Failures by divergence instabilities in rate-independent non-associated material, such as granular matter, can occur from mechanical states described by the plastic stress limit surface, but also from stress states strictly included within this surface. Besides, the failure mode may be localized, with for instance the formation of shear bands, or diffuse with a strain field remaining homogeneous. All these failures can be described in a unique framework where plastic limit stress states appear as particular cases of generalized limit states; and where the effective development of failure is characterized by the unbounded increase of response parameters linked by a failure rule (i.e. a generalized plastic flow rule), together with a bifurcation of the mechanical response from a quasi-static pre-failure response to a dynamic post-failure one. All these features are discussed and highlighted from direct numerical simulations performed with a discrete element model. Moreover, the second order work criterion directly related at the macroscopic scale to divergence instabilities, is shown to be also relevant at the scale of inter-particle contacts.