It is now well recognized that granular media exhibit failure states strictly inside Mohr-Coulomb plastic limit condition. Some of these failure states can be described by the second-order work criterion and correspond to diffuse failure modes (without strain localisation patterns). Bifurcation domains and cones of unstable stress directions are rst obtained numerically and analytically from incrementally nonlinear constitutive relations. In a second part, a discrete element method is used and the behaviour of cubical specimens of 10; 000 spheres is directly simulated. The second-order work is computed from the stress-strain states at the boundaries of the specimen. A bifurcation domain and cones of unstable stress directions are determined and compared with the ones given by the incrementally non-linear relation, calibrated on the same medium. Finally in a third part, a micromechanical model is considered which is based on an elastic-plastic force-displacement relation between two grains, and some homogenisation/localisation techniques. From this model a bifurcation domain and cones of unstable stress directions are also obtained and compared with the previous ones. Material bifurcations appear as an important characteristic of geomaterial behaviour.