Nonlinear dynamic buckling of full-filled fluid sandwich fgm circular cylinder shells

Nội dung text: Nonlinear dynamic buckling of full-filled fluid sandwich fgm circular cylinder shells

1. Vietnam Journal of Mechanics, VAST, Vol.41, No. 2 (2019), pp. 179 – 192 DOI: NONLINEAR DYNAMIC BUCKLING OF FULL-FILLED FLUID SANDWICH FGM CIRCULAR CYLINDER SHELLS Khuc Van Phu1, Le Xuan Doan2,∗ 1Military Academy of Logistics, Hanoi, Vietnam 2Academy of Military Science and Technology, Hanoi, Vietnam ∗E-mail: xuandoan1085@gmail.com Received: 17 November 2018 / Published online: 2 June 2019 Abstract. This paper is concerned with the nonlinear dynamic buckling of sandwich func- tionally graded circular cylinder shells filled with fluid. Governing equations are derived using the classical shell theory and the geometrical nonlinearity in von Karman–Donnell sense is taken into account. Solutions of the problem are established by using Galerkin’s method and Runge–Kutta method. Effects of thermal environment, geometric parameters, volume fraction index k and fluid on dynamic critical loads of shells are investigated. Keywords: dynamic buckling; dynamic critical loads; FGM-sandwich; full-filled fluid; cir- cular cylinder shell. 1. INTRODUCTION In recent years, functionally graded material (FGM) have been widely used in many industry due to outstanding characteristics. Plate and shell structures have received con- siderable attention of scientists in the world. In studies, vibration and dynamic stability of FGM shells are problems interested and achieved encouraging results. On vibration of shells, Bich and Nguyen [1] studied nonlinear responses of a func- tionally graded (FG) circular cylinder shell under mechanical loads. Governing equa- tions were based on improved Donnell shell theory. Kim [2] used an analytical method to study natural frequencies of circular cylinder shells made of FGM partially embedded in an elastic medium with an oblique edge based on the first order shear deformation theory (FSDT). In recent times, Duc et al. investigated nonlinear dynamic responses and vibration of imperfect eccentrically stiffened functionally graded thick circular cylindri- cal shells [3] and the one [4] surrounded on elastic foundation subjected to mechani- cal and thermal loads. The FSDT and the third order shear deformation theory (TSDT) were employed to solve problems. Bahadori and Najafizadeh [5] analyzed free vibra- tion frequencies of two-dimensional FG axisymmetric circular cylindrical shells resting on Winkler–Pasternak elastic foundations. The Navier-Differential Quadrature solution methods was employed to survey. c 2019 Vietnam Academy of Science and Technology
2. 180 Khuc Van Phu, Le Xuan Doan Regarding to dynamic buckling problems, Bich et al. [6] based on the classical shell theory and the smeared stiffeners technique to study nonlinear dynamics responses of eccentrically stiffened FG cylindrical panels. The nonlinear static and dynamic buckling problems of imperfect eccentrically stiffened FG thin circular cylinder shells under ax- ial compression load were solved in [7]. Mirzavand et al. [8] studied the post-buckling behavior of FG circular cylinder shells with surface-bonded piezoelectric actuators un- der the combined action of thermal load and applied actuator voltage. Duc et al. [9, 10] used the TSDT to analyze nonlinear static buckling and post-buckling for imperfect ec- centrically stiffened thin and thick FG circular cylinder shells made of S-FGM resting on elastic foundations under thermal-mechanical loads. Lekhnitsky smeared stiffeners tech- nique and Bubnov–Galerkin method were applied in calculation. By using an analytical approach, based on improved Donnell shell theory with von Karman–Donnell geometri- cal nonlinearity, Bich et al. [11] investigated the buckling and post-buckling of FG circular cylinder shells under mechanical loads including effects of temperature. Nonlinear buck- ling problems of imperfect eccentrically stiffened FG thin circular cylindrical shells sub- jected to axial compression load and surrounded by an elastic foundation were solved by Nam et al. [12]. The classical thin shell theory with the von Karman–Donnell geometrical nonlinearity, initial geometrical imperfection and the smeared stiffeners technique were employed to study. For circular cylindrical shells made of FGM filled with fluid, Sheng et al. [13] based on the FSDT to study free vibration characteristics of FG circular cylinder shells with flowing fluid and embedded in an elastic medium subjected to mechanical and thermal loads. This study was expanded to investigate dynamic characteristics of fluid-conveying FGM circular cylinder shells subjected to dynamic mechanical and thermal loads [14]. Za- far Iqbal et al. [15] examined vibration frequencies of FGM circular cylinder shells filled with fluid using wave propagation approach. Vibration frequencies of shell were ana- lyzed for various boundary conditions taking into account the effect of fluid. Shah et al. [16] based on Love’s thin-shell theory to investigate natural frequencies of full-filled fluid FG circular cylinder shells resting on Winkler and Pasternak elastic foundations. Wave propagation approach was employed to calculate. Silva et al. [17] studied nonlin- ear responses of fluid-filled FG circular cylinder shell under mechanical load. Recently, Hong-Liang Dai et al. [18] analyzed thermos electro elastic behaviors of a fluid-filled functionally graded piezoelectric material cylindrical thin-shell under the combination of mechanical, thermal and electrical loads. By using the classical shell theory and Galerkin method, Khuc et al. [19] considered nonlinear vibration of full-filled fluid circular cylin- der shells made of sandwich-FGM subjected to mechanical loads in thermal environment. To best of the authors’ knowledge, there is no analytical approach on dynamic buck- ling of sandwich FGM circular cylinder shells containing fluid. In this paper, nonlinear dynamic buckling of full-filled fluid sandwich FGM circular cylinder shells subjected to mechanical loads in the thermal environment is investigated. Governing equations are derived by using the classical shell theory with the geometrical nonlinearity in von Karman–Donnell sense. Solution of problem is established by using Galerkin’s method and Runge–Kutta method. Effects of thermal environment, fluid, structures’ geometric
3. Nonlinear dynamic buckling of full-filled fluid sandwich FGM circular cylinder shells 181 parameters and volume fraction index (k) on nonlinear dynamic responses of shell are considered. 2. GOVERNING EQUATIONS Consider a sandwich FGM circular cylinder shell with geometric parameters: R, h, hc, and hm are shown in Fig.1. Suppose that the full-filled fluid circular cylinder shell made of FGM sandwich subjected to an axial compression load N01 = −p (t) h and a uniformly distributed external pressure q (t) varying on time. Nonlinear dynamic buckling of full-filled fluid sandwich-FGM circular cylinder shells 3 Fig.Figure.1. 1. Model Model of of FGM-sandwich FGM-sandwich circular circular cylind cylinderer shell shell With configuration of sandwich FGM as Fig.1, suppose that Vc(z) and Vm(z) are the volumeWith configuration fractions of of ceramicsandwich andFGM metalas figure respectively,.1, suppose that the Vc(z) volume and Vm(z) fraction are the volume of ceramic fractions of ceramic and metal respectively, the volume fraction of ceramic constituent changes constituentaccording to changes the power accordinglaw and can be to expressed the power as law and can be expressed as V = 0, −0.5h ≤ z ≤ − (0.5h − h ), c Vcm=0, − 0,5 h z m −( 0,5 h − h )  k k z + 0.5z+−0,5h − hh hm m (1) Vc = Vc=, , − −(( 0,50.5 h −h h− m) h m z) (≤ 0,5z h −≤ h c()0.5 , kh − 0 hc) , k ≥ 0, (1) h−− h h h − hc −cmhm V = ( h − h ) V≤ =z1,≤ 0,5 h − h z 0,5 h c 1, 0.5 c cc(0.5 . ) ThenThen the the elasticityelasticity modulus modulus E, theE mass, the density mass ρ density and the Poissonρ and ratio the ν Poisson of circular ratio cylinderν of shell circular cylindercan be evaluated shell can as following be evaluated as following EEVEVEEEV= + = + − , E = EmmV mm + cE ccVc m=(E cm + m()E cc − Em) Vc, ρ = =ρ mmVVVV m ++ cρ ccV =c m= +(ρ m c −+ m()ρc c ,− ρm) Vc, (2) (2) νm=mc==νc =constconst. TheThe strain strain componentscomponents of the of circular the circular cylinder cylinder shell are shell are 0 = 0 −zk; 0 =  0 − zk ;  =  0 −0 2 zk (3) εx = εx −xzkx x, ε xy = yεy − yzky, yγxy = xyγxy − xy2zkxy, (3) 2 2 where 0u11  w 0  v w  w 0  u  v  w  w Where: x= + ;;;  y = − +  xy = + + (4) x22  x  y R  y  y  x  x  y ∂u 1  ∂w 2 ∂v w 1  ∂w 2 ∂u ∂v ∂w ∂w 0 = + 0 = − + 0 = + + εx 2w, εy  2 w  2 w , γxy , (4) ∂x 2 k∂=x ;;; k =∂y k =R 2 ∂y ∂y ∂x ∂x ∂y(5) xxy22 y xy xy 0 0 0 In which: x;; y xy are the strains at the middle surface; kx, ky and kxy are curvatures and the twist. By use of Eq (4), the deformation compatibility equation can be written as: 20 2 0 2 0 22 2 2 2   x y xy w  w  w1  w 2+ 2 − = − 2 2 − 2 ; (6) y  xx  y  x  y  x  y R  x
4. 182 Khuc Van Phu, Le Xuan Doan ∂2w ∂2w ∂2w k = , k = , k = , (5) x ∂x2 y ∂y2 xy ∂x∂y 0 0 0 in which εx; εy; γxy are the strains at the middle surface; kx, ky and kxy are curvatures and the twist. By use of Eq. (4), the deformation compatibility equation can be written as 2 0 2 0 2 0  2 2 2 2 2 ∂ ε ∂ εy ∂ γxy ∂ w ∂ w ∂ w 1 ∂ w x + − = − − . (6) ∂y2 ∂x2 ∂x∂y ∂x∂y ∂x2 ∂y2 R ∂x2 For circular cylindrical shell subjected to mechanical load in temperature environ- ment, the Hooke’s law can be defined as E (z) E (z) α (z) ∆T E (z) E (z) α (z) ∆T σ = (ε + νε ) − , σ = (νε + ε ) − , x 1 − ν2 x y 1 − ν y 1 − ν2 x y 1 − ν (7) E (z) τ = γ , xy 2 (1 + ν) xy in which ∆T = T − T0. Internal forces and moment resultants can be defined by integrating stresses compo- nents through the shells’ thickness and can be expressed in matrix form as      0    Nx A11 A12 0 B11 B12 0 εx Φa      0     Ny   A A 0 B B 0   ε   Φa     12 22 12 22   y     N   0 0 A 0 0 B   0   0  xy = 66 66 γxy − , (8)  Mx   B11 B12 0 D11 D12 0   −kx   Φb           My   B12 B22 0 D12 D22 0   −ky   Φb          Mxy 0 0 B66 0 0 D66 −2kxy 0 in which Nx; Ny; Nxy are internal forces, Mx; My; Mxy are moment resultants. Stiffness coefficients and quantities related to thermal load in Eq. (8) are explained in Appendix A. From Eq. (8) the expressions of deformation and moment resultants of sandwich FGM circular cylinder shell can be defined as 0 ∗ ∗ ∗ ∗ ∗ ∗ εx = A22Nx − A12Ny + B11kx + B12ky + Φa (A22 − A12) , 0 ∗ ∗ ∗ ∗ ∗ ∗ εy = −A12Nx + A11Ny + B21kx + B22ky + Φa (A11 − A12) , (9) 0 ∗ ∗ γxy = A66Nxy + 2B66kxy, ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Mx = B11Nx + B21Ny − D11kx − D12ky + [B11 (A22 − A12) + B12 (A11 − A12)] Φa − Φb, ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ My = B12Nx + B22Ny − D21kx − D22ky + [B12 (A22 − A12) + B22 (A11 − A12)] Φa − Φb, ∗ ∗ Mxy = B66Nxy − 2D66kxy, (10) Extended stiffness coefficients in Eq. (9) and Eq. (10) are explained in Appendix B. According to [20], the motion equations of full-filled fluid circular cylinder shell subjected to external pressure q (t) and an axial compression can be given as
5. Nonlinear dynamic buckling of full-filled fluid sandwich FGM circular cylinder shells 183 2 ∂N ∂Nxy ∂ u x + = ρ , ∂x ∂y 1 ∂t2 2 ∂Nxy ∂Ny ∂ v + = ρ , ∂x ∂y 1 ∂t2 2 2 2 2 2 2 2 ∂ Mx ∂ Mxy ∂ My ∂ w ∂ w ∂ w Ny ∂ w ∂w + 2 + + Nx + 2Nxy + Ny + + q − p = ρ + 2ρ ε , ∂x2 ∂x∂y ∂y2 ∂x2 ∂x∂y ∂y2 R L 1 ∂t2 1 ∂t (11) in which ε is the linear damping coefficient and Z h/2 ρcm (h − hc − hm) ρ1 = ρ(z)dz = ρmh + ρcmhc + , −h/2 k + 1 ∂ϕ ∂2w p = −ρ L = M is the dynamic pressure of fluid acting on the shell, L L ∂t L ∂t2 ρLRIn (λm) where ML = 0 is the mass of correspondence fluid to the shell vibration and λm In (λm) mπR λ = [19]. m L Applying the Volmir’s assumption [21] into Eqs. (11) (because of u  w, v  w), the equations of motion can be rewritten as follows ∂N ∂Nxy ∂Nxy ∂Ny x + = 0, + = 0, ∂x ∂y ∂x ∂y 2 2 2 2 2 2 2 ∂ Mx ∂ Mxy ∂ My ∂ w ∂ w ∂ w Ny ∂ w ∂w + 2 + + Nx + 2Nxy + Ny + + q = (ρ + M ) + 2ρ ε . ∂x2 ∂x∂y ∂y2 ∂x2 ∂x∂y ∂y2 R 1 L ∂t2 1 ∂t (12) The first and the second equation of Eqs. (12) are satisfied identically by recommend- ing the stress function: ∂2F ∂2F ∂2F N = , N = , N = − . (13) x ∂y2 y ∂x2 xy ∂x∂y Substituting Eqs. (9) and (13) into Eq. (6), and Eq. (13) into the third equation of Eqs. (12) we obtain the system of two equations ∂4F ∂4F ∂4F A∗ + (A∗ − 2A∗ ) + A∗ 11 ∂x4 66 12 ∂x2∂y2 22 ∂y4 2 ∂4w ∂4w ∂4w 1 ∂2w  ∂2w  ∂2w ∂2w + B∗ + (B∗ + B∗ − 2B∗ ) + B∗ + − + = 0, 21 ∂x4 11 22 66 ∂x2∂y2 12 ∂y4 R ∂x2 ∂x∂y ∂x2 ∂y2 (14)
6. 184 Khuc Van Phu, Le Xuan Doan ∂2w ∂w ∂4w ∂4w ∂4w ∂4F (ρ + M ) + 2ρ ε + D∗ + (D∗ + D∗ + 4D∗ ) + D∗ − B∗ 1 L ∂t2 1 ∂t 11 ∂x4 12 21 66 ∂x2∂y2 22 ∂y4 21 ∂x4 ∂4F ∂4F 1 ∂2F ∂2F ∂2w ∂2F ∂2w ∂2F ∂2w − B∗ − (B∗ + B∗ − 2B∗ ) − − + 2 − + q = 0. 12 ∂y4 11 22 66 ∂x2∂y2 R ∂x2 ∂y2 ∂x2 ∂x∂y ∂x∂y ∂x2 ∂y2 (15) Eqs. (14) and (15) are governing equations used to investigate nonlinear dynamic buckling of full-filled fluid circular cylinder shell made of sandwich FGM. 3. DYNAMIC BUCKLING SOLUTION Suppose that the circular cylinder shell under simply supported at both ends and subjected to axial compression load N01 = −ph. In which p is average axial stress acting on the ends of the shell. Therefor boundary conditions are defined as w = 0, Mx = 0, Nx = N01, Nxy = 0 at x = 0 and x = L. The shells’ deflection satisfying above conditions can be written as mπx ny w = f (t) sin sin , (16) L R where m, n are numbers of half waves in generating line direction and circumference direction, respectively. The solution of stress function F in Eq. (14) can be defined as y2  β2 A∗  x2 = + − − − − 2 + + 12 + F F1 cos 2αx F2 cos 2βy F3 sin αx sin βy N01 ∗ f(t) ψ f(t) N01 ∗ Γ , 2 8A11 A11 2 (17) in which β2 α2 = ∗ 2 = 2 = ∗ 2 = 2 F1 F1 f(t) 2 ∗ f(t), F2 F2 f(t) 2 ∗ f(t), 32α A11 32β A22 4 ∗ 2 ∗ ∗ ∗ 4 ∗ α2 ∗ α B21 + (αβ) (B11 + B22 − 2B66) + β B12 − R F3 = F f(t) = f(t), 3 4 ∗ 2 ∗ ∗
   4 ∗ α A11 + (αβ) A66 − 2A12 + β A22 γη  1  = 2 ∗ − 2 ∗
   ∗ − 2 ∗ + 2 ∗
   + ψ 2 ∗ α A11 β A12 F3 α B21 β B22 , mnπ A11 R ∗ ∗ (A11 − A12) Φa  m   n  mπ nπ Γ = ∗ , γ = (−1) − 1 , η = (−1) − 1 α = , β = . A11 L R Substituting Eq. (16) and Eq. (17) into Eq. (15), then using the Galerkin method we obtain d2 f d f 4γη  A∗  4γη ( + ) + + 3 + 2 + + 12 + = ρ1 ML 2 2ρ1ε H1 f(t) H2 f(t) H3 f(t) 2 ∗ N01 Γ 2 q, dt dt mnπ R A11 mnπ (18)
7. Nonlinear dynamic buckling of full-filled fluid sandwich FGM circular cylinder shells 185 in which 4 2 ∗ ∗ β H1 = 2 (αβ) (F1 + F2 ) + ∗ , 8A11 16γη  α2  1 3 β2 3mnπ2  = − 4 ∗ − ∗ + 4 ∗ ∗ − ( )2 ∗ + + 2 H2 2 4α B21 F1 4β B12F2 αβ F3 ∗ β ψ , 3mnπ R 2 32 RA11 16γη h 4 ∗ 2 ∗ ∗ ∗ 4 ∗ i H3 = α D11 + (αβ) (D12 + D21 + 4D66) + β D22  α2   A∗  4γη + 4 ∗ + ( )2 ( ∗ + ∗ − ∗ ) + 4 ∗ − ∗ − 2 + 12 2 − 2 + α B21 αβ B11 B22 2B66 β B12 F3 α ∗ β N01 Γβ 2 ψ. R A11 mnπ R Eq. (18) is used to investigate the nonlinear dynamic buckling of full-filled fluid FGM sandwich circular cylinder shells under mechanical load in thermal environment. Nonlinear dynamic buckling analysis For dynamic buckling analysis, this paper investigates two cases: - Case 1. Consider a full-filled fluid sandwich FGM circular cylinder shell under lin- ear axial compression load varying on time N01 = −ph with p = c1t (c1-loading speed), q = 0. - Case 2. Consider a full-filled fluid sandwich FGM circular cylinder shell under a pre-axial compression load and an external uniformly distributed pressure varying on time: N01 = const; q = ct (c2-loading speed). In order to analyze the dynamic buckling problem of the considered shells, firstly Eq. (18) is solved for each case respectively to determine the nonlinear dynamic responses; secondarily based on these obtained dynamic responses, the dynamic critical time tcr can be obtained according to Budiansky–Roth criterion [22]. This criterion is based on that for large value of loading speed, the amplitude time curve of obtained displacement re- sponse increases sharply depending on time and this curve obtains a maximum by pass- ing from the slope point and at the corresponding time t = tcr the stability loss occurs. Here t = tcr is called critical time and the load corresponding to this critical time is called dynamic critical buckling load Pcr = c1tcr (case 1) or qcr = c2tcr (case 2). 4. VALIDATION To the best of the author’s knowledge, there is no any publication on the nonlinear dynamic buckling of the sandwich-FGM cylindrical shell containing full filled fluid in thermal environment. Thus, the results in this paper are compared with the fluid-free shell (hc = hm = 0). Authors compare the dynamic critical stress of fluid-free FGM cylindrical shell with the one in publication of Huaiwei Huang, Qiang Han [23] (Tab. 1), for FGM shell made of ZrO2/Ti-6Al-4V and material properties: Em = 122.56 e9Pa, 3 3 ρm = 4429 kg/m , νm = 0.288, Ec = 244.27 e9Pa, ρc = 5700 kg/m , νc = 0.288. Tab.1 shows that, the results of this article are slightly different from the above publication. The cause of this difference is that the authors use different methods, so the results of this article can be reliable.
8. 186 Khuc Van Phu, Le Xuan Doan Table 1. Comparison of critical stress of the compressed cylindrical shell (MPa) k 0.2 1.0 5.0 Huang & Han [23] 194.94 (2, 11) 169.94 (2, 11) 150.25 (2, 11) Present 193.914 (1, 9) 168.685 (1, 9) 149.167 (1, 9) 5. NUMERICAL RESULTS Consider a circular cylindrical shell made of FGM-core with geometric dimensions: h = 0.014 m, hc = h/5, hm = h/5, L/R = 2 and R/h = 200. FGM made of Aluminium 9 2 3 and Alumina with the material properties are Em = 7 × 10 N/m ; ρm = 2702 × 10 3 −5 −1 11 2 3 3 kg/m , αm = 2.3 × 10 C , Ec = 3.8 × 10 N/m ; ρc = 3.8 × 10 kg/m , αc = 5.4 × −6 −1 3 3 10 C , ε = 0.1, the Poisson’s ratio νc = 0.3 the fluid density ρL = 10 kg/m . - Case 1. Consider a full-filled fluid sandwich FGM circular cylinder shell under linear axial compression load varying on time N01 = −ph(p = c1t), q = 0. In this case, the critical time tcr can be obtained according to Budiansky–Roth crite- rion. The dynamic critical force pcr = c1tcr. The nonlinear dynamic responses of shell are shown in Figs.2–7. Nonlinear responses of fluid-filled and fluid-free circular cylinder shell in thermal environment are shown in Figs2–3. From Fig.2 we obtain tcr = 0.065 s and Pcr = 68.1 GPa respectively and from Fig.3, we can see that with fluid-filled cylinder shell, the dynamic critical force Pcr = 68.1 GPa increased by 4.12 times (318%) compared to the dynamic critical force of fluid-free ones Pcr = 16.3 GPa, tcr = 0.015 s, respectively. Doing the same with the next case taking into account the influence of other factors 8must 8 be derived from dynamical responsesPhuPhu V. to VK. determineKand and Doan Doan X the. XL. L critical forces. m=m1;= n1; =n 13; = 13;k = k 1; = R 1; / Rh =/ h 200; = 200; L / RL /= R 2; = 2; o o T =T 50 = 50C ; hC = ; h 0.01 = 0.01 m ;c m1 ;c= 1e12;1 = 1e12; FigFigure. ure. 2. 2.Nonlinear Nonlinear dynamic dynamic response response of of FigFigureure. 3 3.Effect Effects ofs offluid fluid on on dynamic dynamic Fig. 2. Nonlinear dynamic response of fluid- Fig. 3. Effects of fluid on dynamic response of fluidfluid-filled-filled circular circular cylind cylinder ershell shell responseresponse of ofcircular circular cylinder cylinder shell shell filled circular cylinder shell circular cylinder shell FigFigureure.4 .4shows shows nonlinear nonlinear dynamic dynamic responses responses of ofcylind cylinder ershell shell when when volume volume -fraction-fraction index index k k changeschangesFig From4. 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From From the the graph graph it isit isobserved observed that that when when the the temperature temperature increases, increases, the the dynamic dynamic o o 0 0 criticalcritical force force of of shell shell will will decrease. decrease. From From P crP=76,6GPacr=76,6GPa at at0 C0 Cto toPcr P=65GPacr=65GPa at at200 200C.C That. That means, means, the the loadload-bearing-bearing capability capability of ofthe the shell shell will will decrease decrease when when temperature temperature increases. increases. m=m1;= n1; = n 13; = 13; R / R h / = h 200; = 200; 3 3 mm=1;= n1; = n 13; = 13; k = k 1; = 1; LRC/LRC/= 2;= 2; T = T 50 = o 50o ; ; R/R h/== h 200;== 200; L / L R / R 2; 2; hm==0.01 ;c 1e10; hm==0.011 ;c1 1e10; hm==0.01 ;c 1e12; hm==0.011 ;c1 1e12; 3 3 1-k=01-k=0 1- 1ΔT=0- ΔT=00C0 C 2-k=0.52-k=0.5 2- 2ΔT- ΔT =50 =500C0 C 0 0 3-k=13-k=1 3- 3ΔT- ΔT =200 =200C C 2 1 1 2 2 2 1 1 FigFigure ure . 4 4Dynamic. Dynamic response response of offluid fluid-filled-filled Figure.Figure. 5. 5.Temperature Temperature effect effect on on nonlinear nonlinear cylinder shell when k changes dynamicdynamic response responses ofs offluid fluid-filled-filled cylinder cylinder shell shell cylinder shell when k changes TheThe effect effect of ofgeometric geometric parameters parameters (L/R (L/R ratio) ratio) on on nonlinear nonlinear dynamic dynamic response responses ofs ofcylind cylinder ershells shells mademade of ofsandwich sandwich-FGM-FGM filled filled with with fluid fluid is isshown shown in infigure figure. 6 6.The The dynamic dynamic critical critical force force of ofcylind cylinder ershell shell decreasedecreases ws henwhen increas increasinging R/L R/L ratio. ratio. That That means means increas increasinging length length of ofthe the shell shell, the, the stability stability of ofthe the shell shell structurestructure will will decrease. decrease. FigFig.7 .7indicate indicates snonlinear nonlinear dynamic dynamic responses responses of ofcircular circular cylinder cylinder shell shell made made of ofFGM FGM and and sandwichsandwich-FGM-FGM filled filled with with fluid. fluid. For For the the structure structure made made of ofsandwich sandwich-FGM-FGM, the, the critical critical force force is is PcrP=0,496GPacr=0,496GPa, and, and for for FGM FGM ones ones, the, the critical critical force force is isP crP=0,485GPacr=0,485GPa. That. That means, means, with with the the same same geometrygeometry dimens dimensions,ions, the the work workabilityability of ofsandwich sandwich-FGM-FGM cylind cylinder ershell shell is isbetter better than than FGM FGM ones. ones.
9. 8 8 PhuPhu V .V K. Kand and Doan Doan X .X L. L m=1; n = 13; k = 1; R / h = 200; L / R = 2; m=1;o n = 13; k = 1; R / h = 200; L / R = 2; T = 50Co ; h = 0.01 m ;c1 = 1e12; T = 50C ; h = 0.01 m ;c1 = 1e12; Figure. 2. Nonlinear dynamic response of Fig ure. 2. Nonlinear dynamic response of FigFigureure. 3 3.Effect Effects ofs offluid fluid on on dynamic dynamic fluid-filled circular cylinder shell response of circular cylinder shell fluid-filled circular cylinder shell response of circular cylinder shell FigFigureure.4.4 shows shows nonlinear nonlinear dynamic dynamic responses responses of of cylind cylinder ershell shell when when volume volume -fraction-fraction index index k k changeschanges. From. From Fig Fig.4.4 as as can can see see that, that, if ifk increasek increases thes the dynamic dynamic critical critical force force of ofshell shell will will decrease. decrease. That That meansmeans the the load load-bearing-bearing capability capability of of cylind cylinderer shell shell decrease decreases. s. TheThe Effect Effect of of thermal thermal environment environment on on nonlinear nonlinear dynamic dynamic response responses ofs ofcircular circular cylind cylinder ershell shell cancan be be shown shown in in fig figureure.5 5. From From the the graph graph it itis isobserved observed that that when when the the temperature temperature increases, increases, the the dynamic dynamic o o 0 0 criticalcritical force force of of shellNonlinear shell will will dynamicdecrease. decrease. buckling From From of full-filled P crP=76,6GPacr=76,6GPa fluid sandwich at at0 FGM0C Cto circulartoP crP=65GPacr=65GPa cylinder shellsat at200 200C.C That. That means, means, 187 the the loadload-bearing-bearing capability capability of of the the shell shell will will decrease decrease when when temperature temperature increases. increases. mm=1;= n1; =n 13; = 13; R /R h / = h 200;= 200; 3 3 mm=1;= n1; = n 13; = 13; k = k 1; = 1; LRC/LRC/= 2;= 2; T T= 50= 50o o ; ; R/R h/== h 200;== 200; L / L R / R 2; 2; hm==hm0.01==0.01 ;c1 ;c 1e10; 1e10; 1 hm==hm0.01==0.01 ;c1 ;c1 1e12; 1e12; 3 3 1-1k=0-k=0 1-1 ΔT=0- ΔT=00C0 C 2-2k=0.5-k=0.5 2-2 ΔT- ΔT =50 =500C0 C 3-3k=1-k=1 3-3 ΔT- ΔT =200 =2000C0 C 1 1 2 2 2 2 1 1 FigFigure ure . 4. .4 Dynamic. Dynamic response response of of fluid fluid-filled-filled Figure.Figure. 5. 5.T emperatureTemperature effect effect on on nonlinear nonlinear Fig. 4. Dynamiccylindcylinder responseer shell shell when when of fluid-filledk kchanges changes cylin- Fig.dynamicdynamic 5. Temperature response responses ofs effect offluid fluid- onfilled-filled nonlinear cylinder cylinder shell dy- shell der shell when k changes namic responses of fluid-filled cylinder shell TheThe effect effect of of geometric geometric parameters parameters (L/R (L/R ratio) ratio) on on nonlinear nonlinear dynamic dynamic response responses ofs ofcylind cylinder ershells shells mademade of of sandwich sandwich-FGM-FGM filled filled with with fluid fluid is isshown shown in infigure figure. 6 6.T heThe dynamic dynamic critical critical force force of ofcylind cylinder ershell shell The effect of thermal environment on nonlinear dynamic responses of circular cylin- decreasedecreases ws whenhen increas increasinging R/L R/L ratio. ratio. That That means means increas increasinging length length of of the the shell shell, the, the stability stability of ofthe the shell shell der shell is shown in Fig.5. From the graph it is observed that when the temperature structurestructure will will decrease. decrease. increases, the dynamic critical force of shell will decrease. From P = 76.6 GPa at 0◦C to Fig.7 indicates nonlinear dynamic responses of circular cylindercr shell made of FGM and P Fig.7 indicate◦ s nonlinear dynamic responses of circular cylinder shell made of FGM and sandwichcrsandwich= 65 GPa-FGM-FGM at filled 200 filledC. with Thatwith fluid. fluid.means, For For thethe the load-bearing structure structure made made capability of ofsandwich sandwich of the-FGM-FGM shell, the, will the critical critical decrease force force is is whenPcrP=0,496GPacr=0,496GPa temperature, and, and increases.for for FGM FGM ones ones, the, the critical critical force force is isP crP=0,485GPacr=0,485GPa. That. That means, means, with with the the same same geometrygeometryThe effectdimens dimens ofions,ions, geometric the the work workability parametersability of of sandwich sandwich (L/R-FGMratio)-FGM cylind oncylind nonlinearer ershell shell is isbetter dynamic better than than responsesFGM FGM ones. ones. of cylinder shells made of sandwich-FGM filled with fluid is shown in Fig.6. The dynamic critical force of cylinder shell decreases when increasing R/L ratio. That means increas- ing lengthNonlinear ofNonlinear the shell, dynamic dynamic the buckling stability buckling of of offull full the-filled-filled shell fluid fluid structure sandwich sandwich- willFGM-FGM decrease.circular circular cylinder cylinder shells shells 9 9 mm==1;1; n n = =13; 13; k k = =1; 1; R R / h / h = = 200; 200; mm=1;=1; n =n 13;= 13; k k= 1;= 1; T T = =50 500 C0 C TT = = 50 50oCoC ; h ; h = = 0.01 0.01 m m ; ; RR/ h/ h=== 200; 200; L L/ R / R 2; 2; c1c1== 1e12; 1e12; hm==hm==0.010.01 ; ;c 1c1 1e9; 1e9; 1 1 1-1R/L=2-R/L=2 2 2 2 2 2-2R/L=2.2-R/L=2.2 3 3 1-1Sandwich-Sandwich-FGM-FGM 3-3R/L=2.5-R/L=2.5 1 1 2-2FGM-FGM Fig.FigFig 6ure.ure Effect. 6. .6 E. Effect offfect geometric of of geometric geometric parameters parameters parameters on on dy- on Fig.FigFig 7ure.ure Dynamic. 7. .7 Dynamic. Dynamic responses responses responses of of of FGM FGMFGM and and and dynamicnamicdynamic responses responses responses of of fluid-filledof fluid fluid-filled-filled cylinder cyli cylindernder shell shell shell sandwich-FGMsandwichsandwich-FGM-FGM circular circular cylinder cylinder shell shell Fig.7 indicates nonlinear dynamic responses of circular cylinder shell made of FGM CaseCase 2. 2. Consider Consider a afull full-filled-filled fluid fluid sandwich sandwich FGM FGM circular circular cylind cylinderer shell shell under under a auniform uniform pre pre-axial-axial and sandwich-FGM filled with fluid. For the structure made of sandwich-FGM, the crit- compressioncompression load load and and an an external external uniformly uniformly distributed distributed pressure pressure varying varying on on time time: :N N0101=const,=const, q=c q=c2t2 t(c (c2-2 - loadingicalloading force speed speed is P).cr). = 0.496 GPa, and for FGM ones, the critical force is Pcr = 0.485 GPa. That means, with the same geometry dimensions, the workability of sandwich-FGM cylinder TheThe nonlinear nonlinear dynamic dynamic responses responses of of circular circular cylinder cylinder shell shell are are shown shown in in fig figureure.8.8 to to fig figureure. 13. 13 shell is better than FGM ones. 1- Full filled fluid 1 1- Full filled fluid 1 2 2 2-2 -No No fluid fluid mm=1;= n1; =n 13; = 13; k =k 1; = R1; /R h / =h 200;= 200; L /L R / R= 2;= 2; o m=1; n = 13; k = 1; R / h = 200; L / R = 2; T = 50Co ; h = 0.01 m ;c = 1e10; N = 1 e 3 m=1; n = 13; k = 1; R / h = 200; L / R = 2; T = 50C ; h = 0.01 m ;c2 = 1e10; 01N = 1 e 3 o 2 01 T = 50Co ; h = 0.01 m ;c2 = 1e10; N 01 = 1 e 3 T = 50C ; h = 0.01 m ;c2 = 1e10; N 01 = 1 e 3 FigFigureure. 8 8. Nonlinear Nonlinear dynamic dynamic response responses sof of FigFigure.ure. 9 .9 Effect. Effect of of fluid fluid on on dynamic dynamic fullfull-filled-filled fluid fluid circular circular cylind cylinderer shell shell responseresponses sof of circular circular cylind cylinderer shell shell NonlinearNonlinear dynamic dynamic response responses sof of fluid fluid-filled-filled and and fluid fluid-free-free sandwich sandwich FGM FGM circular circular cylind cylinderer shell shell are depicted in figure 8 and figure 9. From fig. 8 we obtain tscr = 0,01 and qcr = 147 MPa respectively, are depicted in figure 8 and figure 9. From fig. 8 we obtain tscr = 0,01 and qcr = 147 MPa respectively, from the fig. 9, it is observed that fluid remarkably increases the dynamic critical force of the shell (from from the fig. 9, it is observed that fluid remarkably increases the dynamic critical force of the shell (from qcr=25 MPa at tscr = 0,002 in case fluid-fee shell to qcr= 147 MPa at tscr = 0,01 in case shell containing qcr=25 MPa at tscr = 0,002 in case fluid-fee shell to qcr= 147 MPa at tscr = 0,01 in case shell containing fluifluid,d i, .ie e the. the critical critical force force increased increased by by 5,88 5,88 times times by by 4 88%).488%). Similarly,Similarly, we we make make other other cases cases when when taking taking into into account account the the influence influence of of other other factors factors derive derive fromfrom dynamic dynamic response response curves curves to to determine determine dynamic dynamic critical critical forces. forces. FigFigureure.10.10 and and fig figureure. .11 11 show show dynamic dynamic responses responses of of circular circular cylind cylinderer shell shell filled filled with with fluid fluid with with variousvarious volume volume-fraction-fraction index index k kand and the the effect effect of of thermal thermal environment environment on on dynamic dynamic responses responses of of circular circular cylindcylinderer shell shells.s .From From the the graph graph as as can can see see that that if if temperature temperature increase increases sthe the dynamic dynamic critical critical force force
10. NonlinearNonlinear dynamic dynamic buckling buckling of offull full-filled-filled fluid fluid sandwich sandwich-FGM-FGM circular circular cylinder cylinder shells shells 9 9 m=m1;= n1; = n 13; = 13; k = k 1; = R 1; / R h / = h 200; = 200; m=1; n = 13; k = 1; T = 500 C0 T = 50oCo ; h = 0.01 m ; m=1; n = 13; k = 1; T = 50 C T = 50C ; h = 0.01 m ; R/ h== 200; L / R 2; c1 c= 1e12;= 1e12; R/ h== 200; L / R 2; 1 hm==0.01 ; c 1e9; hm==0.01 ;1 c1 1e9; 1 1 188 Khuc Van Phu, Le Xuan Doan 1-1R/L=2-R/L=2 2 2 2 2 2-2R/L=2.2-R/L=2.2 3 3 1-1Sandwich-Sandwich-FGM-FGM -3 Case-3R/L=2.5-R/L=2.5 2. Consider a full-filled fluid1 1 sandwich2 FGM-2FGM-FGM circular cylinder shell under a uniform pre-axial compression load and an external uniformly distributed pressure varying on time: N = const, q = c t (c -loading speed). 01 2 2 FigFigureTheure. 6. nonlinear. 6E. ffectEffect of of geometric dynamic geometric parameters responses parameters on of on circular FigFig cylinderureure. 7. .7 Dynamic. shellDynamic are responses responses shown inof of FGM Figs. FGM8 and– and dynamic13dynamic. Nonlinear responses responses dynamic of of fluid fluid- responsesfilled-filled cyli cylinder ofnder fluid-filledshell shell andsandwichsandwich fluid-free-FGM-FGM sandwich circular circular cylinder FGMcylinder circular shell shell cylinder shell are depicted in Figs.8–9. From Fig.8 we obtain tcr = 0.01 s and qcr = 147 MPa respectively, from the Fig.9, it is observed that fluid remarkably increases the CaseCase 2. 2.Consider Consider a fulla full-filled-filled fluid fluid sandwich sandwich FGM FGM circular circular cylind cylinderer shell shell under under a uniforma uniform pre pre-axial-axial dynamic critical force of the shell (from qcr = 25 MPa at tcr = 0.002 s in case fluid-fee compressioncompression load load and and an an external external uniformly uniformly distributed distributed pressure pressure varying varying on on time time: N: 01N=const,01=const, q=c q=c2t 2(ct (c2- 2- loadingshellloading to speed speedqcr ).= ).147 MPa at tcr = 0.01 s in case shell containing fluid, i.e. the critical force increased by 5.88 times by 488%). TheThe nonlinear nonlinear dynamic dynamic responses responses of of circular circular cylinder cylinder shell shell are are shown shown in in fig figureure.8.8 to to fig figureure. 13. 13 1-1 -Full Full filled filled fluid fluid 1 1 2 2 2-2 -No No fluid fluid m=m1;= n1; =n 13; = 13;k =k 1; = R 1; / Rh =/ h 200; = 200; L / LR / = R 2; = 2; o o m=m1;= n1; = n 13; = 13; k = k 1; = R 1; / R h /= h 200; = 200; L / L R / = R 2; = 2; T =T 50 = 50C ; hC = ; h 0.01 = 0.01 m ;c m2 ;c =2 1e10; = 1e10; N 01 N = 01 1 e = 3 1 e 3 o o T =T 50 = 50C ; hC =; h 0.01 = 0.01 m ;c m2 ;c =2 1e10; = 1e10; N 01 N = 01 1 e= 3 1 e 3 FigFigureure. 8 8.Nonlinear Nonlinear dynamic dynamic response responses ofs of FigFigure.ure. 9 .9 Effect. Effect of of fluid fluid on on dynamic dynamic Fig. 8full.full Nonlinear-filled-filled fluid fluid dynamic circular circular responsescylind cylinderer shell ofshell full- Fig. 9. Effectresponseresponse of fluids ofs of circular on circular dynamic cylind cylind responseserer shell shell of filled fluid circular cylinder shell circular cylinder shell NonlinearNonlinear dynamic dynamic response responses ofs of fluid fluid-filled-filled and and fluid fluid-free-free sandwich sandwich FGM FGM circular circular cylind cylinderer shell shell areare depicted depicted in infigure figure 8 and8 and figure figure 9. 9From. From fig. fig. 8 8we we obtain obtain tscr tscr= =0,010,01 and and q crqcr= 147= 147 MPa MPa respectively respectively, , from1010 Similarly, the fig. 9, weit is makeobserved other that cases fluid whenremarkablyPhu taking V. Kincreases and into Doan account the X dynamic. L the influence critical force of other of the fac-shell (from from the fig. 9, it is observed that fluid remarkablyPhu V. increasesK and Doan the Xdynamic. L critical force of the shell (from torsq =25 derive MPa fromat ts dynamic= 0,002 in response case fluid curves-fee shell to determine to q = 147 dynamicMPa at ts critical= 0,01 forces. in case shell containing qcrdecrease=25crdecrease MPsa. s atThat. That tscr crmeans= means0,002 if if the thein temperaturecase temperature fluid-fee increases increases shell to then qthencr=cr the 147the stability stabilityMPa at of tsofcr the crthe= 0,01shell shell structure instructure case shellwill will decrease.containing decrease. fluifluid, di.,e i ethe. the critical critical force force increased increased by by 5,88 5,88 times times by by 4 88%).488%). Similarly,Similarly, we we make make other other cases cases when when taking taking into into account account the the influence influence of of other other factors factors derive derive fromfrom dynamic dynamic1-1 k=0- k=0 response response curves curves3 3 to to determine determine dynamic dynamic critical critical forces. forces. 3 3 2- k=0.5 2- k=0.5 2 2 Fig3Fig-3 urek=1- k=1ure.10 .10 and and fig figureure. 11. 11 show show dynamic dynamic responses responses of of circular circular cylind cylinderer shell shell filled filled with with fluid fluid with with variousvarious volume volume-fraction-fraction index index k andk and the the effect effect of of thermal thermal environment environment1 on1 on dynamic dynamic responses responses of of circular circular cylindcylinder ershell shells. sFrom. From the the graph graph2 2as as can can see see that that if if temperature temperature increase increases sthe the dynamic dynamic critical critical force force 1 1 1-1 ΔT=0- ΔT=0 2-2 ΔT=50- ΔT=50 3-3 ΔT=200- ΔT=200 m=1; n = 13; k = 1; R / h = 200; mm=1;=1; n n= 13;= 13; R R/ h / h= =200; 200; L L/ R / R = =2; 2; m=1; n = 13; k = 1; R / h = 200; o o LL/ R/ R== 2; 2; h h== 0.01 0.01 mm ;c ;c2 =2 1e9;= 1e9; TT = =50 50C ;C h ; h= =0.01 0.01 m ;cm ;c2 =2 1e10;= 1e10; NeNe=13=13 Ne01Ne01=13=13 0101 FigFigureure. 10. 10. Dynamic. Dynamic response responses sof of fluid fluid-filled-filled Figure.Figure. 11 11. E. Effectffect of of thermal thermal on on the the dynamic dynamic Fig. 10circular. circular Dynamic cylinder cylinder responses shell shell with ofwith fluid-filled k kchanges changes cir- Fig. 11. Effectresponseresponse of thermal of of circular circular on thecyli cyli dynamicndernder shells shells re- cular cylinder shell with k changes sponse of circular cylinder shells EffectsEffects of of geometric geometric parameters parameters on on nonlinear nonlinear dynamic dynamic response response of of full full-filled-filled fluid fluid circular circular cylind cylinderer shellsshells are are surveyed surveyed and and presented presented in in fig fig.12 12. D Dynamicynamic critical critical force force of of the the shell shell decrease decrease s sw withith increas increasinging thethe ratio ratio of of length length to to radius radius L/R L/R. That. That means means if if the the length length of of shell shell increases, increases, the the stability stability of of the the shell shell will will decrease.decrease. NonlinearNonlinear responses responses of of FGM FGM and and sandwich sandwich-FGM-FGM circular circular cylinder cylinder shell shell filled filled with with fluid fluid are are shownshown in in figure figure. .13. 13. The The critical critical force force o fo ffull full-filled-filled fluid fluid sandwich sandwich-FGM-FGM circular circular cylind cylinderer shell shell is is higher higher thanthan those those o fo FGMf FGM ones ones. That. That means, means, with with the the same same geometry geometry dimensions, dimensions, sandwich sandwich-FGM-FGM cylind cylinderer shell shell structuresstructures will will work work better better than than FGM FGM ones. ones. 1-1 L/R=2- L/R=2 1-1 FGM- FGM-Core-Core 2- FGM 2-2 L/R=2.2- L/R=2.2 1 1 3 3 2- FGM 3-3 L/R=2.5- L/R=2.5 2 2 1 1 2 2 mm=1;=1; n n= 13;= 13; k k= 1;= 1; R R / h / h== 200; 200; o o m=1; n = 13; k = 0.5; R / h = 200; L / R = 2; TT = =50 50CC ; h ; h= =0.01 0.01 m m;c ;c2 =2 1e10;= 1e10; m=1; n = 13; k = 0.5; R / h = 200; L / R = 2; o o NeNe01 =13=13 01 TT = 100= 100C ;C h ; h= =0.01 0.01 m m;c ;c2 =2 1e9;= 1e9; N N01 01 = 1= e 1 3 e 3 FigFig ure ure. 12. 12. Nonlinear. Nonlinear dynamic dynamic response responses sof of FigFigureure. 13. 13. Effect. Effect of of material material structure structure on on circularcircular cylind cylinderer shell shell with with L/R L/R changes changes dynamicdynamic response response of of shell shell 66. .CONCLUSIONS CONCLUSIONS ThisThis paper paper established established nonlinear nonlinear dynamic dynamic equations equations of of fluid fluid-filled-filled circular circular cylinder cylinder shells shells made made ofof sandwich sandwich-FGM-FGM under under mechanical mechanical load load including including the the effect effect of of temperature. temperature. DDynamicynamic responses responses of of the the simply simply supported supported shell shell are are obtained obtained by by using using Galerkin Galerkin method method and and
11. 10 PPhuhu VV KK andand Doan Doan X X. .L L decreasess ThatThat meansmeans ifif thethe temperaturetemperature increasesincreases thenthen thethe stability stability of of the the shell shell structure structure will will decrease. decrease. 1 k=0 33 33 2- k=0.5 2- k=0.5 22 3 k=1 11 22 11 11 ΔT=0 ΔT=0 22 ΔT=50 ΔT=50 33 ΔT=200 ΔT=200 m=1; n = 13; k = 1; R / h = 200; mm==1;1; n n = = 13; 13; R R / / h h = = 200; 200; L L / / R R = = 2; 2; m=1; n = 13; k = 1; R / h = 200; oo L/ R= 2; h = 0.01 m ;c = 1e9; T = 50C ; h = 0.01 m ;c = 1e10; L/ R= 2; h = 0.01 m ;c2 2 = 1e9; T = 50C ; h = 0.01 m ;c22 = 1e10; Ne=13 NeNe==1313 Ne0101 =13 0101 Figure. 10. Dynamic responses of fluid-filled Figure. 11. Effect of thermal on the dynamic Figure. 10. DynamicNonlinear response dynamics bucklingof fluid of- full-filledfilled fluid sandwichFigure. FGM circular11. Effect cylinder of shells thermal on the dynamic 189 circular cylindercylinder shellshell withwith kk changeschanges responseresponse of of circular circular cyli cylindernder shells shells EffectsFigs. 10 ofof– geometricgeometric11 show dynamicparametersparameters responses onon nonlinear nonlinear of dynamic circulardynamic response cylinderresponse of of shell full full filled filledfilled fluid fluid with circular circular fluid withcylind cylinderer shellsvarious are surveyed volume-fraction and presented index in figk.12.and D theynamic effect critical of thermal force of environmentthe shell decrease ons dynamic with increas re-ing shells are surveyed and presented in fig.12. Dynamic critical force of the shell decrease s with increasing thethesponses ratioratio of of lengthlength circular toto radiusradius cylinder L/RL/R shells.ThatThat meansmeans From ifif the thethe length graphlength of asof shell canshell seeincreases, increases, that if the temperaturethe stability stability of of increasesthe the shell shell will will decrease.the dynamic critical force decreases. That means if the temperature increases then the sta- bility of the shell structure will decrease. Nonlinear responses responses of of FGM FGM and and sandwich sandwich FGMFGM circularcircular cylindercylinder shell shell filled filled with with fluid fluid are are Effects of geometric parameters on nonlinear dynamic response of full-filled fluid shown in figure. 13. The critical force of full-filled fluid sandwich-FGM circular cylinder shell is higher showncircular in figure cylinder. 13. shellsThe critical are surveyed force of full and-filled presented fluid sandwich in Fig. -FGM12. Dynamic circular cylind criticaler shell force is ofhigher thanthanthe thosethose shell ooff decreases FGMFGM onesones. with. ThatThat increasingmeans,means, withwith thethe the same ratiosame geometry geometry of length dimensions, dimensions, to radius sandwichL sandwich/R. That-FGM-FGM means cylind cylind ifer theer shell shell structureslength of willwill shell workwork increases, betterbetter thanthan the FGMFGM stability ones.ones. of the shell will decrease. 1- FGM-Core 11 L/R=2L/R=2 1- FGM-Core 2- FGM 22 L/R=2.2L/R=2.2 11 33 2- FGM 33 L/R=2.5L/R=2.5 22 11 22 mm==1;1; n n = = 13; 13; k k = = 1; 1; R R / / h h = = 200; 200; T = 50ooC ; h = 0.01 m ;c = 1e10; m=1; n = 13; k = 0.5; R / h = 200; L / R = 2; T = 50C ; h = 0.01 m ;c22 = 1e10; m=1; n = 13; k = 0.5; R / h = 200; L / R = 2; o o NeNe01 ==1313 T = 100C ; h = 0.01 m ;c = 1e9; N = 1 e 3 01 T = 100C ; h = 0.01 m ;c22 = 1e9; N 01 01 = 1 e 3 Fig.Fig 12 ure. Nonlinear 1212 NonlinearNonlinear dynamic dynamicdynamic responses responseresponse of circu-ss ofof Fig. 13FigFig. Effectureure. .13 13 of. .Effect Effect material of of material material structure structure structure on dy- on on circular cylindcylinderer shellshell withwith L/RL/R changeschanges dynamicdynamic response response of of shell shell lar cylinder shell with L/R changes namic response of shell Nonlinear responses of FGM and sandwich-FGM circular cylinder shell filled with fluid are shown in Fig. 13. The critical66 CONCLUSIONSCONCLUSIONS force of full-filled fluid sandwich-FGM circular cylinder shell is higher than those of FGM ones. That means, with the same geometry This paperpaper establishedestablished nonlinearnonlinear dynamicdynamic equationsequations ofof fluidfluid filledfilled circular circular cylinder cylinder shells shells made made dimensions, sandwich-FGM cylinder shell structures will work better than FGM ones. of sandwich FGMFGM underunder mechanicalmechanical loadload includingincluding thethe effecteffect of of temperature. temperature. Dynamic responsesresponses ofof thethe simplysimply6. supportedsupported CONCLUSIONS shellshell areare obtainedobtained byby usingusing GalerkinGalerkin methodmethod and and This paper established nonlinear dynamic equations of fluid-filled circular cylinder shells made of sandwich-FGM under mechanical load including the effect of tempera- ture. Dynamic responses of the simply supported shell are obtained by using Galerkin method and Runge–Kutta method. Based on dynamic responses, critical dynamic loads are obtained by using the Budiansky–Roth criterion. Some conclusions can be obtained from the present analysis: - Dynamic critical force of full-filled fluid sandwich-FGM circular cylinder shell is remarkably higher than those of fluid-free ones. That means, the fluid enhances the sta- bility of sandwich-FGM cylinder shell. - Temperature reduces dynamic critical force of sandwich-FGM cylinder shell. That means, temperature reduces stability of shell.
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14. 192 Khuc Van Phu, Le Xuan Doan APPENDIX A Stiffness coefficients and quantities related to thermal load in Eq. (8) h/2 h/2 h/2 Z E E Z νE νE Z E E A = A = dz = 1 ; A = dz = 1 ; A = dz = 1 ; 11 22 1 − ν2 1 − ν2 12 1 − ν2 1 − ν2 66 2 (1 + ν) 2 (1 + ν) −h/2 −h/2 −h/2 h/2 h/2 h/2 Z E.z E Z νEz νE Z Ez E B = B = dz = 2 ; B = dz = 2 ; B = dz = 2 ; 11 22 1 − ν2 1 − ν2 12 1 − ν2 1 − ν2 66 2 (1 + ν) 2 (1 + ν) −h/2 −h/2 −h/2 h/2 h/2 h/2 Z E.z2 E Z νEz2 νE Z Ez2 E D = D = dz = 3 ; D = dz = 3 ; B = dz = 3 ; 11 22 1 − ν2 1 − ν2 12 1 − ν2 1 − ν2 66 2 (1 + ν) 2 (1 + ν) −h/2 −h/2 −h/2 in which h/2 Z E h E = E (z) dz = E h + E h + cm x ; 1 m cm c k + 1 −h/2 h/2 Z E h h E h2 E  h  E h2 E = E (z) zdz = cm c − cm c + cm − h h − cm x ; 2 2 2 k + 1 2 c x (k + 1)(k + 2) −h/2 h/2 Z E  h 2 2E  h  2E E = E (z) z2dz = cm − h h − cm − h h2 + cm h3 3 k + 1 2 c x (k + 1)(k + 2) 2 c x (k + 1)(k + 2)(k + 3) x −h/2 E h3 E hh  h  E  3hh  h  E h i + c c + c c − h + m h3 + m − h + m h3 − 3 (h/2 − h )(h/2 − h ) h ; 3 2 2 c 3 m 2 2 m 3 x m c x h/2 h/2 1 Z 1 Z Φ = E (z) α (z) ∆Tdz, Φ = E (z) α (z) ∆Tzdz. a 1 − ν b 1 − ν −h/2 −h/2 1 If ∆T = const then Φ = P∆T. a 1 − ν For FGM-core: E α h E α h E α h P = E α h + E α h + E α (h − h ) + m cm x + cm m x + cm cm x , m m c c c m m c k + 1 k + 1 2k + 1 where hx = h − hc − hm; Ecm = Ec − Em. APPENDIX B Extended stiffness coefficients in Eq. (9) and Eq. (10) A A A A B − A B ∗ = 11 ∗ = 12 ∗ = 22 ∗ = 22 11 12 12 A11 2 ; A12 2 ; A22 2 ; B11 2 ; A11 A22 − A12 A11 A22 − A12 A11 A22 − A12 A11 A22 − A12 A B − A B A B − A B A B − A B 1 B ∗ = 22 12 12 22 ∗ = 11 12 12 11 ∗ = 11 22 12 12 ∗ = ∗ = 66 B12 2 ; B21 2 ; B22 2 ; A66 ; B66 ; A11 A22 − A12 A11 A22 − A12 A11 A22 − A12 A66 A66 ∗ ∗ ∗ ∗ ∗ ∗ D11 = D11 − B11B11 − B12B21; D12 = D12 − B11B12 − B12B22; ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ D21 = D12 − B12B11 − B22B21; D22 = D22 − B12B12 − B22B22; D66 = D66 − B66B66.