Based on discrete element method simulations, we propose a new form of the constitutive equation for granular flows independent of packing fraction. Rescaling the stress ratio μ by a power of dimensionless temperature Θ makes the data from a wide set of flow geometries collapse to a master curve depending only on the inertial number I. The basic power-law structure appears robust to varying particle properties (e.g., surface friction) in both 2D and 3D systems. We show how this rheology fits and extends frameworks such as kinetic theory and the nonlocal granular fluidity model.

When conducting bonds are occupied randomly in a two-dimensional square lattice, the conductivity of the system increases continuously as the density of those conducting bonds exceeds the percolation threshold. Such a behavior is well known in percolation theory; however, the conductivity behavior has not been studied yet when the percolation transition is discontinuous. Here we investigate the conductivity behavior through a discontinuous percolation model evolving under a suppressive external bias. Using effective medium theory, we analytically calculate the conductivity behavior as a function of the density of conducting bonds. The conductivity function exhibits a crossover behavior from a drastically to a smoothly increasing function beyond the percolation threshold in the thermodynamic limit. The analytic expression fits well our simulation data. The concept of percolation transition has played a central role as a model for the formation of a spanning cluster connecting two opposite edges of a system in Euclidean space as a control parameter p is increased beyond a certain threshold p c [1]. This model has been used to study many phenomena such as metal-insulator transitions and sol-gel transitions. The order parameter P ∞ of percolation transition is defined as the probability that a bond belongs to a spanning cluster, which increases in the form, where p is a control parameter indicating the fraction of occupied bonds and β is the critical exponent related to the order parameter. As an application of percolation model, one can construct a random resistor network in which each occupied bond is regarded as a resistor with unit resistance, and the system is in contact with two bus bars at the opposite edges of the system. When a voltage difference is applied between these two bus bars, the system is in a insulating state for p < p c , but changes to conducting state for p > p c , due to the formation of several conducting paths at p c . Above p c , the conductivity increases continuously as g ∼ (p − p c ) µ , where µ is the conductivity exponent [2].Recently the subject of discontinuous percolation transition (DPT) has been a central issue [3][4][5][6][7][8][9][10][11][12] with, for example, applicability to cascading failures in complex networks [13]. Among others [14][15][16][17][18][19], a model called spanning cluster avoiding (SCA) was introduced [20] aiming to generate a DPT. The DPT of the SCA model is rather trivial, for the percolation threshold is placed at p c = 1 in the thermodynamic limit, but for finite-sized systems p c < 1. Here, we study the conductivity as a function of p in finite-sized systems for the SCA model. Indeed, we find that the conductivity increases drastically just after the percolation threshold and then exhibits a crossover to a smoothly increasing behavior. Such crossover has never been reported, though it is meaningful, as, a drastic change of conductivity in random resistor networks can find application, for example, on resistance switching phenomena in non-volatile memory...