Longitudinal waves in coaxial elastic shells with account for structural damping and with fluid inside

DOI: 10.34759/trd-2021-117-04


Blinkov Y. A.1*, Ivanov S. V.1**, Mogilevich L. I.2***, Popov V. S.2****, Popova E. V.1*****

1. Saratov State University named after N. G. Chernyshevsky, 83, Astrakhanskaya str., Saratov, 410012, Russia
2. Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia

*e-mail: blinkovua@gmail.com
**e-mail: evilgraywolf@gmail.com
***e-mail: mogilevichli@gmail.com
****e-mail: vic_p@bk.ru
*****e-mail: elizaveta.popova.97@bk.ru


This article studies longitudinal deformation waves in coaxial elastic shells with soft cubic nonlinearity, containing a viscous incompressible fluid, both between them and in the inner shell. The structural damping effect of the shell material in both longitudinal and normal directions and the environment surrounding the outer shell on the wave amplitude and speed was accounted for. The article demonstrates that this leads to the need for numerical methods application to study the nonlinear wave process. The numerical study of the model obtained in the course of this work being performed employing a difference scheme for equations similar to the Crank-Nicholson scheme. In the absence of liquid inside the shell, structural damping in the longitudinal direction as well as surrounding elastic medium, the velocity and amplitude of the waves, propagated in the shells, do not change. Computations show that the waves’ movement takes place in the introduced moving coordinates system in the negative direction of the abscissa axis. This means that the found nonlinear addition to the wave velocities in the linear approximation (the speed of sound) decreases the waves velocities and they become subsonic. The result of the computational experiment in this case coincides with the exact solution; therefore, the difference scheme and the system of generalized modified Korteweg – de Vries – Burgers (MCdV-B) equations proposed in this work are adequate. At accounting for the impact of inertia of the liquid motion in the inner shell, a velocity decrease of the deformation waves occurs, while the presence of the elastic medium surrounding the outer shell leads to their velocity increase. The liquid viscous stress in the inner shell and structural damping of the shells’ material in the longitudinal direction leads to the waves amplitudes decrease. Structural damping in the normal direction increases the wave amplitude by a constant value and decreases its velocity.


non-linear waves; elastic cylindrical shells; viscous incompressible fluid; Crank-Nicholson difference scheme


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