  This study examines the flow of dense granular materials under external shear
stress and pressure using discrete element method simulations. In this method,
the material is allowed to strain along all periodic directions and adapt its
solid volume fraction in response to an imbalance between the internal state of
stress and the external applied stress. By systematically varying the external
shear stress and pressure, the steady rheological response is simulated for:
(1) rate-independent quasi-static flow; and (2) rate-dependent inertial flow.
The simulated flow is viscometric with non-negligible first and second normal
stress differences. While both normal stress differences are negative in
inertial flows, the first normal stress difference switches from negative to
slightly positive, and second normal stress difference tends to zero in
quasi-static flows. The first normal stress difference emerges from a lack of
coaxiality between a second-rank contact fabric tensor and strain rate tensor
in the flow plane, while the second normal stress difference is linked to an
excess of contacts in the shear plane compared with the vorticity direction. A
general rheological model of second order (in terms of strain rate tensor) is
proposed to describe the two types of flow, and the model is calibrated for
various values of interparticle friction from simulations on nearly
monodisperse spheres. The model incorporates normal stress differences in both
regimes of flow and provides a complete viscometric description of steady dense
granular flows.
