  We study a multi-body finite element model of a packing of hydrogel particles
using the Flory-Rehner constitutive law to model the deformation of the swollen
polymer network. We show that while the dependence of the pressure, $\Pi$, on
the effective volume fraction, $\phi$, is virtually identical to a monolithic
Flory material, the shear modulus, $\mu$, behaves in a non-trivial way. $\mu$
increases monotonically with $\Pi$ from zero and remains below about $80\%$ of
the monolithic Flory value at the largest $\Pi$ we study here. The local shear
strain in the particles has a large spatial variation. Local strains near the
centers of the particles are all roughly equal to the applied shear strain, but
the local strains near the contact facets are much smaller and depend on the
orientation of the facet. We show that the slip between particles at the facets
depends strongly on the orientation of the facet and is, on average,
proportional to the component of the applied shear strain resolved onto the
facet orientation. This slip screens the stress transmission and results in a
reduction of the shear modulus relative to what one would obtain if the
particles were welded together at the facet. Surprisingly, given the reduction
in the shear modulus arising from the facet slip, and the spatial variations in
the local shear strain inside the particles themselves, the deformation of the
particle centroids is rather homogeneous with the strains of the Delaunay
triangles having fluctuations of only order $\pm 5\%$. These results should
open the way to construction of quantitative estimates of the shear modulus in
highly compressed packings via mean-field, effective-medium type approaches.
