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nism, raise two important questions. First, does the sta- bility criterion δz gen > 0 remain valid in more realistic situations, which necessarily involve finite displacements and anisotropy? More specifically does a granular layer inclined under gravity follow the hypothetical trajectory plotted in blue in the ( z, n$_{m}$ ) space on Figure 1? Second, what is the link between the microscopic role of fully mo- bilized contacts in the failure of the system and the global generalized isostaticity criterion? 

In this paper, we explore these two issues using simula- tion data obtained by Deboeuf et al. [5]. Having precisely isolated the ‘avalanches’ from the ‘quiet periods’, our first immediate observation is that the pile destabilizes when it is still stable according to the generalized isostaticity criterion. This result prompted us to investigate the mi- croscopic role of the fully mobilized (or critical) contacts. These contacts form elongated clusters, the size of which obeys critical scaling, with a characteristic length which approaches system spanning size near the avalanche on- set. These observations, which are consistent with those by Staron et al. [13], allow us to investigate the local and global stability properties of these clusters. Local failure as measured by the number of lost contacts correlates both spatially and temporally with being part of these clusters. Applying then the counting argument at the scale of these clusters, we find that their stability con- dition builds up progressively from a local mechanical property to the generalized isostaticity criterion as they grow in size. Hence the avalanche can be related to a sub- set of the packing becoming marginally stable according to the global criterion, while growing up to the system size. Note however that the above scenario is to be un- derstood as an averaged picture, since these clsuters are permenantly destabilized and renewed. 

The system we study here is a simulation of 2 d pack- ings of grains under gravity. The simulations where per- formed by Deboeuf et al. [5] using the contact dynam- ics code developed by Staron [23], which assumes per- fectly rigid grains interacting at contacts through a hard core repulsion and a Coulomb friction law: the tangen- tial force at contact, f$_{t}$ , is related to the normal force f$_{n}$ by the inequality | f$_{t}$ | ≤ µf$_{n}$ , where µ = 0 . 5 is the fric- tion coefficient. Beyond the fact that contact dynamics treats them as strictly nonsmooth, these contact laws do not differ from those more commonly used in dis- crete simulations [24]. The system consists of 4000 circu- lar grains with diameter uniformly distributed between [ d$_{min}$, d$_{max}$ ] in a way to ensure 20% polydispersity. The length of the box is about 120 d and the height is about 35 d , where d is the mean diameter of the grains. Initially the grains free fall in the box set horizontally. The box is then tilted quasistatically to the desired inclination θ . Deboeuf et al. [5] have run several histories of inclination – including several back and forth oscillations – to inves- tigate the stress anisotropy. In the present case, we have selected the final part of the pile history, starting with a 

<img src='content_image/385810.jpg'>

horizontal pile with θ$_{0}$ = 0 ◦ and tilting it in the direction of positive θ until θ ≈ 30 ◦ . We use 50 independent runs, with different initial conditions and our results are usu- ally shown as an average over these runs. For each run, particle positions, contacts and forces were stored in suc- cessive frames separated by 50 computational time steps, which corresponds to an angle variation of δθ = 0 . 05 ◦ . 

Let us first turn to the identification of the avalanche periods. We are interested in the properties of the pack- ing just before failure occurs, i.e. for θ → θ$_{m}$ , where θ$_{m}$ , the failure angle is, known to show considerable sta- tistical variance [5]. Obtaining a good criterion to pre- cisely identify the start of an avalanche is not straight- forward: one possibility with limited resolution is to con- sider the probability distribution of the kinetic energy (either translational and rotational) and identify as a threshold the value above which the distribution escapes the power law reported in the quiet region (see the arrow in Figure 2a). A more explicit signature of failure is given by considering the probability density of the fraction of contacts lost between two subsequent snapshots. As ob- served in Figure 2(b), the data clearly fall into two dis- tinct sub-populations (note the logarithmic scale on the x-axis), which we associate with the quiet period (left) and the avalanching period (right). The good separation between the two populations allows us to apply a cut- 

