DSpace at VNU: Nonlinear postbuckling of symmetric S-FGM plates resting on elastic foundations using higher order shear deformation plate theory in thermal environments

Nonlinear postbuckling of symmetric S-FGM plates resting onelastic foundations using higher order shear deformation plate theory in thermal environments Vietnam National University, Hano

Nonlinear postbuckling of symmetric S-FGM plates resting on

elastic foundations using higher order shear deformation plate theory

in thermal environments

Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:

Available online 30 January 2013

Keywords:

Functionally graded materials

Nonlinear postbuckling

Third order shear deformation plate theory

Elastic foundation

Imperfection

Thermal environments

a b s t r a c t

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 thermal environments Material properties are graded in the thickness direction according to a Sigmoi power law distribution in terms of the volume fractions of constituents (S-FGM) The formulations are based on third order shear deformation plate theory and stress function taking into account Von Karman nonlinearity, initial geometrical imperfection, temperature 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 The effects of material and geometrical properties, temperature, boundary condi-tions, foundation stiffness and imperfection on the mechanical and thermal buckling and postbuckling loading capacity of the S-FGM plates are analyzed and discussed

Ó 2013 Elsevier Ltd All rights reserved

1 Introduction

Functionally Graded Materials (FGMs) which are

microscopi-cally composites and composed from mixture of metal and ceramic

constituents 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 Wu used the first order

shear deformation theory to obtain closed-form relations of critical

buckling temperatures for simply supported FGM plates[1] Liew

et al.[2,3]used the higher order shear deformation theory in

con-junction with differential quadrature method to investigate the

postbuckling of pure and hybrid FGM plates with and without

imperfection on the point of view that buckling only occurs

for fully clamped FGM plates Based on classical and first order

shear deformation theory, Eslami et al investigated buckling and

post-buckling of FGM plates subjected to mechanical and thermal loads[4–7] The postbuckling behavior of pure and hybrid FGM plates under the combination of various loads were also treated

by Shen using higher order shear deformation theory and two-step perturbation technique taking temperature dependence of mate-rial properties into consideration[8,9] Zhao et al.[10] analyzed the mechanical and thermal buckling of FGM plates using ele-ment-free Ritz method Lee et al.[11]have used element-free Ritz method to analyze the postbuckling 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[12,13] 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 Duc and Tung have studied nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads without elastic foundations with classical[19]and first order shear

0263-8223/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved.

⇑Corresponding author Tel.: +84 4 37547989; fax: +84 4 37547724.

E-mail addresses: ducnd@vnu.edu.vn (N.D Duc), congph_54@vnu.edu.vn (P.H.

Cong).

Contents lists available atSciVerse ScienceDirect

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

plate theory [21] In [20], also Duc and Tung have investigated

mechanical and thermal postbuckling of FGM on elastic foundation

using third order shear deformation plate theory and simple power

law distribution of the volume fraction for metal and ceramic

Comparing to the others, the main difference in the reports by

Duc and Tung[19–21]is the use of the stress function to solve

the 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 supported

by elastic foundations and subjected to in-plane compressive,

ther-mal and thermomechanical loads using Reddy’s third order shear

deformation plate theory, stress function for FGM plate with

Sigmoi power law distribution of the volume of constituents

(S-FGM), taking into account geometrical nonlinearity, initial

geometrical imperfection, temperature and the plate–foundation

interaction is represented by Pasternak model Closed-form

expressions of buckling loads and postbuckling load–deflection

curves for simply supported FGM plates are obtained by Galerkin

method Analysis is carried out to assess the effects of geometrical

and material properties, temperature, boundary conditions,

foun-dation stiffness and imperfection on the buckling and postbuckling

of 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 many

structures 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 and

the core of the structure is made of metal[21] The symmetrical

S-FGM plate considered in this paper is the one example of these

structures

Consider a symmetrical rectangular S-FGM plate that consists of

third layers made of functionally graded ceramic and metal

mate-rials and is midplane-symmetric The outer surface layers of the

plate are ceramic-rich, but the midplane layer is purely metallic

The plate is referred to a Cartesian coordinate system x, y, z, where

xy 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 plate

are a, b and h, respectively (Fig 1)

Unlike[19,20]and other publications, this paper has used the

Sigmoi power-law distribution (S-FGM), the volume fractions of

metal and ceramic, Vmand Vc, are assumed as[21]:

VmðzÞ ¼

2zþh

h

 N

; h=2 6 z 6 0

2zþh

h

 N

; 0 6 z 6 h=2

8

<

: ; VcðzÞ ¼ 1  VmðzÞ ð1Þ

where the volume fraction index N is a nonnegative number that defines the material distribution and can be chosen to optimize the structural response

It is assumed that the effective properties Peffof the functionally graded plate, such as the modulus of elasticity E and the coefficient

of thermal expansiona, vary in the thickness direction z and can be determined by the linear rule of mixture as

Peff ¼ PrmVmðzÞ þ PrcVcðzÞ ð2Þ

where Pr denotes a material property, and the subscripts m and c stand for the metal and ceramic constituents, respectively From Eqs.(1) and (2), the effective properties of the S-FGM plate can be written as follows:

ðE;aÞ ¼ ðEc;acÞ þ ðEmc;amcÞ

2zþh h

 N

; h=2 6 z 6 0

2zþh h

 N

; 0 6 z 6 h=2

8

<

where

Emc¼ Em Ec; amc¼amac ð4Þ

and the Poisson ratiovis assumed constant,v(z) =v 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

2.2 Theoretical formulation The present study uses the Reddy’s third order shear deforma-tion plate theory to establish governing equadeforma-tions and determine the buckling loads and postbuckling paths of the symmetrical S-FGM plates

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

ex

ey

cxy

0 B

1 C

A ¼

e0 x

e0 y

c0 xy

0 B

1 C

A þ z

k1x

k1y

k1xy

0 B

@

1 C

A þ z3

k3x

k3y

k3xy

0 B

@

1 C

cxz

cyz

!

¼ c0 xz

c0 yz

!

þ z2 k2xz

k2yz

!

ð7Þ

where

e0

e0

c0 xy

0 B B

1 C

C¼

u;xþ w2

;x=2

v;yþ w2

;y=2

u;yþv;xþ w;xw;y

0 B

@

1 C A;

k1x

k1y

k1xy

0 B B

1 C

C¼

/x;x /y;y /x;yþ /y;x

0 B

@

1 C A;

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 C

c0 xz

c0 yz

!

¼ /xþ w;x /yþ w;y

!

; k2xz

k2yz

!

¼ 3c1

/xþ w;x /yþ w;y

!

ð8Þ

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

Hooke law for an FGM plate is defined as

ðrx;ry¼ E

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

rxy;rxz;ryz

2ð1 þmÞcxy;cxz;cyz ð9Þ

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

rjð1; 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 ¼ 1

1 m2 ðE1;E2;E4Þ e0

xþme0 y

þ ðE2;E3;E5Þ k 1xþmk1y h

þðE4;E5;E7Þ k 3xþmk3y

 ð1 þmÞð/1;/2;/4Þi

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 þmÞð/1;/2;/4Þi

ðNxy;Mxy;PxyÞ ¼ 1

2ð1 þmÞ ðE1;E2;E4Þc0

xyþ ðE2;E3;E5Þk1xyþ ðE4;E5;E7Þk3xy

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

E1¼ Ech þ Emch

N þ 1;E2¼ 0; E3¼

Ech3

12 þ

Emch3 2ðN þ 1ÞðN þ 2ÞðN þ 3Þ;

E4¼ 0

E5¼Ech

5

80 þ

Emch5

16

1

N þ 1

4

N þ 2þ

6

N þ 3

4

N þ 4þ

1

N þ 5

E7¼Ech

7

448þ

Emch7

64

1

N þ 7

6

N þ 6þ

15

N þ 5

20

N þ 4



þ 15

N þ 3

6

N þ 2þ

1

N þ 1



ð/1;/2;/4Þ ¼

Z h=2

h=2

ð1; z; z3ÞEðzÞaðzÞDTdz ð12Þ

The nonlinear equilibrium equations of a perfect FGM plate resting

on elastic foundations based on the higher order shear deformation

theory are[3,17,18,20,21]:

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

þ 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 Eq.(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), (13d) and (13e) Subsequently, elimination of the variable /x,x+ /y,yfrom two the resulting equations leads to the fol-lowing system of equilibrium equations

Nx;xþ Nxy;y¼ 0

Ny;yþ Nxy;x¼ 0

c2ðD2D5=D4 D3Þr6

w þ ðc1D2=D4þ 1ÞD6r4

w þð1  c1D5=D4Þr2

ðNxw;xxþ 2Nxyw;xyþ Nyw;yy k1w þ k2r2

wÞ

D6=D4ðNxw;xxþ 2Nxyw;xyþ Nyw;yy k1w þ k2r2wÞ ¼ 0 ð14Þ

where

D1¼ E3

1 m2; D2¼ E5

1 m2; D3¼ E7

1 m2

D4¼ D1 c1D2; D5¼ D2 c1D3;

D6¼ 1 2ð1 þmÞ E1 6c1E3þ 9c

2E5

ð15Þ

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

c2ðD2D5=D4 D3Þr6

w þ ðc1D2=D4þ 1ÞD6r4

w þ ð1  c1D5=D4Þr2

 f;yy w;xxþ w

;xx

 2f;xyðw;xyþ w

;xyÞ 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 k2r2

wi

in which w⁄(x, y) is a known function representing initial small imperfection of the plate Note that Eq.(16)gets a complicated form under the third order shear deformation theory which includes the 6th-order partial differential termr6w

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

Nx¼@

2f

@y2; Ny¼@

2f

@x2; Nxy¼  @

2f

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

x¼ 1

E1

ðf;yymf;xxþ /1Þ; e0

y¼1

E1

ðf;xxmf;yyþ /1Þ; c0

xy

¼ 1

Introduction of Eq.(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 symmetric

S-FGM plates on elastic foundations subjected to mechanical,

thermal and thermomechanical loads using the third order shear

deformation plate theory Until now, there is no analytical studies

have been reported in the literature on the postbuckling of thick

S-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 of

boundary 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 freely

movable (FM) The associated boundary conditions are

w ¼ Nxy¼ /y¼ Mx¼ Px¼ 0; Nx¼ Nx0 at x ¼ 0; a

w ¼ Nxy¼ /x¼ My¼ Py¼ 0; Ny¼ Ny0 at 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 Nx0 at x ¼ 0; a

w ¼v¼ /x¼ My¼ Py¼ 0; Ny¼ Ny0 at 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 Nx0 at x ¼ 0; a

w ¼v¼ /x¼ My¼ Py¼ 0; Ny¼ Ny0 at 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, w⁄[7,15]and f[19–21]

satisfy-ing boundary conditions(21)–(23)are assumed to be

ðw; wÞ ¼ ðW;lhÞ sin kmx sin dny ð24aÞ

f ¼ A1cos 2kmx þ A2cos 2dny þ A3sin kmx sin dny þ A4

 cos kmx cos dny þ1

2Nx0y 2

þ1

2Ny0x 2

ð24bÞ

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

is imperfection parameter The coefficients Ai(i = 1  4) are

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

A1¼ E1d

2

n

32k2mWðW þ 2lhÞ; A2¼E1k

2 m 32d2nWðW þ 2lhÞ; A3¼ A4

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

apply-ing the Galerkin procedure for the resultapply-ing equation yield

c2 D2D5

D4

 D3

k2mþ d2

 3

þ D6

c1D2

D4

þ 1

k2mþ d2

 2

þ k 1þ k2k2mþ d2n D6

D4

þ 1 c1D5

D4

k2mþ d2n



W

þE1 16

D6

D4

k4mþ d4

þ 1 c1D5

D4

k6mþ d6þ k2md4þ k4md2

W

 ðW þlhÞðW þ 2lhÞ þ D6

D4

þ 1 c1D5

D4

k2mþ d2

 ½Nx0k2mþ Ny0d2ðW þlhÞ ¼ 0 ð26Þ

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

2.2.1 Mechanical postbuckling analysis Consider a simply supported symmetrical S-FGM plate with all movable edges (all FM) which is rested on elastic foundations and subjected to in-plane edge compressive loads Fx, Fyuniformly dis-tributed on edges x = 0, a and y = 0, b, respectively In this case, pre-bucking force resultants are[6]

Nx0¼ Fxh; Ny0¼ Fyh ð27Þ

and Eq.(26)leads to

Fx¼ e1 W

W þlþ e

where

e1¼16p4ðD2D5 D3D4Þ m2B2

þ n2

þ 3D6B2p2ð4D2þ 3D4Þ m2B2

þ n2

3B2ðm2B2þ bn2Þ½p2ð3D4 4D5Þ m 2B2þ n2

þ 3B2D6 þ

K1B2þ K2p2ðm2B2þ n2Þ

B2D1

B2p2m2B2þ bn2

e1¼

E1m4B4þ n4

p2 16ðm2B2

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

D1¼ E3

1 v2; D2¼ E5

1 v2; D3¼ E7

1 v2

D4¼ D14

3D2; D5¼ D2

4

3D3; D6¼

1 2ð1 þvÞðE1 8E3þ 16E5Þ

ð30Þ

For a perfect FGM plate, Eq.(28)reduces to an equation from which buckling compressive load may be obtained as Fxb¼ e1

2.2.2 Thermal postbuckling analysis

A simply supported FGM plate with all immovable edges (IM) is considered 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[5,8,20,21]

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  w2

;x=2  w;xw

;xþ/1

E1

@v

@y¼

1

E1ðf;xxmf;yyÞ  w2;y=2  w;yw

;yþ/1

E1

ð32Þ

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

(31)give

Nx0¼ 1

8ð1 m2ÞE1 k

2

mþmd2

WðW þ 2lhÞ  /1

1 m

Ny0¼ 1

8ð1 m2ÞE1mk2mþ d2n

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

Nx0¼ Ny0¼  /1

which was derived by Shariat and Eslami[6]by solving the

mem-brane form of equilibrium equations and employing the method

suggested by Meyers and Hyer[23]

Substituting Eq.(33)into Eq.(26)yields the expression of

ther-mal parameter as

/1

1 v¼

c2ðD2D5 D3D4Þ k 2mþ d22

þ D6ðc1D2þ D4Þ k 2mþ d2

D6þ ðD4 c1D5Þ k2

mþ d2

"

þk1þ k2 k

2

mþ d2

k2mþ d2n

# W

W þlhþ

E1k4mþ d4

16 k 2mþ d2n

"

þ E1

8ð1 v2Þ

k4mþ d4þ 2vk2md2

k2mþ d2

# WðW þ 2lhÞ ð35Þ

The S-FGM plate is exposed to temperature environments

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

temperature change DT = Tf Tiis considered to be independent

from thickness variable The thermal parameter /1is obtained from

Eq.(12), and substitution of the result into Eq.(35)yields

DT ¼ e2 W

W þlþ e

where

e2

P 3B2

D6þ ð3D4 4D5Þ m2B2

þ n2

p2

3ðD2D5 D3D4Þ m

2

B2þ n2

B2þ D6ð4D2þ 3D4Þ m2

B2þ n2

þ K1B

2

þ K2p2m2B2þ n2

B2D1ð1 vÞ

PB2p2ðm2B2

þ n2Þ

e2

¼

E1p2

ð1 vÞ m4B4

þ n4

16PB2

m2B2

þ n2

8Pð1 þvÞ

m4B4

þ n4

þ 2vm2n2B2

B2

m2B2

þ n2

ð37Þ

in which

P ¼ EcacþEcamcþ Emcac

N þ 1 þ

Emcamc

By Settingl= 0 Eq.(36)leads to an equation from which

buck-ling temperature change of the perfect FGM plates may be

deter-mined asDTb¼ e2

2.2.3 Thermomechanical postbuckling analysis

The 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 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 Eq.(32)in the second of Eq.(31)as

Ny0¼vNx0 /1þE1

8d

2

Subsequently, Nx0and Ny0are placed in Eq.(26)to give

Fx¼ e3 W

W þlþ e

3WðW þ 2lÞ  Pn

2

DT

m2B2aþvn2 ð40Þ

where

e3¼

16p4ðD2D5 D3D4Þ m 2B2þ n23

þ 3ð4D2þ 3D4ÞD6m2B2þ n22

p2B2 3B2 3D6B2

þ ð3D4 4D5Þ m2B2

þ n2

p2

m2B2

þvn2

þ

K1B2

þ K2p2 m2B2

þ n2

B2D1

B2p2m2B2þvn2

e3¼E1 16

m4B4

þ n4

p2

m2B2þvn2

B2þE1 8

n4p2

B2ðm2B2þvn2Þ

ð41Þ

Eqs.(28), (36) and (40)are explicit expressions of load–deflection curves 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 equations for thick S-FGM plates, i.e., ignoring the third order shear deforma-tions and elastic foundadeforma-tions, gives the corresponding results de-rived by using the first order shear deformation plate theory for S-FGM plates[21]

3 Numerical results and discussion

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[5,8,20,21]:

Em¼ 70 GPa; am¼ 23  106 C1

Ec¼ 380 GPa; am¼ 7:4  106 C1 ð42Þ

and Poisson ratio is chosen to bev= 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

Effects of volume fraction index N on the postbuckling of S-FGM plates under uniaxial compressive load and uniform temperature rise are shown inFigs 2 and 3 In all below figures, it is assumed

Fig 2 Effects of volume fraction index N on the postbuckling of symmetrical

S-that ~ePx¼ Fx Obviously, the mechanical load and the thermal

resistance get better if the volume N increases or the percentage

of ceramic increases It is opposite of the FGM applied simply

power law distribution in [19,20]: Both critical buckling loads

and postbuckling carrying capacity are strongly dropped when N

is increased

Figs 4 and 5show 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-S-FGM plate

Obviously, with the same volume fractions of ceramic–metal, the

critical loads of postbuckling of the S-FGM are different for the first

and third orders Indeed, the critical loads for the third order shear

deformation is smaller than those for the first order shear

deforma-tion For postbuckling of the S-FGM plate,Figs 4 and 5also show

us that the imperfect plate has a better mechanical and thermal

loading capacity than those of the perfect plate

Figs 6 and 7present effects of first and third order shear

defor-mations on buckling and postbuckling of S-FGM plate with various

of thermal and mechanical loads Obviously, with the same volume

fraction of ceramic–metal, the critical loadings of postbuckling of

the S-FGM are different Also, similar to above two figures, the

crit-ical mechancrit-ical and thermal loadings for the third order shear

deformation are smaller than those of the first order shear

deformation

There have been only a few of reports on the buckling and

post-buckling for symmetric S-FGM plate yet We therefore are limited

to compare with the others However, comparing our findings in

Figs 4–7with our previous results[21], it is inferred that there

is a difference between the first and the third of higher order shear

deformation 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 9show the influence of initial imperfections on

post-buckling of S-FGM plate under uniaxial compressive load (all FM

edges) and under uniform temperature (all IM edges).Fig 8shows

us that the critical compressive loads decreases withlin the limit

of the small bending However, it increases withlin the other

lim-it of the large bending, meaning the higher bending-load curve (i.e.,

the better loading ability).Figs 4–9show us that an imperfect FGM

plate has a better mechanical and thermal loading capacity than

the perfect one in postbuckling process This has been shown in

[3,7,8,15,19–21] In particular,Fig 9clearly shows us that an initial

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

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

Fig 6 Effect of first and third order shear deformation on mechanical buckling and

Fig 7 Effect of first and third order shear deformation on critical thermal loads of

buckling and postbuckling of S-FGM plate with various of mechanical loads P x

Fig 9 The influence of imperfections on the stability of symmetrical S-FGM plates

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 of symmetric S-FGM plate under uniform temperature rise (FM on y = 0, b; IM on x = 0, Fig 8 The influence of imperfections on the stability of symmetrical S-FGM plates

under uniaxial compressive load (all FM edges).

Figs 10 and 11present the positive influence of elastic

founda-tions on imperfecfounda-tions on the stability of S-FGM plate under

uniax-ial compressive load (all FM edges) and uniform temperature (all

IM edges) The effect of Pasternak foundation K2on the critical

compressive loads and the thermal resistance of S-FGM is larger

than the Winkler foundation K1 This conclusion has been also

re-ported in[16–18,20]

An investigation of the mechanical–thermal stability has been

determined by(40).Figs 12 and 13have been calculated under

the assumption of the third boundary conditions (Case 3) for the

FM edges x = 0, a and IM edges y = 0, b which are simultaneously

under the compressive uniform loading on the edge x = 0, a

Fig 12shows the effect of the temperature gradient of the

sur-rounding environment on the behavior of an uniaxial compressive

load x The presence of temperature reduces the loading ability (for

both perfect and imperfect plates) Under the non-zero

tempera-ture gradient conditionDT – 0, in the presence of temperature,

the imperfect plate still gets bend immediately even if there is

no mechanical compressive force It is represented by a crossing

point of the dash lines with the axis W/h

Buckling and postbuckling behavior of the S-FGM plate under the

increased uniform temperature gradient fieldDT and the different

values of the uniaxial compressive load Px have been shown in

Fig 13 The presence of the mechanical loading reduces the thermal

loading 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 inFig 14 There are two types of condition for the two edges y = 0, b which are the free motion (FM) and not in motion (IM) conditions The curve for FM edges drawn from(28)with the loading ratio b = 0 (in(30)),

Fig 14shows us that the perfect FGM plate is bended earlier than the imperfect one; however loading capacity of the imperfect plate

is better than perfect one when the bending is large enough in postbuckling process

4 Conclusions This paper presents an analytical investigation on the postbuck-ling behaviors of thick symmetric functionally graded plates

subjected to in-plane compressive, thermal and thermomechanical loads Material properties are graded in the thickness direction according to a Sigmoi power law distribution in terms of the vol-ume fractions of constituents (S-FGM) The formulations are based

on third order shear deformation plate theory and stress function taking into account Von Karman nonlinearity, initial geometrical imperfection, temperature 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 The effects of material and geometrical properties, temperature, boundary conditions, foundation stiffness and imperfection on the postbuckling loading capacity of the S-FGM plates are analyzed and discussed It is easy to realize that the critical mechanical and thermal loadings for third order shear deformation 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 better mechanical and thermal loading capacity

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

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deformation plate theory on buckling and postbuckling of thick

S-FGM plates However, this difference is not much despite of

com-plicated third order. .. compression on the postbuckling of symmetric S-FGM plate under uniform temperature rise (FM on y = 0, b; IM on x = 0, Fig The in? ??uence of imperfections on the stability of symmetrical S-FGM plates< /small>... order shear deformation on thermal buckling and postbuckling of S-FGM plate with various of volume fractions N.

Fig Effect of first and third order shear deformation on mechanical