Thermomechanical postbuckling of thick fgm plates resting on elastic foundations with tangential edge constraints

Vietnam Journal of Mechanics, VAST, Vol. 38, No. 1 (2016), pp. 63 – 79 DOI:10.15625/0866-7136/38/1/7036 THERMOMECHANICAL POSTBUCKLING OF THICK FGM PLATES RESTING ON ELASTIC FOUNDATIONS WITH TANGENTIAL EDGE CONSTRAINTS Hoang Van Tung Hanoi Architectural University, Vietnam ∗E-mail: inter0105@gmail.com Received September 17, 2015 Abstract. This paper investigates the effects of tangential edge constraints and elastic foundations on the buckling and postbuckling behavior of thick FGM rectangular plates resting on elastic foundations and subjected to thermal and thermomechanical loading conditions. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Governing equations are based on the higher order shear deformation plate theory incorporating the von Karman geometrical nonlinearity, initial geometrical imperfection, tangential edge constraints and Pasternak type elastic foundations. Approximate solutions are assumed to satisfy simply supported bound- ary conditions and Galerkin procedure is applied to derive expressions of buckling loads and load-deflection relations. In thermal postbuckling analysis, an iteration algorithm is employed to determine critical buckling temperatures and postbuckling temperature- deflection equilibrium paths. The separate and simultaneous effects of tangential edge restraints, elastic foundations and temperature dependence of material properties on the buckling and postbuckling responses of higher order shear deformable FGM plates are analyzed and discussed. Keywords: Functionally graded materials, tangential edge constraint, temperature depen- dent property, buckling and postbuckling, elastic foundations. 1. INTRODUCTION Functionally graded materials (FGMs) are microscopically composites usually com- posed from a mixture of metal and ceramic constituents and have some advanced fea- tures in comparison with conventional laminated composites. As a result, the stability of FGM structures is an attractive topic for many researchers. Linear buckling of simply supported FGM rectangular plates subjected to compressive and thermal loads has been investigated by Eslami and his co-authors [1–4] and Lanhe [5] using classical, first order and higher order shear deformation theories and an analytical method. Thermal and c© 2016 Vietnam Academy of Science and Technology 64 Hoang Van Tung mechanical linear buckling of FGM plates have also been investigated using numerical methods in works by Zhao et al. [6], Nguyen and his collaborators [7–9] and by Liew and his co-authors [10–12]. Tung and Duc [13, 14] employed Galerkin method on the basis of classical and higher order shear deformation theories to investigate nonlinear stability of simply supported FGM plates subjected to mechanical and thermal loading conditions with and without elastic foundations and temperature independent properties. Nonlin- ear stability and postbuckling of FGM plates have also been addressed in some studies by Woo et al. [15] utilizing an analytical approach, by Shen [16] making use of a two-step perturbation technique and Lee et al. [17] using the element-free kp-Ritz method. Buck- ling and postbuckling of FGM sandwich plates consisting of homogeneous and FGM layers have been analyzed in some works by Zenkour [18], Zenkour and Sobhy [19] em- ploying an analytical method and by Shen and Li [20] and Wang and Shen [21] basing on an semi-analytical approach. In foregoing studies, only two extreme cases of in-plane constraints of edges have been considered. Specifically, the edges of plates are usually assumed to be either unre- strained (free movable edges) or fully restrained (immovable edges). However, in prac- tical situations, the tangential motion of the edges may be partially restrained only. This results from the boundary supports are not completely rigid and they can deform elas- tically. Librescu et al. [22, 23] analyzed the effects of tangential edge constraints on the postbuckling and vibration of laminated flat and curved panels subjected to mechani- cal and thermal loads. These works indicated that, in the static case, the degree of the tangential edge restraint can has great effects on the behavior of plates and shells. In spite of considerable effects of tangential edge constraints and increasing use of FGMs, studies on this subject are comparatively scarce. Recently, the author used an analytical approach based on the classical thin shell theory to analyze the postbuckling behavior of thin FGM cylindrical panels and circular cylindrical shells subjected to mechanical and thermal loads taking the effects of tangential edge constraints into consideration [24, 25] without and with temperature dependent material properties, respectively. It was shown in work [25] that the effects of temperature dependence of material properties on load carrying capacity become more pronounced and deteriorative for partially movable edge FGM cylindrical shells. This paper extends previous works [14, 24] to investigate separate and simultane- ous influences of tangential edge constraints and temperature dependent material prop- erties on the buckling and postbuckling behavior of higher order shear deformable FGM rectangular plates resting on elastic foundations and subjected to thermal and thermo- mechanical loads. The novelty of the present study in comparison with works [14, 24] results from temperature dependence of material properties and varying degree of tan- gential edge constraint. Formulations are based on the higher order shear deformation plate theory taking von Karman geometrical nonlinearity, initial geometrical imperfec- tion, Pasternak type elastic foundations and tangential edge restraints into consideration. Approximate solutions of deflection and stress function are assumed to satisfy simply supported boundary conditions and Galerkin method is employed to determine expres- sions of buckling loads and load-deflection relations. In thermal postbuckling analysis, Thermomechanical postbuckling of thick FGM plates resting on elastic foundations with tangential edge constraints 65 an iteration algorithm is adopted to obtain critical buckling temperatures and postbuck- ling temperature-deflection equilibrium paths. The effects played by the degree of the tangential edge constraints, temperature dependent material properties, stiffness of elas- tic foundations and imperfection on the buckling and postbuckling behavior of thick FGM plates are analyzed and discussed. 2. FGM RECTANGULAR PLATE ON AN ELASTIC FOUNDATION Consider an FGM rectangular plate of plan-form dimensions a and b, and uniform thickness h resting on an elastic foundation. The plate is made from a mixture of ceramics and metals, and is defined in a coordinate system (x, y, z) whose origin is located at the corner on the middle surface of the plate, x and y are in-plane coordinates towards edges a and b, respectively, and z is perpendicular to the middle surface (−h/2 ≤ z ≤ h/2) as shown in Fig. 1. x y z b a k2 k1 h Fig. 1. Geometry and coordinates system of an FGM rectangular plate on an elastic foundation Suppose that the material composition of the plate varies smoothly along the thick- ness in such a way that the bottom surface is metal-rich and the top surface is ceramic-rich by following a simple power law in terms of the volume fractions of the constituents as Vm(z) = ( 2z+ h 2h )N , Vc(z) = 1−Vm(z), (1) where Vm and Vc are the volume fractions of metal and ceramic constituents, respectively, and N ≥ 0 is volume fraction index. Practically, FGMs are most commonly used in high temperature environments, and significant changes in material properties are inherent. Usually, the elasticity modu- lus decreases, and the thermal expansion coefficient increases at elevated temperatures. Therefore, it is essential to account for this temperature dependence for accurate and reliable prediction of the response of thermally loaded FGM structures. It is assumed that the effective properties Pre f f of FGM plates change only in the thickness direction z and can be determined by the linear rule of mixture as Pre f f (z, T) = Prm(T)Vm(z) + Prc(T)Vc(z), (2) 66 Hoang Van Tung where Pr denotes a specific material property assumed to be temperature-dependent in the present study, and subscripts m and c represent the metal and ceramic constituents, respectively. From Eqs. (1) and (2) the effective properties of FGM plates such as the modulus of elasticity E and the coefficient of thermal expansion α can be written in the form E(z, T) = Ec(T) + Emc(T) ( 2z+ h 2h )N , α(z, T) = αc(T) + αmc(T) ( 2z+ h 2h )N , (3) where Emc(T) = Em(T)− Ec(T), αmc(T) = αm(T)− αc(T), (4) and Poisson’s ratio ν is assumed to be constant. In the present study, the FGM plate is fully rested on an elastic foundation and the FGM plate-foundation interaction is repre- sented by Pasternak model as q f = k1w− k2∆w, (5) where ∆ = ∂2/∂x2 + ∂2/∂y2 is Laplace operator, w is the deflection (transverse displace- ment) of the plate; k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of Pasternak model. 3. THEORETICAL FORMULATIONS In the present study, Reddy’s higher order shear deformation plate theory [26] is used to establish governing equations for buckling and postbuckling analysis of thick FGM plates. Based on this theory, normal strains εx, εy, in-plane shear strain γxy and transverse shear deformations γxz,γyz are represented as εxεy γxy  =  ε0xε0y γ0xy + z  k1xk1y k1xy + z3  k3xk3y k3xy  , (γxz γyz ) = ( γ0xz γ0yz ) + z2 ( k2xz k2yz ) , (6) where  ε0xε0y γ0xy  =  u,x + w2,x/2+ w,xw∗,xv,y + w2,y/2+ w∗,yw∗,y u,y + v,x + w,xw,y + w∗,xw,y + w,xw∗,y  ,  k1xk1y k1xy  =  φx,xφy,y φx,y + φy,x  ,  k3xk3y k3xy  = −c1  φx,x + w,xx + w∗,xxφy,y + w,yy + w∗,yy φx,y + φy,x + 2w,xy + 2w∗,xy  , ( γ0xz γ0yz ) = ( φx + w,x + w∗,x φy + w,y + w∗,y ) , ( k2xz k2yz ) = −3c1 ( φx + w,x + w∗,x φy + w,y + w∗,y ) , (7) in which c1 = 4/(3h2) and von Karman nonlinear terms are incorporated. Also, u, v are displacement components along the x, y directions, respectively, and φx, φy are the Thermomechanical postbuckling of thick FGM plates resting on elastic foundations with tangential edge constraints 67 rotations of normal to the midsurface with respect to y and x axes, respectively. Moreover, w∗ is a known function representing initial geometrical imperfection of the plate. Hooke’s law for an FGM plate is defined as σx = E(z, T) 1− ν2 [ εx + νεy − (1+ ν)α(z, T)∆T ] , σy = E(z, T) 1− ν2 [ εy + νεx − (1+ ν)α(z, T)∆T ] ,[ σxy, σxz, σyz ] = E(z, T) 2(1+ ν) [ γxy,γxz,γyz ] , (8) where ∆T is temperature rise from thermal stress-free initial state, and is assumed to be independent of in-plane coordinates x, y. The force and moment resultants of the FGM plate are determined as (Ni, Mi, Pi) = h/2∫ −h/2 σi ( 1, z, z3 ) dz, i = x, y, xy, (Qx,Kx) = h/2∫ −h/2 σxz ( 1, z2 ) dz, ( Qy,Ky ) = h/2∫ −h/2 σyz ( 1, z2 ) dz. (9) Introduction of Eqs. (6), (7) into Eqs. (8) and substitution of the resulting into Eqs. (9) give the constitutive relations as (Nx, Mx, Px) = 1 1− ν2 [ (E1, E2, E4) ( ε0x + νε 0 y ) + (E2, E3, E5) ( k1x + νk 1 y ) + (E4, E5, E7) ( k3x + νk 3 y ) − (1+ ν) (Φ1,Φ2,Φ4) ] ,( Ny, My, Py ) = 1 1− ν2 [ (E1, E2, E4) ( ε0y + νε 0 x ) + (E2, E3, E5) ( k1y + νk 1 x ) + (E4, E5, E7) ( k3y + νk 3 x ) − (1+ ν) (Φ1,Φ2,Φ4) ] ,( Nxy, Mxy, Pxy ) = 1 2 (1+ ν) [ (E1, E2, E4) γ0xy + (E2, E3, E5) k 1 xy + (E4, E5, E7) k 3 xy ] , (Qx,Kx) = 1 2 (1+ ν) [ (E1, E3) γ0xz + (E3, E5) k 2 xz ] , ( Qy,Ky ) = 1 2 (1+ ν) [ (E1, E3) γ0yz + (E3, E5) k 2 yz ] , (10) 68 Hoang Van Tung where (E1, E2, E3, E4, E5, E7) = h/2∫ −h/2 E(z, T) ( 1, z, z2, z3, z4, z6 ) dz, (Φ1,Φ2,Φ4) = h/2∫ −h/2 E(z, T)α(z, T)∆T ( 1, z, z3 ) dz, (11) and specific expressions of temperature dependent coefficients Ei = Ei(T) (i = 1÷ 7) are analogous to those given in the [14] for case of temperature independent properties and are omitted here for sake of brevity. Governing equations of higher order deformable FGM plates on elastic founda- tions have been derived in the [14]. Specifically, nonlinear equilibrium equation has the form c21 (D2D5/D4 − D3)∆3w+ (c1D2/D4 + 1)D6∆2w +(1−c1D5/D4)∆ [ f,yy ( w,xx+w∗,xx )−2 f,xy (w,xy+w∗,xy)+ f,xx (w,yy+w∗,yy)−k1w+k2∆w] −D6/D4 [ f,yy ( w,xx+w∗,xx )−2 f,xy (w,xy+w∗,xy)+ f,xx (w,yy+w∗,yy)−k1w+k2∆w]=0, (12) where D1 = E1E3 − E22 E1 (1− ν2) , D2 = E1E5 − E2E4 E1 (1− ν2) , D3 = E1E7 − E24 E1 (1− ν2) , D4 = D1 − c1D2, D5 = D2 − c1D3, D6 = 12(1+ ν) ( E1 − 6c1E3 + 9c21E5 ) , (13) and strain compatibility equation for an imperfect FGM plate is [14] ∆ f − E1 ( w2,xy − w,xxw,yy + 2w,xyw∗,xy − w,xxw∗,yy − w,yyw∗,xx ) = 0. (14) In Eqs. (12) and (14), f (x, y) is a stress function defined as Nx = f,yy , Ny = f,xx , Nxy = − f,xy . (15) In this study, the FGM plates are assumed to be simply supported at all edges. The associated boundary conditions are [14, 26] w = φy = Mx = Px = 0, Nx = Nx0 at x = 0, a w = φx = My = Py = 0, Ny = Ny0 at y = 0, b (16) In Eqs. (16) Nx0, Ny0 are prebuckling compressive force resultants at freely movable edges where Nxy is zero-valued, and are fictitious compressive edge loads at tangentially restrained edges. For the purpose of the present study, in-plane boundary conditions are assumed to be with varying degrees of tangential edge restraint. The degree of tangential edge restraint considered is bounded by the cases in which the tangential motion of the un- loaded edges of a plate are either unrestrained or completely restrained, respectively, in the in-plane direction perpendicular to the plate edge. For two these cases, the plate Thermomechanical postbuckling of thick FGM plates resting on elastic foundations with tangential edge constraints 69 edges are referred to as movable and immovable edges, respectively. All intermediate cases are referred to herein as partially movable edges and include elastically restrained edge constraints. The average end-shortening displacement ∆1 between edges x = 0 and x = a is related to the corresponding fictitious compressive edge load Nx0 by ∆1s1 = Nx0, (17) where s1 is the average tangential stiffness in the x direction on each opposite edge. Sim- ilarly, for the edges y = 0 and y = b, relation is expressed as ∆2s2 = Ny0, (18) where s2 is the average tangential stiffness in the y direction on each opposite edge. The expressions for the average end-shortening displacements are defined as [22, 23] ∆1 = − 1ab a∫ 0 b∫ 0 ∂u ∂x dydx, ∆2 = − 1ab a∫ 0 b∫ 0 ∂v ∂y dydx. (19) Eqs. (17) and (18) indicate that values of ∆1 = 0 and ∆2 = 0 correspond to immov- able edges at x = 0, a and y = 0, b, respectively. These conditions are satisfied by selecting s1 → ∞ and s2 → ∞, respectively. In addition, values of s1 = 0 and s2 = 0 correspond to movable edges at x = 0, a and y = 0, b, respectively. For these movable edge conditions, fictitious compressive edge loads are zero-valued, i.e. Nx0 = 0 and Ny0 = 0. To satisfy boundary conditions (16), the approximate solutions are assumed as [14] (w,w∗) = (W, µh) sin βmx sin δny, (20) f = A1 cos 2βmx+ A2 cos 2δny+ A3 sin βmx sin δny+ 1 2 Nx0y2 + 1 2 Ny0x2, (21) φx = B1 cos βmx sin δny, φy = B2 sin βmx cos δny, (22) where βm = mpi/a, δn = npi/b, W is amplitude of the deflection and µ is imperfection parameter. The coefficients Ai (i = 1÷ 3) are determined by substitution of Eqs. (20) and (21) into Eq. (14) as A1 = E1δ2n 32β2m W (W + 2µh) , A2 = E1β2m 32δ2n W (W + 2µh) , A3 = 0. (23) Similarly, the coefficients B1, B2 are obtained as procedure described in the [14] as B1 = a12a23 − a22a13 a212 − a11a22 W, B2 = a12a13 − a11a23 a212 − a11a22 W, (24) where (a11, a22, a12) = ( c21D3 + D1 − 2c1D2 ) ( β2m, δ 2 n, νβmδn ) + 1− ν 2 ( c21D3 + D1 − 2c1D2 ) ( δ2n, β 2 m, βmδn ) + D6 (1, 1, 0) , (a13, a23) = c1D5 ( β3m + βmδ 2 n, δ 3 n + δnβ 2 m )− D6 (βm, δn) . (25) 70 Hoang Van Tung Subsequently, Eqs. (20) and (21) are substituted into equilibrium equation (12) and applying Galerkin method for the resulting equation as procedure developed in the [14] yield{ − (D¯2D¯5 − D¯3D¯4) 9D¯6B6h mnpi8 ( m2B2a + n 2)3 + mnpi6 16B4h ( 4 3 D¯2 + D¯4 ) ( m2B2a + n 2)2 + mnpi2D¯1B2a 16B4h ξ¯mn [ B2aK1 + pi 2 (m2B2a + n2)K2] } W + E¯1mnpi6 256B4h ( m4B4a + n 4 ) ξ¯mnW ( W + µ ) ( W + 2µ ) + mnpi4 16B2h ξ¯mn ( m2B2a N¯x0 + n 2N¯y0 ) ( W + µ ) = 0, (26) where m, n are odd numbers, and E¯i = Ei/hi (i = 1÷ 7),W =W/h, Ba = b/a, Bh = b/h, D1 = D1/h3, D2 = D2/h5, D3 = D3/h7, D4 = D4/h3, D5 = D5/h5, D6 = D6/h, N¯x0 = Nx0/h, N¯y0 = Ny0/h,K1 = k1a4 D1 ,K2 = k2a2 D1 , ξ¯mn = (3D¯4 − 4D¯5)pi2 3D¯6B2h ( m2B2a + n 2)+ 1. (27) In what follows, the fictitious compressive edge loads Nx0, Ny0 for the FGM plates under the tangential edge constraints will be specified. From Eqs. (6), (7) and (10) one can obtain the following relations in which Eq. (15) and imperfection have been included ∂u ∂x = 1 E1 ( f,yy − ν f,xx )− E2 E1 φx,x + c1E4 E1 (φx,x + w,xx)− 12w 2 ,x − w,xw∗,x + Φ1 E1 , ∂v ∂y = 1 E1 ( f,xx − ν f,yy )− E2 E1 φy,y + c1E4 E1 ( φy,y + w,yy )− 1 2 w2,y − w,yw∗,y + Φ1 E1 . (28) Introduction of Eqs. (20), (21) and (22) into Eqs. (28) and then substitution of the resulting equations into Eqs. (19) yield the following expressions ∆1 = 1 E1 ( νNy0 − Nx0 )− 4E2B1βm mnpi2E1 + 4c1E4βm mnpi2E1 (B1 + βmW) + β2m 8 W (W + 2µh)− Φ1 E1 , ∆2 = 1 E1 ( νNx0 − Ny0 )− 4E2B2δn mnpi2E1 + 4c1E4δn mnpi2E1 (B2 + δnW) + δ2n 8 W (W + 2µh)− Φ1 E1 . (29) Introduction of Eqs. (29) into Eqs. (17) and (18) and solving obtained equations, the fictitious compressive edge loads can be determined as N¯x0 = e11W + e12W ( W + 2µ )− e13Φ1/h, (30) N¯y0 = e21W + e22W ( W + 2µ )− e23Φ1/h, (31) Thermomechanical postbuckling of thick FGM plates resting on elastic foundations with tangential edge constraints 71 where e11 = − 4eE¯2mnpiBh [s¯1 (E¯1 + s¯2)mBaB¯1 + νs¯1s¯2nB¯2] + 16eE¯4 3mnpiB2h [ s¯1 (E¯1 + s¯2) ( mBaBhB¯1 + pim2B2a ) + νs¯1s¯2 ( nBhB¯2 + pin2 )] , e12 = pi2eE¯1 8B2h [ s¯1 (E¯1 + s¯2)m2B2a + νn 2s¯1s¯2 ] , e13 = e [νs¯1s¯2 + s¯1 (E¯1 + s¯2)] , (32) e21 = − 4eE¯2mnpiB2h [s¯2 (E¯1 + s¯1) nBhB¯2 + νs¯1s¯2mBaBhB¯1] + 16eE¯4 3pimnB2h [ s¯2 (E¯1 + s¯1) ( nBhB¯2 + pin2 ) + νs¯1s¯2 ( mBaBhB¯1 + pim2B2a )] , e22 = pi2eE¯1 8B2h [ νs¯1s¯2m2B2a + s¯2 (E¯1 + s¯1) n 2] , e23 = e [νs¯1s¯2 + s¯2 (E¯1 + s¯1)] , (33) in which s¯1 = s1/h, s¯2 = s2/h, e = 1 (E¯1 + s¯1) (E¯1 + s¯2)− ν2s¯1s¯2 , (34) and B1, B2 are analogous to those given in the [14]. In this study, the FGM plate is entirely exposed to thermal environments uniformly raised from thermal stress free initial state T0 to value T and temperature change ∆T = T − T0 is considered to be independent of thickness variable z. The thermal expression Φ1 is obtained from Eqs. (11) as Φ1/h = H(T)∆T, H(T) = Ec(T)αc(T) + Ec(T)αmc(T) + Emc(T)αc(T) N + 1 + Emc(T)αmc(T) 2N + 1 . (35) 4. STABILITY ANALYSIS 4.1. FGM plate under uniform temperature rise Introduction of Eqs. (35) into Eqs. (30), (31) and then substitution of the results into Eq. (26) give the following relation ∆T = 16B2h mnpi4ξ¯mn (m2B2ae13 + n2e23)H(T) [ b11 W W + µ + b21W + b31W ( W + 2µ )] , (36) where b11 = − (D¯2D¯5 − D¯3D¯4)9D¯6B6h mnpi8 ( m2B2a + n 2)3 + mnpi6 16B4h ( 4 3 D¯2 + D¯4 ) ( m2B2a + n 2)2 + mnpi2D¯1B2a 16B4h ξ¯mn [ B2aK1 + pi 2 (m2B2a + n2)K2] , 72 Hoang Van Tung b21 = mnpi4 16B2h ξ¯mn ( m2B2ae11 + n 2e21 ) , b31 = E¯1mnpi6 256B4h ( m4B4a + n 4 ) ξ¯mn + mnpi4 16B2h ξ¯mn ( m2B2ae12 + n 2e22 ) . (37) It is evident from Eq. (36) that bifurcation type buckling behavior can occur for geo- metrically perfect FGM plates (i.e. µ = 0) and buckling temperature change of thick FGM rectangular plates accounting for effects of elastic foundations and tangential restraints of edges can be predicted as ∆Tb = 16B2hb11 mnpi4ξ¯mn (m2B2ae13 + n2e23)H(T) . (38) Eqs. (36) and (38) are explicit expressions of temperature-deflection relation and buckling temperature change for FGM plates in case of material properties are temper- ature independent. In contrast, as temperature dependence of material properties is in- cluded, Eqs. (36) and (38) are implicit expressions and an iteration algorithm must be uti- lized to obtain critical buckling temperatures and postbuckling temperature-deflection curves. Detailed procedure of iteration process is similar as that suggested in the work [25] and is omitted here for sake of brevity. The error tolerance of iteration in the present study is 0.001. 4.2. FGM plate under uniform uniaxial compression in a thermal environment Consider a thick FGM plate resting on an elastic foundation and exposed to ther- mal environment. Simultaneously, the plate is subjected to uniaxial compressive load F uniformly distributed on two edges x = 0, a assumed to be freely movable, whereas two unloaded edges y = 0, b is tangentially restrained. In this case, N¯x0 = −F [14, 24] and N¯y0 is determined by following the same procedure described in the previous section as N¯y0 = e31N¯x0 + e32W + e33W ( W + 2µ )− e34H(T)∆T, (39) where e34 = s¯2 E¯1 + s¯2 , e31 = νe34, e33 = n2pi2E¯1 8B2h e34, e32 = e34 [ 16E¯4 3mpiBh ( B¯2 + npi Bh ) − 4E¯2B¯2 mpiBh ] . (40) Substitution of N¯x0, N¯y0 into Eq. (26) leads to the following expression F = 16B2h mnpi4ξ¯mn (m2B2a + n2e31) [ b12 W W + µ + b22W + b32W ( W + 2µ )− b42∆T ] , (41) Thermomechanical postbuckling of thick FGM plates resting on elastic foundations with tangential edge constraints 73 where b12 = − (D¯2D¯5 − D¯3D¯4)9D¯6B6h mnpi8 ( m2B2a + n 2)3 + mnpi6 16B4h ( 4 3 D¯2 + D¯4 ) ( m2B2a + n 2)2 + mnpi2D¯1B2a 16B4h ξ¯mn [ B2aK1 + pi 2 (m2B2a + n2)K2] , b22 = mnpi416B2h ξ¯mnn2e32, b32 = E¯1mnpi6 256B4h ( m4B4a + n 4 ) ξ¯mn + mnpi4 16B2h ξ¯mnn2e33, b42 = mnpi4 16B2h ξ¯mnn2e34H(T), (42) It is observed from Eq. (41) that geometrically perfect FGM plates exhibit a bi- furcation type buckling behavior with corresponding buckling compressive load is pre- dicted as Fb = 16B2h mnpi4ξ¯mn (m2B2a + n2e31) (b12 − b42∆T) . (43) To measure the degree of edge constraint in a more convenient way, alternate tan- gential stiffness parameters λ1 and λ2 are introduced such that λ1 = 0 and λ1 = 1 corre- spond to movable and immovable edges at x = 0 and a, respectively. Similarly, λ2 = 0 and λ2 = 1 correspond to movable and immovable edges at y = 0 and b, respectively. Partially restrained edges at x = 0, a and y = 0, b are defined by 0 < λ1 < 1 and 0 < λ2 < 1, respectively. In the present study, these alternate tangential stiffness param- eters are defined by λ1 = s¯1 E¯1(T0) + s¯1 , λ2 = s¯2 E¯1(T0) + s¯2 , (44) in which E¯1(T0) is value of E¯1 calculated at room temperature T0. 5. RESULTS AND DISCUSSION Towards the major purpose of the present study, this section presents numerical re- sults for square plates (a = b) made of functionally graded materials and for deformation modes with half wave numbers m = n = 1. As an example for verification of the present method, a simply supported perfect FGM rectangular plate without foundation interaction and exposed to uniform tempera- ture rise is considered. The plate is immovable at all edges (i.e. λ1 = λ2 = 1) and made of Aluminum (Al) and Alumina (Al2O3) with temperature independent material properties are Em = 70 GPa, αm = 23× 10−6(◦C−1) for Al and Ec = 380 GPa, αc = 7.4× 10−6(◦C−1) for Al2O3, whereas ν = 0.3 for both constituents. Critical buckling temperature change ∆Tcr for Al/Al2O3 FGM plates under uniform temperature rise is calculated by Eq. (38) and presented in Tab. 1 in comparison with results of Javaheri and Eslami [3] using adja- cent equilibrium criterion in linear buckling analysis and results obtained by Loc et al. [9] making use of an isogeometric finite element formulation for thermal buckling analysis. In this table N∗ is volume fraction index for case of Vc(z) = (z/h+ 1/2)N ∗ . As can be seen, an excellent agreement is achieved in this comparison. The remainder of this section presents numerical results for FGM rectangular plates composed of silicon nitride (Si3N4) and stainless steel (SUS304). The material properties 74 Hoang Van Tung Table 1. Critical buckling temperature difference ∆Tcr of perfect Al/Al2O3 FGM square plates under uniform temperature rise b/h Source Power index N∗ 0 0.5 1 4 5 10 10 Ref. [3] 1617.484 - 757.891 - 678.926 692.519 Ref. [9] 1618.7468 923.1991 758.4268 670.4594 679.3379 692.7225 Present 1618.6819 923.1617 758.3956 670.4320 679.3104 692.6948 20 Ref. [3] 421.516 - 196.257 - 178.528 183.141 Present 421.535 239.2399 196.265 175.574 178.535 183.144 40 Ref. [3] 106.492 - 49.500 - 45.213 46.455 Present 106.494 60.363 49.502 44.422 45.214 46.455 Table 2. Temperature-dependent thermo-elastic coefficients for silicon nitride and stainless steel (Reddy and Chin [27]) Materials Properties P0 P−1 P1 P2 P3 Silicon nitride E (Pa) 348.43e+9 0 -3.070e-4 2.160e-7 -8.946e-11 α (1/K) 5.8723e-6 0 9.095e-4 0 0 Stainless steel E (Pa) 201.04e+9 0 3.079e-4 -6.534e-7 0 α (1/K) 12.330e-6 0 8.086e-4 0 0 Pr, such as elasticity modulus E and thermal expansion coefficient α can be expressed as a nonlinear function of temperature [28] Pr = P0 ( P−1T−1 + 1+ P1T + P2T2 + P3T3 ) , (45) in which T = T0 + ∆T and T0 = 300 K (room temperature), P0, P−1, P1, P2 and P3 are the coefficients of temperature T (K) and are unique to the constituent materials. Specific values of these coefficients for E and α of silicon nitride and stainless steel are given by Reddy and Chin [27] and are listed in Tab. 2. Poisson’s ratio is assumed to be a constant ν = 0.3. In addition, temperature-dependent and temperature-independent material properties will be written as T-D and T-ID, respectively, for sake of brevity. The T-ID are material properties calculated at room temperature T0 = 300 K. Tab. 3 indicates that both critical buckling temperature and difference between buckling temperatures in two cases T-D and T-ID are increased as FGM plates become thicker and/or ceramic rich. Tab. 4 shows the effects of the stiffness parameters of elastic foundation and the degree of edge constraint on the critical buckling temperatures of perfect FGM plates under uniform temperature rise. As can be seen, the critical buckling temperatures are enhanced as the stiffness parameters K1,K2 of foundation are increased and the degree of tangential edge constraint λ1,λ2 are reduced. Moreover, the effects of temperature de- pendent material properties on critical buckling temperatures become more pronounced Thermomechanical postbuckling of thick FGM plates resting on elastic foundations with tangential edge constraints 75 Table 3. Effects of power index N and b/h ratio on the critical buckling temperatures Tcr = T0 + ∆Tcr (K) for immovable FGM plates (a/b = 1, K1 = K2 = 0, λ1 = λ2 = 1, T0 = 300 K) N b/h 10 15 20 25 30 40 0 866 a (1082 b) 598 (658) 481 (504) 421 (431) 386 (391) 350 (351) 0.5 964 (1263) 655 (741) 518 (551) 446 (461) 405 (412) 361 (363) 1 1005 (1362) 683 (786) 537 (576) 459 (478) 414 (424) 367 (370) 2 1049 (1475) 713 (837) 557 (605) 474 (496) 425 (437) 373 (377) 5 1111 (1627) 754 (907) 585 (645) 494 (522) 440 (454) 382 (387) 10 1157 (1727) 780 (952) 603 (671) 507 (538) 419 (466) 388 (394) 100 1234 (1879) 822 (1023) 630 (711) 526 (564) 464 (484) 397 (404) a T-D, b T-ID and deteriorative for FGM plates of which boundary edges are partially restrained in tangential direction and/or are supported by stiffer elastic foundations, i.e. with higher values of parameters K1,K2, especially Pasternak type foundations. As degree of tangen- tial constraint is increased, i.e. higher values of parameters λ1,λ2, difference between critical buckling temperatures for T-D and T-ID cases is smaller. These are graphically illustrated in Fig. 2. Table 4. Effects of elastic foundations and degree of edge constraints on critical buckling temperatures of FGM plates Tcr = T0 + ∆Tcr (K) (a/b = 1, b/h = 20, N = 2, T0 = 300 K) K1, K2 λ1,λ2 0.4, 0.4 0.5, 0.5 0.6, 0.6 0.8, 0.8 1, 1 0, 0 900 a (1260 b) 805 (1042) 733 (896) 629 (714) 557 (605) 50, 0 951 (1385) 852 (1138) 775 (974) 663 (768) 586 (645) 50, 10 1118 (1877) 1009 (1519) 921 (1280) 788 (981) 691 (802) 50, 20 1244 (2370) 1136 (1900) 1043 (1586) 897 (1194) 786 (959) 100, 10 1153 (2002) 1044 (1615) 954 (1357) 817 (1035) 716 (841) a T-D, b T-ID The effects of tangential edge constraints and elastic foundations on the thermal postbuckling behavior of geometrically perfect FGM plates are analyzed in Figs. 3 and 4. Fig. 3 shows the effects of tangential constraint of edges on the thermal postbuckling of FGM plates without foundation interaction, i.e. K1 = K2 = 0, in T-D case. As can be observed, increase in the degree of edge constraint has pronounced and deteriorative 76 Hoang Van Tung influences on both critical buckling temperature and postbuckling loading capacity of thermally loaded FGM plates. 16 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 500 1000 1500 2000 2500 3000  1 =  2 T-D T-ID T cr (K) N = 2, a/b = 1, b/h = 20,  = 0, T 0 = 300 K 1: K 1 = 0, K 2 = 0 2: K 1 = 50, K 2 = 10 2 12 1 Fig. 2. Effects of tangential edge constraints and elastic foundations on the buckling temperatures of perfect FGM plates. 0 0.5 1 1.5 200 300 400 500 600 700 800 900 1000 1100 1200 W/h 4:  1 =  2 = 1.0 3:  1 =  2 = 0.7 2:  1 =  2 = 0.5 1:  1 =  2 = 0.3 4 3 2 1 N = 2, a/b = 1, b/h = 20,  = 0 T (K) K 1 = 0, K 2 = 0, T 0 = 300 K, T-D Fig. 3. Effects of edge constraints on the thermal postbuckling of FGM plates without elastic foundations and T-D properties. The simultaneous influences of tangential edge restraint, elastic foundations and temperature dependent material properties on the thermal postbuckling of perfect FGM plates are depicted in Fig. 4. 0 0.5 1 1.5 200 400 600 800 1000 1200 1400 1600 1800 2000 W/h T-D T-ID 1 2 T (K) N = 2, a/b = 1, b/h = 20,  = 0, T 0 = 300 K 1: K 1 = 0, K 2 = 0,  1 = 1.0,  2 = 1.0 2: K 1 = 100, K 2 = 10,  1 = 0.7,  2 = 0.7 Fig. 4. Combined effects of edge constraints, elastic foundations and T-D properties on the thermal postbuckling of FGM plates. 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 6 W/h  = 0  = 0.01 F (GPa) K 1 = 50, K 2 = 10, T 0 = 300 K, T = 0 N = 2, a/b = 1, b/h = 20,  1 = 0 1:  2 = 0 2:  2 = 0.5 3:  2 = 1.0 2 1 3 Fig. 5. Effects of edge constraint at edges 0,y b on the postbuckling of FGM plates under uniform uniaxial compression Fig. 2. Effects of tangential edge constraints and elastic foundations on the buckling tem- peratures of perfect FGM plates 16 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 500 1000 1500 2000 2500 3000  1 =  2 T-D T-ID T cr (K) N = 2, a/b 1, b h 20,  = 0, T 0 = 300 K 1: K 1 = 0, K 2 = 0 2: K 1 = 50, 2 = 10 2 12 1 Fig. 2. Effects of tange tial edg constraints and elastic foundations on the buckling temperatures of perfect FGM plates. 0 0.5 1 1.5 200 300 400 500 600 700 800 900 1000 1100 1200 W/h 4:  1 =  2 = 1.0 3:  1 =  2 = 0.7 2:  1 =  2 = 0.5 1:  1 =  2 = 0.3 4 3 2 1 N = 2, a/b 1, b/h 20,  0 T (K) K 1 = 0, 2 , T 0 300 K, T-D Fig. 3. Effects of edge constraints on the thermal postbuckling of FGM plates without elastic foundations and T-D properties. The simultaneous influences of tangential edge restraint, elastic foundations and temperature dependent material properties on the thermal postbuckling of perfect FGM plates are depicted in Fig. 4. 0 0.5 1 1.5 200 400 600 800 1000 1200 1400 1600 1800 2000 W/h T-D T-ID 1 2 T (K) N = 2, a/b = 1, b/h = 20,  = 0, T 0 = 300 K 1: K 1 = 0, K 2 = 0,  1 = 1.0,  2 = 1.0 2: K 1 = 100, K 2 = 10,  1 = 0.7,  2 = 0.7 Fig. 4. Combined effect of edge constraint , elastic foundations and T-D pr perties on the thermal postbuckling of FGM plates. 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 6 W/h  = 0  = 0.01 F (GPa) K 1 = 50, K 2 = 10, T 0 = 300 K, T = 0 N = 2, a/b = 1, b/h = 20,  1 = 0 1:  2 = 0 2:  2 = 0.5 3:  2 = 1.0 2 1 3 Fig. 5. Effects of edge constraint at edges 0,y b on the postbuckling of FGM plates under uniform uniaxial compression Fig. 3. Effects of edge constraints o the ther- mal postbuckling of FGM plates without elas- tic foundatio s and T-D properties The simultaneous influences of tangential edge restraint, elastic foundations and temperature ependent aterial properties on the thermal postbuckling of perfect FGM plates are depicted in Fig. 4. 0 0.5 1 1.5 200 400 600 800 1000 1200 1400 1600 1800 2000 W/h T-D T-ID 1 2 T (K) N = 2, a/b = 1, b/h = 20,  = 0, T 0 = 300 K 1: K 1 = 0, K 2 = 0,  1 = 1.0,  2 = 1.0 2: K 1 = 100, K 2 = 10,  1 = 0.7,  2 = 0.7 Fig. 4. Combined effects of edge constraints, elastic foundations and T-D properties on the thermal p stbuckling of FGM plates 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 6 W/h  = 0  = 0.01 F (GPa) K 1 = 50, K 2 = 10, T 0 = 300 K, T = 0 N = 2, a/b = 1, b/h = 20,  1 = 0 1:  2 = 0 2:  2 = 0. 3:  2 = 1.0 2 1 3 Fig. 5. Effects of edge constraint at edges y = 0, b on the postbuckling of FGM plates under uniform uniaxial compr ssi As shown, the di ference be ween load-deflection curves in T-D and T-ID cases is smaller for immovable edge FGM plates without the foundation interaction. In con- trast, there is the existence of a sharp difference between postbuckling equilibrium paths Thermomechanical postbuckling of thick FGM plates resting on elastic foundations with tangential edge constraints 77 of partially restrained edge FGM plates resting on elastic foundations for two T-D and T-ID cases. Fig. 5 illustrates the effects of degree of tangential constraint at unloaded edges y = 0, b on the mechanical postbuckling behavior of FGM plates resting on an elas- tic foundation and subjected to uniform uniaxial compression at room temperature, i.e. ∆T = 0. 0 0.2 0.4 0.6 0.8 1 1.2 -1 0 1 2 3 4 5 W/h  = 0  = 0.01 F (GPa) 1 2 3 N = 2, a/b = 1, b/h = 20,  1 = 0 1:  2 = 0 2:  2 = 0.5 3:  2 = 1.0 K 1 = 50, K 2 = 10, T 0 = 300 K, T = 300 K Fig. 6. Effects of constraint of edges y = 0, b on the thermomechanical postbuckling Finally, Fig. 6 plotted as a counterpart of Fig. 5 for case of ∆T = 300 K consid- ers the effects of λ2 parameter on the thermomechanical postbuckling of mechanically compressed FGM plates resting on an elastic foundation and exposed to a thermal envi- ronment. Obviously, critical buckling compressive loads and postbuckling strength are remarkably decreased as unloaded edges y = 0, b are more rigorously restrained, espe- cially at elevated temperature. 6. CONCLUDING REMARKS The thermal and thermomechanical postbuckling behavior of thick FGM plates resting on elastic foundations with tangentially restrained edges have been analyzed. The analysis reveals that the tangential constraints of boundary edges has extremely sensitive influences on the buckling and postbuckling of FGM plates. Specifically, critical buckling loads and postbuckling load capacity of FGM plates are decreased due to the rigorous constraint in tangential motion of edges. The results also shows that deteriorative effects of temperature dependent material properties on the thermal buckling and postbuckling behavior of thick FGM plates are more pronounced as FGM plates are ceramic-rich, rested on stiffer foundations and/or with lower degree of tangential edge constraint. 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