# Waves in Two Coaxial Elastic Cubically Nonlinear Shells with Structural Damping and Viscous Fluid Between Them

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Defining and Resolving Relations of the Physically Nonlinear Theory of Shells

#### 2.2. The Asymptotic Method for Studying the Equations of Shells with a Fluid

#### 2.3. The Study of the Stresses Acting on the Shell from the Side of the Fluid Inside

#### 2.4. The Coaxial Shells Dynamics Equations

#### 2.5. Computing Experiment

## 3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MKdV | modified Korteweg–de Vries equation |

## References

- Zemlyanukhin, A.I.; Mogilevich, L.I. Nonlinear Waves in Inhomogeneous Cylindrical Shells: A New Evolution Equation. Acoust. Phys.
**2001**, 47, 303–307. [Google Scholar] [CrossRef] - Andrejchenko, K.P.; Mogilevich, L.I. On the dynamics of interaction between a compressible layer of a viscous incompressible fluid and elastic walls. Proc. USSR Acad. Sci. Mech. Solid Body
**1982**, 2, 162–172. [Google Scholar] - Amabili, M.; Pellicano, F.; Paidoussis, M.P. Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid. Part IV: Large-amplitude vibrations with flow. J. Sound Vib.
**2000**, 237, 641–666. [Google Scholar] [CrossRef] - Amabili, M.; Pellicano, F.; Paidoussis, M.P. Nonlinear stability of circular cylindrical shells in annular and unbounded axial flow. Trans. ASME J. Appl. Mech.
**2001**, 68, 827–834. [Google Scholar] [CrossRef] - Amabili, M.; Paidoussis, M.P. Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Appl. Mech. Rev.
**2003**, 56, 349–381. [Google Scholar] [CrossRef] - Amabili, M.; Karagiozis, K.; Paidoussis, M.P. Effect of geometric imperfections on nonlinear stability of circular cylindrical shells conveying fluid. Int. J. Non-Linear Mech.
**2009**, 44, 276–289. [Google Scholar] [CrossRef] - Samarskii, A.A. The Theory of Difference Schemes; Marcel Dekker: New York, NY, USA, 2001. [Google Scholar]
- Zemlyanukhin, A.I.; Andrianov, I.V.; Bochkarev, A.V.; Mogilevich, L.I. The generalized Schamel equation in nonlinear wave dynamics of cylindrical shells. Nonlinear Dyn.
**2019**, 98. [Google Scholar] [CrossRef] - Gerdt, V.P.; Blinkov, Y.A. Involution and difference schemes for the Navier-Stokes equations. In Computer Algebra in Scientific Computing; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2009; Volume 5743, pp. 94–105. [Google Scholar]
- Ovcharov, A.A.; Brylev, I.S. Matematicheskaya model’ deformirovaniya nelinejno uprugih podkreplennyh konicheskih obolochek pri dinamicheskom nagruzhenii [Mathematical model of non-linearly deformed elastic reinforced conical shells under dynamic loading]. Sovrem. Probl. Nauk. Obraz.
**2014**, 3. Available online: http://www.science-education.ru/ru/article/viewid=13235 (accessed on 20 February 2020). - Kauderer, H. Nichtlineare Mechanik; Springer: Berlin/Heidelberg, Germany; Göttingen, Germany, 1958. [Google Scholar]
- Fel’dshtejn, V.A. Uprugo Plasticheskie Deformacii Cilindricheskoj Obolochki pri Prodol’nom Udare [Elastic Plastic Deformations of a Cylindrical Shell with a Longitudinal Impact]; Volny v Neuprugih Sredah: Kishinev, Moldova, 1970; pp. 199–204. [Google Scholar]
- Vlasov, V.Z.; Leontiev, N.N. Beams, Plates, and Shells on Elastic Foundation; Israel Program for Scientific Translations: Jerusalem, Israel, 1966. [Google Scholar]
- Mikhasev, G.I.; Sheyko, A.N. O Vliyanii Parametra Uprugoy Nelokal’nosti na Sobstvennyye Chastoty Kolebaniy Uglerodnoy Nanotrubki v Uprugoy Srede [On the Effect of the Elastic Nonlocality Parameter on the Natural Frequencies of a Carbon Nanotube in an Elastic Medium]; Belarusian State Technological University: Minsk, Belarus, 2012; Volume 6, pp. 41–44. [Google Scholar]
- Loytsiansky, L.G. Mechanics of Liquid and Gas; Pergamon Press: Oxford, UK, 1966. [Google Scholar]

**Figure 2.**Checking the adequacy of the difference scheme and the system of resolving equations. Start condition (36) $t=0$, $k=0.2$, ${\sigma}_{0}=0$, ${\sigma}_{1}=0.1$, ${\sigma}_{2}=0$.

**Figure 3.**Start condition (37) $t=0$, $k=0.2$, ${\sigma}_{0}=1$, ${\sigma}_{1}=0.1$, ${\sigma}_{2}=0$.

**Figure 4.**Start condition (37) $t=0$, $k=0.2$, ${\sigma}_{0}=1$, ${\sigma}_{1}=0.1$, ${\sigma}_{2}=0.3$.

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**MDPI and ACS Style**

Mogilevich, L.; Ivanov, S.
Waves in Two Coaxial Elastic Cubically Nonlinear Shells with Structural Damping and Viscous Fluid Between Them. *Symmetry* **2020**, *12*, 335.
https://doi.org/10.3390/sym12030335

**AMA Style**

Mogilevich L, Ivanov S.
Waves in Two Coaxial Elastic Cubically Nonlinear Shells with Structural Damping and Viscous Fluid Between Them. *Symmetry*. 2020; 12(3):335.
https://doi.org/10.3390/sym12030335

**Chicago/Turabian Style**

Mogilevich, Lev, and Sergey Ivanov.
2020. "Waves in Two Coaxial Elastic Cubically Nonlinear Shells with Structural Damping and Viscous Fluid Between Them" *Symmetry* 12, no. 3: 335.
https://doi.org/10.3390/sym12030335