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# 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. 

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: 

where z µ iso = d + 1 is the isostatic value for µ = ∞ fric- 

PACS numbers: 

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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- 

