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The elastic moduli of four numerical random isotropic packings of Hertzianspheres are studied. The four samples are assembled with different preparationprocedures, two of which aim to reproduce experimental compaction by vibrationand lubrication. The mechanical properties of the samples are found to changewith the preparation history, and to depend much more on coordination numberthan on density.  Secondly, the fluctuations in the particle displacements from the averagestrain are analysed, and the way they affect the macroscopic behavior analyzed.It is found that only the average over equally oriented contacts of therelative displacement these fluctuations induce is relevant at the macroscopicscale. This average depends on coordination number, average geometry of thecontact network and average contact stiffness. As far as the separatecontributions from particle displacements and rotations are concerned, theformer is found to counteract the average strain along the contact normal,while the latter do in the tangential plane. Conversely, the tangentialcomponents of the center displacements mainly arise to enforce localequilibrium, and have a small, and generally stiffening effect at themacro-scale.  Finally, the fluctuations and the shear modulus that result from twoapproaches available in the literature are estimated numerically. Theseapproaches are both based on the equilibrium of a small-sized representativeassembly. The improvement of these estimate with respect to the average strainassumption indicates that the fluctuations relevant to the macroscopic behavioroccur with short correlation length.

international journal of solids and structures

On the elastic moduli of three-dimensional assemblies of spheres:  characterization and modeling of fluctuations in the particle displacement  and rotation