  We investigate theoretically how the stress propagation characteristics of
granular materials evolve as they are subjected to increasing pressures,
comparing the results of a two-dimensional scalar lattice model to those of a
molecular dynamics simulation of slightly polydisperse discs.
  We characterize the statistical properties of the forces using the force
histogram and a two-point spatial correlation function of the forces. For the
lattice model, in the granular limit the force histogram has an exponential
tail at large forces, while in the elastic regime the force histogram is much
narrower and has a form that depends on the realization of disorder in the
model. The behavior of the force histogram in the molecular dynamics
simulations as the pressure is increased is very similar to that displayed by
the lattice model. In contrast, the spatial correlations evolve qualitatively
differently in the lattice model and in the molecular dynamics simulations. For
the lattice model, in the granular limit there are no in-plane stress-stress
correlations, whereas in the molecular dynamics simulation significant in-plane
correlations persist to the lowest pressures studied.
