Continuous and discrete modeling of failure in geomaterials

The peak of q, for a loose sand in undrained condition, satisfies neither the Mohr Coulomb plastic limit condition nor a criterion of strain localization. As has been shown, this example shows that the question of failure in geomaterials is still a topical issue. Failure at the peak of q (for a loose sand in undrained condition) is predicted by Hill's condition of stability. The theory of non-associated hardening elastoplasticity constitutes a sufficiently broad framework to describe such failure states (strictly located within the plastic limit surface and the localization condition). Constitutive models, with incrementally octo-linear and nonlinear constitutive relations, as well as direct simulations based on the discrete element method, have shown the existence of a domain in the stress space wherein Hill's condition of stability is not satisfied. The boundary equation of this domain has been given and discussed, as has the equation of cones of unstable stress directions. In this domain, failure is characterized by bifurcation phenomena, loss of uniqueness, and instability in Lyapunov's sense. Occurrence of the diffuse failure mode has been shown with discrete numerical simulations, for bifurcation points detected using Hill's stability condition. These diffuse failure modes can occur only if the stress state is inside the bifurcation domain, if the loading direction is included in the cone of unstable directions, and if the control parameters used allow the development of failure. As shown numerically, these diffuse failure modes can also be interpreted as a loss of sustainability in Nicot's sense, as well as a loss of controllability in Nova's sense. Reproducibility of results with fundamentally differentmodels such as incrementally octo-linear or nonlinear constitutive relations, discrete numerical models, and even micro-mechanical models (such as the micro-directional model [NIC 05]) [NIC 06, SIB 07], show that the sign of the second-order work may constitute an appropriate criterion to detect some failure modes, appearing before the localized failure but for quite particular loading conditions illustrated, among others, in this chapter.