Decoupling of Friction and Solid Fraction in Dense Granular Flow Across Six Orders of Magnitude of Shear Rate

The μ(I) framework for dense granular flows treats the macroscopic friction (μ), defined as the ratio of shear to normal stress transmitted across the sheared assembly, and the solid volume fraction (φ) as single-valued functions of the inertial number (I). To test if these responses evolve together across shear rates, we used 3D discrete-element simulations of ring shear on a monodisperse assembly of spherical glass beads (diameter d = 1 mm), confined between two parallel toothed platens at 100 kPa normal stress. Shear velocity was varied over six orders of magnitude (10^-4 to 10 m/s), spanning a nominal inertial number I ≈ 10^-6 to 10^-1. Throughout, I denotes a bulk value defined from the imposed shear rate v/H and the applied normal stress; the local inertial number varies with height and is reported separately. We found that steady-state friction closely follows the μ(I) law, rising monotonically with rate. In contrast, bulk solid fraction φ is non-monotonic: it falls as rate increases, reaches a minimum near I ≈ 10^-3, and recovers at higher rates, making a single φ(I) curve a poor fit. Height-resolved fields explain this decoupling. At higher rates, shear concentrates near the driving boundary, and strong vertical agitation drives cross-layer mixing that re-compacts the lower bed. Consequently, bulk φ averages a dilating top with a re-densifying base, losing its one-to-one link to I. Instead, over the conditions tested, the contact density (n_c) decreases monotonically-by the same factor whether it is normalised per unit height, per unit volume, or per particle-and so remains in step with the friction where the bulk solid fraction does not. The contact-force distribution broadens, and the fabric anisotropy grows as the flow becomes inertial, so that load is carried through fewer, more strongly oriented chains. These results show that friction and packing can respond to shear rate on different terms, and that bulk φ should be used with care as a state variable in rate-dependent dense flows.