Coulomb’s failure interpreted as a critical transition,and size effects on compressive strength

With his memoir on statics presented to the French Academy of Sciences in 1773, Charles-  Augustin de Coulomb left an indelible mark in the fields of mechanics, civil engineering, and   geophysics. Coulomb proposed, from a combination of intuition and ingenious experimental   data, his celebrated failure criterion for materials loaded under shear or compression. He   postulated that failure occurs along a fault plane when the applied shear stress acting on that   plane overcomes a resistance made of two parts of different nature: a cohesion, which can be   interpreted as an intrinsic shear strength of the material, and a resistance proportional to the   normal pressure. Since the seminal work of Anderson in 1905, Coulomb’s theory of failure   became the standard conceptual tool for the mechanics of earthquakes and faulting, of rocks   and soils, or of granular media. However, despite (or perhaps because of) its age, simplicity,   and ubiquity, the Coulomb’s equation remains nowadays essentially empirical, saying little   about the mechanisms leading to shear failure in disordered media such as rocks, soils, ice,   concrete, or granular media. Classical fracture mechanics concepts are useless, as a shear   crack in an isotropic elastic medium cannot propagate (in mode II) on its own plane. Instead,   shear faulting is associated with multiple instabilities of various sizes and starts from an   initiation stage of randomly distributed microcracks. As deformation proceeds, the rate of   energy released by fracturing increases, whereas damage progressively localizes along a fault   plane, coalescing finally to trigger macroscopic faulting. I will argue that this process can be   interpreted as a critical transition coupling a local threshold mechanics, disorder, and elastic   interactions, and build a formal analogy with the depinning transition of an elastic manifold.   This allows to revisit one of the oldest problem of mechanics, the size effect on strength.   Indeed, this critical transition interpretation naturally entails finite-size scaling laws for the   mean strength and its associated variability. Theoretical predictions are in remarkable   agreement with measurements reported for various materials such as rocks, ice, coal, or   concrete. This formalism, which can also be extended to the flowing instability of granular   media under multiaxial compression, has important practical consequences for future design   rules.