Strain Waves in Two Physically Nonlinear Fluid-Filled Coaxial Shells Featuring Structural Damping and Interacting with the Surrounding Medium

Abstract

The paper investigates longitudinal strain waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid situated both between the shells and within the inner shell. The study takes into account the effects that the following factors have upon the magnitude and velocity of the wave: viscosity, fluid motion inertia, the medium surrounding the outer shell, and structural damping in the shell material. As it is impossible to evaluate the strain wave models proposed via qualitative analysis methods, using numerical methods is necessary. We numerically investigated the model presented using a specific difference scheme to represent its equations, one similar to the Crank --- Nicolson scheme for the thermal conductivity equation. We show that the velocities and amplitudes of strain waves in the shells do not change when disregarding the following factors: the fluid inside the shell, structural damping in the shell material, and the influence of the surrounding elastic medium. The waves evolve in the positive direction of the abscissa axis. At the same time, taking nonlinearity into account leads to increased wave velocity as compared to the linear case, when the wave propagation velocity equals the speed of sound, which means the waves become supersonic. We performed a numerical study of this case and obtained results matching the exact solution. If we take into account the effect that the fluid motion inertia within the inner shell has upon the wave velocity, then a decrease in the strain wave velocity is observed. Taking into account the presence of an elastic medium surrounding the outer shell leads to an increase in velocity. Considering the viscous properties of the fluid within the inner shell and the damping properties of the shell material results in decreased strain wave magnitudes. The models proposed can form the basis for designing modern non-destructive testing devices