Sponge-like rigid structures in frictional granular packings

We show how rigidity emerges in experiments of sheared frictional granular materials by using generalizations of two methods for identifying rigid structures. Both approaches, the force-based dynamical matrix and the topology-based rigidity percolation, agree with each other and identify similar rigid structures. As the system becomes jammed, at a contact number of $z=2.4\pm 0.1$, a rigid backbone interspersed with floppy, particle-filled holes of a broad range of sizes emerges, creating a sponge-like morphology. We also find that the pressure within rigid structures always exceeds the pressure outside the rigid structures, i.e. that the backbone is load-bearing. These findings shows that it is necessary to go beyond mean-field theory to capture the physics of frictional jamming and also suggests that mechanical stability arises through arch structures and hinges at the mesoscale.