  Experiments on isotropic compression of a granular assembly of spheres show
that the shear and bulk moduli vary with the confining pressure faster than the
1/3 power law predicted by Hertz-Mindlin effective medium theories (EMT) of
contact elasticity. Moreover, the ratio between the moduli is found to be
larger than the prediction of the elastic theory by a constant value. The
understanding of these discrepancies has been a longstanding question in the
field of granular matter. Here we perform a test of the applicability of
elasticity theory to granular materials. We perform sound propagation
experiments, numerical simulations and theoretical studies to understand the
elastic response of a deforming granular assembly of soft spheres under
isotropic loading. Our results for the behavior of the elastic moduli of the
system agree very well with experiments. We show that the elasticity partially
describes the experimental and numerical results for a system under
compressional loads. However, it drastically fails for systems under shear
perturbations, particularly for packings without tangential forces and
friction. Our work indicates that a correct treatment should include not only
the purely elastic response but also collective relaxation mechanisms related
to structural disorder and nonaffine motion of grains.
