DSpace at VNU: Mechanical and thermal postbuckling of higher order shear deformable functionally graded plates on elastic foundations

Mechanical and thermal postbuckling of higher order shear deformablefunctionally graded plates on elastic foundations Nguyen Dinh Duca, Hoang Van Tungb,⇑ a University of Engineering and

Mechanical and thermal postbuckling of higher order shear deformable

functionally graded plates on elastic foundations

Nguyen Dinh Duca, Hoang Van Tungb,⇑

a

University of Engineering and Technology, Vietnam National University, Ha Noi, Viet Nam

b

Faculty of Civil Engineering, Hanoi Architectural University, Hanoi, Viet Nam

Article history:

Available online 24 May 2011

Keywords:

Functionally Graded Materials

Postbuckling

Higher order shear deformation theory

Elastic foundation

Imperfection

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 and thermomechanical loads Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents The formulations are based on higher order shear deformation plate theory taking into account Von Karman nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation

By applying Galerkin method, closed-form relations of buckling loads and postbuckling equilibrium paths for simply supported plates are determined Analysis is carried out to show the effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buck-ling and postbuckbuck-ling loading capacity of the plates

Ó 2011 Elsevier Ltd All rights reserved

1 Introduction

Due to high performance heat resistance capacity and excellent

characteristics in comparison with conventional composites,

Functionally Graded Materials (FGMs) which are microscopically

composites and composed from mixture of metal and ceramic

con-stituents have attracted considerable attention recent years By

continuously and gradually varying the volume fraction of

constit-uent materials through a specific direction, FGMs are capable of

withstanding ultrahigh temperature environments and extremely

large 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 structures

widely used in industries Buckling and postbuckling behaviors of

FGM structures under different types of loading are important for

practical applications and have received considerable interest

Eslami and his co-workers used analytical approach, classical and

higher order plate theories in conjunction with adjacent

equilib-rium criterion to investigate the buckling of FGM plates with and

without imperfection under mechanical and thermal loads[1–4]

According to this direction, Lanhe [5] also employed first order

shear deformation theory to obtain closed-form relations of critical

buckling temperatures for simply supported FGM plates Zhao et al

[6]analyzed the mechanical and thermal buckling of FGM plates

using element-free Ritz method Liew et al.[7,8]used the higher

order shear deformation theory in conjunction with differential quadrature method to investigate the postbuckling of pure and hy-brid FGM plates with and without imperfection on the point of view that buckling only occurs for fully clamped FGM plates The postbuckling behavior of pure and hybrid FGM plates under the combination 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 thermal loads

The components of structures widely used in aircraft, reusable space 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 understanding

of the postbuckling behavior of plates and shells Librescu and Lin have extended previous works[14,15] to consider the post-buckling behavior of flat and curved laminated composite panels resting on Winkler elastic foundations[14,15] In spite of practical importance and increasing use of FGM structures, investigation on FGM plates and shells supported by elastic media are limited in number The bending behavior of FGM plates resting on Pasternak type foundation has been studied by Huang et al.[16]using state space method, Zenkour[17]using analytical method and by Shen and Wang[18]making use of asymptotic perturbation technique

To the best of authors’ knowledge, there is no analytical studies have been reported in the literature on the postbuckling of thick FGM plates resting on elastic foundations

0263-8223/$ - see front matter Ó 2011 Elsevier Ltd All rights reserved.

⇑Corresponding author.

E-mail address: hoangtung0105@gmail.com (H.V Tung).

Contents lists available atScienceDirect

Composite Structures

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m p s t r u c t

This paper extends previous work[19]to investigate the

buck-ling and postbuckbuck-ling behaviors of thick functionally graded plates

supported 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 governing

equations taking into account geometrical nonlinearity and initial

geometrical imperfection, and the plate–foundation interaction is

represented by Pasternak model Closed-form expressions of

buck-ling loads and postbuckbuck-ling load–deflection curves for simply

sup-ported FGM plates are obtained by Galerkin method Analysis is

carried out to assess the effects of geometrical and material

prop-erties, in-plane restraint, foundation stiffness and imperfection on

the behavior of the FGM plates

2 Functionally graded plates on elastic foundations

Consider a ceramic–metal FGM plate of length a, width b and

thickness h resting on an elastic foundation A coordinate system

(x, y, z) is established in which (x, y) plane on the middle surface

of the plate and z is thickness direction (h/2 6 z 6 h/2) as shown

inFig 1

The volume fractions of constituents are assumed to vary

through the thickness according to the following power law

distribution

VcðzÞ ¼ 2z þ h

2h

where N is volume fraction index (0 6 N < 1) Effective properties

Preffof FGM plate are determined by linear rule of mixture as

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 expansionaand the coefficient of

ther-mal conduction K gives

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

2h

ð3Þ

where

and Poisson ratiomis 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

where r2= @2/@x2+ @2/@y2, w is the deflection of the plate, k1 is Winkler foundation modulus and k2is the shear layer foundation stiffness 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 FGM plates

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

ex

ey

cxy

0 B B

1 C

e0 x

e0 y

c0 xy

0 B B

1 C

k1x

k1y

k1xy

0 B B

@

1 C C

3

k3x

k3y

k3xy

0 B B

@

1 C C

cxz

cyz

!

xz

c0 yz

!

þ z2 k

2 xz

k2yz

!

ð7Þ

where

e0 x

e0 y

c0 xy

0 B B

@

1 C C

u;xþ w2

;x=2

v;yþ w2

;y=2

u;yþv;xþ w;xw;y

0 B B

1 C

C;

k1x

k1y

k1xy

0 B B B

1 C C

/x;x /y;y

/x;yþ /y;x

0 B B

1 C

C;

k3x

k3y

k3xy

0 B

@

1 C

A ¼ c1

/x;xþ w;xx

/y;yþ w;yy

/x;yþ /y;xþ 2w;xy

0 B

1

c0 xz

c0 yz

!

¼ /xþ w;x /yþ w;y

!

; k2xz

k2yz

!

¼ 3c1

/xþ w;x

/yþ w;y

!

in which c1= 4/3h2,ex,eyare normal strains,cxyis the in-plane shear strain, andcxz,cyzare the transverse shear deformations Also, u,v

are the displacement components along the x, y directions, respec-tively, and /x,/yare the slope rotations in the (x, z) and (y, z) planes, respectively

Hooke law for an FGM plate is defined as

ðrx;ryÞ ¼ E

1 m2½ðex;eyÞ þmðey;exÞ  ð1 þmÞa DTð1; 1Þ ð9Þ

ðrxy;rxz;ryzÞ ¼ E

2ð1 þmÞ cxy;cxz;cyzÞ;

whereDT 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 ¼

Z h=2

h=2

rið1; z; z3Þdz i ¼ x; y; xy

ðQi;RiÞ ¼

Z h=2

h=2

Substitution of Eqs.(6), (7) and (9)into Eqs.(10)yields the consti-tutive relations as[2,3]

x

y

z

h a

shear layer b

ðNx;Mx;PxÞ ¼ 1

1 m2 ðE1;E2;E4Þ e0

xþme0 y

þ ðE2;E3;E5Þ k 1xþmk1y h

þðE4;E5;E7Þ k 3xþmk3y

 ð1 þmÞðU1;U2;U4Þi

ðNy;My;PyÞ ¼ 1

1 m2 ðE1;E2;E4Þ e0

yþme0 x

þ ðE2;E3;E5Þ k 1yþmk1x h

þðE4;E5;E7Þ k 3yþmk3x

 ð1 þmÞðU1;U2;U4Þi

ðNxy;Mxy;PxyÞ ¼ 1

2ð1 þmÞ ðE1;E2;E4Þc0

xyþ ðE2;E3;E5Þk1xy

h

þðE4;E5;E7Þk3xyi

ðQx;RxÞ ¼ 1

2ð1 þmÞ ðE1;E3Þc0

xzþ ðE3;E5Þk2xz

ðQy;RyÞ ¼ 1

2ð1 þmÞ ðE1;E3Þc0

yzþ ðE3;E5Þk2yz

ð11Þ

where

ðE1;E2;E3;E4;E5;E7Þ ¼

Z h=2

h=2

ð1; z; z2;z3;z4;z6ÞEðzÞdz

ðU1;U2;U4Þ ¼

Zh=2

h=2

and specific expressions of coefficients Ei (i = 1–7) are given in

Appendix A

The nonlinear equilibrium equations of a perfect FGM plate

based on the higher order shear deformation theory are[3,21]

Qx;xþ Qy;y 3c1ðRx;xþ Ry;yÞ þ c1ðPx;xxþ 2Pxy;xyþ Py;yyÞ

þ Nxw;xxþ 2Nxyw;xyþ Nyw;yy k1w þ k2r2w ¼ 0 ð13cÞ

Mx;xþ Mxy;y Qxþ 3c1Rx c1ðPx;xþ Pxy;yÞ ¼ 0 ð13dÞ

Mxy;xþ My;y Qyþ 3c1Ry c1ðPxy;xþ Py;yÞ ¼ 0 ð13eÞ

where the plate–foundation interaction has been included The last

three equations of Eqs.(13)may be rewritten into two equations in

terms of variables w and /x,x+ /y,yby substituting Eqs.(8)and(11)

into Eqs (13c)–(13e) Subsequently, elimination of the variable

/x,x+ /y,yfrom two the resulting equations leads to the following

system of equilibrium equations

Nx;xþ Nxy;y¼ 0

Nxy;xþ Ny;y¼ 0

c2ðD2D5=D4 D3Þr6w þ ðc1D2=D4þ 1ÞD6r4w ð14Þ

þ ð1  c1D5=D4Þr2ðNxw;xxþ 2Nxyw;xyþ Nyw;yy k1w þ k2r2wÞ

 D6=D4ðNxw;xxþ 2Nxyw;xyþ Nyw;yy k1w þ k2r2

wÞ ¼ 0

where

D1¼E1E3 E

2

E1ð1 m2Þ; D2¼

E1E5 E2E4

E1ð1 m2Þ ; D3¼

E1E7 E2

E1ð1 m2Þ;

D4¼ D1 c1D2; D5¼ D2 c1D3;

2ð1 þmÞ E1 6c1E3þ 9c

2E5

:

ð15Þ

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

c2ðD2D5=D4 D3Þr6w þ ðc1D2=D4þ 1ÞD6r4w þ ð1  c1D5=D4Þr2

 f;yy w;xxþ w

;xx

 2f;xy w;xyþ w

;xy

þ f;xx w;yyþ w

;yy

h

k1w þ k2r2wi

 D6=D4 f;yy w;xxþ w

;xx

 2f;xy w;xyþ w

;xy

h

þf;xx w;yyþ w

;yy

 k1w þ k2r2wi

in which w⁄(x, y) is a known function representing initial small imperfection of the plate Note that the termsr6w andr4w are un-changed because these terms are obtained from the expressions for bending moments Mijand higher order moments Pijand these mo-ments depend not on the total curvature but only on the change in curvature of the plate[4] Also, f(x, y) is stress function defined by

The geometrical compatibility equation for an imperfect plate is written as

e0 x;yyþe0 y;xxc0 xy;xy¼ w2

;xy w;xxw;yyþ 2w;xyw

;xy w;xxw

;yy

 w;yyw

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

e0

x;e0 y

E1 ðf;yy;f;xxÞ mðf;xx;f;yyÞ  E2k1x;k1y

 E4k3x;k3y

þU1ð1;1Þ

ð19Þ

c0

xy¼ 1

E1

2ð1 þmÞf;xyþ E2k1xyþ E4k3xy

:

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

r4

f  E1 w2

;xy w;xxw;yyþ 2w;xyw

;xy w;xxw

;yy w;yyw

;xx

¼ 0 ð20Þ

which is the same as equation derived by using the classical plate theory[19] Eqs.(16) and (20)are nonlinear equations in terms of variables w and f and used to investigate the stability of thick FGM plates on elastic foundations subjected to mechanical, thermal and thermomechanical loads

Depending on the in-plane restraint at the edges, three cases of boundary 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 freely movable (FM) The associated boundary conditions are

w ¼ Nxy¼ /y¼ Mx¼ Px¼ 0; Nx¼ Nx0at x ¼ 0; a

w ¼ Nxy¼ /x¼ My¼ Py¼ 0; Ny¼ Ny0at y ¼ 0; b: ð21Þ

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

w ¼ u ¼ /y¼ Mx¼ Px¼ 0; Nx¼ Nx0at x ¼ 0; a

w ¼v¼ /x¼ My¼ Py¼ 0; Ny¼ Ny0at y ¼ 0; b: ð22Þ

Case 3 All edges are simply supported Two edges x = 0, a are freely movable and subjected to compressive load in the 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 Nxy¼ /y¼ Mx¼ Px¼ 0; Nx¼ Nx0at x ¼ 0; a

w ¼v¼ /x¼ My¼ Py¼ 0; Ny¼ Ny0at y ¼ 0; b ð23Þ

where Nx0, Ny0are in-plane compressive loads at movable edges (i.e Case 1 and the first of Case 3) or are fictitious compressive edge loads at immovable edges (i.e Case 2 and the second of Case 3)

The approximate solutions of w and f satisfying boundary

con-ditions(21)–(23)are assumed to be[12–15]

f ¼ A1cos 2kmx þ A2cos 2dny þ A3sin kmx sin dny þ1

2Nx0y

2þ1

2Ny0x

2

ð24bÞ /x¼ B1cos kmx sin dny; /y¼ B2sin kmx cos dny ð24cÞ

where km= mp/a, dn= np/b, W is amplitude of the deflection andl

is imperfection parameter The coefficients Ai(i = 1–3) are

deter-mined by substitution of Eqs.(24a,b)into Eq.(20)as

A1¼ E1d

2

n

32k2mWðW þ 2lhÞ; A2¼E1k

2 m

32d2nWðW þ 2lhÞ; A3¼ 0:

ð25Þ

Employing Eqs.(8) and (11)in Eqs.(13d,e)and introduction of

Eqs.(24a,c)into the resulting equations, the coefficients B1,B2are

obtained as

B1¼a12a23 a22a13

a2

12 a11a22

W; B2¼a12a13 a11a23

a2

12 a11a22

where

ða11;a22;a12Þ ¼ c2D3þ D1 2c1D2

k2m;d2n;mkmdn

þ1 m

2D3þ D1 2c1D2

d2;k2m;kmdn

þ D6ð1; 1; 0Þ; ð27Þ

ða13;a23Þ ¼ c1D5k3mþ kmd2;d3þ dnk2m

 D6ðkm;dnÞ:

Subsequently, setting Eqs (24a,b)into Eq.(16) and applying the

Galerkin procedure for the resulting equation yield

c2 D2D5

D4

 D3

k2mþ d2

þ D6

c1D2

D4

þ 1

k2mþ d2

þ k½ 1



þk2k2mþ d2n

1 c1D5

D4

k2mþ d2n

þD6

D4

W

þE1

c1D5

D4

k4md2þ k2

md4þ k6

mþ d6

þD6

D4

k4mþ d4

WðW þlhÞðW þ 2lhÞ þ 1 c1D5

D4

k2mþ d2

þD6

D4

 N x0k2mþ Ny0d2

where m, n are odd numbers This equation will be used to analyze

the buckling and postbuckling behaviors of thick FGM plates under

mechanical, thermal and thermomechanical loads

3.1 Mechanical postbuckling analysis

Consider a simply supported FGM plate with all movable edges

which is rested on elastic foundations and subjected to in-plane

edge compressive loads Fx, Fy (Pascal) uniformly distributed on

edges x = 0, a and y = 0, b, respectively In this case, prebucking

force resultants are[3]

and Eq.(28)leads to

Fx¼ e1 W

where

e1¼16p4ðD2D5 D3D4Þ m2B2

þ n2

 3

þ 3p2B2D6ð4D2þ 3D4Þ m2B2

þ n2

 2

3B2m2B2þ bn2

p2ð3D4 4D5Þ m 2B2þ n2

þ 3B2D6

þ

K1B2þ K2p2m2B2þ n2

D1B2

p2B2m2B2þ bn2 ;

ð31Þ

16B2 m2B2

aþ bn2

p2 3D4 4D5

m2B2

aþ n2

þ 3B2D6

 p23D4 4D5

m4n2B4aþ m2n4B2aþ m6B6aþ n6

h þ3B2hD6m4B4aþ n4i

;

in which

Bh¼ b=h; Ba¼ b=a; W ¼ W=h; b ¼ Fy=Fx;

K1¼k1a

4

D1

; K2¼k2a

2

D1

; Ei¼ Ei=hiði ¼ 1—7Þ;

D1¼E1E3 E

2

E1ð1 m2Þ; D2¼

E1E5 E2E4

E1ð1 m2Þ; D3¼

E1E7 E2

D4¼ D14

3D2; D5¼ D2

4

3D3; D6¼

1 2ð1 þmÞ E1 8E3þ 16E5

:

For a perfect FGM plate, Eq.(30)reduces to an equation from which buckling compressive load may be obtained as Fxb¼ e1

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 and exposed to temperature environments or subjected to through the thickness temperature gradient The in-plane condition on immovability at all edges, i.e u = 0 at x = 0, a andv= 0 at y = 0, b,

is fulfilled in an average sense as[10,12–15,19]

Z b 0

Z a 0

@u

@xdxdy ¼ 0;

Z a 0

Z b 0

@v

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

@u

@x¼

1

E1

ðf;yymf;xxÞ E2

E1

/x;xþc1E4

E1

ð/x;xþ w;xxÞ

2w

2

;x w;xw

;xþU1

E1

@v

1

E1

ðf;xxmf;yyÞ E2

E1

/y;yþc1E4

E1

ð/y;yþ w;yyÞ

2w

2

;y w;yw

;yþU1

E1

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

Nx0¼  U1

1 m

4

mnp2ð1 m2Þ½ðE2 c1E4ÞðkmB1þmdnB2Þ

 c1E4k2mþmd2n

8ð1 m2Þ k

2

mþmd2n

Ny0¼  U1

1 m

4

mnp2ð1 m2Þ½ðE2 c1E4ÞðmkmB1þ dnB2Þ

 c1E4mk2mþ d2

8ð1 m2Þmk2mþ d2

WðW þ 2lhÞ:

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

Nx0¼ Ny0¼  U1

which was derived by Shariat and Eslami[3]by solving the

mem-brane form of equilibrium equations and employing the method

suggested by Meyers and Hyer[20]

Substituting Eqs.(35)into Eq.(28)yields the expression of

ther-mal parameter as

U1

1 m¼

c2ðD2D5 D3D4Þ k2mþ d2n2

þ D6ðc1D2þ D4Þ k 2mþ d2n

ðD4 c1D5Þ k 2mþ d2

þ D6

"

þk1þ k2 k

2

mþ d2

k2mþ d2

# W

4

mnp2ð1 m2Þ k 2mþ d2

 ðE2 c1E4tÞ k3

mB1þmk2mdnB2þmkmd2B1þ d3B2

c1E4k4mþ 2mk2md2þ d4

W

þ E1 ðD4 c1D5Þ k

4

md2nþ k2md4nþ k6mþ d6n

þ D6k4mþ d4n

16 ðD4 c1D5Þ k2mþ d2

þ D6

k2mþ d2

"

þE1 k

4

mþ 2mk2md2þ d4

8ð1 m2Þ k 2mþ d2

#

3.2.1 Uniform temperature rise

The FGM plate is exposed to temperature environments

uni-formly raised from stress free initial state Tito final value Tf, and

temperature changeDT = Tf Tiis considered to be independent

from thickness variable The thermal parameter U1 is obtained

from Eqs.(12), and substitution of the result into Eq.(37)yields

where

L p2ð3D4 4D5Þ m 2B2aþ n2

þ 3B2hD6

16p2

3B2h ðD2D5 D3D4Þ m

2B2

aþ n2

"

þD6ð4D2þ 3D4Þ m 2B2aþ n2

þ

K1B2

aþ K2p2 m2B2

aþ n2

ð1 mÞB2aD1

p2LB2 m2B2

aþ n2

3mnpLð1 þmÞB2hm2B2aþ n2

"

Bhð3E2 4E4Þ:

m3B3aB1þmm2nB2aB2þmmn2BaB1þ n3B2

4pE4 m4B4

aþ 2mm2n2B2

aþ n4

;

e2

16LB2hm2B2aþ n2

p2ð3D4 4D5Þ m 2B2aþ n2

þ 3B2hD6

ð3D4 4D5Þ m4n2B4

aþ m2n4B2

aþ m6B6

aþ n6

h

þ3B2D6 m4B4

aþ n4

þ

E1p2 m4B4

aþ 2mm2n2B2

aþ n4

8Lð1 þmÞB2 m2B2

aþ n2

ð39Þ

in which

L ¼ EmamþEmacmþ Ecmam

Ecmacm

2N þ 1;

B1¼a12a23 a22a13

a2  a11a22

; B2¼a12a13 a11a23

a2  a11a22

Also, specific expressions of a11; a22; a12; a13; a23 can be found in

Appendix A

By Settingl= 0 Eq.(38)leads to an equation from which buck-ling temperature change of the perfect FGM plates may be deter-mined asDTb¼ e2

3.2.2 Through the thickness temperature gradient The metal-rich surface temperature Tmis maintained at refer-ence value while ceramic-rich surface temperature Tcis enhanced and steadily conducted through the thickness direction according

to one-dimensional Fourier equation

d

dT dz

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

TðzÞ ¼ TmþDTr

P5 j¼0 ðr N K cm =K m Þj jNþ1

P5 j¼0 ðK cm =K m Þj jNþ1

ð42Þ

where r = (2z + h)/2h and, in this case of thermal loading,

DT = Tc Tmis defined as the temperature difference between two surfaces of the FGM plate

Substitution of Eq.(42)into Eqs.(12)and setting the resultU1

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

H ¼

P5 j¼0 ðK cm =K m Þj jNþ1

E m a m

jNþ2þE m a cm þE cm a m

ðjþ1ÞNþ2 þ E cm a cm

ðjþ2ÞNþ2

P5 j¼0 ðK cm =K m Þ j

jNþ1

3.3 Thermomechanical postbuckling analysis

The 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 two edges y = 0, b are assumed to be immovable In this case, Nx0= Fxh and fictitious compressive load on immovable edges is determined

by setting the second of Eqs.(34)in the second of Eqs.(33)as

Ny0¼mNx0U1 4dn

mnp2½E2B2 c1E4ðdnþ B2ÞW

þE1d

2 n

Subsequently, Nx0and Ny0are placed in Eq.(28)to give

Fx¼ e3 W

3W þ e3WðW þ 2lÞ  Ln

2

DT

m2B2

aþmn2; ð45Þ

where the coefficients e3;e3;e3are described in detail inAppendix 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 and thermomechanical loads, respectively Specialization of these equations for thin pure FGM plates, i.e ignoring the transverse shear deformations and elastic foundations, gives the correspond-ing results derived by uscorrespond-ing the classical plate theory[19]

4 Results and discussion

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

a simply supported square thick isotropic plate is analyzed The plate is exposed to uniform temperature field with all immovable edges and without foundation interaction.Fig 2gives thermal post-buckling load–deflection curves for perfect and imperfect isotropic plates (m= 0.3) according to the present approach in comparison with 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 postbuckling analysis of thick FGM plates resting on elastic foundations, con-sider a square ceramic–metal plate consisting of aluminum and alumina with the following properties[2–5]

Em¼ 70 GPa;am¼ 23  106 C1; Km¼ 204 W=mK

Ec¼ 380 GPa;ac¼ 7:4  106 C1; Kc¼ 10:4 W=mK; ð46Þ

and Poisson ratio is chosen to bem= 0.3 In this case, the buckling of perfect plates occurs for m = n = 1, and these values of half waves are also used to trace load–deflection equilibrium paths for both perfect and imperfect plates In figures, W/h denotes the dimension-less maximum deflection and the FGM plate–foundation interaction

is ignored, unless otherwise stated

Fig 2 Comparisons of thermal postbuckling load–deflection curves for isotropic

plates.

Fig 3 Effects of volume fraction index on the postbuckling of FGM plates under

uniaxial compressive load (all movable edges).

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

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

Fig 6 Effects of volume fraction index on the postbuckling of FGM plates under

Fig 3shows decreasing trend of postbuckling curves of the FGM

plates with movable edges under uniaxial compressive load as the

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

Fig 4compares the postbuckling behavior of compressed FGM plates under two types of in-plane boundary restraint The plate

is assumed to be freely movable (FM) on all edges (Case 1) and immovable (IM) on two unloaded edges y = 0, b (Case 3) As can

be seen, in spite of lower critical buckling loads, the postbuckling equilibrium paths for Case 3 become higher than those for Case 1

in deep region of postbuckling behavior

Figs 5 and 6 illustrate the variation of thermal postbuckling load–deflection curves for FGM plates with all immovable edges subjected to uniform temperature rise and through the thickness temperature gradient, respectively, with various values of N As ex-pected, the reduction of volume fraction percentage of ceramic constituent makes the capability of temperature resistance of the plates to be decreased In addition, the variation tendency of tem-perature–deflection curves when N increases from 0 to 5 for two cases of thermal loading is not similar

The effects of the elastic foundations on the postbuckling behavior of the FGM plates under two types of thermal loads are depicted inFigs 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 stiffness

K2of Pasternak model has more pronounced influences in compar-ison with foundation modulus K1of Winkler model

Fig 9 shows the thermomechanical postbuckling behavior of FGM plates exposed to temperature field and subjected to uniaxial compression As can be observed, the capacity of mechanical load bearing of the FGM plates is considerably reduced due to the enhancement of pre-existent thermal load

Finally, interactive effects of elastic foundations and tempera-ture gradient on the postbuckling of the FGM plates subjected to uniaxial compressive loads are considered inFig 10 As can be seen, in spite of the raising of ceramic-rich surface temperature, Pasternak type foundations have very beneficial influences on the improvement of thermomechanical loading capacity of the FGM plates

5 Concluding remarks

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

Fig 7 Effects of the elastic foundations on the postbuckling of FGM plates under

uniform temperature rise (all IM edges).

Fig 8 Effects of the elastic foundations on the postbuckling of FGM plates under

temperature gradient (all IM edges).

Fig 9 Effects of the temperature field on the postbuckling of FGM plates under

uniaxial compression (immovable on y = 0, b).

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

postbuckling behaviors of thick FGM plates resting on Pasternak

type 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 carrying

capacity 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 have

pronounced benefit on the stability of FGM plates Furthermore,

volume fraction index, in-plane boundary restraint, imperfection

and temperature conditions also have considerable effects on the

behavior 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 this

financial support

Appendix A

E1¼ Emh þ Ecmh

N þ 1; E2¼

EcmNh2 2ðN þ 1ÞðN þ 2Þ;

E3¼Emh

3

12 þ Ecmh

4ðN þ 1Þ

1

ðN þ 2ÞðN þ 3Þ

;

E4¼Ecmh

4

N þ 1

1

3 4ðN þ 2Þþ

3

ðN þ 3ÞðN þ 4Þ

;

E5¼Emh

5

Ecmh5

N þ 1

1

16

1 2ðN þ 2Þþ

3

ðN þ 2ÞðN þ 3Þ

ðN þ 2ÞðN þ 4ÞðN þ 5Þ

;

E7¼Emh

7

Ecmh7

N þ 1

1

64

6

30 16ðN þ 2ÞðN þ 3Þ

ðN þ 2ÞðN þ 3ÞðN þ 4Þþ

90

ðN þ 2ÞðN þ 3ÞðN þ 4ÞðN þ 5Þ

ðN þ 2ÞðN þ 3ÞðN þ 4ÞðN þ 6ÞðN þ 7Þ

:

ða11; a22; a12Þ ¼p2

B2h

16

9 D3þ D1

8

3D2

m2B2

a;n2;mmnBa

þð1 mÞp2

2B2h

16

9 D3þ D1

8

3D2

n2;m2B2a;mnBa

þ D6ð1; 1; 0Þ;

ða13; a23Þ ¼4p3D5

3B3h m

3B3

aþ mn2Ba;n3

þ m2nB2 a

pD6

Bh ðmBa;nÞ:

e3

¼16p4ðD2D5 D3D4Þ m2B2

þ n2

 3

þ 3p2B2

D6ð4D2þ 3D4Þ m2B2

þ n2

 2

3B2m2B2þmn2

p2ð3D4 4D5Þ m 2B2þ n2

þ 3B2D6

þK1B

2

þ K2p2 m2B2

þ n2

p2B2m2B2þmn2 B2

D1;

2

mpBhm2B2aþmn2 E2B24E4

np

Bh

;

16B2hm2B2aþmn2

p23D4 4D5

m2B2aþ n2

þ 3B2hD6

 p2 3D4 4D5

m4n2B4

aþ m2n4B2

aþ m6B6

aþ n6

h þ3B2hD6m4B4aþ n4i

þ E1p2n4

8B2hm2B2aþmn2

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