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

Composite Structures 100 (2013) 566–574 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Nonlinear postbuckling of symmetric S-FGM plates resting on elastic foundations using higher order shear deformation plate theory in thermal environments Nguyen Dinh Duc ⇑, Pham Hong Cong 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 functionally 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 conditions, 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 Introduction 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 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 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 Based on classical and first order shear deformation theory, Eslami et al investigated buckling and ⇑ Corresponding author Tel.: +84 37547989; fax: +84 37547724 E-mail addresses: ducnd@vnu.edu.vn (N.D Duc), congph_54@vnu.edu.vn (P.H Cong) 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.compstruct.2013.01.006 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 material properties into consideration [8,9] Zhao et al [10] analyzed the mechanical and thermal buckling of FGM plates using element-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 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 [12,13] 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 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 567 N.D Duc, P.H Cong / Composite Structures 100 (2013) 566–574 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 postbuckling behaviors of thick functionally graded plates supported by elastic foundations and subjected to in-plane compressive, thermal 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, foundation stiffness and imperfection on the buckling and postbuckling of the symmetric S-FGM plates 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 Peff of the functionally graded plate, such as the modulus of elasticity E and the coefficient of thermal expansion a, vary in the thickness direction z and can be determined by the linear rule of mixture as Peff ¼ Prm V m zị ỵ Prc V c zị 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: < 2zỵhN ; E; aị ẳ Ec ; ac ị ỵ Emc ; amc ị h N : 2zỵh h 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 environment To increase the ability to adjust to a high temperature, structures 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 materials 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, Àh/2 z 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, Vm and Vc, are assumed as [21]: h ; z h=2 ; V c zị ẳ V m ðzÞ ð1Þ ð3Þ z h=2 ; where Emc ¼ Em À Ec ; amc ¼ am À ac ð4Þ and the Poisson ratio v is assumed constant, v(z) = v The reaction–deflection relation of Pasternak foundation is given by Àh=2 z Àh=2 z qe ¼ k1 w À k2 r2 w Governing equations < 2zỵhN ; V m zị ẳ h N : 2zỵh 2ị 2 ð5Þ 2 where r = @ /@x + @ /@y , w is the deflection of the plate, k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of Pasternak model 2.2 Theoretical formulation The present study uses the Reddy’s third order shear deformation plate theory to establish governing equations and determine the buckling loads and postbuckling paths of the symmetrical SFGM plates The strains across the plate thickness at a distance z from the middle surface are [22] 1 1 kx kx e0x ex C B C B B e C B e0 C 3B C C B @ y A ¼ @ y A þ z@ ky A þ z @ ky A cxy c0xy k k xy cxz cyz ! c0xz c0yz ẳ where ! ỵ z2 kxz ð6Þ xy ! ð7Þ kyz B B e0 B y @ u;x ỵ w2;x =2 1 /x;x C B C B B C C C C B C; B C; k C ¼ B/ C ẳ @ v ;y ỵ w2;y =2 A B A A @ y A @ y;y / þ / cxy u;y þ v ;x þ w;x w;y x;y y;x kxy 1 kx /x;x þ w;xx B C C B ky C ¼ c1 B @ /y;y ỵ w;yy A A @ /x;y ỵ /y;x ỵ 2w;xy k e0x kx xy c0xz c0yz Fig Symmetrical S-FGM plate on elastic foundation ! ẳ ! /x ỵ w;x ; /y ỵ w;y kxz kyz ! ẳ 3c1 /x ỵ w;x /y ỵ w;y ! 8ị 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, y) and (y, z) planes, respectively 568 N.D Duc, P.H Cong / Composite Structures 100 (2013) 566–574 Q x;x ỵ Q y;y 3c1 Rx;x ỵ Ry;y ị ỵ c1 P x;xx ỵ 2Pxy;xy ỵ Py;yy ị þ Nx w;xx Hooke law for an FGM plate is dened as E ẵex ; ey ị ỵ mey ; ex ị ỵ mịaDT1; 1ị m2   À Á E rxy ; rxz ; ryz ẳ cxy ; cxz ; cyz 21 ỵ mị ỵ 2Nxy w;xy ỵ Ny w;yy k1 w ỵ k2 r2 w ẳ rx ; ry ẳ 9ị 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 Z Ni ; M i ; Pi ị ẳ Q i ; Ri ị ẳ Z h=2 ri 1; z; z3 ịdz; i ẳ x; y; xy h=2 10ị h=2 rj 1; z ịdz; i ẳ x; y; j ¼ xz; yz Àh=2     h 1 ðE1 ; E2 ; E4 Þ e0x þ me0y þ ðE2 ; E3 ; E5 Þ kx þ mky 1Àm i   þðE4 ; E5 ; E7 ị Ny ; My ; Py ị ẳ kx ỵm ky ỵ mị/1 ; /2 ; /4 Þ     h 1 E1 ; E2 ; E4 ị e0y ỵ me0x ỵ E2 ; E3 ; E5 ị ky ỵ mkx 1m i   3 ỵE4 ; E5 ; E7 ị ky ỵ mkx ỵ mÞð/1 ; /2 ; /4 Þ ðN xy ; Mxy ; P xy ị ẳ Mx;x ỵ M xy;y Q x ỵ 3c1 Rx c1 Px;x ỵ Pxy;y ị ẳ 13dị Mxy;x ỵ M y;y Q y ỵ 3c1 Ry c1 P xy;x ỵ Py;y Þ ¼ 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,y by substituting Eqs (8) and (11) into Eqs (13c), (13d) and (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 ỵ Nxy;y ẳ Substitution of Eqs (6), (7) and (9) into Eq (10) yields the constitutive relations as: ðNx ; Mx ; Px ị ẳ h i 1 E1 ; E2 ; E4 ịc0xy ỵ E2 ; E3 ; E5 ịkxy ỵ E4 ; E5 ; E7 ịkxy 21 ỵ mị Q x ; Rx ị ẳ h i E1 ; E3 ịc0xz ỵ E3 ; E5 ịkxz 21 ỵ mị Q y ; Ry ị ẳ h i E1 ; E3 ịc0yz ỵ E3 ; E5 ịkyz 21 ỵ mị Ny;y ỵ Nxy;x ẳ c21 D2 D5 =D4 D3 ịr6 w ỵ c1 D2 =D4 ỵ 1ịD6 r4 w ỵ1 c1 D5 =D4 ịr2 Nx w;xx ỵ 2Nxy w;xy ỵ N y w;yy k1 w ỵ k2 r2 wị D6 =D4 Nx w;xx ỵ 2Nxy w;xy ỵ Ny w;yy k1 w ỵ k2 r2 wị ẳ 14ị where E3 E5 E7 ; D2 ¼ ; D3 ¼ À m2 À m2 À m2 D4 ¼ D1 À c1 D2 ; D5 ¼ D2 À c1 D3 ; E1 6c1 E3 ỵ 9c21 E5 D6 ẳ 21 ỵ mị D1 ẳ 15ị For an imperfect FGM plate, Eq (14) are modified into form as ð11Þ c21 ðD2 D5 =D4 À D3 Þr6 w þ ðc1 D2 =D4 þ 1ÞD6 r4 w þ ð1 À c1 D5 =D4 Þr2 h     f;yy w;xx ỵ w;xx 2f ;xy w;xy ỵ w;xy ị ỵ f;xx w;yy ỵ w;yy h     i k1 w ỵ k2 r2 w D6 =D4 f;yy w;xx ỵ w;xx 2f ;xy w;xy þ wÃ;xy where ðE1 ; E2 ; E3 ; E4 ; E5 ; E7 ị ẳ Z   i þf;xx w;yy þ wÃ;yy À k1 w þ k2 r2 w ¼ h=2 ð1; z; z2 ; z3 ; z4 ; z6 ịEzịdz h=2 Emc h E1 ẳ Ec h ỵ ; E2 ẳ 0; Nỵ1 3 Ec h Emc h E3 ẳ ỵ ; 12 2N ỵ 1ịN ỵ 2ịN ỵ 3ị E4 ẳ E5 ẳ 13cị Ec h Emc h ỵ 80 16 ỵ ỵ Nỵ1 Nỵ2 Nỵ3 Nỵ4 Nỵ5 ! /1 ; /2 ; /4 ị ẳ 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 term r6w Also, f(x, y) is stress function defined by Nx ¼ Ec h Emc h 15 20 E7 ẳ ỵ ỵ 448 64 N ỵ N ỵ N ỵ N ỵ ! 15 ỵ ỵ Nỵ3 Nỵ2 Nỵ1 Z @2f ; @y2 Ny ¼ @2f ; @x2 Nxy ¼ À @2f @x@y ð17Þ The geometrical compatibility equation for an imperfect plate is written as e0x;yy ỵ e0y;xx c0xy;xy ẳ w2;xy w;xx w;yy ỵ 2w;xy w;xy w;xx wÃ;yy h=2 ð1; z; z3 ÞEðzÞaðzÞDTdz ð16Þ À w;yy wÃ;xx ð12Þ ð18Þ Àh=2 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]: Nx;x ỵ Nxy;y ẳ 13aị Ny;y ỵ Nxy;x ẳ 13bị From the constitutive relations (11) with the aid of Eq (17) one can write f;yy mf;xx ỵ /1 ị; E1 ẳ 21 ỵ mịf;xy E1 e0x ẳ e0y ẳ f;xx mf;yy ỵ /1 ị; E1 c0xy ð19Þ 569 N.D Duc, P.H Cong / Composite Structures 100 (2013) 566–574 Introduction of Eq (19) into Eq (18) gives the compatibility equation of an imperfect FGM plate as   r4 f À E1 w2;xy À w;xx w;yy þ 2w;xy wÃ;xy À w;xx wÃ;yy À w;yy wÃ;xx ¼ 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 SFGM 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 symmetric 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, and will be considered [8,12,15,20,21]: &     À Á3 À Á2 D2 D5 c1 D2 Àc21 À D3 k2m þ d2n þ D6 þ k2m þ d2n D4 D4   !' Â À ÁÃ D c1 D5 2 km ỵ d2n ỵ W ỵ k1 ỵ k2 km ỵ dn D4 D  4 & ' Á Á E1 D6 À c D5 ỵ km ỵ d6n þ k2m d4n þ k4m d2n W km þ d4n þ À 16 D4 D4   ! Á D6 c D5 km ỵ d2n ỵ W ỵ lhịW ỵ 2lhị ỵ D4 D4 2 26ị ẵN x0 km ỵ Ny0 dn W ỵ lhị ẳ 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, Fy uniformly distributed on edges x = 0, a and y = 0, b, respectively In this case, prebucking force resultants are [6] Nx0 ¼ ÀF x h; Case 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 21ị Nx ẳ Nx0 at x ẳ 0; a w ¼ v ¼ /x ¼ My ¼ Py ¼ 0; Ny ẳ Ny0 at y ẳ 0; b 27ị F x ẳ e11 W W ỵl ỵ e12 WW þ 2lÞ ð28Þ where Case Four edges of the plate are simply supported and immovable (IM) In this case, boundary conditions are w ¼ u ¼ /y ¼ M x ¼ Px ¼ 0; Ny0 ¼ ÀF y h and Eq (26) leads to ð22Þ Case 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  3  2 À16p4 ðD2 D5 D3 D4 ị m2 B2a ỵ n2 ỵ 3D6 B2h p2 4D2 ỵ 3D4 ị m2 B2a ỵ n2   ẳ 3B2h m2 B2a ỵ bn2 ịẵp2 3D4 4D5 ị m2 B2a ỵ n2 ỵ 3B2h D6 h i K B2a ỵ K p2 m2 B2a ỵ n2 ị B2a D1   ỵ B2h p2 m2 B2a ỵ bn2   E1 m4 B4a ỵ n4 p2 e2 ẳ 29ị 16m2 B2a þ bn2 ÞB2h e11 in which where Nx0, Ny0 are in-plane compressive loads at movable edges (i.e., Case and the first of Case 3) or are fictitious compressive edge loads at immovable edges (i.e., Case and the second of Case 3) Bh ¼ b=h; Ba ¼ b=a; W ¼ W=h; b ¼ F y =F x k1 a4 k2 a2 i K1 ¼ ; K2 ¼ ; Ei ¼ Ei =h ði ¼ À 7Þ D1 D1 E3 E5 E7 D1 ¼ ; D2 ¼ ; D3 ¼ À v2 À v2 À v2 4 D4 ¼ D1 À D2 ; D5 ¼ D2 À D3 ; D6 ¼ ðE1 À 8E3 þ 16E5 Þ 3 2ð1 þ v Þ The approximate solutions of w, w⁄ [7,15] and f [19–21] satisfying boundary conditions (21)–(23) are assumed to be For a perfect FGM plate, Eq (28) reduces to an equation from which buckling compressive load may be obtained as F xb ¼ e11 w; w ị ẳ W; lhị sin km x sin dn y 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 = at x = 0, a and v = at y = 0, b is fulfilled in an average sense as [5,8,20,21] w ¼ Nxy ¼ /y ¼ M x ¼ Px ¼ 0; w ¼ v ¼ /x ¼ My ¼ Py ¼ 0; Nx ¼ Nx0 at x ¼ 0; a Ny ¼ Ny0 at y ẳ 0; b 23ị 24aị f ẳ A1 cos 2km x ỵ A2 cos 2dn y ỵ A3 sin km x sin dn y ỵ A4 1 cos km x cos dn y ỵ Nx0 y2 ỵ Ny0 x2 2 ð24bÞ where km = mp/a, dn = np/b, W is amplitude of the deflection and l is imperfection parameter The coefficients Ai(i = À 4) are determined by substitution of Eqs (24a) and (24b) into Eq (20) as A1 ¼ E1 d2n 32k2m ¼0 WðW ỵ 2lhị; A2 ẳ E1 k2m 32d2n 30ị Z b Z a @u dxdy ¼ 0; @x Z a Z b @v dydx ¼ @y ð31Þ From Eqs (8) and (11) one can obtain the following expressions in which Eq (17) and imperfection have been included WW ỵ 2lhị; A3 ẳ A4 25ị Subsequently, setting Eqs (24a) and (24b) into Eq (16) and applying the Galerkin procedure for the resulting equation yield @u / ¼ ðf;yy À mf;xx Þ À w2;x =2 À w;x w;x ỵ @x E1 E1 @v / ẳ f;xx mf;yy ị w;y =2 w;y w;y ỵ @y E1 E1 32ị 570 N.D Duc, P.H Cong / Composite Structures 100 (2013) 566–574 Introduction of Eq (24) into Eq (32) and then the result into Eq (31) give N x0 N y0 F x ẳ e31 / ẳ E1 k2m ỵ md2n WW ỵ 2lhị 81 m ị 1m / ẳ E1 mk2m ỵ d2n WW ỵ 2lhị 81 m2 ị 1Àm ð33Þ When the deflection dependence of fictitious edge loads is ignored, i.e., W = 0, Eq (33) reduce to Nx0 ¼ Ny0 ¼ À Subsequently, Nx0 and Ny0 are placed in Eq (26) to give /1 1Àv ð34Þ which was derived by Shariat and Eslami [6] by solving the membrane form of equilibrium equations and employing the method suggested by Meyers and Hyer [23] Substituting Eq (33) into Eq (26) yields the expression of thermal parameter as W Pn2 DT ỵ e32 WW ỵ 2lị W ỵl m2 B2a ỵ v n2 40ị where  3  2 16p4 D2 D5 D3 D4 ị m2 B2a ỵ n2 ỵ 34D2 ỵ 3D4 ịD6 m2 B2a ỵ n2 p2 B2h h   i  ¼ 3B2h 3D6 B2h ỵ 3D4 4D5 ị m2 B2a ỵ n2 p2 m2 B2a ỵ v n2 h  i K B2a ỵ K p2 m2 B2a ỵ n2 B2a D1   ỵ B2h p2 m2 B2a ỵ v n2   4 E1 m Ba ỵ n p E n p2   ỵ e2 ẳ 16 m2 B2 ỵ v n2 B2 B2h m2 B2a ỵ v n2 ị e31 a h ð41Þ " À Á2 À Á Àc21 D2 D5 D3 D4 ị k2m ỵ d2n ỵ D6 c1 D2 ỵ D4 ị k2m ỵ d2n /1 ẳ 1v D6 ỵ D4 c1 D5 ị k2m ỵ d2n # " k1 ỵ k2 k2m ỵ d2n E1 k4m ỵ d4n W ỵ ỵ W ỵ lh k2m ỵ d2n 16 k2m ỵ d2n # E1 k4m ỵ d4n þ 2v k2m d2n ð35Þ þ WðW þ 2lhÞ 8ð1 v ị k2m ỵ d2n Eqs (28), (36) and (40) are explicit expressions of load–deflection curves for thick S-FGM plates resting on Pasternak elastic foundations and subjected to in-plane compressive, thermal and thermomechanical loads, respectively Specialization of these equations for thick S-FGM plates, i.e., ignoring the third order shear deformations and elastic foundations, gives the corresponding results derived by using the first order shear deformation plate theory for S-FGM plates [21] The S-FGM plate is exposed to temperature environments uniformly raised from stress free initial state Ti to final value Tf, and temperature change DT = Tf À Ti is considered to be independent from thickness variable The thermal parameter /1 is obtained from Eq (12), and substitution of the result into Eq (35) yields Numerical results and discussion DT ẳ e21 W W ỵl ỵ e22 WW ỵ 2lị 36ị 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 [5,8,20,21]: Em ¼ 70 GPa; am ẳ 23 106  C1 42ị where Ec ¼ 380 GPa; am ¼ 7:4 Â 10À6  C1 v ịp2   i e21 ẳ h P 3Bh D6 ỵ 3D4 4D5 ị m2 B2a ỵ n2 p2 " #   p2   16 2 Â À ðD2 D5 D3 D4 ị m2 B2a ỵ n2 ỵ D 4D ỵ 3D ị m B ỵ n a B2h h  i K B2a ỵ K p2 m2 B2a ỵ n2 B2a D1 v ị ỵ PB2h p2 m2 B2a þ n2 Þ   E1 p2 ð1 À v ị m4 B4a ỵ n4 E p2 m4 B4a þ n4 þ 2v m2 n2 B2a     ỵ e22 ẳ 8P1 ỵ v ị 16PB2 m2 B2 ỵ n2 B2 m2 B2 ỵ n2 and Poisson ratio is chosen to be v = 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 Effects of volume fraction index N on the postbuckling of S-FGM plates under uniaxial compressive load and uniform temperature rise are shown in Figs and In all below figures, it is assumed h a h a ð37Þ in which P ẳ Ec ac ỵ Ec amc ỵ Emc ac Emc amc ỵ Nỵ1 2N ỵ 38ị By Setting l = Eq (36) leads to an equation from which buckling temperature change of the perfect FGM plates may be determined as DT b ¼ e21 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 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 = ÀFxh and fictitious compressive load on immovable edges is determined by setting the second of Eq (32) in the second of Eq (31) as Ny0 ẳ v Nx0 /1 ỵ E1 d WW ỵ 2lhị n 39ị Fig Effects of volume fraction index N on the postbuckling of symmetrical SFGM plates under uniaxial compressive load (all FM edges) N.D Duc, P.H Cong / Composite Structures 100 (2013) 566–574 that ~ePx ¼ F x 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 and show effects of first and third order shear deformations on mechanical and thermal buckling and postbuckling of SFGM plate with various volume fractions N of the 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 deformation For postbuckling of the S-FGM plate, Figs and also show us that the imperfect plate has a better mechanical and thermal loading capacity than those of the perfect plate Figs and present effects of first and third order shear deformations 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 critical mechanical 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 postbuckling for symmetric S-FGM plate yet We therefore are limited to compare with the others However, comparing our findings in Figs 4–7 with 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 SFGM plates However, this difference is not much despite of complicated third order shear calculation Figs and show the influence of initial imperfections on postbuckling of S-FGM plate under uniaxial compressive load (all FM edges) and under uniform temperature (all IM edges) Fig shows us that the critical compressive loads decreases with l in the limit of the small bending However, it increases with l in the other limit of the large bending, meaning the higher bending-load curve (i.e., the better loading ability) Figs 4–9 show 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 clearly shows us that an initial imperfection has an useful influence on the thermal resistance of SFGM at the threshold value of the bending Fig Effects of volume fraction index N on the postbuckling of symmetrical SFGM plates under uniform temperature rise (all IM edges) 571 Fig Effect of first and third order shear deformation on mechanical buckling and postbuckling of S-FGM plate with various of volume fractions N Fig Effect of first and third 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 buckling and postbuckling of S-FGM plate with the temperature DT 572 N.D Duc, P.H Cong / Composite Structures 100 (2013) 566–574 Fig 10 Effects of the elastic foundations on the postbuckling of symmetrical SFGM plates under uniaxial compressive load (all FM edges) Fig 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 Px Fig The influence of imperfections on the stability of symmetrical S-FGM plates under uniaxial compressive load (all FM edges) Fig The influence of imperfections on the stability of symmetrical S-FGM plates under uniform temperature rise (all IM edges) Fig 11 Effects of the elastic foundations on the postbuckling of symmetrical SFGM 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, a) N.D Duc, P.H Cong / Composite Structures 100 (2013) 566–574 573 Effect of boundary conditions on postbuckling of symmetric SFGM plate under uniaxial compression is shown in Fig 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 = (in (30)), whereas the result for IM edges drawn from (40) with DT = Fig 14 shows 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 Conclusions Fig 13 Effect of temperature gradient and uniaxial compression on the postbuckling of symmetric S-FGM (FM on y = 0, b; IM on x = 0, a) This paper presents an analytical investigation on the postbuckling behaviors of thick symmetric functionally graded plates resting on elastic foundations in thermal environments and 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 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 conditions, foundation stiffness and imperfection on the postbuckling loading capacity of the SFGM 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 deformation and for the postbuclking period of the S-FGM plate, comparing with a perfect plate, an imperfect plate has a better mechanical and thermal loading capacity Acknowledgment Fig 14 Effect of boundary conditions (FM and IM) on postbuckling of symmetric SFGM plate under uniaxial compression on edges y = 0, b Figs 10 and 11 present the positive influence of elastic foundations on imperfections on the stability of S-FGM plate under uniaxial compressive load (all FM edges) and uniform temperature (all IM edges) The effect of Pasternak foundation K2 on the critical compressive loads and the thermal resistance of S-FGM is larger than the Winkler foundation K1 This conclusion has been also reported in [16–18,20] An investigation of the mechanical–thermal stability has been determined by (40) Figs 12 and 13 have 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 12 shows the effect of the temperature gradient of the surrounding 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 temperature gradient condition DT – 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 field DT 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] This work was supported by Vietnam National University, Hanoi The authors are grateful for this financial support References [1] Wu L Thermal buckling of a simply supported moderately thick rectangular 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Specialization of these equations for thick S-FGM plates, i.e., ignoring the third order shear deformations and elastic foundations, gives the corresponding results derived by using the first order shear