Plane harmonic waves in saturated mixtures of soils

In this paper we consider a saturated mixture for soils consisting of a dilatant granular medium with rigid rotating grains immersed in an incompressible fluid, under the hypothesis that the mass exchange between the phases of the mixture is non-zero, therefore it can be seen as a peculiar case of Giovine (J Solids Struct 187:3–22, 2020). For the granular constituent, we introduce two microstructural fields: the micro-spin vector and the volume fraction; the first describes the rotation of rigid grains, while the scalar models the fluctuations of voids due to the dilatancy of the granular material. For the fluid constituent, the microstructure is only scalar, represented by its volume fraction. The vectorial microstructure seems to be a novelty in these media. We choose as the system of constitutive equations the one suggested in as reported by Giovine (On adsorption and diffusion in microstructured porous media, Springer, Dordrecht, 2005), which includes diffusion and absorption phenomena. Finally, in the system of balance equations, there are four interchange terms concerning: mass, linear and micro-linear momenta and micro-angular momentum; moreover, the rules underlying the formulation of the balance laws for the mixture satisfy the three metaphysical principles of Truesdell (Rational Thermodynamics, McGraw-Hill, New York, 1969). Subsequently, we study the plane harmonic waves in the saturated granular-fluid mixture as non-trivial solutions of the system of linear differential equations, assuming that the volume microstructural wave propagates in the mixture, we obtain three longitudinal waves and three transverse waves, in both cases, two waves are mixed and one is purely microscopic: this last wave does not appear in previous model.