International audienceA model is presented that accurately describes brittle failure in the presence of cohesive forces, with a particular focus on the prediction of non planar crack paths. In comparison with earlier literature, the originality of the procedure lies in the a posteriori computation of the crack advance from the equilibrium, instead of a most common determination beforehand from the stress state ahead of the front. To this aim, a robust way of introducing brittle non-smooth cohesive laws in the X-FEM is presented. Then the a posteriori update algorithm of the crack front is detailed. The crack deflection angle is computed from cohesive quantities exclusively, by introducing equivalent stress intensity factors. The procedure shows good accordance with experiments from the literature

This paper presents a 3D non-locking contact approach, within the eXtended Finite Element Method (X-FEM) framework. X-FEM allows one to introduce interface independently of the mesh. The contact problem on the interface leads to an Augmented Lagrangian formulation derived from the discretization of its continuous formulation. It is shown that a simple choice of the Lagrange multiplier space is not suitable and leads to contact pressure oscillations. An algorithm for the restriction of the Lagrange multiplier approximation space is proposed to stabilize the formulation. The stability of the mixed displacement-contact pressure formulation is discussed in terms of convergence of the energy error. Numerical examples performed with the Finite Element software Code_Aster illustrate this approach while solving three-dimensional problems with contact. RÉSUMÉ. Cet article présente une formulation stabilisée pour les problèmes de contact en 3D, dans le cadre de la méthode des éléments finis étendue (X-FEM), méthode qui autorise des interfaces indépendantes du maillage. Pour la formulation du problème de contact sur l'interface, nous utilisons un Lagrangien Augmenté, qui dérive de la discrétisation du problème de contact écrit sous sa forme continue. Nous montrons qu'un choix simple de l'espace des multiplicateurs de Lagrange n'est pas satisfaisant car cela conduit à des oscillations des pressions de contact. Un algorithme de restriction de l'espace d'approximation des multiplicateurs de Lagrange est proposé afin de stabiliser la formulation. La stabilité de la formulation mixte (déplacement-pression de contact) est démontrée à l'aide des taux de convergence des erreurs. Des exemples numériques réalisés avec le logiciel éléments finis Code_Aster illustrent cette approche pour la résolution de problèmes tridimensionnels avec contact.

SUMMARYThe extended finite element method (X-FEM) has been developed to minimize requirements on the mesh design in a problem with a displacement discontinuity. This advantage, however, still remains limited to the small deformation hypothesis when considering sliding discontinuities. The approach presented in this paper proposes to couple X-FEM with a Lagrangian large sliding frictionless contact algorithm. A new hybrid X-FEM contact element was developed with a contact search algorithm allowing for an update of contacting surfaces pairing. The stability of the contact formulation is ensured by an algorithm for fulfilling Ladyzhenskaya-Babuska-Brezzi (LBB) condition. Several 2D simple examples are presented in this paper in order to prove its efficiency and stability.

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