International audienceThis paper presents a synthesis of the works performed by various teams from France, Italy and Canada around the question of second-order work criterion. Because of the non-associative character of geomaterials plastic strains, it is now recognized that a whole bifurcation domain exists in the stress space with various possible modes of failure. In a first part these failure modes are observed in lab experimental tests and in discrete element modelling. Then a theoretical study of second-order work allows to establish a link with the kinetic energy, giving a basis to explain the transition from a prefailure (quasi)static regime to a postfailure dynamic regime. Eventually the main features of geomaterials failure are obtained by applying second-order work criterion to five different constitutive rate-independent models-three being phenomenological and two micromechanical. As a whole this paper tries to gather together all the elements for a proper understanding and use of second-order work criterion in geomechanics

International audienceThe Petacciato landslide that occurred between 1906 and 1966 along a stretch of the Adriatic coast (Italy) and the Trevoux landslide that appeared in 1983 in France are analysed using Hill's material sufficient stability condition, which is based on the sign of the second-order work. Local and global sufficient stability criteria allow detecting and describing the instability phenomena, respectively, at the material point level and for the whole domain. By considering the nonassociated PLASOL elasto-plastic constitutive model and the second-order work criterion, we show that unstable stress-strain states can be reached in plane strain conditions, strictly inside the plastic limit criterion. The two landslides that occurred in Petacciato and Trevoux are then analysed by modelling unsaturated hydro-mechanical coupling. The analysis is carried out using the LAGAMINE finite element code. This original approach to slope stability analysis makes it possible to successfully exhibit the landslide mechanism observed on these slopes after a long rainy period and that could not be described with the classical methods

SUMMARYThe present paper investigates bifurcation analysis based on the second-order work criterion, in the framework of rate-independent constitutive models and rate-independent boundary-value problems. The approach applies mainly to nonassociated materials such as soils, rocks, and concretes. The bifurcation analysis usually performed at the material point level is extended to quasi-static boundary-value problems, by considering the stiffness matrix arising from finite element discretization. Lyapunov's definition of stability (Annales de la faculté des sciences de Toulouse 1907; 9:203-274), as well as definitions of bifurcation criteria (Rice's localization criterion (Theoretical and Applied Mechanics. Fourteenth IUTAM Congress, Amsterdam, 1976; 207-220) and the plasticity limit criterion are revived in order to clarify the application field of the second-order work criterion and to contrast these criteria. The first part of this paper analyses the second-order work criterion at the material point level. The bifurcation domain is presented in the 3D stress space as well as 3D cones of unstable loading directions for an incrementally nonlinear constitutive model. The relevance of this criterion, when the nonlinear constitutive model is expressed in the classical form (d = Mdε) or in the dual form (dε = N d ), is discussed. In the second part, the analysis is extended to the boundary-value problems in quasi-static conditions. Nonlinear finite element computations are performed and the global tangent stiffness matrix is analyzed. For several examples, the eigenvector associated with the first vanishing eigenvalue of the symmetrical part of the stiffness matrix gives an accurate estimation of the failure mode in the homogeneous and nonhomogeneous boundary-value problem.

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