DSpace at VNU: Vibration and nonlinear dynamic response of imperfect three-phase polymer nanocomposite panel resting on elastic foundations under hydrodynamic loads

Composite Structures 131 (2015) 229–237 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Vibration and nonlinear dynamic response of imperfect three-phase polymer nanocomposite panel resting on elastic foundations under hydrodynamic loads Nguyen Dinh Duc a,⇑, Homayoun Hadavinia b, Pham Van Thu c, Tran Quoc Quan a a b c Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam School of Mechanical & Automotive Engineering, Kingston University, Roehampton Vale, Friars Avenue, London, SW15 3DW, UK Nha Trang University, 44 Hon Ro, Nha Trang, Khanh Hoa, Viet Nam a r t i c l e i n f o Article history: Available online 11 May 2015 Keywords: Nonlinear dynamic Vibration Laminated three-phase polymer nanocomposite panel Hydrodynamics loads Imperfection Elastic foundations a b s t r a c t An investigation on the nonlinear dynamic response and vibration of the imperfect laminated three-phase polymer nanocomposite panel resting on elastic foundations and subjected to hydrodynamic loads is presented in this paper The formulations are based on the classical shell theory and stress function taking into account geometrical nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation Numerical results for dynamic response and vibration of the three-phase polymer composite panel are obtained by Runge–Kutta method The influences of fibers and particles, material and geometrical properties, foundation stiffness, imperfection and hydrodynamic loads on the nonlinear dynamic response and nonlinear vibration of the three-phase composite panel are discussed in detail Ó 2015 Elsevier Ltd All rights reserved Introduction Currently, composite materials have become indispensable in several applications, such as high-performance structures in many fields of civil, marine and aerospace engineering, among others The mechanical behaviors of composite structures, such as bending, vibration, stability, buckling, etc., has attracted attention of many researchers Rango et al [1] presented the formulation of an enriched macro element suitable to analyze the free vibration response of composite plate assemblies Bodaghi et al [2] investigated thermo-mechanical analysis of rectangular shape adaptive composite plates with surface bonded shape memory alloy ribbons Sahoo and Singh [3] used a new trigonometric zigzag theory to research the analysis of laminated composite and sandwich plates Samadpour et al [4] studied nonlinear free vibration of thermally buckled sandwich plate with embedded pre-strained shape memory alloy fibers in temperature dependent laminated composite face sheets Moleiro et al [5] provided an assessment of layerwise mixed models using least-squares formulation for the coupled electromechanical static analysis of multilayered plates Heydarpour et al [6] examined the influences of centrifugal and Coriolis forces on the free vibration behavior of rotating carbon ⇑ Corresponding author E-mail address: ducnd@vnu.edu.vn (N.D Duc) http://dx.doi.org/10.1016/j.compstruct.2015.05.009 0263-8223/Ó 2015 Elsevier Ltd All rights reserved nanotube reinforced composite truncated conical shells Lopatin and Morozov [7] considered free vibrations of a cantilever composite circular cylindrical shell Burgueño et al [8] presented approaches for modifying and controlling the elastic response of axially compressed laminated composite cylindrical shells in the far postbuckling regime The vibration and damping characteristics of free–free composite sandwich cylindrical shell with pyramidal truss-like cores have been conducted by Yang et al [9] using the Rayleigh–Ritz model and finite element method Three-phase composite is a material consisted of matrix, the reinforced fibers and particles which have been investigated by Vanin and Duc [10] Shen et al [11]] analyzed a coated inclusion of arbitrary shape embedded in a three-phase composite plate subjected to anti-plane mechanical and in-plane electrical loadings Lin et al [12] presented a solution of magnetoelastic stresses on a three-phase composite cylinder subjected to a remote uniform magnetic induction Wu et al [13] developed an effective model to bound the effective magnetic permeability of three-phase composites with coated spherical inclusions There are several claims on the deflection and the creep for the three-phase composite laminates in the bending state [14] These findings have shown that optimal three-phase composite can be obtained by controlling the volume ratios of fiber and particles Afonso and Ranalli [15] introduced a new general model to calculate the elastic properties of three-phase composites by means of closed-form analytical solutions is presented Andrianov et al [16] analyzed the 230 N.D Duc et al / Composite Structures 131 (2015) 229–237 three-phase composite model from the viewpoint of the asymptotic homogenization method Recently, Duc et al [17,18] studied nonlinear stability of the three-phase polymer composite plate under thermal and mechanical loads This paper presents an investigation on the nonlinear dynamic response and vibration of the imperfect laminated three-phase polymer nanocomposite panels resting on elastic foundations and subjected to hydrodynamic loads These structures have similar shape with the hydrofoil currently under investigation and manufacturing in Vietnam The formulations are based on the classical shell theory and stress function taking into account geometrical nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation Numerical results for dynamic response and vibration of the three-phase polymer composite panel are obtained by Runge–Kutta method It is noted that the present paper is improvement and supplement of the ideas in proceeding paper which we presented at the Third International Conference on Engineering Mechanics and Automation [22] (ICEMA 3-2014, Hanoi, October-2014), including SEM structures images of two-phase 2D composite (glass fibers, polymer matrix) and the three-phase 2Dm composite (glass fibers, titanium oxide particles and polymer matrix) in order to improve the paper more convinced ỵ E11 ẳ wa Ea ỵ wa ÞE a "  G 2ð1 À wa ịv 1ị ỵ va 1ịv ỵ 2wa ị Ga ẳ ỵ   E11 8G wa ỵ vwa ỵ wa ịva 1ị GGa #)1 v1 wa ị ỵ ỵ wa vị GGa ; ỵ2 v ỵ wa ỵ À wa Þ GGa ( E22  ð1 À w Þðma À m Þ 8Gw a a  ; wa ỵ xwa ỵ wa ịxa 1ị GG m221   G12 ẳ G  G23 ẳ G ẳ m21 ỵ wa ỵ wa ị GGa  wa ỵ þ wa Þ GGa ; v þ wa þ ð1 À wa Þ GGa m23  ; ð1 À wa ịv ỵ ỵ vwa ị GGa E22 "  wa ịx ỵ ỵ wa xị GGa ỵ  E11 8G x ỵ wa ỵ ð1 À wa Þ GG a m221  2ð1 À wa ịx 1ị ỵ xa 1ịx þ 2wa Þ GGa  À wa þ xwa þ ð1 À wa Þðxa À 1Þ GGa  À ma ịwa v ỵ 1ịm  ẳm ; wa ỵ vwa ỵ wa ị va GGa # ; 5ị in which v ẳ À 4m; va ¼ À 4ma : Determination of the elastic modules of three-phase composite In this paper, the algorithm which is successfully applied in Ref [17,18] to determine the elastic modules of three-phase composite has been used According to this algorithm, the elastic modules of 3-phase composites are estimated using two theoretical models of the 2-phase composite consecutively: nDm = Om + nD [17,18] This paper considers 3-phase composite reinforced with particles and unidirectional fibers, so the problem’s model will be: 1Dm = Om + 1D Firstly, the modules of the effective matrix Om which is called ‘‘effective modules’’ are calculated In this step, the effective matrix consists of the original matrix and added particles It is considered to be homogeneous, isotropic and have two elastic modules The next step is estimating the elastic modules for a composite material consists of the effective matrix and unidirectional reinforced fibers Assuming that all the component phases (matrix, fiber and particles) are homogeneous and isotropic, we will use Em ; Ea; Ec ; mm ; ma ; mc ; wm ; wa ; wc to denote Young’s modulus and Poisson ratio and volume fraction for the matrix, fiber and particles, respectively Following [17,18], the modules for the effective composite can be obtained as below  ¼ Gm À wc 5mm ịH ; G ỵ wc 10mm ịH 1ị  ẳ K m ỵ 4wc Gm L3K m ị ; K 4wc Gm L3K m ị1 2ị where Lẳ Kc Km ; K c ỵ 4G3m Hẳ Gm =Gc : 10mm ỵ 5mm Þ GGmc ð3Þ  KÞ  m  as below  can be calculate from (G; E;   ¼ 9K G ; E   ỵG 3K   À 2G 3K m ¼  : 6K À 2G ð4Þ The elastic moduli for three-phase composite reinforced with unidirectional fiber are chosen to be calculated using Vanin’s formulas [20] ð6Þ To verify the validity of these equations, three-phase composite polymer made of polyester AKAVINA (made in Vietnam), glass fibers (made in Korea) and titanium oxide (made in Australia) with the properties shown in Table was investigated [17,18] By using the SEM instrumentation at the Laboratory for Micro-Nano Technology, University of Engineering and Technology, Vietnam National University, Hanoi, Figs and show the images of fabricated samples of composite structures which are made in the Institute of Ship building, Nha Trang University Fig illustrates a SEM image of 2Dm composite polymer two-phase material (glass fibers volume fraction of 25% without particles) and Fig shows a SEM image of 2Dm composite polymer three-phase material (glass fibers volume fraction of 25% and Titanium oxide particles volume fraction of 3%) Obviously, when the particles are doped, the air cavities significantly reduced and the material is finer In other words, particles enhance the stiffness and the penetration resistance of the materials Governing equations Consider a three-phase composite panel subjected to hydrodynamic loads: hydrodynamic lift q1 and drag q2 as shown in Fig The panel is referred to a Cartesian coordinate system x; y; z, where xy is the mid-plane of the panel and z is the thickness coordinator, Àh=2 z h=2 The radii of curvatures, length, width and total thickness of the panel are R; a; b and h, respectively In this study, we assumed that the panel is thin, so the classical laminated shell theory (CLST) is used to establish governing equations and determine the nonlinear response of composite panels In the case of thick panel, we must use higher-order shear deformation theories By choosing of accurate theories can refer to [23] Taking into account the von Karman nonlinearity, the strain– displacement relations are Table Properties of the component phases for three-phase composite Component phase Young modulus E Poisson ratio Matrix polyester AKAVINA (Vietnam) Glass fiber (Korea) Titanium oxide TiO2(Australia) 1,43 GPa 22 GPa 5,58 GPa 0.345 0.24 0.20 m 231 N.D Duc et al / Composite Structures 131 (2015) 229–237 Fig Geometry and coordinate system of three-phase composite panels on elastic foundations Hooke law for a laminated composite panel is defined as 0 rx Q 11 B C B r ¼ @ y A @ Q 12 rxy k Q 016 Q 012 Q 016 10 ex CB C Q 026 A @ ey A ; Q 066 k cxy k Q 022 Q 026 ð9Þ in which k is the number of layers and Q 011 ẳ Q 11 cos4 h ỵ Q 22 sin h ỵ 2Q 12 ỵ 2Q 66 ịsin hcos2 h; 4 Q 012 ¼ Q 12 cos4 h ỵ sin hị ỵ Q 11 ỵ Q 22 À 4Q 66 Þ sin hcos2 h; Fig SEM image of 2Dm composite two-phase material (fibers volume fraction is 25% without particles) Q 012 ¼ Q 12 ðcos4 h ỵ sin hị ỵ Q 11 ỵ Q 22 À 4Q 66 Þ sin hcos2 h; Q 016 ẳ Q 12 Q 22 ỵ 2Q 66 ịsin hcosh ỵ Q 11 Q 12 2Q 66 Þsinhcos3 h; Q 022 ¼ Q 11 sin h ỵ Q 22 cos4 h ỵ 2Q 12 ỵ 2Q 66 ịsin hcos2 h; Q 026 ẳ Q 11 Q 12 2Q 66 ịsin hcosh ỵ Q 12 Q 22 ỵ 2Q 66 ịsinhcos3 h; Q 066 ẳ Q 66 sin h ỵ cos4 hị ỵ ẵQ 11 ỵ Q 22 2Q 12 ỵ Q 66 ị sin hcos2 h; 10ị and Q 11 ¼ Q 12 ¼ E11 À EE22 m212 11 E11 À EE22 m212 11 ¼ ¼ E11 ; À m12 m21 m12 Q 22 ; Q 22 ¼ E22 À EE22 m212 11 ¼ E22 Q ; E11 11 11ị Q 66 ẳ G12 ; where h is the angle between the fiber direction and the coordinate system The force and moment resultants of the laminated composite panels are determined by Ni ¼ Mi ¼ n X Rh k hkÀ1 k¼1 n X R hk hk1 ẵri k dz; i ẳ x; y; xy; 12ị zẵri k dz; i ẳ x; y; xy: kẳ1 Substitution of Eq (7) into Eq (9) and the result into Eq (12) give the constitutive relations as Fig SEM image of 2Dm composite three-phase material (fibers volume fraction is 25% and particles volume fraction is 3%) 1 e0x ex kx B e C B e0 C B C @ y A ¼ @ y A ỵ z@ ky A; cxy kxy cxy Nx ;Ny ;Nxy ẳ A11 ;A12 ;A16 ịe0x ỵA12 ;A22 ;A26 ịe0y ỵA16 ;A26 ;A66 ịc0xy ỵB11 ;B12 ;B16 ịkx ỵB12 ;B22 ;B26 ịky ỵB16 ;B26 ;B66 ịkxy ; Á M x ;M y ;Mxy ¼ ðB11 ;B12 ;B16 ịe0x ỵB12 ;B22 ;B26 ịe0y ỵB16 ;B26 ;B66 ịc0xy þðD11 ;D12 ;D16 Þkx þðD12 ;D22 ;D26 Þky þðD16 ;D26 ;D66 Þkxy ; ð7Þ ð13Þ Table The dependency of hydrodynamic lift and drag on the velocities where 1 e0x u;x ỵ w2;x =2 B e0 C B C @ y A ¼ @ v ;y À w=R ỵ w2;y =2 A; c0xy u;y ỵ v ;x ỵ w;x w;y 1 w;xx kx B C B C @ ky A ¼ @ Àw;yy A; À2w;xy kxy ð8Þ in which u; v are the displacement components along the x; y directions, respectively Velocity (m/s) Lift (N) Drag (N) 10 10.2 10.4 10.6 10.9 12 110537 112730 114204 117836 121090 129529 5550.8 5380.6 5165.6 5243.2 5130.6 4416.1 232 N.D Duc et al / Composite Structures 131 (2015) 229–237 Table Effects of particles volume fraction, fiber volume fraction and elastic foundations on natural frequencies of the three-phase composite polymer panel wa wc 0.2 0.2 0.2 0.2 0.1 0.3 0.1 0.2 0.3 0.2 0.2 0.2 xL rad=sị k1 ; k2 ị ẳ 0; 0ị k1 ; k2 ị ẳ 0:01; 0:002ị 3.3258e3 3.3038e3 3.2981e3 3.2763e3 3.4884e3 3.3975e3 3.1015e3 4.8564e3 4.8414e3 4.8375e3 4.8255e3 5.1195e3 4.9695e3 4.7057e3 Fig Effects of fiber volume fraction wa on the dynamic response of three-phase polymer composite panel Fig Effects of particles volume fraction wc on the dynamic response of threephase polymer composite panel where Aij ¼ n X Q 0ij ịk hk hk1 ị; i; j ẳ 1; 2; 6; k¼1 n X Bij ¼ 2 ðQ Þ ðh À hkÀ1 Þ i; j ¼ 1; 2; 6; k¼1 ij k k Dij ¼ n 1X 3 ðQ Þ ðh À hk1 ị i; j ẳ 1; 2; 6: kẳ1 ij k k ð14Þ The nonlinear motion equation of the composite panels based on CLST with the Volmir’s assumption [19], u