Composite Structures 100 (2013) 566–574

Contents lists available at SciVerse ScienceDirect

Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Nonlinear postbuckling of symmetric S-FGM plates resting onelastic foundations using higher order shear deformation plate theoryin thermal environments

Nguyen Dinh Duc ⇑, Pham Hong CongVietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam

a r t i c l e i n f o a b s t r a c t

Article history:Available online 30 January 2013

Keywords:Functionally graded materialsNonlinear postbucklingThird order shear deformation plate theoryElastic foundationImperfectionThermal environments

0263-8223/$ - see front matter � 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruct.2013.01.006

⇑ Corresponding author. Tel.: +84 4 37547989; fax:E-mail addresses: ducnd@vnu.edu.vn (N.D. Duc),

Cong).

This paper presents an analytical investigation on the postbuckling behaviors of thick symmetric func-tionally graded plates resting on elastic foundations and subjected to thermomechanical loads in thermalenvironments. Material properties are graded in the thickness direction according to a Sigmoi power lawdistribution in terms of the volume fractions of constituents (S-FGM). The formulations are based on thirdorder shear deformation plate theory and stress function taking into account Von Karman nonlinearity,initial geometrical imperfection, temperature and Pasternak type elastic foundation. By applying Galerkinmethod, closed-form relations of buckling loads and postbuckling equilibrium paths for simply supportedplates are determined. The effects of material and geometrical properties, temperature, boundary condi-tions, foundation stiffness and imperfection on the mechanical and thermal buckling and postbucklingloading capacity of the S-FGM plates are analyzed and discussed.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally Graded Materials (FGMs) which are microscopi-cally composites and composed from mixture of metal and ceramicconstituents have attracted considerable attention recent years. Bycontinuously and gradually varying the volume fraction of constit-uent materials through a specific direction, FGMs are capable ofwithstanding ultrahigh temperature environments and extremelylarge thermal gradients. Therefore, these novel materials are cho-sen to use in structure components of aircraft, aerospace vehicles,nuclear plants as well as various temperature shielding structureswidely used in industries.

Buckling and postbuckling behaviors of FGM structures underdifferent types of loading are important for practical applicationsand have received considerable interest. Wu used the first ordershear deformation theory to obtain closed-form relations of criticalbuckling temperatures for simply supported FGM plates [1]. Liewet al. [2,3] used the higher order shear deformation theory in con-junction with differential quadrature method to investigate thepostbuckling of pure and hybrid FGM plates with and withoutimperfection on the point of view that buckling only occursfor fully clamped FGM plates. Based on classical and first ordershear deformation theory, Eslami et al. investigated buckling and

ll rights reserved.

+84 4 37547724.congph_54@vnu.edu.vn (P.H.

post-buckling of FGM plates subjected to mechanical and thermalloads [4–7]. The postbuckling behavior of pure and hybrid FGMplates under the combination of various loads were also treatedby Shen using higher order shear deformation theory and two-stepperturbation technique taking temperature dependence of mate-rial properties into consideration [8,9]. Zhao et al. [10] analyzedthe mechanical and thermal buckling of FGM plates using ele-ment-free Ritz method. Lee et al. [11] have used element-free Ritzmethod to analyze the postbuckling of FGM plates subjected tocompressive and thermal loads.

The components of structures widely used in aircraft, reusablespace transportation vehicles and civil engineering are usually sup-ported by an elastic foundation. Therefore, it is necessary to ac-count for effects of elastic foundation for a better understandingof the postbuckling behavior of plates and shells. Librescu andLin have extended previous works [12,13] to consider the post-buckling behavior of flat and curved laminated composite panelsresting on Winkler elastic foundations [14,15]. In spite of practicalimportance and increasing use of FGM structures, investigation onFGM plates and shells supported by elastic media are limited innumber. The bending behavior of FGM plates resting on Pasternaktype foundation has been studied by Huang et al. [16] using statespace method, Zenkour [17] using analytical method and by Shenand Wang [18] making use of asymptotic perturbation technique.Duc and Tung have studied nonlinear analysis of stability forfunctionally graded plates under mechanical and thermal loadswithout elastic foundations with classical [19] and first order shear

N.D. Duc, P.H. Cong / Composite Structures 100 (2013) 566–574 567

plate theory [21]. In [20], also Duc and Tung have investigatedmechanical and thermal postbuckling of FGM on elastic foundationusing third order shear deformation plate theory and simple powerlaw distribution of the volume fraction for metal and ceramic.Comparing to the others, the main difference in the reports byDuc and Tung [19–21] is the use of the stress function to solvethe buckling and postbuckling problems for FGM plates. Indeed,the others have used the displacement functions.

This paper extends previous work [21] to investigate the post-buckling behaviors of thick functionally graded plates supportedby elastic foundations and subjected to in-plane compressive, ther-mal and thermomechanical loads using Reddy’s third order sheardeformation plate theory, stress function for FGM plate withSigmoi power law distribution of the volume of constituents(S-FGM), taking into account geometrical nonlinearity, initialgeometrical imperfection, temperature and the plate–foundationinteraction is represented by Pasternak model. Closed-formexpressions of buckling loads and postbuckling load–deflectioncurves for simply supported FGM plates are obtained by Galerkinmethod. Analysis is carried out to assess the effects of geometricaland material properties, temperature, boundary conditions, foun-dation stiffness and imperfection on the buckling and postbucklingof the symmetric S-FGM plates.

2. Governing equations

2.1. Symmetric S-functionally graded plates on elastic foundations

In the modern engineering and technology, there are manystructures usually working in a very high heat resistance environ-ment. To increase the ability to adjust to a high temperature, struc-tures with the top and bottom surfaces are made of ceramic andthe core of the structure is made of metal [21]. The symmetricalS-FGM plate considered in this paper is the one example of thesestructures.

Consider a symmetrical rectangular S-FGM plate that consists ofthird layers made of functionally graded ceramic and metal mate-rials and is midplane-symmetric. The outer surface layers of theplate are ceramic-rich, but the midplane layer is purely metallic.The plate is referred to a Cartesian coordinate system x, y, z, wherexy is the midplane of the plate and z is the thickness coordinator,�h/2 6 z 6 h/2. The length, width, and total thickness of the plateare a, b and h, respectively (Fig. 1).

Unlike [19,20] and other publications, this paper has used theSigmoi power-law distribution (S-FGM), the volume fractions ofmetal and ceramic, Vm and Vc, are assumed as [21]:

VmðzÞ ¼2zþh

h

� �N; �h=2 6 z 6 0

�2zþhh

� �N; 0 6 z 6 h=2

8<: ; VcðzÞ ¼ 1� VmðzÞ ð1Þ

Fig. 1. Symmetrical S-FGM plate on elastic foundation.

where the volume fraction index N is a nonnegative number thatdefines the material distribution and can be chosen to optimizethe structural response.

It is assumed that the effective properties Peff of the functionallygraded plate, such as the modulus of elasticity E and the coefficientof thermal expansion a, vary in the thickness direction z and can bedetermined by the linear rule of mixture as

Peff ¼ PrmVmðzÞ þ PrcVcðzÞ ð2Þ

where Pr denotes a material property, and the subscripts m and cstand for the metal and ceramic constituents, respectively.

From Eqs. (1) and (2), the effective properties of the S-FGM platecan be written as follows:

ðE;aÞ ¼ ðEc;acÞ þ ðEmc;amcÞ2zþh

h

� �N; �h=2 6 z 6 0

�2zþhh

� �N; 0 6 z 6 h=2

8<: ð3Þ

where

Emc ¼ Em � Ec; amc ¼ am � ac ð4Þ

and the Poisson ratio v is assumed constant, v(z) = v.The reaction–deflection relation of Pasternak foundation is gi-

ven by

qe ¼ k1w� k2r2w ð5Þ

where r2 = @2/@x2 + @2/@y2, w is the deflection of the plate, k1 isWinkler foundation modulus and k2 is the shear layer foundationstiffness of Pasternak model.

2.2. Theoretical formulation

The present study uses the Reddy’s third order shear deforma-tion plate theory to establish governing equations and determinethe buckling loads and postbuckling paths of the symmetrical S-FGM plates.

The strains across the plate thickness at a distance z from themiddle surface are [22]

ex

ey

cxy

0B@

1CA ¼

e0x

e0y

c0xy

0B@

1CAþ z

k1x

k1y

k1xy

0BB@

1CCAþ z3

k3x

k3y

k3xy

0BB@

1CCA ð6Þ

cxz

cyz

!¼

c0xz

c0yz

!þ z2 k2

xz

k2yz

!ð7Þ

where

e0x

e0y

c0xy

0BBB@

1CCCA ¼

u;x þw2;x=2

v ;y þw2;y=2

u;y þ v ;x þw;xw;y

0BB@

1CCA;

k1x

k1y

k1xy

0BBB@

1CCCA ¼

/x;x

/y;y

/x;y þ /y;x

0BB@

1CCA;

k3x

k3y

k3xy

0BB@

1CCA ¼ �c1

/x;x þw;xx

/y;y þw;yy

/x;y þ /y;x þ 2w;xy

0B@

1CA

c0xz

c0yz

!¼

/x þw;x

/y þw;y

!;

k2xz

k2yz

!¼ �3c1

/x þw;x

/y þw;y

!ð8Þ

in which c1 = 4/3h2, ex, ey are normal strains, cxy is the in-plane shearstrain, and cxz, cyz are the transverse shear deformations. Also, u, vare the displacement components along the x, y directions, respec-tively, and /x, /y are the slope rotations in the (x,y) and (y,z) planes,respectively.

568 N.D. Duc, P.H. Cong / Composite Structures 100 (2013) 566–574

Hooke law for an FGM plate is defined as

ðrx;ry ¼E

1� m2 ½ðex; eyÞ þ mðey; exÞ � ð1þ mÞaDTð1;1Þ�

rxy;rxz;ryz� �

¼ E2ð1þ mÞ cxy; cxz; cyz

� � ð9Þ

where DT is temperature rise from stress free initial state or tem-perature difference between two surfaces of the FGM plate.

The force and moment resultants of the FGM plate are deter-mined by

ðNi;Mi; PiÞ ¼Z h=2

�h=2rið1; z; z3Þdz; i ¼ x; y; xy

ðQ i;RiÞ ¼Z h=2

�h=2rjð1; z2Þdz; i ¼ x; y; j ¼ xz; yz

ð10Þ

Substitution of Eqs. (6), (7) and (9) into Eq. (10) yields the constitu-tive relations as:

ðNx;Mx; PxÞ ¼1

1� m2 ðE1; E2; E4Þ e0x þ me0

y

� �þ ðE2; E3; E5Þ k1

x þ mk1y

� �hþðE4; E5; E7Þ k3

x þ mk3y

� �� ð1þ mÞð/1;/2;/4Þ

i

ðNy;My; PyÞ ¼1

1� m2 ðE1; E2; E4Þ e0y þ me0

x

� �þ ðE2; E3; E5Þ k1

y þ mk1x

� �hþðE4; E5; E7Þ k3

y þ mk3x

� �� ð1þ mÞð/1;/2;/4Þ

i

ðNxy;Mxy; PxyÞ ¼1

2ð1þ mÞ ðE1; E2; E4Þc0xy þ ðE2; E3; E5Þk1

xy þ ðE4; E5; E7Þk3xy

h i

ðQx;RxÞ ¼1

2ð1þ mÞ ðE1; E3Þc0xz þ ðE3; E5Þk2

xz

h i

ðQy;RyÞ ¼1

2ð1þ mÞ ðE1; E3Þc0yz þ ðE3; E5Þk2

yz

h ið11Þ

where

ðE1; E2; E3; E4; E5; E7Þ ¼Z h=2

�h=2ð1; z; z2; z3; z4; z6ÞEðzÞdz

E1 ¼ Echþ EmchN þ 1

; E2 ¼ 0; E3 ¼Ech3

12þ Emch3

2ðN þ 1ÞðN þ 2ÞðN þ 3Þ ;

E4 ¼ 0

E5 ¼Ech5

80þ Emch5

161

N þ 1� 4

N þ 2þ 6

N þ 3� 4

N þ 4þ 1

N þ 5

� �

E7 ¼Ech7

448þ Emch7

641

N þ 7� 6

N þ 6þ 15

N þ 5� 20

N þ 4

�

þ 15N þ 3

� 6N þ 2

þ 1N þ 1

�

ð/1;/2;/4Þ ¼Z h=2

�h=2ð1; z; z3ÞEðzÞaðzÞDTdz ð12Þ

The nonlinear equilibrium equations of a perfect FGM plate restingon elastic foundations based on the higher order shear deformationtheory are [3,17,18,20,21]:

Nx;x þ Nxy;y ¼ 0 ð13aÞ

Ny;y þ Nxy;x ¼ 0 ð13bÞ

Qx;x þ Q y;y � 3c1ðRx;x þ Ry;yÞ þ c1ðPx;xx þ 2Pxy;xy þ Py;yyÞ þ Nxw;xx

þ 2Nxyw;xy þ Nyw;yy � k1wþ k2r2w ¼ 0 ð13cÞ

Mx;x þMxy;y � Q x þ 3c1Rx � c1ðPx;x þ Pxy;yÞ ¼ 0 ð13dÞ

Mxy;x þMy;y � Q y þ 3c1Ry � c1ðPxy;x þ Py;yÞ ¼ 0 ð13eÞ

where the plate–foundation interaction has been included. The lastthree equations of Eq. (13) may be rewritten into two equations interms of variables w and /x,x + /y,y by substituting Eqs. (8) and (11)into Eqs. (13c), (13d) and (13e). Subsequently, elimination of thevariable /x,x + /y,y from two the resulting equations leads to the fol-lowing system of equilibrium equations

Nx;x þ Nxy;y ¼ 0Ny;y þ Nxy;x ¼ 0

c21ðD2D5=D4 � D3Þr6wþ ðc1D2=D4 þ 1ÞD6r4w

þð1� c1D5=D4Þr2ðNxw;xx þ 2Nxyw;xy þ Nyw;yy � k1wþ k2r2wÞ

�D6=D4ðNxw;xx þ 2Nxyw;xy þ Nyw;yy � k1wþ k2r2wÞ ¼ 0 ð14Þ

where

D1 ¼E3

1� m2 ; D2 ¼E5

1� m2 ; D3 ¼E7

1� m2

D4 ¼ D1 � c1D2; D5 ¼ D2 � c1D3;

D6 ¼1

2ð1þ mÞ E1 � 6c1E3 þ 9c21E5

� �ð15Þ

For an imperfect FGM plate, Eq. (14) are modified into form as

c21ðD2D5=D4 � D3Þr6wþ ðc1D2=D4 þ 1ÞD6r4wþ ð1� c1D5=D4Þr2

� f;yy w;xx þw�;xx

� �� 2f ;xyðw;xy þw�;xyÞ þ f;xx w;yy þw�;yy

� �h�k1wþ k2r2w

i� D6=D4 f;yy w;xx þw�;xx

� �� 2f ;xy w;xy þw�;xy

� �h

þf;xx w;yy þw�;yy

� �� k1wþ k2r2w

i¼ 0 ð16Þ

in which w⁄(x,y) is a known function representing initial smallimperfection of the plate. Note that Eq. (16) gets a complicated formunder the third order shear deformation theory which includes the6th-order partial differential term r6w.

Also, f(x,y) is stress function defined by

Nx ¼@2f@y2 ; Ny ¼

@2f@x2 ; Nxy ¼ �

@2f@x@y

ð17Þ

The geometrical compatibility equation for an imperfect plate iswritten as

e0x;yy þ e0

y;xx � c0xy;xy ¼ w2

;xy �w;xxw;yy þ 2w;xyw�;xy �w;xxw�;yy

�w;yyw�;xx ð18Þ

From the constitutive relations (11) with the aid of Eq. (17) one canwrite

e0x ¼

1E1ðf;yy � mf;xx þ /1Þ; e0

y ¼1E1ðf;xx � mf;yy þ /1Þ; c0

xy

¼ � 1E1

2ð1þ mÞf;xy ð19Þ

N.D. Duc, P.H. Cong / Composite Structures 100 (2013) 566–574 569

Introduction of Eq. (19) into Eq. (18) gives the compatibility equa-tion of an imperfect FGM plate as

r4f � E1 w2;xy �w;xxw;yy þ 2w;xyw�;xy �w;xxw�;yy �w;yyw�;xx

� �¼ 0

ð20Þ

which is the same as equation derived by using the classical platetheory [19].

Eqs. (16) and (20) are nonlinear equations in terms of variablesw and f and used to investigate the stability of thick symmetric S-FGM plates on elastic foundations subjected to mechanical,thermal and thermomechanical loads using the third order sheardeformation plate theory. Until now, there is no analytical studieshave been reported in the literature on the postbuckling of thickS-FGM plates using third order shear deformation plate theory.Therefore, the transformations of getting (16) and (20) for the sym-metric S-FGM is one of the most important results in this paper.

Depending on the in-plane restraint at the edges, three cases ofboundary conditions, referred to as Cases 1, 2 and 3 will be consid-ered [8,12,15,20,21]:

Case 1. Four edges of the plate are simply supported and freelymovable (FM). The associated boundary conditions are

w ¼ Nxy ¼ /y ¼ Mx ¼ Px ¼ 0; Nx ¼ Nx0 at x ¼ 0; a

w ¼ Nxy ¼ /x ¼ My ¼ Py ¼ 0; Ny ¼ Ny0 at y ¼ 0; bð21Þ

Case 2. Four edges of the plate are simply supported andimmovable (IM). In this case, boundary conditions are

w ¼ u ¼ /y ¼ Mx ¼ Px ¼ 0; Nx ¼ Nx0 at x ¼ 0; a

w ¼ v ¼ /x ¼ My ¼ Py ¼ 0; Ny ¼ Ny0 at y ¼ 0; bð22Þ

Case 3. All edges are simply supported. Two edges x = 0, a arefreely movable and subjected to compressive load inthe x direction, whereas the remaining two edges y = 0,b are unloaded and immovable. For this case, the bound-ary conditions are defined as

w ¼ Nxy ¼ /y ¼ Mx ¼ Px ¼ 0; Nx ¼ Nx0 at x ¼ 0; a

w ¼ v ¼ /x ¼ My ¼ Py ¼ 0; Ny ¼ Ny0 at y ¼ 0; bð23Þ

where Nx0, Ny0 are in-plane compressive loads at movable edges(i.e., Case 1 and the first of Case 3) or are fictitious compressiveedge loads at immovable edges (i.e., Case 2 and the second ofCase 3).

The approximate solutions of w, w⁄ [7,15] and f [19–21] satisfy-ing boundary conditions (21)–(23) are assumed to be

ðw;w�Þ ¼ ðW;lhÞ sin kmx sin dny ð24aÞ

f ¼ A1 cos 2kmxþ A2 cos 2dnyþ A3 sin kmx sin dnyþ A4

� cos kmx cos dnyþ 12

Nx0y2 þ 12

Ny0x2 ð24bÞ

where km = mp/a, dn = np/b, W is amplitude of the deflection and lis imperfection parameter. The coefficients Ai(i = 1 � 4) are deter-mined by substitution of Eqs. (24a) and (24b) into Eq. (20) as

A1 ¼E1d

2n

32k2m

WðW þ 2lhÞ; A2 ¼E1k

2m

32d2n

WðW þ 2lhÞ; A3 ¼ A4

¼ 0 ð25Þ

Subsequently, setting Eqs. (24a) and (24b) into Eq. (16) and apply-ing the Galerkin procedure for the resulting equation yield

�c21

D2D5

D4� D3

� k2

m þ d2n

� �3 þ D6c1D2

D4þ 1

� k2

m þ d2n

� �2

þ k1 þ k2 k2m þ d2

n

� �� � D6

D4þ 1� c1D5

D4

� k2

m þ d2n

� �� � W

þ E1

16D6

D4k4

m þ d4n

� �þ 1� c1D5

D4

� k6

m þ d6n þ k2

md4n þ k4

md2n

� � W

� ðW þ lhÞðW þ 2lhÞ þ D6

D4þ 1� c1D5

D4

� k2

m þ d2n

� �� �� ½Nx0k

2m þ Ny0d

2n�ðW þ lhÞ ¼ 0 ð26Þ

where m, n are odd numbers. This equation will be used to analyzethe buckling and postbuckling behaviors of thick FGM plates undermechanical, thermal and thermomechanical loads.

2.2.1. Mechanical postbuckling analysisConsider a simply supported symmetrical S-FGM plate with all

movable edges (all FM) which is rested on elastic foundations andsubjected to in-plane edge compressive loads Fx, Fy uniformly dis-tributed on edges x = 0, a and y = 0, b, respectively. In this case, pre-bucking force resultants are [6]

Nx0 ¼ �Fxh; Ny0 ¼ �Fyh ð27Þand Eq. (26) leads to

Fx ¼ e11

W

W þ lþ e1

2WðW þ 2lÞ ð28Þ

where

e11 ¼�16p4ðD2D5 �D3D4Þ m2B2

a þ n2� �3

þ 3D6B2hp2ð4D2 þ 3D4Þ m2B2

a þ n2� �2

3B2hðm2B2

a þ bn2Þ½p2ð3D4 � 4D5Þ m2B2a þ n2

� �þ 3B2

hD6�

þK1B2

a þK2p2ðm2B2a þ n2Þ

h iB2

a D1

B2hp2 m2B2

a þ bn2� �

e12 ¼

E1 m4B4a þ n4

� �p2

16ðm2B2a þ bn2ÞB2

h

ð29Þ

in which

Bh ¼ b=h; Ba ¼ b=a; W ¼W=h; b ¼ Fy=Fx

K1 ¼k1a4

D1; K2 ¼

k2a2

D1; Ei ¼ Ei=hiði ¼ 1� 7Þ

D1 ¼E3

1� v2 ; D2 ¼E5

1� v2 ; D3 ¼E7

1� v2

D4 ¼ D1 �43

D2; D5 ¼ D2 �43

D3; D6 ¼1

2ð1þ vÞ ðE1 � 8E3 þ 16E5Þ

ð30ÞFor a perfect FGM plate, Eq. (28) reduces to an equation from whichbuckling compressive load may be obtained as Fxb ¼ e1

1

2.2.2. Thermal postbuckling analysisA simply supported FGM plate with all immovable edges (IM) is

considered. The plate is also supported by an elastic foundationand exposed to temperature environments or subjected to throughthe thickness temperature gradient. The in-plane condition onimmovability at all edges, i.e., u = 0 at x = 0, a and v = 0 at y = 0, bis fulfilled in an average sense as [5,8,20,21]Z b

0

Z a

0

@u@x

dxdy ¼ 0;

Z a

0

Z b

0

@v@y

dydx ¼ 0 ð31Þ

From Eqs. (8) and (11) one can obtain the following expressions inwhich Eq. (17) and imperfection have been included

@u@x¼ 1

E1ðf;yy � mf;xxÞ �w2

;x=2�w;xw�;x þ/1

E1

@v@y¼ 1

E1ðf;xx � mf;yyÞ �w2

;y=2�w;yw�;y þ/1

E1

ð32Þ

Fig. 2. Effects of volume fraction index N on the postbuckling of symmetrical S-FGM plates under uniaxial compressive load (all FM edges).

570 N.D. Duc, P.H. Cong / Composite Structures 100 (2013) 566–574

Introduction of Eq. (24) into Eq. (32) and then the result into Eq.(31) give

Nx0 ¼1

8ð1� m2Þ E1 k2m þ md2

n

� �WðW þ 2lhÞ � /1

1� m

Ny0 ¼1

8ð1� m2Þ E1 mk2m þ d2

n

� �WðW þ 2lhÞ � /1

1� m

ð33Þ

When the deflection dependence of fictitious edge loads is ignored,i.e., W = 0, Eq. (33) reduce to

Nx0 ¼ Ny0 ¼ �/1

1� v ð34Þ

which was derived by Shariat and Eslami [6] by solving the mem-brane form of equilibrium equations and employing the methodsuggested by Meyers and Hyer [23].

Substituting Eq. (33) into Eq. (26) yields the expression of ther-mal parameter as

/1

1� v ¼�c2

1ðD2D5 � D3D4Þ k2m þ d2

n

� �2 þ D6ðc1D2 þ D4Þ k2m þ d2

n

� �D6 þ ðD4 � c1D5Þ k2

m þ d2n

� �"

þk1 þ k2 k2

m þ d2n

� �k2

m þ d2n

#W

W þ lhþ

E1 k4m þ d4

n

� �16 k2

m þ d2n

� �"

þ E1

8ð1� v2Þk4

m þ d4n þ 2vk2

md2n

k2m þ d2

n

#WðW þ 2lhÞ ð35Þ

The S-FGM plate is exposed to temperature environments uni-formly raised from stress free initial state Ti to final value Tf, andtemperature change DT = Tf � Ti is considered to be independentfrom thickness variable. The thermal parameter /1 is obtained fromEq. (12), and substitution of the result into Eq. (35) yields

DT ¼ e21

W

W þ lþ e2

2WðW þ 2lÞ ð36Þ

where

e21 ¼

ð1� vÞp2

P 3B2hD6 þ ð3D4 � 4D5Þ m2B2

a þ n2� �

p2h i

� �163ðD2D5 � D3D4Þ m2B2

a þ n2� �2 p2

B2h

þ D6ð4D2 þ 3D4Þ m2B2a þ n2

� �" #

þK1B2

a þ K2p2 m2B2a þ n2

� �h iB2

aD1ð1� vÞ

PB2hp2ðm2B2

a þ n2Þ

e22 ¼

E1p2ð1� vÞ m4B4a þ n4

� �16PB2

h m2B2a þ n2

� � þ E1p2

8Pð1þ vÞm4B4

a þ n4 þ 2vm2n2B2a

B2h m2B2

a þ n2� �

ð37Þ

in which

P ¼ Ecac þEcamc þ Emcac

N þ 1þ Emcamc

2N þ 1ð38Þ

By Setting l = 0 Eq. (36) leads to an equation from which buck-ling temperature change of the perfect FGM plates may be deter-mined as DTb ¼ e2

1.

2.2.3. Thermomechanical postbuckling analysisThe S-FGM plate resting on an elastic foundation is uniformly

compressed by Fx (Pascal) on two movable edges x = 0, a and simul-taneously exposed to elevated temperature environments or sub-jected to through the thickness temperature gradient. The twoedges y = 0, b are assumed to be immovable. In this case, Nx0 = �Fxhand fictitious compressive load on immovable edges is determinedby setting the second of Eq. (32) in the second of Eq. (31) as

Ny0 ¼ vNx0 � /1 þE1

8d2

nWðW þ 2lhÞ ð39Þ

Subsequently, Nx0 and Ny0 are placed in Eq. (26) to give

Fx ¼ e31

WW þ l

þ e32WðW þ 2lÞ � Pn2DT

m2B2a þ vn2

ð40Þ

where

e31 ¼�16p4ðD2D5 � D3D4Þ m2B2

a þ n2� �3

þ 3ð4D2 þ 3D4ÞD6 m2B2a þ n2

� �2p2B2

h

3B2h 3D6B2

h þ ð3D4 � 4D5Þ m2B2a þ n2

� �p2

h im2B2

a þ vn2� �

þK1B2

a þ K2p2 m2B2a þ n2

� �h iB2

a D1

B2hp2 m2B2

a þ vn2� �

e32 ¼

E1

16

m4B4a þ n4

� �p2

m2B2a þ vn2

� �B2

h

þ E1

8n4p2

B2hðm2B2

a þ vn2Þ

ð41Þ

Eqs. (28), (36) and (40) are explicit expressions of load–deflectioncurves for thick S-FGM plates resting on Pasternak elastic founda-tions and subjected to in-plane compressive, thermal and thermo-mechanical loads, respectively. Specialization of these equationsfor thick S-FGM plates, i.e., ignoring the third order shear deforma-tions and elastic foundations, gives the corresponding results de-rived by using the first order shear deformation plate theory forS-FGM plates [21].

3. Numerical results and discussion

To illustrate the present approach for buckling and postbucklinganalysis of thick FGM plates resting on elastic foundations, con-sider a square ceramic–metal plate consisting of aluminum andalumina with the following properties [5,8,20,21]:

Em ¼ 70 GPa; am ¼ 23� 10�6 �C�1

Ec ¼ 380 GPa; am ¼ 7:4� 10�6 �C�1ð42Þ

and Poisson ratio is chosen to be v = 0.3. In this case, the buckling ofperfect plates occurs for m = n = 1, and these values of half wavesare also used to trace load–deflection equilibrium paths for bothperfect and imperfect plates. In figures, W/h denotes the dimension-less maximum deflection and the FGM plate–foundation interactionis ignored, unless otherwise stated.

Effects of volume fraction index N on the postbuckling of S-FGMplates under uniaxial compressive load and uniform temperaturerise are shown in Figs. 2 and 3. In all below figures, it is assumed

Fig. 4. Effect of first and third order shear deformation on mechanical buckling andpostbuckling of S-FGM plate with various of volume fractions N.

Fig. 5. Effect of first and third order shear deformation on thermal buckling andpostbuckling of S-FGM plate with various of volume fractions N.

N.D. Duc, P.H. Cong / Composite Structures 100 (2013) 566–574 571

that ~ePx ¼ Fx. Obviously, the mechanical load and the thermalresistance get better if the volume N increases or the percentageof ceramic increases. It is opposite of the FGM applied simplypower law distribution in [19,20]: Both critical buckling loadsand postbuckling carrying capacity are strongly dropped when Nis increased.

Figs. 4 and 5 show effects of first and third order shear deforma-tions on mechanical and thermal buckling and postbuckling of S-FGM plate with various volume fractions N of the S-FGM plate.Obviously, with the same volume fractions of ceramic–metal, thecritical loads of postbuckling of the S-FGM are different for the firstand third orders. Indeed, the critical loads for the third order sheardeformation is smaller than those for the first order shear deforma-tion. For postbuckling of the S-FGM plate, Figs. 4 and 5 also showus that the imperfect plate has a better mechanical and thermalloading capacity than those of the perfect plate.

Figs. 6 and 7 present effects of first and third order shear defor-mations on buckling and postbuckling of S-FGM plate with variousof thermal and mechanical loads. Obviously, with the same volumefraction of ceramic–metal, the critical loadings of postbuckling ofthe S-FGM are different. Also, similar to above two figures, the crit-ical mechanical and thermal loadings for the third order sheardeformation are smaller than those of the first order sheardeformation.

There have been only a few of reports on the buckling and post-buckling for symmetric S-FGM plate yet. We therefore are limitedto compare with the others. However, comparing our findings inFigs. 4–7 with our previous results [21], it is inferred that thereis a difference between the first and the third of higher order sheardeformation plate theory on buckling and postbuckling of thick S-FGM plates. However, this difference is not much despite of com-plicated third order shear calculation.

Figs. 8 and 9 show the influence of initial imperfections on post-buckling of S-FGM plate under uniaxial compressive load (all FMedges) and under uniform temperature (all IM edges). Fig. 8 showsus that the critical compressive loads decreases with l in the limitof the small bending. However, it increases with l in the other lim-it of the large bending, meaning the higher bending-load curve (i.e.,the better loading ability). Figs. 4–9 show us that an imperfect FGMplate has a better mechanical and thermal loading capacity thanthe perfect one in postbuckling process. This has been shown in[3,7,8,15,19–21]. In particular, Fig. 9 clearly shows us that an initialimperfection has an useful influence on the thermal resistance of S-FGM at the threshold value of the bending.

Fig. 3. Effects of volume fraction index N on the postbuckling of symmetrical S-FGM plates under uniform temperature rise (all IM edges).

Fig. 6. Effect of first and third order shear deformation on mechanical buckling andpostbuckling of S-FGM plate with the temperature DT.

Fig. 7. Effect of first and third order shear deformation on critical thermal loads ofbuckling and postbuckling of S-FGM plate with various of mechanical loads Px.

Fig. 9. The influence of imperfections on the stability of symmetrical S-FGM platesunder uniform temperature rise (all IM edges).

Fig. 10. Effects of the elastic foundations on the postbuckling of symmetrical S-FGM plates under uniaxial compressive load (all FM edges).

Fig. 11. Effects of the elastic foundations on the postbuckling of symmetrical S-FGM plates under uniform temperature rise (all IM edges).

Fig. 12. Effect of temperature field and uniaxial compression on the postbuckling ofsymmetric S-FGM plate under uniform temperature rise (FM on y = 0, b; IM on x = 0,a).

Fig. 8. The influence of imperfections on the stability of symmetrical S-FGM platesunder uniaxial compressive load (all FM edges).

572 N.D. Duc, P.H. Cong / Composite Structures 100 (2013) 566–574

Fig. 13. Effect of temperature gradient and uniaxial compression on the postbuck-ling of symmetric S-FGM. (FM on y = 0, b; IM on x = 0, a).

Fig. 14. Effect of boundary conditions (FM and IM) on postbuckling of symmetric S-FGM plate under uniaxial compression on edges y = 0, b.

N.D. Duc, P.H. Cong / Composite Structures 100 (2013) 566–574 573

Figs. 10 and 11 present the positive influence of elastic founda-tions on imperfections on the stability of S-FGM plate under uniax-ial compressive load (all FM edges) and uniform temperature (allIM edges). The effect of Pasternak foundation K2 on the criticalcompressive loads and the thermal resistance of S-FGM is largerthan the Winkler foundation K1. This conclusion has been also re-ported in [16–18,20].

An investigation of the mechanical–thermal stability has beendetermined by (40). Figs. 12 and 13 have been calculated underthe assumption of the third boundary conditions (Case 3) for theFM edges x = 0, a and IM edges y = 0, b which are simultaneouslyunder the compressive uniform loading on the edge x = 0, a.

Fig. 12 shows the effect of the temperature gradient of the sur-rounding environment on the behavior of an uniaxial compressiveload x. The presence of temperature reduces the loading ability (forboth perfect and imperfect plates). Under the non-zero tempera-ture gradient condition DT – 0, in the presence of temperature,the imperfect plate still gets bend immediately even if there isno mechanical compressive force. It is represented by a crossingpoint of the dash lines with the axis W/h.

Buckling and postbuckling behavior of the S-FGM plate under theincreased uniform temperature gradient field DT and the differentvalues of the uniaxial compressive load Px have been shown inFig. 13. The presence of the mechanical loading reduces the thermalloading ability of the perfect and imperfect plates [1,5,10,17,20].

Effect of boundary conditions on postbuckling of symmetric S-FGM plate under uniaxial compression is shown in Fig. 14. Thereare two types of condition for the two edges y = 0, b which arethe free motion (FM) and not in motion (IM) conditions. The curvefor FM edges drawn from (28) with the loading ratio b = 0 (in (30)),whereas the result for IM edges drawn from (40) with DT = 0.Fig. 14 shows us that the perfect FGM plate is bended earlier thanthe imperfect one; however loading capacity of the imperfect plateis better than perfect one when the bending is large enough inpostbuckling process.

4. Conclusions

This paper presents an analytical investigation on the postbuck-ling behaviors of thick symmetric functionally graded plates rest-ing on elastic foundations in thermal environments andsubjected to in-plane compressive, thermal and thermomechanicalloads. Material properties are graded in the thickness directionaccording to a Sigmoi power law distribution in terms of the vol-ume fractions of constituents (S-FGM). The formulations are basedon third order shear deformation plate theory and stress functiontaking into account Von Karman nonlinearity, initial geometricalimperfection, temperature and Pasternak type elastic foundation.By applying Galerkin method, closed-form relations of bucklingloads and postbuckling equilibrium paths for simply supportedplates are determined. The effects of material and geometricalproperties, temperature, boundary conditions, foundation stiffnessand imperfection on the postbuckling loading capacity of the S-FGM plates are analyzed and discussed. It is easy to realize thatthe critical mechanical and thermal loadings for third order sheardeformation are smaller than those for the first order shear defor-mation and for the postbuclking period of the S-FGM plate, com-paring with a perfect plate, an imperfect plate has a bettermechanical and thermal loading capacity.

Acknowledgment

This work was supported by Vietnam National University,Hanoi. The authors are grateful for this financial support.

References

[1] Wu L. Thermal buckling of a simply supported moderately thick rectangularFGM plate. Compos Struct 2004;64:211–8.

[2] Liew KM, Jang J, Kitipornchai S. Postbuckling of piezoelectric FGM platessubject to thermo-electro-mechanical loading. Int J Solids Struct2003;40:3869–92.

[3] Yang J, Liew KM, Kitipornchai S. Imperfection sensitivity of the post-bucklingbehavior of higher-order shear deformable functionally graded plates. Int JSolids Struct 2006;43:5247–66.

[4] Najafizadeh MM, Eslami MR. First-order theory-based thermoelastic stabilityof functionally graded material circular plates. AIAA J 2002;40(7):1444–9.

[5] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates based onhigher order theory. J Therm Stress 2002;25:603–25.

[6] Samsam Shariat BA, Eslami MR. Buckling of thick functionally graded platesunder mechanical and thermal loads. Compos Struct 2007;78:433–9.

[7] Samsam Shariat BA, Eslami MR. Thermal buckling of imperfect functionallygraded plates. Int J Solids Struct 2006;43:4082–96.

[8] Shen HS. Thermal postbuckling behavior of shear deformable FGM plates withtemperature-dependent properties. Int J Mech Sci 2007;49:466–78.

[9] Shen HS. Functionally graded materials. Non linear analysis of plates andshells. London, Newyork: CRC Press, Taylor & Francis Group; 2009.

[10] Zhao X, Lee YY, Liew KM. Mechanical and thermal buckling analysis offunctionally graded plates. Compos Struct 2009;90:161–71.

[11] Lee YY, Zhao X, Reddy JN. Postbuckling analysis of functionally graded platessubject to compressive and thermal loads. Comput Methods Appl Mech Eng2010;199:1645–53.

[12] Librescu L, Stein M. A geometrically nonlinear theory of transversely isotropiclaminated composite plates and its use in the post-buckling analysis. ThinWall Struct 1991;11:177–201.

[13] Librescu L, Stein M. Postbuckling of shear deformable composite flat panelstaking into account geometrical imperfections. AIAA 1992;30(5):1352–60.

574 N.D. Duc, P.H. Cong / Composite Structures 100 (2013) 566–574

[14] Librescu L, Lin W. Postbuckling and vibration of shear deformable flat andcurved panels on a non-linear elastic foundation. Int J Non-Lin Mech1997;32(2):211–25.

[15] Lin W, Librescu L. Thermomechanical postbuckling of geometrically imperfectshear-deformable flat and curved panels on a nonlinear foundation. Int J EngSci 1998;36(2):189–206.

[16] Huang ZY, Lu CF, Chen WQ. Benchmark solutions for functionally graded thickplates resting on Winkler–Pasternak elastic foundations. Compos Struct2008;85:95–104.

[17] Zenkour AM. Hygro-thermo-mechanical effects on FGM plates resting onelastic foundations. Compos Struct 2010;93:234–8.

[18] Shen H-S, Wang Z-X. Nonlinear bending of FGM plates subjected to combinedloading and resting on elastic foundations. Compos Struct 2010;92:2517–24.

[19] Tung HV, Duc ND. Nonlinear analysis of stability for functionally graded platesunder mechanical and thermal loads. Compos Struct 2010;92:1184–91.

[20] Duc ND, Tung HV. Mechanical and thermal postbuckling of higher order sheardeformable functionally graded plates on elastic foundations. J. Compos Struct2011;93:2874–81.

[21] Duc ND, Tung HV. Mechanical and thermal post-buckling of shear-deformableFGM plates with temperature-dependent properties. J Mech Compos Mater2010;46(5):461–76.

[22] Reddy JN. Mechanics of laminated composite plates and shells: theory andanalysis. Boca Raton: CRC Press; 2004.

[23] Meyers CA, Hyer MW. Thermal buckling and postbuckling of symmetricallylaminated composite plates. J Therm Stress 1991;14(4):519–40.