Discontinuous shear thickening for frictional granular particles

We study the rheology of frictional granular particles with analytical modelling and numerical simulations in two dimensions. We derive a phase diagram with a topology different from the well known Liu-Nagel phase diagram for frictionless particles with a zero stress critical point. In contrast to the frictionless scenario, jamming first occurs at finite stress at a critical packing fraction $\phi_C$ while a finite yield stress emerges only at $\phi_{\sigma} > \phi_C$. Remarkably, the flow is reentrant and we observe discontinuous shear thickening in the flow curves for $\phi \in (\phi_C, \phi_{\eta})$ with $\phi_{\eta} > \phi_{\sigma}$. All these features can be rationalized with a simple constitutive equation which contains the frictionless scenario as a limiting case.