Geodesic
Waves of strain in two coaxial cubically nonlinear cylindrical shells with a viscous fluid between them
Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 28 (2020) no. 4, pp. 435-454.

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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.
Keywords: partical directed coherence, coupling, nonlinearity, nonlinear systems, nonlinear coupling, high–dimensional system. 
@article{IVP_2020_28_4_a8,
     author = {L. I. Mogilevich and Yu. A. Blinkov and S. V. Ivanov},
     title = {Waves of strain in two coaxial cubically nonlinear cylindrical shells with a viscous fluid between them},
     journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
     pages = {435--454},
     publisher = {mathdoc},
     volume = {28},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVP_2020_28_4_a8/}
}
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L. I. Mogilevich; Yu. A. Blinkov; S. V. Ivanov. Waves of strain in two coaxial cubically nonlinear cylindrical shells with a viscous fluid between them. Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 28 (2020) no. 4, pp. 435-454. http://geodesic.mathdoc.fr/item/IVP_2020_28_4_a8/