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

Lev Mogilevich,Sergey Ivanov

doi:10.3390/sym12030335

Lev Mogilevich, Sergey Ivanov

Open Access

https://doi.org/10.3390/sym12030335

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Abstract

This article investigates longitudinal deformation waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid between them. The presence of a viscous incompressible fluid between the shells, as well as the influence of the inertia of the fluid motion on the amplitude and velocity of the wave, are taken into account. The mathematical model phenomenon is constructed by means of the method of two-scale asymptotic expansion. Structural damping in the shells and surrounding elastic media did not allow discovery of the exact solution of the problem of the deformation waves propagation. This leads to the need for numerical methods. A numerical study of the model constructed in the course of this work is carried out by using a difference scheme for the equation similar to the Crank–Nicholson scheme for the heat equation. In the absence of the structural damping and surrounding media influences, and under the similar initial conditions for both shells, the velocity and amplitude of the wave do not change. The result of the numerical experiment coincides with the exact solution, which is found in the case of the absence of the structural damping and surrounding media influences; therefore, the difference scheme is adequate to the generalized modified Korteweg–de Vries equations system. There is energy is transferred in the presence of the fluid, between the shells. The presence of inertia of the fluid motion leads to a decrease in the velocity of the deformation wave.

Highlights

The study of the wave processes in elastic shells is widely used in various technical fields.The propagation of deformation waves in elastic, viscoelastic and nonlinear viscoelastic shells was considered in [1]
Two coaxial indefinitely long elastic shells with viscous impressible fluid in the gap between them present the subject of the study
The performed numerical experiments make it possible to evaluate the effects of viscous incompressible fluid presence between the inner and outer shells and the influence of the fluid movement inertia on the behavior of the nonlinear deformation waves

Summary

Introduction

The propagation of deformation waves in elastic, viscoelastic and nonlinear viscoelastic shells was considered in [1]. The case of the interaction of shells with a viscous incompressible fluid is not considered. In [2,3,4,5,6], the interaction of the shell with a viscous incompressible fluid was considered, without taking into account the wave phenomena, and the influence of local terms of inertia was not studied, as well. Various methods are used to solve coupled and uncoupled problems of the interaction of a fluid with an elastic body. At the first stage, when solving the uncoupled problems, the interaction of a fluid with a solid is considered. It is assumed that the body deformation does not influence the fluid motion. At the second stage the obtained parameters are substituted into the equations of elastic body dynamics, Symmetry 2020, 12, 335; doi:10.3390/sym12030335 www.mdpi.com/journal/symmetry

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Discussion

Conclusion