Scalar model of inhomogeneous elastic and granular media

We investigate theoretically how the stress propagation characteristics of granular materials evolve as they are subjected to increasing pressures, co mparing the results of a two-dimensional scalar lattice model to those of a molecular dynamics simulation of slightly polydisperse disks. We character ize the statistical properties of the forces using the force histogram and a two-point spatial correlation function of the forces. For the lattice mod el, in the granular limit the force histogram has an exponential tail at la rge forces, while in the elastic regime the force histogram is much narrowe r, 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 latti ce model. In contrast, the spatial correlations evolve qualitatively differ ently in the lattice model and in the molecular dynamics simulations. For t he 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.