Postbuckling of thick fgm cylindrical panels with tangential edge constraints and temperature dependent properties

Vietnam Journal of Mechanics, VAST, Vol. 38, No. 2 (2016), pp. 123 – 140 DOI:10.15625/0866-7136/38/2/7066 POSTBUCKLING OF THICK FGM CYLINDRICAL PANELS WITH TANGENTIAL EDGE CONSTRAINTS AND TEMPERATURE DEPENDENT PROPERTIES Hoang Van Tung Hanoi Architectural University, Vietnam E-mail: inter0105@gmail.com Received September 21, 2015 Abstract. This paper investigates postbuckling behavior of thick FGM cylindrical panels resting on elastic foundations and subjected to thermal, mechanical and thermomechani- cal 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 higher order shear deformation shell theory incorporating von Karman-Donnell geometrical nonlinear- ity, initial geometrical imperfection, tangential edge constraints and Pasternak type elas- tic 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 cylindrical panels are analyzed and discussed. Keywords: Functionally graded materials, tangential edge constraint, cylindrical panel, buckling and postbuckling, elastic foundations. 1. INTRODUCTION Structural elements in form of cylindrical panels are widely used as major por- tions in the construction of aircraft, missile and aerospace vehicle. Cylindrical panels also find their important applications in building construction, underground structures and shipbuilding industry. Since these panels are frequently subjected to severe loading conditions, their static and dynamic responses are governing problems for accurate pre- dictions and safe design. The appearance of functionally graded material (FGM) with advanced and novel features necessitates more studies on the behavior of FGM struc- tures in general, and the stability of FGM cylindrical panels in particular. As a result, c© 2016 Vietnam Academy of Science and Technology 124 Hoang Van Tung buckling and postbuckling of FGM cylindrical panels are attractive topics for many re- searchers. Nonlinear response and postbuckling behavior of thin and moderately thick FGM cylindrical panels subjected to mechanical and thermomechanical loads have been investigated by Woo et al. [1], Duc and Tung [2–4] using analytical approaches. Basing on asymptotic solutions and an iteration algorithm, Shen and his co-workers have ana- lyzed nonlinear bending and postbuckling of FGM cylindrical panels under mechanical, thermal and electric loadings [5–7]. Geometrically nonlinear response of FGM cylindrical panels under external pressure and axial compression has been addressed by Liew and his collaborators [8-10] using meshless methods. Recently, Shen and Wang [8–10] using meshless methods. Recently, Shen and Wang [11, 12] studied the effects of elastic foun- dations on the nonlinear bending and postbuckling of FGM cylindrical panels utilizing a semi-analytical approach. Effects of stiffeners on the nonlinear static and dynamic re- sponses of thin FGM cylindrical and doubly curved panels without elastic foundations have been analyzed by Bich et al. [13,14] on the basis of thin shell theory. These works are then extended by Duc and Quan [15, 16] for case of thin and curved FGM panels resting on elastic foundations and exposed to thermal environments. In foregoing studies, only two extreme cases of in-plane constraints of edges have been considered. Specifically, the edges of panels are usually assumed to be either unre- strained (movable edges) or fully restrained (immovable edges). However, in practical situations, the tangential motion of the edges may be partially restrained only. As men- tioned by Librescu et al. [17, 18], tangential edge constraints have great and important effects on the nonlinear behavior of laminated composite plate and shells. Recently, Tung used an analytical approach based on the classical shell theory to analyze the postbuck- ling behavior of thin FGM cylindrical panels and circular cylindrical shells subjected to mechanical and thermal loads taking the effects of tangential edge constraints into con- sideration [19, 20] without and with temperature dependent material properties, respec- tively. This paper extends previous works [4,19] to investigate the postbuckling behavior of higher order shear deformable FGM cylindrical panels resting on elastic foundations and subjected to mechanical, thermal and thermomechanical loads. New contributions of the present work are that the separate and combined effects of various degrees of tangential edge constraints and temperature dependence of material properties on the postbuckling behavior of thick FGM cylindrical panels are taken into consideration. 2. FGM CYLINDRICAL PANEL ON AN ELASTICAL FOUNDATION Consider an FGM cylindrical panel with radius of curvature R, thickness h, axial length a and arc length b resting on a Pasternak type elastic foundation as shown in Fig. 1. The panel is made from a mixture of ceramics and metals, and is defined in a coor- dinate system (x, y, z) whose origin is located at the corner on the middle surface of the shell panel, x and y are in the axial and circumferential directions, respectively, and z is perpendicular to the middle surface and points inwards (−h/2 ≤ z ≤ h/2). Suppose that the material composition of the shell panel varies smoothly along the thickness in such a way that the inner surface is metal-rich and the outer surface is ceramic-rich by following a simple power law in terms of the volume fractions of the Postbuckling of thick FGM cylindrical panels with tangential edge constraints and temperature dependent properties 125 h x y z R k2 k1 a b Fig. 1. Configuration and coordinate system of a cylindrical panel on a Pasternak elastic foundation 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 FGM cylindrical panels in thermal environments. It is assumed that the effective properties Pre f f of FGM cylindrical panels 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) 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 panels such as the modulus of elasticity E and the coefficient of thermal expansion α can be written as E(z, T) = Ec(T) + Emc(T) ( 2z+ h 2h )N , α(z, T) = αc(T) + αmc(T) ( 2z+ h 2h )N , ν(z, T) = ν, (3) where Poisson’s ratio ν is assumed to be constant and Emc(T) = Em(T)− Ec(T), αmc(T) = αm(T)− αc(T). (4) In the present study, the FGM cylindrical panel is fully rested on an elastic founda- tion and the FGM panel-foundation interaction is represented by two-parameter 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 panel, k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of Pasternak model. 126 Hoang Van Tung 3. THEORETICAL FORMULATIONS In the present study, higher order shear deformation shell theory developed by Reddy and Liu [21] is used to establish governing equations and derive expressions for nonlinear response analysis of thick FGM cylindrical panels. Based on this theory, nor- mal 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/2v,y − w/R+ w2,y/2 u,y + v,x + w,xw,y  ,  k1xk1y k1xy  =  φx,xφy,y φx,y + φy,x  ,  k3xk3y k3xy  = −c1  φx,x + w,xxφy,y + w,yy φx,y + φy,x + 2w,xy  , ( γ0xz γ0yz ) = ( φx + w,x φy + 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 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 cylindrical panel. Hookes law for an FGM cylindrical panel 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 panel 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) Postbuckling of thick FGM cylindrical panels with tangential edge constraints and temperature dependent properties 127 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) 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 coefficients Ei = Ei(T)(i = 1 ÷ 7) are analogous to those given in [4] for case of temperature independent properties and omitted here for sake of brevity. Nonlinear equilibrium equation of an imperfect FGM cylindrical panel on an elastic foundation on the basis of higher order shear deformation shell theory has the form [4] 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 ) + f,xx/R+ q− k1w+ k2∆w ] − D6/D4 [ f,yy ( w,xx + w∗,xx ) − 2 f,xy ( w,xy + w∗,xy ) + f,xx ( w,yy + w∗,yy ) + f,xx/R+ q− k1w+ k2∆w ] = 0, (12) where q is external pressure uniformly distributed on the upper surface of the panel, f (x, y) is a stress function defined as Nx = f,yy , Ny = f,xx , Nxy = − f,xy , (13) 128 Hoang Van Tung and 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 ) . (14) Strain compatibility equation for an imperfect FGM cylindrical panel is written as [2, 4] ∆2 f − E1 ( w2,xy − w,xxw,yy + 2w,xyw∗,xy − w,xxw∗,yy − w,yyw∗,xx − w,xx/R ) = 0. (15) Eqs. (12) and (15) are governing equations for postbuckling analysis of FGM panels. In this study, the FGM cylindrical panels are assumed to be simply supported at all edges. The associated boundary conditions are [4, 21] 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 mov- able edges where Nxy is zero-valued, and are fictitious compressive edge loads rendering the edges partially movable or immovable. 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 panel are either unrestrained or completely restrained, respectively, in the in-plane direction perpendicular to the panel edge. For these two cases, the panel 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 [17, 18] ∆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 Postbuckling of thick FGM cylindrical panels with tangential edge constraints and temperature dependent properties 129 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 [4] (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. (15) as A1 = E1δ2n 32β2m W (W + 2µh) , A2 = E1β2m 32δ2n W (W + 2µh) , A3 = E1β2m (β2m + δ 2 n) 2 R W, (23) and detail of coefficients B1, B2 in Eq. (22) are given in the work [4]. Subsequently, Eqs. (20), (21) are substituted into equilibrium equation (12) and ap- plying Galerkin method for the resulting equation as procedure developed in the [4] 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 + E¯1m5npi2B6aR2a 16B2h (m 2B2a + n2) 2 ξ¯mn + mnpi2D¯1B2a 16B4h ξ¯mn [ B2aK1 + pi 2 (m2B2a + n2)K2] } W − 2m 4n2pi2E¯1B5aRa 3B3h (m 2B2a + n2) 2 ( 2ξ¯mn − 1 ) W ( W + µ ) − E¯1n 2pi2BaRa 24B3h [ 4pi2m2B2a 3D¯6B2h (3D¯4 − 4D¯5) + 1 ] W ( W + 2µ ) + E¯1mnpi6 256B4h ( m4B4a + n 4 ) ξ¯mnW ( W + µ ) ( W + 2µ ) + mnpi4 16B2h ξ¯mn ( m2B2a N¯x0 + n 2N¯y0 ) ( W + µ )− BaRa Bh N¯y0 − q = 0, (24) where m, n are odd numbers, and E¯i = Ei/hi(i = 1÷ 7), W =W/h, Ba = b/a, Bh = b/h, Ra = a/R, 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. (25) In what follows, the fictitious compressive edge loads Nx0, Ny0 for the FGM panels under the tangential edge constraints will be specified. From Eqs. (6), (7) and (10) one 130 Hoang Van Tung can obtain the following relations in which Eq. (13) 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 + w R . (26) Introduction of Eqs. (20), (21) and (22) into Eqs. (26) and then substitution of the resulting equations into Eqs. (19) yield the following expressions ∆1 = 1 E1 ( νNy0 − Nx0 ) + 4 ( δ2n − νβ2m ) β2m mnpi2R (β2m + δ2n) 2W − 4E2B1βm mnpi2E1 + 4c1E4βm mnpi2E1 (B1 + βmW) + β2m 8 W (W + 2µh)− Φ1 E1 , ∆2 = 1 E1 ( νNx0 − Ny0 ) + 4 ( β2m − νδ2n ) β2m mnpi2R (β2m + δ2n) 2W − 4E2B2δn mnpi2E1 + 4c1E4δn mnpi2E1 (B2 + δnW) + δ2n 8 W (W + 2µh)− Φ1 E1 − 4W mnpi2R . (27) Now, introduction of Eqs. (27) into Eqs. (17), (18) and solving obtained equations, the fictitious compressive edge loads can be determined as N¯x0 = e11W + e12W ( W + 2µ )− e13Φ1/h, (28) N¯y0 = e21W + e22W ( W + 2µ )− e23Φ1/h, (29) where e11 = 4eE¯1mB3aRa npi2Bh (m2B2a + n2) 2 [ s¯1 (E¯1 + s¯2) ( n2 − νm2B2a ) + νs¯1s¯2 ( m2B2a − νn2 )] − 4eE¯2 mnpiBh [s¯1 (E¯1 + s¯2)mBaB¯1 + νs¯1s¯2nB¯2]− 4νes¯1s¯2mnpi2Bh E¯1BaRa + 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)] , (30) e21 = 4eE¯1mB3aRa npi2Bh (m2B2a + n2) 2 [ s¯2 (E¯1 + s¯1) ( m2B2a − νn2 ) + νs¯1s¯2 ( n2 − νm2B2a )] − 4eE¯2 mnpiB2h [s¯2 (E¯1 + s¯1) nBhB¯2 + νs¯1s¯2mBaBhB¯1]− 4es¯2 (E¯1 + s¯1)mnpi2Bh E¯1BaRa + 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)] , (31) Postbuckling of thick FGM cylindrical panels with tangential edge constraints and temperature dependent properties 131 in which the detail of B1, B2 can be found in the work [4], and s¯1 = s1/h, s¯2 = s2/h, e = 1 (E¯1 + s¯1) (E¯1 + s¯2)− ν2s¯1s¯2 . (32) In this study, the FGM cylindrical panel is entirely exposed to thermal environ- ments uniformly raised from thermal stress free initial state T0 to value T and tempera- ture 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 . (33) 4. STABILITY ANALYSIS 4.1. FGM cylindrical panel under uniform external pressure in thermal environment The FGM cylindrical panel with tangentially restrained edges is assumed to be exposed to uniform thermal environment and simultaneously subjected to external pres- sure q uniformly distributed on the outer surface of the panel. Introduction of Eq. (33) and Eqs. (28), (29) into Eq. (24) result in the following expression q = b11W + b21W ( W + µ ) + b31W ( W + 2µ ) + b41W ( W + µ ) ( W + 2µ )− b51∆T, (34) 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 + E¯1m5npi2B6aR2a 16B2h (m 2B2a + n2) 2 ξ¯mn − BaRa Bh e21 + mnpi2D¯1B2a 16B4h ξ¯mn [ B2aK1 + pi 2 (m2B2a + n2)K2] , b21 = mnpi4 16B2h ξ¯mn ( m2B2ae11 + n 2e21 )− 2m4n2pi2E¯1B5aRa 3B3h (m 2B2a + n2) 2 ( 2ξ¯mn − 1 ) , b31 = − E¯1n 2pi2BaRa 24B3h [ 4pi2m2B2a 3D¯6B2h (3D¯4 − 4D¯5) + 1 ] − BaRa Bh e22, b41 = E¯1mnpi6 256B4h ( m4B4a + n 4 ) ξ¯mn + mnpi4 16B2h ξ¯mn ( m2B2ae12 + n 2e22 ) , b51 = H(T) [ mnpi4 16B2h ξ¯mn ( m2B2ae13 + n 2e23 ) ( W + µ )− BaRa Bh e23 ] . (35) 4.2. FGM cylindrical panel under uniform temperature rise Tangentially restrained edge thick FGM cylindrical panel is rested on an elastic foundation and exposed to uniformly elevated temperature field in absence of external 132 Hoang Van Tung pressure, i.e. q = 0. From Eq. (34), relation of temperature-deflection curves can be expressed as ∆T = 1 b51 [ b11W + b21W ( W + µ ) + b31W ( W + 2µ ) + b41W ( W + µ ) ( W + 2µ )] . (36) Eq. (36) indicates that FGM cylindrical panels under bending at the onset of ther- mal loading and temperature-deflection curves have no bifurcation point in general. However, thermally loaded FGM cylindrical panels may exhibit a bifurcation type re- sponse in some particular cases. Specifically, seeing expression of b51, bifurcation type buckling behavior may be predicted for FGM cylindrical panels of which geometry and material parameters and tangential edge constraints satisfy condition mnpi4 16B2h ξ¯mn ( m2B2ae13 + n 2e23 ) µ− BaRa Bh e23 = 0. (37) Eq. (37) shows that geometrically perfect FGM cylindrical panels, i.e. µ = 0, Ra 6= 0, only buckle as e23 = 0 implying s2 = 0 and straight edges y = 0, b are freely mov- able. Also, bifurcation buckling behavior only occurs for imperfect cylindrical panels with tangential edge restraints as the imperfection size µ satisfy condition µ = 16e23BaBhRa mnpi4ξ¯mn (m2B2ae13 + n2e23) . (38) Eq. (36) is explicit expression of temperature-deflection relation for FGM cylindri- cal panels in case of material properties are temperature independent. In contrast, as temperature dependence of material properties is included, Eq. (36) is implicit expres- sion and an iteration algorithm must be utilized to obtain temperature-deflection curves. Detailed procedure of iteration process is similar as that suggested in the work [20] and is omitted here for sake of brevity. The error tolerance of iteration of the present study is 0.001. 4.3. FGM cylindrical panel under uniform axial compression Consider a thick FGM cylindrical panel resting on an elastic foundation and ex- posed to thermal environment in the absence of external pressure. The panel is subjected to axial compressive load F uniformly distributed on two curved edges x = 0, a assumed to be freely movable, whereas two unloaded straight edges y = 0, b are tangentially re- strained. In this case, N¯x0 = −F [4, 19] 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 [ 4E¯1mB3aRa ( m2B2a − νn2 ) npi2Bh (m2B2a + n2) 2 + 16E¯4 3mpiBh ( B¯2 + npi Bh ) − 4E¯2B¯2 mpiBh − 4E¯1BaRa mnpi2Bh ] . (40) Postbuckling of thick FGM cylindrical panels with tangential edge constraints and temperature dependent properties 133 Substitution of N¯x0, N¯y0 into Eq. (24) leads to the following expression F = 1 A¯ [ b12W + b22W ( W + µ ) + b32W ( W + 2µ ) +b42W ( W + µ ) ( W + 2µ )− b52H(T)∆(T)] , (41) 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 + E¯1m5npi2B6aR2a 16B2h (m 2B2a + n2) 2 ξ¯mn + mnpi2D¯1B2a 16B4h ξ¯mn [ B2aK1 + pi 2 (m2B2a + n2)K2]− BaRae32Bh , b22 = mnpi4 16B2h ξ¯mnn2e32 − 2m 4n2pi2E¯1B5aRa 3B3h (m 2B2a + n2) 2 ( 2ξ¯mn − 1 ) , b32 = − E¯1n 2pi2BaRa 24B3h [ 4pi2m2B2a 3D¯6B2h (3D¯4 − 4D¯5) + 1 ] − BaRa Bh e33, b42 = E¯1mnpi6 256B4h ( m4B4a + n 4 ) ξ¯mn + mnpi4 16B2h ξ¯mnn2e33, b52 = mnpi4 16B2h ξ¯mnn2e34 ( W + µ )− BaRa Bh e34, A¯ = mnpi4 16B2h ξ¯mn ( m2B2a + n 2e31 ) ( W + µ )− BaRa Bh e31. (42) It is observed from Eqs. (41) and (42) that axially loaded FGM cylindrical panels will exhibit a bifurcation type buckling behavior as mnpi4 16B2h ξ¯mn ( m2B2a + n 2e31 ) µ− BaRa Bh e31 = 0, (43) with corresponding buckling compressive load is predicted as Fb = 16B2h mnpi4ξ¯mn (m2B2a + n2e31) b12. (44) Eq. (43) indicates that, under compression, perfect FGM cylindrical panel (i.e. Ra 6= 0, µ = 0) only buckle as e31 = 0 deducing the average tangential stiffness s2 = 0 and two straight edges y = 0, b are freely movable. In contrast, imperfect FGM cylin- drical panel subjected to axial compression will experience bifurcation type buckling at tangential stiffness s¯b2 predicted from Eqs. (41) and (42) as s¯b2 = E¯1m3npi4ξ¯mnB2aµ 16νBaRaBh −mnpi4ξ¯mnµ (m2B2a + νn2) . (45) 134 Hoang Van Tung 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 correspond 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, re- spectively. 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 parameters are defined by λ1 = s¯1 E¯1(T0) + s¯1 , λ2 = s¯2 E¯1(T0) + s¯2 , (46) in which E¯1(T0) is value of E¯1 calculated at room temperature T0. 5. RESULTS AND DISCUSSION This section presents numerical results for square plan-form panels (a = b) made of functionally graded materials and for deformation mode with half wave numbers m = n = 1. To validate the proposed approach, nonlinear response of perfect FGM cylindrical panels with movable edges (λ1 = λ2 = 0), without elastic foundations (K1 = K2 = 0) and subjected to uniform external pressure is considered. The cylindrical panel has a = b = 0.2 m, R = 1.0 m, h = 0.01 m and is made of Aluminum (Al) and Zirconia (ZrO2). The temperature independent material properties are Em = 70 GPa, Ec = 151 GPa and ν = 0.3 for both constituents. 0 0.5 1 1.5 0 10 20 30 40 50 60 70 80 90 100 W/h q a 4 /E m h 4 Zhao and Liew [9] Shen and Wang [11] Present 2 a = b = 0.2 m 1: N * = 0.2 2: N * = 2.0 R = 1.0 m, h = 0.01 m 1 Fig. 2. Comparisons of pressure-deflection curves for ZrO2/Al cylindrical panels The load-deflection curves for a simply supported ZrO2/Al cylindrical panel are plotted in Fig. 2 and compared with results of Zhao and Liew [9] using the element-free kp-Ritz method and with results of Shen and Wang [11] utilizing a two-step perturbation technique. In the Fig. 2, N∗ is volume fraction index as the definition in Eq. (1) is given Postbuckling of thick FGM cylindrical panels with tangential edge constraints and temperature dependent properties 135 for case of ceramic and metal constituents are interchanged, i.e. Vc(z) = (1/2 + z/h)N ∗ . It is evident that a very good agreement is obtained in this comparison. The remainder of this section presents numerical results for FGM cylindrical panels composed of silicon nitride (Si3N4) and stainless steel (SUS304). The material properties Pr, such as elasticity modulus E and thermal expansion coefficient α can be expressed as a nonlinear function of temperature [22] Pr = P0 ( P−1T−1 + 1+ P1T + P2T2 + P3T3 ) , (47) 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 [23] and are listed in Tab. 1. 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. In characterizing the behavior of the panels, deformations in which the central region of a panel moves towards the plane that contains the four corners of the panel are referred to as inward deflections, i.e. positive deflection. Deformations in the opposite direction are referred to as outward deflection, i.e. negative deflection. Table 1. Temperature-dependent thermo-elastic coefficients for silicon nitride and stainless steel (Reddy and Chin [23]) 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 Fig. 3 analyzes the effects of parameters λ1,λ2 on the nonlinear response of FGM cylindrical panels supported by an elastic foundation and exposed to a thermal environ- ment. As can be seen, FGM cylindrical panels tangentially restrained at all edges have no bifurcation behavior and are monotonically deflected towards convex side. In contrast, Fig. 4, plotted as counterparts of the Fig. 3 for case of λ2 = 0, shows that FGM cylindrical panels with two freely movable straight edges experience bifurcation type buckling and relatively benign postbuckling curves. These figures indicate sensitive influences of tan- gential constraints of straight edges on the deflection shape of the cylindrical panels and that the difference between temperature-deflection curves for two T-D and T-ID cases become bigger for partially restrained edge FGM cylindrical panels in comparison with immovable edge counterparts. Fig. 5 assesses the effects of power index N and stiffness of elastic foundation on the mechanical postbuckling of axially compressed FGM cylindrical panels with all movable edges. As expected, buckling loads and postbuckling curves are enhanced due to increase 136 Hoang Van Tung -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 W/h T-D T-ID N = 2, a/b = 1, b/h = 20, a/R = 0.2 K 1 = 50, K 2 = 10, T 0 = 300 K,  = 0 1:  1 =  2 = 0.3 2:  1 =  2 = 0.5 3:  1 =  2 = 0.7 4:  1 =  2 = 1.0 1 1 2 2 3 3 4 4 T (K) Fig. 3. Effects of tangential edge constraint on temperature-deflection curves of perfect FGM cylindrical panels 0 0.2 0.4 0.6 0.8 1 1.2 0 1000 2000 3000 4000 5000 6000 7000 W/h T-D T-ID T (K) (K 1 ,K 2 ) = (50,10), T 0 = 300 K N = 2, a/b = 1, b/h = 20, a/R = 0.2  = 0,  2 = 0 1:  1 = 0.3 4:  1 = 1.02:  1 = 0.5 3:  1 = 0.7 1 2 3 4 1 2 3 4 Fig. 4. Counterparts of the Fig. 3 for case of λ2 = 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 3.5 4 W/h  = 0  = 0.01 1 2 3 1: N = 1, (K 1 ,K 2 ) = (0,0) 3: N = 1, (K 1 ,K 2 ) = (50,5) 2: N = 10, (K 1 ,K 2 ) = (0,0) F (GPa) a/b = 1, b/h = 20, a/R = 0.2,  1 =  2 = 0 T 0 = 300 K, T = 0 Fig. 5. Effects of the index N and elastic foun- dations on the postbuckling of FGM panel un- der axial compression -1 -0.8 -0.6 -0.4 -0.2 0 0 1 2 3 4 5 6 W/h N = 2, a/b = 1, b/h = 20, a/R = 0.15 F (GPa) 1 5 5 2 3 1 4 34 1:  2 = 0.1 2:  2 = 0.3 3:  2 = 0.5 4:  2 = 0.7 5:  2 = 1.0 (K 1 ,K 2 ) = (0,0), T 0 = 300 K, T = 0  1 = 0,  = 0.01 2 Fig. 6. Effects of parameter λ2 on the nonlinear response of imperfect FGM cylindrical panels under axial compression in stiffness of foundation and/or percentage of ceramic constituent in FGM cylindrical panel. The effects of imperfection and in-plane constraint of straight edges y = 0, b on the nonlinear response of axially compressed FGM cylindrical panels without foundation in- teraction and thermal environment are analyzed in Figs. 6 and 7. The Fig. 6, plotted with µ = 0.01 and five various values of parameter λ2, shows that the axially compressed FGM cylindrical panels are deflected towards convex side with no bifurcation buckling. Moreover, load-deflection equilibrium paths are dropped as λ2 increases. It seems that constraint of straight edges produces compressive stress and make the curvature of the Postbuckling of thick FGM cylindrical panels with tangential edge constraints and temperature dependent properties 137 cylindrical panels is increased and, as a result, the cylindrical panels are deflected out- wards under axial compression. Conversely, the panel will be deflected inwards and capability of compressive load carrying is enhanced due to increase in λ2 for higher value of imperfection size µ as shown in the Fig. 7 depicted for µ = 0.1. Especially, the axially compressed FGM cylin- drical panel will exhibit a bifurcation type buckling behavior as degree of tangential con- straint λ2 reaches a definite value as predicted by Eqs. (45) and (46). 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 W/h 4 N = 2, a/b = 1, b/h = 20, a/R = 0.15 (K 1 ,K 2 ) = (0,0),  1 = 0,  = 0.1 2 F (GPa) 1 1 5 5 2 3 3 4 T 0 = 300 K, T = 0 1:  2 = 0.1 2:  2 = 0.3 3:  2 = 0.5 4:  2 = 0.7 5:  2 =  2 b = 0.863 Fig. 7. Counterparts of the Fig. 6 for case of µ = 0.1 and λ2 = λb2 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 6 7 W/h 4 5  1 = 0,  = 0.1, (K 1 ,K 2 ) = (50,5) F (GPa) N = 2, a/b = 1, b/h = 20, a/R = 0.1 T 0 = 300 K, T = 300 K 1:  2 = 0 2:  2 = 0.3 3:  2 = 0.6 4:  2 = 0.8 5:  2 = 1.0 4 5 3 3 2 2 1 1 Fig. 8. Effects of parameter λ2 and thermal en- vironment on the thermomechanical postbuck- ling of FGM cylindrical panels The effects of parameter λ2 and thermal environment (∆T = 300 K) on the ther- momechanical postbuckling of imperfect FGM cylindrical panels subjected to uniform axial compression and resting on an elastic foundation are illustrated in Fig. 8. As can be observed, imperfect FGM cylindrical panels experience a bifurcation type buckling due to presence of elevated temperature and constraint of straight edges, i.e. ∆T 6= 0 and λ2 6= 0, and bifurcation point compressive loads are enhanced by virtue of increasing in λ2. Furthermore, a snap-through phenomenon with severe intensity occurs for FGM cylindrical panels rigorously restrained in tangential motion of straight edges, i.e. higher values of λ2. Finally, the effects of tangential edge constraints on the thermomechanical nonlin- ear response of pressure-loaded FGM cylindrical panels exposed to thermal environment without and with foundation interaction are analyzed in Figs. 9 and 10, respectively. It is evident that tangentially restrained edges panel will exhibit a bifurcation buckling behav- ior as exposed to thermal environment (∆T = 200 K), and both bifurcation point pressure and the severity of snap-through instability are considerably enhanced as parameters λ1,λ2 are increased. In addition, pressure-deflection curves are higher and more stable due to the presence of an elastic foundation, although bifurcation buckling pressures, predicted by final term at right hand side of Eq. (34), are unchanged. 138 Hoang Van Tung 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 12 W/h  = 0  = 0.05 2 1:  1 =  2 = 0 q (MPa) N = 2, a/b = 1, b/h = 30 a/R = 0.2, (K 1 ,K 2 ) = (0,0) T 0 = 300 K, T = 200 K 1 1 2 3 3 2:  1 =  2 = 0.5 3:  1 =  2 = 1.0 Fig. 9. Effects of edge constraints on the non- linear response of pressure-loaded FGM cylin- drical panels in thermal environment 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 12 14 16 W/h  = 0  = 0.05 2 3: 1 =  2 = 1.0 N = 2, a/b = 1, b/h = 30 a/R = 0.2, (K 1 ,K 2 ) = (50,5) T 0 = 300 K, T = 200 K 1 1 2 3 3 2: 1 =  2 = 0.5 1: 1 =  2 = 0 q (MPa) Fig. 10. Counterparts of the Fig. 9 for case of K1 = 50, K2 = 5 6. CONCLUDING REMARKS The nonlinear response and postbuckling behavior of thick FGM cylindrical panels subjected to thermal, mechanical and thermomechanical loadings have been analyzed. Formulations are based on the higher order shear deformation shell theory taking geo- metrical nonlinearity, initial imperfection, panel-foundation interaction, tangential edge constraints and temperature dependent material properties into consideration. The anal- ysis reveals that the tangential constraints of boundary edges, especially straight edges, have extremely sensitive influences on the nonlinear response and postbuckling of FGM cylindrical panels, especially for case of these panels subjected to thermal loading and ax- ial compression. The results also show that temperature dependence of material proper- ties has deteriorative and pronounced effects on the thermal buckling and postbuckling behavior of thick FGM cylindrical panels resting on elastic foundations and with tan- gentially restrained edges. Furthermore, elastic foundations, especially Pasternak type foundations, have beneficial influences on the nonlinear stability of FGM panels on both aspects are that increase in load carrying capacity and decrease in the severity of snap- through instability. ACKNOWLEDGEMENT This research is funded by Vietnam National Foundation for Science and Technol- ogy Development (NAFOSTED) under grant number 107.02-2014.09. 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