On Phase Transition and Self-Organized Critical State in Granular   Packings
  We model two-dimensional systems of granular aggregates confined between two planes and demonstrate that at a critical grain volume fraction an abrupt rigidity transition occurs. This transition is observed both in static and shear tests. The grain volume fraction at which the transition occurs, $\nu_c$, decreases with increasing friction between the grains. Densely packed grains, with a volume fraction $\nu> \nu_c$, display an elastic-plastic rheology. Dilute packings, with $\nu<\nu_c$, display gas-like characteristics. Packings with $\nu=\nu_c$ display phase coexistance. It is shown that when volume fraction is allowed to change freely (using constant normal stress boundary condition), it evolves spontaneously to $\nu_c$ under a wide range of boundary conditions, exhibiting 'self-organized criticality'. 
Einat Aharonov and David Sparks