Granular matter instability: A structural rigidity point of view

Stress paths in granular matter often suffer sudden large-scale rearrangements when the system is slightly perturbed, i.e. granular systems are unstable. We show in this paper that the observed instability is due to the minimally rigid, or isostatic, character of the system's contact network. It is first demonstrated that the contact network of a granular packing becomes isostatic (minimally rigid) in any dimension in the limit of large stiffness-to-load ratio. We next show that, in this isostatic limit, the load-stress response function becomes power-law distributed and takes exponentially large (growing as exp(H) where H is the system's height) positive and negative values. Large negative values of the load-stress response function imply instability, since only positive (compressive) stresses are allowed in non-cohesive granular packings. Thus there is an isostatic phase transition in the limit of large stiffness (or small load), and the resulting isostatic phase has an anomalously large susceptibility to perturbation.