Storage and Loss Moduli in an Ideal Aggregate of Elastic Disks, with Lubricated Contacts

We are interested in understanding how elastic waves propagate in a granular material with lubricated contacts. Unlike classical models that attack this problem from a continuum point of view, we choose a micro-mechanical approach, in which the macroscopic stress depends on how particles interact. Because of the complexity of the problem, we present a rather simplified situation, where the aggregate is made of identical elastic disks, isotropically compressed and then incrementally sheared. We assume that particles move with an average rate of deformation and, as they approach, the fluid between them moves out, generating a pressure over the particle surface. We determine this pressure through classical lubrication theory, in which the response to a sinusoidal perturbation provides the storage and loss moduli of the aggregate.