Bridging percolation and particle dynamics models of the granular rigidity transition

Percolation models are routinely used to study the rigidity transition. However, they cannot capture dynamics occurring in granular media. Molecular Dynamics can be used to model the rigidity transition, yet, they cannot represent realistic hard granular materials due to computational constraints. Contact Dynamics can be used instead; but only by assuming perfectly rigid particles where dissipation occurs by friction alone and deformation is excluded. We present an alternative dynamic approach to studying the solid-fluid granular rigidity transition for rigid particles, which is computationally faster than Contact Dynamics, and does not exclude particle deformation. Here we present an adaption of the Granular Medium Lattice Gas model of Károlyi and Kertész (1998) called the Sheared Granular Lattice Gas (SGLG) model which incorporates, for the first time in a lattice gas model, a description of force chains and granular shear. Significantly, we demonstrate that the SGLG model recovers typical experimental measurements such as dilation effects adjacent to the shearing boundary, the functional dependence of the velocity with depth near the transition, and qualitatively, granular anisotropy. The SGLG model also reproduces key results from simulations. Firstly, it exhibits similar functional dependence of particle coordination number on density as observed in Molecular Dynamics simulations. Secondly, the SGLG allows the transition of the percolation network strength to be calculated. It is found to be a first order transition and functionally equivalent to that observed for central force rigidity percolation. However, the exponents characterising the transition differ.