Ứng xử sau vồng của các vỏ cầu thoải sandwich làm từ vật liệu có cơ tính biến đổi

20 T„P CHŠ KHOA H“C KI¦N TR”C - XŸY D¼NG KHOA H“C & C«NG NGHª Postbuckling behavior of functionally graded sandwich shallow spherical shells Ứng xử sau vồng của các vỏ cầu thoải sandwich làm từ vật liệu có cơ tính biến đổi Hoàng Văn Tùng Tóm tắt Bài báo giới thiệu một cách tiếp cận giải tích để nghiên cứu ứng xử sau vồng của các vỏ cầu thoải sandwich với các lớp mặt làm từ vật liệu cơ tính biến đổi, tựa trên nền đàn hồi và chịu áp lực ngoài phân bố đều. Các tính chất vật liệu của các lớp mặt được biến đổi qua chiều dày theo quy luật hàm lũy thừa dưới dạng các tỷ lệ thể tích của các vật liệu thành phần. Các phương trình cơ bản được thiết lập dựa trên lý thuyết vỏ biến dạng trượt bậc nhất có kể đến tính phi tuyến hình học và nền đàn hồi loại Pasternak. Nghiệm xấp xỉ được chọn để thỏa mãn điều kiện biên ngàm cứng và phương pháp Galerkin được áp dụng để dẫn ra các biểu thức hiển của liên hệ tải-độ võng và từ biểu thức này ứng xử sau vồng của vỏ được phân tích Từ khóa: Vỏ cầu thoải, Cấu trúc sandwich, Nền đàn hồi, Ứng xử sau vồng Abstract This paper presents an analytical approach to investigate the postbuckling behavior of Sandwich Shallow Spherical Shell (SSSS) with functionally graded face sheets resting on elastic foundations and subjected to uniform external pressure. Effective material properties of face sheets are graded in the thickness direction according to a simple power law distribution in terms of volume fractions of constituents. Governing equations are based on first order shear deformation shell theory taking into account geometrical nonlinearity and Pasternak elastic foundations. Approximate solutions are assumed to satisfy immovably clamped boundary condition and Galerkin method is applied to derive explicit expressions of load-deflection relation from which the postbuckling behavior the shells is analyzed. Keywords: Shallow spherical shell, Sandwich structures, Postbuckling behavior. TS. Hoàng Văn Tùng Faculty of Civil Engineering Hanoi Architectural University Email: inter0105@gmail.com 1. Introduction Sandwich-type structures exhibit a number of exceptional features such as increased bending stiffness with little resultant weight penalty, excellent thermal and sound insulation, and extended operational life. Due to these outstanding properties, the sandwich-type constructions play a great role as major portions in the construction of advanced supersonic and hypersonic space vehicles [1]. The sandwich structures are also used widely in building constructions and shipbuilding industry. Nonlinear response and postbuckling of anisotropic and laminated flat and curved sandwich panels have received researching interest in past years [2]. Functionally Graded Material (FGM) is advanced composite material with many excellent characters. The effective properties of the FGM are varied smoothly and continuously across the thickness direction of the structures. Thus, FGM can avoid huge stress concentration and interface problems of conventional laminated composites. The nonlinear response and postbuckling of FGM sandwich plates and shells are important problems and should be addressed. Linear buckling behaviors of FGM sandwich plates under compressive and thermal loads have been investigated by Zenkour [3] and Zenkour and Sobhy [4] using an analytical method. Shen and Li [5] employed a semi-analytical approach based on a two-step perturbation technique to deal with the postbuckling behavior of FGM sandwich plates under mechanical and thermomechanical loads. Tung [6] investigated the thermal and thermomechanical postbuckling behavior of FGM sandwich plates making use of Galerkin method and an iteration algorithm. Structural elements in the form of spherical shells are widely used in engineering structures. Tung [7] analyzed the nonlinear response of FGM spherical shells under uniform external pressure in thermal environments taking into account temperature dependence of material properties. Recently, Tung [8] presented an analytical study on the nonlinear stability of FGM shallow spherical shells subjected to external pressure with tangential edge constraints. This paper extends the previous works [7,9] to investigate the postbuckling behavior of FGM sandwich shallow spherical shells resting on elastic foundations and subjected to uniform external pressure. Analytical solutions are assumed and Galerkin method is applied to obtain explicit expression of load-deflection relation from which the nonlinear stability of the shells are analyzed. 2. Sandwich Shallow Spherical Shell (SSSS) on an elastic foundation Consider a FGM SSSS of radius of curvature R, base radius a, total thickness h and rise of shell H. The shell is immovably clamped at boundary edge and rested on a Pasternak elastic foundation as shown in Fig. 1. The SSSS is constructed from two functionally graded material (FGM) face sheets (i.e. skins) separated by a thicker core layer made of metal material. h h0 h1 h2 h3 k1 ϕ,u z,w k2 a FGM FGM R r H hf hf METAL P Fig. 1. Geometry and coordinate system of a FGM SSSS on an elastic foundation. 21 S¬ 27 - 2017 It is assumed that core and face layers are perfectly bonded and the thickness of each face sheet is . The top skin varies from a ceramic-rich surface ( ) to a metal-rich interface, whereas the bottom skin is graded from a metal- rich interface to ceramic-rich surface ( ). Such a configuration of FGM sandwich shell is mid-plane symmetric and the volume fraction of the metal constituent is obtained by power law distribution as 0 1 0 ( ) n m z h V z h h  − =  −  , 0 1[ , ]z h h∈ top skin ( ) 1mV z = , 1 2[ , ]z h h∈ core layer (1) 3 2 3 ( ) n m z h V z h h  − =  −  , 2 3[ , ]z h h∈ bottom skin Herein h0=-h/2, h1=-h/2+hf, h3=h/2 and n≥0 is volume fraction index. Volume fraction of ceramic constituent is defined as Vc(z)=1-Vm(z). Effective modulus of elasticity of the FGM SSSS is determined by linear rule of mixture as ( ) ( ) ( ) ( )m m c c c mc mE z E V z E V z E E V z= + = + (2) where Emc=Em-Ec and Em, Ec are elastic moduli of metal and ceramic constituents, respectively. Poisson ratio is assumed to be a constant in the present study. The FGM SSSS is rested on an elastic foundation and shell-foundation interaction is represented by Pasternak model as 1 2fq k w k w= − ∆ (3) where 2 2 2 2/ /x y∆ = ∂ ∂ + ∂ ∂ is Laplace’s operator, w is deflection of shell; k1 is modulus of Winkler foundation, k2 is stiffness of shear layer of Pasternak foundation model. 3. Formulations The first order shear deformation shell theory (FSDT) is used to formulate for the present study. The FGM SSSS is assumed to be under axisymmetric deformation and displacement components , ,u v w in , , zϕ θ directions, respectively, at a distance z from the middle surface are represented as [7] ( , ) ( ) z ( )u r z u r rψ= + , ( , ) 0v r z = , ( , ) ( )w r z w r= (4) in which sinr R ϕ= , u is displacement in the meridional direction at the middle surface, w is the deflection of the shell, and ψ is the rotation of a normal to the middle surface. Due to shallowness of the shell, it is approximately assumed that cos 1ϕ = , Rd drϕ = and 2 / (2 )R a H= . The non-zero strain components are 0r r rzε ε χ= + , 0 zθ θ θε ε χ= + , ,rz rwε ψ= + (5) where ,( ) ( ) /r d dr= , and the strains at the middle surface 0 0,r θε ε and curvatures ,r θχ χ are related to the displacements and rotation in the form 2 0 , ,/ / 2r r ru w R wε = − + , 0 / /u r w Rθε = − , ,r rχ ψ= , / rθχ ψ= . (6) Based on Hooke law, stress-strain relations are ( )2 ( ) 1r r E z θσ ε νεν = + − , ( )2 ( ) 1 r E z θ θσ ε νεν = + − , 2(1 )rz rz E σ ε ν = + . (7) The force and moment resultants are expressed in terms of the stress components ( ) ( ) /2 /2 , , h r r h N N dzθ θσ σ − = ∫ , ( ) ( ) /2 /2 , , h r r h M M zdzθ θσ σ − = ∫ , /2 /2 h r S rz h Q K dzσ − = ∫ (8) where is shear correction coefficient. Introduction of Eqs. (7) into Eqs. (8) gives the expressions of force and moment resultants as [ ] [ ] ( ) [ ] ( )1 2 2 30 02 2 , , , 1 1r r r r E E E E N M θ θε νε χ νχν ν = + + + − − , [ ] [ ] ( ) [ ] ( )1 2 2 30 02 2 , , , 1 1r r E E E E N Mθ θ θ θε νε χ νχν ν = + + + − − , ( )1 ,2(1 ) S r r K E Q wψ ν = + + . (9) where [ ] /2 2 1 2 3 /2 , , ( ) 1, , h h E E E E z z z dz −  =  ∫ (10) The nonlinear equilibrium equations of a FGM SSSS resting on an elastic foundation based on the FSDT are [7,8] ( ), 0r rrN Nθ− = , ( ), 0r rrrM M rQθ− − = , ( ) ( ) ( ) ( ),, , 0r r r r fr r r rQ N N rN w r P q R θ + + + + − = (11) where is external pressure uniformly distributed on the outer surface of shell. The FGM SSSS is assumed to be clamped and immovable in the meridional direction at the boundary edge, and under axisymmetric deformation. The symmetry condition at the center and boundary conditions at are expressed as [6,7] 0ψ = at 0r = ; 0w = , 0ψ = , 0u = at r a= . (12) To satisfy conditions (12), the following approximate solutions are assumed [7,9] 2 ( )r a r u U a − = , 2 2 3 ( )r a r a ψ − = Ψ , ( )22 2 4 a r w W a − = (13) where are coefficients to be determined, is the deflection amplitude. Now, solutions (13) are substituted into Eqs. (11) and Galerkin method is applied for the resulting equations 1 0 ( ) 0 a L r a r dr− =∫ , 2 2 2 0 ( ) 0 a L r a r dr− =∫ , 2 2 2 3 0 ( ) 0 a L a r dr− =∫ (14) where 1 2 3, ,L L L are the resulting expressions obtained 22 T„P CHŠ KHOA H“C KI¦N TR”C - XŸY D¼NG KHOA H“C & C«NG NGHª by substituting (13) into the left sides of Eqs. (11), respectively. Eqs. (14) are system of nonlinear algebra equations in term of U, Ψ, W. Next, eliminating U and Ψ from these equations yields the following nonlinear relation of external pressure P and deflection amplitude W as 2 3 31 32 33P e W e W e W= + + (15) where /W W h= and coefficients 31 32 33, ,e e e can be found in the work [7]. 4. Results and discussion This section presents numerical results for FGM SSSS made of aluminum (metal) and alumina (ceramic) with the following properties 70mE = GPa and 380cE = GPa, and Poisson ratio 0.3ν = for both constituents. Reliability of proposed approach has been verified in the previous work [7]. Fig. 2 examines the effects of volume fraction index n(=0 ,1,3 and 10) on the postbuckling of FGM SSSSs without foundation interaction. It is evident that FGM SSSSs exhibit extreme-type buckling behavior and an unstable postbuckling response with relatively intense snap-through phenomenon. Specifically, buckling loads, load-deflection curves and intensity of snap-though instability (measured by difference between upper and lower pressures at limit points) are all increased. Next, the effects of face sheet thickness hf-to-total thickness h ratio (hf/h=0.1, 0.15, 0.2 and 0.25) on the postbuckling of FGM SSSSs are shown in Fig. 3. As can be seen, buckling loads and postbuckling strength of FGM SSSSs are remarkably improved when the thickness of FGM face sheet is increased. The effects of rise-to-base radius H/a ratio and non- dimensional stiffness of elastic foundations K1, K2 on the postbuckling behavior of FGM SSSSs subjected to uniform external pressure are considered in figures 4 and 5, herein 2 4 21 1 112(1 ) /K a k E hν= − and 2 2 2 2 2 112(1 ) /K a k E hν= − . Fig. 4 indicates that H/a ratio has very sensitive influences on the nonlinear response of FGM SSSSs. Specifically, external pressure-deflection curves are pronouncedly enhanced as H/a ratio is increased. However, this increase in (limit-type) buckling loads is accompanied by a severe snap-through instability in postbuckling region. Finally, 0 1 2 3 40 20 40 60 80 100 120 W/h P (M Pa ) 2: H/a = 0.125 3: H/a = 0.15 4: H/a = 0.175 5: H/a = 0.2 1: H/a = 0.1n = 2, a/h = 20, hf/h = 0.2 (K1,K2) = (0,0) 1 2 3 4 5 0 1 2 3 40 10 20 30 40 50 60 W/h P (M Pa ) 1: (K1,K2) = (0,0) 2: (K1,K2) = (50,0) n = 2, a/h = 20, H/a = 0.175, hf/h = 0.2 2 1 3 4 3: (K1,K2) = (80,0) 4: (K1,K2) = (60,5) Fig. 4. Effects of the H/a ratio on the postbuckling of the SSSSs. Fig. 5. Effects of elastic foundations on the postbuckling of the SSSSs. 0 1 2 3 40 10 20 30 40 50 60 W/h P (M Pa ) a/h = 20, H/a = 0.15 hf/h = 0.2, (K1,K2) = (0,0) 2 3 4 1: n = 0 2: n = 1 4: n = 10 3: n = 3 1 0 1 2 3 40 5 10 15 20 25 30 35 40 45 W/h P (M Pa ) 1: hf/h = 0.1 a/h = 20, H/a = 0.15, n = 1 (K1,K2) = (0,0) 4 3 2 1 2: hf/h = 0.15 3: hf/h = 0.2 4: hf/h = 0.25 Fig. 2. Effects of volume fraction index on the postbuckling of the SSSSs. Fig. 3. Effects of the hf/h ratio on the postbuckling of the SSSSs. (xem tiếp trang 26)