The question of failure for geomaterials, and more generally for nonassociative materials, is revisited through the second-order work criterion defining, for such media, a whole domain of bifurcation included in the plastic limit surface. In a first theoretical part of the chapter, relations between the vanishing of the second-order work, the existence of limit states and the occurrence of failures characterized by a transition from a quasi-static pre-failure regime to a dynamic post-failure regime, are presented and illustrated from discrete element computations. Then boundaries of the bifurcation domain and cones of unstable loading directions are given in fully three-dimensional loading conditions for a phenomenological incrementally non-linear relation, and in axisymmetric loading conditions for a numerical discrete element model. Finally, conditions for the triggering and the development of failure inside the bifurcation domain are described and emphasized from direct simulations with the discrete element method for proportional stress loading paths.