Force-based ensembles in granular materials

It has long been known that athermal materials, including real granular materials with friction, exhibit highly heterogeneous patterns of force transmission. I will describe two routes to understanding this phenomenon, both arising from the consideration of statistical mechanical ensembles in a quasi two-dimensional experiment of disks floating on a gentle cushion of air. First, force network ensembles examine the configuration-space of allowed interparticle forces, given a fixed particle configuration. We perform experiments tabulating up to 1000 realizations of the force network in the same particle packing. Using the tools of network science, we evaluate the causes of individual particles being on the strong network, as well as whether or not particles are reliably on the strong network. Second, the Edwards-like force-moment ensemble predicts the existence of a temperature-like variable which would provide an equation of state. To test the efficacy of this approach, we perform experiments under three different loading conditions. From measurements of the interparticle forces, we find that the distributions of the normal component of the coarse-grained force-moment tensor are exponential-tailed, while the deviatoric component is Gaussian-distributed. This implies that the correct ensemble will need to consider both force-moment tensor (responsible for the linear term in the exponent) and the Maxwell-Cremona force-tiling area (the quadratic term). These two terms lead to two candidate variables of state, correspondingly the tensorial angoricity and a new temperature-like quantity associated with the force-tile area which we name keramicity. We observe an equation of state in which each of these is inversely proportional to the global confining pressure; however only keramicity exhibits the protocol-independence expected of a state variable, while angoricity behaves as a variable of process.