Nonlinear dynamic buckling of full filled fluid sandwich FGM circular cylinder shells

Vietnam Journal of Mechanics, VAST, Vol 41, No (2019), pp 179 – 192 DOI: https://doi.org/10.15625/0866-7136/13306 NONLINEAR DYNAMIC BUCKLING OF FULL-FILLED FLUID SANDWICH FGM CIRCULAR CYLINDER SHELLS Khuc Van Phu1 , Le Xuan Doan2,∗ Military Academy of Logistics, Hanoi, Vietnam Academy of Military Science and Technology, Hanoi, Vietnam ∗ E-mail: xuandoan1085@gmail.com Received: 17 November 2018 / Published online: June 2019 Abstract This paper is concerned with the nonlinear dynamic buckling of sandwich functionally 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; circular cylinder shell 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 considerable 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 functionally graded (FG) circular cylinder shell under mechanical loads Governing equations 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 cylindrical shells [3] and the one [4] surrounded on elastic foundation subjected to mechanical and thermal loads The FSDT and the third order shear deformation theory (TSDT) were employed to solve problems Bahadori and Najafizadeh [5] analyzed free vibration 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 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 axial 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 under 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 eccentrically stiffened thin and thick FG circular cylinder shells made of S-FGM resting on elastic foundations under thermal-mechanical loads Lekhnitsky smeared stiffeners technique and Bubnov–Galerkin method were applied in calculation By using an analytical approach, based on improved Donnell shell theory with von Karman–Donnell geometrical nonlinearity, Bich et al [11] investigated the buckling and post-buckling of FG circular cylinder shells under mechanical loads including effects of temperature Nonlinear buckling problems of imperfect eccentrically stiffened FG thin circular cylindrical shells subjected 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] Zafar Iqbal et al [15] examined vibration frequencies of FGM circular cylinder shells filled with fluid using wave propagation approach Vibration frequencies of shell were analyzed 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 nonlinear 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 cylinder 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 buckling 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 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 GOVERNING EQUATIONS Consider a sandwich FGM circular cylinder shell with geometric parameters: R, h, hc , and hm are shown in Fig 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 Fig Model circular cylinder Figure.1 ModelofofFGM-sandwich FGM-sandwich circular cylinder shellshell With configuration of sandwich FGM as Fig 1, suppose that Vc (z) and Vm (z) are With configuration sandwich and FGM metal as figure.1, suppose thatthe Vc(z)volume and Vm(z)fraction are the volume the volume fractions ofofceramic respectively, of ceramic fractions of ceramic and metal respectively, the volume fraction of ceramic constituent changes constituent changes according to the power law and can be expressed as according to the power law and can be expressed as Vc = 0, −0.5h ≤ zVc≤= − 0, (0.5h − 0, 5h−  zhm−)(, 0, 5h − hm ) k  k z + 0.5h − hm Vc = Vc =  z + 0, 5h − hm  , , − h z ) (≤ ≤hc()0.5h − ((0,0.5h 5h − h− 0, 5zh − , k − hc ) , k ≥ 0, m ) m h c− − hc −hhmm   h  h− Vc = 1, (0.5h − hc ) V≤c =z1,≤ (0.5h 0, 5h − hc )  z  0, 5h (1) (1) Then the modulus thedensity massρdensity ρ andratio the νPoisson ν of circular Then theelasticity elasticity modulus E, theE, mass and the Poisson of circularratio cylinder shell cylinder shell can be evaluated as following can be evaluated as following E =EEmmV EcVEc c=VEc m= + (E Ecm−+ Em () V E= Vmm++ Ecc, − Em ) Vc ,  =ρ V +  V =  +  −  V ( ) ρ= V + ρ V = ρ + ρ ( m mm mm c cc c m mc cc ,− ρm ) Vc ,  =  = const νm = νcc = const m (2) (2) (3) (3) strain components components of the cylindercylinder shell are shell are TheThe strain of circular the circular  y0 zk − zk,y ; γ  xy == xy0γ−0 zk ε x = ε0x −x =zkxx ,− zkεxy; = yε0y= − y xy xy −xy2zk xy , u  w  v w  w  where Where:  x0 = +   ;  y0 = − +   ;  xy0 x 2  x  y R  y  ∂v w ∂w ∂u ∂w ε0x = + , ε = −  2+ , y 2 w 2 w w ∂x k x∂x ∂y = ; k y = ∂y ; k xy =R ; x xy y = u v w w + + ; y x x y ∂u ∂v γ0xy = + ∂y (4) ∂w ∂w + , ∂x ∂x ∂y(5) In which:  x0 ;  y0 ;  xy are the strains at the middle surface; kx, ky and k xy are curvatures and the twist By use of Eq (4), the deformation compatibility equation can be written as:   x0 y +   y0 x −   xy0  2 w  2 w 2 w 2 w = − ;  − xy  xy  x y R x (6) (4) 182 Khuc Van Phu, Le Xuan Doan kx = ∂2 w , ∂x2 ky = ∂2 w , ∂y2 k xy = ∂2 w , ∂x∂y (5) in which ε0x ; ε0y ; γ0xy are the strains at the middle surface; k x , k y and k xy are curvatures and the twist By use of Eq (4), the deformation compatibility equation can be written as ∂2 ε0y ∂2 γ0xy ∂2 ε0x + − = ∂y2 ∂x2 ∂x∂y ∂2 w ∂x∂y − ∂2 w ∂2 w ∂2 w − 2 ∂x ∂y R ∂x2 (6) For circular cylindrical shell subjected to mechanical load in temperature environment, the Hooke’s law can be defined as E (z) E (z) α (z) ∆T (ε x + νε y ) − , 1−ν 1−ν E (z) = γxy , (1 + ν ) σx = τxy σy = E (z) E (z) α (z) ∆T (νε x + ε y ) − , 1−ν 1−ν (7) in which ∆T = T − T0 Internal forces and moment resultants can be defined by integrating stresses components through the shells’ thickness and can be expressed in matrix form as        ε N A A B B Φ         x x a 11 12 11 12                         ε N A A B B Φ         y a 22 22 12 12 y                Nxy 0 A66 0 B66 γ xy = − , (8) Mx  B11 B12 D11 D12  Φb      −k x                      My  D12 D22     B12 B22   −k y    Φb                  Mxy 0 B66 0 D66 −2k xy 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 ∗ ∗ ∗ ∗ ∗ ∗ ε0x = A22 Nx − A12 Ny + B11 k x + B12 k y + Φ a ( A22 − A12 ), ∗ ∗ ∗ ∗ ∗ ∗ ε0y = − A12 Nx + A11 Ny + B21 k x + B22 k y + Φ a ( A11 − A12 ), (9) ∗ ∗ γ0xy = A66 Nxy + 2B66 k xy , ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Mx = B11 Nx + B21 Ny − D11 k x − D12 k y + [ B11 ( A22 − A12 − A12 ) + B12 ( A11 )] Φ a − Φb , ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ My = B12 Nx + B22 Ny − D21 k x − D22 k y + [ B12 ( A22 − A12 − A12 ) + B22 ( A11 )] Φ a − Φb , ∗ ∗ Mxy = B66 Nxy − 2D66 k xy , (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 Nonlinear dynamic buckling of full-filled fluid sandwich FGM circular cylinder shells 183 ∂Nxy ∂Nx ∂2 u + = ρ1 , ∂x ∂y ∂t ∂Nxy ∂Ny ∂2 v + = ρ1 , ∂x ∂y ∂t ∂2 Mxy ∂ My Ny ∂2 w ∂2 w ∂2 w ∂2 w ∂w ∂2 M x +2 + + Nx + 2Nxy + Ny + + q − p L = ρ1 + 2ρ1 ε , 2 ∂x∂y ∂x∂y R ∂t ∂x ∂y ∂x ∂y ∂t (11) in which ε is the linear damping coefficient and h/2 ρ1 = −h/2 p L = −ρ L where ML = ρ(z) dz = ρm h + ρcm hc + ρcm (h − hc − hm ) , k+1 ∂ϕ L ∂2 w = ML is the dynamic pressure of fluid acting on the shell, ∂t ∂t ρ L RIn (λm ) is the mass of correspondence fluid to the shell vibration and λm In (λm ) mπR [19] L Applying the Volmir’s assumption [21] into Eqs (11) (because of u equations of motion can be rewritten as follows λm = w, v w), the ∂Nxy ∂Nxy ∂Ny ∂Nx + = 0, + = 0, ∂x ∂y ∂x ∂y ∂ My ∂2 Mxy Ny ∂2 w ∂2 w ∂2 M x ∂2 w ∂2 w ∂w + + N + q = ρ + M + + N + 2N + + 2ρ1 ε ( ) y x xy L 2 2 ∂x∂y ∂x∂y R ∂t ∂x ∂y ∂x ∂y ∂t (12) The first and the second equation of Eqs (12) are satisfied identically by recommending the stress function: Nx = ∂2 F , ∂y2 Ny = ∂2 F , ∂x2 Nxy = − ∂2 F ∂x∂y (13) 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 ∗ A11 ∂4 F ∂4 F ∗ ∗ ∗ ∂ F + A + A − 2A ( ) 66 22 12 ∂x2 ∂y2 ∂x4 ∂y4 ∗ + B21 ∂4 w ∂4 w ∂2 w ∗ ∗ ∗ ∗ ∂ w + B + B − 2B + B + − ( ) 22 66 11 12 ∂x2 ∂y2 R ∂x2 ∂x4 ∂y4 ∂2 w ∂x∂y + ∂2 w ∂2 w = 0, ∂x2 ∂y2 (14) 184 Khuc Van Phu, Le Xuan Doan ( ρ1 + M L ) ∗ − B12 4 ∂4 w ∂2 w ∂w ∗ ∗ ∗ ∗ ∂ F ∗ ∂ w ∗ ∂ w + 2ρ ε + D + D + D + 4D + D − B ) ( 66 22 11 12 21 21 ∂t ∂t2 ∂x2 ∂y2 ∂x4 ∂y4 ∂x4 ∂4 F ∂4 F ∂2 F ∂2 F ∂2 w ∂2 F ∂2 w ∂2 F ∂2 w ∗ ∗ ∗ − 2 +2 − 2 + q = − ( B11 + B22 − 2B66 ) 2− R ∂x ∂x∂y ∂x∂y ∂x ∂y ∂y ∂x ∂x ∂y ∂y (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 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 = at x = and x = L The shells’ deflection satisfying above conditions can be written as w = f (t) sin ny mπx sin , L R (16) 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 F = F1 cos 2αx + F2 cos 2βy − F3 sin αx sin βy − N01 A∗ y2 β2 − − ∗ f (2t) + ψ f (+t) N01 12 ∗ +Γ 8A11 A11 x2 , (17) in which F1 = F1∗ f (2t) = F3 = F3∗ f (t) = β2 ∗ f (t) , 32α2 A11 F2 = F2∗ f (2t) = α2 ∗ f (t) , 32β2 A22 ∗ + ( αβ )2 ( B∗ + B∗ − 2B∗ ) + β4 B∗ − α4 B21 22 66 11 12 ∗ + ( αβ )2 A∗ − 2A∗ + β4 A∗ α4 A11 66 22 12 α2 R f (t) , γη ∗ ∗ ∗ ∗ α2 A11 − β2 A12 F3∗ − α2 B21 + β2 B22 + , ∗ mnπ A11 R ∗ − A∗ ) Φ ( A11 mπ nπ a 12 Γ= , γ = (−1)m − , η = (−1)n − α = ,β = ∗ A11 L R ψ= Substituting Eq (16) and Eq (17) into Eq (15), then using the Galerkin method we obtain ( ρ1 + M L ) d2 f df 4γη + 2ρ1 ε + H1 f (3t) + H2 f (2t) + H3 f (+t) dt dt mnπ R ∗ A12 ∗ N01 + Γ A11 = 4γη q, mnπ (18) Nonlinear dynamic buckling of full-filled fluid sandwich FGM circular cylinder shells 185 in which H1 = (αβ)2 ( F1∗ + F2∗ ) + H2 = − 16γη 3mnπ β4 ∗ , 8A11 ∗ 4α4 B21 − α2 R ∗ ∗ F1∗ + 4β4 B12 F2 − β2 3mnπ 2 β ψ , + (αβ)2 F3∗ + ∗ 32 RA11 16γη ∗ ∗ ∗ ∗ ∗ H3 = α4 D11 + (αβ)2 ( D12 + D21 + 4D66 ) + β4 D22 ∗ ∗ ∗ ∗ ∗ − + α4 B21 + (αβ)2 ( B11 + B22 − 2B66 ) + β4 B12 A∗ α2 F3∗ − α2 + 12 ∗ β R A11 N01 − Γβ2 + 4γη ψ 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 Consider a full-filled fluid sandwich FGM circular cylinder shell under linear axial compression load varying on time N01 = − ph with p = c1 t (c1 -loading speed), q = - Case 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 response increases sharply depending on time and this curve obtains a maximum by passing 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 = c1 tcr (case 1) or qcr = c2 tcr (case 2) 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, ρm = 4429 kg/m3 , νm = 0.288, Ec = 244.27 e9Pa, ρc = 5700 kg/m3 , νc = 0.288 Tab 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 186 Khuc Van Phu, Le Xuan Doan Table Comparison of critical stress of the compressed cylindrical shell (MPa) k 0.2 1.0 5.0 Huang & Han [23] Present 194.94 (2, 11) 193.914 (1, 9) 169.94 (2, 11) 168.685 (1, 9) 150.25 (2, 11) 149.167 (1, 9) 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 = and R/h = 200 FGM made of Aluminium and Alumina with the material properties are Em = × 109 N/m2 ; ρm = 2702 × 103 kg/m3 , αm = 2.3 × 10−5 C−1 , Ec = 3.8 × 1011 N/m2 ; ρc = 3.8 × 103 kg/m3 , αc = 5.4 × 10−6 C−1 , ε = 0.1, the Poisson’s ratio νc = 0.3 the fluid density ρ L = 103 kg/m3 - Case Consider a full-filled fluid sandwich FGM circular cylinder shell under linear axial compression load varying on time N01 = − ph( p = c1 t), q = In this case, the critical time tcr can be obtained according to Budiansky–Roth criterion The dynamic critical force pcr = c1 tcr 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 Figs 2–3 From Fig 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 Phu V.to K determine and Doan X.the L Lcritical forces Phu V K and Doan X be derived from dynamical responses m =m1;=n1;=n13; k =k1;=R1;/ R h =/ h200; L/R = 13; = 200; L /=R2;= 2;  T=T50=o50 C;ohC=; h0.01 m;cm = 1e12; = 0.01 ;c = 1e12; m =m1;=n1;=n13; L/R = k13;=k1;=R1;/ hR =/ h200; = 200; L /=R2;= 2;  T=T50=o50 C;ohC=; h0.01 m;cm = 1e12; = 0.01 ;c = 1e12; Figure 2 Nonlinear dynamic response of of Figure 3 Effects of of fluid onon dynamic Figure Nonlinear dynamic response Figure Effects fluid dynamic Fig fluid-filled Nonlinear circular dynamic response of fluidFig Effects of fluid on dynamic response of cylinder shell fluid-filled circular cylinder shell response of circular cylinder shell response of circular cylinder shell filled circular cylinder shell circular cylinder shell Figure.4 shows nonlinear dynamic responses of of cylinder shell when volume-fraction index k k Figure.4 shows nonlinear dynamic responses cylinder shell when volume-fraction index changes Fig.4 as can seesee that, if if k increases thethe dynamic critical force of of shell will decrease That changes Fig.4 as can that, k increases dynamic critical force shell will decrease That Fig.From 4From shows nonlinear dynamic responses of cylinder shell when volume-fraction means the load-bearing capability cylinder decreases means load-bearing capability of cylinder shell decreases index k the changes It can be seenofthat, if k shell increases the dynamic critical force of shell will The Effect ofof thermal environment onon nonlinear responses of of circular cylinder shell The Effect thermal environment nonlinear dynamic responses circular cylinder shell decrease That means the load-bearing capability ofdynamic cylinder shell decreases can bebe shown in in figure.5 From thethe graph it is observed that when thethe temperature increases, thethe dynamic can shown figure.5 From graph it is observed that when temperature increases, dynamic 0 critical force ofof shell will decrease From PcrP=76,6GPa at at oC to to PcrP=65GPa at at 200 C.C That means, thethe critical force shell will decrease From 0oC 200 That means, cr=76,6GPa cr=65GPa load-bearing capability ofof thethe shell will decrease when temperature increases load-bearing capability shell will decrease when temperature increases m= R /Rh /=h200; m1;=n1;=n13; = 13; = 200; o o L /LR/ =R2;  T C; C; = 2; =T50 = 50 h =h0.01 m;c =11e10; = 0.01 m1;c = 1e10; 33 mm = 1;=n1;=n13; k =k1;= 1; = 13; R /Rh/=h 200; L /LR/ = = 200; R 2; = 2; h =h 0.01 m;c = 0.01 m1;c=11e12; = 1e12; Figure 2.2.Nonlinear dynamic response ofof Figure Nonlinear dynamic response fluid-filled circular cylinder shell fluid-filled circular cylinder shell Figure 3 Effects ofof fluid onon dynamic Figure Effects fluid dynamic response of circular cylinder shell response of circular cylinder shell Figure.4 ofof cylinder shell when volume-fraction index k k Figure.4shows showsnonlinear nonlineardynamic dynamicresponses responses cylinder shell when volume-fraction index changes Fig.4 asas can see that, if if k increases the dynamic critical force ofof shell will decrease That changes.From From Fig.4 can see that, k increases the dynamic critical force shell will decrease That means capability ofof cylinder shell decreases meansthe theload-bearing load-bearing capability cylinder shell decreases The nonlinear dynamic responses ofof circular cylinder shell TheEffect Effectofofthermal thermalenvironment environmentonon nonlinear dynamic responses circular cylinder shell can bebe shown inin figure.5 From the graph it it is is observed that when thethe temperature increases, thethe dynamic can shown figure.5 From the graph observed that when temperature increases, dynamic 0 critical ofof shell will decrease From PcrP=76,6GPa at at 0oFGM CoC tocircular PcrP=65GPa at at 200 C.C That means, criticalforce force shell will decrease From to 200 That means, cr=76,6GPa cr=65GPa Nonlinear dynamic buckling of full-filled fluid sandwich cylinder shells 187 thethe load-bearing capability ofof the shell will decrease when temperature increases load-bearing capability the shell will decrease when temperature increases mm = 1; R /Rh/ =h 200; =n 1;= n 13; = 13; = 200; o L /LR/ R = 2; T = 50 = 2; T = 50Co;C; h= m;c h 0.01 = 0.01 m1;c=1 1e10; = 1e10; mm = 1; k= =n 1;= n13; = 13; k 1; = 1; R /Rh/ = 200; L / h = 200; LR/ = R 2; = 2; h= 0.01 m ;c = 1e12; h = 0.01m1;c1 = 1e12; 33 33 1-k=0 1-k=0 2-k=0.5 2-k=0.5 3-k=1 3-k=1 0 1-1ΔT=0 CC ΔT=0 0 2-2ΔT =50 CC ΔT =50 0 ΔT =200 3-3ΔT =200 CC 11 22 Figure Dynamic response fluid-filled Figure 4.4.Dynamic response ofof fluid-filled Fig Dynamic response of fluid-filled cylinder shell when changescylincylinder shell when kk changes der shell when k changes 2 1 Figure Temperature effect nonlinear Figure 5 Temperature effect onon nonlinear Fig Temperature on nonlinear dydynamic responses of fluid-filled cylinder shell dynamic responses ofeffect fluid-filled cylinder shell namic responses of fluid-filled cylinder shell The effect geometric parameters (L/R ratio) nonlinear dynamic responses cylinder shells The effect ofof geometric parameters (L/R ratio) onon nonlinear dynamic responses of of cylinder shells made of sandwich-FGM filled with fluid is shown in figure The dynamic critical force of cylinder made of sandwich-FGM filled with fluid is shown in figure The dynamic critical force of cylinder shell The effect of thermal environment on nonlinear dynamic responses of circular cylin- shell decreaseswhen when increasing R/L ratio That means increasing length shell, stability shell decreases increasing R/L ratio That means increasing length ofof thethe shell, thethe stability of of thethe shell der shell will iswill shown in Fig From the graph it is observed that when the temperature structure decrease structure decrease ◦ increases, the dynamic critical force of shell will decrease From Pcr = 76.6 GPa at C to Fig.7indicates indicates nonlineardynamic dynamicresponses responsesofofcircular circularcylinder cylindershell shell madeof ofFGM FGMand and ◦ nonlinear Pcrsandwich-FGM = 65 Fig.7 GPa at 200 C That means, thethe load-bearing capability of the shellmade will critical decrease filled withfluid fluid For structuremade made sandwich-FGM, forceis is sandwich-FGM filled with For the structure ofofsandwich-FGM, thethecritical force when temperature P=0,496GPa, andincreases forFGM FGMones, ones,the thecritical criticalforce forceis isPcrP=0,485GPa That means, with same cr=0,496GPa, cr=0,485GPa Pcr and for That means, with thethe same geometry dimensions, the workability sandwich-FGM cylinder shell better than FGM ones The effect of geometric parameters (L/R ratio)cylinder on nonlinear dynamic responses of geometry dimensions, the workability ofof sandwich-FGM shell is is better than FGM ones cylinder shells made of sandwich-FGM filled with fluid is shown in Fig The dynamic critical force of cylinder shell decreases when increasing R/L ratio That means increasNonlinear dynamic bucklingofof full-filled fluid sandwich-FGM circularcylinder cylindershells shells dynamic full-filled fluid sandwich-FGM circular 99 ing length Nonlinear of the shell, the buckling stability of the shell structure will decrease mm== 1;1; n n== 13; k k== 1;1; RR/ h/ h==200; 13; 200; oC oC  TT==50 ; h; h==0.01 mm ;; 50 0.01 c1c1== 1e12; 1e12; 0 mm= 1; 5050 CC = 1;n = n 13; = 13;k k= 1; = 1;TT= = R R/ h/ h= =200; 200;L L/ R/ R= =2;2; h= mm ; ;c1c=1 1e9; h =0.01 0.01 = 1e9; 11 1-R/L=2 1-R/L=2 2-R/L=2.2 2-R/L=2.2 3-R/L=2.5 3-R/L=2.5 22 33 11 Figure 6.6.Effect ofofgeometric parameters Figure Effect geometric parameters on Fig Effect of geometric parameters on on dydynamic responses of fluid-filled cylinder shell dynamic responses of fluid-filled cylinder shell namic responses of fluid-filled cylinder shell 22 1-Sandwich-FGM 1-Sandwich-FGM 2-FGM 2-FGM Figure 7.7.Dynamic responses Figure Dynamic responses FGMand and Fig Dynamic responses ofofofFGM FGM and sandwich-FGM circular cylinder sandwich-FGMcircular circularcylinder cylindershell shell sandwich-FGM shell Fig.Case indicates nonlinear dynamic responses of circular cylinder shell amade of FGM Case2.2.Consider Considera afull-filled full-filledfluid fluidsandwich sandwichFGM FGMcircular circularcylinder cylindershell shellunder under auniform uniformpre-axial pre-axial and sandwich-FGM filled with fluid For the structure made of sandwich-FGM, the critcompression =const, 0101 2t2t(c(c 2-2compressionload loadand andananexternal externaluniformly uniformlydistributed distributedpressure pressurevarying varyingon ontime: time:NN =const,q=c q=c ical force is Pcr = 0.496 GPa, and for FGM ones, the critical force is Pcr = 0.485 GPa That loading speed) loading speed) means,The with the same geometry dimensions, the workabilityareofshown sandwich-FGMtocylinder Thenonlinear nonlineardynamic dynamicresponses responsesofofcircular circularcylinder cylindershell shell are shownininfigure.8 figure.8 tofigure figure.13 13 shell is better than FGM ones 1-1-Full Fullfilled filledfluid fluid 1 2-2-No fluid No fluid 22 m= k =k1;=R m1;=n1;=no13; = 13; 1;/Rh/=h 200; = 200; T Co;C h;=h0.01 m;m; = T50 = 50 = 0.01 c1 c= 1e12; = 1e12; m =m1;= 1; n =n13; k =k1;= 1; T=T 50 = 13; = 50C0 C R /Rh/=h200; L /LR/ =R2;= 2; = 200; h =h0.01 m ; c = 0.01m;1 c=11e9; = 1e9; 11 188 Khuc Van Phu, Le Xuan Doan 22 22 1-R/L=2 1-R/L=2 33 2-R/L=2.2 2-R/L=2.2 1-Sandwich-FGM 1-Sandwich-FGM 1 sandwich 2-FGM -3-R/L=2.5 Case Consider a full-filled fluid FGM circular cylinder shell under 3-R/L=2.5 2-FGM a uniform pre-axial compression load and an external uniformly distributed pressure varying on time: N01 = const, q = c2 t (c2 -loading speed) Figure Effect ofof geometric parameters on The6.nonlinear dynamic responses ofoncircular cylinder shell areresponses shown in Figs 8– Figure Effect geometric parameters Figure 7.7 Dynamic ofofFGM Figure Dynamic responses FGMand and dynamic responses of fluid-filled cylinder shell dynamic responses of fluid-filled cylinder shell sandwich-FGM circular shell 13 Nonlinear dynamic responses of fluid-filled and fluid-free sandwich FGM circular sandwich-FGM circularcylinder cylinder shell cylinder shell are depicted in Figs 8–9 From Fig we obtain tcr = 0.01 s and qcr = 147 MPa respectively, from the Fig 9, it is observed that fluid remarkably increases the Case 2 Consider a full-filled fluid sandwich FGM circular cylinder shell under a uniform pre-axial Consider fluid sandwich FGM circular shell a uniform pre-axial dynamicCase critical force ofa full-filled the shell (from qcr = 25 MPa at tcrcylinder = 0.002 s under in case fluid-fee compression load and anan external uniformly distributed pressure onon time: q=c 2t 2(c 2- 2compression load and external uniformly distributed pressurevarying varying time:N01 N=const, q=c t (c 01=const, shell to qcr = 147 MPa at tcr = 0.01 s in case shell containing fluid, i.e the critical force loading speed) loading speed) increased by 5.88 times by 488%) The nonlinear dynamic responses ofof circular cylinder shell are shown inin figure.8 The nonlinear dynamic responses circular cylinder shell are shown figure.8totofigure figure.1313 1-1-Full filled fluid Full filled fluid 1 2-2-NoNo fluid fluid m =m1;=n1;=n13; k =k1;=R1;/ R h =/ h200; L/L R /=R2;= 2; = 13; = 200;  T=T50=o50 C;ohC=; h0.01 m;cm2 ;c = 21e10; N01N=011e=31e3 = 0.01 = 1e10; 22 m =m1;=n1;=n13; k =k1;=R1;/Rh /=h200; L / LR/=R2;= 2; = 13; = 200;  T=T50 C;o hC=; h0.01 m;cm2;c=21e10; N01N=011e=31e3 = o50 = 0.01 = 1e10; Figure Effect fluidonondynamic dynamic Figure 9.9 Effect ofof fluid Figure Nonlinear dynamic responses Figure 8 Nonlinear dynamic responses ofof Fig Nonlinear dynamic responses of fullFig Effect of fluid on dynamic responses responses circular cylinder shellof responses ofof circular cylinder shell full-filled fluid circular cylinder shell full-filled fluid circular cylinder shell filled fluid circular cylinder shell circular cylinder shell Nonlinear dynamic responses fluid-filled and fluid-free sandwichFGM FGMcircular circularcylinder cylindershell shell Nonlinear dynamic responses ofof fluid-filled and fluid-free sandwich 0,01 = 147 MParespectively, depicted figure and figure From fig obtaintcrtcr= = andqcrqcr= 147 respectively, 0,01 s sand MPa areare depicted inin figure and figure 9.9 From fig 88 wewe obtain Similarly, we make other cases when taking intoDoan account the influence of other facPhuV.V andDoan X from the fig it is observed that fluid remarkably increases dynamic critical force the shell (from 1010the from fig 9, 9, it is observed that fluid remarkably increases thethe dynamic critical force ofof the shell (from Phu KK and X LL derive from dynamic response curves totodetermine dynamic critical forces t = 0,002 s t = 0,01 s qdecreases =25 MPa at in case fluid-fee shell to q = 147 MPa at in case shell containing t = 0,002 s t = 0,01 s qcrtors =25 MPa at in case fluid-fee shell q = 147 MPa at in case shell containing cr cr cr cr cr cr means crthe decreases.That That meansififthe thetemperature temperatureincreases increasesthen thenthe thestability stabilityofof theshell shellstructure structurewill willdecrease decrease fluid, critical force increased 5,88 times 488%) fluid, i.e.i.e thethe critical force increased byby 5,88 times byby 488%) Similarly, make other cases when taking into account the influence otherfactors factorsderive derive Similarly, wewe make other cases when taking into account the influence ofofother k=0response 1-1-k=0 3 toto from dynamic response curves determine dynamic critical forces 3 from dynamic curves determine dynamic critical forces k=0.5 2-2-k=0.5 2 Figure.10 and figure show dynamic responses circularcylinder cylinder shellfilled filledwith withfluid fluidwith with 3-k=1 k=1 and Figure.10 figure 1111 show dynamic responses ofof circular shell 3various volume-fraction index k and the effect thermal environment dynamic responses circular various volume-fraction index k and the effect ofof thermal environment dynamic responses ofof circular on on cylindershells shells.From Fromthethegraph graph cansee seethat thatif iftemperature temperatureincreases increasesthe thedynamic dynamiccritical criticalforce force cylinder 2asascan 11 = 1; = 13; 200; mm = 1; nn = 13; RR / h/ h == 200; LL / R/ R == 2;2; o o 0.01 = 1e10;  TT == 5050 C ;Ch; h == 0.01 m;mc;2c= 1e10; N 01= =e13e3 N 01 Figure.10 10.Dynamic Dynamicresponses responsesofoffluid-filled fluid-filled Figure circular cylinder shellwith with changescirFig 10.circular Dynamic responses of fluid-filled cylinder shell k kchanges cular cylinder shell with k changes ΔT=0 1-1-ΔT=0 ΔT=50 2-2-ΔT=50 ΔT=200 3-3-ΔT=200 = 1; = 13; = 1; 200; mm = 1; nn = 13; k k= 1; RR / h/ h == 200; /R 0.01 = 1e9; LL /R == 2;2; hh == 0.01 mm ;c;c 1e9; = =e13e3 NN =1 0101 Figure.11 11.Effect Effectofofthermal thermalon onthe thedynamic dynamic Figure Fig 11 Effect of thermal on the dynamic reresponse circular cylinder shells response ofofcircular cylinder shells sponse of circular cylinder shells Effectsofofgeometric geometricparameters parameterson onnonlinear nonlineardynamic dynamicresponse responseofoffull-filled full-filledfluid fluidcircular circularcylinder cylinder Effects shellsare aresurveyed surveyedand andpresented presentedininfig.12 fig.12.Dynamic Dynamiccritical criticalforce forceofofthe theshell shelldecreases decreaseswith withincreasing increasing shells theratio ratioofoflength lengthtotoradius radiusL/R L/R.That Thatmeans meansififthe thelength lengthofofshell shellincreases, increases,the thestability stabilityofofthe theshell shellwill will the decrease decrease Nonlinear Nonlinearresponses responsesofofFGM FGMand andsandwich-FGM sandwich-FGMcircular circularcylinder cylindershell shellfilled filledwith withfluid fluidare are shown shownininfigure figure.13 13.The Thecritical criticalforce forceofoffull-filled full-filledfluid fluidsandwich-FGM sandwich-FGMcircular circularcylinder cylindershell shellisishigher higher than thanthose thoseofofFGM FGMones ones.That Thatmeans, means,with withthe thesame samegeometry geometrydimensions, dimensions,sandwich-FGM sandwich-FGMcylinder cylindershell shell structures structureswill willwork workbetter betterthan thanFGM FGMones ones 11 m m ==1;1;nn ==13; 13;RR//hh==200; 200;LL//RR==2;2; o TT ==50 50oCC;;hh ==0.01 0.01mm;c ;c22 ==1e10; 1e10; N N0101 ==11ee33 11-ΔT=0 ΔT=0 22-ΔT=50 ΔT=50 33-ΔT=200 ΔT=200 mm==1;1;nn==13; 13;kk==1;1;RR/ /hh==200; 200; LL/ /RR==2;2;hh==0.01 mm;c;c2 2==1e9; 0.01 1e9; NN0101==1e1e33 Figure 10 responses of Figure of thermal 10 Dynamic Dynamic responses of fluid-filled fluid-filled Figure 11.Effect Effect ofshells thermalon onthe thedynamic dynamic Nonlinear dynamic buckling of full-filled fluid sandwich FGM 11 circular cylinder 189 circular cylinder shell with k response circular cylinder shell with k changes changes responseof ofcircular circularcylinder cylindershells shells Figs 10–11 show dynamic of dynamic circular cylinder shell filledfluid withcircular fluid with Effects of parameters on full-filled cylinder Effects of geometric geometric parametersresponses onnonlinear nonlinear dynamicresponse responseof of full-filled fluid circular cylinder various volume-fraction index k and the effect of thermal environment on dynamic reshells are surveyed and presented in fig.12 Dynamic critical force of the shell decreases with increasing surveyed and presented in fig.12 Dynamic critical force of the shell decreases with increasing sponses circular cylinder shells From the graph as can see that if temperature increases the ratio of of length to radius L/R That means if the length of shell increases, the stability of the shell length to radius L/R That means if the length of shell increases, the stability of the shellwill will the dynamic critical force decreases That means if the temperature increases then the stadecrease bility of the shell structure will decrease Nonlinear Nonlinear responses responses of of FGM FGM and and sandwich-FGM sandwich-FGM circular circular cylinder cylinder shell shell filled filled with with fluid fluidare 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 figure 13 The critical force of full-filled fluid sandwich-FGM circular cylinder shell is higher circular cylinder shells are surveyed and presented in Fig 12 Dynamic critical force of than FGM That means, the cylinder shell ofdecreases FGM ones ones.with Thatincreasing means,with withthe thesame samegeometry geometry dimensions, sandwich-FGM cylinder shell thethose shellof ratio of lengthdimensions, to radius sandwich-FGM L/R That means if the structures will work better ones length of shell increases, theFGM stability will work better than than FGM ones.of the shell will decrease 11- L/R=2 L/R=2 22- L/R=2.2 L/R=2.2 33- L/R=2.5 L/R=2.5 11 11-FGM-Core FGM-Core 22-FGM FGM 33 11 22 22 m m ==1;1;nn ==13; 13;kk ==1;1;RR//hh==200; 200; o TT ==50 50oCC;;hh==0.01 0.01m m;c ;c22 ==1e10; 1e10; N N0101 ==11ee33 Figure 12 dynamic responses of 12 Nonlinear Nonlinear dynamic responses of Fig 12 Nonlinear dynamic responses of circucircular cylinder shell with L/R changes circular cylinder changes lar cylinder shellshell withwith L/RL/R changes mm==1;1;nn==13; 13;kk==0.5; 0.5;RR/ h/ h==200; 200;LL/ R / R==2;2; oo TT==100 mm;c;c2 2==1e9; 100CC; h; h==0.01 0.01 1e9;NN0101==1e13e3 Figure ofofmaterial structure Figure 13 Effect material structure on Fig 13 Effect 13 of Effect material structure on dy-on dynamic response ofofshell dynamic response shell namic response of shell Nonlinear responses of FGM and sandwich-FGM circular cylinder shell filled with fluid are shown in Fig 13 The critical force of full-filled fluid sandwich-FGM circular 6 CONCLUSIONS CONCLUSIONS cylinder shell is higher than those of FGM ones That means, with the same geometry This paper paper established established nonlinear nonlinear dynamic dynamic equations equations of of fluid-filled fluid-filledcircular circularcylinder cylindershells shellsmade made dimensions, sandwich-FGM cylinder shell structures will work better than FGM ones of sandwich-FGM under mechanical load including the effect of temperature sandwich-FGM under mechanical load including the effect of temperature Dynamic shell Dynamic responses responses of of the the simply simply supported shell are are obtained obtained by by using usingGalerkin Galerkinmethod methodand and supported CONCLUSIONS This paper established nonlinear dynamic equations of fluid-filled circular cylinder shells made of sandwich-FGM under mechanical load including the effect of temperature 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 stability of sandwich-FGM cylinder shell - Temperature reduces dynamic critical force of sandwich-FGM cylinder shell That means, temperature reduces stability of shell 190 Khuc Van Phu, Le Xuan Doan - When the volume-fraction index k increases (it means the volume fraction of metal increases), the critical force decreases (the stability of the shell structure will decrease) - Dynamic critical force of the shell decreases when increasing ratio of length to radius (L/R) On the other hand, length of shell decreases stability of shell - With the same geometry dimensions, sandwich-FGM circular cylinder shell structures will work better than FGM one ACKNOWLEDGEMENTS This research is funded by National Foundation for Science and Technology Development of Vietnam (NAFOSTED) under grant number 107.02-2018.324 REFERENCES [1] D H Bich and N X Nguyen Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations Journal of Sound and Vibration, 331, (25), (2012), pp 5488–5501 https://doi.org/10.1016/j.jsv.2012.07.024 [2] Y W Kim Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge Composites Part B: Engineering, 70, 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and shells McGraw-Hill, New York, (1975) [21] A S Volmir The nonlinear dynamics of plates and shells Science edition, Moscow, (1975) [22] B Budiansky and R S Roth Axisymmetric dynamic buckling of clamped shallow spherical shells NASA Technical Note, 510, (1962), pp 597–606 [23] H Huang and Q Han Nonlinear dynamic buckling of functionally graded cylindrical shells subjected to time-dependent axial load Composite Structures, 92, (2), (2010), pp 593–598 https://doi.org/10.1016/j.compstruct.2009.09.011 192 Khuc Van Phu, Le Xuan Doan APPENDIX A Stiffness coefficients and quantities related to thermal load in Eq (8) h/2 E1 E dz = ; A12 = − ν2 − ν2 A11 = A22 = −h/2 h/2 B11 = B22 = −h/2 h/2 E2 E.z dz = ; B12 = − ν2 − ν2 D11 = D22 = −h/2 E.z2 1−ν dz = h/2 h/2 νE νE1 dz = ; A66 = − ν2 − ν2 −h/2 h/2 νEz νE2 dz = ; B66 = − ν2 − ν2 −h/2 h/2 E3 ; D12 = − ν2 −h/2 νEz2 1−ν dz = −h/2 h/2 Ez E2 dz = ; (1 + ν ) (1 + ν ) −h/2 h/2 νE3 ; B66 = − ν2 E E1 dz = ; (1 + ν ) (1 + ν ) −h/2 Ez2 E3 dz = ; (1 + ν ) (1 + ν ) in which h/2 E1 = E (z) dz = Em h + Ecm hc + −h/2 h/2 E2 = E (z) zdz = −h/2 h/2 Ecm h2c Ecm Ecm hc h − + 2 k+1 E (z) z2 dz = E3 = −h/2 Ec hhc Ec h3c + + Ecm h x ; k+1 Ecm k+1 h − hc Φa = 1−ν h − hc + hx − hx − h/2 E (z) α (z) ∆Tdz, h − hm Ecm h2x ; ( k + 1) ( k + 2) h − hc 2Ecm ( k + 1) ( k + 2) Em 3hhm hm + −h/2 h − hc + h2x + 2Ecm h3 ( k + 1) ( k + 2) ( k + 3) x Em h x − (h/2 − hm ) (h/2 − hc ) h x ; Φb = 1−ν h/2 E (z) α (z) ∆Tzdz −h/2 P∆T If ∆T = const then Φ a = 1−ν For FGM-core: P = Em αm h + Ec αc hc + Em αm (h − hc ) + Ecm αm h x Ecm αcm h x Em αcm h x + + , k+1 k+1 2k + where h x = h − hc − hm ; Ecm = Ec − Em APPENDIX B Extended stiffness coefficients in Eq (9) and Eq (10) A11 A12 A22 A22 B11 − A12 B12 ; A∗ = ; A∗ = ; B∗ = ; A11 A22 − A212 12 A11 A22 − A212 22 A11 A22 − A212 11 A11 A22 − A212 A B − A12 B22 ∗ A B − A12 B11 ∗ A B22 − A12 B12 ∗ B66 ∗ B12 = 22 12 ; B21 = 11 12 ; B22 = 11 ; A66 = ; B∗ = ; A66 66 A66 A11 A22 − A212 A11 A22 − A212 A11 A22 − A212 ∗ ∗ ∗ ∗ ∗ ∗ ; ; D12 = D12 − B11 B12 − B12 B22 D11 = D11 − B11 B11 − B12 B21 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ D21 = D12 − B12 B11 − B22 B21 ; D22 = D22 − B12 B12 − B22 B22 ; D66 = D66 − B66 B66 ∗ A11 = ... on time Nonlinear dynamic buckling of full- filled fluid sandwich- FGM circular cylinder shells Fig Model circular cylinder Figure.1 ModelofofFGM -sandwich FGM -sandwich circular cylinder shellshell... increasNonlinear dynamic bucklingofof full- filled fluid sandwich- FGM circularcylinder cylindershells shells dynamic full- filled fluid sandwich- FGM circular 99 ing length Nonlinear of the shell, the buckling. .. of fluid -filled cylinder shell 22 1 -Sandwich- FGM 1 -Sandwich- FGM 2 -FGM 2 -FGM Figure 7.7 .Dynamic responses Figure Dynamic responses FGMand and Fig Dynamic responses ofofofFGM FGM and sandwich- FGM