DSpace at VNU: Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external p...
International Journal of Non-Linear Mechanics 46 (2011) 1195–1204 Contents lists available at ScienceDirect International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects Dao Huy Bich a, Hoang Van Tung b,à a b Vietnam National University, Ha Noi, Viet Nam Faculty of Civil Engineering, Hanoi Architectural University, Ha Noi, Viet Nam a r t i c l e i n f o abstract Article history: Received April 2011 Received in revised form 23 May 2011 Accepted 26 May 2011 Available online June 2011 This paper presents an analytical approach to investigate the non-linear axisymmetric response of functionally graded shallow spherical shells subjected to uniform external pressure incorporating the effects of temperature 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 shallow spherical shells are derived by using the classical shell theory and specialized for axisymmetric deformation with both geometrical nonlinearity and initial geometrical imperfection are taken into consideration One-term deflection mode is assumed and explicit expressions of buckling loads and load–deflection curves are determined due to Galerkin method Stability analysis for a clamped spherical shell shows the effects of material and geometric parameters, edge restraint and temperature conditions, and imperfection on the behavior of the shells & 2011 Elsevier Ltd All rights reserved Keywords: Nonlinear response Functionally graded materials Imperfection Temperature effects Shallow spherical shells Introduction Shallow spherical shells constitute an important portion in many engineering structures They can find applications in the aircraft, missile and aerospace components These shell elements also be widely used in other industries such as shipbuilding, underground structures and building constructions As a result, the problems relating to buckling and postbuckling behaviors bring major importance in the design of this type of shell structure and have attracted attention of many researchers Huang [2] reported an investigation on unsymmetrical buckling of thin isotropic shallow spherical shells under external pressure He pointed out that unsymmetrical deformation may be the source of discrepancy in critical pressures between axisymmetrical buckling theory and experiment Tillman [3] investigated the buckling behavior of clamped shallow spherical caps under a uniform pressure load He also considered the effect of shell geometric parameters on the axisymmetric and asymmetric postbifurcation behavior of a clamped perfect and imperfect isotropic shallow spherical cap under uniform external pressure both theoretically and experimentally Uemura [4] employed a twoterm approximation of deflection to treat axisymmetrical snap buckling of a clamped imperfect isotropic shallow spherical shell à Corresponding author E-mail address: hoangtung0105@gmail.com (H Van Tung) 0020-7462/$ - see front matter & 2011 Elsevier Ltd All rights reserved doi:10.1016/j.ijnonlinmec.2011.05.015 subjected to uniform external pressure This work also assessed the effect of shell shallowness on the non-uniqueness of solution by using the second variation of total potential energy Non-linear static and dynamic responses of spherical shells with simply supported and clamped immovable edge have been analyzed by Nath and Alwar [5] by making use of Chebyshev series expansion Non-linear free vibration response, static response under uniformly distributed load, and the maximum transient response under uniformly distributed step load of orthotropic thin spherical caps on elastic foundations have been obtained by Dumir [6] Static and dynamic axisymmetric snap-through buckling of orthotropic shallow spherical shells subjected to uniform pressure have also been investigated by Chao and Lin [7] using the classical thin shell theory and finite difference method Buckling and postbuckling behaviors of laminated spherical caps subjected to uniform external pressure have been analyzed by Xu [8] and Muc [9] The former employed non-linear shear deformation theory and a Fourier-Bessel series solution to determine load– deflection curves of spherical shell under axisymmetric deformation, whereas the latter applied the classical shell theory and Rayleigh–Ritz procedure to obtain upper and lower pressures and postbuckling equilibrium paths without considering axisymmetry Alwar and Narasimhan [10] used method of global interior collocation to study axisymmetric non-linear behavior of laminated orthotropic annular spherical shells Subsequently, a static and dynamic non-linear axisymmetric analysis of thick shallow spherical and conical orthotropic caps has been reported by Dube 1196 D Huy Bich, H Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204 et al [11] employing Galerkin method and the first order shear deformation theory Also, Nie [12] proposed the asymptotic iteration method to treat non-linear buckling of externally pressurized isotropic shallow spherical shells with various boundary conditions incorporating the effects of imperfection, edge elastic restraint and elastic foundation Recently, Eslami et al [13] and Shahsiah and Eslami [14] used an analytical approach to study linear buckling of isotropic shallow spherical shells with and without imperfection under thermal and mechanical loads In these works the equilibrium and linear stability equations are derived using the variational method and analytically solved to obtain closed-form solutions of critical buckling loads for simply supported shallow spherical shells Li et al [15] adopted the modified iteration method to solve non-linear stability problem of shear deformable isotropic shallow spherical shells under uniform external pressure Due to advanced characteristics in comparison with traditional metals and conventional composites, Functionally Graded Materials (FGMs) consisting of metal and ceramic constituents have received increasingly attention in structural applications recent years Smooth and continuous change in material properties enable FGMs to avoid interface problems and unexpected thermal stress concentrations By high performance heat resistance capacity, FGMs are now chosen to use as structural components exposed to severe temperature conditions such as aircraft, aerospace structures, nuclear plants and other engineering applications Despite the evident importance in practical applications, it is fact from the open literature that investigations on the buckling and postbuckling behaviors of FGM spherical shells are comparatively scarce Shahsiah et al [16] extended their previous works for isotropic material to analyze linear stability of FGM shallow spherical shells subjected to three types of thermal loading This study is performed on the point of view of small deflection and the existence of type-bifurcation buckling of thermally loaded spherical shells Recently, the non-linear axisymmetric dynamic stability of clamped FGM shallow spherical shells has been analyzed by Prakash et al [17] and Ganapathi [18] using the first order shear deformation theory and finite element method To best of authors’ knowledge, there is no analytical investigation on the non-linear stability of FGM shallow spherical shells In this paper, the non-linear axisymmetric response of FGM shallow spherical shells under uniform external pressure with and without the effects of temperature is investigated by an analytical approach The properties of constituent materials are assumed to be temperature-independent and the effective properties of FGMs are graded in thickness direction according to a power law function of thickness coordinate Equilibrium and compatibility equations of a spherical shell are established by using the classical shell theory Then these equations are specialized for axisymmetrically deformed shallow spherical shells taking into account geometric non-linearity and initial geometrical imperfection One-term approximation of deflection is assumed and explicit expressions of extremum buckling loads and load–deflection curves for a clamped spherical shell are determined by Galerkin method An analysis is carried out to assess the effects of material and geometric parameters, edge restraint and temperature conditions, and imperfection on the non-linear response of the shells Functionally graded shallow spherical shells Consider a functionally graded shallow spherical shell with radius of curvature R, base radius a and thickness h as shown in Fig The shell is made from a mixture of ceramics and metals, and is defined in coordinate system ðj, y,zÞ whose origin is located Fig Configuration and the coordinate system of a shallow spherical shell on the middle surface of the shell, j and y are in the meridional and circumferential directions, respectively, of the shell and z is perpendicular to the middle surface and points outwards ðÀh=2 r z rh=2Þ Suppose that the material composition of the shell 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 constituents as k 2z ỵ h Vc zị ¼ , Vm ðzÞ ¼ 1ÀVc ðzÞ, ð1Þ 2h where Vc and Vm are the volume fractions of ceramic and metal constituents, respectively, and volume fraction index k is a nonnegative number that defines the material distribution It is assumed that the effective properties Preff of FGM spherical shell change in the thickness direction z and can be determined by the linear rule of mixture as Preff zị ẳ Prc Vc zị ỵ Prm Vm zị, 2ị where Pr denotes a temperature-independent material property, and subscripts m and c represent the metal and ceramic constituents, respectively From Eqs (1) and (2) the effective properties of FGM shallow spherical shell such as modulus of elasticity E, the coefficient of thermal expansion a, and the coefficient of thermal conduction K can be dened as k 2z ỵh , 3ị ẵEzị, azị,Kzị ẳ ẵEm , am ,Km ỵ ẵEcm , acm ,Kcm 2h whereas Poisson ratio n is assumed to be constant and Ecm ¼ Ec ÀEm , acm ¼ ac am , Kcm ẳ Kc Km : 4ị It is evident that E ¼ Ec , a ¼ ac , K ¼ Kc at z ¼h/2 and E ¼ Em , a ¼ am , K ¼ Km at z¼ Àh/2 Governing equations In the present study, the classical shell theory is used to obtain the equilibrium and compatibility equations as well as expressions of buckling loads and non-linear load–deflection curves of FGM shallow spherical shells For a shallow spherical shell it is convenient to introduce a variable r referred to as the radius of parallel circle and defined by r ¼ Rsinj Moreover, due to shallowness of the shell it is approximately assumed that cosj ¼ and Rdj ¼ dr The strains across the shell thickness at a distance z from the mid-plane are er ¼ erm Àzwr , ey ¼ eym Àzwy , gry ¼ grym Àzwry , ð5Þ where erm and eym are the normal strains, grym is the shear strain at the middle surface of the spherical shell, whereas wr , wy , wry are the change of curvatures and twist D Huy Bich, H Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204 According to the classical shell theory, the strains at the middle surface and the change of curvatures and twist are related to the displacement components u,v,w in the j, y,z coordinate directions, respectively, as [1] erm ¼ u,r grym ẳ r w ỵ w , R ,r v r ,r ỵ eym ẳ v, y ỵu w ỵ w2, y , r R 2r u, y ỵ w,r w, y , r r wr ¼ w,rr , wy ¼ w, yy w,r , ỵ r r2 6ị r wry ẳ w,ry w, y , r2 7ị where geometrical non-linearity in case of small strain and moderately small rotation is accounted for, also, subscript (,) indicates the partial derivative Hooke law for a spherical shell including temperature effect is dened as sr , sy ị ẳ E ẵer , ey ị ỵ ney , er ị1 ỵ nịaDT 1,1ị, 1n2 E sry ẳ g , 21 ỵ nị ry 1 r rM r ị,rr ỵ2 Mry,ry ỵ Mry, y ỵ My, yy My,r ỵ Nr þ Ny Þ r r R þ rN r w,r ỵ Nry w, y ị,r ỵ Nry w,r ỵ Ny w, y ỵ qr ẳ 0, r ,y where q is uniform external pressure positive inwards The first two of Eqs (13) are identically satisfied by introducing a stress function f as f 1 14ị Nr ẳ f,r ỵ f, yy , Ny ¼ f,rr , Nry ¼ À , y : r r ,r r Introduction of Eqs (10), (11) and (14) into the third of Eqs (13) gives the following equation: 1 1 f,r ỵ f, yy w,rr ỵ f,ry f, y w,ry DD2 wÀ Df À R r r r r r 1 1 f,rr w, yy f,rr w,r ỵ 2 f, y À f,ry w, y Àq ¼ 0, 15ị r r r r r 8ị Dẳ E1 E3 E22 , E1 1n2 ị r Dị ẳ ị,rr þ ðÞ,r þ ðÞ : r , yy 1 1 Dw ỵ w2ry wr wy : e erm,r ỵ r eym,r ị,r r grym ị,ry ẳ R r rm, yy r r r ð17Þ Introduction of Eqs (3), (5) and (8) into Eqs (9) gives the constitutive relations as From the constitutive relations (10) one can write Nr ¼ E1 E2 Fm erm ỵ neym ị w ỵ nwy ÞÀ , 1Àn 1Àn2 1Àn2 r ðerm , eym Þ ẳ Ny ẳ E1 E2 Fm e ỵ nerm ị w ỵ nwr ị , 1n 1n2 ym 1n2 y grym ¼ E1 E g À w : Nr y ẳ 21 ỵ nị rym ỵ n ry Mr ẳ E2 E3 F erm ỵ neym ị w þ nwy ÞÀ b , 1Àn 1Àn2 1Àn2 r My ẳ E2 E3 F e ỵ nerm ị w ỵ nwr ÞÀ b , 1Àn 1Àn2 ym 1Àn2 y E2 E g w , 21ỵ nị rym ỵ n ry 10ị 11ị where E1 ẳ Em h ỵ Ecm h=k ỵ 1ị, E2 ẳ Ecm h2 ẵ1=k ỵ 2ị1=2kỵ 2ị, E3 ẳ Em h3 =12 ỵEcm h3 ẵ1=k ỵ 3ị1=k ỵ 2ị ỵ 1=4k ỵ4ị, k # 2z ỵh Em ỵ Ecm Fm , Fb ị ẳ 2h h=2 " k # 2z ỵ h am ỵ acm DT1,zị dz: 2h h=2 16ị Eq (15) is the equilibrium equation of FGM shallow spherical shells in terms of two dependent unknowns, that is deflection of shell w and stress function f To obtain a second equation relating these two unknowns, the compatibility equation may be used The geometrical compatibility equation of a shallow spherical shell is written as Àh=2 Z ð13Þ where where DT denotes the change of environment temperature from stress free initial state or temperature difference between the surfaces of FGM spherical shell The force and moment resultants of an FGM spherical shell are expressed in terms of the stress components through the thickness as Z h=2 Nij ,Mij ị ẳ sij 1,zị dz, ij ẳ r, y,ry: 9ị Mry ẳ 1197 " ð12Þ The non-linear equilibrium equations of a perfect shallow spherical shell based on the classical shell theory are given by [1,16] ẵNr ,Ny ịnNy ,Nr ị ỵ E2 wr , wy ị ỵ Fm 1,1ị, E1 ẵ1 ỵ nịNry ỵE2 wry : E1 18ị Substituting the above equations into Eq (17), with the aid of Eqs (7) and (14), leads to the compatibility equation of a perfect FGM shallow spherical shell as 2 Dw 1 1 ỵ w,ry w, y w,rr w, yy ỵ w,r : D f ¼À ð19Þ E1 R r r r r Eqs (15) and (19) are equilibrium and compatibility equations, respectively, of an FGM shallow spherical shell in the case of asymmetric deformation As shown by Huang [2], unsymmetrical buckling may occur for shallow spherical shells and the presence of unsymmetric deformation reduces the critical pressures and has influences on the postbuckling behavior of the shell However, the analysis of such a problem is very complicated Therefore, with the purpose of obtaining explicit expressions of buckling pressures and load–deflection curves, the present study restricts to considering axisymmetrically deformed FGM shallow spherical shells Specialization of Eqs (15) and (19) for an FGM shallow spherical shell under axisymmetric deformation gives equilibrium equation 1 DD2s wÀ Ds f À f w00 À f 00 w0 q ẳ R r r 20ị and compatibility equation rN r ị,r ỵ Nry, y Ny ẳ 0, Dw D f ¼ À s À w0 w00 , E1 s R r ðrN ry ị,r ỵ Ny, y ỵNry ẳ 0, where Ds ị ẳ ị00 ỵ ị0 =r and prime indicates differentiation with respect to r, i.e ị0 ẳ dị=dr 21ị 1198 D Huy Bich, H Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204 For an imperfect spherical shell, let wn(r) denotes a known small axisymmetric imperfection This parameter represents a small initial deviation of the shell surface from a spherical shape When imperfection is accounted for, Eqs (20) and (21) are, respectively, modified into forms as 1 DD2s w Ds f f w00 ỵ w00n ị f 00 w0 ỵ w0n ịq ẳ 0, R r r ð22Þ Dw 1 D f ¼ À s À w0 w00 À w00 w0n À w0 w00n : E1 s R r r r ð23Þ Eqs (22) and (23) are the basic equations used to investigate the non-linear axisymmetric stability of functionally graded shallow imperfect spherical shells These are non-linear equations in terms of two dependent unknowns w and f Stability analysis In this section, an analytical approach is used to investigate the non-linear axisymmetric response of FGM shallow spherical shells under uniform external pressure with and without the effects of temperature The FGM spherical shells are assumed to be clamped along the periphery and subjected to external pressure uniformly distributed on the outer surface of the shells and, in some cases, exposed to temperature conditions Depending on the in-plane behavior at the edge, two cases of boundary conditions, labeled Cases (1) and (2), will be considered [4,15] Case (1) The edge is clamped and freely movable (FM) in the meridional direction The associated boundary conditions are w0 ¼ 0, r ¼ 0, w ¼ W, r ¼ a, w ¼ w0 ¼ 0, Nr ¼ 0: w ¼ W, r ¼ a, w ẳ w0 ẳ 0, 25ị where W is the amplitude of deflection (i.e radial maximum displacement) and Nr0 is the fictitious compressive edge load rendering the edge immovable With the consideration of the boundary conditions (24) and (25), the deflection w is approximately assumed as follows w¼W ða2 Àr Þ2 : a4 ð26Þ Also considering the boundary conditions (24) and (25), the imperfections of the shallow spherical shells are assumed to be the same form of the deection as wn ẳ mh a2 r ị2 a4 E1 W E1 WW ỵ 2mhị rỵ r ỵ Nr0 r, 3R 2a2 ð27Þ where the small coefficient m (i.e À1 r m r1) represents imperfection size Solution form (26) is similar to first part of two-term solution presented in [4] Introduction of Eqs (26), (27) into Eq (23) and integration of the resulting equation give stress function f with E1 W r a2 r E1 WW ỵ 2mhị r 2a2 r f0 ẳ ỵ a4 r a R a C1 r C2 r C3 ln r ỵ ỵ , 28ị þ 2 r where C1,C2,C3 are constants of integration Due to the finiteness of the strains and resultants at the apex of shallow spherical shell, i.e at r ¼0, the constants C1 and C3 must be zero After determining the constant C2 from in-plane restraint condition on the ð29Þ where Nr0 ¼0 for the spherical shells with movable clamped edge Substituting Eqs (26), (27) and (29) into Eq (22) and applying Galerkin method for the resulting equation yield 64D 3E1 976E1 409E1 qẳ WW ỵ mhị WW þ 2mhÞ þ WÀ a4 7R2 693a2 R 693a2 R þ 848E1 40Nr0 WðW þ mhÞðW þ 2mhÞ þ W ỵ mhị Nr0 : R 429a4 7a2 30ị Eq (30) is used to determine the buckling loads and non-linear equilibrium paths of FGM shallow spherical shells under uniform external pressure with and without the effects of temperature conditions 4.1 Mechanical stability analysis The clamped FGM shallow spherical shell with freely movable edge (that is, Case (1)) is assumed to be subjected to external pressure q (in Pascals) uniformly distributed on the outer surface of the shell in the absence of temperature conditions In this case Nr0 ¼0 and Eq (30) leads to ! 64D 3E E1 ỵ W W 1385W ỵ1794mị qẳ R4h R4a 7R2h 693R3h R2a ỵ 848E W W ỵ mịW ỵ2mị, 429R4h R4a 31ị where Rh ¼ R=h, w ¼ 0, Nr ¼ Nr0 , À ð24Þ Case (2) The edge is clamped and immovable (IM) For this case, the boundary conditions are r ¼ 0, boundary, i.e Nr ðr ¼ aÞ ¼ Nr0 , the stress function f is obtained such that E1 W r a2 r E1 WW ỵ2mhị r 2a2 r ỵa f0 ẳ r a R a8 Ra ¼ a=R, D ¼ D=h3 , E ¼ E1 =h, W ¼ W=h: 32ị For a perfect spherical shell, i.e m ẳ 0, it is deduced from Eq (31) that ! 64D 3E 1385E 848E ỵ W W ỵ W : 33ị qẳ R4h R4a 7R2h 693R3h R2a 429R4h R4a It is evident from Eqs (31) and (33) that qðW Þ curves originate from coordinate origin and there is no bifurcation-type buckling for both perfect and imperfect FGM shallow spherical shells under uniform external pressure This indicates that externally pressure-loaded shallow spherical shells are axisymmetrically deflected at the onset of loading and buckling, if occurs, only may be extremum-type buckling For perfect spherical shells, extremum points of qW ị curves are obtained from condition dq ẳ A2BW ỵ CW ẳ 0, dW 34ị which yields p B2 AC W 1,2 ẳ B C 35ị provided B2 AC 40, 36ị where Aẳ 64D 3E ỵ , R4h R4a 7R2h B¼ 1385E , 693R3h R2a Cẳ 848E : 143R4h R4a 37ị It is easy to examine that if condition (36) is satisfied qðW Þ curve of perfect shell reaches maximum at W and minimum at W D Huy Bich, H Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204 with respective load values are qu ẳ qW ị ẳ ẵB3AC2B2 ị ỵ2B2 ACị3=2 , 3C ỵ 38ị 1199 2637n4331ịE W W ỵ 2mị 27721nịR3h R2a 42 83327 103nịE W W ỵ mịW ỵ2mị 90091nịR4h R4a ! 40P DT 40m P DT W ỵ , Rh 7R2h R2a 1n 71nịR2h R2a ỵ ql ẳ qW ị ẳ ẵB3AC2B2 ị2B2 ACị3=2 : 3C ð39Þ Eqs (38) and (39) represent upper and lower limit buckling loads of perfect FGM spherical shell, respectively, under uniform external pressure This analysis immediately indicates that if material and geometric parameters of the shell satisfy the condition (36), the shell will exhibit a snap-through behavior whose intensity is measured by difference between upper and lower buckling loads, that is by 4ðB2 ÀACÞ3=2 =3C If B2 ¼AC, load–deflection curve has only an inflexion and for the shells such that B2 oAC, the equilibrium paths are stable, that is, the deflection is monotonically increased when the pressure increases A similar behavior trend of imperfect FGM spherical shells also may be predicted This nonlinear axisymmetric response of FGM shallow spherical shells is relatively similar to non-linear behavior of FGM cylindrical panels under uniform external pressure presented in [19] 43ị where P ẳ Em am ỵ Em acm ỵ Ecm am Ecm acm ỵ , kỵ 2kỵ E ẳ E2 =h2 : 44ị 4.2 Thermomechanical stability analysis 4.2.2 Through the thickness temperature gradient In this case, the temperature through the thickness is governed by the one-dimensional Fourier equation of steady-state heat conduction established in spherical coordinate system whose origin is the center of complete sphere as d dT Kzị dT Kzị ỵ ẳ 0, dz dz z dz Tz ẳ Rh=2ị ẳ Tm , Tz ẳ R ỵ h=2ị ẳ Tc , 45ị A clamped FGM shallow spherical shell with immovable edge (that is, Case (2)) under simultaneous action of uniform external pressure q (in Pascals) and thermal load is considered The condition expressing the immovability on the boundary edge, i.e u ¼0 on r¼a, is fulfilled on the average sense as Z pZ a @u r dr dy ẳ 0: 40ị 0 @r where Tc and Tm are temperatures at ceramic-rich and metal-rich surfaces, respectively In Eq (45), z is radial coordinate of a point which is distant z from the shell middle surface with respect to the center of sphere, i.e z ẳ R ỵz and Rh=2 r z rR þ h=2 The solution of Eq (45) can be obtained as follows: Z z DT dz Tzị ẳ Tm ỵ R R ỵ h=2 , 46ị dz Rh=2 z KðzÞ RÀh=2 From Eqs (6) and (10) one can obtain the following relation in which Eq (14), imperfection and axisymmetry have been included @u f0 E2 w Fm ẳ nf 00 ỵ w00 w0 ị2 w0 w0n ỵ ỵ : 41ị @r E1 r R E1 E1 Substituting Eqs (26), (27) and (29) into Eq (41) and then putting the result into Eq (40) give Fm 5n7ịE1 2E2 Nr0 ẳ ỵ W 1n 361nịR 1nịa2 ỵ 3513nịE1 WW ỵ 2mhÞ 72ð1ÀnÞa2 ð42Þ which represents the compressive stress making the edge immovable In what follows, specific expressions of thermomechanical load–deflection curves of FGM shallow spherical shells under uniform external pressure and two types of thermal loads will be determined 4.2.1 Uniform temperature rise Environment temperature is assumed to be uniformly raised from initial value Ti, at which the shell is thermal stress free, to final one Tf and temperature change DT ¼ Tf ÀTi is independent to thickness variable The thermal parameter Fm can be expressed in terms of the DT due to Eqs (12) Subsequently, employing this expression Fm in Eq (42) and then substitution of the result into Eq (30) lead to " # 64D ð103À89nÞE 4E qẳ 4 ỵ ỵ W Rh Ra 1261nịR2h ð1ÀnÞR3h R2a " # ð1526nÀ1746ÞE 80E À W W ỵ mị ỵ 6931nịR3h R2a 71nịR4h R4a z Kzị where, in this case of thermal loading, DT ¼ Tc ÀTm is defined as the temperature difference between ceramic-rich and metal-rich surfaces of the FGM spherical shell Due to mathematical difficulty, this section only considers linear distribution of metal and ceramic constituents, i.e kẳ1, and 2zRị ỵ h Kzị ẳ Km ỵKcm : 47ị 2h Introduction of Eq (47) into Eq (46) gives temperature distribution across the shell thickness as ( DTh 4Kcm Kc ỵ Km ịh ỵ 2Kcm z Tzị ẳ Tm ỵ ln I 2hK m Kc ỵ Km 2Kcm Rh ị2 h 2R ỵzị 22z ỵ hị ln ỵ , 48ị 2Rh Kc ỵ Km 2Kcm Rh ịR ỵ zị2Rhị where z has been replaced by z ỵR after integration, and Iẳ 4Kcm Kc 2Rh 1ị ln Kc ỵ Km 2Kcm Rh ị2 Km 2Rh ỵ 1ị : ỵ Kc ỵ Km À2Kcm Rh Þð4R2h À1Þ ð49Þ Assuming the metal surface temperature as reference temperature and substituting Eq (48) into Eq (12) give Fm ¼ DThL=I, where Kcm b b L ẳ ẵxRh ỵ 1=2ị1 x 2Rh À1 2J J b Kc Kcm d þ Km ÀKc þ Kc ln À ½Rh ÀðR2h À1=4Þx Km J J d d Kc Km Kc2 ỵ2Km Kc ln 1Rh xị J Km 2J Kcm " !# Kcm Ecm acm 4Rh 4Rh 2Ecm acm ỵ þ À x þ þ 6ð2Rh À1Þ 3 J J2 1200 D Huy Bich, H Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204 Ecm acm Kc ỵRh R2h x þ 2 4ðKm ÀKc3 Þ þ3Kc ðKc2 þ 3Km ịln Km 9J Kcm 50ị and J ẳ Kc ỵ Km 2Kcm Rh , x ẳ ln 2Rh ỵ1 , 2Rh b ẳ ac ỵ am ịEc ỵ Em ị, d ẳ Ecm ac ỵ am ị ỵ acm Ec ỵ Em ị: To illustrate the proposed approach, we consider a ceramicmetal functionally graded spherical shell that consists of aluminum and alumina with the following properties [20]: Em ¼ 70 GPa, am ¼ 23  10À6 J CÀ1 , Km ¼ 204 W=mK, Ec ¼ 380 GPa, ac ¼ 7:4  10À6 J CÀ1 , Kc ¼ 10:4 W=mK, ð52Þ ð51Þ By following the same procedure as the preceding loading case we obtain thermomechanical qðW Þ curves for the case of through the thickness temperature gradient as Eq (43), provided P is replaced by L/I Such a expression is omitted here for sake of brevity Eq (43) and its temperature gradient counterpart are explicit expressions of external pressure–deflection curves incorporating the effects of temperature These expressions can be used to consider the non-linear axisymmetric response of immovably clamped FGM spherical shells subjected to external pressure and exposed to temperature conditions Inversely, the temperature difference DT may be obtained in terms of q,W as well as material and geometric properties due to these expressions whereas Poisson’s ratio is chosen to be 0.3 The effects of material and geometric parameters on the non-linear response of the perfect and imperfect FGM shallow spherical shells with movable clamped edge under uniform external pressure are considered in Figs 3–5 It is noted that in all figures W/h denotes the dimensionless maximum deflection of the shell Fig shows the effects of volume fraction index k (¼0, and 5) on the non-linear response of FGM spherical shells subjected to external pressure As can be seen, the load–deflection curves become lower when k increases This is expected because the volume percentage of ceramic constituent, which has higher 0.04 perfect 0.035 imperfect (μ = 0.1) Results and discussion 0.03 R/h = 80, a/R = 0.3 0.025 q (GPa) In this section, the non-linear response of the axisymmetrically deformed FGM shallow spherical shells is analyzed The shells are assumed to be clamped along boundary edge and, unless otherwise specified, edge is freely movable In characterizing the behavior of the spherical shells, deformations in which the central region of a shell moves toward the plane that contains the periphery of the shell are referred to as inward deflections (positive deflections) Deformations in the opposite direction are referred to as outward deflection (negative deflections) As part of the validation of the present method, the buckling behavior of an isotropic thin shallow spherical shell under uniform external pressure is analyzed, which was also considered by Li et al [15] using an updated iteration method and first order shear deformation shell theory The dimensionless upper buckling loads of a clamped immovable shell versus a2/Rh ratio are compared in Fig with result of Ref [15] As can be seen, a good agreement is achieved in this comparison study 0.02 k=0 0.015 k=1 0.01 0.005 k=5 0 W/h Fig Effects of volume fraction index k on the non-linear response of FGM shallow spherical shells 100 0.06 Present Li et al [15] perfect 60 imperfect (μ = 0.1) 0.05 isotropic thin shallow spherical shell clamped immovable edge, ν = 0.3 a/R = 0.3, k = 1.0 1: R/h = 60 2: R/h = 70 3: R/h = 80 0.04 q (GPa) qua4/Eh4 80 40 0.03 0.02 20 0.01 0 10 12 14 a2/Rh Fig Comparison of dimensionless upper buckling loads for clamped immovable isotropic thin shallow spherical shells W/h Fig Effects of curvature radius-to-thickness ratio on the non-linear response of FGM shallow spherical shells D Huy Bich, H Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204 upper and lower loads are almost identical and non-linear behavior of shells is predicted to be very mild However, when a/R ratio become higher, i.e for deeper shells, the difference between upper and lower loads to be bigger This shows that deeper spherical shells experience an unstable postbuckling behavior and a snap-through phenomenon with increasing intensity Although the buckling loads of immovable edge shells are higher, variation tendency of buckling loads is very similar for both movable and immovable edge shells The effects of environment temperature on the thermomechanical behavior of FGM shallow spherical shells with immovable clamped edge are analyzed in Fig In this case, the shells are exposed to temperature field prior to applying external pressure It is evident that the spherical shells exhibit a bifurcation-type buckling behavior due to the presence of temperature field Specifically, no deflection occurring until external pressure reaches a bifurcation 0.05 perfect imperfect (μ = 0.1) 0.04 R/h = 80, a/R = 0.3, k = 1.0 1: FM q (GPa) elasticity modulus, is dropped with increasing values of k However, the increase in the extremum-type buckling load and load carrying capacity of the ceramic-rich spherical shells is paid by a more severe snap-through behavior, i.e a bigger difference between upper and lower buckling loads Fig depicts the effects of curvature radius-to-thickness ratio R/h ( ¼60, 70 and 80) on the non-linear behavior of the externally pressurized FGM spherical shells As can be observed, the load bearing capability of the spherical shell is considerably enhanced as R/h ratio decreases Furthermore, the increase in R/h ratio is accompanied by a drop of non-linear equilibrium paths and a more severe snap-through response Fig analyzes the effects of radius of base-to-curvature radius ratio a/R ( ¼0.3, 0.4 and 0.5) on the non-linear response of FGM spherical shells subjected to uniform external pressure It is shown that the non-linear response of spherical shells is very sensitive with change of a/R ratio characterizing the shallowness of spherical shell Specifically, the enhancement of the upper buckling loads and the load carrying capacity in small range of deflection as a/R increases is followed by a very severe snapthrough behavior In other words, in spite of possessing higher limit buckling loads, deeper spherical shells exhibit a very unstable response from the postbuckling point of view Fig considers the effects of in-plane restraint conditions on the behavior of FGM spherical shells under external pressure In this figure the load–deflection curves of clamped shells with freely movable (FM) edge are plotted in comparison with those of shells with immovable (IM) edge As can be seen, although the spherical shells with immovable clamped edge have a comparatively higher capability of carrying external pressure, their response is unstable That is, these shells experience a snapthrough with much higher intensity than their movable clamped edge counterparts This trend of response is very similar for both perfect and imperfect shells Furthermore, above figures also show that the effect of initial imperfection on the non-linear response of FGM spherical shells under external pressure is not very pronounced This indicates the imperfection insensitivity of externally pressure-loaded spherical shells Fig illustrates the variation of the upper and lower buckling loads versus radius of base-to-curvature radius ratio a/R for both movable and immovable edge spherical shells As can be observed, in small range of a/R, i.e for very shallow shells, the 1201 0.03 2: IM 0.02 0.01 0 W/h Fig Effects of in-plane restraint on the non-linear response of FGM shallow spherical shells under uniform external pressure 0.25 Movable 0.06 0.2 perfect 0.05 imperfect (μ = 0.1) R/h = 80, k = 1.0 0.04 0.15 1: a/R = 0.3 qu & ql (GPa) q (GPa) 2: a/R = 0.4 3: a/R = 0.5 0.03 0.02 Immovable R/h = 80, k = 1.0 0.1 qu 0.05 0.01 ql −0.05 −0.1 −0.01 −0.15 W/h Fig Effects of radius of base-to-curvature radius ratio a/R on the non-linear response of FGM shallow spherical shells 0.2 0.4 0.6 0.8 a/R Fig Effects of the a/R ratio on the upper and lower buckling loads of FGM shallow spherical shells 1202 D Huy Bich, H Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204 value represented by intersection of curves with q axis The bifurcation point external pressure depending on the temperature difference may be predicted by the last term in Eq (43) This behavior of the spherical shells can be explained as follows The temperature field makes the shell to deflect outwards (negative deflection) prior to application of mechanical load With the action of uniform external pressure, the outward deflection is reduced and when external pressure exceeds bifurcation point load an inward deflection occurs The enhancement of temperature difference is accompanied by the increase in bifurcation points, load bearing capability in the small region of deflection and the intensity of snapthrough behavior of the spherical shells In addition, it is interesting to note that there exists an intersection point where all qðW Þ curves with various values of temperature difference pass The coordinates n n of this node can be determined from Eq (43) are ðW ,qðW ÞÞ, where n W ¼ 7Rh Ra =20Àm which is independent of thermal loading conditions Fig analyzes the effects of through the thickness temperature gradient on the non-linear response of clamped immovable FGM shallow spherical shells In this figure, the curves are plotted for three various values of temperature at ceramic surface Tc (¼ 27, 400 and 800 1C), whereas temperature at metal surface is retained at Tm ¼27 1C (room temperature) It seems that bifurcation points are lower and the intensity of snap-through is weaker under temperature gradient in comparison with their uniform temperature counterparts Subsequently, Fig 10 assesses the effects of the side of gradient on the behavior of immovable edge FGM perfect spherical shells under uniform external pressure Beside the present model of FGM spherical shell, another type of FGM shell with interchanged properties whose inner surface (i.e concave side) is ceramic-rich and outer surface (i.e convex side) is metal-rich is considered The temperature is conducted through the thickness from ceramic pure surface while metal surface temperature is maintained at reference value It is shown that 0.08 0.1 ceramic−rich outer surface ceramic−rich inner surface perfect imperfect (μ = 0.1) 0.08 0.06 R/h = 80, a/R = 0.3, k = 1.0 2: ΔT = 200°C 0.04 q (GPa) q (GPa) R/h = 80, a/R = 0.3, k = 1.0, μ = 1: ΔT = 0.06 3: ΔT = 400°C 1: Tc = 400°C 0.04 2: Tc = 800°C 0.02 0.02 0 −0.02 −0.02 W/h Fig Effects of temperature field on the non-linear response of FGM shallow spherical shells under uniform external pressure Fig 10 Effects of the side of temperature gradient on the non-linear response of FGM shallow spherical shells (immovable edge) 0.03 0.1 R/h = 80, a/R = 0.3, k = 1.0 perfect imperfect (μ = 0.1) 0.08 1: μ = − 0.3 0.025 2: μ = − 0.2 R/h = 80, a/R = 0.3, k = 1.0 0.06 0.02 1: Tc = 27°C q (GPa) q (GPa) W/h 2: Tc = 400°C 0.04 3: Tc = 800°C 3: μ = − 0.1 0.015 4: μ = 54 21 7: μ = 0.3 0.01 0.02 5: μ = 0.1 6: μ = 0.2 0.005 −0.02 W/h Fig Effects of temperature gradient on the non-linear response of FGM shallow spherical shells under external pressure W/h Fig 11 Effects of imperfection on the non-linear response of FGM shallow spherical shells (movable edge) D Huy Bich, H Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204 0.08 0.07 R/h = 80, a/R = 0.3, k = 1.0 1: μ = 0, ΔT = 1: μ = − 0.3 0.06 0.05 2: μ = 0.1, ΔT = 200 0.06 2: μ = − 0.2 3: μ = 0.2, ΔT = 400 3: μ = − 0.1 0.04 0.03 4: μ = − 0.1, ΔT = 200 4: μ = q (GPa) q (GPa) 1203 5: μ = 0.1 6: μ = 0.2 0.04 5: μ = − 0.2, ΔT = 400 0.02 7: μ = 0.3 0.02 0.01 R/h = 80, a/R = 0.3, k = 1.0 −0.02 0 W/h both bifurcation point loads and the intensity of snap-through behavior are increased when temperature is conducted from concave side In contrast, bifurcation points become lower and the postbuckling behavior of the shells is milder when conduction is performed from the convex side These thermomechanical nonlinear behaviors of FGM shallow spherical shells are similar in trend to the response of FGM cylindrical panels subjected to simultaneous action of external pressure and temperature conditions reported in [19] The effects of initial imperfection on the non-linear axisymmetric response of externally pressurized FGM spherical shells with movable and immovable clamped edge are illustrated in Figs 11 and 12, respectively In these figures, the non-linear equilibrium paths are plotted for various values of imperfection size m in which negative or positive imperfections produce perturbations in the spherical shell geometry that move the central region of a shell outward or inward, respectively [21] As can be seen, the load carrying capacity of shell in the small range of deflection is reduced when positive imperfection increases In contrast, the capability of load bearing of spherical shells is remarkably improved due to the presence of negative initial imperfection However, accompanied with this enhancement of upper buckling loads and load–deflection curves is an unstable postbuckling behavior Fig 13 depicts the interactive effects of imperfection and temperature field on the thermomechanical response of FGM spherical shells It is shown that the perfect spherical shells without the effect of environment temperature exhibit a more benign snap-through response and a more stable postbuckling behavior In contrast, the presence of temperature field is followed by both higher bifurcation buckling loads and an increase in the intensity of snap-through, even though the shells possessing inward imperfection Specially, the bifurcation buckling external pressures of the negative imperfect spherical shells are considerably higher than their positive imperfect counterparts in the same temperature condition and the amplitude of imperfection Nevertheless, this increase in the buckling load is paid by a very severe snap-through response and, as a result, very unstable equilibrium paths from the postbuckling point of view This fact may be explained that the initial imperfection in conjunction with temperature field produces an effective imperfection [21] and the amplitude of this effective imperfection is enhanced due to high Fig 13 Interactive effects of imperfection and temperature field on the non-linear response of FGM shallow spherical shells (immovable edge) 2500 perfect imperfect (μ = 0.1) 2000 R/h = 80, a/R = 0.3, k = 1.0 ΔT (°C) Fig 12 Effects of imperfection on the non-linear response of FGM shallow spherical shells (immovable edge) W/h 1500 1000 1: q = 2: q = 0.05 (GPa) 500 3: q = 0.1 (GPa) −3 −2.5 −2 −1.5 W/h −1 −0.5 Fig 14 Effects of pre-existent external pressure on the thermal non-linear response of FGM shallow spherical shells (immovable edge) temperature difference and negative initial imperfection It seems that the real curvature of spherical shell is increased because of combination between temperature field and negative imperfection Finally, Fig 14 assesses the effects of pre-existent external pressure on the thermal non-linear response of immovable edge FGM spherical shells exposed to temperature field Shown in this figure is a monotonically increasing non-linear response with outward deflection of the shells The shells exhibit a bifurcation buckling behavior when they are subjected to external pressure prior to the application of thermal load Both the bifurcation buckling loads and the capability of temperature resistance are enhanced with the increase in pre-existent uniform external pressure However, with all values of mechanical load, the thermal postbuckling behavior is very stable, i.e without a snap-through In fact, the shell deflects inwards under external pressure and when temperature reaches a specific value, i.e bifurcation point temperature, the shell surface returns to initial 1204 D Huy Bich, H Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204 state When the temperature exceeds bifurcation point, the spherical shell is monotonically deflected outwards In addition, Fig 14 also shows that the effect of initial imperfection on the non-linear behavior of the shells under this condition of loading is almost immaterial Concluding remarks This paper presents an analytical approach to investigate the non-linear axisymmetric response of FGM shallow spherical shells subjected to uniform external pressure with and without including the effects of temperature conditions The effective properties of functionally graded material are expressed as power functions of thickness variable whereas the properties of constituents are assumed to be temperature-independent Formulation for axisymmetrically deformed spherical shells is based on the classical shell theory with both geometrical non-linearity and initial geometrical imperfection are incorporated One-term deflection mode is approximately assumed and explicit expressions of buckling loads and load–deflection curves for a clamped spherical shell under mentioned loads are determined by applying Galerkin procedure From these explicit expressions, the nonlinear axisymmetric response of the shells is analyzed and the results are illustrated in graphic form The results shows that the non-linear response of the FGM spherical shells is complex and greatly influenced by the material and geometric parameters and in-plane restraint The study also reveals important role of preexistent temperature conditions and weak effect of initial imperfection on the non-linear response of FGM shallow spherical shells under uniform external pressure Acknowledgements This work was supported by the National Foundation for Science and Technology Development of Vietnam – NAFOSTED, project code 107.02.07.09 The authors are grateful for this financial support References [1] D.O Brush, B.O Almroth, Buckling of Bars, Plates and Shells, McGraw-Hill, New York, 1975 [2] N.C Huang, Unsymmetrical buckling of thin shallow spherical shells, J Appl Mech Trans ASME 31 (1964) 447–457 [3] S.C Tillman, On the buckling behavior of shallow spherical caps under a uniform pressure load, Int J Solids Struct (1970) 37–52 [4] M Uemura, Axisymmetrical buckling of an initially deformed shallow spherical shell under external pressure, Int J Nonlinear Mech (1971) 177–192 [5] N Nath, R.S Alwar, Non-linear static and dynamic response of spherical shells, Int J Nonlinear Mech 13 (1978) 157–170 [6] P.C Dumir, Nonlinear axisymmetric response of orthotropic thin spherical caps on elastic foundations, Int J Mech Sci 27 (1985) 751–760 [7] C.C Chao, I.S Lin, Static and dynamic snap-through of orthotropic spherical caps, Composite Struct 14 (1990) 281–301 [8] C.S Xu, Buckling and post-buckling of symmetrically laminated moderately thick spherical caps, Int J Solids Struct 28 (1991) 1171–1184 [9] A Muc, Buckling and postbuckling behavior of laminated shallow spherical shells subjected to external pressure, Int J Nonlinear Mech 27 (3) (1992) 465–476 [10] R.S Alwar, M.C Narasimhan, Axisymmetric non-linear analysis of laminated orthotropic annular spherical shells, Int J Nonlinear Mech 27 (4) (1992) 611–622 [11] G.P Dube, S Joshi, P.C Dumir, Nonlinear analysis of thick shallow spherical and conical orthotropic caps using Galerkin’s method, Appl Math Modelling 25 (2001) 755–773 [12] G.H Nie, Asymptotic buckling analysis of imperfect shallow spherical shells on non-linear elastic foundation, Int J Mech Sci 43 (2001) 543–555 [13] M.R Eslami, H.R Ghorbani, M Shakeri, Thermoelastic buckling of thin spherical shells, J Therm Stresses 24 (2001) 1177–1198 [14] R Shahsiah, M.R Eslami, Thermal and mechanical instability of an imperfect shallow spherical cap, J Therm Stresses 26 (7) (2003) 723–737 [15] Q.S Li, J Liu, J Tang, Buckling of shallow spherical shells including the effects of transverse shear deformation, Int J Mech Sci 45 (2003) 1519–1529 [16] R Shahsiah, M.R Eslami, R Naj, Thermal instability of functionally graded shallow spherical shell, J Therm Stresses 29 (8) (2006) 771–790 [17] T Prakash, N Sundararajan, M Ganapathi, On the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps, J Sound Vibrat 299 (2007) 36–43 [18] M Ganapathi, Dynamic stability characteristics of functionally graded materials shallow spherical shells, Composite Struct 79 (2007) 338–343 [19] N.D Duc, H Van Tung, Nonlinear response of pressure-loaded functionally graded cylindrical panels with temperature effects, Composite Struct 92 (2010) 1664–1672 [20] H Van Tung, N.D Duc, Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads, Composite Struct 92 (2010) 1184–1191 [21] L Librescu, M.P Nemeth, J.H Starnes, W Lin, Nonlinear response of flat and curved panels subjected to thermomechanical loads, J Therm Stresses 23 (2000) 549–582 ... Fig Effects of temperature field on the non-linear response of FGM shallow spherical shells under uniform external pressure Fig 10 Effects of the side of temperature gradient on the non-linear response. .. Fig Effects of temperature gradient on the non-linear response of FGM shallow spherical shells under external pressure W/h Fig 11 Effects of imperfection on the non-linear response of FGM shallow. .. used to investigate the non-linear axisymmetric response of FGM shallow spherical shells under uniform external pressure with and without the effects of temperature The FGM spherical shells are assumed