Brittle failure of rocks under compressive shear has features in common with a thermodnamicel phase transition: deformation localizes quickly at grain scale from the departure of reversible elastic regime, staying at first homogeneous at large scales. Whereas individual microcracks still grow in a stable manner, some large scale structures (shear bands) appear around peak stress, which spontaneaously break both translational and rotational symmetries of the problem. Analyzing stress/strain data obtained on different rocks and load conditions, we show that they present a high degree of universality: they follow a law $\tau_c-\tau = \alpha (\epsilon_{i_c} - \epsilon_i)^{2.2 \pm 0.3}$, where $\tau$ is the axial stress, $\epsilon_i$ inelastic axial deformation, and $\tau_c, \epsilon_{i_c}$ two parameters very close to the values of $\tau$ and $\epsilon_i$ at peak stress.The fact that this exponent seems independent of rock, together with the symmetry breaking and appearance of a macroscopic structuration, is typical of a critical point. Accordingly, we develop from simple postulate of maximization of entropy, a statistical physics formalism adapted to this problem. We so get a probability distribution on crack configurations, as a function of the energy necessary for their formation. From a model for these energies, based on elastic interactions, we obtain a system exhibiting two phase transitions: the first one corresponds to the onset of microcracks activity, the second one to the macroscopic localization. This model predicts the quadratic behaviour observed in the data at the approach of the second transition, whereas the material's effectice modulus remains finite.
