Influence of vibration amplitude on dynamic triggering of slip in sheared granular layers

We perform a systematic statistical investigation of the effect of harmonic boundary vibrations on a sheared granular layer undergoing repetitive, fully dynamic stick-slip motion. The investigation is performed using two-dimensional discrete element method simulations. The main objective consists of improving the understanding of dynamic triggering of slip events in the granular layer. Here we focus on how the vibration amplitude affects the statistical properties of the triggered slip events. The results provide insight into the granular physical controls of dynamic triggering of failure in sheared granular layers.

Figure 1

Rendering of a horizontal segment of the 2D discrete element method model. The $X$ direction corresponds to the shear load direction. A normal confining load is applied along the $Y$ direction. Periodic boundary conditions are applied along the $X$ direction.

Figure 2

Examples of stick-slip events from the spontaneous slip catalog and associated energy release. (a) Time series for the macroscopic friction coefficient ${\mu }_f$ used for identifying slip events. The round markers identify the onset $t_{b,i}$ and end $t_{e,i}$ of one slip event ($i\mathrm{th}$ event), defined as a decrease in the ${\mu }_f$ value. The duration of the stick period for that $i\mathrm{th}$ event $\Delta t_{\mathrm{st},i}\equiv t_{b,i}-t_{e,i-1}$ is highlighted by the horizontal double arrow and corresponds to one value of the time-to-failure variable. (b) Total kinetic energy of the granular layer $E_{\mathrm{tot}}^{\mathrm{tot}}$ for the same temporal segment in (a). The round markers identify the onset and end of the event, while the vertical double arrow indicates $E_r$, marking the sharp increase in $E_{\mathrm{tot}}^{\mathrm{tot}}$, used in this work as a proxy of the energy release during the slip event.

Figure 3

Example of dynamically triggered slip events relative to spontaneous slip events. (a) Macroscopic friction coefficient ${\mu }_f$ time series: solid black line (first from the right side) and circles, reference run, as in Fig. 2a; dashed blue line (second from the right) and squares, corresponding perturbed run with ac vibration peak amplitude $A=0.004$; solid red line (third from right) and triangles, perturbed run with $A=0.01$; dashed green line (fourth from the right) and stars, perturbed run with $A=0.016$. The markers identify the beginning and the end of each slip event. The arrow indicates the order of curves for increasing $A$. (b) Boundary ac vibration displacement signal $u_Y$ imposed at the substrate's bottom. Curves of different colors (online) correspond to different ac vibration amplitude values as in inset (a). (c) Total kinetic energy $E_{\mathrm{tot}}^{\mathrm{tot}}$ signal for the granular layer. Here as well the colors (online) correspond to different ac vibration amplitude values as in (a). The markers mirror those of inset (a) for highlighting the beginning and the end of each slip event. The arrow indicates the order for increasing $A$.

Figure 4

(a) Time to failure for one spontaneous slip event $\Delta t_{\mathrm{st}}(A=0)$ (solid black line and circles) [this is the same event as in Figs. 2a and 3a] and the corresponding dynamically perturbed event at the ac vibration amplitude $A=0.016$, $\Delta t_{\mathrm{st}}(A=0.016)$ (dashed green line and stars). (b) Relative change $\Gamma$ in the time to failure (duration of the stick phase $\Delta t_{\mathrm{st}}$) with each ac vibration peak amplitude value $A$ compared with the reference run ($A=0$). See Eq. (7) for the definition of $\Gamma \left(A\right)$. The markers indicate average values over the population of the 99 events, while the error bars indicate the standard deviation. Negative values of $\Gamma$ indicate a decrease of the time to failure in correspondence with the application of ac vibration.

Figure 5

Weibull plot for the slip event energy release $E_r$. Different markers refer to different catalogs of slip events. The stars refer to the spontaneous slip catalog, while the other markers refer to the dynamically perturbed slip catalogs, corresponding to the values of the peak vibration amplitude value $A$. Each catalog represents the statistical ensemble for $E_r$ as a random variable. Here $G_{E_r}\left({\tilde{E}}_r\right)$ is the complementary cumulative distribution function (CCDF) of $E_r$ as calculated from the corresponding catalog and it represents the probability for the event $\{E_r>{\tilde{E}}_r\}$ as estimated from the catalog. Each dashed line is the Weibull best fit curve of the corresponding CCDF obtained by a nonlinear least squares best fitting procedure. The arrow indicates the increasing $A$.

Figure 6

Histograms for the relative change in energy release $\epsilon \left(A\right)\equiv \frac{\Delta E_r}{E_r}$ when comparing perturbed slip and spontaneous slip catalogs. Different insets refer to different ac vibration peak amplitude values: (a) $A=0.004$, (b) $A=0.006$, (c) $A=0.007$, (d) $A=0.0085$, (e) $A=0.009$, (f) $A=0.01$, (g) $A=0.012$, (h) $A=0.014$, (i) $A=0.016$, and (j) $A=0.018$. With increasing $A$, on average $\epsilon$ increases and the number of dynamically triggered slip events with large energy release increases as well.

Figure 7

Average value (marker) and standard deviation (error bar) for the relative change in slip energy release $\epsilon \left(A\right)$ for each perturbed slip catalog, i.e., in correspondence with each ac vibration peak amplitude value $A$.

Figure 8

Scatter plot for the relative difference in energy release $\epsilon \left(A\right)$ vs the relative difference in time to failure $\Gamma \left(A\right)$ for each perturbed event compared with the corresponding spontaneous one. Different markers correspond to three different slip catalogs with different $A$ values.

Figure 9

Example of delayed, dynamically triggered event, taking place later in time compared to its spontaneous counterpart: solid line, spontaneous slip event; dashed line, triggered event in correspondence with the ac vibration peak amplitude $A=0.004$. The vertical dashed lines delimit the vibration time interval. The events are described by their friction coefficient signals ${\mu }_f\left(t\right)$.