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

 Vietnam Journal of Mechanics, VAST, Vol.41, No. 2 (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,∗
 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
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 i ... 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 
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.
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 ra-
dius (L/R). On the other hand, length of shell decreases stability of shell.
 - With the same geometry dimensions, sandwich-FGM circular cylinder shell struc-
tures will work better than FGM one.
 ACKNOWLEDGEMENTS
 This research is funded by National Foundation for Science and Technology Devel-
opment of Vietnam (NAFOSTED) under grant number 107.02-2018.324.
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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.