Local Coulomb versus Global Failure Criterion for Granular Packings Silke Henkes, 1 Carolina Brito, 1, 2 Olivier Dauchot, 2 and Wim van Saarloos 1 $^{1}$Instituut-Lorentz, LION, Leiden University, P.O. Box 9506, 2300 RA Leiden, Netherlands $^{2}$GIT, SPEC, CEA Saclay, 91191 Gif sur Yvette, France Contacts at the Coulomb threshold are unstable to tangential perturbations and thus contribute to failure at the microscopic level. How is such a local property related to global failure, beyond the effective picture given by a Mohr-Coulomb type failure criterion? Here, we use a simulated bed of frictional disks slowly tilted under the action of gravity to investigate the link between the avalanche process and a global generalized isostaticity criterion. The avalanche starts when the packing as a whole is still stable according to this criterion, underlining the role of large heterogeneities in the destabilizing process: the clusters of particles with fully mobilized contacts concentrate local failure. We demonstrate that these clusters, at odds with the pile as a whole, are also globally marginal with respect to generalized isostaticity. More precisely, we observe how the condition of their stability from a local mechanical proprety progressively builds up to the generalized isostaticity criterion as they grow in size and eventually span the whole system when approaching the avalanche. PACS numbers: Understanding the failure of granular packings is of tremendous importance from both practical and theoret- ical aspects. Practically, avalanches are clearly of spe- cial interest for industrial and natural processes. From a more fundamental point of view, the mechanical rigidity of granular packings is related to the recently explored field of rheology close to dynamical arrest [1–3], as well as to the nature of the jamming transition for frictional par- ticles [4–7]. Despite many studies both from a continuum and a microscopic point of view (see, for example, [8–17]), the mechanisms of failure in frictional granular media are still unclear. Macroscopically, the application of the well known Coulomb criterion [18] requires the knwowledge of an ef- fective friction coefficient, which remains out of reach of most recent developments. From a microscopic perspec- tive, Maxwell derived a global stability criterion based on counting the number of independent contact force components, which has to exceed the number of de- grees of freedom for a packing to be mechanically sta- ble [19]. Recently, this isostaticity criterion has been gen- eralized for frictional packings by including contacts ex- actly at the Coulomb threshold in the above counting ar- gument [20, 21]. Such fully mobilized contacts are prone to tangential slipping, and it was indeed shown by Staron et al. [13, 15] that they play a key role in the destabiliza- tion process. A frictional packing of N particles with mean contact number z and mean number of fully mo- bilized contacts per particle n$_{m}$ has only Ndz/ 2 − Nn$_{m}$ independent force components, due to the additional re- strictions on the tangential forces. For this packing to be stable, this number has to exceed Nd ( d + 1) / 2, the number of degrees of freedom and the the generalized isostaticity criterion in d dimensions reads: tional packings [20]. It can be represented in a ( z, n$_{m}$ ) phase diagram (see Figure 1) where a line of marginal stability divides the stable from the unstable regions of phase space. Recent molecular dynamics simulations us- ing an isotropic compression protocol [20] have shown that frictional packings unjam (in the sense that p → 0) close to the generalized isostaticity line. The final state is characterized by z ( µ ) and n$_{m}$ ( µ ) and the linear response properties of these packings suggest that it is the distance to the line of marginal stability, δz gen = z − z gen iso , which controls stability [22]. These promising results, regarding both the global sta- bility criterion and the details of the microscopic mecha- where z µ iso = d + 1 is the isostatic value for µ = ∞ fric- z ≥ z µ iso + 2 n$_{m}$ d ≡ z gen iso , (1) FIG. 1: a - Generalized isostaticity phase diagram in 2D. In red is the line of marginal stability (Eq. 1). Frictional pack- ings under isotropic compression unjam – when compression is released (schematic green arrow) – at different positions close to this line (crosses) depending on friction µ [20, 22]. Does a pile slowly inclined under gravity towards avalanche follow the hypothesized blue line? b - The system during the avalanche, at an inclination of 24 ◦ , for the details of the coloring scheme please refer to Fig. 3. arXiv:1001.2202v1 [cond-mat.soft] 13 Jan 2010