Vibration and Instability of Third?Order Shear Deformable FGM Sandwich Cylindrical Shells Conveying Fluid

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1.Key Laboratory of Structural Dynamics of Liaoning Province，College of Sciences，Northeastern University，Shenyang 110819，P.R.China；

2.School of Aerospace Engineering，Shenyang Aerospace University，Shenyang 110136，P.R.China

Abstract: The vibration and instability of functionally graded material（FGM）sandwich cylindrical shells conveying fluid are investigated. The Navier-Stokes relation is used to describe the fluid pressure acting on the FGM sandwich shells. Based on the third-order shear deformation shell theory，the governing equations of the system are derived by using the Hamilton’s principle. To check the validity of the present analysis，the results are compared with those in previous studies for the special cases. Results manifest that the natural frequency of the fluid-conveying FGM sandwich shells increases with the rise of the core-to-thickness ratio and power-law exponent，while decreases with the rise of fluid density，radius-to-thickness ratio and length-to-radius ratio. The fluid-conveying FGM sandwich shells lose stability when the non-dimensional flow velocity falls in 2.1—2.5，which should be avoided in engineering application.

Key words：FGM sandwich shell；fluid；third-order shear deformation shell theory；vibration；stability

0 Introduction

Pipes conveying fluid are found in numerous in?dustrial applications，in particular in water conser?vancy project and submarine oil transport［1-2］. For pipes containing fluid，couple vibrations are a major problem due to fluid flow［3-4］. The dynamics of fluidconveying pipes were extensively reviewed in Refs.［5-7］. One of the earliest studies in the area of dy?namics and stability of pipes conveying fluid was proposed by Paidoussis et al.［8］. Zhang et al.［9］inves?tigated the multi-pulse chaotic dynamics of pipes conveying pulsating fluid in parametric resonance.Ding et al.［10］studied the nonlinear vibration isola?tion of pipes conveying fluid using quasi-zero stiff?ness characteristics. Tan et al.［11］studied the para?metric resonances of pipes conveying pulsating highspeed fluid based on the Timoshenko beam theory.Selmane et al.［12］discussed the effect of flowing flu?id on the vibration characteristics of an open，aniso?tropic cylindrical shell submerged and subjected si?multaneously to internal and external flow. Amabili et al.［13］investigated the non-linear dynamics and stability of simply supported，circular cylindrical shells containing inviscid and incompressible fluid flow.

Functionally gradient materials（FGMs）have some prominent advantages such as avoiding crack，avoiding delamination，reducing stress concentra?tion，eliminating residual stress，etc.［14］. Due to these superiorities，vibrations and dynamics stability of structures with FGM properties have attracted much attention［15-21］. Chen et al.［22］studied the free vibration of simply supported，fluid-filled FGM cy?lindrical shells with arbitrary thickness based on the three-dimensional elasticity theory. Sheng et al.［23］studied dynamic characteristics of fluid-conveying FGM cylindrical shells under mechanical and ther?mal loads. Park et al.［24］presented vibration charac?teristics of fluid-conveying FGM cylindrical shells resting on Pasternak elastic foundation with an oblique edge.

As one of the most prevalent composite struc?tures applied in aerospace，naval，automotive and nuclear engineering，sandwich structures have at?tracted tremendous interests from academic and in?dustrial communities［25-33］. Note that the use of FGMs in sandwich shells can mitigate the interfacial shear stress concentration. Thus，dynamics studies of FGM sandwich shells have been carried out by several researchers. Based on the Donnell’s shell theory，Dung et al.［34］studied the nonlinear buckling and post-bucking behavior of FGM sandwich circu?lar cylindrical shells. Chen et al.［35］presented the free vibration analysis of FGM sandwich doublycurved shallow shells under simply supported condi?tion. Fazzolari et al.［36］studied the free vibration of FGM sandwich shells using the Ritz minimum ener?gy method. Tornabene et al.［37］studied the free vi?bration of rotating FGM sandwich shells with vari?able thicknesses.

Due to the good thermal insulation of sandwich shells，they can be used as pipelines for transporting petroleum to prevent the paraffin in the crude oil from depositing on the pipe wall after the oil temper?ature is lowered. The core layer usually uses a mate?rial with good heat insulation properties，such as ce?ramics［38-40］. However，sandwich shells have the in?terfacial shear stress concentration and are prone to some accidents［41］. Nowadays， FGM sandwich shells can solve this problem and have promising ap?plications in submarine oil transport.

In the present study，we deal with the vibration and instability of FGM sandwich shells conveying fluid. The Navier-Stokes relation is used to describe the fluid pressure acting on the shells. The third-or?der shear deformation shell theory is used to model the present system. Then，the governing equations and boundary conditions are derived by using the Hamilton’s principle. Finally，the frequency and stability results are presented for FGM sandwich shells conveying fluid under various conditions.

1 Theoretical Formulation

1.1 FGM sandwich cylindrical shell

As shown in Fig.1，a fluid-conveying FGM sandwich cylindrical shell made up of three layers，namely，Layer 1，Layer 2 and Layer 3，is consid?ered. The thicknesses of the three layers areh1，h2andh3，respectively. Layer 2 is the pure ceramic lay?er，and Layer 1 and Layer 3 are FGM layers. The material properties of Layer 1 and Layer 3 change from pure metal at the outer and inner surfaces to pure ceramic. The dimensions of the shell are denot?ed by the lengthL，the middle-surface radiusRand the thicknessh. A cylindrical coordinate system（x，θ，z）is chosen，wherex- andθ-axes define the mid?dle-surface of the shell andz-axis denotes the out-ofsurface coordinate.

Fig.1 FGM sandwich cylindrical shell conveying fluid

For the FGM sandwich shell，the effective ma?terial properties of layerj（j= 1，2，3）can be ex?pressed as［42］

wherePmandPcdenote the material properties of metal and ceramic，respectively；the volume frac?tionV（j）（j= 1，2，3）through the thickness of the sandwich shell follows a power law while it equals unity in the core layer，which reads［43］

wherek∈[0，∞)is the power-law exponent.

Therefore，the Poisson’s ratioν(j)(z)，Young’s modulusE(j)(z) and mass densityρ(j)(z)（j= 1，2，3）of the FGM sandwich shell are expressed as

whereνm，ρm，Emare the Poisson ratio，mass densi?ty and Young’s modulus of metal，respectively；νc，ρc，Ecthe Poisson ratio，mass density and Young’s modulus of ceramic，respectively.

1.2 Fluid?shell interaction

The fluid inside the shell is assumed to be in?compressible，isentropic and time independent. To simplify the problem，we ignore the influence of the deformation and vibration of the shell on the liquid flow，the shear force transferred from the flow，the flow separation and the Reynold number. The mo?mentum-balance equation for the fluid motion can be described by the well-known Navier-Stokes equa?tion［44］

wherev≡(vr，vθ，vx)is the flow velocity with com?ponents in ther，θandxdirections；Pandμare the pressure and the viscosity of the fluid，respectively；ρfis the mass density of the internal fluid；?2the Laplacian operator andFbodythe body forces. In this paper，we neglect the action of body forces and con?sider Newtonian fluid，i.e.，the viscosity is time-in?dependent.

At the interface between the fluid and the shell，the velocity of the fluid is equal to the shell in the radial direction. These relationships can be writ?ten as［3］

whereris the distance from the center of the shell to an arbitrary point in the radial direction，and

whereUis the mean flow velocity.

Consider the fluid as inviscid. By substituting Eqs.（7，8）into Eq.（6），the fluid pressurePis ob?tained as［3］

1.3 Third?order shear deformation theory

According to the third-order shear deformation shell theory，the displacement fields of the fluid-con?veying FGM sandwich shell are expressed as［45］

wherec1=4/3h2；u，vandware the displacements of an arbitrary point of the shell；u0，v0andw0the displacements of a generic point of the middle sur?face；φxandφθthe rotations of a normal to the midsurface aboutθandxaxes，respectively.

The strain components at a distancezfrom the mid-plane are［45］

where

whereεx，εθ，γxθ，γxzandγθzare the strains of an ar?bitrary point；the strains of a generic point of the middle surface；k1x，k1θ，k1xθ，k3x，the curvatures of a generic point of the middle surface.

The relationship between stresses and strains of the fluid-conveying FGM sandwich shell is stated as

1.4 Governing equations and solution

The strain energy of the FGM sandwich shell can be expressed as

where the resultant forcesNx，NθandNxθ，momentsMx，MθandMxθ，shear forcesQxzandQθz，and high?er-order stress resultantsPx，Pθ，Pxθ，KxzandKθzare defined by

The kinetic energy of the FGM sandwich shell is written as

In addition，the potential energy associated to the fluid pressure is given by［4］

By using the Hamilton principle

and then equating the coefficients ofδu0，δv0，δw0，δφxandδφθto zero，the motion equations of the flu?id-conveying FGM sandwich shell can be obtained as

where the inertiasand（i= 1，2，3，4，5）are defined by

By substituting Eq.（23）into Eqs.（27—31），the equations of motion can be rewritten as follows

The simply supported boundary condition is considered in this study.It is given by［46］

The solutions to Eqs.（38—42）and Eq.（43）can be separated into a function of time and position as follows［46］

whereλm=mπ/L；umn(t)，vmn(t)，wmn(t)，φmn(t)and(t)represent generalized coordinates.

Substituting Eqs.（44—48）into Eqs.（38—42）yields

wherem= 1，2，…，；i= 1，2，…，；n= 1，2，…，N；m≠iandm±iare odd numbers.MijandKijare integral coefficients.

Eqs.（49—53） can be written in the matrix form as

whereM，C，Kdenote the mass，damping and stiff?ness matrices，respectively；Xis a 5××Ncol?umn vector consisting ofumn(t)，vmn(t)，wmn(t)，φmn(t)and(t).

Eq.（54）is solved in the state space by settingX=eΛtq，which gives the following eigenvalue equation

where {q Λq}Tis the state vector. It should be noted that eigenvalueΛis a non-zero complex num?ber. The imaginary part ofΛis the frequency and the real part is the damping.

2 Numerical Results and Discus?sion

The natural frequency is obtained by finding the eigenvalues of the matrixIn order to demonstrate the accuracy of the present analysis，a comparison investigation related to a liq?uid-filled homogenous cylindrical shell is carried out. The used parameters are：h/R= 0.01，ρf=1 000 kg/m3，iron densityρ= 7 850 kg/m3，L/R= 2 andν =0.3. For convenience，the non-di?mensional axial flow velocityVis defined asV=U/{(π/L2)[D/(ρh)]1/2} withD=Eh3/[12(1-ν2)]；the non-dimensional eigenvalueΩis intro?duced asΩ=Λ/{(π2/L2)[D/(ρh)]1/2}. As can be seen from Fig.2，the result obtained from the cur?rent analysis is in good agreement with Amabili et al.［47］. The small difference between them is because the rotational inertia termsI2φ?xandI2φ?θwere ne?glected by Amabili et al.［47］. It is worth mentioning that the real part represents the natural frequency in Ref.［47］，which is caused by the use of different so?lutions.

Fig.2 Non-dimensional eigenvalue Ω versus non-dimension?al flow velocity V

Next，we investigate the stability and free vi?bration of FGM sandwich shells conveying fluid.The fluid is considered as crude oil，with a mass densityρf= 0.81 g/cm3［48］. Here，the ceramic and metal forming the FGM sandwich shell shown in Fig.1 are considered as Zirconia and Aluminum，re?spectively.Their properties are［49］

Aluminum：Em= 70 GPa，νm= 0.3，ρm=2 707 kg/m3

Zirconia：Ec= 151 GPa，νc= 0.3，ρc=3 000 kg/m3

Fig.3 plots the non-dimensional natural fre?quency versus the circumferential wave numbernof FGM sandwich shell conveying fluid. An obvious trend can be found that the frequency first decreases and then increases with the circumferential wave numbern. As a result，the fundamental natural fre?quency of the system happens at mode（n= 5，m= 1 forV= 0）. WhenV≠0，the modes are coupled for the same circumferential wave numbern，and we taken= 5 as the representative circum?ferential wave number for analysis.

Fig.3 Non-dimensional natural frequency versus cir?cumferential wave number n of FGM sandwich shell conveying fluid (m=1, L/R = 2.0, V =0, k = 1, R/h = 80, h2/h = 0.2)

Fig.4 shows the first two non-dimensional natu?ral frequencies versus the non-dimensional flow ve?locity of FGM sandwich shell conveying fluid，where= 2 is adopted because more axial modes have no effect on the first two natural frequencies.As the flow velocity increases，it is found that both frequencies decrease at first. If the flow velocity reaches certain value，the first frequency vanishes.This velocity is named the critical velocity，at which the system loses its stability due to the divergence via a pitchfork bifurcation. After a small range of in?stability，the first frequency increases and then coin?cides with the second frequency，and the system re?covers its stability. Moreover，when the flow veloci?ty is between about 2.1—2.5，the fluid-conveying FGM sandwich shell loses stability，which should be avoided in real application.

Fig.4 The first two non-dimensional natural frequen?cies versus non-dimensional flow velocity of FGM sandwich shell conveying fluid (n = 5,L/R = 2.0, = 2, k = 1, R/h = 80,h2/h = 0.2)

Fig.5 gives the first non-dimensional natural frequency versus the length-to-radius ratio of fluidconveying FGM sandwich shell with different pow?er-law exponents. It is shown that as the power-law exponent increases，the first non-dimensional natu?ral frequency shows an increasing trend. But the ef?fect of power-law exponent becomes more and more insignificant with increasing length-to-radius ratio. It is also found that as the length-to-radius ratioL/Rincreases，the first non-dimensional natural frequen?cy decrease. When the length-to-radius ratio is small，the first non-dimensional natural frequency changes obviously. However，when this ratio is large，the first non-dimensional natural frequency is no more sensitive to the length-to-radius ratio.

Fig.5 The first non-dimensional natural frequency versus length-to-radius ratio L/R of fluid-con?veying FGM sandwich shell with different pow?er-law exponents (n = 5, V = 1, = 2,R/h = 80, h2/h = 0.2)

Fig.6 illustrates the first non-dimensional natu?ral frequency versus the radius-to-thickness ratio of fluid-conveying FGM sandwich shell with different power-law exponents. It is shown that as radius-tothickness ratioR/hincreases，the first non-dimen?sional natural frequency of the fluid-conveying FGM sandwich shell decreases.

Fig.6 The first non-dimensional natural frequency ver?sus radius-to-thickness ratio R/h of fluid-con?veying FGM sandwich shell with different pow?er-law exponents (n = 5, L/R = 2.0, = 2,V = 1, h2/h = 0.2)

Fig.7 presents the first non-dimensional natural frequency versus the mass densityρfof fluid-convey?ing FGM sandwich shell with different power-law exponents. The first non-dimensional natural fre?quency of the fluid-conveying FGM sandwich shell decreases with increasing fluid mass density. This is reasonable because the FGM sandwich shell vi?brates as though its mass is increased by the mass of fluid，which is defined as added virtual mass effect.

Fig.7 The first non-dimensional natural frequency versus the fluid density ρf of fluid-conveying FGM sand?wich shell with different power-law exponents (n =5, L/R = 2.0, = 2, V = 1, h2/h = 0.2,R/h = 80)

Fig.8 plots the first non-dimensional natural fre?quency versus the core-to-thickness ratioh2/hof flu?id-conveying FGM sandwich shell with different power-law exponents. It is found that the first nondimensional natural frequency increases gradually as the core-to-thickness ratioh2/hincreases. This can be understood because the increase of core-to-thick?ness ratio enhances the stiffness of the structure.

Fig.8 The first non-dimensional natural frequency versus core-to-thickness ratio h2/h of fluid-conveying FGM sandwich shell with different power-law ex?ponents (n = 5, L/R = 2.0, = 2, V = 1,R/h = 80)

3 Conclusions

The vibration and instability of FGM sandwich shells conveying fluid are investigated based on the third-order shear deformation shell theory. By using the Hamilton’s principle，the governing equations of the present system are derived. Results show that as the flow velocity increases，the natural frequen?cies of the fluid-conveying FGM sandwich shells de?crease at first. When reaching the critical velocity，the first frequency vanishes and the system loses its stability. Moreover，the fluid-conveying FGM sand?wich shells lose stability when the non-dimensional flow velocity falls in 2.1—2.5，which should be avoided in submarine oil transport. Besides，the first non-dimensional natural frequency decreases with the rise of fluid density，radius-to-thickness ratio and length-to-radius ratio while increases with the raise of core-to-thickness ratio and power-law expo?nent. The fluid viscosity has insignificant effect on the first non-dimensional natural frequency of the fluid-conveying FGM sandwich shells.

AcknowledgementsThis work was supported by the Na?tional Natural Science Foundation of China（Nos.11922205 and 12072201），and the Fundamental Research Fund for the Central Universities（No.N2005019）.

AuthorMr. LI Zhihang received the B.S. degree in solid mechanics from Northeastern University，Shenyang，China，in 2021. He is now pursuing the Ph.D. degree in Wuhan Uni?versity. His current research interests include vibration and stability of fluid-conveying pipe，and structural health moni?toring.

Author contributionsMr.LI Zhihang designed the study，complied the models，conducted the analysis，interpreted the results and wrote the manuscript. Prof. ZHANG Yufei and Prof. WANG Yanqing contributed to the discussion and background of the study. All authors commented on the manuscript draft and approved the submission.

Competing interestsThe authors declare no competing interests.