Statistical mechanics of dry granular materials: Between fragile solid (jamming) and dry fluid (rheology)

Dry granular matter, with infinite tangential friction, is modeled as a connected graph of grains linked by purely repulsive contacts. The degrees of freedom of a grain are non-slip rotation on, and disconnection from another. The material stability under shear (jamming) is ensured by odd circuits of grains in contact that prevent the grains from rolling on each other. A dense hard granular material has two possible states: fragile solid, blocked by odd circuits, and dry fluid or bearing, in the absence of odd circuits, that flows under shear by creation and glide of a pair of dislocations as in plasticity of continuous media. We did introduce the notions of blob, a region of the material containing only even circuits, and of critical contact that closes an odd circuit. The granular material is then represented, at low energies and critical applied shear, as a chain of blobs connected by critical contacts. The entropy is the logarithm of the number of spanning trees constrained to go through critical link...