Avalanche precursors in a frictional model

We present a one-dimensional numerical model based on elastically coupled sliders on a frictional incline of variable tilt. This very simple approach makes it possible to study the precursors to the avalanche and to provide a rationalization of different features that have been observed in experiments. We provide a statistical description of the model leading to master equations describing the state of the system as a function of the angle of inclination. Our central results are the reproduction of large-scale regular events preceding the avalanche, on the one hand, and an analytical approach providing an internal threshold for the outbreak of rearrangements before the avalanche in the system, on the other hand.

Figure 1

Sketch of the system under study.

Figure 2

Behavior of a small system without global coupling upon inclination. (a) Hot map of the displacements $\mathrm{\Delta }{\tilde{x}}_i$ vs angle $\alpha$. Each horizontal line corresponds to one slider. The darker the points are, the larger is the displacement of the slider. The vertical dashed line corresponds to the avalanche angle and $\mathrm{\Delta }\mu ={\overline{\mu }}_s-{\mu }_d$. (b) Tangential force ${\tilde{F}}_t$ (${\tilde{F}}_t={\tilde{f}}_{n+1\to n}+{\tilde{f}}_{n-1\to n}+{\tilde{f}}_{C\to n}+sin\alpha$) vs angle $\alpha$. (The slider number is indicated by the color bar.) The upper dashed line is the stability limit ${\overline{\mu }}_{s}cos\alpha$. The lower dashed line is the arrest limit $(2{\mu }_d-{\overline{\mu }}_s)cos\alpha$. The dash-dotted line is $sin\alpha$ ($N=10, \xi =0, l=10, {\mu }_d=0.55, {\overline{\mu }}_s=0.6$, and ${\sigma }_{\mu }=0.01$).

Figure 3

Behavior of a large system without global coupling upon inclination. (a) Hot map of the displacements $\mathrm{\Delta }{\tilde{x}}_i$ vs angle $\alpha$. (b) Tangential force ${\tilde{F}}_t$ vs angle $\alpha$. The upper dashed line is the stability limit ${\overline{\mu }}_{s}cos\alpha$. The lower dashed line is the arrest limit $(2{\mu }_d-{\overline{\mu }}_s)cos\alpha$ ($N=400, \xi =0, l=10, {\mu }_d=0.55, {\overline{\mu }}_s=0.6$, and ${\sigma }_{\mu }=0.01$).

Figure 4

Behavior of a large system with global coupling upon inclination. (a) Hot map of the displacements $\mathrm{\Delta }{\tilde{x}}_i$ vs angle $\alpha$. (b) Tangential force ${\tilde{F}}_t$ vs angle $\alpha$. The upper dashed line is the stability limit ${\overline{\mu }}_{s}cos\alpha$. The lower dashed line is the arrest limit $(2{\mu }_d-{\overline{\mu }}_s)cos\alpha$ ($N=10, \xi =0.001, l=10, {\mu }_d=0.55, {\overline{\mu }}_s=0.6$, and ${\sigma }_{\mu }=0.01$).

Figure 5

Tangential force ${\tilde{F}}_t$ vs angle $\alpha$. The upper dashed line is the stability limit ${\overline{\mu }}_{s}cos\alpha$. The lower dashed line is the arrest limit $(2{\mu }_d-{\overline{\mu }}_s)cos\alpha$. Inset: Sketch of the system under study (${\mu }_d=0.55$ and ${\mu }_d=0.6$).

Figure 6

Probability distribution function of the tangential force ${\tilde{F}}_t$ upon inclination, at different values of the inclination angle. Each distribution corresponds to an average, at a given $\alpha$ (indicated by the color scale), over 20 numerical runs. Inset: Enlargement of the distribution for ${\tilde{F}}_t<0.4$ ($N=100, \xi =0, \tilde{l}=10, {\mu }_d=0.55, {\overline{\mu }}_s=0.6$, and ${\sigma }_{\mu }=0.01$).

Figure 7

Probability distribution function of the ratio ${\tilde{F}}_t/{\tilde{F}}_t^{*}$ for the sliders that moved upon inclination. The data are from the same 20 numerical runs as in Fig. 6 [($N=100, \xi =0, \tilde{l}=10, {\mu }_d=0.55, {\overline{\mu }}_s=0.6$, and ${\sigma }_{\mu }=0.01$).

Figure 8

Schematic representation of the histogram equivalent to the probability distribution function of the tangential force ${\tilde{F}}_t$ at a given inclination angle $\alpha$. The arrows indicate the main modification of the distribution upon further inclination by $\delta \alpha$ (see Sec. 4b2).

Figure 9

Height $P_i$ vs angle $\alpha$ for the data of Fig. 6. The dashed straight lines correspond to prediction of the crude model [Eq. (19)]. The exponential decay of $P_{\nu }\left(\alpha \right)$ is given in Eq. (18). The dotted line is a guide for the eye. The vertical solid line is the threshold from Eq. (21) ($N=100, \xi =0, \tilde{l}=10, {\mu }_d=0.55, {\overline{\mu }}_s=0.6$, and ${\sigma }_{\mu }=0.01$).

Figure 10

(a) Cumulated displacement $A$ vs angle $\alpha$. Upon inclination, $A$ increases exponentially in spite of the linear increase of the number of sliders involved in the rearrangements with $\alpha$. (b) Average displacement during a rearrangement $\langle \mathrm{\Delta }\tilde{x}\rangle$ vs angle $\alpha$ ($N=100, \xi =0, \tilde{l}=10, {\mu }_d=0.55, {\overline{\mu }}_s=0.6$, and ${\sigma }_{\mu }=0.01$).