Geometric origin of mechanical properties of granular materials

Model granular assemblies in which grains are assumed rigid and frictionless at equilibrium under some prescribed external load, are shown to possess, under generic conditions, several remarkable mechanical properties, related to isostaticity and potential energy minimization. Isostaticity-the uniqueness of the contact forces, once the list of contacts is known-is established in a quite general context, and the important distinction between isostatic problems under given external loads and isostatic (rigid) structures is presented. Complete rigidity is only guaranteed, on stability grounds, in the case of spherical cohesionless grains. Otherwise, the network of contacts might deform elastically in response to small load increments, even though grains are perfectly rigid. In general, one gets an upper bound on the contact coordination number. The approximation of small displacements that is introduced and discussed allows analogies to be drawn with other model systems studied in statistical mechanics, such as minimum paths on a lattice. It also entails the uniqueness of the equilibrium state (the list of contacts itself is geometrically determined) for cohesionless grains, and thus the absence of plastic dissipation in rearrangements of the network of contacts. Plasticity and hysteresis are related to the lack of such uniqueness, which can be traced back, apart from intergranular friction, to nonreversible rearrangements of small but finite extent, in which the system jumps between two distinct potential energy minima in configuration space, or to bounded tensile forces, deriving from a nonconvex potential, in the contacts. Properties of response functions to load increments are discussed. On the basis of past numerical studies, it is argued that, provided the approximation of small displacements is valid, displacements due to the rearrangements of the rigid grains in response to small load increments, once averaged on the macroscopic scale, are solutions to elliptic boundary value problems (similar to the Stokes problem for viscous incompressible flow).