Composite Structures 93 (2011) 2874–2881

Contents lists available at ScienceDirect

Composite Structures

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

Mechanical and thermal postbuckling of higher order shear deformablefunctionally graded plates on elastic foundations

Nguyen Dinh Duc a, Hoang Van Tung b,⇑a University of Engineering and Technology, Vietnam National University, Ha Noi, Viet Namb Faculty of Civil Engineering, Hanoi Architectural University, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:Available online 24 May 2011

Keywords:Functionally Graded MaterialsPostbucklingHigher order shear deformation theoryElastic foundationImperfection

0263-8223/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruct.2011.05.017

⇑ Corresponding author.E-mail address: hoangtung0105@gmail.com (H.V.

a b s t r a c t

This paper presents an analytical investigation on the buckling and postbuckling behaviors of thick func-tionally graded plates resting on elastic foundations and subjected to in-plane compressive, thermal andthermomechanical loads. Material properties are assumed to be temperature independent, and graded inthe thickness direction according to a simple power law distribution in terms of the volume fractions ofconstituents. The formulations are based on higher order shear deformation plate theory taking intoaccount Von Karman nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation.By applying Galerkin method, closed-form relations of buckling loads and postbuckling equilibrium pathsfor simply supported plates are determined. Analysis is carried out to show the effects of material andgeometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buck-ling and postbuckling loading capacity of the plates.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Due to high performance heat resistance capacity and excellentcharacteristics in comparison with conventional composites,Functionally Graded Materials (FGMs) which are microscopicallycomposites and composed from mixture of metal and ceramic con-stituents 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 ofFGM structures under different types of loading are important forpractical applications and have received considerable interest.Eslami and his co-workers used analytical approach, classical andhigher order plate theories in conjunction with adjacent equilib-rium criterion to investigate the buckling of FGM plates with andwithout imperfection under mechanical and thermal loads [1–4].According to this direction, Lanhe [5] also employed first ordershear deformation theory to obtain closed-form relations of criticalbuckling temperatures for simply supported FGM plates. Zhao et al.[6] analyzed the mechanical and thermal buckling of FGM platesusing element-free Ritz method. Liew et al. [7,8] used the higher

ll rights reserved.

Tung).

order shear deformation theory in conjunction with differentialquadrature method to investigate the postbuckling of pure and hy-brid FGM plates with and without imperfection on the point ofview that buckling only occurs for fully clamped FGM plates. Thepostbuckling behavior of pure and hybrid FGM plates under thecombination of various loads were also treated by Shen [9,10]using two-step perturbation technique taking temperature depen-dence of material properties into consideration. Recently, Lee et al.[11] made of use element-free Ritz method to analyze the post-buckling of FGM plates subjected to compressive and thermalloads.

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 [14,15] 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.To the best of authors’ knowledge, there is no analytical studieshave been reported in the literature on the postbuckling of thickFGM plates resting on elastic foundations.

N.D. Duc, H.V. Tung / Composite Structures 93 (2011) 2874–2881 2875

This paper extends previous work [19] to investigate the buck-ling and postbuckling behaviors of thick functionally graded platessupported by elastic foundations and subjected to in-plane com-pressive, thermal and thermomechanical loads. Reddy’s higher or-der shear deformation theory is used to establish governingequations taking into account geometrical nonlinearity and initialgeometrical imperfection, and the plate–foundation interaction isrepresented by Pasternak model. Closed-form expressions of buck-ling loads and postbuckling load–deflection curves for simply sup-ported FGM plates are obtained by Galerkin method. Analysis iscarried out to assess the effects of geometrical and material prop-erties, in-plane restraint, foundation stiffness and imperfection onthe behavior of the FGM plates.

2. Functionally graded plates on elastic foundations

Consider a ceramic–metal FGM plate of length a, width b andthickness h resting on an elastic foundation. A coordinate system(x,y,z) is established in which (x,y) plane on the middle surfaceof the plate and z is thickness direction (�h/2 6 z 6 h/2) as shownin Fig. 1.

The volume fractions of constituents are assumed to varythrough the thickness according to the following power lawdistribution

VcðzÞ ¼2zþ h

2h

� �N

; VmðzÞ ¼ 1� VcðzÞ ð1Þ

where N is volume fraction index (0 6 N <1). Effective propertiesPreff of FGM plate are determined by linear rule of mixture as

Preff ðzÞ ¼ PrmðzÞVmðzÞ þ PrcðzÞVcðzÞ ð2Þ

where Pr denotes a temperature independent material property,and subscripts m and c stand for the metal and ceramic constitu-ents, respectively.

Specialization of Eqs. (1) and (2) for the modulus of elasticity E,the coefficient of thermal expansion a and the coefficient of ther-mal conduction K gives

½EðzÞ;aðzÞ;KðzÞ� ¼ ½Em;am;Km� þ ½Ecm;acm;Kcm�2zþ h

2h

� �N

ð3Þ

where

Ecm ¼ Ec � Em;acm ¼ ac � am;Kcm ¼ Kc � Km ð4Þ

and Poisson ratio m is assumed to be constant. It is evident from Eqs.(3), (4) that the upper surface of the plate (z = h/2) is ceramic-rich,while the lower surface (z = �h/2) is metal-rich.

The reaction–deflection relation of Pasternak foundation is gi-ven by

qe ¼ k1w� k2r2w ð5Þ

x

y

z

ha

shear layerb

Fig. 1. Geometry and coordinate system of an FGM plate on elastic foundation.

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.

3. Theoretical formulation

The present study uses the Reddy’s higher order shear defor-mation plate theory to establish governing equations and deter-mine the buckling loads and postbuckling paths of the FGMplates.

The strains across the plate thickness at a distance z from themiddle surface are [21]

ex

ey

cxy

0BBB@

1CCCA ¼

e0x

e0y

c0xy

0BBB@

1CCCAþ z

k1x

k1y

k1xy

0BBBB@

1CCCCAþ z3

k3x

k3y

k3xy

0BBBB@

1CCCCA ð6Þ

cxz

cyz

!¼

c0xz

c0yz

!þ z2 k2

xz

k2yz

!ð7Þ

where

e0x

e0y

c0xy

0BBBB@

1CCCCA ¼

u;x þw2;x=2

v ;y þw2;y=2

u;y þ v ;x þw;xw;y

0BBB@

1CCCA;

k1x

k1y

k1xy

0BBBBB@

1CCCCCA ¼

/x;x

/y;y

/x;y þ /y;x

0BBB@

1CCCA;

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 ð8Þ

c0xz

c0yz

!¼

/x þw;x

/y þw;y

!;

k2xz

k2yz

!¼ �3c1

/x þw;x

/y þw;y

!

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,z) and (y,z) planes,respectively.

Hooke law for an FGM plate is defined as

ðrx;ryÞ ¼E

1� m2 ½ðex; eyÞ þ mðey; exÞ � ð1þ mÞaDTð1;1Þ� ð9Þ

ðrxy;rxz;ryzÞ ¼E

2ð1þ mÞ ðcxy; cxz; cyzÞ;

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 Eqs. (10) yields the consti-tutive relations as [2,3]

2876 N.D. Duc, H.V. Tung / Composite Structures 93 (2011) 2874–2881

ðNx;Mx; PxÞ ¼1

1� m2 ðE1; E2; E4Þ e0x þ me0

y

� �þ ðE2; E3; E5Þ k1

x þ mk1y

� �h

þðE4; E5; E7Þ k3x þ mk3

y

� �� ð1þ mÞðU1;U2;U4Þ

i

ðNy;My; PyÞ ¼1

1� m2 ðE1; E2; E4Þ e0y þ me0

x

� �þ ðE2; E3; E5Þ k1

y þ mk1x

� �h

þðE4; E5; E7Þ k3y þ mk3

x

� �� ð1þ mÞðU1;U2;U4Þ

i

ðNxy;Mxy; PxyÞ ¼1

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

xy

h

þðE4; E5; E7Þk3xy

i

ðQ x;RxÞ ¼1

2ð1þ mÞ ðE1; E3Þc0xz þ ðE3; E5Þk2

xz

h i

ðQ y;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

ðU1;U2;U4Þ ¼Z h=2

�h=2ð1; z; z3ÞEðzÞaðzÞDTðzÞdz ð12Þ

and specific expressions of coefficients Ei (i = 1–7) are given inAppendix A.

The nonlinear equilibrium equations of a perfect FGM platebased on the higher order shear deformation theory are [3,21]

Nx;x þ Nxy;y ¼ 0 ð13aÞ

Nxy;x þ Ny;y ¼ 0 ð13bÞ

Q x;x þ Qy;y � 3c1ðRx;x þ Ry;yÞ þ c1ðPx;xx þ 2Pxy;xy þ Py;yyÞþ Nxw;xx þ 2Nxyw;xy þ Nyw;yy � k1wþ k2r2w ¼ 0 ð13cÞ

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

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

where the plate–foundation interaction has been included. The lastthree equations of Eqs. (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)–(13e). Subsequently, elimination of the variable/x,x + /y,y from two the resulting equations leads to the followingsystem of equilibrium equations

Nx;x þ Nxy;y ¼ 0Nxy;x þ Ny;y ¼ 0

c21ðD2D5=D4 � D3Þr6wþ ðc1D2=D4 þ 1ÞD6r4w ð14Þþ ð1� c1D5=D4Þr2ðNxw;xx þ 2Nxyw;xy þ Nyw;yy � k1wþ k2r2wÞ� D6=D4ðNxw;xx þ 2Nxyw;xy þ Nyw;yy � k1wþ k2r2wÞ ¼ 0

where

D1 ¼E1E3 � E2

2

E1ð1� m2Þ ; D2 ¼E1E5 � E2E4

E1ð1� m2Þ ; D3 ¼E1E7 � E2

4

E1ð1� m2Þ ;

D4 ¼ D1 � c1D2; D5 ¼ D2 � c1D3;

D6 ¼1

2ð1þ mÞ E1 � 6c1E3 þ 9c21E5

� �:

ð15Þ

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

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

� f;yy w;xx þw�;xx

� �� 2f ;xy w;xy þw�;xy

� �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

� �hþf;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 the termsr6w andr4w are un-changed because these terms are obtained from the expressions forbending moments Mij and higher order moments Pij and these mo-ments depend not on the total curvature but only on the change incurvature of the plate [4]. Also, f(x,y) is stress function defined by

Nx ¼ f;yy; Ny ¼ f;xx; Nxy ¼ �f;xy: 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 ;e

0y

� �¼ 1

E1ðf;yy;f;xxÞ�mðf;xx;f;yyÞ�E2 k1

x ;k1y

� ��E4 k3

x ;k3y

� �þU1ð1;1Þ

h ið19Þ

c0xy ¼ �

1E1

2ð1þ mÞf;xy þ E2k1xy þ E4k3

xy

h i:

Introduction of Eqs. (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

ð20Þwhich is the same as equation derived by using the classical platetheory [19]. Eqs. (16) and (20) are nonlinear equations in terms ofvariables w and f and used to investigate the stability of thickFGM plates on elastic foundations subjected to mechanical, thermaland thermomechanical loads.

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 [12–15].

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; b:ð21Þ

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; b:ð22Þ

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 compressive edgeloads at immovable edges (i.e. Case 2 and the second of Case 3).

N.D. Duc, H.V. Tung / Composite Structures 93 (2011) 2874–2881 2877

The approximate solutions of w and f satisfying boundary con-ditions (21)–(23) are assumed to be [12–15]

ðw;w�Þ ¼ ðW;lhÞ sin kmx sin dny ð24aÞ

f ¼ A1 cos 2kmxþ A2 cos 2dnyþ A3 sin kmx sin dnyþ 12

Nx0y2 þ 12

Ny0x2

ð24bÞ

/x ¼ B1 cos kmx sin dny; /y ¼ B2 sin kmx cos dny ð24cÞ

where km = mp/a, dn = np/b, W is amplitude of the deflection and lis imperfection parameter. The coefficients Ai (i = 1–3) are deter-mined by substitution of Eqs. (24a,b) into Eq. (20) as

A1 ¼E1d

2n

32k2m

WðW þ 2lhÞ; A2 ¼E1k

2m

32d2n

WðW þ 2lhÞ; A3 ¼ 0:

ð25Þ

Employing Eqs. (8) and (11) in Eqs. (13d,e) and introduction ofEqs. (24a,c) into the resulting equations, the coefficients B1,B2 areobtained as

B1 ¼a12a23 � a22a13

a212 � a11a22

W; B2 ¼a12a13 � a11a23

a212 � a11a22

W ð26Þ

where

ða11; a22; a12Þ ¼ c21D3 þ D1 � 2c1D2

� �k2

m; d2n; mkmdn

� �þ 1� m

2c2

1D3 þ D1 � 2c1D2� �

d2n; k

2m; kmdn

� �þ D6ð1;1; 0Þ; ð27Þ

ða13; a23Þ ¼ c1D5 k3m þ kmd2

n; d3n þ dnk

2m

� �� D6 km; dnð Þ:

Subsequently, setting Eqs. (24a,b) into Eq. (16) and applying theGalerkin 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 k1½�

þk2 k2m þ d2

n

� �1� c1D5

D4

� �k2

m þ d2n

� �þ D6

D4

��W

þ E1

161� c1D5

D4

� �k4

md2n þ k2

md4n þ k6

m þ d6n

� �þ D6

D4k4

m þ d4n

� � �

�WðW þ lhÞðW þ 2lhÞ þ 1� c1D5

D4

� �k2

m þ d2n

� �þ D6

D4

�

� Nx0k2m þ Ny0d

2n

� �ðW þ lhÞ ¼ 0 ð28Þ

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.

3.1. Mechanical postbuckling analysis

Consider a simply supported FGM plate with all movable edgeswhich is rested on elastic foundations and subjected to in-planeedge compressive loads Fx, Fy (Pascal) uniformly distributed onedges x = 0, a and y = 0, b, respectively. In this case, prebuckingforce resultants are [3]

Nx0 ¼ �Fxh; Ny0 ¼ �Fyh ð29Þ

and Eq. (28) leads to

Fx ¼ e11

WW þ l

þ e12WðW þ 2lÞ ð30Þ

where

e11¼�16p4ðD2D5�D3D4Þ m2B2

aþn2� �3

þ3p2B2hD6ð4D2þ3D4Þ m2B2

aþn2� �2

3B2h m2B2

aþbn2� �

p2ð3D4�4D5Þ m2B2aþn2

� �þ3B2

hD6

h i

þK1B2

aþK2p2 m2B2aþn2

� �h iD1B2

a

p2B2h m2B2

aþbn2� � ;

ð31Þ

e12 ¼

p2E1

16B2h m2B2

a þ bn2� �

p2 3D4 � 4D5� �

m2B2a þ n2

� �þ 3B2

hD6

h i� p2 3D4 � 4D5

� �m4n2B4

a þm2n4B2a þm6B6

a þ n6� �h

þ3B2hD6 m4B4

a þ n4� �i

;

in which

Bh¼ b=h; Ba¼b=a; W ¼W=h; b¼ Fy=Fx;

K1¼k1a4

D1; K2¼

k2a2

D1; Ei ¼Ei=hiði¼1—7Þ;

D1¼E1E3�E2

2

E1ð1�m2Þ; D2¼

E1E5�E2E4

E1ð1�m2Þ; D3¼

E1E7�E24

E1ð1�m2Þ; ð32Þ

D4¼D1�43

D2; D5¼D2�43

D3; D6¼1

2ð1þmÞ E1�8E3þ16E5� �

:

For a perfect FGM plate, Eq. (30) reduces to an equation from whichbuckling compressive load may be obtained as Fxb ¼ e1

1.

3.2. Thermal postbuckling analysis

A simply supported FGM plate with all immovable edges is con-sidered. The plate is also supported by an elastic foundation andexposed 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, b,is fulfilled in an average sense as [10,12–15,19]Z b

0

Z a

0

@u@x

dxdy ¼ 0;Z a

0

Z b

0

@v@y

dydx ¼ 0: ð33Þ

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 �

E2

E1/x;x þ

c1E4

E1ð/x;x þw;xxÞ

� 12

w2;x �w;xw�;x þ

U1

E1

@v@y¼ 1

E1ðf;xx � mf;yyÞ �

E2

E1/y;y þ

c1E4

E1ð/y;y þw;yyÞ

� 12

w2;y �w;yw�;y þ

U1

E1: ð34Þ

Introduction of Eqs. (24) into Eqs. (34) and then the result intoEqs. (33) give

Nx0 ¼ �U1

1� m� 4

mnp2ð1� m2Þ ðE2 � c1E4ÞðkmB1 þ mdnB2Þ½

� c1E4 k2m þ md2

n

� �þ E1

8ð1� m2Þ k2m þ md2

n

� �WðW þ 2lhÞ; ð35Þ

Ny0 ¼ �U1

1� m� 4

mnp2ð1� m2Þ ðE2 � c1E4ÞðmkmB1 þ dnB2Þ½

� c1E4 mk2m þ d2

n

� �W þ E1

8ð1� m2Þ mk2m þ d2

n

� �WðW þ 2lhÞ:

When the deflection dependence of fictitious edge loads is ig-nored, i.e. W = 0, Eqs. (35) reduce to

2878 N.D. Duc, H.V. Tung / Composite Structures 93 (2011) 2874–2881

Nx0 ¼ Ny0 ¼ �U1

1� mð36Þ

which was derived by Shariat and Eslami [3] by solving the mem-brane form of equilibrium equations and employing the methodsuggested by Meyers and Hyer [20].

Substituting Eqs. (35) into Eq. (28) yields the expression of ther-mal parameter as

U1

1� m¼�c2

1 D2D5 � D3D4ð Þ k2m þ d2

n

� �2 þ D6ðc1D2 þ D4Þ k2m þ d2

n

� �ðD4 � c1D5Þ k2

m þ d2n

� �þ D6

"

þk1 þ k2 k2

m þ d2n

� �k2

m þ d2n

#W

W þ lh� 4

mnp2ð1� m2Þ k2m þ d2

n

� �� ðE2 � c1E4tÞ k3

mB1 þ mk2mdnB2 þ mkmd2

nB1 þ d3nB2

� � �c1E4 k4

m þ 2mk2md2

n þ d4n

� �W

þE1 ðD4 � c1D5Þ k4

md2n þ k2

md4n þ k6

m þ d6n

� �þ D6 k4

m þ d4n

� � 16 ðD4 � c1D5Þ k2

m þ d2n

� �þ D6

k2

m þ d2n

� �"

þE1 k4

m þ 2mk2md2

n þ d4n

� �8ð1� m2Þ k2

m þ d2n

� �#

WðW þ 2lhÞ: ð37Þ

3.2.1. Uniform temperature riseThe 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 U1 is obtainedfrom Eqs. (12), and substitution of the result into Eq. (37) yields

DT ¼ e21

WW þ l

þ e22W þ e2

3WðW þ 2lÞ ð38Þ

where

e21 ¼

ð1� mÞp2

L p2ð3D4 � 4D5Þ m2B2a þ n2

� �þ 3B2

hD6

h i��16p2

3B2h

ðD2D5 � D3D4Þ m2B2a þ n2

� �2"

þD6ð4D2 þ 3D4Þ m2B2a þ n2

� �#

þK1B2

a þ K2p2 m2B2a þ n2

� �h ið1� mÞB2

aD1

p2LB2h m2B2

a þ n2� � ;

e22 ¼ �

4

3mnpLð1þ mÞB2h m2B2

a þ n2� �

"Bhð3E2 � 4E4Þ:

m3B3aB1 þ mm2nB2

aB2 þ mmn2BaB1 þ n3B2

� ��4pE4 m4B4

a þ 2mm2n2B2a þ n4

� �i;

e23 ¼

E1p2ð1� mÞ16LB2

h m2B2a þ n2

� �p2ð3D4 � 4D5Þ m2B2

a þ n2� �

þ 3B2hD6

h i� p2ð3D4 � 4D5Þ m4n2B4

a þm2n4B2a þm6B6

a þ n6� �h

þ3B2hD6 m4B4

a þ n4� �i

þE1p2 m4B4

a þ 2m m2n2B2a þ n4

� �8Lð1þ mÞB2

h m2B2a þ n2

� �ð39Þ

in which

L ¼ Emam þEmacm þ Ecmam

N þ 1þ Ecmacm

2N þ 1;

B1 ¼�a12�a23 � �a22�a13

�a212 � �a11�a22

; B2 ¼�a12�a13 � �a11�a23

�a212 � �a11�a22

: ð40Þ

Also, specific expressions of �a11; �a22; �a12; �a13; �a23 can be found inAppendix A.

By Setting l = 0 Eq. (38) leads to an equation from which buck-ling temperature change of the perfect FGM plates may be deter-mined as DTb ¼ e2

1.

3.2.2. Through the thickness temperature gradientThe metal-rich surface temperature Tm is maintained at refer-

ence value while ceramic-rich surface temperature Tc is enhancedand steadily conducted through the thickness direction accordingto one-dimensional Fourier equation

ddz

KðzÞdTdz

�¼ 0; Tðz ¼ �h=2Þ ¼ Tm; Tðz ¼ h=2Þ ¼ Tc: ð41Þ

Using K(z) defined in Eq. (3), the solution of Eq. (41) may be found interms of polynomial series, and the first seven terms of this seriesgives the following approximation [1,3,5,19]

TðzÞ ¼ Tm þ DTrP5

j¼0ð�rN Kcm=KmÞj

jNþ1P5j¼0ð�Kcm=KmÞj

jNþ1

ð42Þ

where r = (2z + h)/2h and, in this case of thermal loading,DT = Tc � Tm is defined as the temperature difference between twosurfaces of the FGM plate.

Substitution of Eq. (42) into Eqs. (12) and setting the result U1

into Eq. (37) yield a closed-form expression of temperature–deflec-tion curves which is similar to Eq. (38), providing L is replaced by Hdefined as

H ¼P5

j¼0ð�Kcm=KmÞj

jNþ1EmamjNþ2 þ

EmacmþEcmamðjþ1ÞNþ2 þ Ecmacm

ðjþ2ÞNþ2

h iP5

j¼0ð�Kcm=KmÞj

jNþ1

: ð43Þ

3.3. Thermomechanical postbuckling analysis

The FGM plate resting on an elastic foundation is uniformlycompressed 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 Eqs. (34) in the second of Eqs. (33) as

Ny0 ¼ mNx0 �U1 �4dn

mnp2 ½E2B2 � c1E4ðdn þ B2Þ�W

þ E1d2n

8WðW þ 2lhÞ: ð44Þ

Subsequently, Nx0 and Ny0 are placed in Eq. (28) to give

Fx ¼ e31

W

W þ lþ e3

2W þ e33WðW þ 2lÞ � Ln2DT

m2B2a þ mn2

; ð45Þ

where the coefficients e31; e

32; e

33 are described in detail in Appendix A

and L is replaced by H in the case of the FGM plates subjected to com-bined action of uniaxial compressive load and temperature gradient.

Eqs. (30), (38) and (45) are explicit expressions of load–deflec-tion curves for thick FGM plates resting on Pasternak elastic foun-dations and subjected to in-plane compressive, thermal andthermomechanical loads, respectively. Specialization of theseequations for thin pure FGM plates, i.e. ignoring the transverseshear deformations and elastic foundations, gives the correspond-ing results derived by using the classical plate theory [19].

4. Results and discussion

In the verification of the present formulation for the buckling andpostbuckling behaviors of thick FGM plates, thermal postbuckling of

Fig. 2. Comparisons of thermal postbuckling load–deflection curves for isotropicplates.

Fig. 3. Effects of volume fraction index on the postbuckling of FGM plates underuniaxial compressive load (all movable edges).

Fig. 4. Effects of in-plane restraint on the postbuckling of FGM plate under uniaxialcompression.

Fig. 5. Effects of volume fraction index on the postbuckling of FGM plates underuniform temperature rise (all IM edges).

N.D. Duc, H.V. Tung / Composite Structures 93 (2011) 2874–2881 2879

a simply supported square thick isotropic plate is analyzed. Theplate is exposed to uniform temperature field with all immovableedges and without foundation interaction. Fig. 2 gives thermal post-buckling load–deflection curves for perfect and imperfect isotropicplates (m = 0.3) according to the present approach in comparisonwith Shen’s results [10] using asymptotic perturbation technique.As can be seen, a good agreement is obtained in this comparison.

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 [2–5]

Em ¼ 70 GPa; am ¼ 23� 10�6 �C�1; Km ¼ 204 W=mK

Ec ¼ 380 GPa; ac ¼ 7:4� 10�6 �C�1; Kc ¼ 10:4 W=mK; ð46Þ

and Poisson ratio is chosen to be m = 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.

Fig. 6. Effects of volume fraction index on the postbuckling of FGM plates undertemperature gradient (all IM edges).

Fig. 7. Effects of the elastic foundations on the postbuckling of FGM plates underuniform temperature rise (all IM edges).

Fig. 8. Effects of the elastic foundations on the postbuckling of FGM plates undertemperature gradient (all IM edges).

Fig. 9. Effects of the temperature field on the postbuckling of FGM plates underuniaxial compression (immovable on y = 0, b).

Fig. 10. Interactive effects of elastic foundation and temperature gradient on thepostbuckling of FGM plates under uniaxial compression (immovable on y = 0, b).

2880 N.D. Duc, H.V. Tung / Composite Structures 93 (2011) 2874–2881

Fig. 3 shows decreasing trend of postbuckling curves of the FGMplates with movable edges under uniaxial compressive load as the

volume fraction index N increases. Both critical buckling loads andpostbuckling carrying capacity are strongly dropped when N is in-creased from 0 to 1, and a slower variation is observed when N isgreater than 1.

Fig. 4 compares the postbuckling behavior of compressed FGMplates under two types of in-plane boundary restraint. The plateis assumed to be freely movable (FM) on all edges (Case 1) andimmovable (IM) on two unloaded edges y = 0, b (Case 3). As canbe seen, in spite of lower critical buckling loads, the postbucklingequilibrium paths for Case 3 become higher than those for Case 1in deep region of postbuckling behavior.

Figs. 5 and 6 illustrate the variation of thermal postbucklingload–deflection curves for FGM plates with all immovable edgessubjected to uniform temperature rise and through the thicknesstemperature gradient, respectively, with various values of N. As ex-pected, the reduction of volume fraction percentage of ceramicconstituent makes the capability of temperature resistance of theplates to be decreased. In addition, the variation tendency of tem-perature–deflection curves when N increases from 0 to 5 for twocases of thermal loading is not similar.

The effects of the elastic foundations on the postbucklingbehavior of the FGM plates under two types of thermal loads aredepicted in Figs. 7 and 8. Obviously, both buckling loads and post-buckling loading bearing capability are enhanced due to the pres-ence of elastic foundations. Furthermore, the shear layer stiffnessK2 of Pasternak model has more pronounced influences in compar-ison with foundation modulus K1 of Winkler model.

Fig. 9 shows the thermomechanical postbuckling behavior ofFGM plates exposed to temperature field and subjected to uniaxialcompression. As can be observed, the capacity of mechanical loadbearing of the FGM plates is considerably reduced due to theenhancement of pre-existent thermal load.

Finally, interactive effects of elastic foundations and tempera-ture gradient on the postbuckling of the FGM plates subjected touniaxial compressive loads are considered in Fig. 10. As can beseen, in spite of the raising of ceramic-rich surface temperature,Pasternak type foundations have very beneficial influences on theimprovement of thermomechanical loading capacity of the FGMplates.

5. Concluding remarks

This paper presents an analytical approach to investigate themechanical, thermal and thermomechanical buckling and

N.D. Duc, H.V. Tung / Composite Structures 93 (2011) 2874–2881 2881

postbuckling behaviors of thick FGM plates resting on Pasternaktype elastic foundations. The formulations are based on the Red-dy’s higher order shear deformation theory to obtain accurate pre-dictions for buckling loads and postbuckling loading carryingcapacity of thick plates. In addition, obtained closed-form expres-sions of load–deflection curves have practical significance in anal-ysis and design. The results reveal that elastic foundations havepronounced benefit on the stability of FGM plates. Furthermore,volume fraction index, in-plane boundary restraint, imperfectionand temperature conditions also have considerable effects on thebehavior of the plates.

Acknowledgements

This paper was supported by the National Foundation for Sci-ence and Technology Development of Vietnam - NAFOSTED, pro-ject code 107.02-2010.08. The authors are grateful for thisfinancial support.

Appendix A

E1 ¼ Emhþ EcmhN þ 1

; E2 ¼EcmNh2

2ðN þ 1ÞðN þ 2Þ ;

E3 ¼Emh3

12þ Ecmh3 1

4ðN þ 1Þ �1

ðN þ 2ÞðN þ 3Þ

�;

E4 ¼Ecmh4

N þ 118� 3

4ðN þ 2Þ þ3

ðN þ 3ÞðN þ 4Þ

�;

E5 ¼Emh5

80þ Ecmh5

N þ 11

16� 1

2ðN þ 2Þ þ3

ðN þ 2ÞðN þ 3Þ

� 12ðN þ 2ÞðN þ 4ÞðN þ 5Þ

�;

E7 ¼Emh7

448þ Ecmh7

N þ 11

64� 6

32ðN þ 2Þ þ30

16ðN þ 2ÞðN þ 3Þ

� 15ðN þ 2ÞðN þ 3ÞðN þ 4Þ þ

90ðN þ 2ÞðN þ 3ÞðN þ 4ÞðN þ 5Þ

� 360ðN þ 2ÞðN þ 3ÞðN þ 4ÞðN þ 6ÞðN þ 7Þ

�:

ð�a11; �a22; �a12Þ ¼p2

B2h

169

D3 þ D1 �83

D2

� �m2B2

a ; n2; mmnBa

� �

þ ð1� mÞp2

2B2h

169

D3 þ D1 �83

D2

� �n2;m2B2

a ;mnBa

� �þ D6ð1;1; 0Þ;

ð�a13; �a23Þ ¼4p3D5

3B3h

m3B3a þmn2Ba;n3 þm2nB2

a

� �� pD6

BhðmBa;nÞ:

e31¼�16p4ðD2D5�D3D4Þ m2B2

aþn2� �3

þ3p2B2hD6ð4D2þ3D4Þ m2B2

aþn2� �2

3B2h m2B2

aþmn2� �

p2ð3D4�4D5Þ m2B2aþn2

� �þ3B2

hD6

h i

þK1B2

aþK2p2 m2B2aþn2

� �p2B2

h m2B2aþmn2

� � B2aD1;

e32 ¼ �

4n2

mpBh m2B2a þ mn2

� � E2B2 �4E4

3B2 þ

npBh

� �" #;

e33 ¼

p2E1

16B2h m2B2

a þ mn2� �

p2 3D4 � 4D5� �

m2B2a þ n2

� �þ 3B2

hD6

h i� p2 3D4 � 4D5

� �m4n2B4

a þm2n4B2a þm6B6

a þ n6� �h

þ3B2hD6 m4B4

a þ n4� �i

þ E1p2n4

8B2h m2B2

a þ mn2� �

References

[1] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates. AIAA2002;40(1):162–9.

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

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

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

[5] Lanhe W. Thermal buckling of a simply supported moderately thickrectangular FGM plate. Compos Struct 2004;64:211–8.

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

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

[8] 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.

[9] Shen H-S. Postbuckling of FGM plates with piezoelectric actuators underthermo–electro-mechanical loadings. Int J Solids Struct 2005;42:6101–21.

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

[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. Thin-Wall 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.

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

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