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Accepted Manuscript Nonlinear stability analysis of thin FGM annular spherical shells on elastic foundations under external pressure and thermal loads Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc PII: S0997-7538(14)00149-1 DOI: 10.1016/j.euromechsol.2014.10.004 Reference: EJMSOL 3130 To appear in: European Journal of Mechanics / A Solids Received Date: 21 May 2014 Accepted Date: 13 October 2014 Please cite this article as: Anh, V.T.T., Bich, D.H., Duc, N.D., Nonlinear stability analysis of thin FGM annular spherical shells on elastic foundations under external pressure and thermal loads, European Journal of Mechanics / A Solids (2014), doi: 10.1016/j.euromechsol.2014.10.004 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain ACCEPTED MANUSCRIPT Nonlinear stability analysis of thin FGM annular spherical shells on elastic foundations under external pressure and thermal loads Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc* Vietnam National University, Hanoi – 144 Xuan Thuy – Cau Giay – Hanoi – Vietnam Tel: +84-4-3754 79 78; Fax: +84-4-3754 77 24 RI PT Abstract: To increase the thermal resistance of various structural components in hightemperature environments, the present research deals with nonlinear stability analysis of thin annular spherical shells made of functionally graded materials (FGM) on elastic foundations under external pressure and temperature Material properties are graded in the thickness SC direction according to a simple power law distribution in terms of the volume fractions of constituents Classical thin shell theory in terms of the shell deflection and the stress function M AN U is used to determine the buckling loads and nonlinear response of the FGM annular spherical shells Galerkin method is applied to obtain closed – form of load – deflection paths An analysis is carried out to show the effects of material, geometrical properties, elastic foundations and combination of external pressure and temperature on the nonlinear stability of the annular spherical shells Keywords: Nonlinear stability analysis, FGM annular spherical shells, elastic foundations, TE D external pressure, temperature effects Introduction Nowadays, with the development of aesthetics, architectures and designs are EP becoming diversified and abundant Thus, it requires study of shape and material of structures to be cared AC C A considerable number of published researches in recent years have focused on the thermo-elastic, dynamic and buckling analyses of functionally graded material (FGM) This is mainly due to the increasing use of FGM as the components of structures in the advanced engineering FGM consisting of metal and ceramic constituents have received remarkable attention in structural applications Smooth and *Corresponding author: e-mail: Duc N.D Email: ducnd@vnu.edu.vn ACCEPTED MANUSCRIPT continuous change in material properties enable FGM to avoid interface problems and unexpected thermal stress concentrations By high performance heat resistance capacity, FGM is now chosen to use as structural components exposed to severe temperature conditions such as aircraft, aerospace structures, nuclear plants and other engineering applications RI PT As a result, the problems relating to the thermo-elastic, dynamic and buckling analyses of plates and shells made of FGMs have attracted attention of many researchers, especially the FGM spherical shells Shahsiah et al (2006) extended their previous works for isotropic material to analyze linear stability of FGM shallow SC spherical shells subjected to three types of thermal loading Ganapathi (2007) studied the problem, which is performed on the point of view of small deflection and the M AN U existence of type-bifurcation buckling of thermally loaded spherical shells The nonlinear axisymmetric dynamic stability of clamped FGM shallow spherical shells has been analyzed by Prakash et al (2007) using the first order shear deformation theory and finite element method Bich and Tung (2011) have studied the nonlinear axisymmetric response of functionally graded shallow spherical shells under uniform TE D external pressure including temperature effects Huang (1964) reported an investigation on unsymmetrical buckling of thin isotropic shallow spherical shells under external pressure Tillman (1970) investigated the buckling behavior of clamped shallow spherical caps under a uniform pressure load Uemura (1971) employed a two EP term approximation of deflection to treat axisymmetrical snap buckling of a clamped imperfect isotropic shallow spherical shell subjected to uniform external pressure AC C Nonlinear static and dynamic responses of spherical shells with simply supported and clamped immovable edge have been analyzed by Nath and Alwar (1978) by making use of Chebyshev’s series expansion Nonlinear 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 foundation have been obtained by Dumir (1985) Buckling and postbuckling behaviors of laminated spherical caps subjected to uniform external pressure also have been analyzed by Xu (1991) and Muc (1992) Duc et al (2014) investigated nonlinear axisymmetric response of FGM shallow spherical shells on elastic foundations ACCEPTED MANUSCRIPT The annular spherical shell is one of the special shapes of the spherical shells Despite the evident importance in practical applications, it is a fact from the open literature that investigations on the thermo-elastic, dynamic and buckling analyses of FGM annular spherical shell is comparatively scarce There has been recently a few of publications on the annular shells The most difficult in annular shell problems is RI PT complex calculations Alwar and Narasimhan (1992) investigated the axisymmetric nonlinear analysis of laminated orthotropic annular spherical shells, the object of this investigation is to give analytical solutions of large axisymmetric deformation of laminated orthotropic SC spherical shells including asymmetric laminates Wu and Tsai (2004) studied the asymptotic DQ solutions of functionally graded annular spherical shells by combining M AN U the method of differential quadrature (DQ) with the asymptotic expansion approach Dumir et al (2005) analyzed axisymmetric dynamic buckling analysis of laminated moderately thick shallow annular spherical cap under central ring load and uniformly distributed transverse load, applied statically or dynamically as a step function load Kiani and Eslami (2013) studied an exact solution for thermal buckling of annular TE D FGM plates on an elastic medium, Bagri and Eslami (2008) generalized coupled thermo-elasticity of functionally graded annular disk considering the Lord – Shulman theory EP To the best of our knowledge, there has been recently no publication on solution of the nonlinear stability analysis (buckling and post-buckling) of thin FGM annular AC C spherical shells on elastic foundations under temperature In this study, by using the classical thin shell theory, an approximate solution, which was proposed by Agamirov (1990)and was used by Sofiyev (2010) for truncated conical shells, the authors tried to give analytical solutions to the problem of nonlinear stability analysis of FGM thin annular spherical shells on elastic foundations under uniform external pressure and temperature Governing equations Consider an annular spherical shell made of FGM resting on elastic foundations with radius of curvature R, base radii r1 , r0 and thickness h The FGM ACCEPTED MANUSCRIPT annular spherical shell is subjected to external pressure q uniformly distributed on the RI PT outer surface as shown in Fig.1 SC Fig Configuration of a FGM annular spherical shell M AN U The annular spherical shell is made from a mixture of ceramics and metals, and is defined in coordinate system (ϕ ,θ , z) , where ϕ and θ are in the meridional and circumferential direction of the shells, respectively and z is perpendicular to the middle surface positive inwards Suppose that the material composition of the shell varies smoothly along the TE D thickness by a simple power law in terms of the volume fractions of the constituents as 2z + h k h h ) ,− ≤ z ≤ , 2h 2 Vm (z) = − Vc ( z ) Vc ( z ) = ( EP (1) where k (volume fraction index) is a non-negative number that defines the material AC C distribution, subscripts m and c represent the metal and ceramic constituents, respectively The effective properties of FGM shallow spherical shell such as modulus of elasticity, the coefficient of thermal expansion, the coefficient of thermal conduction of FGM annular spherical shell can be defined as [ E ( z), α (z), K ( z)] = [ Em ,α m , K m ] + [ Ecm ,α cm , Kcm ] ( 2z + h k h h ) ,− ≤ z ≤ 2h 2 The Poisson ratio ν is assumed to be constant v( z ) = const and Ecm = Ec − Em (2) ACCEPTED MANUSCRIPT The reaction-deflection relation of Pasternak foundation is given by (Dumir 1985, Duc et al 2014) qe = k1w − k ∆w ∂ w ∂w ∂ w + where ∆w = + is a Laplace’s operator, w is the deflection of the ∂r r ∂r r ∂θ RI PT annular spherical shell, k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of Pasternak model In the present study, the classical shell theory is used to obtain the equilibrium SC and compatibility equations as well as expressions of buckling loads and nonlinear load–deflection curves of thin FGM annular spherical shells For a thin annular M AN U spherical shell it is convenient to introduce a variable r , referred as the radius of parallel circle with the base of shell and defined by r = R sin ϕ Moreover, due to shallowness of the shell it is approximately assumed that cos ϕ = 1, Rdϕ = dr 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 TE D the ϕ ,θ , z coordinate directions, respectively, taking into account Von Karman – Donnell nonlinear terms as (Bich and Tung 2011; Dumir 1985; Xu 1991; Duc et al 2014) AC C EP ∂u w ∂w − + ( ) , ∂r R ∂r ∂v u w ∂w + − + ( )2 , ε θ0 = r ∂θ R R 2r ∂θ ∂v ∂u v ∂w ∂w γ rθ = + − + , ∂r r ∂θ r r ∂r ∂θ ε r0 = ∂2w χr = , ∂r ∂w ∂ w χθ = , + r ∂r r ∂θ ∂ w ∂w − χ rθ = r ∂r∂θ r ∂θ (3) where ε r0 and ε θ0 are the normal strains, γ rθ is the shear strain at the middle surface of the spherical shell, χ r , χθ , χ rθ are the changes of curvatures and twist The strains across the shell thickness at a distance z from the mid-plane are: ε r = ε r0 − z χr ; εθ = εθ0 − z χθ ; γ rθ = γ rθ − z χrθ (4) ACCEPTED MANUSCRIPT Using Eqs.(3) and (4), the geometrical compatibility equation of an shallow spherical shell is written as (Bich 2011; Duc, 2014) ∂ ε r0 ∂ε r0 ∂ ∂ε θ0 ∂2 ∆w − + + χ r2θ − χ r χθ , (r )− ( rγ rθ ) = − 2 r ∂θ r ∂r r ∂r ∂r r ∂r ∂θ r (5) The stress – strain relationships for annular spherical shell including temperature RI PT effect are defined by the Hooke law E( z) ( ε r , εθ ) + ν ( εθ , ε r ) − (1 + v)α∆T (1,1) , −ν E( z) σ rθ = γ rθ 2(1 + ν ) (6) SC (σ r , σ θ ) = where σ r and σ θ are the normal stress, σ rθ is the shear stress at the middle M AN U surface of the spherical shell in spherical system coordinate and ∆T denotes the increments of temperature from a surface to another one of FGM annular spherical shell The force and moment resultants of an FGM spherical shell are expressed in terms of the stress components through the thickness as: TE D h /2 ( N ij , M ij ) = ∫ σ ij (1, z ) dz , ij = ( rr ,θθ , rθ ) (7) − h /2 In case of (i = j = r ) or (i = j = θ ) for simplicity denoted N rr = N r , Nθθ = Nθ , EP M rr = M r , M θθ = M θ By using Eqs (4), (6), and (7) the constitutive relations can be given as ( E1 , E2 ) ε +νε + ( E2 , E3 ) χ + νχ − (Φ m , Φb ) , θ ) θ) ( r ( r AC C ( Nr , M r ) = −ν −ν 1− v (E ,E ) (E ,E ) (Φ , Φ ) ( Nθ , M θ ) = 22 (εθ0 + νε r0 ) + 23 ( χθ +νχ r ) − m b , −ν −ν 1− v (E ,E ) (E ,E ) ( N rθ , M rθ ) = γ r0θ + χ rθ 2(1 + ν ) +ν From the relations one can write (8) where: E1 = ∫ [ Ec + Ecm ( − h /2 2z + h k hE ) ]dz = hEm + cm , h k +1 h /2 E2 = ∫ z[ Ec + Ecm ( − h /2 RI PT M AN U h /2 (9) 2z + h k 1 ) ]dz = h Ecm ( − ), h k + 2k + h3 Em h3 Ecm 2z + h k E3 = ∫ z [ Ec + Ecm ( + ) ]dz = , h 12 2(k + 1)(k + 2)(k + 3) − h /2 h /2 2z + h k 2z + h k Ec + Ecm ( h ) × α c + α cm ( h ) ∆T (1, z )dz , − h /2 h /2 ∫ E1E3 − E22 E1 (1 −ν ) EP D= (11) TE D (Φ m , Φ b ) = (10) SC ACCEPTED MANUSCRIPT Φ E ε r0 = ( N r −ν Nθ ) + χ r + m , E1 E1 E1 E Φ ε θ0 = ( Nθ −ν N r ) + χθ + m , E1 E1 E1 2(1 + ν ) 2E γ r0θ = N rθ + χ rθ , E1 E1 Φ E M r = N r − D ( χ r + νχθ ) − b , E1 1− v E Φ M θ = Nθ − D ( χθ + νχ r ) − b , E1 1− v E M rθ = N rθ − D (1 −ν ) χ rθ E1 The nonlinear equilibrium equations of a perfect shallow spherical shell based on AC C the classical shell theory (Xu, 1991; Muc 1992) ∂N r ∂N rθ N r Nθ + + − = 0, ∂r r ∂θ r r ∂Nθ ∂N rθ N r θ + + = 0, r ∂θ ∂r r ∂ M r ∂M r ∂ M rθ ∂M rθ ∂ M θ ∂M θ + + 2( + ) + − + ( N r + Nθ ) ∂r r ∂r r ∂r ∂θ r ∂θ r ∂θ r ∂r R ∂ ∂w ∂w ∂ ∂w N θ ∂w + ( rN r + N rθ )+ ( N rθ + ) + q − k1 w + k ∆ w = r ∂r ∂r ∂θ r ∂θ ∂r r ∂θ (12) (13) (14) The Eqs (12) , (13) are identically satisfied by introducing a stress function F as ACCEPTED MANUSCRIPT ∂F ∂ F ∂2F ∂2F ∂F Nr = + , Nθ = , N rθ = − + r ∂r r ∂θ ∂r r ∂r∂θ r ∂θ (15) Substituting Eqs (3), (9), (15) into the Eqs (5) and substituting Eqs (3), (10), (15) into Eq (14) leads to (16) RI PT ∆w ∂ w ∂w ∂ w ∂w ∂ w ∆∆F = − +( − ) − 2( + ), E1 R r ∂r ∂θ r ∂θ ∂r r ∂r r ∂θ ∆F ∂F ∂ F ∂ w ∂w ∂ w ∂ F −( + ) −( + ) R r ∂r r ∂ 2θ ∂r r ∂r r ∂θ ∂r ∂2F ∂F ∂ w ∂w + 2( − )( − ) = q − k1w + k2 ∆w r ∂r ∂θ r ∂θ r ∂r ∂θ r ∂θ D∆∆w − (17) SC Regularly, the stress function F should be determined by the substitution of deflection function w into compatibility equation (16) and solving resulting equation M AN U However, such a procedure is very complicated in mathematical treatment because obtained equation is a variable coefficient partial differential equation Accordingly, integration to obtain exact stress function F ( r ,θ ) is extremely complex Similarly, the problem of solving the equilibrium is in the same situation Therefore one should find a transformation to lead Eqs (16), (17) into constant coefficient differential equations TE D Suppose such a transformation w = w(ς ), F = F0 (ς ) e 2ς , where r = r0 eς ; ς = ln r r0 (18) Substituting Eq (18) into Eqs (16), (17) and establishing a lot of calculations EP lead to the transformed equations AC C ∂ F0 ∂ F0 ∂ F0 ∂ F0 ∂ F0 ∂ F0 ∂ F0 + + + + + + E1 ∂ς ∂ς ∂ς ∂ς∂θ ∂ς 2∂θ ∂θ ∂θ r02 ∂ w ∂ w ∂ w ∂w ∂ w ∂w ∂w ∂ w = − ( + ) + 4ς ( − ) + 4ς ( − )( + ); R ∂ς ∂θ e ∂ς∂θ ∂θ e ∂ς ∂ς ∂ς ∂θ ∂4w ∂3w ∂2w ∂3w ∂4w ∂2w ∂4w D( − + − +2 2 +4 + 4)+ ∂ς ∂ς ∂ς ∂ς∂θ ∂ς ∂θ ∂θ ∂θ 4ς 2 r0 e ∂ F0 ∂F0 ∂ F0 ∂F0 ∂ F0 ∂ w ∂w 2ς − ( +4 + F0 + )−( + F0 + )( − )e + R ∂ς ∂ς ∂θ ∂ς ∂θ ∂ς ∂ς ∂2F ∂F ∂w ∂ w 2ς ∂ F0 ∂F0 ∂ w ∂w 2ς − ( 20 + F0 + )( + )e + 2( + )( − )e + ∂ς ∂ς ∂ς ∂ς∂θ ∂ς∂θ ∂θ ∂ς∂θ ∂θ (19) (20) ACCEPTED MANUSCRIPT ∂ w ∂ w 2ς 4ς 4ς − qr0 e + k1 wr0 e − k2 ( + )r0 e = ∂ς ∂θ Eqs (19) and (20) are the basic equations used to investigate the nonlinear buckling of FGM annular spherical shells These are nonlinear equations in terms of two dependent unknowns w(ς ) and F0 (ς ) RI PT Stability analysis In this section, an analytical approach is used to investigate the nonlinear stability analysis of FGM annular spherical shell under mechanical loads including the SC effects of temperature The FGM annular spherical shell is assumed to be simply supported along the periphery and subjected to mechanical loads uniformly distributed M AN U on the outer surface and the base edges of the shell Depending on the in-plane behavior at the edge of boundary conditions will be considered in case the edges are simply supported and immovable For this case, the boundary conditions are u = 0, w = 0, ∂ w ∂w − = 0, N r = N0 , N rθ = 0, with ς = (i.e at r = r0 ) ∂ς ∂ς (21) TE D where N is the fictitious compressive load rendering the immovable edges The boundary conditions (21) can be satisfied when the deflection w is approximately assumed as follows (Agamirov, 1990; Sofiyev, 2010) EP w = Weς sin( β1ς )sin(nθ ), β1 = mπ r , a = ln a r0 (22) AC C where W is the maximum amplitude of deflection and m, n are the numbers of half waves in meridional and circumferential direction, respectively The form of this approximate solution was proposed by Agamirov (1990) and it was used by Sofiyev (2010) for FGM truncated conical shells Introduction of Eqs (22) into Eq (19) gives ∂3F ∂2F ∂ F0 ∂ F0 ∂ F0 ∂ F0 ∂ F0 ( + 30 + 20 + + + + ) E1 ∂ς ∂ς ∂ς ∂ς∂θ ∂ς ∂θ ∂θ ∂θ =− ς r eW [(1 − β12 − n )sin( β1ς ) + 2β1 cos( β1ς )]sin(nθ ) + R (23) parameter Φ m ACCEPTED MANUSCRIPT can be expressed in terms of the ∆T : Φ m = Ph∆T Subsequently, employing this expression Φ m in Eq (34) and then substitution of the result N into Eq (28) lead to q= − A0 P∆TRh P∆TA5 Rh2 * − W + (1 − v) B1 (1 − v) B1 R02 M AN U SC RI PT D* R A E * A R + K D* A K D* A R A0 ( E2*3 A9 Rh3 + R02 E1* A10 Rh2 ) h h W * + + + h + + 2a B1 B1 R0 (−1 + v) ( −1 + e ) π B1 R0 B1 R0 E * A R ( A5 A9 E2* Rh4 + E1* A5 A10 Rh3 R02 ) A0 E1* A8 Rh3 h (W * ) + + + + 2a 2a (−1 + v) ( −1 + e ) π B1 R0 (−1 + v) ( −1 + e ) π B1 R0 B1 R0 E* A R4 E1* A5 A8 Rh4 1 h (W * )3 + + ( 35) 2a (−1 + v) ( −1 + e ) π R0 B1 B1 R0 E α + Emcα c Emcα mc + where: P = Ecα c + c mc k +1 2k + * Eq (35) shows that when thermal load ( ∆T ) ≠ the deflection curve q(W ) starting − A0 P∆TRh (1 − v) B1 TE D from a pressure point on the axis q defined by the term: 3.2.2 Through the thickness temperature gradient The metal-rich surface temperature Tm is maintained at stress free initial value EP while ceramic-rich surface temperature Tc is elevated and in this case, the temperature through the thickness is governed by the one-dimensional Fourier equation of steady- AC C state heat conduction established in spherical coordinate system whose origin is the center of complete sphere as (Bich and Tung 2011) () d dT K z dT K z + = 0; T | h = Tm ; T | h = Tc ; z = R− z=R+ d z d z zd z 2 () (36) where 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 = R + z and R − h / ≤ z ≤ R + h / The solution of Eq (36) can be obtained as follows 13 ACCEPTED MANUSCRIPT z ∆T dτ T z = Tm + h ∫h τ K (τ ) R+ dz () ∫ h R− 2 () z K z R− (37) where, in this case of thermal loading, ∆T = Tc − Tm is the temperature difference between ceramic-rich and metal-rich surfaces of the FGM spherical shell Due ceramic constituents, i.e k = , and () K z = K m + K cm 2( z − R ) + h 2h RI PT to mathematical difficulty, this section only considers linear distribution of metal and (38) SC Substituting Eq (38) into Eq (37) gives temperature distribution across the shell thickness as M AN U ( K c + K m ) h + K cm z K cm 2( R + z ) − ln ln + 2hK m 2R − h ∆T ( K c + K m − K cm Rh ) h T ( z ) = Tm + I 2(2 z + h) + ( K + K − K R ) ( R + z )2 R − h c m cm h (39) where z has been replaced by z + R after integration TE D Assuming the metal surface temperature as reference temperature and substituting Eq (39) into Eq (11) give Φ m = h∆TL I EP The explicit analytical expressions of L, I are calculated and given in the Appendix By following the same procedure as the preceding loading case we obtain AC C thermo-mechanical q(W* ) curves for the case of through the thickness temperature gradient as Eq (35) provided is replaced by L / I Such a expression is omitted here for sake of brevity Results and discussion In this section, the nonlinear response of the FGM annular spherical shell is analyzed The shell is assumed to be simply supported along boundary edges and, unless otherwise specified, edges are freely movable In characterizing the behavior of the spherical shell, deformations in which the central region of a shell moves toward 14 ACCEPTED MANUSCRIPT 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) The following properties of the FGM shell are chosen (Bich and Tung 2011; Duc Em = 70GPa , α m = 23 × 10 −6 o C −1 , K m = 204 W Ec = 380GPa α m = 7.4 × 10 −6 ; mK o C −1 , K m = 10.4 W mK SC where Poisson’s ratio is chosen to be v = 0.3 RI PT et al 2014): The effects of material and geometric parameters on the nonlinear response of the FGM annular spherical shells under mechanical loads (without effect temperature M AN U and elastic foundations K1 = K = ) are presented in Figs 2–5 It is noted that in all EP TE D figures W / h denotes the dimensionless maximum deflection of the shell Fig Effects of curvature radius- AC C Fig Effects of volume fraction index k on the nonlinear response of the FGM thickness ratio on the nonlinear response annular spherical shell under external of FGM annular spherical shells under pressure external pressure Fig.2 shows the effects of volume fraction index k (0,1,5, +∞) on the nonlinear response of the FGM annular spherical shell subjected to external pressure (mode ( m , n ) = (1,11) ) 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 elasticity modulus, is dropped with increasing values of k 15 ACCEPTED MANUSCRIPT Fig.3 depicts the effects of curvature radius - thickness ratio R / h (200, 300, 400, and 500) on the nonlinear behavior of the external pressure of the FGM annular spherical shells (mode (m, n) = (1,11) ) From Fig we can conclude that when the annular spherical shells get thinner - corresponding with R / h getting bigger, the critical buckling loads will get smaller RI PT Fig.4 analyzes the effects of base-curvature radius ratio r1 / r0 on the nonlinear response of FGM annular spherical shells subjected to uniform external pressure It is shown that the nonlinear response of annular spherical shells is very sensitive with SC change of r1 / r0 ratio characterizing the shallowness of annular spherical shell Specifically, the enhancement of the upper buckling loads and the load carrying M AN U capacity in small range of deflection as r1 / r0 increases is followed by a very severe snap - through behaviors In other words, in spite of possessing higher limit buckling loads, deeper spherical shells exhibit a very unstable response from the post-buckling point of view Furthermore, in the same effects of base-curvature radius ratio r1 / r0 the load of the nonlinear response of FGM annular spherical shells is higher when the TE D shallowness of annular spherical shell ( H ) is smaller, where H is the distance between two radius r1 , r0 , and calculated by 2 r0 r1 H ( r1 , r0 ) = R − r − R − r = R − − − R R 2 AC C EP 16 ACCEPTED MANUSCRIPT Fig.4 Effects of radius of base-curvature radius ratio r1 / r0 on the nonlinear response of FGM annular spherical shells Figure examines the dependence of the nonlinear response of FGM annular spherical shells on the mode ( m , n ) It is easily recognized that with m = , the more increased the value of n , the higher increasing of the value of extreme point, RI PT corresponding to the higher load capacity of the shells Note that, when m is even or m ≥ , the graphic consists of symmetric curves through the origin of the coordinate TE D M AN U SC system and the extreme point does not exist in the load-deflection curves Fig Effects of the elastic foundations nonlinear response of FGM annular ( K1 , K2 ) on the nonlinear response of FGM spherical shells annular spherical shells EP Fig Effects of mode ( m , n ) on the AC C The effects of temperature and elastic foundations on the nonlinear response of the FGM annular spherical shells under uniform external pressure are presented in Figs 6–10 Effects of the elastic foundations ( K1 , K ) on the nonlinear response of FGM annular spherical shells are shown in Fig Obviously, elastic foundations played positive role on nonlinear static response of the FGM annular spherical shell: the large K1 and K coefficients are, the larger loading capacity of the shells is It is clear that the elastic foundations can enhance the mechanical loading capacity for the FGM 17 ACCEPTED MANUSCRIPT annular spherical shells, and the effect of Pasternak foundation K on critical uniform external pressure is bigger than the Winkler foundation K1 Fig.7 presents the effects of temperature and elastic foundations on the nonlinear response of FGM annular spherical shells under uniform external pressure As shown in Fig.7, the temperature makes the annular spherical shell to be deflected outward RI PT prior to mechanical loads acting on it Under mechanical loads, outward deflection of the shell is reduced, and external pressure exceeds bifurcation point of load, an inward deflection occurs In this context, Fig.7 also shows the bad effect of temperature on the nonlinear response of the FGM annular spherical shells Indeed, the mechanical TE D M AN U SC loading ability of the system has been reduced in the presence of temperature Fig Effects of temperature gradient on rise and elastic foundations on the the nonlinear response of FGM annular nonlinear response of FGM annular spherical shell AC C EP Fig Effects of uniform temperature spherical shells The effects of temperature gradient through the thickness on the nonlinear response of FGM annular spherical shells under uniform external pressure are shown in Fig In this case, the metal-rich surface temperature Tm is maintained at initial value Tm = 27 o C (room temperature) and temperature is transmitted from ceramicrich surface through the thickness of annular In Fig and Fig 8, we can realize that the buckling load capacities of the shell in the cases of uniform temperature rise are better than the one in cases of gradient temperature through the thickness of the shell 18 ACCEPTED MANUSCRIPT Fig.9 shows the effects of volume fraction index k on the thermal nonlinear response of FGM annular spherical shells under uniform temperature rise As can be observed, the annular spherical shells deflect outwards under thermal loads and the response of the annular spherical shells are relatively benign, i.e there is no snapthrough phenomenon and bifurcation type buckling RI PT Fig.10 presents the effects of curvature radius-thickness ratio on the thermal nonlinear response of FGM annular spherical shells under uniform temperature rise It is evident that the effects of curvature radius-thickness ratio on the thermal nonlinear response is not considerable, i.e the thermal load – deflection curves approach TE D M AN U SC contiguous when the ratio R / h changes Fig 10 Effects of curvature radius- k on the thermal nonlinear response of thickness ratio on the thermal nonlinear FGM annular spherical shells response of FGM annular spherical shells EP Fig Effects of volume fraction index AC C Figure 11 examines the dependence of the thermal nonlinear response of FGM annular spherical shells on the mode ( m , n ) in the presence of temperature It is easily recognized that with m = , the more increased the value of n the higher increasing of the thermal nonlinear response Fig 12 investigates the effects of the pre-existent external pressure on the thermal loading ability of the annular spherical shells in the presence of temperature Shown in this figure is a monotonically increasing nonlinear response with outward deflection of the annular spherical shells The annular spherical shells exhibit a bifurcation buckling behavior when they are subjected to external pressure prior to the 19 ACCEPTED MANUSCRIPT 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 RI PT point temperature, the shell surface returns to initial state When the temperature TE D M AN U SC exceeds bifurcation point, the spherical shell is monotonically deflected outwards Fig 11 Effects of mode ( m , n ) on the Fig 12 Effects of pre-existent external thermal nonlinear response of FGM pressure on the thermal nonlinear response annular spherical shells of FGM shallow spherical shells EP Concluding remarks AC C Due to practical importance of FGM annular spherical shells and the lack of investigations on stability of these structures, the present paper aims to propose an analytical approach to study the problem of nonlinear stability analysis of FGM thin annular spherical shells on elastic foundations under uniform external pressure and temperature Based on the classical shell theory, the equilibrium and compatibility equations are derived in terms of the shell deflection and the stress function This system of equations has been transformed into another system of more simple equations Galerkin method is used to get the explicit expression of postbuckling load – deflection curves of the shells The effects of material, geometrical properties, elastic foundations and combination of external pressure and temperature on the 20 ACCEPTED MANUSCRIPT nonlinear buckling and postbuckling of the FGM annular spherical shells are analyzed AC C EP TE D M AN U SC RI PT and discussed 21 ACCEPTED MANUSCRIPT Acknowledgement This paper was supported by the Grant in Mechanics “Nonlinear analysis on stability and dynamics of functionally graded shells with special shapes” code QG.14.02 of Vietnam National University, Hanoi The authors are grateful for this support RI PT References Agamirov, V.I., 1990 Dynamic problems of nonlinear shells theory Science Edition, Moscow (in Russian) SC Alwar, R.S., Narasimhan, M.C., 1992 The axisymmetric non-linear analysis of laminated orthotropic annular spherical shells, Int J of Nonlinear Mech 27(4), 611 – 622 M AN U Bagri, A., Eslami, M.R., 2008 Generalized coupled thermoelasticity of functionally graded annular disk considering the Lord–Shulman theory Composite Structure 83, 168-179 Bich, D.H., Tung, H.V., 2011 Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects Int J.of Nonlinear Mech 46, 1195 – 2004 Duc, N.D., Anh, V.T.T., Cong, P.H., 2014 Nonlinear axisymmetric response of FGM shallow spherical shells on elastic foundations under uniform external pressure and temperature J Eur J of Mech – A/Solids 45, 80-89 TE D Dumir, P.C., 1985 Nonlinear axisymmetric response of orthotropic thin spherical caps on elastic foundations Int J Mech Sci 27, 751–760 EP Dumir, P.C., Dube, G.P., Mallick A., 2005 Axisymmetric buckling of laminated thick annular spherical cap Nonlinear Science and Numerical Simulation 10, 191-204 Ganapathi, M., 2007 Dynamic stability characteristics of functionally graded materials shallow spherical shells J Compos Struct 79, 338-343 AC C Huang, N.C., 1964 Unsymmetrical buckling of thin shallow spherical shells, J Appl Mech Trans ASME 31, 447–457 Kiani, Y., Eslami, M.R., 2013 An exact solution for thermal buckling of annular FGM plates on an elastic medium Composites Part B: Engineering 45, 101-110 Muc, A., 1992 Buckling and postbuckling behavior of laminated shallow spherical shells subjected to external pressure Int J Nonlinear Mech 27(3), 465–476 Nath, N., Alwar, R,S,, 1978 Non-linear static and dynamic response of spherical shells Int J Nonlinear Mech 13, 157–170 22 ACCEPTED MANUSCRIPT Prakash, T., Sundararajan, N., Ganapathi M., 2007 On the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps J Sound Vibrat 299, 36-43 Shahsiah, R., Eslami, M.R., Naj, R., 2006 Thermal instability of functionally graded shallow spherical shell J Therm Stresses 29(8), 771–790 RI PT Sofiyev, A.H., 2010 The buckling of FGM truncated conical shells subjected to axial compressive load and resting on Winkler-Pasternak foundations Int.J.of Pressure Vesels and Piping 87, 753-761 Tillman, S.C., 1970 On the buckling behavior of shallow spherical caps under a uniform pressure load Int J Solids Struct 6, 37–52 Uemura, M., 1971 Axisymmetrical buckling of an initially deformed shallow spherical shell under external pressure Int J Nonlinear Mech 6, 177–192 SC Wu, C.P., Tsai, Y.H., 2004 The asymptotic DQ solutions of functionally graded annular spherical shells Eur J of Mech A/Solids 23, 283 – 299 Appendix I a3 = 16( β14 − β12 ), b3 = 32 β13 , a4 = 16( β14 − β12 + β12 n − n + n ), a5 = 32( β13 + β1n ), b4 = 32( β13 + β1n ), b5 = 16( β1 − β12 + β12 n − n + n ), b6 = 16( β14 − β12 ), c3 = 0.5( β12 n − β12 ), c4 = 0.25( β1 n − β1 + β13 ), EP TE D a = 32 β13 , Appendix II M AN U Xu, C.S., 1991 Buckling and post-buckling of symmetrically laminated moderately thick spherical caps Int J Solids Struct 28,1171–1184 −2mπ a (−1 + e5 a (−1)m ) C1 = a , C2 = b5 , B1 = , n(25a + m 2π ) c5 = 0.5β12 , c6 = − c , A0 = m 2π (1 − e6 a ) ( 9a + m 2π ) ; AC C mπ mπ mπ mπ A1 = h11 + h12 t3 + t4 − 2t4 n + h13 t6 + t5 + h14 t5 + t6 + h15 t4 − t3 − 2t3 n + a a a a 2m 2π −2m 2π 2m 2π 3mπ 3mπ 3mπ + h16 t + t − t + h t + t + t + h t5 + t6 − t5 + 3 17 4 18 2 a a a a a a −2m 2π 3mπ 2mπ 2mπ + h19 t6 + t5 + t6 + h110 t3 − t4 + h111 t4 + t3 , a a a a 23 ACCEPTED MANUSCRIPT mπ 5mπ m 2π 5mπ 2mπ m 2π A2 = h21 −6t1 + t1 + t2 + h22 6t2 − t2 + t1 + h23 −t3 + t4 + t3 + n 2t3 + a a a a a a 2 m 2π − m 2π 2mπ m 2π 2mπ 2mπ + h24 t3 + t3 − t4 + n 2t4 + h25 t5 + t6 − t5 + h26 t6 + t5 + t6 + a a a a a a mnπ mnπ mπ mπ + h27 + h28 2nt1 − t2 + h29 2nt2 + t1 + h210 −3t1 + t2 + n 2t1 + h211 3t2 + t1 − n 2t2 , a a a a ( )( ) A6 = 24a (9a + m π ) ( 2 ) + mπ ( e 6a −8m3π −3m 2π − a + 3e5 a m 2π ( −1) + 7e5 a ( −1) a m m 3an(625a + 250 m π a + m π ) m 2π ( e6 a − 1) 24 ( 9a + m 2π ) ; A7 = 2 4 )( − −9t2 a − 6t1mπ a + t2 m 2π + t2 n a 8a (9a + m π ) ), 2 SC A5 = )( m 2π ( −m 2π + 3a e a − 3a + e a m 2π ) M AN U A4 = ( −m 2π e6 a − −9t1a + 6t2 mπ a + t1m 2π + t1n a RI PT 2a 2 4 2 2 4 m π e − a + 2m π a + m π − 2n a + 2m n π a + n a A3 = , a ( a + m 2π ) 16a ( 4a + m 2π ) ; TE D m 2π ( t6 a + t5 mπ a − vt6 a + 2vt6 m 2π − 3vt5 mπ a ) − + −π m −1 + ea ( −1) m a + m 2π 16a ; A = A8 = ; mπ ( t5 a − t6 mπ a − vt5 a + 2vt5 m 2π + 3vt6 mπ a ) an + 2 π a a + m ( ) ( ) h12 = h13 = AC C EP aπ m ( −1 + e3a (−1) m ) + 2 n ( 9a + m π ) 3a m 2 2 2 mπ ( −1 + e ( −1) )( −t2 mπ a + 3t1a − n t1a + vm π t1 + 5vt2 mπ a − 6vt1a ) A10 = − + an ( 9a + m 2π ) 3a m 2 2 2 ( −1 + e (−1) )( mπ t1a + 3t2 a − n t2 a − 5vmπ t1a + vt2 m π − 6vt2 a ) + 2 n ( 9a + m π ) m 3π ( e a − 1) 4a 2π m ( 3n − 1) + ( n − 1)( m 3π − 2a ) π m (e a − 1) ( a − π m ) h11 = + ; 512a ( n − 1)(4a + m 2π ) 256a ( n − 1)( a + m 2π ) −π m ( e a − 1) ( 2π m − a ) 32 ( a + m 2π ) π m2 ( e a − 1) ( 2π m − a ) 16a ( a + m2π ) + + π m ( e a − 1)(π m − 2a ) 32a (4a + 5a m 2π + m 4π ) 9m7 nπ 10 ( −1 + e a ) ; 16a (4a + 5a m 2π + m4π ) h14 = −2h12 ; h15 = − h13 ; 24 ; ), ACCEPTED MANUSCRIPT h17 = h18 = −3aπ m (−1 + e ) π m ( −1 + e a ) ; + ( a + 5a m 2π + 4m 4π ) ( a + 4m 2π ) 2a π m ( e a − 1) 2π m3 − a 2π m + a3 + am2π ( a + 4m 2π )( a + m 2π ) π m (−1 + e a ) ( 3aπ m − 2a − 2π m ) ( a + 4m 2π )( a + m 2π ) ; π m ( e a − 1) −4π m3 + 2a 2π m + a + am 2π ; h19 = ( a + 4m 2π )( a + m2π ) ( 8am π ( a − mπ ) −1 + ( −1) e −π nm ( −1 + e a ) nπ m (−1 + e a ) = ; h = ; h21 = 111 2 2 2 16a (a + m π ) 16(a + m π ) 9n 9a + 10a m2π + m 4π h24 = ( 4amπ ( −1 + (−1) m e3a )( −2amπ + m2π + 3a ) 9n ( 9a + 10a m 2π + m 4π ) −8aπ m ( −1 + (−1) m e5 a )( 25a − 3m 2π ) 3n(625a + 250a m π + 9m π ) 2 4 h26 = −h24 , h28 = , h211 = 3(625a + 250a m 2π + 9m 4π ) 4m 2π ( −1 + (−1)m e5a )( 3m 2π + 25a ) h210 = 160 a m 2π , 3n (625 a + 250 a m 2π + m 4π ) , h25 = 3(625a + 250a m 2π + 9m 4π ) −8m 2π ( −1 + (−1) m e5a )( −5a + 10a mπ + 3m3π ) 3an(625a + 250a m 2π + 9m 4π ) , 4π m ( −1 + (−1) m e5 a )( 50a mπ − 25a − 3am 2π + 16m3π ) 3n(625a + 250a m 2π + 9m 4π ) t1 { A(1 − β = t2 {B(1 − β = t3 {C (−β n = − n ) + B β1} − n ) + Aβ1} ( A2 + B ) 1 t4 , AC C ( A2 + B ) C1C2 + (32β13 + 32β1n ) {C β = t5 = , + β1 − β13 ) − 64( β13 + β1n ) β12 } , , TE D h29 = 12n(25a + m 2π ) ) ), 160a m 2π (−1 + (−1) m e5 a ) , 3n(625a + 250a m 2π + 9m 4π ) −40π m3 a ( −1 + ( −1)m e5 a ) EP h27 = amπ ( −1 + ( −1) m e5 a ) , h23 = m 3a SC h22 = M AN U h110 RI PT h16 = , t6 = , + 16( β13 + β1n )(− β1n + β1 − β13 )} C1C2 + (32β13 + 32β1n ) , 4{( β14 − β12 )( β1n − β1 + β13 ) − 4β13 ( β12 n − β12 )} (16β14 − 16β12 ) + 1024 β16 , (16β14 − 16β12 )2 + 1024 β16 , 8{( β14 − β12 )( β12 n − β12 ) + ( β1n − β1 + β13 ) β13 } ( + Rh ) ; η = ln K c ; J = K + K − K R ; c m cm Km h ( − Rh ) δ = (α c + α m )( Ec + Em ) ; υ = Ecm (α c + α m ) + α cm ( Ec + Em ) ψ = ln 25 ; ACCEPTED MANUSCRIPT δ 2h ψ − + 2R − h J − K cmυ R R − − ψ J h h + K cm E cm α cm J2 + E cm α cm K m3 − K c3 + K c K c2 + K m2 η ; J K c2m (K m υ ψ υ − 1 − − J R h J K cm 1 4R2 R E cm α cm − + + ψ + 3h h J 6 ( ) ( K c (2 R − h ) K cm 8h2 ln + J2 K m (2 R + h ) J R − h (K m ) − K c2 + 2η K m K c + h R R + −ψ + h h ( R − h ) ) EP TE D M AN U SC ( − Kc +η Kc )+ ) AC C I = R δ ψ h + − − J RI PT K cm δ J2 L = 26 ACCEPTED MANUSCRIPT Highlights For the manuscript “Nonlinear stability analysis of thin FGM annular spherical shells on elastic foundations under external pressure and thermal loads” RI PT by Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc* • We investigated nonlinear stability analysis of FGM annular spherical shells • The shell resting on elastic foundations SC • The shell subjected by external pressure and thermal loads • The classical thin shell theory and the stress function are used AC C EP TE D M AN U • The nonlinear response of the shell is analyzed and discussed ... analysis of FGM thin annular spherical shells on elastic foundations under uniform external pressure and temperature Governing equations Consider an annular spherical shell made of FGM resting on elastic. .. Effects of temperature gradient on rise and elastic foundations on the the nonlinear response of FGM annular nonlinear response of FGM annular spherical shell AC C EP Fig Effects of uniform temperature... radius- k on the thermal nonlinear response of thickness ratio on the thermal nonlinear FGM annular spherical shells response of FGM annular spherical shells EP Fig Effects of volume fraction index