  We consider a random aggregate of identical frictionless elastic spheres that
has first been subjected to an isotropic compression and then sheared. We
assume that the average strain provides a good description of how stress is
built up in the initial isotropic compression. However, when calculating the
increment in the displacement between a typical pair of contaction particles
due to the shearing, we employ force equilibrium for the particles of the pair,
assuming that the average strain provides a good approximation for their
interactions with their neighbors. The incorporation of these additional
degrees of freedom in the displacement of a typical pair relaxes the system,
leading to a decrease in the effective moduli of the aggregate. The
introduction of simple models for the statistics of the ordinary and
conditional averages contributes an additional decrease in moduli. The
resulting value of the shear modulus is in far better agreement with that
measured in numerical simulations.
