Signature of Transition between Granular Solid and Fluid: Rate-Dependent Stick Slips in Steady Shearing

Despite extensive studies on either smooth granular-fluid flow or the solidlike deformation at the slow limit, the change between these two extremes remains largely unexplored. By systematically investigating the fluctuations of tightly packed grains under steady shearing, we identify a transition zone with prominent stick-slip avalanches. We establish a state diagram, and propose a new dimensionless shear rate based on the speed dependence of interparticle friction and particle size. With fluid-immersed particles confined in a fixed volume and forced to “flow” at viscous numbers $J$ decades below reported values, we answer how a granular system can transition to the regime sustained by solid-to-solid friction that goes beyond existing paradigms based on suspension rheology.

Figure 1

(a) Schematics for the setup and measurements, from left to right: the time-dependent torque and three net forces as determined from sensors around the base; cross-sectional view of the main setup (with a stack of acrylic rings alternating with ball bearings to minimize resistance along $\theta$); 3D illustration of the roughened cones; close-up of a typical fluorescent image; and the differential image $\delta \mathfrak{I}$ computed from two frames with $\delta \theta =8\pi \times {10}^{-4}$. (b) Statistical distributions of torque ${\tau }_z$ measured over time, at different rotation rates $\mathrm{\Omega }/2\pi$ (in revolutions per second, rps) denoted by symbols. Dashed lines show curves $\sim \exp \{-0.5{[(S-S_c)/W]}^2-b{[(S-S_c)/W]}^3\}$, where $S\equiv {\tau }_z/(2\pi R^3/3)$ with empirical fit parameters indicated on the graph $\phi =0.60$. (c) Time series of $S$ for three states in (b), all plotted against the angular displacement, which can also be converted to effective $\mathrm{strain}\equiv \mathrm{\Omega }t/2\tan \xi$. The high-frequency noises are smoothed by 40 ms. (d) Time-averaged stress ${\sigma }_S$, plotted against the shear rate $\dot{\gamma }\equiv \mathrm{\Omega }/2\tan \xi =13.7(\mathrm{\Omega }/2\pi )$. (e) Counts of LD per unit strain, plotted against the viscous number $J$, as defined in main texts. Data in (d) and (e) involve experiments with three different viscosity $\eta$ in the fluid, at the same $\phi =0.56$. (f) Sequence of differential images $\delta \mathfrak{I}$ with $\delta \theta =2\pi \times {10}^{-4}$, accompanying one of the large stress jumps in state $T$. Pixels in positive (negative) values are displayed in blue (red). The boundary movement $\theta (t)$ is indicated in reference to a fixed ${\theta }_1$.

Figure 2

Cumulative distribution of torque (stress) drops with $-\mathrm{\Delta }{\tau }_z>{\mathrm{\Delta }}_{\min }$, for experiments at different driving rates (rps). Event counts are normalized by the strain accumulated. Data cover experiments using two batches of particles ($A2$ and $C$) in the same fluid. $\phi =0.56$. In the inset, ${\mathrm{\Delta }}_{\min }$ is normalized by the mean $⟨{\tau }_z⟩$ for each driving rate, while logarithmic slopes of $-0.7$, $-2.5$, and $-3$ are indicated by straight lines and a truncated power law $\sim \{{({\mathrm{\Delta }}_{\min }/⟨{\tau }_z⟩)}^{-0.7}\exp -{[({\mathrm{\Delta }}_{\min }/⟨{\tau }_z⟩)/0.54]}^{2.8}\}$ by the dash-dotted curve—see Appendix D in Supplemental Material [28] for further analyses. In the main plot, two dotted lines illustrate exponential decays with the characteristic stress being 0.2 and 0.4 kPa, respectively. Although statistics from the two batches of particles do not coincide exactly in terms of the corresponding revolutions per second (due to different histories of immersion—see Appendix C3 in Supplemental Material), the general trend over the wide range of driving rates is consistent.

Figure 3

(a) Illustration of the motion of densely packed particles under an imposed shear rate $\dot{\gamma }$; (b) schematics and results of measurements on the average tangential force $f_x$, normalized by the normal force $f_z$, for a localized contact between PDMS surfaces with a radius of curvature $R$. Data are plotted as functions of the sliding speed, for experiments at multiple pressing depths (controlled by the distance $S$ between the base planes) up to about $0.1d$. Data reveal a critical velocity $V_c$ that appears insensitive to the pressing depth $2R\text{−}S$. (c) State diagram summarizing data from experiments with different values of $\phi$ and $\eta$. On the main graph, the vertical coordinate shows the time-averaged shear stress ${\sigma }_S$, plotted against the dimensionless shear rate $d\dot{\gamma }/V_c$ and the normal stress ${\sigma }_N$. On the right, we display a schematic projection onto the ${\sigma }_N-d\dot{\gamma }/V_c$ plane, with contours of constant viscous number $J$ represented by dashed lines indicating ${10}^{-4}$, ${10}^{-6}$, and in turn to values as small as ${10}^{-8}$ back on the main graph. Reported values of $J$ are based on Ref. [1].