Frictional Weakening of Vibrated Granular Flows

We computationally study the frictional properties of sheared granular media subjected to harmonic vibration applied at the boundary. Such vibrations are thought to play an important role in weakening flows, yet the independent effects of amplitude, frequency, and pressure on the process have remained unclear. Based on a dimensional analysis and DEM simulations, we show that, in addition to a previously proposed criterion for peak acceleration that leads to breaking of contacts, weakening requires the absolute amplitude squared of the displacement to be sufficiently large relative to the confining pressure. The analysis provides a basis for predicting flows subjected to arbitrary external vibration and demonstrates that a previously unrecognized second process that is dependent on dissipation contributes to shear weakening under vibrations.

Figure 1

(a) Depiction of the simulations; real simulations have rough walls. (b) Plots of $F_{\tau }/F_p$ versus shear strain $\gamma$ for two simulations with $I={10}^{-5}$ after transients have subsided. The curves have $\mathrm{\Gamma }=0.004$, $\tilde{A}=0.0003$ (dark gray), and $\mathrm{\Gamma }=0.4$, $\tilde{A}=0.03$ (red). Dashed lines show the average, $\mu$. (c) $\mu$ versus $I$ for varying $\mathrm{\Gamma }$ and $\tilde{A}$, with $e_n=0.2$ and $\tilde{p}={10}^{-3}$. (d) ${\mu }_s/{\mu }_{s,0}$ versus $\mathrm{\Gamma }$ for varying $\tilde{f}$. Solid and open symbols correspond to $\tilde{p}={10}^{-5}$ and $\tilde{p}={10}^{-4}$, respectively; experimental data is from Fig. 2 of Dijksman et al. [17] using the smallest shear rate shown.

Figure 2

(a) ${\mu }_s/{\mu }_{s,0}$ versus $\mathrm{\Gamma }$, where $f$ is varied and $\tilde{A}$ is held constant. We use the data between the vertical dashed lines to characterize the dependence of ${\mu }_s$ on $\tilde{A}$ and $\tilde{p}$; see text for discussion. (b) ${\mu }_s/{\mu }_{s,0}$ versus $\tilde{f}$ for the same data shown in (a), showing that $\tilde{f}>1$ corresponds to the rise of ${\mu }_s$ at high $\mathrm{\Gamma }$.

Figure 3

(a) ${\mu }_{\min }$, measured between the dashed lines in Fig. 2, is plotted as a function of ${\tilde{A}}^2$ for different $\tilde{p}$ and $e_n$. Dashed black line shows a fit to a sigmoidlike function, ${\mu }_{\min }=({\mu }_{s,0}/2)\{1-\tanh [{\log }_{10}(\tilde{A}/{\tilde{A}}^{*}{)}^2]\}$, to the data for $\tilde{p}={10}^{-4}$ and $e_n=0.8$. The fit estimates ${\tilde{A}}^{*}$ as the value of $\tilde{A}$ where ${\mu }_s/{\mu }_{s,0}=1/2$, which we use as the characteristic value of $\tilde{A}$ for frictional weakening. (b) $({\tilde{A}}^{*}{)}^2$ versus $\tilde{p}$ for $e_n=0.2$ (circles), $e_n=0.5$ (triangles), and $e_n=0.8$ (squares). Color denotes $\tilde{p}$; large and small symbols have different values of $E$, confirming that $\tilde{p}$ captures the scaling. All data are approximately captured by $({\tilde{A}}^{*}{)}^2\propto \tilde{p}$. (c–e) ${\mu }_{\min }/{\mu }_{s,0}$ versus ${\tilde{A}}^2/\tilde{p}$ for (c) $e_n=0.2$, (d) $e_n=0.5$, and (e) $e_n=0.8$, with the same symbol convention as in (b).

Figure 4

A phase diagram of ${\mu }_s$ as a function of $\mathrm{\Gamma }$ and ${\tilde{A}}^2/\tilde{p}$. Symbol shapes correspond to different $e_n$ and are slightly shifted for visibility, with circles for $e_n=0.2$ (shifted down), triangles for $e_n=0.5$, and squares for $e_n=0.8$ (shifted up). Symbol size and color represents the amount of frictional weakening. Dashed lines show $\mathrm{\Gamma }=1$ and ${\tilde{A}}^2/\tilde{p}=1$.