Storage and Loss Moduli in an Ideal Aggregate of Elastic Disks, with Lubricated Contacts Giuseppina Recchia, James T. Jenkins, and Luigi La Ragione Abstract 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 a random 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. Keywords Granular material · Acoustic waves · Micro-Mechanics · Visco-Elasticity Mathematics Subject Classification (2000) Granularity 74E20; Lubrication Theory 76D08, Micromechanical theories 74A60 1 Introduction The incremental response of a fully saturated random aggregate of particles exhibits both elastic and viscous behavior. A fundamental contribution to the understanding of this problem is proposed by Biot (e.g. [1]) in a series of papers in which wave propagation in porous materials, whose pores are full of water, is studied. Biot G. Recchia · L. La Ragione () DICAR-Politecnico di Bari, Bari, Italy e-mail: giuseppina.recchia@poliba.it; luigi.laragione@poliba.it J. T. Jenkins School of Civil Engineering, Cornell University, Ithaca, NY, USA e-mail: jtj2@cornell.edu © Springer Nature Switzerland AG 2018 P. Giovine et al. (eds.), Micro to MACRO Mathematical Modelling in Soil Mechanics, Trends in Mathematics, https://doi.org/10.1007/978-3-319-99474-1_27 267