Reynolds Pressure and Relaxation in a Sheared Granular System

TL;DR

This study investigates the pressure and relaxation behaviors of frictional granular disks under shear, revealing divergence in a shear modulus-like parameter and slow logarithmic pressure relaxation towards limit cycles.

Contribution

It introduces a novel shear apparatus avoiding shear band formation and characterizes the coupling between shear strain and pressure, including the divergence of a Reynolds coefficient near jamming.

Findings

Reynolds coefficient diverges as packing fraction approaches jamming

Pressure relaxes logarithmically under cyclic shear

Coupling between shear strain and pressure is characterized by Reynolds pressure

Abstract

We describe experiments that probe the evolution of shear jammed states, occurring for packing fractions $\phi_S \leq \phi \leq \phi_J$, for frictional granular disks, where above $\phi_J$ there are no stress-free static states. We use a novel shear apparatus that avoids the formation of inhomogeneities known as shear bands. This fixed $\phi$ system exhibits coupling between the shear strain, $\gamma$, and the pressure, $P$, which we characterize by the `Reynolds pressure', and a `Reynolds coefficient', $R(\phi) = (\partial ^2 P/\partial \gamma ^2)/2$. $R$ depends only on $\phi$, and diverges as $R \sim (\phi_c - \phi)^{\alpha}$, where $\phi_c \simeq \phi_J$, and $\alpha \simeq -3.3$. Under cyclic shear, this system evolves logarithmically slowly towards limit cycle dynamics, which we characterize in terms of pressure relaxation at cycle $n$: $\Delta P \simeq -\beta \ln(n/n_0)$. $\beta$ depends only on the shear cycle amplitude, suggesting an activated process where $\beta$ plays a temperature-like role.