A nonlocal constitutive law for slow granular flows

The local $\mu(I)$ rheology gives the stress ratio $\mu = \tau / P$ as a function of the inertial number $I = \gamma_{\mathrm{dot}} d / \sqrt{P / \rho_s}$ (shear rate $\gamma_{\mathrm{dot}}$, grain diameter $d$, pressure $P$, grain density $\rho_s$). It fails as $I \to 0$: creep below the yield ratio $\mu_s$, thickness-dependent stopping height $h_{\mathrm{stop}}$ of flowing layers, and flow induced by distant shear. Find a constitutive law (for example a fluidity model with cooperativity length $\xi \sim \mid \mu - \mu_s \mid^{-1/2}$) that predicts all three effects quantitatively for frictional spheres, with parameters derived from grain-scale physics.

A law with grain-scale-derived parameters that predicts creep profiles, $h_{\mathrm{stop}}(\theta)$ and remote-shear fluidization in several geometries within experimental error.