The stress field in granular systems: Loop forces and potential formulation

The transmission of stress through a marginally stable granular pile in two dimensions is exactly formulated in terms of a vector field of loop forces, and thence in terms of a single scalar potential. The loop force formulation leads to a local constitutive equation coupling the stress tensor to {\it fluctuations} in the local geometry. For a disordered pile of rough grains (even with simple orientational order) this means the stress tensor components are coupled in a frustrated manner, analogous to a spin glass. If the local geometry of the pile of rough grains has long range staggered order, frustration is avoided and a simple linear theory follows. Known exact lattice solutions for rough grains fall into this class. We show that a pile of smooth grains (lacking friction) can always be mapped into a pile of unfrustrated rough grains. Thus it appears that the problems of rough and smooth grains may be fundamentally distinct.