Jamming is a phenomenon occurring in systems as diverse as traffic, colloidal suspensions and granular materials. A theory on the reversible elastic deformation of jammed states is presented.First, an explicit granular stress-strain relation is derived that captures many relevant features of sand, including especially the Coulomb yield surface and a third-order jamming transition. Then this approach is generalized, and employed to consider jammed magneto-and electro-rheological fluids, again producing results that compare well to experiments and simulations.

A complete continuum mechanical theory for granular media, including explicit expressions for the energy current and the entropy production, is derived and explained. Its underlying notion is: granular media are elastic when at rest, but turn transiently elastic when the grains are agitatedsuch as by tapping or shearing. The theory includes the true temperature as a variable, and employs in addition a granular temperature to quantify the extent of agitation. A free energy expression is provided that contains the full jamming phase diagram, in the space spanned by pressure, shear stress, density and granular temperature. We refer to the theory as GSH, for granular solid hydrodynamics. In the static limit, it reduces to granular elasticity, shown previously to yield realistic static stress distributions. For steady-state deformations, it is equivalent to hypoplasticity, a state-of-the-art engineering model.

A self-contained elastic theory is derived which accounts both for mechanical yield and shear-induced volume dilatancy. Its two essential ingredients are thermodynamic instability and the dependence of the elastic moduli on compression.