A nonlocal constitutive law for slow granular flows
In plain words
A simple local friction law describes fast flows of grains well but fails when the flow is slow: grains creep below the expected stopping point and thin layers stop earlier than thick ones. The correct replacement law, and its basis in grain-scale physics, is not established.
Precise statement
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.
What would settle it
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.
Status in the literature
Unverified note
Nonlocal fluidity models introduced from 2012 fit several geometries, but their microscopic basis and the cooperativity exponent remain debated (2026).