The effect of scale and criticality in rock slope stability

The apparent shear strength of rock discontinuities is considerably smaller than that of small scale samples. At the same time, the sliding behavior is characterized, in situ, by marked instabilities, with the typical features of Critical Phenomena. Contact mechanics permits to calculate normal and tangential forces at any point, and to follow the stick-slip transition for arbitrary loading histories. On the other hand, the above aspects are not captured by the classical theories, including those based on roughness indices. We argue that the multiscale topology of contact domains plays a fundamental role in determining the behavior of rock joints. In particular, experiments and numerical simulations show that these domains are lacunar sets with fractal dimension smaller than 2.0. This provides peculiar scaling of normal and tangential pressures at the interface, and the consequent size-dependence of the apparent friction coefficient. Moreover, we implement Renormalization Group to determine the critical point (e.g. the critical shear force) when rock sliding occurs. We show that the critical force is less than the one predicted by the classical Coulomb's theory, and that it depends on the specimen size and on the topology of the interface. The same reasoning can be extended to other phenomena, e.g., to the rupture of brittle materials.