Force distributions and stress response in granular materials

Despite the apparent simplicity of static collections of grains, our understanding of the physics required to describe them remains surprisingly incomplete. We study two aspects of this problem: the distribution of forces within a granular medium, and the transmission of force through the bulk of the material.; We study a lattice model of underconstrained contact forces between grains. We study the distribution of forces in the ensemble with all noncohesive force balanced configurations of the lattice given equal weight. We find that regular lattices subject to hydrostatic pressure possess force distributions that decay faster than exponentially. Under contact dilution the distribution broadens but remains faster than exponential. Under increasingly anisotropic confining pressure the distribution develops an exponential tail, in agreement with experimental results of Majmudar and Behringer. We also study spatial fluctuations in the pressure. We find that fluctuations possess no characteristic length scale. The flat spectrum is shown to be a consequence of the restriction to noncohesive contact forces and persists in anisotropic lattices.; Secondly, we study the stress response to a boundary force of a two-dimensional hexagonally anisotropic material in nonlinear elasticity. We find that nonlinear effects become prominent as the rigidity of the grains increases. We develop corrections to the Boussinesq result of linear elasticity in an expansion in inverse powers of the distance from the boundary force. Free parameters in the calculation are fixed by fitting to the data of Geng and Behringer for hexagonal packings of photoelastic disks. The model reproduces the multiple peaks observed at shallow depth and predicts a crossover at large depths to the single-peaked linearly broadening Boussinesq profile.; Thirdly, we study the directed force chain network theory, a continuum model for the stresses in a static packing of rigid grains that describes a collection of propagating, interacting chains of force within the material. We extend previous calculations to the case of a continuous set of propagation directions for homogeneous isotropic materials. We identify a sum rule for the density of chains and relate two model parameters to measurable quantities. The chain density satisfies a nonlinear integro-differential equation whose solution awaits further study.