Nonlinear vibration of FG-CNTRC cylindrical panels with corrugated core using the FSDT

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In this paper, a new design of cylindrical panels is proposed with two functionally graded carbon nanotube-reinforced composite (FG-CNTRC) face sheets and a corrugated core. The corrugated core layer is modelled by applying the homogeneous technique according to the firstorder shear deformation theory (FSDT). The nonlinear vibration behaviour of first-order shear deformable cylindrical panels with geometric nonlinearities is analysed in the present paper.

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Nội dung Text: Nonlinear vibration of FG-CNTRC cylindrical panels with corrugated core using the FSDT

82 99 Tuyển tập công trình Hội nghị Cơ học toàn quốc lần thứ XI, Hà Nội, 02-03/12/2022 Nonlinear vibration of FG-CNTRC cylindrical panels with corrugated core using the FSDT Vu Minh Duc1,*, Tran Ngoc Hung1, Nguyen Thi Phuong2 1 Institute of Transport Technology, University of Transport Technology, Hanoi 100000, Vietnam 2 Faculty of Civil Engineering, University of Transport Technology, Hanoi 100000, Vietnam *Email: ducvm@utt.edu.vn Abstract. In this paper, a new design of cylindrical panels is proposed with two functionally graded carbon nanotube-reinforced composite (FG-CNTRC) face sheets and a corrugated core. The corrugated core layer is modelled by applying the homogeneous technique according to the first- order shear deformation theory (FSDT). The nonlinear vibration behaviour of first-order shear deformable cylindrical panels with geometric nonlinearities is analysed in the present paper. The stress function is considered and the Galerkin method is used to formulate the nonlinear motion equation system. Nonlinear dynamic responses of panels can be achieved by using the fourth-order Runge-Kutta method. Numerical investigations can show the very large effects of corrugated core, the volume fraction of carbon nanotube, and the type of carbon nanotube distribution on the nonlinear vibration behaviour of sandwich FG-CNTRC cylindrical panels. Keywords: Nonlinear vibration; First-order shear deformation theory; FG-CNTRC; Corrugated core; Galerkin method. 1. Introduction The functionally graded material (FGM) is becoming an important material for the main structures subjected to complex loading types in modern technologies of excellent thermo-mechanical properties. The linear and nonlinear vibration behaviour of many form of FGM plates and cylindrical panels were analysed by using different shell theories and different methods [1-3]. Zenkour [1] presented a comprehensive analysis of FGM plates in the problems of buckling and free vibration. The nonlinear vibration of a coating-FGM-substrate cylindrical panel subjected to a temperature gradient was studied by Liew et al. [2]. By using the higher-order shear deformation theory and the FGM smeared stiffener technique, Dong and Dung [3] investigated the nonlinear vibration analysis of stiffened FGM sandwich plates, cylindrical panels, and doubly curved shallow shells with four material models. Functionally graded carbon nanotube reinforced composite (FG-CNTRC) is a new kind of nanocomposite material, where the carbon nanotubes (CNTs) are reinforced in an isotropic matrix and its volume fraction is continuously varied from one surface of the structure to another. Wang and Shen [4] investigated the nonlinear vibration of FG-CNTRC plates in thermal environments by using the higher-order shear deformation theory and the two-step perturbation technique. Similarly, Yu and Shen [5] investigated the effects of positive and negative Poisson's ratios on the nonlinear vibration of hybrid FG-CNTRC/metal laminated plates. Free vibration analyses of FG-CNTRC plates were also mentioned by Zhu et al. [6] using the finite element method with the first-order shear deformation theory (FSDT). Wu and Li [7] considered the free vibration of FG-CNTRC plates with various boundary conditions using the three-dimensional elasticity theory. For FG-CNTRC cylindrical panels, Shen and Xiang [8] presented the nonlinear vibration of FG-CNTRC cylindrical panels resting on the elastic foundations in thermal environments. The stiffened FG-CNTRC cylindrical panels with FG-CNTRC stiffeners were mentioned in the nonlinear vibration problems by Dong et al. [9]. For the corrugated structures, Liew et al. [10, 11] studied the nonlinear bending and vibration behaviours using the mesh-free method and mesh-free Galerkin method, respectively, by developing a homogeneous technique for first shear deformable corrugations.
83 100 Duc V.M. et al. A homogeneous technique for the corrugated core is applied for FSDT corrugated core and combined appropriately with the FSDT of shells. The Galerkin procedure is applied to obtain the nonlinear motion equations of panels, and the dynamic responses can be archived by using the fourth- order Runge-Kutta method. Numerical examples show the large effects of corrugated core, CNT volume fraction, CNT distribution laws and geometrical parameters on the nonlinear vibration behaviours of FG-CNTRC cylindrical panels. 2. FG-CNTRC cylindrical panel with corrugated core and analytical solution Fig. 1. Geometrical properties of cylindrical panels, corrugated cores, and CNT distribution laws Consider the moderately thick cylindrical panels with FG-CNTRC face sheets and corrugated core with the geometrical properties presented in Fig. 1. The panel is subjected to the time-dependent harmonic pressure q Q sin Ωt (N/m2) uniformly distributed over the top surface. The corrugated core = is designed in trapezoidal or sinusoidal forms. Assuming that the panel is reinforced by CNTs longitudinal direction, while, the corrugations are also in the longitudinal direction of the panel. Three CNT distribution laws are presented in Fig. 1.
84 Nonlinear vibration of FG-CNTRC cylindrical panels with corrugated core … 101 The CNT volume fractions for the upper face sheet ( − h 2 ≤ z ≤ − hcore 2 ) are distributed through the following function the thickness, as * - UD law: VCNT = VCNT , (1)  h + 2z  * - FG-X law: VCNT = 2  core  VCNT , (2)  hcore − h   h + 2z  * - FG-O law: VCNT = 2  core  VCNT . (3)  h − hcore  The CNT volume fractions for the lower face sheet ( hcore 2 ≤ z ≤ h 2 ) are distributed through the following function the thickness, as * UD law: VCNT = VCNT , (4)  h − 2z  * FG-X law: VCNT = 2  core  VCNT , (5)  hcore − h   h − 2z  * FG-O law: VCNT = 2  core  VCNT , (6)  h − hcore  The Hooke’s law for the orthotropic panels is expressed as  σ x   Q11 Q12 0 0 0   εx     ε   σ y  Q12 Q22 0 0 0  y  σ  =  0 0 Q66 0 0   γ xy  , (7)  xy      σ xz   0 0 0 Q44 0   γ xz      σ yz   0 0 0 0 Q55   γ yz        where the reduced stiffnesses Qij are presented as E1 E2 ν 21 E1 =Q11 = = = G23 , Q= G12 . , Q22 , Q12 = G13 , Q66 , Q44 1 − ν 21ν12 1 − ν 21ν12 1 − ν 21ν12 55 The forces and moments of a cylindrical panel according to the FSDT can be expressed in the general form, as  Nx   A A12 0 0 0 0  ε0 x    11    Ny   A A22 0 0 0  0  εy 0   12 N   0  γ0   xy  0 A66 0 0 0  xy   =  , (8)  Mx   0 0 0 D11 D12 0   φx ,x   My   0 0 0 D12 D22 0   φy ,y        M xy   0 D66  φ + φ      x ,y 0 0 0 0  y,x  and the shear forces are expressed by
85 102 Duc V.M. et al.  Qx     H 44 γ xz     H 44 φ x + H 44 w, x    = K SC    = K SC   , (9)  Qy      H 55 γ yz     H 55 φy + H 55 w,y   where K SC = 5 6 is the shear correction factor, ε0 , ε0 , γ 0 are the strains at the mid-plane of panels, x y xy φ x , φy are rotations, and the total stiffnesses of panels are determined by summing the stiffenesses of the core and the face sheets, as (= ( A A ,D ) ij ij ij , Dij )+(A CNTRC ij CNTRC , Dij ), (10) ( = ( H 44 , H 55 ) + ( H 44 , H 55 ) , H 44 , H 55 ) CNTRC CNTRC (11) with (ACNTRC ij CNTRC , Dij )= ∑ ∫Q CNTRC ij (1, z ) dz, 2 (12) i =1,2 Γ i = CNTRC H 44 CNTRC 44 = 1,= 1,2 Γ i 2Γ CNTRC 55 i ∑ ∫ Q dz, H = ∑ ∫Q CNTRC 55 dz , (13) i i In this paper, the homogeneous technique for the corrugated structures of Liew et al. [10, 11] using the FSDT is chosen to model the effects of the corrugated core. The corrugated core is homogenized to an anisotropic panel with the stiffnesses determined as follows 2c Emtc3 l c Emtc A11 = A12 =22 = 66 = , , ν m A11 , A Emtc , A I 2 12 c l 2 (1 + ν m ) c E t3 Emt 3 E t Emt 3 D11 = m c 2 , D12 =11 , D 22 = c 2 + m c αT , D 66 = c , νm D (14) ( l 12 1 − ν m ) 12 1 − ν m c ( 24 (1 + ν m ) ) D 66 = H 44 = k1 D 66 , H 55 k2 and the stiffnesses for sinusoidal corrugated core are presented as Emtc Emtc Emtc 3 E t f2 =A11 = , A66 = , D22 + mc , (15) αS 2 (1 + ν m ) 12 1 − ν m 2 2 ( ) with other expressions for stiffnesses of sinusoidal corrugated core are the same as for trapezoidal corrugated core, l is the half-length of one corrugation unit, and  π 2 1 + γ 2 nc 2  3πε0 nc   π  π  ( ) 2 2 2 = k1 1 + γ 2 nc E  k ,  , = 2 k2  1 +   F  k ,  + γ nc − 1 E  k ,   ,   2 2 2 π  2 3πε0 nc 2 2  1 + γ nc  2 2   2   (16) 2πf 4 π 2t c 2 γnc =γ = , ε0 2 = , k , L 12 L2 1 + γ 2 nc 2
86 Nonlinear vibration of FG-CNTRC cylindrical panels with corrugated core … 103 π π 2 2  π  π dθ E  k, =  2  ∫ 0 1 − k 2 sin 2 θdθ , F  k , =  2  ∫ 0 1 − k 2 sin 2 θ , (17) 4f3  f   l2 2 l 2πl  αT = f 2 c − 3 tan θ t  ( , αS = 1 +   6 1 − ν2  2 − 2πc sin )c  , c The motion equation system of cylindrical panels according to the FSDT, is presented as N x , x + N xy ,= I1u,tt + I 2 φ x ,tt , y N xy, x + N y ,= I1v,tt + I 2 φy ,tt , y Ny N x w, xx + N y w,yy + 2 N xy w, xy + + q + Qx , x + Qy ,y = w,tt , I1 (18) R M x , x + M xy ,y − Qx I 2 u,tt + I 3φ x ,tt , = M xy , x + M y ,y − Qy I 2 v,tt + I 3φy ,tt , = The compatibility equation of the FSDT is also applied as w, xx ε0 ,yy − γ 0 , xy + ε0 , xx = − x xy y − w, xx w,yy + w,2xy . (19) R In the case of simply supported boundary condition, the approximate solution of deflection, and rotations can be chosen in the forms, as w = ( t ) sin αx sin βy, φ x =x ( t ) cos αx sin βy, φy = ( t ) sin αx cos βy, W Φ Φy (20) where α = mπ / a, β = nπ / b ; m and n are the vibration modes; W (t ) and Φ x (t ), Φ y (t ) are the amplitude of deflection and rotations. By introducing the stress function ψ satisfies the conditions N x = ψ ,yy , N xy = −ψ , xy , N y = ψ , xx . (21) Substituting the stress function into the first two equations of (18), the expressions of u,tt and v,tt can be obtained, then, substituting those into the last two equations of (18) and rewriting the last three equation of (18), leads to ψ , xx w,yy − 2ψ , xy w, xy + ψ ,yy w, xx + K SC H 44 w, xx + K SC H 55 w,yy ψ , xx (22) + K SC H 44 φ x , x + K SC H 55 φy ,y + + q =w,tt , I1 R * ( * * * )  D11φ x , xx + D12 + D66 φy , xy + D66 φ x ,yy − K SC H 44 w, x − K SC H 44 φ x = I 3φ x ,tt , (23) (D * 21 * + D66 )φ x , xy * *  + D22 φ y ,yy + D66 φ y , xx − K SC H 55 w,y − K SC H 55 φ y = I 3φ y ,tt . (24) Similarly, the compatibility equation can be rewritten by w, xx R * * * ( * ) + w, xx w,yy − w,2xy + A11ψ , xxxx + −2 A12 + A66 ψ , xxyy + A22 ψ ,yyyy = . 0 (25)
87 104 Duc V.M. et al. Substituting the solution forms (20) into the compatibility equation (25), the form of stress function f can be obtained in the form as N y0 x 2 N x 0 y2 =ψ + + ψ1 cos 2αx + ψ 3 sin αx sin βy + ψ 2 cos 2βy. (26) 2 2 Then, substituting the solution forms (20) and obtained stress function into the system (22, 23, 24), the Galerkin procedure is applied, leads to   e5W 2 + h21W + h22 Φ x + h23Φ y = I 3Φ x ,   e6W 2 + h31W + h32 Φ x + h33Φ y = I 3Φ y , (27) s2W 2 + s3W 3 + s4 q + h11W + h12 Φ x + h13Φ y + h14 Φ xW ( ) I  + h15 Φ yW − N x 0 α 2 + N y 0β2 W + s4 N y 0 R =1W . In this paper, four freely movable edges are considered, leads to N x 0 = 0 and N y 0 = 0 . If the   inertia terms Φ x and Φ y are infinitesimal, the equation of motion is obtained as s4 q + g1W + g2W 2 + g3W 3 =, I1W (28) The free and linear frequency expression of panels is obtained by ωmn = − g1 I1 , (29) The fundamental frequency of panels can be determined by minimizing the free and linear frequency for all vibration modes. The harmonically forced external pressure is applied with = Q sin Ωt into Eq. (28) and the Runge-Kutta method is applied to obtain the dynamic responses of q panels. 3. Results and discussions The results of dimensionless free and linear frequencies of FG-X cylindrical panels without core are compared with the previous results of Shen and Xiang [8] in Table 1. In this comparison, the environment temperature is fixed at the room temperature ( T =300K). The validation is performed with four first modes and three cases of CNT volume fraction. As can be seen, the present results agree well with the previous results.  Table 1. Comparisons of dimensionless free and linear frequencies ωmn =mn b 2 h ω ( ) ρ0 E0 of FG- = = = = X cylindrical panels ( a b a R 1, b h 20, h 1mm ) ( m, n ) (1,1) (1,2) (1,3) (2,1) * VCNT = 0.28 Shen and Xiang [8] 32.1718 38.7997 55.0382 79.1357 Present 32.5061 37.7993 53.6805 81.7969 * VCNT = 0.17 Shen and Xiang [8] 27.3541 32.7020 47.7171 71.2389 Present 27.5863 32.5771 47.3858 73.0110 * VCNT = 0.12 Shen and Xiang [8] 22.0781 26.1749 37.3250 56.9005 Present 22.6016 25.8795 36.7549 56.9776 The present approach is illustrated by numerical examples for FG-CNTRC cylindrical panels with corrugated core where the material parameters of FG-CNTRC are chosen according to Shen et al. [4,8].
88 Nonlinear vibration of FG-CNTRC cylindrical panels with corrugated core … 105 The corrugated cores are made from Poly methyl methacrylate (PMMA) which is the same material as the matrix of FG-CNTRC. The environment temperature is fixed at the room temperature ( T =300K). Unless otherwise noted, the geometrical parameters in this paper are selected as follows a= b= 0.21 m, R = 1.05 m, h = 0.007 m, hCNTRC = 0.001 m, tc = 0.0012 m, f = 0.0019 m, c = 4 f , θ = π 4 . The fundamental frequencies of FG-CNTRC cylindrical panels with corrugated core and with solid core are presented in Table 2. The solid core thickness is calculated so that the volume of the solid core is equal to the corresponding corrugated core, respectively. The very large effects of corrugated core in comparison with the solid core can be observed. Additionally, the fundamental frequency of panel strongly increases when the CNT volume fraction increases. Oppositely, the differences in fundamental frequency are not too large with three distribution laws (UD, FG-X, and FG-O) in all investigated cases. Nonlinear dynamic responses of FG-CNTRC cylindrical panels with trapezoidal and sinusoidal corrugated core and with solid core are investigated in Figs. 2 and 3. As can be seen that the vibration amplitudes of panels with corrugated core are much smaller than those of panels with solid cores. Fig. 4 presents the effects of CNT volume fraction on the dynamic responses of cylindrical panels with trapezoidal corrugated core. Clearly, the vibration amplitude strongly decreases when the CNT volume fraction increases. Effects of CNT distributed law on the nonlinear dynamic responses of FG-CNTRC cylindrical panels with sinusoidal corrugated core are presented in Fig. 5. The recognized results showed the small differences of vibration amplitudes between three considered CNT distribution laws. Table 2. Fundamental frequencies (rad/s) of FG-CNTRC cylindrical panels with corrugated core and with solid core CNT volume Core type UD FG-X FG-O fraction Trapezoidal corrugation 4455.96 4629.23 4276.93 * Solid core 2124.33 2328.02 1898.89 VCNT = 0.12 Sinusoidal corrugation 4508.56 4683.85 4327.45 Solid core 2086.45 2291.83 1858.50 Trapezoidal corrugation 5449.12 5673.49 5228.21 * Solid core 2556.40 2806.08 2283.43 VCNT = 0.17 Sinusoidal corrugation 5512.93 5739.91 5289.46 Solid core 2511.81 2763.51 2235.86 Trapezoidal corrugation 6555.56 6859.54 6323.55 * Solid core 3118.10 3448.73 2766.02 VCNT = 0.28 Sinusoidal corrugation 6631.18 6938.66 6396.52 Solid core 3062.41 3395.97 2705.98
89 106 Duc V.M. et al. Fig. 2. Dynamic responses of FG-CNTRC Fig. 5. Effects of CNT distribution laws on the cylindrical panels with trapezoidal corrugated dynamic responses of cylindrical panels with * * core and with solid core (FG-X, VCNT = 0.12 , sinusoidal corrugated core ( VCNT = 0.28 , q = 104 sin ( 500t ) N/m2) q = 104 sin ( 500t ) N/m2) Fig. 3. Dynamic responses of FG-CNTRC Fig. 6. Effects of longitudinal radius on the cylindrical panels with sinusoidal corrugated core dynamic responses of cylindrical panels with * * and with solid core (UD, VCNT = 0.17 , trapezoidal corrugated core (FG-X, VCNT = 0.28 , q = 104 sin ( 500t ) N/m2) q = 104 sin ( 500t ) N/m2) Fig. 4. Effects of CNT volume fraction on the Fig. 7. Effects of excited amplitude on the dynamic responses of cylindrical panels with dynamic responses of cylindrical panels with * trapezoidal corrugated core (FG-X, sinusoidal corrugated core (FG-X, VCNT = 0.17 , q = 104 sin ( 500t ) N/m2) R = 0.21 m, q = 104 sin ( 500t ) N/m2)
90 Nonlinear vibration of FG-CNTRC cylindrical panels with corrugated core … 107 Fig. 8. Effects of excitation frequency on the Fig. 9. The amplitude-velocity curve of dynamic responses of cylindrical panels with cylindrical panels with corrugated core(FG-X, * trapezoidal corrugated core (FG-X, VCNT = 0.12 , VCNT = 0.12 , q = 104 sin ( 4600t ) N/m2) * Q = 104 N/m2) The large effects of longitudinal radius and excited amplitude on the dynamic responses of cylindrical panels with corrugated core are presented in Figs. 6 and 7. The results show that just a slight curvature of the panel significantly reduces the amplitude of the vibration. When the excitation frequencies are near to free and linear frequencies, the interesting phenomenon is observed like the harmonic beat phenomenon of a linear vibration (presented in Fig. 8). These results show that the amplitude of beats of panels increases rapidly when the excitation frequency approaches the free and linear frequency. Fig. 9 presents the closed form of the amplitude-velocity curve. The amplitude and velocity are equal to zero at the initial time and final time of beat and the contour of this relation corresponds to the period which has the greatest amplitude of beat. 4. Conclusion A semi-analytical approach of nonlinear vibration of FG-CNTRC cylindrical panels with corrugated core, under harmonically external pressure has been established and numerically illustrated. A homogeneous technique for first-order shear deformable corrugation has been applied which is suitable for the FSDT of cylindrical panels. Numerical results present the very large effects of corrugated core on the free and linear frequency and vibration amplitude in comparisons for the solid core can be observed. The large effects of geometrical and material properties of panels are obtained in the investigated examples. References [1] A. M. Zenkour. A comprehensive analysis of functionally graded sandwich plates: Part 2 - Buckling and free vibration. International Journal of Solids and Structures, 42, (2005), pp. 5243–5258. [2] K. M. Liew, J. Yang, Y. F. Wu. Nonlinear vibration of a coating-FGM-substrate cylindrical panel subjected to a temperature gradient. Computer Methods in Applied Mechanics and Engineering, 195, (2006), pp. 1007- 1026. [3] D. T. Dong, D. V. Dung. A third order shear deformation theory for nonlinear vibration analysis of stiffened FGM sandwich doubly curved shallow shells with four material models. Journal of Sandwich Structures and Materials, 21, (4), (2019), pp. 1316–1356. [4] Z. X. Wang, S. H. Shen. Nonlinear vibration of nanotube-reinforced composite plates in thermal environments. Computational Materials Science, 50, (8), (2011), pp. 2319-2330. [5] Y. Yu, H. S. Shen. A comparison of nonlinear vibration and bending of hybrid CNTRC/metal laminated plates with positive and negative Poisson's ratios. International Journal of Mechanical Sciences, 183, (2020), 105790.
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