A 3D HIGHER-ORDER GRADIENT MODEL FOR A HALF-SPACE GEOMEDIUM

Abstract

In this paper, a structural model of a geomedium (a soil) is proposed in the form of a simple cubic lattice of spherical particles (blocks) possessing three translational and three rotational degrees of freedom. The force and couple interactions between the particles are modeled by elastic springs of five types. A long-wavelength continuum mathematical model of the soil has been elaborated by the method of structural modeling. An analytical relationship between its macroconstants and microstructure parameters has been found. It is shown that the obtained model differs from the model of the reduced Cosserat medium, which is often employed to describe granular media. In the low-frequency approximation, it can be reduced to the equations of the higher-order gradient theory of elasticity, which are similar to the Cosserat continuum equations with constrained rotation of particles. These equations contain both terms with mixed derivatives with respect to time and coordinate that take into account the inertia of particles rotation in the medium and lead to the wave dispersion, and terms proportional to the spatial derivatives of the fourth order, which describe the contribution of stresses caused by bending of the medium to the potential energy. In the framework of the higher-order gradient model, expressions for the classical and couple stresses in the considered geomedium are found. Boundary conditions are set that consist in the absence of normal and shear stresses on the upper platform of a half-space medium. A condition for the microstructure parameters has been revealed, under which the couple stresses can be neglected. The proposed here higher-order gradient model with given boundary conditions can be used to investigate the propagation and interaction of elastic waves in a semi-infinite geomedium, which are generated by the high-speed movement of trains, as well as to identify and study potentially dangerous effects caused by such movement.