Determining Nonlocal Granular Rheology from Discrete Element Simulations

Abstract

We determine the constitutive equation of simple granular materials considering them as continuous fluids. Based on discrete element simulations, we propose two rheological models with different Rivlin-Ericksen tensor orders. In the first-order model, we identify that rescaling the shear-to-stress ratio $\mu$ by a power function of dimensionless granular temperature $\Theta$ makes the data from many different flow geometries collapse to a single curve which depends only on the inertial number $I$. The basic power-law structure appears robust to varying surface friction in both 2D and 3D systems. We also observe that $\phi$ is a function of $\mu$, which connects our rheology to kinetic theory and the nonlocal granular fluidity model. In order to describe stress anisotropy and secondary flows, we extend our model by including the second-order Rivlin Ericksen tensor. Using DEM data, we find the equations for three model parameters $\mu_1$, $\mu_2$, and $\mu_3$ as functions of $I$ and $\Theta$. We observe similar power-law scaling in $\mu_1$ and $\mu_2$ while $\mu_3$ distributes near zero for small $I$. The first and second normal stress differences $N_1$ and $N_2$ are also measured and discussed. We validate the models by running finite difference method simulations of inclined chute flows. We show that the second-order model predicts all the velocity components including secondary flows while the first-order model predicts velocity in the downstream direction only. Both models successfully predict the exponentially decaying velocity as $\Theta$ is included in the model parameters.