Keywords: multiscale strategy, crack propagation, XFEM, homogenization, macroenrichment, microenrichment.

Today, structural analysis involving cracking is being reconsidered in the light of emerging new methods, such as the Strong Discontinuity Approach (SDA) introduced by Simo, Oliver and Armero [1]. Other approaches, such as the eXtended Finite Element Method (XFEM) [2] and the Generalized Finite Element Method (GFEM) [3], make use of the Partition of Unity Method (PUM) first introduced by Melenk and Babuška in 1996 [4]. By enabling one to enrich the kinematics of continuous media, these techniques allow the introduction of discontinuities into the displacement field using only a relatively small number of degrees of freedom. One of the main advantages in this case is that the mesh does not have to conform to the crack's geometry. These techniques greatly simplify the meshing and remeshing processes which, despite the improvement of meshing tools, remain tedious tasks for engineers confronted with crack propagation situations. However, these techniques do not completely incorporate the multiscale aspect induced by the localization of strains in the cracked zone. Generally, they require further remeshing around the crack: thus, the remeshing problem is only partially resolved. Moreover, conditioning difficulties remain because of the treatment of multiscale phenomena without separation.

To overcome these two difficulties, the strategy we developed in cooperation with Dassault Aviation is based on a two-scale approach in which the enrichment is introduced on the microscale. The process involved is a combination of two techniques. The first technique stems from work done at LMT-Cachan, which has been developing computational strategies with a strong mechanical meaning which makes them efficient. More precisely, this technique consists in applying the recently developed micro-macro approach [5] based on a homogenization technique. The microscale is associated with local phenomena which occur around the crack. This is a much smaller scale than the macroscale, which corresponds to the whole structure. This multiscale approach ensures a correct global-local interaction between the macroscale and the microscale. The second technique, known as the PUM, is used to define a proper representation of the local solution (in terms of discontinuity and solution at the crack's tip) on the microscale. The integration of enrichment functions is obtained by the XFEM. With this scale separation, the macroproblem keeps the same structure throughout the calculation while the whole numerical effort is directed towards the microlevel [6].

In the micro-macro approach, the fact that a crack affects both the local level and the global level raises the question of the kinematics and the description of forces on the two scales. The simplest solution consists in keeping the "usual" macroscopic description. Some examples showing the feasibility, but also the disadvantages, of such a method are presented. Consequently, different means of enriching the macroscale to improve the description of the macrokinematics are studied and illustrated. The integration of the PUM on the microlevel will be the subject of a subsequent paper.

1

J. Oliver, A. E. Huespe, M.D.G. Pulido, E. Chaves, "From continuum mechanics to fracture mechanics: the strong discontinuity approach", Engineering Fracture Mechanics, 69, 113-136, 2002. doi:10.1016/S0013-7944(01)00060-1

2

C. Daux, N. Moës, J. Dolbow, N. Sukumar, T. Belytschko, "Arbitrary branched and intersecting cracks with the extended finite element method", International Journal for Numerical Methods in Engineering, 48, 1741-1760, 2000. doi:10.1002/1097-0207(20000830)48:12<1741::AID-NME956>3.0.CO;2-L

3

T. Strouboulis, K. Copps, I. Babuška, "The generalized finite element method", Computer Methods in Applied Mechanics and Engineering, 190, 4081-4193, 2001. doi:10.1016/S0045-7825(01)00188-8

4

J.M. Melenk, I. Babuška, "The Partition of Unity Finite Element Method : Basic Theory and Applications", Computer Methods in Applied Mechanics and Engineering, 139, 289-314, 1996. doi:10.1016/S0045-7825(96)01087-0

5

P. Ladevèze, O. Loiseau, D. Dureisseix, "A micro-macro and parallel computational strategy for highly heterogeneous structures", International Journal for Numerical Methods in Engineering, 52(1-2), 121-138, 2001. doi:10.1002/nme.274

6

P.-A. Guidault, O. Allix, L. Champaney, "Structural multiscale global-local analysis : a micro-macro approach for crack propagation with local enrichment (in French)", Technical report for Dassault Aviation, 1 and 2, 2003.

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