Microstructure and modeling of granular materials

Granular materials are ubiquitous in natural and technological settings, but a predictive theory linking the microscopic grain-interactions with observed behavior remains elusive. Of particular interest are theories of constitutive relations, which challenge conventional models of statistical physics due to the athermal and amorphous nature of granular media. Here we explore properties of granular materials undergoing shear deformation, emphasizing how macroscopic properties arise from the microscopic interactions between grains. This is carried out using numerical simulations, which confirm that there is indeed a bulk rheology, independent of boundary conditions, that can be modeled using only the characteristics of the granular packing. In these simulations we measure spatial force correlations to demonstrate that long-range correlation exists and arises from clusters of simultaneously contacting grains in dense regimes. The size of the clusters defines an important microscopic length-scale xi that diverges at the jamming transition, where the material first acquires a yield stress, and reveals the nature of grain-interactions. For small xi grains interact solely through binary collisions whereas for large xi we observe that clusters of simultaneous contacts, along with complex force-chain networks, spontaneously emerge. This network transition occurs at a well-defined value of xi and is accompanied by a dramatic transformation in the distribution of contact forces between grains that has been observed in previous simulations and experiments.; These basic results regarding the microscopic grain-interactions are generic to granular media and have important consequences for constitutive modeling. In particular we show that kinetic theories, which assume binary collisions, only apply below the network transition. In this regime we show that Enskog kinetic theory agrees with data from the simulations. We then proceed to introduce two analytical theories that use the observed microscopic grain-interactions to make predictions. First we propose a new constitutive model---the Force-Network model---that quantitatively predicts constitutive relations using properties of the force-networks for all values of xi. Second we demonstrate that STZ theory, which predicts constitutive relations by assuming certain dynamical correlations in amorphous materials, is in agreement with both the microscopic motion of grains and measured constitutive relations for large xi.