Criticality of the ``critical state'' of granular media: Dilatancy angle in the Tetris model

The dilatancy angle describes the propensity of a granular medium to dilate under an applied shear. Using a simple spin model (the Tetris model) which accounts for geometrical ``frustration'' effects, we study such a dilatancy angle as a function of density. An exact mapping can be drawn with a directed percolation process which proves that there exists a critical density ${\ensuremath{\rho}}_{c}$ above which the system expands and below which it contracts under shear. When applied to packings constructed by a random deposition under gravity, the dilatancy angle is shown to be strongly anisotropic, and it constitutes an efficient tool to characterize the texture of the medium.