Study On Localization Of Waves In Granular System

Granular systems have attracted considerable attention in recent years due to their rich dynamical behaviors and interesting properties.They have potential applications in passive shock mitigation designs,in energy containment,impact protection,and in a gravity-driven dense granular flow target,in which heavy metal grains are chosen as the spallation target material.Granular materials are densely packed arrays of elastic-plastic beads that interact nonlinearly through Hertzian contacts.The unique property of the waves propagating in the granular material is the attenuation or the localization due to the contribution of the randomness and the plasticity(dissipation)of the beads.The localization or the attenuation of the waves due to either the disorder or the dissipation in granular matter is reported.However,the theory on this kind of phenomena is still lacking.Present paper will set up a general three dimensional wave equation which can explain the localization or attenuation of the waves in granular materials.The effects of both the dissipation and the randomness of the granular medium are described by this wave equation.The theory suggest that the localization length depends on several parameters such as the wave frequency,the bead radius,the magnitude of the initial prestress,the Young's modulus and the bead mass.It is also noted that the localization length depend on the permutation of the bead as well as the kinds of the waves,for example,whether it is the longitudinal wave or shear wave.Several examples are shown in the present paper.Comparisons are given between the analytical results from the wave equation and the numerical ones from the original equation.Good agreements are observed.