂w ^ y; ¼ ψ y ¼ ψ x ¼ Nxy ¼ 0; Ny ¼ N w¼ ∂y   ðfor case SC1Þ at y ¼ 0; Ly ^x w ¼ ψ y ¼ N xy ¼ M x ¼ P x ¼ 0; N x ¼ N

ð2f Þ

ðat x ¼ 0; Lx Þ

  ∂w ^ y at y ¼ 0; Ly ¼ ψ y ¼ ψ x ¼ Nxy ¼ 0; Ny ¼ N w¼ ∂y

ð2gÞ w ¼ ψ y ¼ N xy ¼ Δx ¼ 0; M x ¼  z0 ∂∂yφ2 ; P x ¼ c0 ∂∂yφ2 ; ðat x ¼ 0; Lx Þ 2

i

j

k

l

1Þ Bðijkl cos

ði  kÞπx Lx

cos

ðj  lÞπy Ly

ði þ kÞπx ðj þ lÞπy ði  kÞπx ðjþ lÞπy 3Þ cos þ Bðijkl cos cos Lx Ly Lx Ly ði þ kÞπx ðj  lÞπy y2 x2 ð5dÞ cos cos þ N x þN y Lx Ly 2 2

2Þ cos þ Bðijkl 4Þ þ Bðijkl

Note that Eqs. (5a–c) satisfy simply supported boundary condition w¼ψ x ¼ 0 (at y¼ 0, Ly) and w¼ψ y ¼0 (at x ¼0, Lx) identically. For symmetry problems about the plate with four edges clamped, it can be assumed that    m X 2iπx 2jπy aij 1  cos 1  cos j ¼ miþ1 ð6aÞ Wm ¼ Lx Ly i¼1 Similarly, ψ x , ψ y and φ can be determined as  X X ð1Þ 2iπx 2jπy ð2Þ 2iπx ψx ¼ cij sin cos þ cij sin Lx Ly Lx i j ψy ¼

 X X ð1Þ 2iπx 2jπy 2jπy ð2Þ dij cos sin þ dij sin Lx Ly Ly i j

φ ¼ ð1  ν4 ÞA11 2

ðfor case SC2Þ

PPPPn

PPPPn i

j

k

l

1Þ Bðijkl cos

2ði  kÞπx Lx

cos

ðfor case SC3Þ ð2hÞ

ð6cÞ

2ðj  lÞπy Ly

2Þ þ Bðijkl cos

2ði þkÞπx 2ðj þlÞπy 2ði kÞπx 2ðj þ lÞπy 3Þ cos þ Bðijkl cos cos Lx Ly Lx Ly

4Þ þ Bðijkl cos

2ði þ kÞπx 2ðj lÞπy 2iπx 2lπy 5Þ cos þ Bðijkl cos cos Lx Ly Lx Ly

6Þ þ Bðijkl cos

2ði þ kÞπx 2lπy 2ði kÞπx 2lπy 7Þ cos þ Bðijkl cos cos Lx Ly Lx Ly

2

  ∂w ¼ ψ y ¼ ψ x ¼ Nxy ¼ Δy ¼ 0; aty ¼ 0; Ly w¼ ∂y

ð6bÞ

D.-G. Zhang, H.-M. Zhou / Thin-Walled Structures 89 (2015) 142–151

2iπx 2ðjþ lÞπy 2iπx 2ðj  lÞπy 9Þ cos þBðijkl cos cos Lx Ly Lx Ly

8Þ þ Bðijkl cos

þ Nx

y2 x2 þ Ny 2 2

ð6dÞ

For symmetry problems about the plate with two opposite edges simply supported in the x direction and two opposite edges clamped in the y direction, it can be assumed that   2m 1 X iπx 2jπy Wm ¼ ð2j ¼ 2m  i þ1Þ ð7aÞ aij sin 1  cos Lx Ly i ¼ 1;3⋯

XX i

dij sin

j

φ ¼ ð1  ν4

2

iπx 2jπy sin Lx Ly

n ÞA11 P P P P i

2Þ þ Bðijkl cos

j

k

l

1Þ Bðijkl cos

ð7b; cÞ ði  kÞπx Lx

cos

2ðj  lÞπy Ly

ðiþ kÞπx 2ðj þ lÞπy ði kÞπx 2ðjþ lÞπy 3Þ cos þ Bðijkl cos cos Lx Ly Lx Ly

ðiþ kÞπx 2ðj  lÞπy ði kÞπx 2lπy 5Þ cos þ Bðijkl cos cos Lx Ly Lx Ly ðiþ kÞπx 2lπy y2 x2 ð7dÞ þ Nx þ Ny cos cos Lx Ly 2 2

4Þ þ Bðijkl cos 6Þ þ Bðijkl

All symbols used in Eqs. (5a–d)–(7a–d) are given in detail in Appendix A, B and C. Nx and Ny are average membrane forces in Eqs. (5a–d)–(7a–d). For movable in-plane boundary conditions Z Z 1 Ly ^ 1 Lx ^ N x dy; N y ¼ N y dx Nx ¼ ð8a; bÞ Ly 0 Lx 0 and for immovable in-plane boundary conditions, N x and N y are caused by thermal loads and can be expressed by h    i RL RL n 2 4c0 ∂2 w ∂ψ x x N x ¼ Lx1Ly 0 y 0 x A11 12 ∂w þ z0 ∂ψ ∂x ∂x  3h2 ∂x2 þ ∂x "   )  # ∂ψ y 4c0 ∂2 w ∂ψ y 1 ∂w 2  N ð8cÞ  2 þ z0 þ þ A12 T dxdy 2 ∂y ∂y 3h ∂y2 ∂y N y ¼ Lx1Ly

R Ly R Lx n 0

0

A21

h   1 ∂w 2 2 ∂x

4c0 x þ z0 ∂ψ ∂x  2 3h



∂2 w ∂ψ x þ ∂x ∂x2

iterative method or other equivalent methods. Substituting these coefficients back into Eqs. (5a–d)–(7a–d), w, ψ x , ψ y and φ may then be completely determined. In addition, Eqs. (5a–d)–(7a–d) are suitable for analysis of FGM plates if critical buckling mode is (1, 1), and other modes are not discussed, critical mechanical and thermal buckling loads can be easily obtained by making solutions of deflection coefficients aij approach to zero.

4. Results and discussion Numerical results are presented in this section for postbuckling of FGM rectangular plates. The effective material properties P of FGMs, such as Young’s modulus E, thermal conductivity κ and thermal expansion coefficient α, can be expressed as

ψ x , ψ y and φ can be determined as  X X ð1Þ iπx 2jπy iπx cij cos cos þ cðij2Þ cos ; ψx ¼ Lx Ly Lx i j ψy ¼

145

i

"   )  # ∂ψ y 4c0 ∂2 w ∂ψ y 1 ∂w 2   NT dxdy þ z0 þ þ A22 2 ∂y ∂y 3h2 ∂y2 ∂y

P ¼ Pc V c þ Pm V m

ð10Þ

in which Lx =h ¼ Ly =h ¼ 20 and V c are the metal and ceramic volume fractions and are related by V m þ V c ¼ 1, P m and P c denote the temperature-dependent properties of metal and ceramic plate, respectively, and may be expressed as a nonlinear function of temperature [62] P ¼ P 0 ðP  1 T  1 þ 1 þ P 1 T þ P 2 T 2 þ P 3 T 3 Þ

in which T ¼ T 0 þΔT and T 0 ¼300 K (room temperature), P  1 , P 0 , P 1 , P 2 and P 3 are the coefficients of temperature T (K) and are unique to the constituent materials. A Si3N4/SUS304 is selected as an example. Typical values for Young’s modulus E (in Pa), thermal expansion coefficient α (in K  1), and thermal conductivity κ(in W/ mK) are listed in Table 1 from Reddy and Chin [63]. Poisson’s ratio ν is assumed to be a constant, and ν ¼ 0:28. One dimensional temperature field is assumed to be constant in the x–y plane of the layer in respect that temperature variation is uniform or occurs in the thickness direction only. In such a case, the temperature distribution along the thickness can be obtained by solving a steady-state heat transfer equation

 d dT  κðz; TÞ ¼0 ð12Þ dz dz This equation can be solved by imposing boundary condition of T ¼ T t at the top surface (z ¼  h=2) and T ¼ T b at bottom surface (z ¼ h=2). The solution of this equation is

Rz

ð8dÞ

1  2h κðz;TÞdz

T ¼ T t  ðT t  T b ÞR h

2

 2h

Substituting w, φ, ψ x and ψ y into the following expression ∂Π ¼0 ∂aij

ð11Þ

ð9Þ

Cubic algebraic equations about aij can be obtained. As for FGM plate ^ n , thermal loads ΔT) with given loads (like normal boundary loads N and other known coefficients, aij can be solved by Newton–Raphson

ð13Þ

1 κðz;TÞdz

Note that the temperature field is uniform when T t ¼ T b . In addition, the present paper extends convergence studies of the previous works [52,61] to mechanical and thermal postbuckling analysis of FGM rectangular plates, in consideration of both simplicity and convergence, M ¼ 3 is used in all the following calculations.

Table 1 Temperature-dependent coefficients for ceramic and metals (from Reddy and Chin [63]). Material

Properties

P 1

P0

P1

P2

P3

Si3N4

E (Pa) α(K  1) κ(W/mK) E (Pa) α(K  1) κ(W/mK)

0 0 0 0 0 0

348.43e þ 9 5.8723e  6 13.723 201.04e þ9 12.330e 6 15.379

 3.070e 4 9.095e  4  1.032e  3 3.079e  4 8.086e  4  1.264e  3

2.160e  7 0 5.466e  7  6.534e  7 0 2.092e  6

 8.946e  11 0  7.876e  11 0 0  7.223e  10

SUS304

146

D.-G. Zhang, H.-M. Zhou / Thin-Walled Structures 89 (2015) 142–151

4 Bhimaraddi and Chandashekhara [64] Shen [65] Duc and Tung [58] Present

ΔT/ΔTcr

3

2

1

0 0.0

0.4

0.8

1.2

wcenter /h Fig. 2. Comparisons of thermal post-buckling load–deflection curves for isotropic square plates with SS3.

present method, two examples are solved for thermal buckling and post-buckling analysis of isotropic and FGM square plates. Example 1. The thermal post-buckling load–deflection curves for isotropic square plates (Lx =h ¼ Ly =h ¼ 10, ν ¼ 0:3) with SS3 in uniform temperature rise fields are compared in Fig. 2 with the analytical solutions of Bhimaraddi and Chandashekhara [64], perturbation solutions of Shen [65] based on higher order shear deformation theory, approximate solutions of Duc and Tung [58] using Galerkin method, where the dimensionless critical temperature is λnT ¼ αΔT cr  104 ¼ 119:783. Example 2. Critical thermal buckling loads of Si3N4/SUS304 FGM square plates with CC2 subjected to different temperature fields are calculated and compared with results of Bateni et al. [66] in Table 2. The present results agree well with most results of Bateni et al. [66], although some discrepancies are seen to exist in heat conduction fields. In summary, acceptable agreement can be seen from Fig. 2 and Table 2, and thus the validity and accuracy of the present method can be confirmed. 4.2. Parametric studies

Table 2 Comparisons of critical thermal buckling loads of Si3N4/SUS304 square plates with CC2 subjected to different temperature fields. Lx =h

N

Uniform temperature rise

Heat conduction

Bateni et al. [66]

Bateni et al. [66]

Present

Present

100

0 0.5 1 2 3 5

44.136 32.771 29.553 27.340 26.446 25.501

44.188 32.859 29.667 27.455 26.545 25.576

88.944 71.025 64.180 58.576 55.989 53.149

87.782 70.601 63.967 58.526 55.955 53.114

25

0 0.5 1 2 3 5

519.840 409.123 376.250 353.515 344.294 334.265

519.154 409.261 376.805 354.099 344.676 334.311

1100.199 932.028 854.879 782.418 744.626 700.558

996.646 910.157 853.660 795.811 763.190 722.806

15

λ*

10

A parametric study was undertaken for nonlinear bending of Si3N4/SUS304 square plates with  Lx =h ¼ LyN=h ¼ 50, the volume fraction V c is defined by V c ¼ 1=2  z=h . The dimensionless foundation stiffnesses are (k1 , k2 , k3 )¼(50, 0, 0) for the Winkler elastic foundation, (k1 , k2 , k3 )¼ (50, 5, 0) for the Pasternak elastic foundation, (k1 , k2 , k3 )¼(50, 0, 25) and (k1 , k2 , k3 )¼ (50, 5, 25) for 3 the nonlinear elastic foundation, where k1 ¼ K 1 Lx 4 =E0 h , k2 ¼ K 2 Lx 2 3 4 =E0 h and k3 ¼ K 3 Lx =E0 h, E0 is Young’s modulus of SUS304 at reference temperature. The top surface is ceramic-rich, whereas the bottom surface is metal-rich, hence T t ¼ T c and T b ¼ T m for heat conduction. Post-buckling deflections of Si3N4/SUS304 FGM square plates resting on elastic foundations with various boundary conditions subjected to biaxial compression in room temperature fields are calculated, see Figs. 3–8, in which the non-dimensional compres^ x Lx 2 =E0 h3 ¼  N ^ y Ly 2 =E0 h3 , and the nonsion is defined by λn ¼  N dimensional central deflection is wcenter =h. It can be observed that bifurcation of bucking occurs and critical bucking loads decrease with increasing the value of volume fraction index N for FGM plates with SS2, SC2 and CC1, while bifurcation of bucking can’t occur for FGM plates with SS1 and SC1. Thermal post-buckling behaviors for Si3N4/SUS304 FGM square plates resting on elastic foundations in uniform temperature rise and heat conduction fields are calculated, see Figs. 9–20. It can be found that bifurcation 12

5

T =300K, T =300K

Si N SS1

( k , k , k )=(50, 5, 25)

N=0.5 SS1 N=0.5 SS2 N=2 SS1 N=2 SS2 SUS304 SS1

0

( k , k , k )=(50, 5, 0) SS1

8

( k , k , k )=(50, 5, 0) SS2 ( k , k , k )=(50, 0, 0) SS1

λ*

L /h=50

T =300K, T =300K N=2, L /h=50

0.0

-0.4

-0.8

( k , k , k )=(50, 0, 0) SS2

-1.2

wcenter /h

4 ( k , k , k )=(50, 0, 25) SS1

Fig. 3. Post-buckling load-deflection curves for Si3N4/SUS304 square plates with SS1 and SS2 resting on nonlinear elastic foundation subjected to biaxial compression.

( k , k , k )=(50, 0, 25) SS2 ( k , k , k )=(50, 5, 25) SS1 ( k , k , k )=(50, 5, 25) SS2

4.1. Comparison studies The accuracy and effectiveness of the present method for the rectangular plates were examined by many comparison studies in Refs. [52,61], in order to further ensure the effectiveness of the

0 0.0

-0.4

-0.8

-1.2

wcenter /h Fig. 4. Effect of elastic foundations on post-buckling behaviors of Si3N4/SUS304 square plates with SS1 and SS2.

D.-G. Zhang, H.-M. Zhou / Thin-Walled Structures 89 (2015) 142–151

147

18

18

( k , k , k )=(50, 0, 0) ( k , k , k )=(50, 5, 0) ( k , k , k )=(50, 0, 25)

15

( k , k , k )=(50, 5, 25)

λ*

λ*

12

6

T =300K, T =300K

Si N SC1

( k , k , k )=(50, 5, 25)

N=0.5 SC1 N=0.5 SC2 N=2 SC1 N=2 SC2 SUS304 SC1

L /h=50

12 T =300K, T =300K N=2 L /h=50

9

6

0 0.0

-0.4

-0.8

-1.2

0.0

0.4

0.8

1.2

wcenter /h

wcenter /h Fig. 5. Post-buckling load–deflection curves for Si3N4/SUS304 square plates with SC1 and SC2 resting on nonlinear elastic foundation subjected to biaxial compression.

Fig. 8. Effect of elastic foundations on mechanical post-buckling behaviors of Si3N4/SUS304 square plates with CC1 subjected to biaxial compression.

400

15

T =300K, T =300K

T =300K+ΔT, T =300K+ΔT

N=2, L /h=50

( k , k , k )=(50, 5, 25) L /h=50

300

( k , k , k )=(50, 5, 0) CS2 ( k , k , k )=(50, 0, 0) CS1

ΔT

( k , k , k )=(50, 5, 0) CS1

10

200

λ*

( k , k , k )=(50, 0, 0) CS2 Si N N=0.5 N=2 SUS304

100

5

(k

, k )=(50, 0, 25) CS1

( k , k , k )=(50, 0, 25) CS2 ( k , k , k )=(50, 5, 25) CS1

0

( k , k , k )=(50, 5, 25) CS2

0.0

0 0.0

-0.4

-0.8

0.4

Fig. 6. Effect of elastic foundations on post-buckling behaviors of Si3N4/SUS304 square plates with CS1 and CS2.

1.2

wcenter /h

-1.2

wcenter /h

0.8

Fig. 9. Thermal post-buckling behaviors for Si3N4/SUS304 square plates with SS3 resting on nonlinear elastic foundation subjected to uniform temperature rise.

300

20

18

T =300K, T =300K

Si N

( k , k , k )=(50, 5, 25)

N=0.5 N=2 SUS304

L /h=50

T =300K+ΔT, T =300K+ΔT N=2, L /h=50

200

λ*

ΔT

16

14

100

( k , k , k )=(50, 0, 0) ( k , k , k )=(50, 5, 0)

12

( k , k , k )=(50, 0, 25) ( k , k , k )=(50, 5, 25)

10 0.0

0.4

0.8

1.2

wcenter /h Fig. 7. Post-buckling load–deflection curves for Si3N4/SUS304 square plates with CC1 resting on nonlinear elastic foundation subjected to biaxial compression.

of bucking can’t occur for FGM plates with SS3 and SC3, while bifurcation of bucking could occur for the plate with CC2 and critical thermal bucking loads decrease with increasing the value of volume fraction index N. Particularly, Bifurcation of bucking

0 0.0

0.5

1.0

1.5

wcenter /h Fig. 10. Effect of uniform temperature rise and elastic foundations on thermal postbuckling behaviors of Si3N4/SUS304 square plates with SS3.

can’t occur for metal or ceramic plates with SS3 and SC3 in heat conduction fields, bifurcation of bucking could occur for metal or ceramic plates with SS3 and SC3 in uniform temperature rise fields, and occur for metal or ceramic plates with CC2 in both uniform temperature rise and heat conduction fields. It is worth

148

D.-G. Zhang, H.-M. Zhou / Thin-Walled Structures 89 (2015) 142–151

400

400

300

T =300K+ΔT, T =300K+ΔT

( k , k , k )=(50, 0, 0)

( k , k , k )=(50, 5, 25)

T =300K+ΔT, T =300K+ΔT

( k , k , k )=(50, 5, 0)

L /h=50

N=2, L /h=50

( k , k , k )=(50, 0, 25) ( k , k , k )=(50, 5, 25)

ΔT

ΔT

300 200 Si N

200

N=0.5 N=2 SUS304

100

0

100 0.0

0.4

0.8

0.0

1.2

0.4

0.8

1.2

wcenter /h

wcenter /h Fig. 11. Thermal post-buckling behaviors for Si3N4/SUS304 square plates with SC3 resting on nonlinear elastic foundation subjected to uniform temperature rise.

Fig. 14. Effect of uniform temperature rise and elastic foundations on thermal postbuckling behaviors of Si3N4/SUS304 square plates with CC2.

300 600

T =300K+ΔT, T =300K+ΔT

T =300K+ΔT

N=2, L /h=50

, T =300K

( k , k , k )=(50, 5, 25)

200

L /h=50

ΔTc-m

ΔT

400

100

( k , k , k )=(50, 0, 0)

Si N

200

( k , k , k )=(50, 5, 0)

N=0.5 N=2 SUS304

( k , k , k )=(50, 0, 25) ( k , k , k )=(50, 5, 25)

0 0.0

0.5

1.0

1.5

0

wcenter /h

0.0

Fig. 12. Effect of uniform temperature rise and elastic foundations on thermal post-buckling behaviors of Si3N4/SUS304 square plates with SC3.

-0.4

-0.8

-1.2

wcenter /h Fig. 15. Thermal post-buckling behaviors for Si3N4/SUS304 square plates with SS3 resting on nonlinear elastic foundation subjected to heat conduction.

600

450 T =300K+ΔT, T =300K+ΔT

Si N

( k , k , k )=(50, 5, 25)

N=0.5 N=2 SUS304

L /h=50

375

T =300K+ΔT

, T =300K

N=2, L /h=50

ΔTc-m

ΔT

400

300

200

( k , k , k )=(50, 0, 0)

225

( k , k , k )=(50, 5, 0) ( k , k , k )=(50, 0, 25) ( k , k , k )=(50, 5, 25)

0

150 0.0

0.4

0.8

1.2

0.0

-0.5

-1.0

-1.5

wcenter /h

wcenter /h Fig. 13. Thermal post-buckling behaviors for Si3N4/SUS304 square plates with CC2 resting on nonlinear elastic foundation subjected to uniform temperature rise.

Fig. 16. Effect of heat conduction and elastic foundations on thermal post-buckling behaviors of Si3N4/SUS304 square plates with SS3.

noting that critical bucking loads for the plates with Winkler elastic foundation is lower than the plates with Pasternak elastic foundation, and the effect of nonlinear elastic foundation is small at the pre-buckling and initial post-buckling state and is significant with increasing deflection at the deep post-buckling state.

5. Conclusions A model of the FGM rectangular plates resting on nonlinear elastic foundations is successfully established by physical neutral surface and high-order shear deformation theory. In post-bucking

D.-G. Zhang, H.-M. Zhou / Thin-Walled Structures 89 (2015) 142–151

800

800 T =300K+ΔT

, T =300K

( k , k , k )=(50, 5, 25)

600

L /h=50

( k , k , k )=(50, 0, 0)

T =300K+ΔT

( k , k , k )=(50, 5, 0)

N=2, L /h=50

, T =300K

( k , k , k )=(50, 0, 25)

600

( k , k , k )=(50, 5, 25)

ΔTc-m

ΔTc-m

149

400 Si N

400

N=0.5 N=2 SUS304

200

0

200 0.0

-0.4

-0.8

-1.2

0.0

0.4

wcenter /h Fig. 17. Thermal post-buckling behaviors for Si3N4/SUS304 square plates with SC3 resting on nonlinear elastic foundation subjected to heat conduction.

600 T =300K+ΔT

0.8

1.2

wcenter /h Fig. 20. Effect of heat conduction and elastic foundations on thermal post-buckling behaviors of Si3N4/SUS304 square plates with CC2.

physical neutral surface, and thermal post-buckling behaviors are also different for the plates under uniform temperature rise and heat conduction fields. In addition, it is worth noting that the effect of nonlinear elastic foundation is significant with increasing deflection at the deep post-buckling state.

, T =300K

N=2, L /h=50

ΔTc-m

400

Acknowledgements

200

( k , k , k )=(50, 0, 0) ( k , k , k )=(50, 5, 0) ( k , k , k )=(50, 0, 25) ( k , k , k )=(50, 5, 25)

0 0.0

-0.5

-1.0

-1.5

wcenter /h Fig. 18. Effect of heat conduction and elastic foundations on thermal post-buckling behaviors of Si3N4/SUS304 square plates with SC3.

T =300K+ΔT

, T =300K

Si N

Ly ði  kÞ þ Lx ðj  lÞ

N=0.5 N=2 SUS304

( k , k , k )=(50, 5, 25) L /h=50

    2 2 Lx 2 Ly 2 ijkl  i2 l aij akl Lx 2 Ly 2 ijkl þ i2 l aij akl ð3Þ ¼h i2 ; Bijkl ¼ h i2 ; 2 2 2 2 Ly 2 ðiþ kÞ þ Lx 2 ðj þ lÞ Ly 2 ði kÞ þ Lx 2 ðj þ lÞ

2Þ Bðijkl

700

ΔTc-m

Appendix A In Eqs. (5a–d)  2  Lx 2 Ly 2 i2 l  ijkl aij akl 1Þ ð1Þ Bðijkl ¼

2 ; ðif i ¼ k and j ¼ l; Bijkl ¼ 0Þ; 2 2 2 2

900 800

This research was supported by a grant of the Fund of the National Natural Science Foundation of China (nos. 11172285, 10802082) and the Natural Science Foundation of Zhejiang Province (nos. LR13A020002). The authors would like to express their sincere appreciation to these supports.

600

  2 Lx 2 Ly 2 ijkl þ i2 l aij akl ¼h i2 ; 2 2 Ly 2 ðiþ kÞ þ Lx 2 ðj  lÞ

4Þ Bðijkl

500 400

ðA:1Þ

where

300 0.0

0.4

0.8

1.2

cij ¼

wcenter /h Fig. 19. Thermal post-buckling behaviors for Si3N4/SUS304 square plates with CC2 resting on nonlinear elastic foundation subjected to heat conduction.

analysis, influences played by different supported boundaries, thermal environmental conditions and volume fraction index are discussed in detail. It can be concluded that mechanical postbuckling behaviors are different for FGM plates subjected to boundary compression acting on geometric middle surface and

ð1Þ

ð2Þ

bij C ðij22Þ  bij C ðij12Þ C ðij11Þ C ðij22Þ 

C ðij11Þ ¼

i2 π 2 Lx

2



þ

ð1 Þ

ð 2Þ

ð1Þ

bij C ðij11Þ  bij C ðij12Þ C ðij11Þ C ðij22Þ 



C ðij12Þ

! 1 ν j2 π 2 ~ D 11 þ A~ 44 ; C ðij22Þ ¼ 2 2 Ly

C ðij12Þ ¼ C ðij21Þ ¼ bij ¼

C ðij12Þ

2 ; dij ¼

2 ; j2 π 2 Ly

2

þ

! 1  ν i2 π 2 ~ D 11 þ A~ 44 ; 2 2 Lx

1 þ ν ijπ ~ D 11 ; 2 Lx Ly 2

" ! # " ! # iπ 4F 11 i2 π 2 j2 π 2 jπ 4F 11 i2 π 2 j2 π 2 ð2 Þ þ 2  A~ 44 aij ; bij ¼ þ 2  A~ 44 aij ; 2 2 2 2 Lx 3h Ly 3h Lx Ly Lx Ly

ðA:2Þ

150

D.-G. Zhang, H.-M. Zhou / Thin-Walled Structures 89 (2015) 142–151

Appendix B In Eqs. (6a–d)   2 Lx 2 Ly 2 i2 l  ijkl aij akl ð1Þ ð1Þ Bijkl ¼ h i2 ðif i ¼ k and j ¼ l; Bijkl ¼ 0Þ; 2 2 2 2 Ly ði  kÞ þ Lx ðj lÞ     2 2 Lx 2 Ly 2 i2 l  ijkl aij akl Lx 2 Ly 2 ijkl þ i2 l aij akl ð2Þ ð3Þ Bijkl ¼ h i2 ; Bijkl ¼  h i2 ; 2 2 2 2 Ly 2 ðiþ kÞ þ Lx 2 ðj þ lÞ Ly 2 ði  kÞ þ Lx 2 ðj þlÞ   2 2 Lx 2 Ly 2 ijkl þ i2 l aij akl 4Lx 2 Ly 2 i2 l aij akl ð5Þ ¼ h i2 ; Bijkl ¼   2 ; 2 2 2 Ly 2 ði þ kÞ þ Lx 2 ðj lÞ Ly 2 i2 þ Lx 2 l

4Þ Bðijkl

2

2

6Þ ¼h Bðijkl

2Lx 2 Ly 2 i2 l aij akl ð7Þ i2 ; Bijkl ¼ h i2 ; 2 2 2 2 Ly 2 ðiþ kÞ þ Lx 2 l Ly 2 ði kÞ þ Lx 2 l

8Þ Bðijkl ¼h

2Lx 2 Ly 2 i2 l aij akl 2Lx 2 Ly 2 i2 l aij akl ð9Þ i2 ; Bijkl ¼ h i2 ; 2 2 Ly 2 i2 þ Lx 2 ðj lÞ Ly 2 i2 þ Lx 2 ðj þ lÞ

2Lx 2 Ly 2 i2 l aij akl

2

bðij1Þ C ðij22Þ  bðij2Þ C ðij12Þ C ðij11Þ C ðij22Þ 

4i2 π 2

C ðij11Þ ¼

Lx 2

þ

C ðij12Þ ¼ C ðij21Þ ¼



C ðij12Þ

ðB:1Þ

ð1Þ

2 ; dij ¼

bðij2Þ C ðij11Þ  bðij1Þ C ðij12Þ C ðij11Þ C ðij22Þ 



C ðij12Þ

! 1  ν 4j2 π 2 ~ D 11 þ A~ 44 ; C ðij22Þ ¼ 2 Ly 2

ð2Þ 2 ; cij ¼

4j2 π 2 Ly 2

þ

ð3Þ

bij

ð2Þ

; dij ¼ ð33Þ

C ij

ð4Þ

bij

C ðij44Þ

! 1  ν 4i2 π 2 ~ D 11 þ A~ 44 ; 2 Lx 2

1 þ ν 4ijπ 2 ~ 4i2 π 2 ~ D 11 ; C ðij33Þ ¼ D 11 þ A~ 44 ; 2 Lx Ly Lx 2

4j2 π 2 ~ D 11 þ A~ 44 ; Ly 2 " !# 2iπ ~ 4F 11 4i2 π 2 4j2 π 2 ð1Þ aij ; þ A 44  2 bij ¼ Lx Lx 2 Ly 2 3h " !# 2jπ ~ 4F 11 4i2 π 2 4j2 π 2 ð2Þ bij ¼ þ A 44  2 aij ; Ly Lx 2 Ly 2 3h ! ! 2iπ 4F 11 4i2 π 2 ~ 2jπ 4F 11 4j2 π 2 ~ ð3Þ ð4Þ bij ¼  A 44 aij ; bij ¼  A 44 aij ; Lx 3h2 Lx 2 Ly 3h2 Ly 2 C ðij44Þ ¼

ðB:2Þ

Appendix C In Eqs. (7a–d)   2 Lx 2 Ly 2 4ijkl  4i2 l aij akl 1Þ ð1Þ Bðijkl ¼

2 ; ðif i ¼ k and j ¼ l; Bijkl ¼ 0Þ; 2 2 2 2 Ly ði  kÞ þ 4Lx ðj  lÞ

2Þ Bðijkl

4Þ Bðijkl

    2 2 Lx 2 Ly 2 4ijkl  4i2 l aij akl Lx 2 Ly 2 4ijkl þ 4i2 l aij akl ð3Þ ¼ h i2 ; Bijkl ¼  h i2 ; 2 2 2 2 Ly 2 ði þ kÞ þ 4Lx 2 ðj þ lÞ Ly 2 ði  kÞ þ 4Lx 2 ðj þ lÞ

  2 2 Lx 2 Ly 2 4ijkl þ 4i2 l aij akl 8Lx 2 Ly 2 i2 l aij akl ð5Þ ¼h i2 ; Bijkl ¼ h i2 ; 2 2 2 2 Ly 2 ði þ kÞ þ 4Lx 2 ðj  lÞ Ly 2 ði  kÞ þ 4Lx 2 l

6Þ ¼ h Bðijkl

2

8Lx 2 Ly 2 i2 l aij akl 2

2

Ly 2 ði þ kÞ þ 4Lx 2 l

i2 ;

ðC:1Þ

where cðij1Þ ¼

ð1Þ

ð 2Þ

bij C ðij22Þ  bij C ðij12Þ C ðij11Þ C ðij22Þ 



C ðij12Þ

i2 π 2

1 þ ν 2ijπ 2 ~ i2 π 2 ~ ~ D 11 ; C ðij33Þ ¼  2 D 11  A 44 ; 2 Lx Ly Lx " ! # iπ 4F 11 i2 π 2 4j2 π 2 ð1Þ ~ þ bij ¼  A 44 aij ; Lx 3h2 Lx 2 Ly 2 " !# 2jπ ~ 4F 11 i2 π 2 4j2 π 2 bijð2Þ ¼ þ A 44  2 aij ; Ly Lx 2 Ly 2 3h ! iπ ~ 4F 11 i2 π 2 ð3Þ bij ¼ A 44  2 aij ; 2 Lx 3h Lx C ðij12Þ ¼ C ðij21Þ ¼

ðC:2Þ

2

where cðij1Þ ¼

! 1  ν 4j2 π 2 ~ ¼ þ D 11  A~ 44 ; 2 Ly 2 Lx 2 ! 4j2 π 2 1 ν i2 π 2 ~ ð22Þ C ij ¼  þ D 11  A~ 44 ; 2 Lx 2 Ly 2 C ðij11Þ

2 ; dij ¼

ð2Þ

ð1Þ

bij C ðij11Þ  bij C ðij12Þ C ðij11Þ C ðij22Þ 



C ðij12Þ

ð2Þ 2 ; cij ¼

ð3Þ

bij

C ðij33Þ

;

References [1] Javaheri R, Eslami MR. Buckling of functionally graded plates under in-plane compressive loading. ZAMM 2002;82:277–83. [2] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates. AIAA J 2002;40:162–9. [3] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates based on higher order theory. J Therm Stresses 2002;25:603–25. [4] Najafizadeh MM, Eslami MR. Buckling analysis of circular plates of functionally graded materials under uniform radial compression. Int J Mech Sci 2002;44:2479–93. [5] Najafizadeh MM, Eslami MR. First-order-theory-based thermoelastic stability of functionally graded material circular plates. AIAA J 2002;40:1444–50. [6] Ma LS, Wang TJ. Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings. Int J Solids Struct 2003;40:3311–30. [7] Ma LS, Wang TJ. Axisymmetric post-buckling of a functionally graded circular plate subjected to uniformly distributed radial compression. Mater Sci Forum 2003;423–425:719–24. [8] Ma LS, Wang TJ. Thermal post-buckling and bending behavior of circular plates with temperature dependent material properties. Key Eng Mater 2003;243–244:195–200. [9] Wu LH. Thermal buckling of a simply supported moderately thick rectangular FGM plate. Compos Struct 2004;64:211–8. [10] Liew KM, Yang J, Kitipornchai S. Post-buckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading. Int J Solids Struct 2003;40:3869–92. [11] Liew KM, Yang J, Kitipornchai S. Thermal post-buckling of laminated plates comprising functionally graded materials with temperature-dependent properties. J Appl Mech: Trans ASME 2004839–50. [12] Na KS, Kim JH. Three-dimensional thermal buckling analysis of functionally graded materials. Composites Part B Eng 2004;35:429–37. [13] Na KS, Kim JH. Thermal post-buckling investigations of functionally graded plates using 3-D finite element method. Finite Elem Anal Des 2006;42:749–56. [14] Na KS, Kim JH. Three-dimensional thermomechanical buckling analysis of functionally graded composite plates. Compos Struct 2006;73:413–22. [15] Woo J, Meguid SA, Stranart JC, Liew KM. Thermomechanical post-buckling analysis of moderately thick functionally graded plates and shallow shells. Int J Mech Sci 2005;47:1147–71. [16] Park JS, Kim JH. Thermal post-buckling and vibration analyses of functionally graded plates. J Sound Vib 2006;289:77–93. [17] Li SR, Zhang JH, Zhao YG. Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection. Thin Walled Struct 2007;45:528–36. [18] Shariat BAS, Eslami MR. Thermal buckling of imperfect functionally graded plates. Int J Solids Struct 2006;43:4082–96. [19] Mohammadi M, Saidi AR, Jomehzadeh E. Levy solution for buckling analysis of functionally graded rectangular plates. Appl Compos Mater 2010;17:81–93. [20] Bouazza M, Tounsi A, Adda-Bedia EA, Megueni A. Thermoelastic stability analysis of functionally graded plates: an analytical approach. Comput Mater Sci 2010;49:865–70. [21] Nguyen-Xuan H, Tran-Loc V, Nguyen-Thoi T, Vu-Do HC. Analysis of functionally graded plates using an edge-based smoothed finite element method. Compos Struct 2011;93:3019–39. [22] Zenkour AM, Sobhy M. Thermal buckling of functionally graded plates resting on elastic foundations using the trigonometric theory. J Therm Stresses 2011;34:1119–38. [23] Lal A, Singh HN, Shegokar NL. FEM model for stochastic mechanical and thermal postbuckling response of functionally graded material plates applied to panels with circular and square holes having material randomness. Int J Mech Sci 2012;62:18–33. [24] Lee CY, Kim J-H. Hygrothermal postbuckling behavior of functionally graded plates. Compos Struct 2013;95:278–82.

D.-G. Zhang, H.-M. Zhou / Thin-Walled Structures 89 (2015) 142–151

[25] Lei ZX, Liew KM, Yu JL. Buckling analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method. Compos Struct 2013;98:160–8. [26] Thai H-T, Vo TP. A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates. Appl Math Modell 2013;37:3269–81. [27] Valizadeh N, Natarajan S, Gonzalez-Estrada OA, Rabczuk T, Bui TQ, Bordas SPA. NURBS-based finite element analysis of functionally graded plates: static bending, vibration, buckling and flutter. Compos Struct 2013;99:309–26. [28] Yaghoobi H, Yaghoobi P. Buckling analysis of sandwich plates with FGM face sheets resting on elastic foundation with various boundary conditions: an analytical approach. Meccanica 2013;48:2019–35. [29] Lal A, Jagtap KR, Singh BN. Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties. Appl Math Modell 2013;37:2900–20. [30] Foroughi H, Azhari M. Mechanical buckling and free vibration of thick functionally graded plates resting on elastic foundation using the higher order B-spline finite strip method. Meccanica 2014;49:981–93. [31] Kiani Y, Eslami MR. Thermal buckling and post-buckling response of imperfect temperature-dependent sandwich FGM plates resting on elastic foundation. Arch Appl Mech 2012;82:891–905. [32] Shen H-S. Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments. Int J Mech Sci 2002;44:561–84. [33] Leissa AW. Conditions for laminated plates to remain flat under inplane loading. Compos Struct 1986;6:261–70. [34] Qatu MS, Leissa AW. Buckling or transverse deflections of unsymmetrically laminated plates subjected to in-plane loads. AIAA J 1993;31:189–94. [35] Shen H-S. Post-buckling of FGM plates with piezoelectric actuators under thermo-electro-mechanical loadings. Int J Solids Struct 2005;42:6101–21. [36] Naderi A, Saidi AR. Exact solution for stability analysis of moderately thick functionally graded sector plates on elastic foundation. Compos Struct 2011;93:629–38. [37] Chen XL, Liew KM. Buckling of rectangular functionally graded material plates subjected to nonlinearly distributed in-plane edge loads. Smart Mater Struct 2004;13:1430–7. [38] Ma LS, Wang TJ. Relationships between axisymmetric bending and buckling solutions of FGM circular plates based on third-order plate theory and classical plate theory. Int J Solids Struct 2004;111:85–101. [39] Samsam-Shariata BA, Javaherib R, Eslamia MR. Buckling of imperfect functionally graded plates under in-plane compressive loading. Thin Walled Struct 2005;43:1020–36. [40] Zhao X, Lee YY, Liew KM. Mechanical and thermal buckling analysis of functionally graded plates. Compos Struct 2009;90:161–71. [41] Saidi AR, Rasouli A, Sahraee S. Axisymmetric bending and buckling analysis of thick functionally graded circular plates using unconstrained third-order shear deformation plate theory. Compos Struct 2009;89:110–9. [42] Zhang D-G, Zhou Y-H. A theoretical analysis of FGM thin plates based on physical neutral surface. Comput Mater Sci 2008;44:716–20. [43] Ma LS, Lee DW. A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading. Compos Struct 2011;93:831–42. [44] Ma LS, Lee DW. Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading. Eur J Mech A Solids 2012;31:13–20.

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[45] Prakash T, Singha MK, Ganapathi M. Influence of neutral surface position on the nonlinear stability behavior of functionally graded plates. Comput Mech 2009;43:341–50. [46] Singha MK, Prakash T, Ganapathi M. Finite element analysis of functionally graded plates under transverse load. Finite Elem Anal Des 2011;47:453–60. [47] Fekrar A, Houari MSA, Tounsi A, Mahmoud SR. A new five-unknown refined theory based on neutral surface position for bending analysis of exponential graded plates. Meccanica 2014;49:795–810. [48] Abrate S. Functionally graded plates behave like homogeneous plates. Composites Part B Eng 2008;39:151–8. [49] Zhang D-G. Nonlinear bending analysis of FGM beams based on physical neutral surface and high order shear deformation theory. Compos Struct 2013;100:121–6. [50] Zhang D-G. Thermal post-buckling and nonlinear vibration analysis of FGM beams based on physical neutral surface and high order shear deformation theory. Meccanica 2014;49:283–93. [51] Zhang D-G, Zhou H-M. Nonlinear bending and thermal post-buckling analysis of FGM beams resting on nonlinear elastic foundations. CMES—Comput Modell Eng 2014;100:201–22. [52] Zhang D-G. Modeling and analysis of FGM rectangular plates based on physical neutral surface and high order shear deformation theory. Int J Mech Sci 2013;68:92–104. [53] Zhang D-G. Nonlinear static analysis of FGM infinite cylindrical shallow shells based on physical neutral surface and high order shear deformation theory. Appl Math Modell (in press). [54] Yang J, Shen H-S. Nonlinear analysis of functionally graded plates under transverse and in-plane loads. Int J Non Linear Mech 2003;38:467–82. [55] Shen H-S, Wang Z-X. Nonlinear bending of FGM plates subjected to combined loading and resting on elastic foundations. Compos Struct 2010;92:2517–24. [56] Wang Z-X, Shen H-S. Nonlinear analysis of sandwich plates with FGM face sheets resting on elastic foundations. Compos Struct 2011;93:2521–32. [57] Shen H-S, Zhu ZH. Postbuckling of sandwich plates with nanotube-reinforced composite face sheets resting on elastic foundations. Eur J Mech A Solids 2012;35:10–21. [58] Duc ND, Tung HV. Mechanical and thermal postbuckling of higher order shear deformable functionally graded plates on elastic foundations. Compos Struct 2011;93:2874–81. [59] Sepahi O, Forouzan MR, Malekzadeh P. Large deflection analysis of thermomechanical loaded annular FGM plates on nonlinear elastic foundation via DQM. Compos Struct 2010;92:2369–78. [60] Zhang D-G. Nonlinear bending analysis of FGM elliptical plates resting on twoparameter elastic foundations. Appl Math Modell 2013;37:8292–309. [61] Zhang D-G. Nonlinear bending analysis of FGM rectangular plates with various supported boundaries resting on two-parameter elastic foundations. Arch Appl Mech 2014;84:1–20. [62] Touloukian YS. Thermophysical properties of high temperature solid materials. 1st ed.. New York, NY: Macmillan; 1967. [63] Reddy JN, Chin CD. Thermoelastical analysis of functionally graded cylinders and plates. J Therm Stresses 1998;21:593–626. [64] Bhimaraddi A, Chandashekhara K. Nonlinear vibrations of heated antisymmetric angle-ply laminated plates. Int J Solids Struct 1993;30:1255–68. [65] Shen H-S. Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties. Int J Mech Sci 2007;49:466–78. [66] Bateni M, Kiani Y, Eslami MR. A comprehensive study on stability of FGM plates. Int J Mech Sci 2013;75:134–44.