1. Introduction

To take into account the rubbing contact on the lips of the crack with X- FEM, we chose the framework of the continuous method [bib 1], [bib 2].

Unlike discrete approaches where the contact problem is taken into account by assembling nodal forces, the equations are here discretized by the finite element method and the contact problem is taken into account by an assembly of elementary contributions. This approach has been developed by numerous authors, such as Alart et al. [bib 3], Laursen*et al.* [bib 8], Wriggers [bib 9], Curnier et al. [bib 10], Pietrzak [bib 11].

In this « continuous » approach, contact conditions are seen as an interface law and not as boundary conditions. To the concept of the law of interface between deformable bodies, we can associate the concept of contact element during the transition from continuous formalism to the discrete model. The exact, and therefore rigorous, resolution of contact laws (which cannot be distinguished) can be carried out via a hybrid contact element including contact efforts in the unknowns of the problem. In this framework, the hybrid formulation leads to a tangent matrix that is not symmetric, not defined, positive, and poorly conditioned, with zeros on the diagonal. The difficulty of the problem lies in the non-differentiability of the system to be solved [bib 12].

The method proposed by Ben Dia is very similar, but we choose to eliminate the non-differentiability of contact by an algorithm of active constraints, and that of friction by a fixed point problem on friction in order to obtain a series of regular problems using resolution methods whose convergence is established. Numerous variants, depending on the algorithms adopted and their arrangement, exist, but the convergence of the overall scheme is not ensured.

We recall that one starts from a Lagrangian formalist from the problem of contact between two deformable solids, which one introduces into the principle of virtual work. A mixed variational displacement-pressure formulation is deduced by incorporating weak formulations of contact laws. The equations are discretized by the finite element method. The choice of finite element discretization spaces as well as the integration schemes (contact terms) are explained.

With X- FEM, the crack lips are treated as a single geometric surface of discontinuity that may be internal to the finite elements. The integration of contact terms on this (non-meshed) surface then uses the quantities carried by the nodes of the elements crossed by it. In small displacements, no pairing is necessary because the points on the surface of the crack facing each other are intrinsically linked (they correspond to the same geometric entity). The displacement jump is expressed as a function of the discontinuous enrichment degrees of freedom introduced by X- FEM.

This document is organized around 6 main sections, including this introduction which takes the place of section 1. The friction contact problem and the equations involved are introduced in paragraph [§ 2]. Starting from a Lagrangian approach to contact introduced into the principle of virtual work, paragraph [§ 3] results in an expression of the mixed variational formulation displacement-pressure. Paragraph [§ 4] refers to the choice of discretizations (finite elements) of contact action fields. The resolution strategy is specified in paragraph [§ 5]. The expressions of the elementary terms of contact and friction resulting from the X- FEM approach are detailed in paragraph [§ 5.6]. Paragraph [§ 6] focuses on a particular condition of compatibility of the fields of movement and pressure. An algorithm for determining an adequate contact pressure multiplier space (i.e. complying with condition LBB) is explained.