Volume-weighted mixture theory for granular materials

In the present work we treat granular materials as mixtures composed of a solid and a surrounding void continuum, proposing then a continuum thermodynamic theory for it. In contrast to the common mass-weighted balance equations of mass, momentum, energy and entropy for mixtures, the volume-weighted balance equations and the associated jump conditions of the corresponding physical quantities are derived in terms of volume-weighted field quantities here. The evolution equations of volume fractions, volume-weighted velocity, energy, and entropy are presented and explained in detail. By virtue of the second law of thermodynamics, three dissipative mechanisms are considered which are specialized for a simple set of linear constitutive equations. The derived theory is applied to the analysis of reversible and irreversible compaction of cohesionless granular particles when a vertical oscillation is exerted on the system. In this analysis, a hypothesis for the existence of a characteristic depth within the granular material in its closely compacted state is proposed to model the reversible compaction.