# 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

- Sheng, G.; Wang, X. Thermomechanical vibration analysis of a functionally graded shell with flowing fluid. Eur. J. Mech. A Solids
**2008**, 27, 1075–1087. [Google Scholar] [CrossRef] - Bagherizadeh, E.; Kiani, Y.; Eslami, M. Thermal buckling of functionally graded material cylindrical shells on elastic foundation. AIAA J.
**2012**, 50, 500–503. [Google Scholar] [CrossRef] - Sofiyev, A.; Kuruoglu, N. Torsional vibration and buckling of the cylindrical shell with functionally graded coatings surrounded by an elastic medium. Compos. Part B Eng.
**2013**, 45, 1133–1142. [Google Scholar] [CrossRef] - Tornabene, F.; Fantuzzi, N.; Bacciocchi, M.; Dimitri, R. Dynamic analysis of thick and thin elliptic shell structures made of laminated composite materials. Compos. Struct.
**2015**, 133, 278–299. [Google Scholar] [CrossRef] - Tornabene, F.; Fantuzzi, N.; Bacciocchi, M. The GDQ method for the free vibration analysis of arbitrarily shaped laminated composite shells using a NURBS-based isogeometric approach. Compos. Struct.
**2016**, 154, 190–218. [Google Scholar] [CrossRef] - Civalek, Ö. Discrete singular convolution method for the free vibration analysis of rotating shells with different material properties. Compos. Struct.
**2017**, 160, 267–279. [Google Scholar] [CrossRef] - Wang, Y.; Wu, D. Free vibration of functionally graded porous cylindrical shell using a sinusoidal shear deformation theory. Aerosp. Sci. Technol.
**2017**, 66, 83–91. [Google Scholar] [CrossRef] - Pourasghar, A.; Moradi-Dastjerdi, R.; Yas, M.; Ghorbanpour Arani, A.; Kamarian, S. Three-dimensional analysis of carbon nanotube-reinforced cylindrical shells with temperature-dependent properties under thermal environment. Polym. Compos.
**2018**, 39, 1161–1171. [Google Scholar] [CrossRef] - Vuong, P.M.; Duc, N.D. Nonlinear vibration of FGM moderately thick toroidal shell segment within the framework of Reddy’s third order-shear deformation shell theory. Int. J. Mech. Mater. Des.
**2019**, 16, 1–20. [Google Scholar] [CrossRef] - Ghasemi, A.R.; Mohandes, M.; Dimitri, R.; Tornabene, F. Agglomeration effects on the vibrations of CNTs/fiber/polymer/metal hybrid laminates cylindrical shell. Compos. Part B Eng.
**2019**, 167, 700–716. [Google Scholar] [CrossRef] - Ebrahimi, F.; Dabbagh, A.; Rastgoo, A. Vibration analysis of porous metal foam shells rested on an elastic substrate. J. Strain Anal. Eng. Des.
**2019**, 54, 199–208. [Google Scholar] [CrossRef] - Karimiasl, M.; Ebrahimi, F.; Mahesh, V. Nonlinear forced vibration of smart multiscale sandwich composite doubly curved porous shell. Thin Walled Struct.
**2019**, 143, 106152. [Google Scholar] [CrossRef] - Ebrahimi, F.; Hafezi, P.; Dabbagh, A. Buckling analysis of embedded graphene oxide powder-reinforced nanocomposite shells. Available online: https://doi.org/10.1016/j.dt.2020.02.010 (accessed on 2 January 2020).
- Allahkarami, F.; Tohidi, H.; Dimitri, R.; Tornabene, F. Dynamic Stability of Bi-Directional Functionally Graded Porous Cylindrical Shells Embedded in an Elastic Foundation. Appl. Sci.
**2020**, 10, 1345. [Google Scholar] [CrossRef] [Green Version] - Kögl, M. Free vibration analysis of anisotropic solids with the boundary element method. Eng. Anal. Bound. Elem.
**2003**, 27, 107–114. [Google Scholar] [CrossRef] - Towfighi, S.; Kundu, T. Elastic wave propagation in anisotropic spherical curved plates. Int. J. Solids Struct.
**2003**, 40, 5495–5510. [Google Scholar] [CrossRef] - Batra, R.; Qian, L.; Chen, L. Natural frequencies of thick square plates made of orthotropic, trigonal, monoclinic, hexagonal and triclinic materials. J. Sound Vib.
**2004**, 270, 1074–1086. [Google Scholar] [CrossRef] - Demasi, L. Quasi-3D analysis of free vibration of anisotropic plates. Compos. Struct.
**2006**, 74, 449–457. [Google Scholar] [CrossRef] - Lü, C.; Huang, Z.; Chen, W. Semi-analytical solutions for free vibration of anisotropic laminated plates in cylindrical bending. J. Sound Vib.
**2007**, 304, 987–995. [Google Scholar] [CrossRef] - Jansen, E. The effect of geometric imperfections on the vibrations of anisotropic cylindrical shells. Thin Walled Struct.
**2007**, 45, 274–282. [Google Scholar] [CrossRef] - Ferreira, A.; Fasshauer, G.; Batra, R. Natural frequencies of thick plates made of orthotropic, monoclinic, and hexagonal materials by a meshless method. J. Sound Vib.
**2009**, 319, 984–992. [Google Scholar] [CrossRef] - Paiva, W.P.; Sollero, P.; Albuquerque, E.L. Modal analysis of anisotropic plates using the boundary element method. Eng. Anal. Bound. Elem.
**2011**, 35, 1248–1255. [Google Scholar] [CrossRef] - Tornabene, F. 2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution. Compos. Struct.
**2011**, 93, 1854–1876. [Google Scholar] [CrossRef] - Singhal, P.; Bindal, G. Generalised differential quadrature method in the study of free vibration analysis of monoclinic rectangular plates. Am. J. Comput. Appl. Math.
**2012**, 2, 166–173. [Google Scholar] [CrossRef] - Shen, H.-S. Boundary layer theory for the nonlinear vibration of anisotropic laminated cylindrical shells. Compos. Struct.
**2013**, 97, 338–352. [Google Scholar] [CrossRef] - Kumar, Y. Differential transform method to study free transverse vibration of monoclinic rectangular plates resting on Winkler foundation. Appl. Comput. Mech.
**2013**, 7, 145–154. [Google Scholar] - Mirzaei, M.; Asadi, M.T.; Akbari, R. On vibrational behavior of pulse detonation engine tubes. Aerosp. Sci. Technol.
**2015**, 47, 177–190. [Google Scholar] [CrossRef] - Ahmadi, H.; Rasheed, H.A. Lateral torsional buckling of anisotropic laminated thin-walled simply supported beams subjected to mid-span concentrated load. Compos. Struct.
**2018**, 185, 348–361. [Google Scholar] [CrossRef] - Bahrami, K.; Afsari, A.; Janghorban, M.; Karami, B. Static analysis of monoclinic plates via a three-dimensional model using differential quadrature method. Struct. Eng. Mech.
**2019**, 72, 131–139. [Google Scholar] - Heyliger, P.R.; Asiri, A. A total Lagrangian elasticity formulation for the nonlinear free vibration of anisotropic beams. Int. J. Non Linear Mech.
**2020**, 118, 103286. [Google Scholar] [CrossRef] - Malekan, M.; Khosravi, A.; Zanin, A.; Aghababaei, R. On the vibrational responses of thin FGM tubes subjected to internal sequential moving pressure. J. Braz. Soc. Mech. Sci. Eng.
**2020**, 42, 220. [Google Scholar] [CrossRef] - Akbaş, Ş.D. Wave propagation of a functionally graded beam in thermal environments. Steel Compos. Struct.
**2015**, 19, 1421–1447. [Google Scholar] [CrossRef] - Dorduncu, M.; Apalak, M.K.; Cherukuri, H. Elastic wave propagation in functionally graded circular cylinders. Compos. Part B Eng.
**2015**, 73, 35–48. [Google Scholar] [CrossRef] - Janghorban, M.; Nami, M.R. Wave propagation in functionally graded nanocomposites reinforced with carbon nanotubes based on second-order shear deformation theory. Mech. Adv. Mater. Struct.
**2017**, 24, 458–468. [Google Scholar] [CrossRef] - Fourn, H.; Atmane, H.A.; Bourada, M.; Bousahla, A.A.; Tounsi, A.; Mahmoud, S. A novel four variable refined plate theory for wave propagation in functionally graded material plates. Steel Compos. Struct.
**2018**, 27, 109–122. [Google Scholar] - Gul, U.; Aydogdu, M. Wave propagation analysis in beams using shear deformable beam theories considering second spectrum. J. Mech.
**2018**, 34, 279–289. [Google Scholar] [CrossRef] - Ebrahimi, F.; Seyfi, A.; Dabbagh, A.; Tornabene, F. Wave dispersion characteristics of porous graphene platelet-reinforced composite shells. Struct. Eng. Mech.
**2019**, 71, 99–107. [Google Scholar] - Bouanati, S.; Benrahou, K.H.; Atmane, H.A.; Yahia, S.A.; Bernard, F.; Tounsi, A.; Bedia, E.A.A. Investigation of wave propagation in anisotropic plates via quasi 3D HSDT. Geomech. Eng.
**2019**, 18, 85–96. [Google Scholar] - Dravinski, M.; Niu, Y. Three-dimensional time-harmonic Green’s functions for a triclinic full-space using a symbolic computation system. Int. J. Numer. Methods Eng.
**2002**, 53, 445–472. [Google Scholar] [CrossRef] - Ebrahimi, F.; Seyfi, A. Wave propagation response of multi-scale hybrid nanocomposite shell by considering aggregation effect of CNTs. Mech. Based Des. Struct. Mach.
**2019**, 47, 1–22. [Google Scholar] [CrossRef] - Rabani Bidgoli, M.; Saeed Karimi, M.; Ghorbanpour Arani, A. Nonlinear vibration and instability analysis of functionally graded CNT-reinforced cylindrical shells conveying viscous fluid resting on orthotropic Pasternak medium. Mech. Adv. Mater. Struct.
**2016**, 23, 819–831. [Google Scholar] [CrossRef] - Ke, L.; Wang, Y.; Reddy, J. Thermo-electro-mechanical vibration of size-dependent piezoelectric cylindrical nanoshells under various boundary conditions. Compos. Struct.
**2014**, 116, 626–636. [Google Scholar] [CrossRef] - Barati, M.R.; Zenkour, A.M. Vibration analysis of functionally graded graphene platelet reinforced cylindrical shells with different porosity distributions. Mech. Adv. Mater. Struct.
**2019**, 26, 1580–1588. [Google Scholar] [CrossRef]

**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