Extended constraints, arches and soft modes in granular materials

A granular material is a network of fixed length edges linking two grains in contact, and flexible hinges if the grains can rotate without sliding on each other. The network is then dynamically unfrustrated: it can deform freely under shear and behaves like a dry fluid. A sufficient condition for non-frustration in 3D is that all circuits of grains in contact are even. Frustration is concentrated on odd circuits, and the ‘odd vorticity’ forms loops in 3D. Thus, some constraints in the network cannot be accounted for by local mean-field analysis. These extended constraints and the means of accounting for them are discussed. A granular packing can be described as a graph, with a connection carried by the edges. If all circuits of grains in contact are even, the connection is pure gauge, independent of the path between grains. When shear stress is applied over a large scale, it can break the frustrating odd circuits by disconnecting selected grains in contact, leaving all other contacts and even circuits unaffected. The granular material deforms freely like dry quicksand, as occurs in silent earthquakes, soil liquefaction, wet sand under foot (dilatancy) and, to some extent, in geological faults.