Forced Vibrations of a System Consisting of a Pre-strained Highly Elastic Plate under Compressible Viscous Fluid Loading

Abstract

The forced vibration of the system consisting of the pre-stretched plate made of highly-elastic material and half-plane filled by barotropic compressible Newtonian viscous fluid is considered. It is assumed that this forced vibration is caused by the lineal located time-harmonic force acting on the free face plane of the plate. The motion of the pre-stretched plate is written by utilizing of the linearized exact equations of the theory of elastic waves in the initially stressed bodies, but the motion of the compressible viscous fluid is described by the linearized Navier-Stokes equations. The elastic relations of the plate material are described with the use of the harmonic potential. Moreover, it is assumed that the velocities and stresses of the constituents are continuous on the contact plane between the plate and fluid. The dimensionless parameters which characterize the compressibility, viscosity of the fluid and elastic constants of the plate material are introduced. The plane strain state in the plate is considered and the corresponding boundary- and contact-value problem is solved by employing exponential Fourier transformation with respect to the coordinate directed along the interface line and the inverse of this transformation is determined numerically by employing the Sommerfeld contour. Numerical results on the interface stresses and velocities and the influence of the problem parameters such as initial strains and thickness of the plate, the compressibility and viscosity of the plate, as well the magnitude of the frequency of the external forces on these results are presented and discussed. Numerical results are examined in the case where the fluid is Glycerin and the values of the elastic constants which enter into the mentioned above harmonic potential and the density of the plate material are taken as Lame's constants and density of the Plexiglass (Lucite).