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

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

Composite Structures 93 (2011) 2874–2881 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/co...

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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 deformable functionally graded plates on elastic foundations Nguyen Dinh Duc a, Hoang Van Tung b,⇑ a b

University of Engineering and Technology, Vietnam National University, Ha Noi, Viet Nam 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 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 functionally 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 buckling and postbuckling 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 constituents have attracted considerable attention recent years. By continuously and gradually varying the volume fraction of constituent materials through a specific direction, FGMs are capable of withstanding ultrahigh temperature environments and extremely large thermal gradients. Therefore, these novel materials are chosen 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 equilibrium 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 ⇑ Corresponding author. E-mail address: [email protected] (H.V. Tung). 0263-8223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2011.05.017

order shear deformation theory in conjunction 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. 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 dependence of material properties into consideration. Recently, Lee et al. [11] made of use 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 supported by an elastic foundation. Therefore, it is necessary to account 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 postbuckling 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.

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This paper extends previous work [19] to investigate the buckling and postbuckling behaviors of thick functionally graded plates supported by elastic foundations and subjected to in-plane compressive, thermal and thermomechanical loads. Reddy’s higher order 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 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, in-plane restraint, foundation stiffness and imperfection on the behavior of the FGM plates.

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 in Fig. 1. The volume fractions of constituents are assumed to vary through the thickness according to the following power law distribution

V m ðzÞ ¼ 1  V c ðzÞ

ð1Þ

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

Preff ðzÞ ¼ Prm ðzÞV m ðzÞ þ Prc ðzÞV c ðzÞ

ð2Þ

where Pr denotes a temperature independent material property, and subscripts m and c stand for the metal and ceramic constituents, respectively. Specialization of Eqs. (1) and (2) for the modulus of elasticity E, the coefficient of thermal expansion a and the coefficient of thermal conduction K gives

 N 2z þ h ½EðzÞ; aðzÞ; KðzÞ ¼ ½Em ; am ; K m  þ ½Ecm ; acm ; K cm  2h

ð3Þ

where

Ecm ¼ Ec  Em ; acm ¼ ac  am ; K cm ¼ K c  K m

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 given by

qe ¼ k1 w  k2 r2 w

ð5Þ

z

shear layer

b a

y

3. Theoretical formulation The present study uses the Reddy’s higher order shear deformation plate theory to establish governing equations and determine 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]

0

2. Functionally graded plates on elastic foundations

 N 2z þ h V c ðzÞ ¼ ; 2h

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

B B B @

ex

1

0

0

1

e0x

1

kx

1

0

3

kx

1

C C B B C B C C B 1C B B 0C 3B 3 C C ey C C ¼ B ey C þ zB B ky C þ z B ky C

cxy cxz cyz

A

@

!

0 xy

c

@

!

c0xz c0yz

¼

A

1 kxy 2

þz

kxz

2

A

@

3 kxy

ð6Þ

A

! ð7Þ

2

kyz

where

0 B B B B @ 0

e0x e0y c0xy

1

0

C B C B C¼B C @ A

1

u;x þ w2;x =2

C C C; A

v ;y þ w2;y =2 u;y þ v ;x þ w;x w;y

0

1

1 0 /x;x C B C B C B C B 1C B /y;y C; B ky C ¼ B C @ A B A @ 1 / þ / x;y y;x kxy 1

kx

1

0 1 /x;x þ w;xx B 3C C B ky C ¼ c1 B /y;y þ w;yy @ A A @ 3 / þ / þ 2w ;xy x;y y;x kxy 3

kx

c0xz c0yz

!

/x þ w;x

¼

/y þ w;y

!

2

;

kxz 2

kyz

! ¼ 3c1

ð8Þ

/x þ w;x

!

/y þ w;y

in which c1 = 4/3h2, ex,ey are normal strains, cxy is the in-plane shear strain, and cxz, cyz are the transverse shear deformations. Also, u, v are the displacement components along the x, y directions, respectively, 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 ; ry Þ ¼

E ½ðex ; ey Þ þ mðey ; ex Þ  ð1 þ mÞaDTð1; 1Þ 1  m2

ðrxy ; rxz ; ryz Þ ¼

ð9Þ

E ðc ; c ; c Þ; 2ð1 þ mÞ xy xz yz

where DT is temperature rise from stress free initial state or temperature difference between two surfaces of the FGM plate. The force and moment resultants of the FGM plate are determined by

h ðNi ; M i ; Pi Þ ¼

Z

h=2

ri ð1; z; z3 Þdz i ¼ x; y; xy

h=2

x

ðQ i ; Ri Þ ¼

Z

h=2

rj ð1; z2 Þdz i ¼ x; y; j ¼ xz; yz:

ð10Þ

h=2

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

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

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    1 h 1 1 ðE1 ; E2 ; E4 Þ e0x þ me0y þ ðE2 ; E3 ; E5 Þ kx þ mky 2 1m i   3 3 þðE4 ; E5 ; E7 Þ kx þ mky  ð1 þ mÞðU1 ; U2 ; U4 Þ

ðNx ; M x ; Px Þ ¼

    1 h 1 1 ðE1 ; E2 ; E4 Þ e0y þ me0x þ ðE2 ; E3 ; E5 Þ ky þ mkx 2 1m i   3 3 þðE4 ; E5 ; E7 Þ ky þ mkx  ð1 þ mÞðU1 ; U2 ; U4 Þ

ðNy ; M y ; Py Þ ¼

ðNxy ; Mxy ; Pxy Þ ¼ 3

h 1 1 ðE1 ; E2 ; E4 Þc0xy þ ðE2 ; E3 ; E5 Þkxy 2ð1 þ mÞ i

þðE4 ; E5 ; E7 Þkxy

c21 ðD2 D5 =D4  D3 Þr6 w þ ðc1 D2 =D4 þ 1ÞD6 r4 w þ ð1  c1 D5 =D4 Þr2 h        f;yy w;xx þ w;xx  2f ;xy w;xy þ w;xy þ f;xx w;yy þ w;yy i h     k1 w þ k2 r2 w  D6 =D4 f;yy w;xx þ w;xx  2f ;xy w;xy þ w;xy i   þf;xx w;yy þ w;yy  k1 w þ k2 r2 w ¼ 0 ð16Þ in which w⁄(x, y) is a known function representing initial small imperfection of the plate. Note that the terms r6w and r4w are unchanged because these terms are obtained from the expressions for bending moments Mij and higher order moments Pij and these moments 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

Nx ¼ f;yy ; Ny ¼ f;xx ; Nxy ¼ f;xy :

h i 1 2 ðE1 ; E3 Þc0xz þ ðE3 ; E5 Þkxz 2ð1 þ mÞ h i 1 2 ðQ y ; Ry Þ ¼ ðE1 ; E3 Þc0yz þ ðE3 ; E5 Þkyz 2ð1 þ mÞ ðQ x ; Rx Þ ¼

The geometrical compatibility equation for an imperfect plate is written as

e0x;yy þ e0y;xx  c0xy;xy ¼ w2;xy  w;xx w;yy þ 2w;xy w;xy  w;xx w;yy ð11Þ

where

ðE1 ; E2 ; E3 ; E4 ; E5 ; E7 Þ ¼

Z

Z

 w;yy w;xx :

ð1; z; z2 ; z3 ; z4 ; z6 ÞEðzÞdz





e0x ; e0y ¼

h=2

ð1; z; z3 ÞEðzÞaðzÞDTðzÞdz

ð18Þ

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

h=2

h=2

ðU1 ; U2 ; U4 Þ ¼

ð17Þ

ð12Þ

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]

Nx;x þ Nxy;y ¼ 0

ð13aÞ

Nxy;x þ Ny;y ¼ 0

ð13bÞ

c0xy ¼ 

i     1h 1 1 3 3 ðf;yy ;f;xx Þ  mðf;xx ;f;yy Þ  E2 kx ;ky  E4 kx ;ky þ U1 ð1;1Þ E1 ð19Þ

i 1h 1 3 2ð1 þ mÞf;xy þ E2 kxy þ E4 kxy : E1

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





r4 f  E1 w2;xy  w;xx w;yy þ 2w;xy w;xy  w;xx w;yy  w;yy w;xx ¼ 0 ð20Þ

Q x;x þ Q y;y  3c1 ðRx;x þ Ry;y Þ þ c1 ðPx;xx þ 2Pxy;xy þ Py;yy Þ þ Nx w;xx þ 2Nxy w;xy þ Ny w;yy  k1 w þ k2 r2 w ¼ 0

ð13cÞ

M x;x þ M xy;y  Q x þ 3c1 Rx  c1 ðPx;x þ Pxy;y Þ ¼ 0

ð13dÞ

M xy;x þ M y;y  Q y þ 3c1 Ry  c1 ðPxy;x þ Py;y Þ 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,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 following system of equilibrium equations

Nx;x þ Nxy;y ¼ 0 Nxy;x þ Ny;y ¼ 0 c21 ðD2 D5 =D4  D3 Þr6 w þ ðc1 D2 =D4 þ 1ÞD6 r4 w

ð14Þ

þ ð1  c1 D5 =D4 Þr2 ðNx w;xx þ 2N xy w;xy þ Ny w;yy  k1 w þ k2 r2 wÞ  D6 =D4 ðNx w;xx þ 2Nxy w;xy þ N y w;yy  k1 w þ k2 r2 wÞ ¼ 0 where

D1 ¼

E1 E3  E22 ; E1 ð1  m2 Þ

D2 ¼

E1 E5  E2 E4 ; E1 ð1  m2 Þ

D4 ¼ D1  c1 D2 ; D5 ¼ D2  c1 D3 ;   1 E1  6c1 E3 þ 9c21 E5 : D6 ¼ 2ð1 þ mÞ

D3 ¼

E1 E7  E24 ; E1 ð1  m2 Þ

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 considered [12–15]. Case 1. Four edges of the plate are simply supported and freely movable (FM). The associated boundary conditions are

w ¼ Nxy ¼ /y ¼ M x ¼ Px ¼ 0;

Nx ¼ Nx0 at x ¼ 0; a

w ¼ Nxy ¼ /x ¼ M y ¼ Py ¼ 0;

Ny ¼ Ny0 at y ¼ 0; b:

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

w ¼ u ¼ /y ¼ Mx ¼ Px ¼ 0;

N x ¼ Nx0 at x ¼ 0; a

w ¼ v ¼ /x ¼ M y ¼ P y ¼ 0;

Ny ¼ Ny0 at y ¼ 0; b:

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

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 boundary conditions are defined as

w ¼ Nxy ¼ /y ¼ M x ¼ Px ¼ 0; ð15Þ

ð21Þ

w ¼ v ¼ /x ¼ M y ¼ P y ¼ 0;

Nx ¼ Nx0 at x ¼ 0; a Ny ¼ Ny0 at y ¼ 0; b

ð23Þ

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 edge loads at immovable edges (i.e. Case 2 and the second of Case 3).

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

ðw; w Þ ¼ ðW; lhÞ sin km x sin dn y

e11

ð24aÞ

1 1 f ¼ A1 cos 2km x þ A2 cos 2dn y þ A3 sin km x sin dn y þ N x0 y2 þ Ny0 x2 2 2 ð24bÞ /x ¼ B1 cos km x sin dn y;

/y ¼ B2 sin km x cos dn y

ð24cÞ

ð31Þ

p2 E1   h   i  2 m2 Ba þ bn2 p2 3D4  4D5 m2 B2a þ n2 þ 3B2h D6 h     p2 3D4  4D5 m4 n2 B4a þ m2 n4 B2a þ m6 B6a þ n6  i þ3B2h D6 m4 B4a þ n4 ;

e12 ¼

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

A1 ¼

E1 d2n 32k2m

WðW þ 2lhÞ;

A2 ¼

E1 k2m 32d2n

WðW þ 2lhÞ;

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

a12 a23  a22 a13 W; a212  a11 a22

B2 ¼

a12 a13  a11 a23 W a212  a11 a22

ð26Þ

where

ða11 ; a22 ; a12 Þ ¼ þ



c21 D3

þ D1  2c1 D2



k2m ; d2n ;

mkm dn

ð27Þ

  ða13 ; a23 Þ ¼ c1 D5 k3m þ km d2n ; d3n þ dn k2m  D6 ðkm ; dn Þ: Subsequently, setting Eqs. (24a,b) into Eq. (16) and applying the Galerkin procedure for the resulting equation yield

      3  2 D2 D5 c 1 D2 c21  D3 k2m þ d2n þ D6 þ 1 k2m þ d2n þ ½k1 D4 D4

   2    D6 c D 1 5 k2m þ d2n þ W 1 þk2 km þ d2n D4 D4 

  D6  4  E1 c1 D5  4 2 þ km dn þ k2m d4n þ k6m þ d6n þ km þ d4n 1 16 D4 D4

    D6 c 1 D5 k2m þ d2n þ WðW þ lhÞðW þ 2lhÞ þ 1  D4 D4   2 2  Nx0 km þ Ny0 dn ðW þ lhÞ ¼ 0 ð28Þ 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.

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]

Ny0 ¼ F y h

ð29Þ

and Eq. (28) leads to

F x ¼ e11 where

W þ e12 WðW þ 2lÞ W þl

k1 a4 k2 a2 i ; K2 ¼ ; Ei ¼ Ei =h ði ¼ 1—7Þ; D1 D1 E1 E3  E22 E1 E5  E2 E4 E1 E7  E24 ; D2 ¼ ; D3 ¼ ; ð32Þ D1 ¼ 2 2 E1 ð1  m Þ E1 ð1  m Þ E1 ð1  m2 Þ   4 4 1 D4 ¼ D1  D2 ; D5 ¼ D2  D3 ; D6 ¼ E1  8E3 þ 16E5 : 3 3 2ð1 þ mÞ K1 ¼

3.2. Thermal postbuckling analysis A simply supported FGM plate with all immovable edges 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 and v = 0 at y = 0, b, is fulfilled in an average sense as [10,12–15,19]

Z 0

b

Z

a

0

@u dxdy ¼ 0; @x

Z 0

a

Z 0

b

@v dydx ¼ 0: @y

ð33Þ

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

@u 1 E2 c 1 E4 ¼ ðf;yy  mf;xx Þ  /x;x þ ð/x;x þ w;xx Þ @x E1 E1 E1 1 U1  w2;x  w;x w;x þ 2 E1 @v 1 E2 c1 E4 ð/y;y þ w;yy Þ ¼ ðf;xx  mf;yy Þ  /y;y þ @y E1 E1 E1 1 U1 :  w2;y  w;y w;y þ 2 E1

ð34Þ

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

3.1. Mechanical postbuckling analysis

Nx0 ¼ F x h;

Bh ¼ b=h; Ba ¼ b=a; W ¼ W=h; b ¼ F y =F x ;

For a perfect FGM plate, Eq. (30) reduces to an equation from which buckling compressive load may be obtained as F xb ¼ e11 .



  1  m 2 c1 D3 þ D1  2c1 D2 d2n ; k2m ; km dn þ D6 ð1; 1; 0Þ; 2

16B2h

in which

A3 ¼ 0: ð25Þ

B1 ¼

 3  2 16p4 ðD2 D5  D3 D4 Þ m2 B2a þ n2 þ 3p2 B2h D6 ð4D2 þ 3D4 Þ m2 B2a þ n2  h   i ¼ 3B2h m2 B2a þ bn2 p2 ð3D4  4D5 Þ m2 B2a þ n2 þ 3B2h D6 h  i K 1 B2a þ K 2 p2 m2 B2a þ n2 D1 B2a   þ ; p2 B2h m2 B2a þ bn2

ð30Þ

U1 4  ½ðE2  c1 E4 Þðkm B1 þ mdn B2 Þ 1  m mnp2 ð1  m2 Þ    2  E1  c1 E4 k2m þ md2n þ k þ md2n WðW þ 2lhÞ; 8ð1  m2 Þ m

Nx0 ¼ 

ð35Þ

U1 4  ½ðE2  c1 E4 Þðmkm B1 þ dn B2 Þ 1  m mnp2 ð1  m2 Þ    2  E1  c1 E4 mk2m þ d2n W þ mkm þ d2n WðW þ 2lhÞ: 8ð1  m2 Þ

Ny0 ¼ 

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

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Nx0 ¼ Ny0 ¼ 

U1

ð36Þ

1m

which was derived by Shariat and Eslami [3] by solving the membrane form of equilibrium equations and employing the method suggested by Meyers and Hyer [20]. Substituting Eqs. (35) into Eq. (28) yields the expression of thermal parameter as

"  2   c21 ðD2 D5  D3 D4 Þ k2m þ d2n þ D6 ðc1 D2 þ D4 Þ k2m þ d2n U1 ¼  2  1m ðD4  c1 D5 Þ km þ d2n þ D6  2 # k1 þ k2 km þ d2n W 4 þ    W þ lh mnp2 ð1  m2 Þ k2m þ d2n k2m þ d2n

   ðE2  c1 E4 tÞ k3m B1 þ mk2m dn B2 þ mkm d2n B1 þ d3n B2   c1 E4 k4m þ 2mk2m d2n þ d4n W "  4 2    E1 ðD4  c1 D5 Þ km dn þ k2m d4n þ k6m þ d6n þ D6 k4m þ d4n þ

    16 ðD4  c1 D5 Þ k2m þ d2n þ D6 k2m þ d2n #  4  E1 km þ 2mk2m d2n þ d4n ð37Þ þ   WðW þ 2lhÞ: 8ð1  m2 Þ k2m þ d2n

3.2.1. Uniform temperature rise The FGM plate is exposed to temperature environments uniformly raised from stress free initial state Ti to final value Tf, and temperature change DT = Tf  Ti is 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

DT ¼ e21

W þ e22 W þ e23 WðW þ 2lÞ W þl

ð38Þ

11 ; a 22 ; a 12 ; a 13 ; a 23 can be found in Also, specific expressions of a Appendix A. By Setting l = 0 Eq. (38) leads to an equation from which buckling temperature change of the perfect FGM plates may be determined as DT b ¼ e21 . 3.2.2. Through the thickness temperature gradient The metal-rich surface temperature Tm is maintained at reference value while ceramic-rich surface temperature Tc is enhanced and steadily conducted through the thickness direction according to one-dimensional Fourier equation

d dT KðzÞ ¼ 0; dz dz

Tðz ¼ h=2Þ ¼ T m ;

j P NK cm =K m Þ r 5j¼0 ðr jNþ1 TðzÞ ¼ T m þ DT P 5 ðK cm =K m Þj

j¼0

¼

ð42Þ

jNþ1

where r = (2z + h)/2h and, in this case of thermal loading, DT = Tc  Tm is defined as the temperature difference between two surfaces 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–deflection curves which is similar to Eq. (38), providing L is replaced by H defined as

P5

j¼0



ðK cm =K m Þj jNþ1

h

Em am jNþ2

P5

j¼0

acm þEcm am Ecm acm þ Emðjþ1ÞNþ2 þ ðjþ2ÞNþ2

i

ðK cm =K m Þj jNþ1

ð43Þ

:

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 simultaneously exposed to elevated temperature environments or subjected 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

h  i K 1 B2a þ K 2 p2 m2 B2a þ n2 ð1  mÞB2a D1   ; þ p2 LB2h m2 B2a þ n2 " 4   Bh ð3E2  4E4 Þ: e22 ¼  3mnpLð1 þ mÞB2h m2 B2a þ n2   m3 B3a B1 þ mm2 nB2a B2 þ mmn2 Ba B1 þ n3 B2  i 4pE4 m4 B4a þ 2mm2 n2 B2a þ n4 ; e23

ð41Þ

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]

where

ð1  mÞp2   i e21 ¼ h 2 L p ð3D4  4D5 Þ m2 B2a þ n2 þ 3B2h D6 "  2 16p2 ðD2 D5  D3 D4 Þ m2 B2a þ n2  2 3Bh #   2 2 2 þD6 ð4D2 þ 3D4 Þ m Ba þ n

Tðz ¼ h=2Þ ¼ T c :

Ny0 ¼ mN x0  U1  þ

4dn ½E2 B2  c1 E4 ðdn þ B2 ÞW mnp2

E1 d2n WðW þ 2lhÞ: 8

ð44Þ

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

F x ¼ e31

E1 p2 ð1  mÞ

 h   i 16LB2h m2 B2a þ n2 p2 ð3D4  4D5 Þ m2 B2a þ n2 þ 3B2h D6 h    p2 ð3D4  4D5 Þ m4 n2 B4a þ m2 n4 B2a þ m6 B6a þ n6  i þ3B2h D6 m4 B4a þ n4   E1 p2 m4 B4a þ 2m m2 n2 B2a þ n4   þ 8Lð1 þ mÞB2h m2 B2a þ n2 ð39Þ

W W þl

þ e32 W þ e33 WðW þ 2lÞ 

Ln2 DT m2 B2a

þ mn2

;

ð45Þ

where the coefficients e31 ; e32 ; e33 are described in detail in Appendix A and L is replaced by H in the case of the FGM plates subjected to combined action of uniaxial compressive load and temperature gradient. Eqs. (30), (38) and (45) are explicit expressions of load–deflection curves for thick FGM plates resting on Pasternak elastic foundations 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 corresponding results derived by using the classical plate theory [19].

in which

Em acm þ Ecm am Ecm acm þ ; Nþ1 2N þ 1 12 a 23  a 13 12 a 13  a 23 22 a 11 a a a B1 ¼ 2 ; B2 ¼ 2 : 12  a 22 12  a 22 11 a 11 a a a L ¼ Em a m þ

4. Results and discussion

ð40Þ

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

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

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

2879

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

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. 2 gives thermal postbuckling 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, consider a square ceramic–metal plate consisting of aluminum and alumina with the following properties [2–5]

Em ¼ 70 GPa; am ¼ 23  106  C1 ; K m ¼ 204 W=mK Ec ¼ 380 GPa; ac ¼ 7:4  106  C 1 ; K c ¼ 10:4 W=mK;

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 compression.

ð46Þ

and Poisson ratio is chosen to be m = 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 dimensionless maximum deflection and the FGM plate–foundation interaction is ignored, unless otherwise stated.

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

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

volume fraction index N increases. Both critical buckling loads and postbuckling carrying capacity are strongly dropped when N is increased from 0 to 1, and a slower variation is observed when N is greater than 1. Fig. 4 compares 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 expected, 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 temperature–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 in Figs. 7 and 8. Obviously, both buckling loads and postbuckling loading bearing capability are enhanced due to the presence of elastic foundations. Furthermore, the shear layer stiffness K2 of Pasternak model has more pronounced influences in comparison with foundation modulus K1 of 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 temperature gradient on the postbuckling of the FGM plates subjected to uniaxial compressive loads are considered in Fig. 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.

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

5. Concluding remarks

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

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

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

postbuckling behaviors of thick FGM plates resting on Pasternak type elastic foundations. The formulations are based on the Reddy’s higher order shear deformation theory to obtain accurate predictions for buckling loads and postbuckling loading carrying capacity of thick plates. In addition, obtained closed-form expressions of load–deflection curves have practical significance in analysis 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.

"

 # 4n2 4E4 np 3   E2 B 2  ; e2 ¼  B2 þ 3 Bh mpBh m2 B2a þ mn2

p2 E 1   h   i  2 2 2 2 m Ba þ mn p 3D4  4D5 m2 B2a þ n2 þ 3B2h D6 h     p2 3D4  4D5 m4 n2 B4a þ m2 n4 B2a þ m6 B6a þ n6

e33 ¼

16B2h

 i þ3B2h D6 m4 B4a þ n4 þ

Acknowledgements This paper was supported by the National Foundation for Science and Technology Development of Vietnam - NAFOSTED, project code 107.02-2010.08. The authors are grateful for this financial support. Appendix A 2

Ecm h Ecm Nh ; E2 ¼ ; Nþ1 2ðN þ 1ÞðN þ 2Þ

3 Em h 1 1 3 ;  þ Ecm h E3 ¼ 4ðN þ 1Þ ðN þ 2ÞðN þ 3Þ 12

4 Ecm h 1 3 3 ; E4 ¼  þ N þ 1 8 4ðN þ 2Þ ðN þ 3ÞðN þ 4Þ

E1 ¼ Em h þ

5 5

Em h Ecm h 1 1 3  þ þ 80 N þ 1 16 2ðN þ 2Þ ðN þ 2ÞðN þ 3Þ 12 ;  ðN þ 2ÞðN þ 4ÞðN þ 5Þ

E5 ¼

7 7

Em h Ecm h 1 6 30  þ þ 448 N þ 1 64 32ðN þ 2Þ 16ðN þ 2ÞðN þ 3Þ 15 90 þ  ðN þ 2ÞðN þ 3ÞðN þ 4Þ ðN þ 2ÞðN þ 3ÞðN þ 4ÞðN þ 5Þ 360 :  ðN þ 2ÞðN þ 3ÞðN þ 4ÞðN þ 6ÞðN þ 7Þ

E7 ¼

11 ; a 22 ; a 12 Þ ¼ ða



  8 m2 B2a ; n2 ; mmnBa D þ D  D 3 1 2 2 9 3 Bh    ð1  mÞp2 16 8 D3 þ D1  D2 n2 ; m2 B2a ; mnBa þ 2 9 3 2Bh

p2 16

þ D6 ð1; 1; 0Þ;

23 Þ ¼ 13 ; a ða

e31

 pD 4p 3 D 5  3 3 6 m Ba þ mn2 Ba ; n3 þ m2 nB2a  ðmBa ; nÞ: 3 Bh 3Bh

 3  2 16p4 ðD2 D5  D3 D4 Þ m2 B2a þ n2 þ 3p2 B2h D6 ð4D2 þ 3D4 Þ m2 B2a þ n2  h   i ¼ 3B2h m2 B2a þ mn2 p2 ð3D4  4D5 Þ m2 B2a þ n2 þ 3B2h D6   K 1 B2a þ K 2 p2 m2 B2a þ n2   B2a D1 ; þ p2 B2h m2 B2a þ mn2

8B2h

E p2 n4  1  m2 B2a þ mn2

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