基于三阶剪切变形理论功能梯度夹层输流圆柱壳的振动与稳定性研究

李志航; 张宇飞; 王延庆; LI Zhihang; ZHANG Yufei; WANG Yanqing

Transactions of Nanjing University of Aeronautics & Astronautics

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

- ORCID：
LI Zhihang ¹
✉
- ORCID：
ZHANG Yufei ²
- ORCID：
WANG Yanqing ¹

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

CLC： TB12

Updated：2022-03-14

DOI：10.16356/j.1005-1120.2022.01.005

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.

Keywords

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

0 Introduction

Pipes conveying fluid are found in numerous industrial applications， in particular in water conservancy project and submarine oil transport^［

1-2］. For pipes containing fluid， couple vibrations are a major problem due to fluid flow^{［Reference 3-4}3-4］. The dynamics of fluid-conveying pipes were extensively reviewed in Refs.［5-7］. One of the earliest studies in the area of dynamics and stability of pipes conveying fluid was proposed by Paidoussis et al.^{［Reference 8

Baidu Scholar}8］. Zhang et al.^{［Reference 9

Baidu Scholar}9］ investigated the multi-pulse chaotic dynamics of pipes conveying pulsating fluid in parametric resonance. Ding et al.^{［Reference 10

Baidu Scholar}10］ studied the nonlinear vibration isolation of pipes conveying fluid using quasi-zero stiffness characteristics. Tan et al.^{［Reference 11

Baidu Scholar}11］ studied the parametric resonances of pipes conveying pulsating high-speed fluid based on the Timoshenko beam theory. Selmane et al.^{［Reference 12

Baidu Scholar}12］ discussed the effect of flowing fluid on the vibration characteristics of an open， anisotropic cylindrical shell submerged and subjected simultaneously to internal and external flow. Amabili et al.^{［Reference 13

Baidu Scholar}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 concentration， eliminating residual stress， etc.^［

14］. Due to these superiorities， vibrations and dynamics stability of structures with FGM properties have attracted much attention^{［Reference 15-21}15-21］. Chen et al.^{［Reference 22

Baidu Scholar}22］ studied the free vibration of simply supported， fluid-filled FGM cylindrical shells with arbitrary thickness based on the three-dimensional elasticity theory. Sheng et al.^{［Reference 23

Baidu Scholar}23］ studied dynamic characteristics of fluid-conveying FGM cylindrical shells under mechanical and thermal loads. Park et al.^{［Reference 24

Baidu Scholar}24］ presented vibration characteristics of fluid-conveying FGM cylindrical shells resting on Pasternak elastic foundation with an oblique edge.

As one of the most prevalent composite structures applied in aerospace， naval， automotive and nuclear engineering， sandwich structures have attracted tremendous interests from academic and industrial 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.^{［Reference 34

Baidu Scholar}34］ studied the nonlinear buckling and post-bucking behavior of FGM sandwich circular cylindrical shells. Chen et al.^{［Reference 35

Baidu Scholar}35］ presented the free vibration analysis of FGM sandwich doubly-curved shallow shells under simply supported condition. Fazzolari et al.^{［Reference 36

Baidu Scholar}36］ studied the free vibration of FGM sandwich shells using the Ritz minimum energy method. Tornabene et al.^{［Reference 37

Baidu Scholar}37］ studied the free vibration of rotating FGM sandwich shells with variable 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 temperature is lowered. The core layer usually uses a material with good heat insulation properties， such as ceramics^［

38-40］. However， sandwich shells have the interfacial shear stress concentration and are prone to some accidents^{［Reference 41

Baidu Scholar}41］. Nowadays， FGM sandwich shells can solve this problem and have promising applications 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-order 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 considered. The thicknesses of the three layers are h₁， h₂ and h₃， respectively. Layer 2 is the pure ceramic layer， 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 denoted by the length L， the middle-surface radius R and the thickness h. A cylindrical coordinate system （x， θ， z） is chosen， where x- and θ-axes define the middle-surface of the shell and z-axis denotes the out-of-surface coordinate.

Fig.1 FGM sandwich cylindrical shell conveying fluid

For the FGM sandwich shell， the effective material properties of layer j （j = 1， 2， 3） can be expressed as^［

42］

P^{(j)} (z) = (P_{c} - P_{m}) V^{(j)} (z) + P_{c}

(1)

where P_m and P_c denote the material properties of metal and ceramic， respectively； the volume fraction V^（^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］

\{\begin{array}{l} V^{(1)} (z) = {(\frac{z - z_{0}}{z_{1} - z_{0}})}^{k} z \in [z_{0}, z_{1}] \\ V^{(2)} (z) = 1 z \in [z_{1}, z_{2}] \\ V^{(3)} (z) = {(\frac{z - z_{3}}{z_{2} - z_{3}})}^{k} z \in [z_{2}, z_{3}] \end{array}

(2)

where $k \in [0, \infty)$ is the power-law exponent.

Therefore， the Poisson’s ratio $ν^{(j)} (z)$ ， Young’s modulus $E^{(j)} (z)$ and mass density $ρ^{(j)} (z)$ （j = 1， 2， 3） of the FGM sandwich shell are expressed as

ν^{(j)} (z) = (ν_{c} - ν_{m}) V^{(j)} (z) + ν_{m}

(3)

E^{(j)} (z) = (E_{c} - E_{m}) V^{(j)} (z) + E_{m}

(4)

ρ^{(j)} (z) = (ρ_{c} - ρ_{m}) V^{(j)} (z) + ρ_{m}

(5)

where $ν_{m}$ ， $ρ_{m}$ ， $E_{m}$ are the Poisson ratio， mass density and Young’s modulus of metal， respectively； $ν_{c}$ ， $ρ_{c}$ ， $E_{c}$ the 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 incompressible， 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 momentum-balance equation for the fluid motion can be described by the well-known Navier-Stokes equation^［

44］

ρ_{f} \frac{d v}{d t} = - \nabla P + μ \nabla^{2} v + F_{b o d y}

(6)

where $v \equiv (v_{r}, v_{θ}, v_{x})$ is the flow velocity with components in the r， θ and x directions； P and μ are the pressure and the viscosity of the fluid， respectively； $ρ_{f}$ is the mass density of the internal fluid； $\nabla^{2}$ the Laplacian operator and F_body the body forces. In this paper， we neglect the action of body forces and consider Newtonian fluid， i.e.， the viscosity is time-independent.

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 written as^［

3］

v_{r} = \frac{d w}{d t}

(7)

where r is the distance from the center of the shell to an arbitrary point in the radial direction， and

\frac{d}{d t} = \frac{\partial}{\partial t} + U \frac{\partial}{\partial x}

(8)

where U is the mean flow velocity.

Consider the fluid as inviscid. By substituting Eqs.（7，8） into Eq.（6）， the fluid pressure P is obtained as^［

3］

P = ρ_{f} (\frac{\partial^{2} w}{\partial t^{2}} + 2 U \frac{\partial^{2} w}{\partial x \partial t} + U^{2} \frac{\partial^{2} w}{\partial x^{2}})

(9)

1.3 Third‑order shear deformation theory

According to the third-order shear deformation shell theory， the displacement fields of the fluid-conveying FGM sandwich shell are expressed as^［

45］

\{\begin{array}{l} u (x, θ, z, t) = u_{0} (x, θ, t) + \\ z ϕ_{x} (x, θ, t) - c_{1} z^{3} (ϕ_{x} + \frac{\partial w_{0}}{\partial x}) \\ v (x, θ, z, t) = v_{0} (x, θ, t) + \\ z ϕ_{θ} (x, θ, t) - c_{1} z^{3} (ϕ_{θ} + \frac{1}{R} \frac{\partial w_{0}}{\partial θ}) \\ w (x, θ, z, t) = w_{0} (x, θ, t) \end{array}

(10)

where $c_{1} = 4 / 3 h^{2}$ ； u， v and w are the displacements of an arbitrary point of the shell； u₀， v₀ and w₀ the displacements of a generic point of the middle surface； ϕ_x and ϕ_θ the rotations of a normal to the mid-surface about $θ$ and x axes， respectively.

The strain components at a distance z from the mid-plane are^［

45］

(\begin{matrix} ε_{x} \\ ε_{θ} \\ γ_{x θ} \end{matrix}) = (\begin{matrix} ε_{x}^{0} \\ ε_{θ}^{0} \\ γ_{x θ}^{0} \end{matrix}) + z (\begin{matrix} k_{x}^{1} \\ k_{θ}^{1} \\ k_{x θ}^{1} \end{matrix}) + z^{3} (\begin{matrix} k_{x}^{3} \\ k_{θ}^{3} \\ k_{x θ}^{3} \end{matrix})

(11)

(\begin{matrix} γ_{x z} \\ γ_{θ z} \end{matrix}) = (\begin{matrix} γ_{x z}^{0} \\ γ_{θ z}^{0} \end{matrix}) + z^{2} (\begin{matrix} k_{x z}^{2} \\ k_{θ z}^{2} \end{matrix})

(12)

where

(\begin{matrix} ε_{x}^{0} \\ ε_{θ}^{0} \\ γ_{x θ}^{0} \end{matrix}) = (\begin{matrix} \frac{\partial u_{0}}{\partial x} \\ \frac{1}{R} \frac{\partial v_{0}}{\partial θ} + \frac{w_{0}}{R} \\ \frac{1}{R} \frac{\partial u_{0}}{\partial θ} + \frac{\partial v_{0}}{\partial x} \end{matrix})

(13)

(\begin{matrix} k_{x}^{1} \\ k_{θ}^{1} \\ k_{x θ}^{1} \end{matrix}) = (\begin{matrix} \frac{\partial ϕ_{x}}{\partial x} \\ \frac{1}{R} \frac{\partial ϕ_{θ}}{\partial θ} \\ \frac{1}{R} \frac{\partial ϕ_{x}}{\partial θ} + \frac{\partial ϕ_{θ}}{\partial x} \end{matrix})

(14)

(\begin{matrix} k_{x}^{3} \\ k_{θ}^{3} \\ k_{x θ}^{3} \end{matrix}) = - c_{1} (\begin{matrix} \frac{\partial ϕ_{x}}{\partial x} + \frac{\partial^{2} w_{0}}{\partial x^{2}} \\ \frac{1}{R} \frac{\partial ϕ_{θ}}{\partial θ} + \frac{1}{R^{2}} \frac{\partial^{2} w_{0}}{\partial θ^{2}} \\ \frac{1}{R} \frac{\partial ϕ_{x}}{\partial θ} + \frac{\partial ϕ_{θ}}{\partial x} + \frac{2}{R} \frac{\partial^{2} w_{0}}{\partial x \partial θ} \end{matrix})

(15)

(\begin{matrix} γ_{x z}^{0} \\ γ_{θ z}^{0} \end{matrix}) = (\begin{matrix} ϕ_{x} + \frac{\partial w_{0}}{\partial x} \\ ϕ_{θ} + \frac{1}{R} \frac{\partial w_{0}}{\partial θ} \end{matrix})

(16)

(\begin{matrix} k_{x z}^{2} \\ k_{θ z}^{2} \end{matrix}) = - 3 c_{1} (\begin{matrix} ϕ_{x} + \frac{\partial w_{0}}{\partial x} \\ ϕ_{θ} + \frac{1}{R} \frac{\partial w_{0}}{\partial θ} \end{matrix})

(17)

where $ε_{x}, ε_{θ}, γ_{x θ}, γ_{x z} a n d γ_{θ z}$ are the strains of an arbitrary point； $ε_{x}^{0}, ε_{θ}^{0}, γ_{x θ}^{0}, γ_{x z}^{0} a n d γ_{θ z}^{0}$ the strains of a generic point of the middle surface； $k_{x}^{1}, k_{θ}^{1}, k_{x θ}^{1}, k_{x}^{3},$ $k_{θ}^{3}, k_{x θ}^{3}, k_{x z}^{2} a n d k_{θ z}^{2}$ 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

\{\begin{matrix} σ_{x}^{(j)} \\ σ_{θ}^{(j)} \\ τ_{x θ}^{(j)} \\ τ_{x z}^{(j)} \\ τ_{θ z}^{(j)} \end{matrix}\} = [\begin{matrix} Q_{11}^{(j)} & Q_{12}^{(j)} & 0 & 0 & 0 \\ Q_{21}^{(j)} & Q_{22}^{(j)} & 0 & 0 & 0 \\ 0 & 0 & Q_{66}^{(j)} & 0 & 0 \\ 0 & 0 & 0 & Q_{44}^{(j)} & 0 \\ 0 & 0 & 0 & 0 & Q_{55}^{(j)} \end{matrix}] \{\begin{matrix} ε_{x} \\ ε_{θ} \\ γ_{x θ} \\ γ_{x z} \\ γ_{θ z} \end{matrix}\}

(18)

where Q_ijis given by

Q_{11}^{(j)} = \frac{E^{(j)} (z)}{1 - v^{(j)} {(z)}^{2}}, Q_{12}^{(j)} = \frac{v^{(j)} (z) E^{(j)} (z)}{1 - v^{(j)} {(z)}^{2}}

(19)

Q_{21}^{(j)} = \frac{v^{(j)} (z) E^{(j)} (z)}{1 - v^{(j)} {(z)}^{2}}, Q_{22}^{(j)} = \frac{E^{(j)} (z)}{1 - v^{(j)} {(z)}^{2}}

(20)

Q_{44}^{(j)} = Q_{55}^{(j)} = Q_{66}^{(j)} = \frac{E^{(j)} (z)}{2 [1 + v^{(j)} (z)]}

(21)

1.4 Governing equations and solution

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

\begin{array}{l} Π_{S} = \frac{1}{2} \int_{0}^{2 π} \int_{0}^{L} \sum_{j = 1}^{3} \int_{z_{j - 1}}^{z_{j}} (σ_{x}^{(j)} ε_{x} + σ_{θ}^{(j)} ε_{θ} + τ_{x θ}^{(j)} γ_{x θ} + τ_{x z}^{(j)} γ_{x z} + τ_{θ z}^{(j)} γ_{θ z}) R d x d θ = \frac{1}{2} \int_{0}^{2 π} \int_{0}^{L} \{[(N_{x} \frac{\partial u_{0}}{\partial x} + \frac{N_{x θ}}{R} \frac{\partial u_{0}}{\partial θ}) + \\ (N_{x θ} \frac{\partial v_{0}}{\partial x} + \frac{N_{θ}}{R} \frac{\partial v_{0}}{\partial θ}) + [Q_{x z} \frac{\partial w_{0}}{\partial x} + \frac{Q_{θ z}}{R} \frac{\partial w_{0}}{\partial θ} - \frac{N_{θ}}{R} w_{0} - 3 c_{1} (K_{x z} \frac{\partial w_{0}}{\partial x} + \frac{K_{θ z}}{R} \frac{\partial w_{0}}{\partial θ}) + \\ c_{1} (P_{x}^{2} \frac{\partial^{2} w_{0}}{\partial x^{2}} + \frac{P_{θ}^{2}}{R^{2}} \frac{\partial^{2} w_{0}}{\partial θ^{2}} + 2 \frac{P_{x θ}^{2}}{R} \frac{\partial^{2} w_{0}}{\partial x \partial θ})] + [M_{x} \frac{\partial ϕ_{x}}{\partial x} + \frac{M_{x θ}}{R} \frac{\partial ϕ_{x}}{\partial θ} - Q_{x z} ϕ_{x} + 3 c_{1} K_{x z} ϕ_{x} - \\ c_{1} (P_{x} \frac{\partial ϕ_{x}}{\partial x} + \frac{P_{x θ}}{R} \frac{\partial ϕ_{x}}{\partial θ})] + [M_{x θ} \frac{\partial ϕ_{θ}}{\partial x} + \frac{M_{θ}}{R} \frac{\partial ϕ_{θ}}{\partial θ} - Q_{θ z} ϕ_{θ} + 3 c_{1} K_{θ z} ϕ_{θ} - c_{1} (P_{x θ} \frac{\partial ϕ_{θ}}{\partial x} + \frac{P_{θ}}{R} \frac{\partial ϕ_{θ}}{\partial θ})]\} R d x d θ \end{array}

(22)

where the resultant forces N_x， N_θand N_xθ， moments M_x， M_θ and M_xθ， shear forces Q_xzand Q_θz， and higher-order stress resultants P_x， P_θ， P_xθ， K_xz and K_θz are defined by

(N_{x}, N_{θ}, N_{x θ}) = \sum_{j = 1}^{3} \int_{z_{j - 1}}^{z_{j}} (σ_{x}^{(j)}, σ_{θ}^{(j)}, τ_{x θ}^{(j)}) d z

(23‑1)

(M_{x}, M_{θ}, M_{x θ}) = \sum_{j = 1}^{3} \int_{z_{j - 1}}^{z_{j}} (σ_{x}^{(j)}, σ_{θ}^{(j)}, τ_{x θ}^{(j)}) z d z

(23‑2)

(Q_{x z}, Q_{θ z}) = \sum_{j = 1}^{3} \int_{z_{j - 1}}^{z_{j}} (τ_{x z}^{(j)}, τ_{θ z}^{(j)}) d z

(23‑3)

(P_{x}, P_{θ}, P_{x θ}) = \sum_{j = 1}^{3} \int_{z_{j - 1}}^{z_{j}} (σ_{x}^{(j)}, σ_{θ}^{(j)}, τ_{x θ}^{(j)}) z^{3} d z

(23‑4)

(K_{x z}, K_{θ z}) = \sum_{j = 1}^{3} \int_{z_{j - 1}}^{z_{j}} (τ_{x z}^{(j)}, τ_{θ z}^{(j)}) z^{2} d z

(23‑5)

The kinetic energy of the FGM sandwich shell is written as

\begin{array}{l} Π_{T} = \frac{1}{2} \int_{V} \sum_{j = 1}^{3} \int_{z_{j - 1}}^{z_{j}} ρ (z) \{[\frac{\partial u_{0}}{\partial t} + z \frac{\partial ϕ_{x}}{\partial t} - \\ {c_{1} z^{3} (\frac{\partial ϕ_{x}}{\partial t} + \frac{\partial w_{0}^{2}}{\partial x \partial t})]}^{2} + [\frac{\partial v_{0}}{\partial t} + z \frac{\partial ϕ_{θ}}{\partial t} - \\ {c_{1} z^{3} (\frac{\partial ϕ_{θ}}{\partial t} + \frac{1}{R} \frac{\partial w_{0}^{2}}{\partial θ \partial t})]}^{2} + {(\frac{\partial w_{0}}{\partial t})}^{2}\} d V \end{array}

(24)

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

4］

Π_{p} = \int_{0}^{2 π} \int_{0}^{L} P R d x d θ

(25)

By using the Hamilton principle

δ \int_{t_{1}}^{t_{2}} (Π_{T} - Π_{S} - Π_{p}) d t = 0

(26)

and then equating the coefficients of $δ u_{0}$ ， $δ v_{0}$ ， $δ w_{0}$ ， $δ ϕ_{x}$ and $δ ϕ_{θ}$ to zero， the motion equations of the fluid-conveying FGM sandwich shell can be obtained as

\frac{\partial N_{x}}{\partial x} + \frac{1}{R} \frac{\partial N_{x θ}}{\partial θ} = {\bar{I}}_{1} {\ddot{u}}_{0} + {\bar{I}}_{2} {\ddot{ϕ}}_{x} - {\bar{I}}_{3} \frac{\partial \ddot{w}}{\partial x}

(27)

\frac{\partial N_{x θ}}{\partial x} + \frac{1}{R} \frac{\partial N_{θ}}{\partial θ} = {\tilde{I}}_{1} {\ddot{v}}_{0} + {\tilde{I}}_{2} {\ddot{ϕ}}_{θ} - {\tilde{I}}_{3} \frac{1}{R} \frac{\partial \ddot{w}}{\partial θ}

(28)

\begin{array}{l} \frac{\partial Q_{x z}}{\partial x} + \frac{1}{R} \frac{\partial Q_{θ z}}{\partial θ} - \frac{N_{θ}}{R} - 3 c_{1} (\frac{\partial K_{x z}}{\partial x} + \frac{1}{R} \frac{\partial K_{θ z}}{\partial θ}) + \\ c_{1} (\frac{\partial^{2} P_{x}}{\partial x^{2}} + \frac{1}{R^{2}} \frac{\partial^{2} P_{θ}}{\partial θ^{2}} + \frac{2}{R} \frac{\partial^{2} P_{x θ}}{\partial x \partial θ}) = \\ {\bar{I}}_{3} \frac{\partial \ddot{u}}{\partial x} + {\bar{I}}_{5} \frac{\partial {\ddot{ϕ}}_{x}}{\partial x} + {\tilde{I}}_{3} \frac{1}{R} \frac{\partial \ddot{v}}{\partial θ} + {\tilde{I}}_{5} \frac{1}{R} \frac{\partial {\ddot{ϕ}}_{θ}}{\partial θ} + I_{1} \ddot{w} - \\ c_{1}^{2} I_{7} (\frac{\partial^{2} \ddot{w}}{\partial x^{2}} + \frac{1}{R^{2}} \frac{\partial^{2} \ddot{w}}{\partial θ^{2}}) + P \end{array}

(29)

\frac{\partial M_{x}}{\partial x} + \frac{1}{R} \frac{\partial M_{x θ}}{\partial θ} - Q_{x z} + 3 c_{1} K_{x z} - c_{1} (\frac{\partial P_{x}}{\partial x} + \frac{1}{R} \frac{\partial P_{x θ}}{\partial θ}) = {\bar{I}}_{2} \ddot{u} + {\bar{I}}_{4} {\ddot{ϕ}}_{x} - {\bar{I}}_{5} \frac{\partial \ddot{w}}{\partial x}

(30)

\frac{\partial M_{x θ}}{\partial x} + \frac{1}{R} \frac{\partial M_{θ}}{\partial θ} - Q_{θ z} + 3 c_{1} K_{θ z} - c_{1} (\frac{\partial P_{x θ}}{\partial x} + \frac{1}{R} \frac{\partial P_{θ}}{\partial θ}) = {\tilde{I}}_{2} \ddot{v} + {\tilde{I}}_{4} ϕ_{θ} - {\tilde{I}}_{5} \frac{1}{R} \frac{\partial \ddot{w}}{\partial θ}

(31)

where the inertias ${\bar{I}}_{i}$ and ${\tilde{I}}_{i}$ （i = 1， 2， 3， 4， 5） are defined by

{\bar{I}}_{1} = I_{1}, {\tilde{I}}_{1} = I_{1} + \frac{2}{R} I_{2}

(32)

{\bar{I}}_{2} = I_{2} - c_{1} I_{4}, {\tilde{I}}_{2} = I_{2} + \frac{1}{R} I_{3} - c_{1} I_{4} - \frac{c_{1}}{R} I_{5}

(33)

{\bar{I}}_{3} = c_{1} I_{4}, {\tilde{I}}_{3} = c_{1} I_{4} + \frac{c_{1}}{R} I_{5}

(34)

{\bar{I}}_{4} = I_{3} - 2 c_{1} I_{5} + c_{1}^{2} I_{7}, {\tilde{I}}_{4} = I_{3} - 2 c_{1} I_{5} + c_{1}^{2} I_{7}

(35)

{\bar{I}}_{5} = c_{1} I_{5} - c_{1}^{2} I_{7}, {\tilde{I}}_{5} = c_{1} I_{5} - c_{1}^{2} I_{7}

(36)

(I_{1}, I_{2}, I_{3}, I_{4}, I_{5}, I_{7}) = \sum_{j = 1}^{3} \int_{z_{j - 1}}^{z_{j}} ρ^{(j)} (z) (1, z, z^{2}, z^{3}, z^{4}, z^{6}) d z

(37)

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

L_{1} (u_{0}, v_{0}, w_{0}, ϕ_{x}, ϕ_{θ}) = {\bar{I}}_{1} {\ddot{u}}_{0} + {\bar{I}}_{2} {\ddot{ϕ}}_{x} - {\bar{I}}_{3} \frac{\partial {\ddot{w}}_{0}}{\partial x}

(38)

L_{2} (u_{0}, v_{0}, w_{0}, ϕ_{x}, ϕ_{θ}) = {\tilde{I}}_{1} {\ddot{v}}_{0} + {\tilde{I}}_{2} {\ddot{ϕ}}_{θ} - {\tilde{I}}_{3} \frac{1}{R} \frac{\partial {\ddot{w}}_{0}}{\partial θ}

(39)

\begin{matrix} L_{3} (u_{0}, v_{0}, w_{0}, ϕ_{x}, ϕ_{θ}) = {\bar{I}}_{3} \frac{\partial {\ddot{u}}_{0}}{\partial x} + {\bar{I}}_{5} \frac{\partial {\ddot{ϕ}}_{x}}{\partial x} + \\ {\tilde{I}}_{3} \frac{1}{R} \frac{\partial {\ddot{v}}_{0}}{\partial θ} + {\tilde{I}}_{5} \frac{1}{R} \frac{\partial {\ddot{ϕ}}_{θ}}{\partial θ} + I_{1} {\ddot{w}}_{0} - \\ c_{1}^{2} I_{7} (\frac{\partial^{2} {\ddot{w}}_{0}}{\partial x^{2}} + \frac{1}{R^{2}} \frac{\partial^{2} {\ddot{w}}_{0}}{\partial θ^{2}}) + P \end{matrix}

(40)

L_{4} (u_{0}, v_{0}, w_{0}, ϕ_{x}, ϕ_{θ}) = {\bar{I}}_{2} {\ddot{u}}_{0} + {\bar{I}}_{4} {\ddot{ϕ}}_{x} - {\bar{I}}_{5} \frac{\partial {\ddot{w}}_{0}}{\partial x}

(41)

L_{5} (u_{0}, v_{0}, w_{0}, ϕ_{x}, ϕ_{θ}) = {\tilde{I}}_{2} {\ddot{v}}_{0} + {\tilde{I}}_{4} ϕ_{θ} - {\tilde{I}}_{5} \frac{1}{R} \frac{\partial {\ddot{w}}_{0}}{\partial θ}

(42)

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］

\{\begin{array}{l} ϕ_{θ} (0, θ, t) = ϕ_{θ} (L, θ, t) = 0 \\ N_{x} (L, θ, t) = N_{x} (0, θ, t) = \\ M_{x} (L, θ, t) = M_{x} (0, θ, t) = 0 \\ v_{0} (L, θ, t) = v_{0} (0, θ, t) = \\ w_{0} (0, θ, t) = w_{0} (L, θ, t) = 0 \end{array}

(43)

u_{0} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} u_{m n} (t) c o s (λ_{m} x) c o s (n θ)

(44)

v_{0} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} v_{m n} (t) s i n (λ_{m} x) s i n (n θ)

(45)

w_{0} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} w_{m n} (t) s i n (λ_{m} x) c o s (n θ)

(46)

ϕ_{x} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} ϕ_{m n} (t) c o s (λ_{m} x) c o s (n θ)

(47)

ϕ_{θ} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} {\bar{ϕ}}_{m n} (t) s i n (λ_{m} x) s i n (n θ)

(48)

where $λ_{m} = m π / L$ ； $u_{m n} (t)$ ， $v_{m n} (t)$ ， $w_{m n} (t)$ ， $ϕ_{m n} (t)$ and ${\bar{ϕ}}_{m n} (t)$ represent generalized coordinates.

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

\begin{array}{l} M_{11} {\ddot{u}}_{m n} (t) + M_{13} {\ddot{w}}_{m n} (t) + M_{14} {\ddot{ϕ}}_{m n} (t) + \\ K_{11} u_{m n} (t) + K_{12} v_{m n} (t) + K_{13} w_{m n} (t) + \\ K_{14} ϕ_{m n} (t) + K_{15} {\bar{ϕ}}_{m n} (t) = 0 \end{array}

(49)

\begin{array}{l} M_{22} {\ddot{v}}_{m n} (t) + M_{23} {\ddot{w}}_{m n} (t) + M_{25} {\ddot{\bar{ϕ}}}_{m n} (t) + \\ K_{21} u_{m n} (t) + K_{22} v_{m n} (t) + K_{23} w_{m n} (t) + \\ K_{24} ϕ_{m n} (t) + K_{25} {\bar{ϕ}}_{m n} (t) = 0 \end{array}

(50)

\begin{array}{l} M_{31} {\ddot{u}}_{m n} (t) + M_{32} {\ddot{v}}_{m n} (t) + M_{33} {\ddot{w}}_{m n} (t) + \\ M_{34} {\ddot{ϕ}}_{m n} (t) + M_{35} {\ddot{\bar{ϕ}}}_{m n} (t) + K_{31} u_{m n} (t) + \\ \frac{8 ρ_{f} m U}{L} \sum_{i = 1}^{\bar{m}} \frac{i}{m^{2} - i^{2}} {\dot{w}}_{i n} (t) + K_{32} v_{m n} (t) + \\ K_{33} w_{m n} (t) + K_{34} ϕ_{m n} (t) + K_{35} {\bar{ϕ}}_{m n} (t) = 0 \end{array}

(51)

\begin{array}{l} M_{41} {\ddot{u}}_{m n} (t) + M_{43} {\ddot{w}}_{m n} (t) + M_{44} {\ddot{ϕ}}_{m n} (t) + \\ K_{41} u_{m n} (t) + K_{42} v_{m n} (t) + K_{43} w_{m n} (t) + \\ K_{44} ϕ_{m n} (t) + K_{45} {\bar{ϕ}}_{m n} (t) = 0 \end{array}

(52)

\begin{array}{l} M_{52} {\ddot{v}}_{m n} (t) + M_{53} {\ddot{w}}_{m n} (t) + M_{55} {\ddot{\bar{ϕ}}}_{m n} (t) + \\ K_{51} u_{m n} (t) + K_{52} v_{m n} (t) + K_{53} w_{m n} (t) + \\ K_{54} ϕ_{m n} (t) + K_{55} {\bar{ϕ}}_{m n} (t) = 0 \end{array}

(53)

where m = 1， 2， …， $\bar{m}$ ； i = 1， 2， …， $\bar{m}$ ； n = 1， 2， …， N； m ≠ i and m ± i are odd numbers. M_ijand K_ij are integral coefficients.

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

M \ddot{X} + C \dot{X} + K X = 0

(54)

where M， C， K denote the mass， damping and stiffness matrices， respectively； X is a 5× $\bar{m}$ ×N column vector consisting of $u_{m n} (t)$ ， $v_{m n} (t)$ ， $w_{m n} (t)$ ， $ϕ_{m n} (t)$ and ${\bar{ϕ}}_{m n} (t)$ .

Eq.（54） is solved in the state space by setting $X = e^{Λ t} q$ ， which gives the following eigenvalue equation

Λ \{\begin{matrix} q \\ Λ q \end{matrix}\} = [\begin{matrix} 0 & I \\ - M^{- 1} K & - M^{- 1} C \end{matrix}] \{\begin{matrix} q \\ Λ q \end{matrix}\}

(55)

where ${\{\begin{matrix} q & Λ q \end{matrix}\}}^{T}$ is the state vector. It should be noted that eigenvalue $Λ$ is a non-zero complex number. The imaginary part of $Λ$ is the frequency and the real part is the damping.

2 Numerical Results and Discussion

The natural frequency is obtained by finding the eigenvalues of the matrix $[\begin{matrix} 0 & I \\ - M^{- 1} K & - M^{- 1} C \end{matrix}]$ .In order to demonstrate the accuracy of the present analysis， a comparison investigation related to a liquid-filled homogenous cylindrical shell is carried out. The used parameters are： h/R = 0.01， ρ_f= 1 000 kg/m³， iron density ρ = 7 850 kg/m³， L/R = 2 and ν = 0.3. For convenience， the non-dimensional axial flow velocity V is defined as $V = U / {(π / L^{2}) [D / {(ρ h)]}^{1 / 2}}$ with $D = E h^{3} / [12 (1 - ν^{2})]$ ； the non-dimensional eigenvalue Ω is introduced as $Ω = Λ / {(π^{2} / L^{2}) [D / {(ρ h)]}^{1 / 2}}$ . As can be seen from Fig.2， the result obtained from the current analysis is in good agreement with Amabili et al.^［

47］. The small difference between them is because the rotational inertia terms

I_{2} {\ddot{ϕ}}_{x}

and

I_{2} {\ddot{ϕ}}_{θ}

were neglected by Amabili et al.^{［Reference 47

Baidu Scholar}47］. It is worth mentioning that the real part represents the natural frequency in Ref.［47］， which is caused by the use of different solutions.

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

Next， we investigate the stability and free vibration of FGM sandwich shells conveying fluid. The fluid is considered as crude oil， with a mass density ρ_f= 0.81 g/cm^3［

48］. Here， the ceramic and metal forming the FGM sandwich shell shown in Fig.1 are considered as Zirconia and Aluminum， respectively. Their properties are^{［Reference 49

Baidu Scholar}49］

Aluminum： E_m = 70 GPa， ν_m = 0.3， ρ_m = 2 707 kg/m³

Zirconia： E_c = 151 GPa， ν_c = 0.3， ρ_c = 3 000 kg/m³

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

Fig.3 Non-dimensional natural frequency versus circumferential 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 natural frequencies versus the non-dimensional flow velocity of FGM sandwich shell conveying fluid， where $\bar{m}$ = 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 instability， the first frequency increases and then coincides with the second frequency， and the system recovers its stability. Moreover， when the flow velocity 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 frequencies versus non-dimensional flow velocity of FGM sandwich shell conveying fluid (n = 5, L/R = 2.0, $\bar{m}$ = 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 fluid-conveying FGM sandwich shell with different power-law exponents. It is shown that as the power-law exponent increases， the first non-dimensional natural frequency shows an increasing trend. But the effect 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 ratio L/R increases， the first non-dimensional natural frequency 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-conveying FGM sandwich shell with different power-law exponents (n = 5, V = 1, $\bar{m}$ = 2, R/h = 80, h₂/h = 0.2)

Fig.6 illustrates the first non-dimensional natural frequency versus the radius-to-thickness ratio of fluid-conveying FGM sandwich shell with different power-law exponents. It is shown that as radius-to-thickness ratio R/h increases， the first non-dimensional natural frequency of the fluid-conveying FGM sandwich shell decreases.

Fig.6 The first non-dimensional natural frequency versus radius-to-thickness ratio R/h of fluid-conveying FGM sandwich shell with different power-law exponents (n = 5, L/R = 2.0, $\bar{m}$ = 2, V = 1, h₂/h = 0.2)

Fig.7 presents the first non-dimensional natural frequency versus the mass density ρ_f of fluid-conveying FGM sandwich shell with different power-law exponents. The first non-dimensional natural frequency of the fluid-conveying FGM sandwich shell decreases with increasing fluid mass density. This is reasonable because the FGM sandwich shell vibrates 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 ρ_f of fluid-conveying FGM sandwich shell with different power-law exponents (n = 5, L/R = 2.0, $\bar{m}$ = 2, V = 1, h₂/h = 0.2, R/h = 80)

Fig.8 plots the first non-dimensional natural frequency versus the core-to-thickness ratio h₂/h of fluid-conveying FGM sandwich shell with different power-law exponents. It is found that the first non-dimensional natural frequency increases gradually as the core-to-thickness ratio h₂/h increases. This can be understood because the increase of core-to-thickness 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 exponents (n = 5, L/R = 2.0, $\bar{m}$ = 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 frequencies of the fluid-conveying FGM sandwich shells decrease at first. When reaching the critical velocity， the first frequency vanishes and the system loses its stability. Moreover， 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 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 exponent. The fluid viscosity has insignificant effect on the first non-dimensional natural frequency of the fluid-conveying FGM sandwich shells.

Contributions Statement

Mr.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.

Acknowledgements

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

Conflict of Interest

The authors declare no competing interests.

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