DSpace at VNU: MECHANICAL AND THERMAL POSTBUCKLING OF SHEAR-DEFORMABLE FGM PLATES WITH TEMPERATURE-DEPENDENT PROPERTIES

DSpace at VNU: MECHANICAL AND THERMAL POSTBUCKLING OF SHEAR-DEFORMABLE FGM PLATES WITH TEMPERATURE-DEPENDENT PROPERTIES...

Mechanics of Composite Materials, Vol 46, No 5, 2010 MECHANICAL AND THERMAL POSTBUCKLING OF SHEAR-DEFORMABLE FGM PLATES WITH TEMPERATURE-DEPENDENT PROPERTIES N D Duca,* and H V Tungb Keywords: postbuckling, functionally graded materials, temperature-dependent properties, imperfection An analytical approach to investigating the stability of simply supported rectangular functionally graded plates under in-plane compressive, thermal, and combined loads is presented The material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of constituents The equilibrium and compatibility equations for the plates are derived by using the first-order shear deformation theory of plates, taking into account both the geometrical nonlinearity in the von Karman sense and initial geometrical imperfections The resulting equations are solved by employing the Galerkin procedure to obtain expressions from which the postbuckling load–deflection curves can be traced by an iterative procedure A stability analysis performed for geometrically midplane-symmetric FGM plates shows the effects of material and geometric parameters, in-plane boundary conditions, temperature-dependent material properties, and imperfections on the postbuckling behavior of the plates Introduction Functionally graded materials (FGMs) are microscopically inhomogeneous composites usually made of a mixture of metals and ceramics By gradually varying the volume fraction of their constituents, it can be achieved that the effective properties of FGMs exhibit a smooth and continuous change from one surface to another, thus eliminating interface problems and mitigating thermal stress concentrations Due to the high heat resistance, FGMs are used as structural components operating in ultrahigh-temperature environments and subjected to extremely high thermal gradients, such as aircraft, space vehicles, nuclear plants, and other engineering applications These new materials pose interesting topics for structural mechanics, e.g., problems relating to the buckling and postbuckling behavior of FGM structures subjected to mechanical, thermal, and thermomechanical loads Eslami and his co-workers [1-7] have treated a series of problems relating to the linear buckling of simply supported rectangular FGM plates, with and without imperfections, under mechanical and thermal loads By using an analytical approach, they obtained a University of Engineering and Technology, Vietnam National University, Ha Noi, Viet Nam bFaculty of Civil Engineering, Hanoi Architectural University, Ha Noi, Viet Nam Russian translation published in Mekhanika Kompozitnykh Materialov, Vol 46, No 5, pp 679-700, September-October, 2010 Original article submitted February 19, 2010 *Corresponding author; e-mail: ducnd@vnu.edu.vn 0191-5665/10/4605-0461 © 2010 Springer Science+Business Media, Inc 461 closed-form expressions for buckling loads Following this direction, Lanhe [8] examined the thermal postbuckling of rectangular moderately thick FGM plates But the effects of prebuckling deformation, the temperature dependence of material properties, and postbuckling behavior of FGM plates have not been considered in these works Shen [9-11] investigated the postbuckling behavior of FGM plates subjected to transverse and in-plane [9], thermoelectromechanical [10], and thermal [11] loads By using Reddy’s higher-order shear deformation theory in conjunction with a two-step perturbation technique, he successfully analyzed the postbuckling of FGM plates with both temperature-dependent material properties and initial imperfections accounted for Liew et al analyzed the postbuckling of FGM plates under simultaneous actions of various loads [12] and uniform temperature changes [13] by using a higher-order shear deformation theory They also investigated the postbuckling of cylindrical panels under combined thermomechanical loads within the framework of the classical shell theory [14] By employing differential quadrature method and an iteration technique, the postbuckling curves for fully clamped plates were traced in these studies Na and Kim [15] investigated the thermal postbuckling of FGM plates by using the three-dimensional finite-element method Zhao et al [16] and Zhao and Liew [17] analyzed the buckling and postbuckling behavior of FGM plates by invoking the element-free kp-Ritz method It is evident from the literature that investigations considering the temperature dependence of material properties are few in number However, in practice, the FGM structures are usually exposed to high-temperature environments, where significant changes in material properties are unavoidable Therefore, the temperature dependence of their properties should be considered for an accurate and reliable prediction of deformation behavior of the composites This paper presents an analytical approach to investigating the buckling and postbuckling behavior of simply supported rectangular FGM plates subjected to in-plane compressive, thermal, and thermomechanical loads Their material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of constituents The governing equations are derived within the framework of the first-order shear deformation theory, with account of both the von Karman nonlinearity and initial imperfections The resulting equations are solved by the Galerkin method to obtain expressions from which the buckling loads and postbuckling curves are determined by an iteration procedure A stability analysis carried out for geometrically in-plane symmetric FGM plates shows the effects of material and geometric parameters, in-plane boundary conditions, temperature-dependent material properties, and imperfections on the postbuckling behavior of the plates Functionally Graded Plates Consider a rectangular plate that consists of two 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 length, width, and total thickness of the plate are a, b, and h, respectively 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 coordinate, -h £ z £ h By applying a simple power-law distribution, the volume fractions of metal and ceramic,Vm andVc , are assumed as ìỉ 2z + h k ùỗ ữ , -h Ê z £ 0, ïè h ø Vm ( z ) = í k ï ỉ -2 z + h , Ê z Ê h 2, ỗ ữ ù îè h ø (1) where the volume fraction index k 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 [2, 3] 462 Peff = Prm Vm ( z ) + Prc Vc ( z ), (2) where Pr denotes a temperature-dependent 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 FGM plate can be written as follows: ìỉ 2z + h k ùỗ ữ , -h Ê z £ 0, ïè h ø ( E , a ) = ( E c , a c ) + ( E mc , a mc ) í k ï ỉ -2 z + h ÷ , £ z Ê h 2, ùỗ ợố h ứ (3) where E mc = E m - E c , a mc = a m - a c , and the Poisson ratio n is assumed constant, n( z ) = n Obviously, due to the temperature-dependent properties of constituents, the effective properties E and a of the FGM plate are both temperature- and position-dependent Governing Equations In the present study, the first-order shear deformation theory is used to obtain the equilibrium and compatibility equations, as well as expressions for determining the buckling loads and the postbuckling load–deflection curves of FGM plates The strain–displacement relations taking into account the von Karman nonlinear terms are ỉ e x ỉ e xm ç ÷ ç ç e y ÷ = ç e ym ỗg ữ ỗg ố xy ứ ố xym ổ kx ữ ỗ ữ ữ + zỗ k y ữ , ữ ỗk ữ ứ ố xy ứ æ g xz ö æ w ,x + fx ö ç ÷ =ç ÷ ç g yz ÷ ç w , y + f y ÷ è ø è ø (4) with ổ e xm ỗ ỗ e ym ỗg è xym u ,x + w ,2x ư÷ ổỗ ữ ữ = ỗ v , y + w ,2y ữ , ữ ỗu + v + w w ữ ,x ,x , y ữ ứ ỗố , y ø ỉ kx ỉ fx , x ç ÷ ç ÷ ç k y ÷ = ç f y, y ữ, ỗk ữ ỗf + f ữ y,x ø è xy ø è x, y (5) where e xm and e ym are the normal strains, g xym is the shear strain on the midplane of the plate, and g xz and g yz are the transverse shear strains; u, v, and w are the midplane displacement components along the x, y, and z axes; fx and f z are the rotation angles in the xz and yz planes, respectively; (, ) indicates a partial derivative Hooke’s law for the plate, including the thermal effects, is sx = E 1- n [ e x + ne y - (1+ n )aDT ] , s y = s xy = E 1- n [ e y + ne x - (1+ n )aDT ] , (6) E E E g xy , s xz = g xz , s yz = g yz 2(1+ n ) 2(1+ n ) 2(1+ n ) The force and moment resultants of the plate can be expressed in terms of stress components across the plate thickness as 463 h2 (N i , M i ) = ò s i (1, z )dz, i = x, y, xy, Qi = -h h2 ò s j dz, i = x, y; j = xz , yz -h (7) Inserting Eqs (3), (4), and (6) into Eqs (7) gives the constitutive relations (N x , M x ) = (N y , M y ) = 1- n 1- n [( E1 , E )( e xm + ne ym ) + ( E , E )( k x + nk y ) - (1+ n )( F m , F b )] , [( E1 , E )( e ym + ne xm ) + ( E , E )( k y + nk x ) - (1+ n )( F m , F b )] , ( N xy , M xy ) = (8) [( E1 , E )g xy + ( E , E )k xy ] , 2(1+ n ) (Qx , Q y ) = E1 ( g xz , g yz ) , 2(1+ n ) where E1 = E c h + E mc h E h3 E mc h , E = 0, E = c , + 12 2( k + 1)( k + 2)( k + 3) k +1 (9) h2 (F m , F b ) = ò E ( z , T )a ( z , T )DT (1, z )dz -h Within the framework of the first-order shear deformation theory, with the assumption that the temperature field is raised uniformly, the nonlinear equilibrium equations for a perfect plate can be written in terms of deflection w and force resultants as [7, 8] N x,x + N xy, y = 0, N xy,x + N DÑ w + y, y = 0, (10) 2(1+ n )D Ñ ( N x w ,xx + 2N xy w ,xy + N y w , yy ) - ( N x w ,xx + 2N xy w ,xy + N y w , yy ) = 0, E1 where Ñ = ¶2 ¶x + ¶2 ¶y and D = E1 E - E 22 E1 (1- n ) For an imperfect plate, let w* ( x, y )denote a known small imperfection This parameter represents a small initial deviation of the plate plane from a flat shape When the imperfection is considered, equilibrium equations (10) obtain the form [7] N x,x + N xy, y = 0, N xy,x + N DÑ w + y, y = 0, 2(1+ n )D Ñ [ N x ( w ,xx + w*,xx ) + 2N xy ( w ,xy + w*,xy ) + N y ( w , yy + w*, yy )] E1 - [ N x ( w ,xx + w*,xx ) + 2N xy ( w ,xy + w*,xy ) + N y ( w , yy + w*, yy )] = Considering the first two of Eqs (11), a stress function f may be defined as 464 (11) N x = f , yy , N y = f ,xx , N xy = - f ,xy (12) Inserting Eq (12) into the third of Eqs (11) leads to DÑ w + 2(1+ n )D Ñ [ f , yy ( w ,xx + w*,xx ) - f ,xy ( w ,xy + w*,xy ) + f ,xx ( w , yy + w*, yy )] E1 - [ f , yy ( w ,xx + w*,xx ) - f ,xy ( w ,xy + w*,xy ) + f ,xx ( w , yy + w*, yy )] = (13) Equation (13) includes two dependent unknowns, w and f To obtain a second equation, relating the unknowns, the geometrical compatibility [18] e xm , yy + e ym ,xx - g xym ,xy = w ,2xy - w ,xx w , yy (14) can be used For an imperfect plate, this equation may be transformed to the form e xm , yy + e ym ,xx - g xym ,xy = w ,2xy - w ,xx w , yy + 2w ,xy w*,xy - w ,xx w*, yy - w , yy w*,xx (15) From constitutive relations (8), with E = 0, one can write ( e xm , e ym ) = 2(1+ n ) N xy [( N x , N y ) - n( N y , N x ) + F m (1, 1)] , g xym = E1 E1 Inserting the previous equations into Eq (15), with account of Eq (14), leads to the compatibility equation for an imperfect FGM plate Ñ f - E1 ( w ,2xy - w ,xx w , yy + 2w ,xy w*,xy - w ,xx w*, yy - w , yy w*,xx ) = (16) Equations (13) and (16) are the basic relations used to investigate the buckling and postbuckling of FGM plates They are nonlinear in the dependent unknowns w and f Stability Analysis In this section, an analytical approach is used to investigate the stability of FGM plates subjected to mechanical, thermal, and combined loads Three cases of boundary conditions, labeled Cases 1, 2, and 3, will be considered [19] Case Plate edges are simply supported and freely movable (FM) The associated boundary conditions are w = f y = M xx = N xy = 0, N x = N x0 , x = 0, a, w = fx = M yy = N xy = 0, N y =N y0 , (17) y = 0, b Case The edges are simply supported and immovable (IM) The associated boundary conditions are w = u = f y = M xx = 0, N x = N x0 , x = 0, a, w = v = fx = M yy = 0, N y =N y0 , (18) y = 0, b Case The edges are simply supported, and uniaxial edge loads operate in the x-coordinate direction The edges x = 0, a and x = 0, b are considered freely movable, but the other two are load-free and immovable For this case, the boundary conditions are w = f y = M xx = N xy = 0, N x = N x0 , x = 0, a, (19) 465 w = v = fx = M yy = 0, N y =N y0 , y = 0, b, (19) where N x0 and N y0 are the prebuckling force resultants in the x and y directions, respectively, for Case and the first of Case 3, and are fictitious compressive edge loads rendering the edges immovable for Case and the second of Case To solve Eqs (13) and (16) for the unknowns w and f, with consideration of boundary conditions (17)-(19), we assume that [19, 20] w = W sin l m x sin m n y, (20) 1 f = A1 cos 2l m x + A cos 2m n y + A cos 2l m x cos 2m n y + A sin l m x sin m n y + N x0 y + N 2 y0 x , where l m = mp a and m n = np b (m, n = 1, 2, ) are the numbers of half-waves in the x and y directions, respectively, and W is the deflection amplitude; A i (i = 1-4) are the coefficients to be determined Considering boundary conditions (17)-(19), the imperfections of the plate are assumed in the form [5-7, 18] w* = mh sin l m x sin m n y; m, n = 1, 2, , (21) where the coefficient m, varying between and 1, represents the size of the imperfections After substituting Eqs (20) and (21) into Eq (16), the coefficients A i are found: A1 = E1m 2n 32l2m W (W + 2mh ), A = E1 l2m 32m 2n W (W + 2mh ), A = A = (22) Introducing Eqs (20)-(22) into Eq (13) and applying the Galerkin method to the resulting equation, we obtain é (1+ v )D[ 5l2 m ( l2 + m ) + l6 + m ] m n m n m n D ( l2m + m n2 ) W + ê êë + ù ( l4m + m 4n )úW (W + mh )(W + 2mh ) ú 16 û E1 é 2(1+ v )D ù +ê ( l m + m n2 ) + 1ú ( l2m N x0 + m 2n N ë E1 û y0 )(W + mh ) = (23) This equation, derived for odd values of mand n, is used to determine the buckling loads and postbuckling curves for rectangular FGM plates under mechanical, thermal, and combined loads 4.1 Analysis of mechanical stability A simply supported FGM plate with freely movable edges (Case 1) is assumed to be under in-plane compressive loads Px and P y uniformly distributed along the edges x = 0, a and y = 0, b, respectively The material properties are assumed to be temperature-independent in this case The prebuckling force resultants are [1, 18] N x0 = -Px h, N y0 = -P y h The introduction of Eq (24) into Eq (23) gives Px = 466 p D ( m2 B a2 + n ) B h2 ( m2 B a2 + bn )Lmn × é p E ( m4 B + n ) W a +ê W + m ê16B ( m2 B a2 + bn )Lmn h ë (24) + p (1+ v )D [ 5m2 n B a2 ( m2 B a2 + n ) + m6 B a6 + n ] ù úW (W + 2m ), ú 8B h4 ( m2 B a2 + bn )Lmn û (25) where E b b D W B a = , B h = , D = , E1 = , W = , a h h h h Lmn = 2(1+ v )p D E1 B h2 ( m2 B a2 + n ) + 1, b = Py Px Equation (25) may be used to trace the postbuckling load–deflection curves for FGM plates subjected to in-plane compressive loads For a perfect plate, m = 0, Eq (25) leads to an equation from which the buckling compressive load Pxb may be obtained: Pxb = p D ( m2 B a2 + n ) B h2 ( m2 B a2 + bn )Lmn (26) The critical buckling load Pxcr is found for the values of m and n which make the preceding expression minimum In contrast, when m ¹ 0, the imperfection sensitivity of the plates may be predicted The counterparts of Eqs (25) and (26) corresponding to the classical theory can also be easily obtained, but the pertinent expressions are omitted for the sake of brevity 4.2 Analysis of thermal stability A simply supported FGM plate with temperature-dependent material properties and immovable edges (Case 2) under a thermal load is considered The condition expressing the immovability of edges, u = at x = 0, a and v = at y = 0, b, is fulfilled on the average [11, 19]: ba ịị 00 ¶u dxdy = 0, ¶x àb ¶v ò ò ¶y dydx = 00 (27) From Eqs (5) and (8), the following relations can be obtained in which Eq (12) and the imperfection have been included: F ¶u 1 = ( f , yy - nf ,xx ) - w ,2x - w ,x w ,*x + m , ¶x E1 E1 (28) F ¶v 1 = ( f ,xx - nf , yy ) - w ,2y - w , y w ,*y + m ¶y E1 E1 Introducing Eqs (20) and (21) into Eqs (28) and then into Eqs (27) yields N x0 = - N y0 =- Fm E1 + ( l2m + nm 2n )W (W + 2mh ), 1- n 8(1- n ) (29) Fm E1 + (m 2n + nl2m )W (W + 2mh ) 1- n 8(1- n ) Equations (29) express the compressive stresses making the edges immovable These stresses depend on the thermal parameter F m and the prebuckling deflection If the prebuckling deflection is ignored, Eq (29) leads to N x0 = N y0 =- Fm 1- n 467 This equation may also be derived by using the membrane form of equilibrium equations and the method proposed by Meyers and Hyer [21] Such an approach was employed by Eslami and his co-workers [2-4, 6, 7] and by Lanhe [8] in investigating pure FGM plates Inserting Eqs (29) into Eq (23) gives the following expression for the thermal parameter: Fm = p D (1- v )( m2 B a2 + n ) × b Lmn W W + mh é p (1- v )D[ 5m2 n B ( m2 B + n ) + m6 B + n ] a a a +ê 2 8b ( m B a + n )Lmn êë + + p E1 (1- v )( m4 B a4 + n ) 16b ( m2 B a2 + n )Lmn p E1 ( m4 B a4 + n + 2vm2 n B a2 ) ù úW (W + 2mh ) 8(1+ v )b ( m2 B a2 + n ) úû (30) In this study, it is assumed that the temperature is uniformly raised from an initial value Ti to a final one T f and the temperature difference DT = T f - Ti is constant By using Eqs (9), the thermal parameter F m can be expressed in terms of DT: F m = PhDT , P = E c a c + E c a mc + E mc a c E mc a mc + k +1 2k + (31) Although DT is included in the expression for P due to the temperature dependence of material properties (T = T0 + DT ), one may formally express DT from Eqs (30) and (31) as follows: DT = p D (1- v )( m2 B a2 + n ) B h2 PLmn × W W +m é p (1- v )D [ 5m2 n B ( m2 B + n ) + m6 B + n ] a a a +ê 2 ê 8B h P ( m B a + n )Lmn ë + + p (1- v )E1 ( m4 B a4 + n ) 16B h2 P ( m2 B a2 + n )Lmn p E1 ( m4 B a4 + n + 2vm2 n B a2 ) ù úW (W + 2m ), ú 8(1+ v )B h2 P ( m2 B a2 + n ) û (32) which is a closed-form expression for thermal postbuckling in the case of temperature-independent material properties Conversely, when material properties are temperature-dependent, an iteration procedure must be used to determine the thermal postbuckling curves of FGM plates Specifically, for given material and geometric parameters and specific values of W h and m, the temperature difference is gradually increased from an initial value DT = (T = T0 ) to the final one DT * (T = T0 + DT * ) at which the difference in values between both sides of Eq (32) reaches a small value prescribed When the initial imperfection is ignored, Eq (32) leads to an equation from which the buckling temperature difference DTb may be derived as 468 DTb = p D (1- v )( m2 B a2 + n ) B h2 PLmn (33) An iterative process is adopted to determine the critical values of DTb when material properties are temperature-dependent When m ¹ 0, no bifurcation buckling point exists, and the imperfection sensitivity of the plate may be predicted Moreover, the counterparts of Eqs (32) and (33) corresponding to the classical theory can readily be derived 4.3 Analysis of thermomechanical stability Let us consider a simply supported plate, with movable edges x = 0, a and immovable ones y = 0, b (Case 3), subjected to the simultaneous action of a thermal field and an in-plane compressive load Px distributed uniformly along the edges x = 0, a From the first of Eqs (24) and the second of Eqs (27) and (28), we have N x0 = -Px h, N y0 = vN x0 - F m + E1m 2n W (W + 2mh ) Employing these relations in Eq (23) yields Px = p D ( m2 B a2 + n ) B h2 ( m2 B a2 + )Lmn × W W +m ì p (1+ v )D [ 5m2 n B ( m2 B + n ) + m6 B + n ] ï a a a +í 2 8B h ( m B a + )Lmn ïỵ + p E1 ( m4 B a4 + n + 2Lmn n ) üï n PDT W (W + 2m ) ý 16B h2 ( m2 B a2 + )Lmn ùỵ m2 B a2 + (34) Equation (34) is utilized to trace the postbuckling curves of FGM plates under combined mechanical and thermal loads Specifically, it is used to determine the in-plane compressive loads as functions of total deflection (for a given uniform temperature rise) or the variation in temperature as a function of total deflection (for a given compressive edge load) This equation shows that, when the temperature dependence of properties is accounted for, the function Px (W )is affected by the temperature field in all its terms and is not merely shifted along the Px axis by the amount DPx = - n PDT m2 B a2 + as its temperature-independent counterpart From Eq (34), DT can be formally expressed in terms of the remaining members The resulting equation may be immediately used to trace the thermal postbuckling paths, and the DT (W )curves can be displaced along the DT axis by the amount - ( m2 B a2 + )Px when the temperature dependence of material properties is ignored In contrast, when the properties are n2P temperature-dependent, an iterative process must be used to determine the thermal postbuckling curves for given compressive loads Results and Discussion Here, several numerical examples will be presented for perfect and imperfect simply supported midplane-symmetric FGM plates The silicon nitride and stainless steel are regarded as constituents of the FGM plates A material property Pr, such as the elastic modulus and the thermal expansion coefficient, can be expressed as a nonlinear function of temperature [22]: Pr = P0 ( P-1T -1 + 1+ P1T + P2 T + P3 T ), (35) 469 TABLE Temperature-Dependent Coefficients of the Silicon Nitride and Stainless Steel [23] Properties E, Pà a, 1/K Materials P-1 P0 P1 Silicon nitride 348.43 ×10 –3.070 ×10 2.160 ×10 Stainless steel 201.04 ×109 3.079 ×10-4 –6.534 ×10-7 Silicon nitride -6 -4 0 Stainless steel -4 0 5.8723×10 -6 12.330 ×10 P2 -4 9.095 ×10 8.086 ×10 P3 -7 –8.946 ×10-11 where T = T0 + DT and T0 = 300 Ê (room temperature); P0 , P-1 , P1 , P2 , and P3 are temperature-dependent coefficients characterizing the constituent materials The typical values of the coefficients of the materials mentioned are listed in Table In what follows, the temperature-dependent and temperature-independent properties will be referred to as T-D and T-ID, respectively Moreover, Poisson’s ratio is chosen to be n = 0.3, and the T-ID properties are those computed by formula (35) at T = T0 in this study, unless stated otherwise To validate the present formulation for buckling and postbuckling of FGM plates under mechanical, thermal, and combined loads, the postbuckling of a midplane-symmetric FGM plate with T-D properties under a uniform temperature rise is considered This problem was also analyzed by Shen [11], which used the asymptotic perturbation method and Reddy’s higher-order shear deformation theory The plate is simply supported and immovable at all edges (Case 2) The postbuckling load–deflection curves of FGM plates with and without an initial imperfection are compared with Shen’s results in Fig It is evident that a good agreement has been achieved in this comparison study Table shows the effects of the volume fraction index k and the aspect ratio a b on the difference of buckling temperature of FGM plates in comparison with Shen’s results [11] It is seen that the plates buckle sooner when their material properties are temperature-dependent, and the buckling temperature increases when k increases and a b decreases, as expected Table shows the effects of temperature field on the buckling loads of FGM plates under uniaxial compression Obviously, an increasing environment temperature lowers the buckling load The effects of material and geometric parameters, as well as temperature field, on the postbuckling behavior of FGM plates are depicted in Figs 2-10 In the case of mechanical stability, a simply supported square FGM plate under uniaxial compression was considered as an example Here, the critical buckling load of perfect plates corresponds to m = n =1, which is the first buckling mode Figure shows the effects of the volume fraction index k on the postbuckling behavior of FGM plates with movable edges and T-ID properties under a uniaxial compressive load As expected, the postbuckling strength of the plates increased with k, i.e., with the volume content of silicon nitride in the FGM plates The effects of in-plane boundary conditions on the postbuckling behavior of FGM plates under a uniaxial compression are illustrated in Fig Two types of in-plane conditions at the edges y = 0, b, referred to as freely movable (FM) and immovable (IM) edges, were considered In this figure, the postbuckling curves for the cases of FM and IM edges were traced by using Eq (25) with b =0 and Eq (34) with DT = 0, respectively As seen, the postbuckling strength of the plates increases when the edges y = 0, b are made immovable and the deflection is sufficiently large In the case of thermal stability, the perfect FGM plates buckled at m= n = for any aspect ratio a b Figure shows the pronounced effect of T-D material properties on the thermal postbuckling behavior of the FGM plates It is evident that the postbuckling load-carrying capacity of the plates decreases drastically when the temperature dependence of material properties is taken into consideration Figure demonstrates the increased thermal postbuckling load-carrying capability of the FGM plates with T-D material properties when k increases Figure shows the effects of the aspect ratio a b on the thermal postbuckling behavior of the FGM plates with T-ID and T-D material properties As can be seen, the postbuckling load-deflection curves become lower when k increases and the temperature dependence of material properties is taken into account 470 800 DT, K 600 400 200 W/h 0.5 1.0 1.5 Fig Comparison of thermal buckling curves for FGM plates with T-D material properties at a b = 1.0, b h = 20, k = and m = ( , l) and 0.05 ( -, k): according to the present theory (lines) and [11] (dots) TABLE Buckling Temperature DT of FGM Plates under a Uniform Temperature Rise (b h = 20 and n = 0.29) ab k=0 k=1 k=2 k=5 1.0 205.6 (206.8)* 313.7 (315.1) 351.1 (352.6) 378.0 (388.8) 2.0 129.1 (129.9) 197.0 (198.1) 220.5 (221.7) 243.0 (244.4) T-ID T-D * 1.0 182.4 (184.6) 265.3 (268.2) 292.0 (295.1) 316.1 (319.5) 2.0 118.9 (120.4) 174.9 (176.9) 193.3 (195.6) 210.4 (212.9) The numbers in parentheses are Shen’s results [11] TABLE Critical Buckling Loads Pxcr (GPa) of FGM Plates in Uniaxial Compression (a b = 1,0 and b h = 20 at y = 0, b) DT , K k=0 k=1 k=2 k=5 1.427 2.015 2.133 2.199 100 1.150 1.738 1.866 1.944 200 0.838 1.443 1.583 1.675 The effect of in-plane boundary condition on the thermal postbuckling behavior of the FGM plates with T-D properties is depicted in Fig In this figure, the postbuckling paths are plotted for two cases: with all edges immovable [governed by Eq (32)] and with the edges x = 0, a freely movable [governed by Eq (34) with Px = 0] As can be observed, the thermal postbuckling load-carrying capacity is considerably reduced when all edges are immovable, and the longer the immovable edges, the lower the postbuckling curves The thermomechanical postbuckling behavior of the FGM plates with T-D material properties are illustrated in Figs and Figure exhibits the decreased postbuckling strength of the FGM plates under the action of a uniaxial compressive load 471 Px, GPa W/h 0.5 1.0 1.5 Fig Postbuckling curves of FGM plates at a b = 1.0, b h = 20, b = 0, and m = ( ( -) under a uniaxial compressive load Px vs W h; k = (1), (2), and (3) ) and 0.1 Px, GPa 3 2 W/h 0.5 1.0 1.5 Fig Effect of in-plane boundary conditions FM (1, 2) and IM (3, 4) on the postbuckling of FGM plates at a b = 1.0, b h = 20, k = 1, and m = (1, 3) and 0.1 (2, 4) in uniaxial compression DT, K 1200 1000 800 600 400 200 W/h 0.5 1.0 1.5 Fig Thermal postbuckling behavior of FGM plates with T-ID (1) and T-D (2) material properties at a b = 1,0, b h = 20, k = 1, and m = ( ) and 0.1 ( -) 472 1000 DT, K 800 600 400 200 W/h 0.5 1.0 1.5 Fig Thermal postbuckling curves of FGM plates with T-D material properties at a b = 1.0, b h = 20, m = ( ) and 0.1 ( -); k = (1), (2), and (3) 1500 DT, K 1000 500 W/h 0.5 1.0 1.5 Fig Thermal postbuckling curves of FGM plates with T-ID (1,3) and T-D (2,4) material properties at b h = 20, k = 2, a b = 1.0 (1, 2) and 2.0 (3, 4); m = ( ) and 0.1 ( -) when the environment temperature is increased Similarly, Fig shows the effect of uniaxial compression on the thermal postbuckling behavior of FGM plates with T-D and T-ID material properties Obviously, the presence of a compressive load and the T-D material properties make the postbuckling curves lower In general, (prestressed) preheated FGM plates exhibit a decreased postbuckling load-carrying capacity when they are subjected to the action of (thermal) compressive loads, as already mentioned Finally, the effect of an initial imperfection on the thermal postbuckling behavior of FGM plates with T-ID and T-D material properties are depicted in Fig 10 This figure demonstrates the increasing postbuckling load-carrying capability of FGM plates when the imperfection size m increases and the deflection exceeds a specific value This behavior holds for FGM plates both with T-ID and T-D material properties Concluding Remarks The paper presents an analytical approach, in conjunction with an iterative procedure, to investigating the buckling and postbuckling behavior of simply supported midplane-symmetric rectangular FGM plates under in-plane compressive, 473 1200 DT, K 1000 800 600 400 200 W/h 0.5 1.0 1.5 Fig Effect of in-plane boundary conditions FM (0, à) (1,2) and IM (3) on the thermal postbuckling behavior of FGM plates with T-D material properties at b h = 20, k = 1, and a b = 1.0 (1,3) and 1.5 (2); m = ( ) and 0.1 ( -) Px, GPa 3 W/h 0.5 1.0 1.5 Fig Effect of temperature rise on the postbuckling behavior of FGM plates at a b = 1.0, b h = 20, k = 2., and m = ( ) and 0.1 ( -); DT = (1), 200 (2), and 300 K (3) 2000 DT, K 1500 1000 500 W/h 0.5 1.0 1.5 Fig Effect of a compressive load Px on the thermal postbuckling behavior of FGM plates with T-ID (1, 2) and T-D (3, 4) material properties at a b = 1.0, b h = 20, k = 1, and m = 0; Px = 0.5 (1, 3) and 1.5 GPa (2, 4) 474 1400 DT, K 1200 1000 800 600 400 200 W/h 0.5 1.0 1.5 Fig 10 Postbuckling curves of FGM plates with T-ID (1-3) and T-D (4-6) material properties at a b = 1.0, b h = 20, and k = 1; m = (1, 4), 0.1 (2, 5), and 0.2 (3, 6) 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