# Propagation of Flexural Waves in Anisotropic Fluid-Conveying Cylindrical Shells

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Anisotropic Materials

_{ij}, C

_{ijkl}, and ε

_{kl}denote components of Cauchy stress, elasticity, and strain tensors, respectively. The elastic components of four anisotropic materials, namely monoclinic, triclinic, trigonal, and hexagonal materials which are utilized in the present study are presented below.

^{3}.

_{1}- and the rotated x

_{2}-axis [39], where x

_{1}and x

_{2}refer to the x-axis and y-axis in the Cartesian coordinate. The elastic components of triclinic materials which are utilized here can be defined as follows [17]:

^{3}.

^{3}.

^{3}.

## 3. First-Order Shear Deformation Shell Theory

_{x}, and θ

_{ψ}denote axial, circumferential, and lateral displacements and the rotation elements about axial and circumferential directions, respectively, and t denotes time. Therefore, the nonzero strains of a cylindrical shell can be written in the following form [40]:

_{S}, Π

_{K}, and Π

_{W}represent strain energy, kinetic energy, and work done by an external force, respectively. The variation of strain energy for an elastic solid can be written as follows:

_{f}and P stand for density and pressure of the fluid, respectively. Owing to the mutual identity between the acceleration and speed of the fluid and cylindrical shell in the contact points, the following relations can be extended as follows:

_{x}denotes the mean flow velocity. Shear stress (τ) and viscosity (μ

_{f}) relations can be written in the following form:

_{f}and μ

_{f}are taken 1100 Kg/m

^{3}and 0.25 cP.

_{r}, N

_{x}, and N

_{ψ}are radial, axial, and circumferential loadings, respectively, which have not been regarded. Therefore, to attain the motion equations of cylindrical shell, Equations (11), (12), and (16) are inserted into Equation (10) and the obtained equations can be stated as follows:

_{s}denotes shear correction factor.

_{x}and Θ

_{ψ}are the rotation amplitudes. Moreover, β

_{x}and β

_{n}represent longitudinal and circumferential wave numbers, respectively, and ω

_{n}is circular frequency. By substituting u, v, w, θ

_{x}, and θ

_{ψ}from Equation (33) in Equations (28)–(32), the following equation is obtained:

## 4. Numerical Results

_{x}= 0 and v

_{x}= 1000, k = 8 is related to v

_{x}= 2000 and v

_{x}= 3000, and k = 9 is related to v

_{x}= 4000; in monoclinic shell, k = 8 is related to v

_{x}= 0 and v

_{x}= 1000, k = 9 is related to v

_{x}= 2000 and v

_{x}= 3000, and k = 11 is related to v

_{x}= 4000; and in trigonal shell, k = 9 is related to v

_{x}= 0, v

_{x}= 1000 and v

_{x}= 2000 and k = 10 is related to v

_{x}= 3000 and v

_{x}= 4000; and finally, in hexagonal shell, k = 9 is related to all of the flow velocities. Besides, flow velocity possesses a negative effect on the variation of phase velocity values and this negative effect is owing to the aforementioned damping influence.

_{x}= 2000 and based upon these diagrams it can be expressed that at a certain flow velocity, by varying radius to thickness ratio, critical wave number changes. For more explanation, it can be said that critical flow velocity can happen at various wave number and it depends on the amount of radius to thickness ratio. Also, as same as other illustrations, triclinic, monoclinic, trigonal and hexagonal possess the lowest value of phase velocity, respectively.

## 5. Conclusions

- Wave frequency and phase velocity of anisotropic cylindrical shells can be reduced by increasing flow velocity amount;
- There is a critical flow velocity that occurs for cylindrical shells at various wave numbers and it can be different for various radius to thickness ratios and different anisotropic materials;
- Hexagonal, trigonal, monoclinic, and triclinic materials experience the highest wave frequency, respectively;
- With an increase in radius to thickness ratio there is a decreasing effect on the value of wave frequency and phase velocity of anisotropic fluid-conveying cylindrical shells.

- Conducting wave propagation analysis of anisotropic fluid-conveying truncated conical shell;
- Performing wave propagation analysis of anisotropic joined conical–conical shells;
- Analyzing the wave propagation behavior of anisotropic joined conical–cylindrical–conical shells.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Variation of wave frequency against circumferential wave number for various flow velocities for different anisotropic materials.

**Figure 3.**Variation of wave frequency against circumferential wave number for various radius to thickness ratios for different anisotropic materials (v

_{x}= 2000).

**Figure 4.**Variation of phase velocity against wave number for various flow velocities for different anisotropic materials.

**Figure 5.**Variation of phase velocity against wave number for various radius to thickness ratios for different anisotropic materials (v

_{x}= 2000).

**Figure 6.**Variation of wave frequency against longitudinal wave number for various flow velocities and radius to thickness ratios for different anisotropic materials (β

_{n}= 10).

**Table 1.**Comparison of dimensionless natural frequencies of cylindrical shells for both S–S and C–C boundary conditions ($\overline{\omega}=R\omega \sqrt{\frac{\rho \left(1-{\upsilon}^{2}\right)}{E}},\frac{h}{R}=0.01,\frac{L}{R}=20$).

Boundary conditions | n | ||||||||
---|---|---|---|---|---|---|---|---|---|

1 | Error (%) | 2 | Error (%) | 3 | Error (%) | 4 | Error (%) | ||

S–S | [42] | 0.01608 | 0.124 | 0.00938 | 0 | 0.02210 | 0.136 | 0.04209 | 0.285 |

[43] | 0.01610 | 0 | 0.00938 | 0 | 0.02210 | 0.136 | 0.04208 | 0.261 | |

Present | 0.01610 | 0.00938 | 0.02207 | 0.04197 | |||||

C–C | [42] | 0.03276 | 7.173 | 0.01389 | 3.960 | 0.02267 | 0.176 | 0.04221 | 0.261 |

[43] | 0.03278 | 7.108 | 0.01390 | 3.885 | 0.02267 | 0.176 | 0.04221 | 0.261 | |

Present | 0.03511 | 0.01444 | 0.02271 | 0.04210 |

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**MDPI and ACS Style**

Ebrahimi, F.; Seyfi, A.
Propagation of Flexural Waves in Anisotropic Fluid-Conveying Cylindrical Shells. *Symmetry* **2020**, *12*, 901.
https://doi.org/10.3390/sym12060901

**AMA Style**

Ebrahimi F, Seyfi A.
Propagation of Flexural Waves in Anisotropic Fluid-Conveying Cylindrical Shells. *Symmetry*. 2020; 12(6):901.
https://doi.org/10.3390/sym12060901

**Chicago/Turabian Style**

Ebrahimi, Farzad, and Ali Seyfi.
2020. "Propagation of Flexural Waves in Anisotropic Fluid-Conveying Cylindrical Shells" *Symmetry* 12, no. 6: 901.
https://doi.org/10.3390/sym12060901