Reynolds Pressure and Relaxation in a Sheared Granular System

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.