Three-dimensional discrete element modeling of triggered slip in sheared granular media

This paper reports results of a three-dimensional discrete element method modeling investigation of the role of boundary vibration in perturbing stick-slip dynamics in a sheared granular layer. The focus is on the influence of vibration within a range of amplitudes and on the fact that above a threshold early slip will be induced. We study the effects of triggering beyond the vibration interval and their origins. A series of perturbed simulations are performed for 30 large slip events selected from different reference runs, in the absence of vibration. For each of the perturbed simulations, vibration is applied either about the middle of the stick phase or slightly before the onset of a large expected slip event. For both cases, a suppression of energy release is on average observed in the perturbed simulations, within the short term following the vibration application. For cases where vibration is applied in the middle of the stick phase, a significant clock advance of the large slip event occurs. In the long term after vibration, there is a recovery period with higher-energy release and increased activity in the perturbed simulations, which compensates for the temporary suppression observed within the short term.

Figure 1

The 3D DEM model comprised of the driving block (top), a granular gouge layer (center), and a substrate block (bottom).

Figure 2

Particle size distribution (PSD) of the roughness layer of driving block and substrate and the granular gouge layer.

Figure 3

Triggering influence with an increase of the vibration amplitude: time series of (a) friction coefficient, (b) total kinetic energy, and (c) slipping contact ratio. Two insets for (a) and (c) show a more detailed look at the signals during and shortly after the vibration interval. The vibration interval is indicated by vertical dashed lines and a shadowed area in all panels. Black lines in all panels refer to the reference run, while colored lines refer to perturbed runs with a range of amplitudes shown in the figure's color bar.

Figure 4

Characteristic behavior when vibration is applied at about the middle of the stick phase: (a) friction coefficient signal, (b) total kinetic energy signal, (c) cumulative energy release $\tilde{E}\left(t\right)$ for the reference and perturbed runs, (d) difference between the cumulative energy release of the perturbed run and that of the reference run (${\tilde{E}}_{\mathrm{pert}}-{\tilde{E}}_{\mathrm{ref}}$), and (e) recurrence time for slip events with $E\ge 5\times {10}^{-7}{M}_0{L}_0^2{t}_0^{-2}$. The vibration amplitude is $A=7\times {10}^{-6}{L}_0$.

Figure 5

(a) Average results of the difference between the cumulative energy release of the perturbed run and that of the reference run (${\tilde{E}}_{\mathrm{pert}}-{\tilde{E}}_{\mathrm{ref}}$) for 30 triggered large events when vibration is applied at about the middle of the stick phase for each case (arrow shows the approximate location of vibration in the perturbed runs). The vibration amplitude is $A=7\times {10}^{-6}{L}_0$. (b) Percentage of realizations $\frac{N_{\mathrm{pos}}}{N_{\mathrm{tot}}}$ where ${\tilde{E}}_{\mathrm{pert}}-{\tilde{E}}_{\mathrm{ref}}>0$. (c) Percentage of realizations $\frac{N_{\mathrm{neg}}}{N_{\mathrm{tot}}}$ where ${\tilde{E}}_{\mathrm{pert}}-{\tilde{E}}_{\mathrm{ref}}<0$.

Figure 6

Complementary cumulative size distribution $G$ of the (a) event energy size $E$ and (b) recurrence time $T_R$ for the period of energy release suppression in the perturbed and reference runs for 30 triggered large events. Also shown is the $G$ of the (c) event energy size $E$ and (d) recurrence time $T_R$ for the perturbed and reference runs during the recovery period of perturbed runs. The results are for when the vibration is applied at about the middle of the stick phase for each event. The dashed lines (black for reference and blue for perturbed) in (b) and (d) show the exponential distributions of the type $G\left(T_R\right)=e^{-T_R/\tau }$, where $T_R$ is the recurrence time and $\tau$ is the average recurrence time of the distribution. These exponential distributions are best fit to the recurrent time $G$. The vibration amplitude is $A=7\times {10}^{-6}{L}_0$ and only events with $E\ge 5\times {10}^{-7}{M}_0{L}_0^2{t}_0^{-2}$ are considered in the calculations.

Figure 7

Characteristic behavior when vibration is applied slightly before a large event onset: (a) friction coefficient signal, (b) total kinetic energy signal, (c) cumulative energy release $\tilde{E}\left(t\right)$ for the reference and perturbed runs, (d) difference between the cumulative energy release of the perturbed run and that of the reference run (${\tilde{E}}_{\mathrm{pert}}-{\tilde{E}}_{\mathrm{ref}}$), and (e) recurrence time for slip events with $E\ge 5\times {10}^{-7}{M}_0{L}_0^2{t}_0^{-2}$. The vibration amplitude is $A=7\times {10}^{-6}{L}_0$.

Figure 8

(a) Average results of the difference between the cumulative energy release of the perturbed run and that of the reference run (${\tilde{E}}_{\mathrm{pert}}-{\tilde{E}}_{\mathrm{ref}}$) for 30 triggered large events when vibration is applied slightly before the onset of a large event for each case (arrow indicates the approximate location of vibration in the perturbed runs). The vibration amplitude is $A=7\times {10}^{-6}{L}_0$. (b) Percentage of realizations $\frac{N_{\mathrm{pos}}}{N_{\mathrm{tot}}}$ where ${\tilde{E}}_{\mathrm{pert}}-{\tilde{E}}_{\mathrm{ref}}>0$. (c) Percentage of realizations $\frac{N_{\mathrm{neg}}}{N_{\mathrm{tot}}}$ where ${\tilde{E}}_{\mathrm{pert}}-{\tilde{E}}_{\mathrm{ref}}<0$.

Figure 9

Complementary cumulative distribution function $G$ of (a) event energy size $E$ and (b) recurrence time $T_R$ for the period of higher energy release in the perturbed runs and their respective periods in the reference runs for 30 triggered large events. Also shown is the $G$ of (c) event energy size $E$ and (d) recurrence time $T_R$ for the perturbed and reference runs during the period of higher energy release in the perturbed runs and the respective time periods in the reference runs. The results are for when the vibration is applied slightly before the onset of a large event for each case. The dashed lines (black for reference and blue for perturbed) in (b) and (d) show the exponential distributions of the type $G\left(T_R\right)=e^{-T_R/\tau }$, where $T_R$ is the recurrence time and $\tau$ is the average recurrence time of the distribution. The vibration amplitude is $A=7\times {10}^{-6}{L}_0$ and only events with $E\ge 5\times {10}^{-7}{M}_0{L}_0^2{t}_0^{-2}$ are considered in the calculations.