Critical scaling near the yielding transition in granular media

We show that the yielding transition in granular media displays second-order critical-point scaling behavior. We carry out discrete element simulations in the low inertial number limit for frictionless, purely repulsive spherical grains undergoing simple shear at fixed nondimensional shear stress $\Sigma$ in two and three spatial dimensions. To find a mechanically stable (MS) packing that can support the applied $\Sigma$, isotropically prepared states with size $L$ must undergo a total strain $\gamma_{\rm ms}(\Sigma,L)$. The number density of MS packings ($\propto \gamma_{\rm ms}^{-1}$) vanishes for $\Sigma > \Sigma_c \approx 0.11$ according to a critical scaling form with a length scale $\xi \propto |\Sigma - \Sigma_c|^{-\nu}$, where $\nu \approx 1.7-1.8$. Above the yield stress ($\Sigma>\Sigma_c$), no MS packings that can support $\Sigma$ exist in the large system limit, $L/\xi \gg 1$. MS packings generated via shear possess anisotropic force and contact networks, suggesting that $\Sigma_c$ is associated with an upper limit in the degree to which these networks can be deformed away from those for isotropic packings.