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Accepted Manuscript Post-Buckling of Sigmoid-Functionally Graded Material Toroidal Shell Segment Surrounded by an Elastic Foundation Under Thermo-Mechanical Loads Dao Huy Bich, Dinh Gia Ninh PII: DOI: Reference: S0263-8223(15)01047-8 http://dx.doi.org/10.1016/j.compstruct.2015.11.044 COST 6999 To appear in: Composite Structures Please cite this article as: Bich, D.H., Ninh, D.G., Post-Buckling of Sigmoid-Functionally Graded Material Toroidal Shell Segment Surrounded by an Elastic Foundation Under Thermo-Mechanical Loads, Composite Structures (2015), doi: http://dx.doi.org/10.1016/j.compstruct.2015.11.044 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 POST-BUCKLING OF SIGMOID-FUNCTIONALLY GRADED MATERIAL TOROIDAL SHELL SEGMENT SURROUNDED BY AN ELASTIC FOUNDATION UNDER THERMO-MECHANICAL LOADS Dao Huy Bich1, Dinh Gia Ninh2* Vietnam National University, Hanoi, Vietnam School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam *Corresponding author Tel: +84 988 287 789 Email address: ninhdinhgia@gmail.com and ninh.dinhgia@hust.edu.vn Abstract: The nonlinear buckling and post-buckling of ceramic-metal-ceramic layers (SFGM) toroidal shell segment surrounded by elastic foundation under thermo-mechanical loads are investigated with an analytical approach in this paper Based on the classical thin shell theory with geometrical nonlinearity in von Karman-Donnell sense, Stein and McElman assumption and Pasternak foundation model, the governing equations of nonlinear buckling of S-FGM toroidal shell segment are analyzed The static critical buckling loads and the postbuckling analyses in two cases - movable and immovable boundary conditions including temperature effects are obtained Furthermore, the effects of geometry ratios, characteristic of materials, elastic foundation and thermal environment on the nonlinear buckling of shells are presented Key words: Sigmoid Functionally graded material; nonlinear buckling; toroidal shell segments; elastic foundation; thermo-mechanical load Introduction Japanese scientists firstly founded the functionally graded materials in Sendai area in 1984 [1] Ever since then, a myriad of studies on this material have been published, attracting scholarly attention worldwide Functionally graded materials are composite materials composed of two phases: ceramic and metallic constituent materials Mechanical and physical features of FGMs are better than fiber reinforced laminated composite materials because of such properties as stress concentration, oxidation resistance, high toughness, and heat-resistance Hence, FGMs are used to manufacture heat-resistant and lightweight structures in the aerospace industry, mechanical and medical industry and so forth Therefore, the nonlinear buckling and post-buckling problem of FGM structures have fueled a great deal of research Many FGM structures on elastic media have been studied for a long time by many scientists The simplest model for the elastic foundation is Winkler [2], which features a series of separated springs without coupling effects between each other Then the model was expanded by Pasternak [3] to incorporate a shear layer Bagherizadehet al [4] investigated the mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation Theoretical formulations were presented based on a higherorder shear deformation shell theory Sofiyev [5-8] studied the buckling of FGM shells on elastic foundation Moreover, the post-buckling of FGM cylindrical shells surrounded by an elastic medium was presented by Shen [9-11] Bich et al [12-13] gave an analytical approach to present the nonlinear vibration and buckling for FGM shell on the elastic foundation The static, dynamic and vibration analyses of FGM doubly curved panel resting on Pasternak-type elastic foundation based on the first order shear deformation and the modified Sanders shell theories using the Navier type solution and the Laplace transform were investigated by Y Kiani et al [14] Noda [15], Praveen et al [16] first discovered the heat-resistant FGM structures and studied temperature-dependent material properties in thermo-elastic analyses The postbuckling analysis of axially-loaded functionally graded cylindrical shells in thermal environments using the classical shell theory with von Kármán-Donnell-type of kinematic nonlinearity was investigated by Shen [17] In Shen [18], the post-buckling analysis of imperfect stiffened laminated cylindrical shell under combined external pressure and thermal loading using the formulation based on a boundary layer theory of shell buckling including the effects of nonlinear prebuckling deformations, nonlinear large deflection in the postbuckling range and initial geometrical imperfections of the shell was studied Kadoli and Ganesan [19] studied the linear thermal buckling and free vibration for functionally graded cylindrical shells subjected to a clamped-clamped boundary condition with temperaturedependent material properties Furthermore, the analytical approach to investigate the nonlinear axisymmetric response of functionally graded shallow spherical shells subjected to uniform external pressure incorporating the effects of temperature was given by Bich and Tung [20] M S Boroujerdy and M R Eslami [21] investigated the thermomechanical instability of FGM shallow spherical shells and surface-bonded piezoelectric actuators based on the classical shell theory and Sanders nonlinear kinematics equations An analytic approach for thermoelastic bending of FGM cylindrical shell under a uniform transverse mechanical load and non-uniform thermal loads using the equations with the radial defection and horizontal displacement was discussed by H L Dai and T Dai [22] In addition, M R Eslami et al [23] pointed out bifurcation behavior of heated FGM conical shell The heat conduction equation of the shell was resolved based on an iterative generalized differential quadrature method General nonlinear equilibrium equations and the associated boundary conditions were obtained using the virtual displacement principle in the Donnell sense Huang and Han [24-25] investigated the nonlinear postbuckling behaviors of functionally graded cylindrical shells under uniform radial pressure using the nonlinear large deflection theory of cylindrical shells with the temperature-dependent material properties taken into account In the analysis, the nonlinear strain-displacement relations of large deformation and the Ritz energy method were used while by taking the temperature-dependent material properties into account; various effects of the external thermal environment were also investigated Dung and Hoa [26] studied the nonlinear buckling of eccentrically stiffened functionally graded circular cylindrical shells under external pressure, using approximate three-terms solution of deflection and Galerkin’s method to give explicit expression for critical load and postbuckling load-deflection curves Duc and Thang [27] researched the nonlinear response of imperfect eccentrically stiffened S-FGM thin circular cylindrical shells surrounded on elastic foundation under uniform radial load The approximate solution of deflection in this paper, however, was only one-term with linear buckling shape and obtained postbuckling equilibrium paths incompletely illustrated the nonlinear response of the shell The more correct on can be seen in [26] Based on third order shear deformation shell theory, the buckling analysis of a two dimensional FGM cylindrical shell embedded in an outer elastic medium under combined axial and transverse loading was investigated by Allahkarami et al [28] Francesco et al [29] analyzed recovery of through-the-thickness transverse normal and shear strains and stresses in statically deformed FG doubly-curved sandwich shell structures and shells of revolution using the generalized zigzag displacement field and the Carrera Unified Formulation Three different through-the-thickness distributions of the volume fractions of constituents and two different homogenization techniques were employed to deduce the effective moduli of linear elastic isotropic materials Toroidal shell segment has been used in such applications as satellite support structures, fusion reactor vessels, rocket fuel tanks, diver’s oxygen tanks and underwater toroidal pressure hull Today, FGMs consisting of metal and ceramic constituents have received remarkable attention in structural applications The smooth and continuous change in material properties enables FGMs to avoid interface problems and unexpected thermal stress concentrations Some components of the above-mentioned structures may be made of FGM Stein and McElman [30] investigated the homogenous and isotropic toroidal shell segments with the buckling problem McElman [31] carried out the eccentrically stiffened shallow shells of double curvature with the static and dynamic behaviors in NASA technical note The initial post-buckling behavior of toroidal shell segments subject to several loading conditions based on the basic of Koiter’s general theory was studied by Hutchinson [32] Recently, there have been some new publications about toroidal shell segment structures Bich et al [33] studied the buckling of eccentrically stiffened functionally graded toroidal shell segment under axial compression, lateral pressure and hydrostatic pressure based on the classical thin shell theory, the smeared stiffeners technique and the adjacent equilibrium criterion Furthermore, the nonlinear buckling and post-buckling problems of ES-FGM surrounded by an elastic medium under torsional load based on the analytical approach are investigated by Bich et al [34] To the best of the authors’ knowledge, this is the first time an analytical approach to the nonlinear buckling of Sigmoid FGM toroidal shell segments subjected to lateral pressure surrounded by an elastic foundation and in a thermal environment has studied In the present paper, the nonlinear buckling and post-buckling behaviors of S-FGM toroidal shell segments surrounded by an elastic medium under lateral pressure loads including temperature effects are investigated The governing equations are derived based on the classical shell theory with the nonlinear strain-displacement relation of large deflection Moreover, the three-term solution of deflection including the linear buckling and nonlinear buckling shape is chosen The Galerkin method is used for the nonlinear buckling analysis of shells to give closed-form expressions of the critical buckling load and the relation between deflection and lateral load The specific features on the critical buckling loads and postbuckling curves for convex and concave shell in two cases - movable and immovable boundary conditions including temperature effects are investigated Furthermore, the influence of mechanical properties of two material structures S-FGM and P-FGM can be analyzed Effects of buckling modes, geometrical parameters, volume fraction index, elastic foundation and temperature on the nonlinear buckling behavior of shells are also considered Sigmoid Functionally Graded Material (S-FGM) Denote Vm and Vc as volume - fractions of metal and ceramic phases respectively, where Vm + Vc = and  z + h  k h  , k ≥ 0, − ≤ z ≤   Vm is espressed as Vm ( z ) =  h  , k   − z + h  ,0 ≤ z ≤ h   h  (1) where h is the thickness of thin – walled structure, k is the volume – fraction exponent (k ≥ 0); z is the thickness coordinate and varies from –h/2 to h/2; the subscripts m and c refer to the metal and ceramic constituents respectively Fig.1 describes the material characteristic of Sigmoid FGM According to the mentioned law, the Young modulus E (z ) and the thermal expansion coefficient α (z ) can be expressed in the form  z + h  k h  , k ≥ 0,− ≤ z ≤ 0,   h  , E ( z ) = E mV m + E cVc = E c + ( E m − E c ) k − z + h h     ,0 ≤ z ≤   h   z + h  k h  , k ≥ 0,− ≤ z ≤ 0,   α ( z ) = α mVm + α cVc = α c + (α m − α c )  h  k   − z + h  ,0 ≤ z ≤ h   h  (2) the Poisson’s ratio ν is assumed to be constant Fig The material characteristic of Sigmoid Functionally Graded Material Governing equations Consider a funtionally graded toroidal shell segment of thickness h, length L, which is formed by rotation of a plane circular arc of radius R about an axis in the plane of the curve as shown in Fig For the middle surface of a toroidal shell segment, from the figure: r = a − R (1 − sin ϕ ) (3) where a is the equator radius and ϕ is the angle between the axis of revolution and the normal to the shell surface For a sufficiently shallow toroidal shell in the region of the equator of the torus, the angle ϕ is approximately equal to π/2, thus sin ϕ ≈ ; cos ϕ ≈ and r =a [30] The form of governing equation is simplified by putting: dx = Rd ϕ , dx2 = adθ (4) The radius of arc R is positive with convex toroidal shell segment and negative with concave toroidal shell segment (a) (b) (c) Fig Configuration and coordinate system of toroidal shell segments: (a) convex shell; (b) concave shell and (c) toroidal shell segment on elastic foundation The strains across the shell thickness at a distance z from the mid-surface are: ε = ε 10 − zχ1 ; ε = ε 20 − zχ ; γ 12 = γ 120 − zχ 12 , (5) where ε10 and ε 20 are normal strains, γ 120 is the shear strain at the middle surface of the shell and χ ij are the curvatures According to the classical shell theory the strains at the middle surface and curvatures are related to the displacement components u, v, w in the x1, x2, z coordinate directions as [35]: ∂v w  ∂w ∂u w  ∂w   ; ε 20 = − +  ε = − +  ∂x a  ∂x ∂x1 R  ∂x1  χ2 = 2 ∂ w ∂u ∂v ∂w ∂w   ; γ 120 = + + ; χ1 = ; ∂x ∂x1 ∂x1 ∂x ∂x1  ∂2w ∂ 2w ; χ = 12 ∂x1∂x ∂x 22 (6) From Eqs (5) the strains must be satisfied in the deformation compatibility equation ∂ ε 10 ∂ 2ε 20 ∂ 2γ 120 ∂2w ∂2w  ∂2w  ∂ 2w ∂ 2w   + − = − − + − ∂x 22 ∂x12 ∂x1∂x R∂x 22 a∂x12  ∂x1 ∂x  ∂x12 ∂x 22 (7) The constitutive stress-strain equations by Hooke law for the shell material are given E( z) (ε + νε ) − E ( z )α ( z )∆T , −ν −ν E ( z) σ2 = (ε + νε ) − E ( z )α ( z )∆T , −ν −ν E(z) σ 12 = γ 12 , 2(1 + ν ) σ1 = (8) where α (z) is the thermal expansion coefficient Results and discussions 5.1 Validation of the present study The buckling behavior of a homogenous isotropic toroidal shell segment under uniform lateral pressure is investigated This was also considered earlier by Hutchinson [32] using the Koiter’s general theory Eq (37) degenerates into a linear buckling load of isotropic toroidal shell segments without elastic foundation This coincides excellently with the classical linear result reported by Hutchinson [32] as follows:    nL  a    m +     2 2 π  a R     π  πa    nL  Eh 12 z    qa = m + +         2 π4  12(1 −ν ) L2  nL     πa     nL   m +       πa      where z = − ν 5.2 L2 ah Results of nonlinear buckling of S-FGM toroidal shell segments To illustrate the proposed approach we consider Sigmoid-Functionally graded toroidal shell segments that consist of Aluminum and Alumina with the following properties: Em = 70 × 10 N / m ; α m = 23 ×10 −6 0C −1 Ec = 380 × 10 N / m ; α c = 7.4 × 10 −6 0C −1 where Poison’s ratio is chosen to be 0.3 The effects of material and geometric parameters, elastic foundation and thermal environment on the non-linear response of Sigmoid-Functionally graded material (S-FGM) 20 toroidal shell segments under uniform lateral pressure are considered in Figs – It is noted that in all figures Wmax/h denotes the dimensionless maximum deflection of the shell The effects of the R/h on q~Wmax/h post-buckling curves of FGM convex and concave toroidal shell segment on elastic medium (K1 = 2.5×108 N/m3, K2 = 5×105 N/m) are described in Figs and 4, respectively As can be seen, the load-deflection curves of S-FGM convex toroidal shell segment become lower, whereas those of S-FGM concave shell become higher when increasing the ratio R/h The load bearing capability of convex toroidal shells is considerably enhanced as ratio decreases, but conversely for concave ones Table shows the comparison of the critical buckling loads of two types S-FGM and PFGM toroidal shells As can be observed, with the same geometrical parameters, the critical buckling loads of the S-FGM with ceramic-metal-ceramic layers are higher than those of the P-FGM with metal-ceramic layers 21 Fig.3 Effect of R/h ratio on the post-buckling curves of S-FGM convex shell on elastic foundation Fig.4 Effect of R/h ratio on the post-buckling curves of S-FGM concave shell on elastic foundation Table Comparison of the critical buckling loads of two types S-FGM and P-FGM shells qcr (×107Pa) (m, n) = (1, 5) R/h = 300 Types R/h = 400 R/h = -300 R/h = -400 Upper Lower Upper Lower Upper critical Lower Upper critical Lower critical load critical load critical load critical load load critical load load critical load calculated calculated calculated calculated calculated by calculated calculated by calculated by Eq (37) by Eq by Eq (37) by Eq Eq (37) by Eq Eq (37) by Eq (35)&(36) S-FGM qcr (×107Pa) (m, n) = (1, 2) 2.7218 2.4404 (35)&(36) 2.1828 2.0969 (35)&(36) 8.9610 8.6021 (35)&(36) 10.257 9.1613 22 P-FGM 2.7205 2.4381 2.1815 2.0951 8.9591 8.5993 10.255 9.1579 The effect of L/R ratio on the critical buckling load of S-FGM convex and concave shell on Pasternak foundation is illustrated in Table and Table 3, respectively The data of problem: h = 0.01m; a = 100h; R = ±500h Table Effect of L/R ratio on the critical buckling load (MPa) of S-FGM convex shell on elastic foundation with various modes (m =1) n L/R 1.5 2.5 24.758 24.511 24.398 24.337 15.928 15.827 15.780 15.755 11.192 11.143 11.120 11.108 8.3569 8.3302 8.3179 8.3112 6.5251 6.5093 6.5020 6.4981 5.2730 5.2630 5.2584 5.2559 Table Effect of L/R ratio on the critical buckling load (MPa) of S-FGM concave shell on elastic foundation with various modes (m =1) n L/R -1.5 -2 -2.5 -3 23.417 23.750 23.908 23.996 15.374 15.513 15.579 15.615 10.924 10.991 11.023 11.040 23 8.2117 8.2483 8.2654 8.2748 6.4399 6.4613 6.4713 6.4767 5.2197 5.2330 5.2392 5.2426 It is observed that when the L/R ratio increases, the critical buckling loads of S-FGM convex shell decrease while ones of S-FGM concave shell increase It means that the critical loads of shorter convex shells are greater than those of longer ones while the concave shells are completely on the contrary For instance, when L/R ratio increases from 1.5 to 3, the critical loads slightly fall by about 0.3 ~ 1.7% for the convex shell and increase by about 0.4 ~ 2.5% for the concave shell Fig illustrates the effect of L/a ratio on the post-buckling curves of S-FGM convex shell on elastic foundation It is shown that the non-linear response of toroidal shell segment is sensitive with change of L/a ratio characterizing the shallowness of toroidal shell segment Specifically, the enhancement of the upper buckling loads and the load carrying capacity in small range of deflection as L/a decrease is followed by a very severe snap-through behavior In other words, in spite of possessing higher limit buckling loads, the toroidal shells with smaller ratio L/a exhibit a very unstable response from the post-buckling point of view 24 Fig.5 Effect of L/a ratio on the post-buckling curves of S-FGM convex shell on elastic foundation 25 Fig.6 Effect of volume-fraction k on the post-buckling curves of S-FGM convex shell on elastic foundation Fig shows the influence of the volume fraction index k on the post-buckling behavior of the S-FGM convex shell subjected to lateral pressure.As can be seen, the load capacity of the shell becomes higher when volume-fraction k increases This is expected because the volume percentage of ceramic constituent, which has higher elasticity modulus, is raised with increasing values k However, the load carrying capacity of ceramic-rich toroidal shell is paid by a more severe snap-through behavior, i.e a bigger difference between upper and lower buckling loads Fig considers the effect of elastic foundation on the behavior of S-FGM toroidal shell segments under lateral pressure Obviously, the elastic foundation enhances the load carrying capability of the shells, especially Pasternak type foundations have beneficial influences on the nonlinear stability of toroidal shell segments on both aspects that are increase in load carrying capacity and decrease in the severity of snap-through instability 26 Fig.7 Effect of elastic foundation on the post-buckling curves of S-FGM convex shell on elastic foundation Based on Eqs (40)-(43), the effects of environment temperature on the thermo-mechanical behavior of S-FGM toroidal shell segments with immovable simply-supported edges are analyzed in Figs 8-9 In this case, the shells are exposed to a temperature field prior to applying lateral pressure Figs and represent the non-linear responses of S-FGM convex and concave toroidal shells subjected to lateral pressure, respectively under various temperature changes ∆T ( = 0, 800, 1600, 2400) It is evident that the nonlinear response of S-FGM toroidal shell segments with immovable simply-supported edges under thermal-mechanical loads is quite different from that of movable simply-supported S-FGM toroidal shell segments subjected to only mechanical loads The post-buckling paths intersect q axis, this behavior of the toroidal shell segments can be explained as follows The temperature field makes the shell to deflect 27 outwards (negative deflection) before it is affected the mechanical load, where the more temperature change introduces the more negative deflection With the action of uniform lateral pressure, the outward deflection is reduced and when lateral pressure exceed bifurcation point load an inward deflection occurs The enhancement of temperature change is accompanied by the increase in bifurcation points, load bearing capacity in small region of deflection and the intensity of snap-through behavior for convex shell, but in negative side for concave shell Fig.8 Effect of temperature field on post-buckling curves of S-FGM convex shell 28 Fig.9 Effect of temperature field on post-buckling curves of S-FGM concave shell Concluding remarks The nonlinear response and postbuckling behavior of S-FGM toroidal shell segments with ceramic-metal-ceramic layers surrounded by elastic foundations under external pressure in thermal environment have been analyzed Formulations are based on the thin shell theory taking geometrical nonlinearity, panel-foundation interaction, Stein and McElman assumption into consideration Some concluding remarks can be drawn from the obtained results - The more correctly chosen three-term solution of deflection including the linear and nonlinear buckling shapes and Galerkin’s method provide to obtain closeform expressions to determine critical buckling loads and post-buckling load maximal deflection curves, that allow more detail investigating behavior of the shell 29 - The upper critical load (corresponding bifurcation point) of the shell coincides with the linear critical load - The load bearing capacity of the ceramic-metal-ceramic layers S-FGM toroidal shell segment is higher than that of the metal-ceramic layers P-FGM toroidal shell segment with the same geometrical parameters - Geometric parameters, elastic foundations and the thermal environment greatly influence on the nonlinear buckling of S-FGM toroidal shell segments, but convex and concave shells behave differently, moreover contrarily in some characteristics when changing geometrical ratios and temperature rise That is a specific feature of toroidal shell segments Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2014.09 References [1] M Koizumi, The concept of FGM, Ceramic Transactions, Functionally Gradient Materials 34 (1993) 3–10 [2] E Winkler Die Lehre von der Elasticitaet und Festigkeit, Dominicus, Prague;1867 [3] P L Pasternak, On a new method of analysis of an elastic foundation by meansof two foundation constants Gos.Izd.Lit.po strait i Arkh, Moscow, Russia; 1954 [In Russian] [4] E Bagherizadeh, Y Kiani, M R Eslami, Mechanical buckling of functionally graded material cylindrical shellssurrounded by Pasternak elastic foundation, Composite Structures 93 (2011) 3063-3071 30 [5] A H Sofiyev, 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buckling and post-buckling of ceramic-metal-ceramic layers (SFGM) toroidal shell segment surrounded by elastic foundation under thermo-mechanical loads are investigated with an analytical approach... toroidal shell segment and negative with concave toroidal shell segment (a) (b) (c) Fig Configuration and coordinate system of toroidal shell segments: (a) convex shell; (b) concave shell and (c) toroidal