DSpace at VNU: Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads

Composite Structures 92 (2010) 1184–1191 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads Hoang Van Tung a,*, Nguyen Dinh Duc b a b Faculty of Civil Engineering, Hanoi Architectural University, Ha Noi, VietNam College of Technology, Vietnam National University, Ha Noi, VietNam a r t i c l e i n f o Article history: Available online 20 October 2009 Keywords: Nonlinear analysis Functionally graded materials Postbuckling Imperfection a b s t r a c t This paper presents a simple analytical approach to investigate the stability of functionally graded plates under in-plane compressive, thermal and combined 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 Equilibrium and compatibility equations for functionally graded plates are derived by using the classical plate theory taking into account both geometrical nonlinearity in von Karman sense and initial geometrical imperfection The resulting equations are solved by Galerkin procedure to obtain explicit expressions of postbuckling load–deflection curves Stability analysis of a simply supported rectangular functionally graded plate shows the effects of the volume fraction index, plate geometry, in-plane boundary conditions, and imperfection on postbuckling behavior of the plate Ó 2009 Elsevier Ltd All rights reserved Introduction Functionally Graded Materials (FGMs) are microscopically inhomogeneous composites usually made from a mixture of metals and ceramics By gradually varying the volume fraction of constituent materials, their material properties exhibit a smooth and continuous change from one surface to another, thus eliminating interface problems and mitigating thermal stress concentrations By high performance heat resistance capacity, FGMs are now developed for general use as structure components in ultrahigh temperature environments and extremely large thermal gradients such as aircraft, space vehicles, nuclear plants, and other engineering applications Buckling and postbuckling behaviors are one of main interest in design of structural components such as plates, shells and panels for optimal and safe usage Therefore, it is important to study the buckling and postbuckling behaviors of FGM plates under mechanical, thermal and combined thermomechanical loads for accurate and reliable design Some works about the stability of FGM structures relating to present study are introduced in the following Javaheri and Eslami [2–4] and Shariat and Eslami [5] reported mechanical and thermal buckling of rectangular functionally graded plates by using the classical plate theory [2,3] and higher order shear deformation plate theory [4,5] They used energy method to derive governing equations that analytically solved to obtain the closed-form solutions of critical loading The same * Corresponding author E-mail address: htung0105@gmail.com (H.V Tung) 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd All rights reserved doi:10.1016/j.compstruct.2009.10.015 authors and Shariat [6–8] extended their these studies when influences of initial geometrical imperfection on the critical buckling loading are taken into consideration Lanhe [9] used the first order shear deformation theory to derive closed-form relations for buckling temperature difference of simply supported moderately thick rectangular FGM plates Three dimensional thermal buckling analysis of functionally graded composite plates, using finite element method, is reported by Na and Kim [10] The research on thermoelastic stability of FGM cylindrical shells is introduced by Eslami and his co-workers [12–14] and Lanhe et al [15] Except [10], above mentioned works used analytical approach to study buckling of FGM plates and shells Furthermore, by linear buckling analysis effects of prebuckling deformation and postbuckling behavior have not been considered in these works Recently, Darabi et al [16] presented nonlinear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading by analytical approach Some investigations about postbuckling behavior of functionally graded plates are also reported by Liew et al [17,18] using differential quadrature method, Shen [19,20] using perturbation asymptotic method, and Zhao and Liew [21] using the element-free kp-Ritz method The influences of shear deformation, initial imperfection, piezoelectric actuators, and temperature-dependent properties on postbuckling behavior of FGM plates are also taken into consideration in these works This paper presents a simple analytical approach to investigate buckling and postbuckling behaviors of functionally graded plates subjected to in-plane compressive, thermal, and combined loads The motivation of this study results from practical significance of 1185 H.V Tung, N.D Duc / Composite Structures 92 (2010) 1184–1191 relatively simple closed-form expressions of buckling load and postbuckling load–deflection curves in the design Formulation is based on the classical plate theory with both von Karman type of kinematic nonlinearity and initial geometrical imperfection are accounted for By Galerkin procedure, the resulting equations are solved to obtain closed-form expressions of postbuckling equilibrium paths Stability analysis is carried out for a rectangular FGM plate simply supported on all edges and effects of material and geometric parameters, in-plane boundary conditions, and imperfection on the postbuckling behavior are discussed where geometrical nonlinearity in von Karman sense is accounted for and subscript (,) indicates the partial derivative Hooke law for a plate is dened as rx ; ry ị ẳ ẵE=1 m2 ịẵex ; ey ị ỵ mey ; ex ị ỵ mịaDT1; 1ị; rxy ẳ ẵE=21 þ mފcxy The force and moment resultants of a plate are expressed in terms of the stress components through the thickness as Nij ; M ij ị ẳ Consider a rectangular functionally graded plate of length a, width b, and thickness h, referred to the rectangular Cartesian coordinates ðx; y; zÞ, where ðx; yÞ plane coincides with middle surface of the plate and z is the thickness coordinate ðÀh=2 z h=2Þ By applying a simple power law distribution, the volume fractions of metal and ceramic, V m and V c , are obtained as follows [3,4,9,11]:  k 2z ỵ h ; 2h V m zị ẳ À V c ðzÞ ð1Þ where 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 P eff of functionally graded plate, such as the modulus of elasticity E, the coefficient of thermal expansion a, and the coefficient of thermal conduction K, change in the thickness direction z and can be determined by the linear rule of mixture as [3,4,9,11] Peff ẳ P c V c zị ỵ Pm V m ðzÞ ð2Þ where P denotes a temperature-independent material property, and subscripts m and c stand for the metal and ceramic constituents, respectively From Eqs (1) and (2), the effective properties of FGM plate can be written as follows in which Poisson’s ratio m is assumed to be constant  k 2z ỵ h ẵEzị; azị; Kzị ẳ ẵEm ; am ; K m ỵ ẵEcm ; acm ; K cm 2h h=2 rij 1; zịdz; ij ẳ x; y; xy Nx ẳ E1 E2 Um exm ỵ meym ị ỵ kx ỵ mky ị m2 À m2 1Àm Ny ¼ E1 E2 Um ðeym þ mexm Þ þ ðky þ mkx Þ À À m2 À m2 1Àm Nxy ¼ E1 E c ỵ kxy 21 ỵ mị xym ỵ m Mx ẳ E2 E3 Ub exm ỵ meym ị þ ðkx þ mky Þ À À m2 m2 1m My ẳ E2 E3 Ub eym ỵ mexm ị ỵ ky ỵ mkx ị m2 À m2 1Àm M xy ¼ acm ¼ ac À am ; K cm ¼ K c À K m 10ị E2 E c ỵ kxy 21 ỵ mị xym ỵ m where E1 ẳ Em h ỵ Ecm h=k ỵ 1ị; E2 ẳ Ecm h ẵ1=k ỵ 2ị 1=2k ỵ 2ị; E3 ẳ Em h =12 ỵ Ecm h ẵ1=k ỵ 3ị 1=k ỵ 2ị ỵ 1=4k ỵ 4ị;  k # Z h=2 " 2z ỵ h Em ỵ Ecm Um ; Ub ị ẳ 2h h=2 "  k # 2z ỵ h DT1; zịdz am ỵ acm 2h 11ị 4ị Nx;x ỵ Nxy;y ẳ Nxy;x ỵ Ny;y ẳ 12ị M x;xx ỵ 2M xy;xy þ M y;yy þ Nx w;xx þ 2Nxy w;xy þ Ny w;yy ¼ Governing equations In the present study, the classical plate theory is used to obtain the equilibrium and compatibility equations as well as expressions of buckling loads and postbuckling equilibrium paths of FGM plates The strains across the plate thickness at a distance z from the mid-plane are [1] ex ẳ exm ỵ zkx ; ey ẳ eym ỵ zky ; cxy ẳ cxym ỵ 2zkxy ð5Þ where exm and eym are the normal strains, cxym is the shear strain at the middle surface of the plate, and kij are the curvatures In the framework of classical plate theory, the strains at the middle surface and the curvatures are related to the displacement components u; v ; w in the coordinates as [1] exm ¼ u;x þ w2;x =2; eym ¼ v ;y þ w2;y =2; cxym ẳ u;y ỵ v ;x ỵ w;x w;y ; ky ẳ w;yy ; 9ị The nonlinear equilibrium equations of a perfect plate based on the classical plate theory are given by where kx ẳ w;xx ; 8ị Substituting Eqs (3), (5) and (7) into Eq (8) gives the constitutive relations 3ị mzị ẳ m Ecm ẳ Ec Em ; Z Àh=2 Functionally graded plates V c ðzÞ ¼ ð7Þ kxy ¼ Àw;xy ð6Þ If the temperature distributes uniformly in x and y directions and when Eqs (9) and (10) are substituted into Eq (12), the equilibrium equations can be written in terms of deflection variable w and force resultants as Nx;x ỵ Nxy;y ẳ Nxy;x ỵ Ny;y ẳ 13ị Dr w Nx w;xx ỵ 2N xy w;xy ỵ N y w;yy ị ẳ where r2 ẳ @ =@x2 ỵ @ =@y2 , and D¼ E1 E3 À E22 E1 ð1 À m2 Þ ð14Þ For an imperfect plate, let wà ðx; yÞ denotes a known small imperfection This parameter represents a small initial deviation of the plate plane from a flat shape When imperfection is considered, the equilibrium Eq (13) is modified into form as [6–8] 1186 H.V Tung, N.D Duc / Composite Structures 92 (2010) 11841191 N x;x ỵ N xy;y ¼ w ¼ M xx ¼ Nxy ẳ 0; N xy;x ỵ Ny;y ẳ h i Dr4 w Nx w;xx ỵ w;xx ị ỵ 2Nxy w;xy ỵ w;xy ị ỵ Ny w;yy ỵ w;yy ị ¼ w ¼ v ¼ Myy ¼ 0; ð15Þ Considering the first two of Eqs (15), a stress function f may be defined as Nx ¼ f;yy ; Ny ¼ f;xx ; Nxy ¼ Àf;xy ð16Þ Substituting Eq (16) in the third of Eqs (15) leads to h i Dr4 w f;yy w;xx ỵ w;xx ị 2f ;xy w;xy ỵ w;xy ị ỵ f;xx w;yy ỵ w;yy Þ ¼ ð17Þ The Eq (17) includes two dependent unknowns, w and f To obtain a second equation relating these two unknowns, the compatibility equation may be used The geometrical compatibility equation is written as [1] exm;yy ỵ eym;xx cxym;xy ẳ w2;xy w;xx w;yy 18ị For a imperfect plate, the above equation may be modified into form as exm;yy ỵ eym;xx cxym;xy ẳ w2;xy w;xx w;yy ỵ 2w;xy w;xy w;xx w;yy w;yy w;xx ð19Þ From the constitutive relations (9), one can write ðexm ; eym ị ẳ cxym ẵNx ; Ny ị À mðNy ; Nx Þ À E2 ðkx ; ky ị ỵ Um 1; 1ị; E1 ẳ ẵ1 ỵ mịNxy E2 kxy E1 N y ẳ Ny0 on x ẳ 0; a 24ị on y ẳ 0; b where Nx0 ; Ny0 are prebuckling force resultants in directions x and y, respectively, for Case (1) and the first of Case (3), and are fictitious compressive edge loads rendering the edges immovable for Case (2) and the second of Case (3) To solve two Eqs (17) and (21) for unknowns w and f, and with the consideration of the boundary conditions (22)–(24), we assume the following approximate solutions [22–24] w ¼ W sin km x sin ln y f ẳ A1 cos 2km x ỵ A2 cos 2ln y ỵ A3 cos 2km x cos 2ln y 1 ỵ A4 sin km x sin ln y ỵ N x0 y2 ỵ Ny0 x2 2 25ị where km ¼ mp=a; ln ¼ np=b; m; n ¼ 1; 2; are number of half waves in x and y directions, respectively, and W is amplitude of deection Also, Ai i ẳ 4ị are coefcients to be determined Considering the boundary conditions (22)–(24), the imperfections of the plate are assumed as [1,6–8] wà ¼ lh sin km x sin ln y; m; n ¼ 1; 2; ð26Þ where the coefficient l varying between and represents imperfection size By substituting Eqs (25) and (26) into Eq (21), the coefficients Ai are determined as E1 l2n A1 ẳ 20ị Nx ẳ Nx0 32k2m WW ỵ 2lhị; A2 ẳ E1 k2m WW ỵ 2lhị; 32l2n A3 ẳ A4 ẳ 27ị Introduction of Eqs (25) and (26) into Eq (17) and applying Galerkin method for the resulting equation yield Substituting the above equations in Eq (19), with the aid of Eqs (6) and (16), leads to the compatibility equation of an imperfect FGM plate as Dk2m ỵ l2n ị2 W ỵ 2k2m l2n A1 ỵ A2 ị ỵ k2m Nx0 ỵ l2n Ny0 W ỵ lhị ẳ   r4 f E1 w2;xy w;xx w;yy ỵ 2w;xy w;xy À w;xx wÃ;yy À w;yy wÃ;xx ¼ Eq (28), derived for odd values of m; n, is used to determine buckling loads and postbuckling curves of rectangular FGM plates under mechanical, thermal, and combined loads ð21Þ Eqs (17) and (21) are the basic equations used to investigate the stability of functionally graded plates They are nonlinear equations in terms of two 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 Depending on the in-plane behavior at the edges, three cases of boundary conditions, labelled Cases (1), (2) and (3) will be considered [22] Case (1) The edges are simply supported and freely movable (FM) The associated boundary conditions are w ¼ M xx ¼ N xy ¼ 0; w ¼ M yy ¼ Nxy ¼ 0; Nx ¼ Nx0 ; Ny ¼ Ny0 ; on x ¼ 0; a on y ¼ 0; b Nx ¼ Nx0 on x ¼ 0; a w ¼ v ¼ M yy ¼ 0; Ny ¼ N y0 on y ¼ 0; b 4.1 Mechanical stability analysis The simply supported FGM plate with freely movable edges (that is, Case (1)) is assumed to be under in-plane compressive loads Px and Py (in Pascals), uniformly distributed along the edges x ¼ 0; a and y ¼ 0; b, respectively The prebuckling force resultants are [1,2] Nx0 ¼ ÀPx h; N y0 ¼ ÀPy h ð29Þ Introduction of Eqs (27) and (29) into Eq (28) gives  Px ẳ p2 D m2 B2a ỵ n2 2  p2 E1 m4 B4a ỵ n4 W      WW ỵ 2lị ỵ B2h m2 B2a ỵ bn2 W ỵ l 16B2h m2 B2a þ bn2 ð22Þ Case (2) The edges are simply supported and immovable (IM) The associated boundary conditions are w ¼ u ẳ Mxx ẳ 0; 28ị 23ị Case (3) The edges are simply supported Uniaxial edge loads are applied in the direction of the x-coordinate The edges x ¼ 0; a are considered freely movable, the remaining two edges being unloaded and immovable For this case, the boundary conditions are 30ị where Ba ẳ b=a; Bh ẳ b=h; D ¼ D=h ; b ¼ Py =Px E1 ¼ E1 =h; W ẳ W=h; 31ị For a perfect plate, l ¼ 0, Eq (30) leads to equation from which buckling compressive load P xb may be obtained as  Pxb ẳ p2 D m2 B2a ỵ n2 2   B2h m2 B2a ỵ bn2 32ị 1187 H.V Tung, N.D Duc / Composite Structures 92 (2010) 1184–1191 The above equation has been reported by Javaheri and Eslami [2] when they analyze linear buckling of perfect FGM plates under in-plane compressive loadings The critical buckling loads P xcr is obtained for values of m and n that make the preceding expression a minimum In contrast, when l – 0, imperfection sensitivity of the plates may be predicted Specifically, no bifurcation-type buckling occurs, and the plates start to deflect at the onset of compression Eq (30) may be used to trace postbuckling load–deflection curves of FGM plates subjected to in-plane compressive loads 4.2.1 Uniform temperature rise Under mentioned boundary conditions, temperature can be uniformly raised from initial value T i to final one T f and temperature difference DT ¼ T f À T i is a constant The thermal parameter Um can be expressed in terms of the DT from Eq (11) and then introduction of the result into Eq (37) one obtains DT ¼ 4.2 Thermal stability analysis A simply supported FGM plate with immovable edges (that is, Case (2)) under thermal loads is considered The condition expressing the immovability on the edges, u ¼ (on x ¼ 0; a) and v ¼ (on y ¼ 0; b), is fulfilled on the average sense as [20,22] Z b Z a @u dxdy ¼ 0; @x Z a Z 0 b ð33Þ @u E2 Um ẳ f;yy mf;xx ị ỵ w;xx w2;x w;x w;x ỵ @x E1 E1 E1 @v E2 Um ẳ f;xx mf;yy ị ỵ w;yy w2;y w;y w;y ỵ @y E1 E1 E1 Um ỵ Ny0 ẳ 1m ỵ Um ð36Þ 1Àm which is derived by Javaheri and Eslami [3] by solving the membrane form of equilibrium equations and using the method proposed by Meyers and Hyer [25] By substituting Eqs (27) and (35) into Eq (28) we obtain the following expression for thermal parameter   W W ỵ lh b   4E2 m4 B4a ỵ 2mm2 n2 B2a ỵ n4   W ỵ mn1 ỵ mịb m2 B2a ỵ n2 Um ẳ h ỵ p2 E1 m2 ịm4 B4a ỵ n4 ị þ 4mm2 n2 B2a 16ð1 þ mÞb ðm2 B2a ỵ n2 ị E2 ẳ E2 =h 39ị  p2 D1 mị m2 B2a ỵ n2  40ị B2h P This equation has been derived by Javaheri and Eslami [3] when they analyze linear buckling of perfect FGM plates under uniform temperature rise When minimization methods are carried, the critical buckling temperature difference of perfect plates is obtained for m ¼ n ¼ In addition, with this buckling mode Eq (38) may be used to trace postbuckling curves of FGM plates subjected to thermal load under consideration 4.2.2 Nonlinear temperature change across the thickness In this case, the temperature through thickness is governed by the one-dimensional Fourier equation of steady-state heat conduction Eq (35) represents the compressive stresses making the edges immovable and depending on thermal parameter and prebuckling deflection It should be noted that when prebuckling deflection is ignored Eq (35) leads to p2 Dð1 À mị m2 B2a ỵ n2 Em acm ỵ Ecm am Ecm acm ỵ ; kỵ1 2k ỵ 35ị 4E2 l ỵ mk2m W mnp2 m2 ị n E1 l ỵ mk2m WW ỵ 2lhị 81 m2 ị n Nx0 ẳ N y0 ẳ 38ị When imperfection is not taken into consideration, Eq (38) leads to expression from which bifurcation-type buckling temperature difference DT b may be obtained as DT b ẳ 4E2 k ỵ ml2n W mnp2 m2 ị m E1 k2 ỵ ml2n ịWW ỵ 2lhị 81 m2 ị m Um ỵ ỵ P ẳ Em am ỵ 34ị Substituting Eqs (25) and (26) into Eq (34) and then into Eq (33) yield 1m W W ỵl   4E2 m4 B4a þ 2mm2 n2 B2a þ n4   W þ mn1 ỵ mịB2h P m2 B2a ỵ n2 h   i p2 E1 ð3 À m2 Þ m4 B4a þ n4 þ 4mm2 n2 B2a WðW þ 2lÞ þ 161 ỵ mịB2h Pm2 B2a ỵ n2 ị B2h P where @v dydx ¼ @y From Eqs (6) and (9) one can obtain the following relations in which Eq (16) and imperfection have been accounted for Nx0 ¼ À p2 D1 mịm2 B2a ỵ n2 ị ! d dT Kzị ẳ 0; dz dz Tz ẳ h=2ị ẳ T c ; n P k cm =K m Þ r r Knkỵ1 Tzị ẳ T m ỵ DT Pnẳ0 K cm =K m ịn nẳ0 37ị By Eq (37) the postbuckling behavior of rectangular FGM plates under two types of thermal loads will be analyzed 42ị nkỵ1 where r ẳ 2z ỵ hị=2h and DT ẳ T c À T m is defined as the temperature difference between ceramic-rich and metal-rich surfaces of the plate By following the same procedure as the preceding loading case, and assuming the metal surface temperature as reference temperature, yields DT ¼ WW ỵ 2lhị 41ị where T c and T m are temperatures at ceramic-rich and metal-rich surfaces, respectively The solution of Eq (41) can be obtained by means of polynomial series Taking the first seven terms of the series, the solution for temperature distribution across the plate thickness becomes [3,7,9]  i Tz ẳ h=2ị ẳ T m p2 D1 mị m2 B2a ỵ n2  W W ỵl   4 2 4E2 m Ba þ 2mm n Ba þ n4   W þ mn1 ỵ mịB2h H m2 B2a ỵ n2 h   i p2 E1 ð3 À m2 Þ m4 B4a þ n4 þ 4mm2 n2 B2a   WðW þ 2lị ỵ 161 ỵ mịB2h H m2 B2a ỵ n2 B2h H ð43Þ 1188 H.V Tung, N.D Duc / Composite Structures 92 (2010) 1184–1191 where P5 ðÀK cm =K m ịn nẳ0 nkỵ1 Hẳ h Em am nkỵ2 P5 acm þEcm am Ecm acm þ Emðnþ1Þkþ2 þ ðnþ2Þkþ2 Results and discussion i 44ị K cm =K m ịn nẳ0 nkỵ1 It is similar to preceding loading case, when initial imperfection is ignored Eq (43) is reduced to expression from which buckling temperature change may be obtained as Eq (40), provided P is replaced by H Such a result has been reported by Javaheri and Eslami [3] by linear buckling analysis of the perfect FGM plates under nonlinear temperature gradient The imperfection sensitivity of the plates under thermal loads may be predicted from Eqs (38) and (43), that is, postbuckling curves originate from coordinate origin because no bifurcation buckling point exists when l – 4.3 Thermomechanical stability analysis A simply supported plate with movable edges x ¼ 0; a and immovable edges y ¼ 0; b (that is, Case (3)) and subjected to the simultaneous action of a thermal field and an uniaxial compressive loading Px, uniformly distributed along the edges x = 0, a is considered Employing N x0 ¼ ÀP x h, Eq (27) and the second of Eqs (33), (34) in Eq (28) yields  Px ẳ p2 D m2 B2a ỵ n2 2 W 4E n3    W ỵ 2 m2 Ba ỵ mn2 W ỵ l Bh m m2 B2a þ mn2   p2 E1 m4 B4a þ 3n4 n2 P DT   WW ỵ 2lị ỵ 2 m B2a ỵ mn2 16Bh m2 Ba þ mn2 To validate the present formulation in buckling and postbuckling of plates under mechanical, thermal and combined loads, the postbuckling of a homogeneous isotropic plate under uniaxial compression is considered, which was also analyzed by Shen [19] using the perturbation asymptotic method and Reddy’s higher-order shear deformation theory The plate is simply supported on all edges (Case (1)) The postbuckling loaddeection curves of an isotropic plate m ẳ 0:326ị with and without initial imperfection are compared in Fig with Shen’s results It is evident that good agreement is achieved in this comparison study As a second comparison study, postbuckling of a simply supported isotropic plate with all immovable edges (Case (2)) under uniform temperature rise is considered, which was also analyzed by Bhimaraddi and Chandrashekhara [26] using the single mode approach and the parabolic shear deformation theory The postbuckling paths of perfect ΔT/ΔT cr B2h ð45Þ 1.5 Eq (45) is employed to trace postbuckling curves of the FGM plates under combined mechanical and thermal loads Specifically, it is used to determine the dependence of the in-plane compressive edge loads vs total deflection (for given uniform temperature rise) and conversely, the variation of the temperature rise vs total deflection (for given compressive edge load) Obviously, temperature changes can shift Px ðWÞ curves along the Px - axis by an amount DPx ẳ n2 P DT=m2 B2a ỵ mn2 Þ and conversely, DTðWÞ curves can be displaced along the DT- axis an amount m2 B2a ỵ mn2 ịP x =ðn2 PÞ due to the presence of axial compressive load P /P x 2.5 b/a = 1.0, b/h = 10 0.5 0 0.2 0.4 W/h 0.6 0.8 Fig Comparisons of postbuckling curves for isotropic plates under uniform temperature rise x 1.6 Shen [19], μ = Shen [19], μ = 0.1 Present, μ = Present, μ = 0.1 1.4 1.2 1.5 perfect imperfect (μ=0.1) k=0 b/a = 1.0, b/h = 40, β = 0.8 k=1 0.6 isotropic thin plate (ν = 0.326) b/a = 1.0, (m,n) = (1,1) 0.5 isotropic thick plate P (GPa) xcr Ref [26], μ = Ref [26], μ = 0.1 Present, μ = Present, μ = 0.1 2.5 0.4 k=5 0.2 0.5 W/h 1.5 Fig Comparisons of postbuckling curves for isotropic thin plates under uniaxial compression 0 0.5 W/h 1.5 Fig Postbuckling curves of FGM plates under uniaxial compressive load vs k 1189 H.V Tung, N.D Duc / Composite Structures 92 (2010) 1184–1191 and imperfect isotropic plates are compared in Fig with results in Ref [26] As can be observed, a good agreement is obtained in this comparison study To illustrate the present approach, we consider a ceramic–metal functionally graded plate that consist of aluminum and alumina with the following properties [3,5,9] Em ¼ 70 GPa; am ¼ 23:10À6  CÀ1 ; Ec ¼ 380 GPa; ac ¼ 7; 4:10 À6  K m ¼ 204 W=mK À1 C ; K c ¼ 10; W=mK ð46Þ In the case of mechanical stability, a simply supported square FGM plate under uniaxial compression is considered as a example In this case, the critical buckling load of perfect plates corresponds to m ¼ n ¼ 1, which is the first buckling mode Fig shows variation of postbuckling equilibrium paths of a FGM plate with side-to-thickness ratio b=h ¼ 40 under uniaxial compressive load vs three different values of volume fraction in- P (GPa) x 1.4 ΔT (oC) 1000 1: FM, μ=0.0 2: FM, μ=0.1 3: IM, μ=0.0 4: IM, μ=0.1 1.2 b/a = 1.0, b/h = 40 k=1 400 0.4 0.2 0.5 W/h 1.5 W/h 1.5 Fig Postbuckling curves of FGM plates under nonlinear temperature change vs k perfect imperfect (μ = 0.1) 800 b/a = 1.0, b/h = 40 k=0 b/h = 40, k = 1.0 600 2: b/a = 1.5 400 k=5 100 0.5 W/h 1: b/a = 1.0 k=1 200 0.5 ΔT ( C) 1000 perfect imperfect (μ = 0.1) 300 o ΔT (oC) 500 400 k=5 200 Fig Effect of in-plane boundary conditions on postbuckling behavior of FGM plates under uniaxial compression k=0 600 0.6 perfect imperfect (μ = 0.1) 800 b/a = 1.0, b/h = 40, k = 1.0 0.8 dex k (=0, 1, 5) As can be seen, the postbuckling curves become lower as the k increases as expected, and postbuckling curves of imperfect plates are lower than those of perfect plates when deflection is small Effects of in-plane boundary conditions on postbuckling behavior of FGM plates under uniaxial compression are illustrated in Fig Two types of in-plane conditions on edges y ¼ 0; b, referred to as freely movable (FM) and immovable (IM) edges, are considered In Fig 4, postbuckling curves of the FM and IM cases are traced by Eq (30) with b ¼ and Eq (45) with DT ¼ 0, respectively It is shown that postbuckling strength of the plate is increased when the edges y ¼ 0; b are immovable and the deflection is sufficiently large In the case of thermal stability, the perfect FGM plates buckle when m ¼ n ¼ for arbitrary aspect ratio b=a Figs and give postbuckling temperature–deflection curves of a square FGM plate with three various values of k and under two types of thermal loadings As can be seen, postbuckling curves to be lower with increas- 3: b/a = 2.0 200 1.5 Fig Postbuckling curves of FGM plates under uniform temperature rise vs k 0 0.5 W/h 1.5 Fig Postbuckling curves of FGM plates under uniform temperature rise vs b=a 1190 H.V Tung, N.D Duc / Composite Structures 92 (2010) 1184–1191 Px (GPa) 2.5 Px (GPa) perfect imperfect (μ = 0.1) 0.8 b/a = 1.0, b/h = 30, k = 0.5 0.6 1.5 0.4 0.5 1: ΔT = 2: ΔT = 100 (oC) 0.2 o 3: ΔT = 200 ( C) 0 0.5 1: μ = 2: μ = 0.1 3: μ = 0.2 4: μ = 0.3 W/h 1.5 b/a = 1.0, b/h = 40, k = 1.0 0 0.5 W/h 1.5 Fig 10 Postbuckling curves of FGM plates under uniaxial compression vs l Fig Effects of temperature rise on postbuckling behavior of FGM plates under uniaxial compression o ΔT ( C) 350 ΔT (oC) 1000 800 600 perfect imperfect (μ = 0.1) 250 150 100 200 1: P = x 0 3: Px = 0.4 GPa W/h 1.5 b/a = 1.0, b/h = 40, k = 1.0 50 2: Px = 0.2 GPa 0.5 1: μ = 2: μ = 0.1 3: μ = 0.2 4: μ = 0.3 200 b/a = 1.0, b/h = 30, k = 0.5 400 0 300 0.5 W/h 1.5 Fig 11 Postbuckling curves of FGM plates under uniform temperature rise vs l Fig Effects of compressive load on postbuckling behavior of FGM plates under uniform temperature rise ing values of k as above, and postbuckling loading carrying capability of the plate under nonlinear temperature gradient is higher than that of plate under uniform temperature rise Furthermore, postbuckling strength of imperfect plates is higher than that of perfect plates when the deflection is sufficiently large Effects of aspect ratio b=a on thermal postbuckling behavior of FGM plates are depicted in Fig It is seen that the postbuckling strength of the plates under uniform temperature rise is considerably increased when b=a ratio increases Fig shows effects of temperature field on postbuckling behavior of FGM plates under uniaxial compression Conversely, the effects of in-plane compressive load on postbuckling behavior of FGM plates under uniform temperature rise are depicted in Fig It is shown in these Figs that the (prestressed) preheated FGM plates exhibit a decreasing tendency in postbuckling loading carrying capacity when they are subjected to action of (thermal) compressive loads as mentioned Finally, the effects of initial imperfection on postbuckling behavior of FGM plates with all FM edges subjected to uniaxial compressive loads are depicted in Fig 10 It is shown that postbuckling loading capacity of the plates is reduced with increasing values of imperfection size l when the deflection is small However, a inverse trend occurs when the deflection is sufficiently large Similarly, variation of postbuckling curves of FGM plates under uniform temperature rise vs different values of l is plotted in Fig 11 As can be observed, when the deflection exceeds a specific value, the curves become higher when l is increased In other words, initial imperfection makes FGM plates more stable under temperature field Concluding remarks The paper presents a simple analytical approach to investigate buckling and postbuckling behaviors of functionally graded plates under in-plane edge compressive, thermal, and combined loads The formulation is based the classical plate theory with both von Karman nonlinear terms and initial imperfection are incorporated By using Galerkin method, closed-form expressions of postbuck- H.V Tung, N.D Duc / Composite Structures 92 (2010) 1184–1191 ling load–deflection curves of a simply supported FGM plate are determined for all mentioned types of load with and without imperfection From these explicit expressions, closed-form relations of buckling loads of perfect plates, obtained in foregoing works by linear buckling analysis, may be derived as particular cases The results show that postbuckling behavior of FGM plates are greatly influenced by material and geometric parameters, and in-plane boundary conditions Furthermore, it is also shown that initial imperfection has significant effects on postbuckling behavior of FGM plates Acknowledgement The authors would like to express their science 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gradient is higher than that of plate under uniform temperature rise Furthermore, postbuckling strength of imperfect plates is higher than that of. .. curves of FGM plates under uniform temperature rise vs l Fig Effects of compressive load on postbuckling behavior of FGM plates under uniform temperature rise ing values of k as above, and postbuckling... ỵ mn2 To validate the present formulation in buckling and postbuckling of plates under mechanical, thermal and combined loads, the postbuckling of a homogeneous isotropic plate under uniaxial