Rayleigh Waves in the Cosserat Half-Space (Reduced Model) and Half-Space of Damaged Material

Abstract

The peculiarities of propagation of Rayleigh surface waves along the free boundary of the half-space of the Cosserat medium (reduced model), as well as along the free boundary of the half-space made of damaged material, are studied. It is shown that, in contrast to the classical Rayleigh surface wave, a wave propagating along the boundary of the Cosserat half-space has dispersion. In the plane “phase velocity – frequency” for such waves, there are two dispersion branches: lower (“acoustic”) and upper (“optical”). As the frequency increases, the phase velocity of the wave related to the lower dispersion branch decreases. The phase velocity of the wave belonging to the upper dispersion branch increases with increasing frequency. The phase velocity of the surface wave in the entire frequency range exceeds the phase velocity of the bulk shear wave. The stresses and displacements arising in the zone of propagation of a surface wave are calculated. For an isotropic elastic half-space with damage of its material, a self-consistent problem is formulated that includes the dynamic equation of elasticity theory and the kinetic equation of damage accumulation in the material of the medium. This system of equations with boundary conditions expressing the absence of stresses at the boundary of the half-space reduces to a complex dispersion equation. A surface wave propagating along the boundary of a damaged half-space decays in the direction of propagation, and low-frequency perturbations have frequency-dependent dissipation and dispersion. It is shown that the dispersion has an anomalous character. It has been established that with a decrease in the value of the damage factor, in the region of high frequencies, the value of the phase velocity increases, and the group velocity decreases. At very low frequencies, both velocities increase as the damage factor decreases.