Modeling the evolution of solitary strain waves in two coaxial shells of incompressible material with combined nonlinearity, containing a viscous fluid between them and in the inner shell

Аuthors

Popova E. V.

Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia

e-mail: elizaveta.popova.97@bk.ru

Abstract

The article deals with the hydroelasticity problem formulation for the two coaxial cylindrical shells of the Kirchhoff-Love type, containing a viscous incompressible fluid in the annular gap and in the inner shell. The material of the shells is considered as incompressible and of a nonlinear law of stress-strain relationship and strain intensity. The shell dynamics equations are obtained for the case when this law has a tough combined nonlinearity in the form of a function with a fractional exponent and a quadratic function. The viscous fluid dynamics are being considered in the framework of the hydrodynamic theory of lubrication, i.e., the fluid motion is assumed to be creeping. The asymptotic analysis of the formulated problem of hydroelasticity is performed applying the two-scale perturbation method. As the result, a system of two evolution equations for modeling the onlinear longitudinal strain waves propagation in shells is obtained. It demonstrated that in the case of incompressible shell material, the presence of viscous fluid in the inner shell does not affect the wave process. The equations of the system represent the Korteweg-de Vries-Schamel equations. The exact partial solution for the obtained system of evolution equations in the form of a solitary wave with an arbitrary wave number is found for the case when that wave propagates in each of the shells. The new difference scheme for the nonlinear system of two generalized Korteweg-de Vries-Schamel equations based on the application of the Gröbner basis technique is derived for numerical simulations. Computational experiments have been perforomed to study the evolution of solitary longitudinal strain waves excited in shells. Numerical modeling has revealed that solitary nonlinear strain waves in the shells are supersonic solitons, as well as the presence of the energy transfer from one shell to another due to the fluid viscosity between them.

Keywords:

mathematical modeling, nonlinear strain waves, coaxial shells, viscous fluid, incompressible material, combined nonlinearity, computational experiment

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