Mechanics of granular media under vibration action: The methods of description and mathematical modeling

The article presents a review of the modern methods of computational investigation and modeling of dynamics of granular media under vibration action. Attention is focused on typical states of granular material, realized depending on intensity of vibration action and mechanical characteristics of material. Three «physical states» are distinguished: «elastic-plastic» solid - if mutual arrangement of material particles is changed little, viscous «granular liquid» - if most particles are shifted without breaking contacts, and «granular gaseous» - when time of contact interaction between particles is small in comparison with their free time of flight. Also, three groups of approaches to modeling are distinguished: microscopical - considering material at the level of separate particles, statistical mechanics / kinetic theory - generalizing kinetic theory of gases with respect to inelastic interactions, phenomenological models and continuum mechanical models. A range of applicability and possibilities are specified for each group of models. Application of the described approaches is illustrated by a number of cases of specific behavior of granular materials: Leidenfrost granular effect, granular convection, oscillons, etc.