Thermal buckling of imperfect functionally graded cylindrical shells according to wan-Donnell model

Vietnam Journal of Mechanics, VAST, Vol. 30, No. 3 (2008), pp. 185 – 194 THERMAL BUCKLING OF IMPERFECT FUNCTIONALLY GRADED CYLINDRICAL SHELLS ACCORDING TO WAN-DONNELL MODEL Hoang Van Tung Hanoi Architectural University, Vietnam Nguyen Dinh Duc Vietnam National University, Hanoi Abstract. A thermal buckling analysis of imperfect circular cylindrical shells of func- tionally graded material is considered. The material properties are assumed varying as a power form of thickness coordinate variable. The Donnell equilibrium and stability equations are considered and the Wan-Donnell model for initial geometrical imperfec- tion is adopted. The thermal loads include the uniform temperature rise and nonlinear temperature change across the thickness of shell. A closed form solution for the thermal buckling of simply supported cylindrical FG shell under the described thermal loads is obtained. The influences of the relative thickness, the imperfection size and the power law index on buckling thermal loads are all discussed. 1. INTRODUCTION The initial geometric imperfections are inherent in many real structures. Therefore, many investigations are conducted on the stability analysis of imperfect structures. Elastic, plastic, and creep buckling of imperfect cylinders under mechanical and thermal loads is studied by Eslami and Shariyat [3]. Mossavarali et al. studied the thermoelastic buckling of isotropic and orthotropic plates with imperfections [8, 9]. Murphy and Ferreira [10] investigated thermal buckling analysis of clamped rectangular plates based on the energy consideration. They determined the ratio of the critical temperature for a perfect flat plate to that one for an imperfect plate as a function of the initial imperfection size. The study includes experimental results. Eslami and Shahsiah [2] reported thermal buckling of imperfect circular cylindrical shells made of isotropic materials. They used the Wan- Donnell and Koiter models to describle initial geometrical imperfection. The development of new materials with new constitutive models have necessitated more research in the area of stability analysis. Functionally graded materials (FGMs) are of these new and high-temperature resistant materials in which material constitution vary continuously across the thickness of a structure. Some works about the stability of FGM structures are introduced in the following. Javaheri and Eslami [4, 5, 6] reported mechanical and thermal buckling of rectangular functionally graded plates. They used energy method and mainly reached to the closed-form solutions. The research on thermal buckling of functionally graded cylindrical shells is introduced by Shahsiah and Eslami [12, 13]. Shen [14] represented thermal postbuckling behavior of functionally graded cylindrical shells with dependent temperature properties. He considered initial geometrical imperfections in the analysis. Wu et al. [16] studied the thermoelastic stability of functionally graded 186 Hoang Van Tung, Nguyen Dinh Duc cylindrical shells by using the Donnell shell theory. They obtained closed-form solutions for the critical buckling loads with three types of thermal loads being uniform temperature rise, linear temperature change and nonlinear thermal gradient through the thickness direction of shell. In the present article, mainly motivated by Eslami and Shahsiah [2], the influence of geometrical imperfections on thermal instability of FG shells is investigated. The shell is graded through the thickness direction according to a power law function. The Donnell stability equations are considered and the Wan-Donnell model for axisymmetric imperfec- tion is adopted. The shell is assumed to be simply supported at two ends. The buckling of shell under two types of thermal loads are obtained. The thermal loads are assumed to be uniform temperature rise and nonlinear thermal gradient through the thickness direction. Closed form solutions are given for two types of the assumed thermal loads. The influences of imperfection size, the relative thickness and the power law index on the critical thermal loads are considered and discussed. 2. FUNCTIONALLY GRADED SHELL Functionally graded materials (FGMs) are microscopically inhomogeneous materials in which the mechanical properties vary smoothly and continuously through the thickness. This is achieved by gradually changing the volume fraction of the constituent materials. These materials are made from a mixture of ceramic and metal or a combination of dif- ferent metals [11]. The ceramic part provides high-temperature resistance due to its low thermal conductivity and metal part prevents fracture because of thermal loadings. FGMs are able to withstand ultra high temperature environments and extremly large thermal gradients while maintaining their structural integrity. FGMs have found diverse engineer- ing applications in aerospace structures, nuclear reactors, chemical plants, etc. The functionally graded (FG) cylindrical shell is assumed to be a thin shell of length L, thickness h and radius R. The x-axis is taken along a generatrix, the circumferential length subtends an angle θ, and the z-axis is directed radially inwards. We assume that the modulus of elasticity E, the coefficient of thermal expansion α and conductivity K change in the thickness direction z, while the Poisson ratio ν is assumed to be constant. The material properties of FG shell are introduced as [13, 16]. E(z) = Em +Ecm ( 2z + h 2h )k , α(z) = αm + αcm ( 2z + h 2h )k , K(z) = Km +Kcm ( 2z + h 2h )k , ν(z) = ν (1) where Ecm = Ec −Em , αcm = αc − αm , Kcm = Kc −Km , (2) and subscripts “m” and “c” refer to the metal and ceramic constituents, respectively. The variable z is the thickness coordinate (−h/2 6 z 6 h/2), where h is the thickness of the shell and k is the power law index which takes values greater than or equal to zero. The Thermal buckling of imperfect functionally graded cylindrical... 187 variation of the composition of ceramic and metal is linear for k = 1. The value of k equal to zero represents a fully ceramic shell. 3. ANALYSIS The constitutive relations are written as [16] Nx = E1 1− ν2 (εxm + νεθm) + E2 1− ν2 (kx + νkθ)− Φ 1− ν , Nθ = E1 1− ν2 (εθm + νεxm) + E2 1− ν2 (kθ + νkx)− Φ 1− ν , Nxθ = E1 2(1 + ν) γxθm + E2 1 + ν kxθ. (3) Mx = E2 1− ν2 (εxm + νεθm) + E3 1− ν2 (kx + νkθ)− Θ 1− ν , Mθ = E2 1− ν2 (εθm + νεxm) + E3 1− ν2 (kθ + νkx)− Θ 1− ν , Mxθ = E2 2(1 + ν) γxθm + E3 1 + ν kxθ. (4) where E1 = Emh+ Ecmh k + 1 , E2 = kEcmh 2 2(k + 1)(k + 2) , E3 = 1 12 Emh 3 +Ecmh 3 [ 1 k + 3 − 1 k + 2 + 1 4(k + 1) ] , Φ = ∫ h/2 −h/2 [ Em +Ecm ( 2z + h 2h )k ][ αm + αcm ( 2z + h 2h )k ] ∆T (x, y, z)dz , Θ = ∫ h/2 −h/2 [ Em +Ecm ( 2z + h 2h )k ][ αm + αcm ( 2z + h 2h )k ] ∆T (x, y, z)zdz . (5) In the above equations,Nij andMij are force and moment resultants, respectively, εxm, εθm are the extension strains and γxθm is shear strain at the middle surface of the shell. The curvature changes are shown by kij. The strain displacement relations according to Donnell’s assumption [1] as follows. εxm = u,x + 1 2 w2,x, εθm = (v,θ +w) /R+w 2 ,θ/2R 2, γxθm = u,θ/R+ v,x +w,xw,θ/R, kx = −w,xx, kθ = −w,θθ/R 2, kxθ = −w,xθ/R, (6) where u, v, w are the axial, circumferential, and deflection displacements of the shell and the comma symbols partial derivative. 188 Hoang Van Tung, Nguyen Dinh Duc By using Donnell shell theory [1], the equilibrium equations of FG cylindrical shell are derived as RNx,x +Nxθ,θ = 0 , RNxθ,x +Nθ,θ = 0 , D44w + 1 R Nθ − [ Nxw,xx + 2 R Nxθw,xθ + 1 R2 Nθw,θθ ] = 0 , (7) where D = E1E3 −E 2 2 E1(1− ν2) , 4 = ∂2 ∂x2 + 1 R2 ∂2 ∂θ2 . (8) For a slightly imperfect shell, let w∗(x, θ) denotes a known small imperfection. This pa- rameter represents a small deviation of the shell middle plane from a flat shape. According to [1], equilibrium equations for the imperfect FG shell are as RNx,x +Nxθ,θ = 0, RNxθ,x +Nθ,θ = 0, D44w + 1 R Nθ − [ Nx ( w,xx +w ∗ ,xx ) + 2 R Nxθ ( w,xθ +w ∗ ,xθ ) + 1 R2 Nθ ( w,θθ +w ∗ ,θθ ) ] = 0 . (9) The stability equations of the shell may be derived by the adjacent equilibrium criterion [1] as follows RNx1,x +Nxθ1,θ = 0, RNxθ1,x +Nθ1,θ = 0 , D44w1 + 1 R Nθ1 −Nx0w1,xx − 2 R Nxθ0w1,xθ − 1 R2 Nθ0w1,θθ−[ Nx1 ( w0,xx +w ∗ ,xx ) + 2 R Nxθ1 ( w0,xθ +w ∗ ,xθ ) + 1 R2 Nθ1 ( w0,θθ + w ∗ ,θθ ) ] = 0 . (10) In Eqs. (10), terms with the subscript ‘0’ are related to the state of equilibrium and terms with the subscript ‘1’ are those characterizing the state of stability. The Wan-Donnell model for the radial imperfection is [1, 2] w∗ = K − 1 2 w , (11) where the coefficient K is a constant value 0 6 K 6 1. The value of K = 1 represents a perfect shell. The imperfectionw∗ is thus defined as a function of w, the lateral deflection of the FG cylindrical shell. Considering an axisymmetric imperfection, due to the dependency of w to w∗, the lateral deflection w must be assumed to be axisymmetric. This assumption results in an axisymmetric buckling mode of the FG cylindrical shell, such that w∗ = w∗(x) , w0 = w0(x) , w1 = w1(x). (12) The prebuckling force resultants of FG shell under thermal load are determined as [2, 12] Nx0 = − Φ 1− ν , Nθ0 = 0 , Nxθ0 = 0. (13) Putting Eqs. (12), (13) into Eqs. (9) and (10) gives the equilibrium equation Dw0,xxxx−Nx0 ( w0,xx + w ∗ ,xx ) = 0 , (14) and the stability equation Dw1,xxxx + 1 R f,xx −Nx0w1,xx − f,yy ( w0,xx +w ∗ ,xx ) = 0 (15) Thermal buckling of imperfect functionally graded cylindrical... 189 of the axisymmetric imperfect FG cylindrical shell, where a change of variable y = Rθ and a stress function f(x, y) has been used, such as Nx1 = f,yy , Ny1 = f,xx , Nxy1 = −f,xy. (16) Equation (15) is the Wan-Donnell stability equation of an imperfect FG cylindrical shell. This equation includes two dependent functions, w1 and f . To obtain a second equation relating the dependent functions w1 and f , the compatibility equation may be used, as follows εxm1,yy + εym1,xx − γxym1,xy = 0, (17) where εxm1, εym1, γxym1 denote parts of the strain components which are linear in stability state displacements u1, v1, w1. These strains may be written in terms of the displacement components, using Eqs. (6) with consideration of the imperfection term w∗, as εxm1 = u1,x + (w0,x +w ∗ ,x)w1,x , εym1 = v1,y + w1/R+ (w0,y +w ∗ ,y)w1,y , γxym1 = u1,y + v1,x + (w0,x+ w ∗ ,x)w1,y + (w0,y +w ∗ ,y)w1,x. (18) Moreover, from the constitutive relations (3), one can write for stability state as εxm1 = 1 E1 (Nx1− νNy1 − E2kx1 + Φ) , εym1 = 1 E1 (Ny1 − νNx1 − E2ky1 + Φ) , γxym1 = 2 E1 [ (1 + ν)Nxy1 − E2kxy1 ] . (19) From Eqs. (17), (18), (19), taking into consideration Eqs. (12), (16), we obtain equation 1 E1 44f = 1 R w1,xx. (20) Equations (14), (15) and (20) are the basic equations used to obtain the critical buckling loads of imperfect FG cylindrical shells. By putting k = 0, these equations turn into respective ones in [2] for imperfect isotropic and homogeneous shells. 3.1. Buckling of imperfect functionally graded shells under uniform tempera- ture rise Consider a imperfect cylindrical shell of thickness h and length L made of functionally graded material. The ends are assumed to be simply supported, where the boundary conditions at x = 0 and x = L are w = w,xx = 0 (21) The lateral imperfection of the shell may be assumed to be w∗ = (K − 1)ζ 2 sin mpix L , (22) where ζ is the imperfection size. Substituting Nx0 from (13) into (14) and assuming a solution in the form w0 = Asin mpix L , (23) 190 Hoang Van Tung, Nguyen Dinh Duc which satisfies the simply supported edge conditions at x = 0 and L, the constant A is obtained and the final approximate solution of Eq. (14) is w0 = (K − 1)Φζ 2D(1− ν) (mpi L )2 − 2Φ sin mpix L . (24) Substituting w0 and w∗ into Eqs. (15) and (20) yields R1 ≡ Dw1,xxxx+ 1 R f,xx + [ Φ 1− ν w1,xx + (A+H) (mpi L )2 f,yy sin mpix L ] = 0 R2 ≡ 1 E1 44f − 1 R w1,xx = 0 , (25) where H ≡ (K − 1)ζ 2 To solve the system of Eqs. (25), with the consideration of the boundary conditions, the approximate solutions are assumed w1 = Em sin mpix L , f = Fm sin mpix L cos y R , 0 6 x 6 L ; 0 6 y 6 2piR , (26) where Em, Fm are constant coefficients. Applying the Galerkin mehtod to the system of equation (25). The system of Eqs. (25) yields∫ 2piR 0 ∫ L 0 R1 sin mpix L cos y R dxdy = 0 , ∫ 2piR 0 ∫ L 0 R2 sin mpix L dxdy = 0 (27) where R1, R2 designate the left sides of Eq. (25) known as the residues of Galerkin method. The determinant of the system of Eqs. (27) for the coefficients Em and Fm is set to zero, which yields Φmin = D(1− ν)pi L2 ( 8H 3R + pi ) , (28) obtained with m = 1. The smallest value of ∆T is obtained using Eqs. (5) and (28) ∆Tcr = D(1− ν)pi PL2 ( 8H 3R + pi ) , (29) where P = Emαmh + (Emαcm + Ecmαm) h k + 1 +Ecmαcm h 2k + 1 . (30) 3.2. Buckling of imperfect FG shell under nonlinear temperature change across the thickness The steady state heat conduction equation and the related boundary conditions d dz ( K(z) dT dz ) = 0 , T (z) ∣∣∣ z=−h/2 = Tm , T (z) ∣∣∣ z=h/2 = Tc , (31) where Tc and Tm are the temperature of ceramic-rich and metal-rich surfaces, respectively. The solution of Eq. (31) is obtained by means of polynomial series. Taking the first seven Thermal buckling of imperfect functionally graded cylindrical... 191 terms of the series [16], the solution for temperature distribution across the shell thickness becomes T (z) = Tm + ∆T.r. ∑ 6 n=0 [ ( −rk.Kcm/Km )n /(nk + 1) ] ∑ 6 n=0 [ (−Kcm/Km)n/(nk + 1) ] , (32) where ∆T = Tc − Tm and r = (2z + h)/2h. Putting (32) into (5) and from (28) we obtain ∆Tcr = D(1− ν)pi QL2 ( 8H 3R + pi ) , (33) where Q = n=6∑ n=0 [ (−Kcm/Km) n/(nk + 1) ] h n=6∑ n=0 (−Kcm) n Knm(nk + 1) [Emαm nk + 2 + Emαcm +Ecmαm (n+ 1)k+ 2 + Ecmαcm (n+ 2)k + 2 ] (34) 4. RESULTS AND DISCUSSION The thermal buckling loads of the imperfect functionally graded cylindrical shell are obtained in closed form solutions for the assumed thermal loadings and represented by Eqs. (29) and (33). These equations indicate that the critical buckling temperature change of an imperfect FG shell is decreased in comparision with a perfect one. The decrease in ∆Tcr is expressed by an negative imperfection term 8H/3R, which directly depends on the imperfection size ζ. As an example, consider an imperfect ceramic-metal FG shell that consists of aluminum and alumina with following properties [4, 12, 16] Em = 70GPa, αm = 23× 10 −6/0C,Km = 204W/mK, Ec = 380GPa, αc = 7.4× 10 −6/0C,Kc = 10.4W/mK, ν = 0.3. (35) The shell is assumed to be simply supported on two ends. The variation of∆Tcr versus the relative thickness h/R for two types of thermal loadings is shown in Figs. 1 and 2. These Figs. indicate that the critical thermal loads are increased when the thickness increases. The pure ceramic shell (k = 0) is remarkably more stable than graded shells, moreover, the greater power law index k to be, the smaller value of critical thermal loads become for FG shells. This is expected because ceramic-rich shell withstands temperature better than metal-rich one. Especially, FG shell under nonlinear thermal gradient is considerably more stable than under uniform temperature rise. In Fig. 3, a variation of critical thermal loads versus relative thickness with different imperfection sizes for uniform temperature rise is plotted. It is observed that the critical thermal loads decrease when imperfection sizes to be larger. This is good agreement, in tend, of present result with those reported in [2] for imperfect isotropic circular cylindrical shell under thermal loads. 192 Hoang Van Tung, Nguyen Dinh Duc Fig. 1. Critical buckling temperature change of FG shell under uniform temper- atue rise versus h/R and power law index Fig. 2. Critical buckling temperature change of FG shell under nonlinear thermal gradient across the thickness versus h/R and power law index 5. CONCLUSIONS In the present paper, equilibrium, stability, and compability equations for an imperfect functionally graded cylindrical shell are derived. Derivations are based on the Donnell shell theory and the assumption of power law composition for the constituent materials. The imperfection of the shell is assumed to be axisymmetric and follows Wan-Donnell model. Thermal buckling of imperfect functionally graded cylindrical... 193 Fig. 3. Critical buckling temperature change of FG shell under uniform temper- atue rise versus h/R and imperfection amplitude The buckling analysis of such a shell under two types of thermal loadings is investigated. The shell buckles at first mode and the followings are concluded: (1) The critical buckling temperature change, ∆Tcr, of an imperfect FG shell is smaller than a perfect one. This decrease is directly indicated in closed form solutions of critical loads. (2) ∆Tcr of a FG shell decreases with the increase of imperfection size ζ. (3) ∆Tcr of an imperfect FG cylindrical shell increases with the increase of the relative thick- ness h/R. In contrast, ∆Tcr of imperfect FG shell is reduced when the power law index k increases. The reduction from k = 0 to k = 1 is considerable. However, for k > 1, it is marginal. (4) ∆Tcr of the shell subjected to nonlinear temperature change across the thickness is con- siderably greater than the one subjected to uniform temperature rise. ACKNOWLEDGMENT The results of researching presented in the paper have been performed according to scientific research project of Vietnam National University, Hanoi, coded QGTĐ.08.07. The authors gratefully acknowledge this financial support. REFERENCES 1. D. O. Brush, B. O. Almroth, Buckling of Bars, Plates and Shells, McGraw-Hill, New York, 1975. 2. M. R. Eslami, R. Shahsiah, Thermal buckling of imperfect cylindrical shells, Journal of Thermal Stresses 24 (1) (2001) 71-89. 3. M. R. Eslami, M. Shariyat, Elastic, plastic and creep buckling of imperfect cylinders under mechanical and thermal loadings, Journal of Pressure Vessel Technology, Transactions of the ASCE 119 (1) (1997) 27-36. 4. R. Javaheri, M. R. Eslami, Thermal buckling of functionally graded plates, AIAA Journal 40 (1) (2002) 162-169. 194 Hoang Van Tung, Nguyen Dinh Duc 5. R. Javaheri, M. R. Eslami, Buckling of functionally graded plates under in-plane compressive loading, ZAMM Journal 82 (4) (2002) 277-283. 6. R. 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Các phương trình cân bằng và ổn định được thiết lập dựa trên lý thuyết vỏ Donnell và mô hình Wan-Donnell cho imperfect hình học ban đầu được xét. Hai trường hợp tải nhiệt được nghiên cứu là sự tăng nhiệt độ đồng nhất của môi trường chứa vỏ trụ và sự truyền nhiệt qua chiều dày của vỏ trụ. Biểu thức giải tích cho nhiệt độ tới hạn của hai trường hợp tải nhiệt được đưa ra đối với vỏ trụ tựa bản lề ở hai đầu. Tính chất của vật liệu chức năng được giả thiết biến đổi theo chiều dày vỏ theo một hàm mũ. Những ảnh hưởng của độ dày tương đối của vỏ, cỡ imperfect ban đầu và chỉ số luật mũ lên tải nhiệt tới hạn được nghiên cứu và thảo luận.