1. Introduction

Numerical simulation in fracture mechanics is essentially based on the use of the finite element method (MEF). Conventionally, for a crack propagation problem, a first calculation is carried out on an initial mesh, then a new mesh is determined, taking into account the progress of the crack (according to the propagation law chosen). A new calculation results, and the method is iterated for each propagation step.

A major and immediate disadvantage is the need to re-mesh at each propagation step. The remeshing process can be easily automated in 2D [bib 1], and in some cases in 3D [bib 2], but high-quality 3D remeshing is costly in terms of time (human supervision) and money. Indeed, with an automatic mesh, appropriate local refinement at the level of the cracking zone often leads to an excessive number of elements throughout the rest of the structure. A process of refinement by layers is generally useful, such as the introduction of a torus at the bottom of the crack, an efficient solution (also facilitating the installation of Barsoum elements [bib 3]) but requiring costly human intervention; especially since the geometric shape of the crack is very complex (helical cracks for example [bib 4]). The problem becomes almost inconceivable in 3D multi-cracking. In addition to these practical difficulties, the projection of quantities (constraints, internal variables) from one mesh to another poses fundamental theoretical problems (verification of equations for the conservation of energy, momentum, mass) [bib 5].

In addition to the difficulties associated with propagation, mesh methods are not very effective for parametric studies where we are interested in the influence of the position and shape of the crack.

So-called « Meshless » methods [bib 6] have been proposed to overcome mesh-related constraints. Meshless are based on only nodal discretization, without connectivity, and shape functions are built from the nodal configuration. Initially introduced in the late 70s for problems without borders (Smoothed Particle Hydrodynamics method in the field of astrophysics), Meshless methods were later extended to mechanical problems and are now available in various variants: methods MLS (Moving Least Square), Diffuse Element Method (DEM), Diffuse Element Method (), Element Free Galerkin (EFG), Reproducing Kernel Particle Method (RKPM), hp clouds, and many more. The common element of all these methods is the concept of Partition of Unity, which is a set of functions whose sum is equal to one at each point in the field in question. The main disadvantage of these methods is that they require a greater numerical effort than those based on a mesh (time CPU in particular). In particular, the evaluation of form functions is far from being as trivial, numerical integration schemes are generally richer and therefore more expensive and the resulting global system of equations has a greater width of the band compared to MEF [bib 7]. Even in recent versions of Meshless methods combining physical and mathematical supports [bib 8], the imposition of boundary conditions remains a problem.

Other methods are based on a division of the unit within the framework of standard finite elements, where only the definition of form functions differs. This choice of unit partition avoids the problems of integrating stiffness, which is considerably expensive for Meshless methods (EFG, DEM, RKPM in particular). In addition, the use of a unit partition based on finite elements allows easy implementation of Dirichlet-type boundary conditions (unlike techniques using a unit partition based on least squares). These methods, which are first found in their mesh versions under the name of Partition of Unity Finite Element Method (PUFEM) or Generalized Finite Element Method (GFEM) [bib 9], make it easy to enrich the space of form functions, thanks to the a priori knowledge of the properties of the solution of the problem. Combining GFEM and hp-cloud, Duarte et al. [bib 10] proposes a mixed unit partition (finite elements and Shepard), which makes it possible to consider a non-meshed crack, with enriched shape functions.

Even closer to the classical finite element framework, the extended finite element method (X- FEM) has aroused one of the greatest interest if we refer to the evolution of the number of publications on this subject since its appearance and the place reserved for it in international conferences. This method uses the partition of the unit to enrich the base of the shape functions in order to represent a jump in the field of displacement at the level of the crack lips, as well as the singularity at the bottom of the crack. Two enhancements are then introduced: an enrichment by a jump function that makes it possible to manage the discontinuity through the lips of the crack, and an enrichment by asymptotic functions, which allows a faithful representation of the physical phenomena taking place at the crack bottom level. Historically, the precursors are due to Belytschko and Black [bib 11], who apply the unit partition [bib 23] to the mechanics of rupture by incorporating the analytical formulas of asymptotic fields into the approximation of the displacement. The addition of the generalized Heaviside function [bib 24] makes it possible to write the enriched approximation of displacement in its final form, an expression that gives rise to the extended finite element method. Compared to GFEM, it offers less dependence on knowing the shape of solutions, which provides greater flexibility [bib 15]. The field of application of X- FEM is constantly expanding, this approach having been used in very varied frameworks: Reissner-Mindlin plates, 3D fracture shells, multi-cracking, cohesive zones, modeling holes and bi-materials, modeling holes and bi-materials, modeling holes and bi-materials, incompressible formulations and large transformations, incompressible formulations and large transformations, crack nucleation, contact cracking, dynamic failure, plasticity…

In addition, the use of the level set method coupled with X- FEM greatly facilitated the treatment of cracks in 2D ([bib 15], [bib 14] and [bib 17]) and in 3D [bib 18]. Initially introduced in the context of fluid mechanics to represent the evolution of interfaces, the level set method considers the interface as the iso-zero of a distance function. This method is particularly effective for the propagation of a crack in 3D [bib 12], coupled with the use of the Fast Marching Method [bib 16].

This document is organized around 4 sections, including this introduction which takes the place of section 5.

Section 2 is devoted to using the level set method for cracking. After a brief theoretical review, we specify the calculation of the level sets, which are practical for determining the position of the crack in 3D and of the crack bottom, and which make it possible to define a local base at the bottom of the crack.

Section 3 shows the cracking problem addressed with X- FEM. In paragraph [§ 3.2], the approximation of the displacement written in a database of rich form functions is introduced. The addition of discontinuous functions through the interface leads to a sub-division procedure detailed in paragraph [§ 3.3], prior to the phase of integrating the terms of rigidity and the second member, the implementation of which is explained in paragraph [§ 3.5].

Section 5 focuses on the post-treatment of the energy restoration rate and stress intensity factors in linear fracture mechanics. The \(G\) -theta method makes it possible to calculate the local energy return rate. For this, a theta field is introduced, representing a virtual extension of the crack. The local \(G\) is then the solution of a variational equation, using the \(J\) integral in the form of domain integrals. The discretization choices of \(G\) and the theta field lead to a linear system, whose resolution leads to the values of \(G\) along the crack bottom [bib 48]. The G-theta method also makes it possible to determine the stress intensity factors along the crack bottom. Instead of using the \(J\) integral, we use the bilinear form of \(G\), which leads to domain integrals combining solution fields and analytic asymptotic fields (also called interaction integrals). So, as for \(G\), \(K\) are then solutions of a variational equation, using interaction integrals. The discretization choices of \(K\) and the theta field lead to a linear system, whose resolution leads to the values of \(K\) along the crack bottom [bib 49].