Propagation problem of a Rayleigh surface wave in the half-space of a Cosserat medium in the case of homogeneous and elastically constrained boundary condition

The problem of propagation of a Rayleigh surface wave in an infinite half-space is studied within the framework of the micropolar theory of elasticity. It is assumed that the deformed state of the medium is described by independent vectors of displacement and rotation (a Cosserat medium). A general solution describing the propagation of a surface Rayleigh wave is obtained. Using the method of constructing majorants, it is shown that there are no surface Rayleigh waves when boundary conditions are specified on the surface corresponding to the main problems of the classical theory of elasticity: “rigid embedding”, “sliding embedding”, and “rigid mesh”. For the cases of boundary conditions “free surface” and “elastic constraint,” corresponding to the problems of the classical theory of elasticity, it is shown by the method of constructing majorants that there is a surface Rayleigh wave when moment stresses are zero on the surface, while the phase velocity of the wave tends to a finite limit at high wave frequencies; when the rotation vector is equal to zero on the surface, sufficient conditions are found for the parameters of the Cosserat medium for the existence of surface Rayleigh waves, while the phase velocity of the wave tends to a finite limit at high wave frequencies. A qualitative analysis of the obtained dispersion relations showed that the Rayleigh surface wave has dispersion; the elastic constraint leads to the absence of a surface wave at low frequencies. In the case of a micropolar medium made of polyurethane foam, numerical values of the parameters of the wave and deformation of the medium are constructed. The attenuation of the displacement vector with depth in the micropolar theory of elasticity is slower than the attenuation in the classical theory of elasticity. A significant difference in the values of the displacement vector in the classical and micropolar environments is observed in the direction of elastic constraint.