Stick-slip transition at the granular critical state

We study the force on a flat plate (3.8~cm width, 7.0~cm depth) dragged at constant velocity $v$ through the surface of a granular medium (250~$\mu$m glass beads) as a function of volume fraction $0.57<\phi<0.63$. The dynamics of the drag force $F_d$ are sensitive to $\phi$: we find a sharp transition in the form of $F_d$ at a critical volume fraction $\phi_c=0.605$. For $\phi<\phi_c$, $F_d$ increases with time and saturates, while for $\phi>\phi_c$ $F_d$ exhibits an initial peak followed by periodic oscillations at frequency $f$ about a constant mean. The standard deviation in force (a measure of the fluctuations) shows a sharp transition at $\phi_c$. The force oscillations suggest that the granular media periodically jams and flows as the plate is horizontally translated. Examining the bed surface we observe a spatially periodic scalloped feature of length $\lambda$ which is equal to $v/f$, independent of $v$, and increases linearly with $\phi$ for $\phi>\phi_c$. By measuring the displaced volume after the drag $\Delta V$, we observe a transition from media compaction ($\Delta V<0$) for $\phi<\phi_c$ to dilation ($\Delta V>0$) for $\phi>\phi_c$.