  Physical experiments can characterize the elastic response of granular
materials in terms of macroscopic state-variables, namely volume (packing)
fraction and stress, while the microstructure is not accessible and thus
neglected. Here, by means of numerical simulations, we analyze dense,
frictionless, granular assemblies with the final goal to relate the elastic
moduli to the fabric state, i.e., to micro-structural averaged contact network
features as contact number density and anisotropy.
  The particle samples are first isotropically compressed and later
quasi-statically sheared under constant volume (undrained conditions). From
various static, relaxed configurations at different shear strains, now
infinitesimal strain steps are applied to "measure" the effective elastic
response; we quantify the strain needed so that plasticity in the sample
develops as soon as contact and structure rearrangements happen. Because of the
anisotropy induced by shear, volumetric and deviatoric stresses and strains are
cross-coupled via a single anisotropy modulus, which is proportional to the
product of deviatoric fabric and bulk modulus (i.e. the isotropic fabric).
Interestingly, the shear modulus of the material depends also on the actual
stress state, along with the contact configuration anisotropy.
  Finally, a constitutive model based on incremental evolution equations for
stress and fabric is introduced. By using the previously measured dependence of
the stiffness tensor (elastic moduli) on the microstructure, the theory is able
to predict with good agreement the evolution of pressure, shear stress and
deviatoric fabric (anisotropy) for an independent undrained cyclic shear test,
including the response to reversal of strain.
