ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Mogilevich L. I., Blinkov Y. A., Ivanov S. V. Waves of strain in two coaxial cubically nonlinear cylindrical shells with a viscous fluid between them. Izvestiya VUZ. Applied Nonlinear Dynamics, 2020, vol. 28, iss. 4, pp. 435-454. DOI: 10.18500/0869-6632-2020-28-4-435-454

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 94)
Language: 
Russian
Article type: 
Article
UDC: 
532.516:539.3:517.957

Waves of strain in two coaxial cubically nonlinear cylindrical shells with a viscous fluid between them

Autors: 
Mogilevich Lev Ilich, Saratov State University
Blinkov Yu.  A., Saratov State University
Ivanov S. V., Saratov State University
Abstract: 

Subject of the study. Longitudinal deformation waves are investigated in physically nonlinear coaxial elastic shells with a viscous incompressible fluid between its. There are taken into account effect on the amplitude and speed of inertia wave of the fluid and the environment, and as well as the damping properties of the structural materials, from which the waveguides are made. It is impossible to study the models of deformation waves using the methods of qualitative analysis in the case of filling the shell with a viscous incompressible fluid and in the presence of structural damping in the longitudinal direction. This leads to the need for numerical methods. Methods. To construct a mathematical model of the phenomenon, the asymptotic method of two-scale decompositions is used. Constructed in the course of this work model is studied numerically using a difference scheme for an equation similar to scheme of Crank–Nicholson for heat equation. Results. At the absence of structural damping in longitudinal direction, the speed and amplitude of the wave does not change. Result of the computational experiment coincides with the exact solution; therefore, the difference scheme and the resolving equations are adequate. In the presence of viscous incompressible fluid between the shells, energy is transfered between ones. Due to the environment the wave speed is increased in the outer shell. At structural damping in the normal direction the speed of the wave is decrease in the outer shell. The presence of structural damping in the longitudinal direction leads to decrease of wave amplitudes.

Reference: 
  1. Zemlyanuhin A.I., Mogilevich L.I. Nonlinear deformation waves in cylindrical shells. Izvestiya VUZ. Applied Nonlinear Dynamics, 1995, vol. 3, no. 1, pp. 52–58.

  2. Erofeev V.I., Klyueva N.V. Solitons and nonlinear periodic strain waves in rods, plates, and shells (A Review). Acoustical Physics, vol. 48, no. 6, 2002, pp. 643–655.

  3. Zemlyanuhin A.I., Mogilevich L.I. Nonlinear waves in inhomogeneous cylindrical shells: A new evolution equation. Acoustical Physics, vol. 47, no. 3, 2001, pp. 303–307.

  4. 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. Vol. 98, no. 1. DOI:10.1007/s11071-019-05181-5

  5. Nariboli G.A.. Nonlinear longitudiinal waves in elastic rods // Journal of Mathematical and Physical Sciences. 1970. Vol. 4. P. 64–73.

  6. Nariboli G.A., Sedov A. Burgers’s – Korteveg–de Vries equation for viscoelastic rods and plates // Journal of Mathematical Analysis and Applications. 1970. Vol. 32. P. 661–677.

  7. Ageev R.V., Kuznetsova E.L., Kulikov N.I., Mogilevich L.I., Popov V.S. Mathematical model of movement of a pulsing layer of viscous liquid in the channel with an elastic wall. Perm National Research Polytechnic University. Mechanics Bulletin, 2014, no. 3, pp. 17–35. DOI: 10.15593/perm.mech/2014.3.02 

  8. Lekomcev S.V. Finite-elemental algorithms for calculating the natural oscillations of threedimensional shells. Computational Continuum Mechanics, 2012, vol. 5, no. 2, pp. 233–243. 

  9. Bochkarev S.A., Matveenko V.P. Stability of coaxial cylindrical shells containing rotating fluid flow. Computational Continuum Mechanics, 2013, vol. 6, no. 1, pp. 94–102. 

  10. Vol’mir A.S. Shells in the flow of liquid and gas: Problems of hydroelasticity. Moscow, Nauka, 1979, p. 320.

  11. Andrejchenko K.P., Mogilevich L.I. On the dynamics of interaction between a compressible layer of a viscous incompressible fluid and elastic walls. Proceedings of the USSR Academy of Sciences. Mechanics of a solid body, 1982, no 2, pp. 162–172. 

  12. Blinkov Yu.A., Ivanov S.V., Mogilevich L.I. Mathematical and computer simulation of nonlinear deformation waves in the shell, containing viscous fluid. Bulletin of RUDN University. Series: Mathematics. Computer science. Physics. 2012. vol. 3. pp. 52–60.

  13. Blinkov Yu.A., Kovaleva I.A., Mogilevich L.I. Modeling dynamics of nonlinear waves in coaxial geometrically and physically nonlinear shells containing a viscous incompressible fluid between them. Bulletin of RUDN University. Series: Mathematics. Computer science. Physics., 2013, vol. 3, pp. 42–51.

  14. Blinkov Yu.A., Mesyanzhin A.V., Mogilevich L.I. Spread nonlinear waves in coaxial physically nonlinear cylindrical shells filled with viscous liquid. Bulletin of RUDN University. Series: Mathematics. Computer science. Physics. 2017. vol. 25, no. 1, pp. 19–35. DOI:10.22363/2312-9735-2017-25-1-19-35.

  15. Blinkov Yu.A., Evdokimova E.V., Mogilevich L.I., Rebrina A.Yu. Modeling wave processes in two coaxial shells, filled with a viscous fluid and surrounded by an elastic medium. Bulletin of RUDN University. Series: Mathematics. Computer science. Physics. 2018. no. 3, pp. 203–215.

  16. Samarskij A.A., Mihajlov A.P. Mathematical Modeling: Ideas. Methods Examples. 2 ed. Mosсow, Fizmatlit, 2001.

  17. Samarskii A.A. The Theory of Difference Schemes. New York, Marcel Dekker, 2001.

  18. Gorokhov V.A., Kazakov D.A., Kapustin S.A., Churilov Yu.A. Algorithms for numerical simulation of structures deformation and fracture within relations of damaged medium mechanics. Perm National Research Polytechnic University Mechanics Bulletin, 2016, no. 4, pр. 86–105. DOI:10.15593/perm.mech/2016.4.06

  19. Blinkov Yu.A., Mozzhilkin V.V. Generation of difference schemes for the burgers equation by constructing Grobner bases // Programming and Computer Software. 2006. Vol. 32, no. 2. P. 114–117.

  20. Il’yushin A.A. Continuum Mechanics. Moscow, Izd-vo MGU, 1990.

  21. Ovcharov A.A., Brylev I.S. Mathematical model of non-linearly deformed elastic reinforced conical shells under dynamic loading. Sovremennye problemy nauki i obrazovaniya, 2014, no. 3, URL: http://www.science-education.ru/ru/article/viewid=13235

  22. Kauderer H. Nihtlineare Mechanic. Berlin, Springer-verlag, 1958, 777 p. 

  23. Fel’dshtejn V.A. Elastic plastic deformations of a cylindrical shell with a longitudinal impact. Volny v Neuprugih Sredah. Kishinev, 1970, pp. 199–204.

  24. Vlasov B.Z., Leontiev N.N. Beams, Plates and Shells on an Elastic Base. Moscow, State. ed. Phys.-Math. literature, 1960, 490 p.

  25. Mikhasev G.I., Sheiko A.N.. On the effect of the elastic nonlocality parameter on the natural frequencies of a carbon nanotube in an elastic medium. Transactions of Belarusian State Technological University. Minsk: Belarusian State Technological University, 2012, vol. 153, no. 6, pp. 41–44.

Received: 
22.02.2020
Accepted: 
27.05.2020
Published: 
31.08.2020